research article supply chain bilateral coordination with option contracts...

Research ArticleSupply Chain Bilateral Coordination with OptionContracts under Inflation Scenarios

Nana Wan and Xu Chen

School of Management and Economics University of Electronic Science and Technology of China Chengdu 611731 China

Correspondence should be addressed to Xu Chen xchenxchen263net

Received 28 January 2015 Accepted 21 April 2015

Academic Editor Chuanxi Qian

Copyright copy 2015 N Wan and X ChenThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

There exist obvious changes in price and demand during the inflationary period both of which are regarded as the key factorsleading to supply chain uncertainty In this paper we focus our discussion on price increase and demand contraction caused byinflation integrate the effect of inflation and option contracts within themodel framework and analyze how to use option contractsto achieve supply chain coordination under inflation scenarios We consider a one-period two-stage supply chain consisting ofone supplier and one retailer and explore the effect of inflation on the optimal ordering and production decisions under threedifferent types of contracts wholesale price contracts option contracts and portfolio contracts Moreover we explore the impactof option contracts on the supply chain through using wholesale price contracts model as the benchmark We find that the retailerprefers adopting portfolio contracts but the supplier prefers providing option contracts under inflation scenarios Ultimatelyoption contracts will be implemented owing to the supplierrsquos market dominant position In addition we discuss the supply chainbilateral coordination mechanism with option contracts from the perspectives of two members and derive that option contractscan coordinate the supply chain and achieve Pareto improvement under inflation scenarios

1 Introduction

Along with the continued development of economic global-ization economy in one country is more susceptible to whathappens in other countries Owing to the global financialcrisis inflation has been emerging in recent years This istrue not only in developing countries but also in developedcountries According to the latest reports by Statistics theglobal inflation rate from 2004 to 2014 has always deviatedfar from 2 inflation target [1] which suggests that inflationhas a negative impact on the enterprise operation and posesa threat to the real economics As we have seen a lot ofissues seem to appear as the inflationary pressure spreadingto all aspects of the economy The most obvious changes thatoccur during inflation are price and demand both of whichare considered as the significant factors impacting supplychain risk We can observe that there exists a remarkableincrease in price due to the effect of inflation Over the pastfew years food prices have climbed to previously unknownheights andnonfoodprices have gone updramatically aroundthe world People need to spend more money to buy thesame goods and services than they did in the past Moreover

we can also observe that there exists a remarkable decreasein demand due to the effect of inflation In recent yearssales of many items such as clothing and cars continue toshrink and a large number of firms are trapped in the survivecrisis The main reason for this feature is because wagesfail to keep up with rising price under inflation scenarioswhich results in the relative decline of individual incomes andthe direct reduction of purchasing powers Since companiesare inseparable from their partners the effect of inflationcan exert great influence on the daily operations of thesupply chain Thus how to manage these two risks justmentioned becomes the key issue that needs to be addressedurgently in supply chainmanagement applications So far theeffect of inflation has been studied in inventory managementapplications but it has not been addressed in supply chainmanagement applicationsMotivated by this we plan to studyhow the supply chain members make the optimal decisionpolicies under inflation scenarios

Options have been demonstrated to be a viable instru-ment to protect against the effect of inflation in financialapplications [2] We introduce option contracts into supplychain management to hedge the risk of price increase and

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 502021 16 pageshttpdxdoiorg1011552015502021

2 Discrete Dynamics in Nature and Society

demand contraction caused by inflation It is worth notingthat options in supply chain management applications dif-ferent from options in financial applications are consideredas real options In addition option contracts are alwaysclassified into three categories [3 4] and we limit ourdiscussion to call option contracts in this paper Optioncontracts can help the demand-side obtain the goods after themarket demand is realized and provide the demand-side withmore flexibility to accommodate changing market Optioncontracts can also help the supply-side make a productionplan that maximizes its own profit and provide the supply-side with more flexibility to reduce the production cost Inthe real world many famous companies such as HP [5] Sun[6] and IBM [7] adapt derivatives of option contracts topurchase various inputs such as memory chips and scannerassemblies Since then extensive attentions from scholars arepaid to option contracts So far there aremany papers relatingto option contacts under various scenarios in supply chainmanagement applications However all these papers do notconsider the effect of inflation Motivated by this we plan tostudy whether option contracts are an efficient tool to resolvechannel conflicts and achieve channel coordination underinflation scenarios

On the other hand the coordination problem has alwaysbeen an important part in the study on supply chain man-agement As we know there exists double marginalizationphenomenon under the decentralized decision-making [8]The primary purpose of channel coordination is to promptthe members involved to carry out the actions which areconsistent with the supply chainrsquos objective [9] To the bestof our knowledge except for some scenarios such as randomyield the supply-side is always assumed to adopt the make-to-order production policy and commit tomanufacturing theproducts up to the order requirement of the demand-sideunder various contract types such as buyback contracts andrevenue sharing contracts In this case supply chain coordi-nation can be achieved only when the demand-side decisionunder the decentralized situation is the same as the integrateddecision under the centralized situation At this momenthow to make a nonintegrated supply chain coordinate issimplified to the supply chain unilateral coordination fromthe demand-side perspective However the supply-side has astrong incentive to decide the production quantity accordingto the profit maximization rule in the presence of optioncontracts The major reason is because option contracts givethe right not the obligation to obtain the items and so thedemand-side may not exercise all the options purchasedObviously this poses a challenge on the implementationof supply chain unilateral coordination mechanism in thepresence of option contracts Motivated by this we designthe supply chain bilateral coordination mechanism from theperspective of both the demand-side and the supply-side inthe presence of option contracts

In this paper we consider one-period two-stage supplychain which contains one supplier and one retailer Consid-ering price increase and demand contraction due to the effectof inflation we introduce option contracts into supply chaindecision-making in order to hedge against these risks justmentionedThis paper mainly solves the following problems

(1) What are the optimal ordering and production poli-cies for the supply chain members in the presence ofoption contracts under inflation scenarios

(2) What are the optimal expected profits for the supplychain members in the presence of option contractsunder inflation scenarios

(3) What effect does inflation have on the optimal order-ing and production policies

(4) What effect do option contracts have on supply chainunder inflation scenarios

(5) How should option contracts be set to achieve supplychain coordination under inflation scenarios

The main contributions of our work are as follows

(1) To the best of our knowledge there are no publishedpapers that study the use of option contracts to protectagainst the effect of inflation in supply chain manage-ment applications We develop supply chain modelsthat incorporate the effect of inflation and optioncontracts in this paper Our objective is to providemanagement insights into the effect of inflation andoption contracts on the supply chain

(2) We explore the effect of inflation on the retailerrsquosoptimal ordering policy and the supplierrsquos optimalproduction policy and gain many management inter-esting results

(3) We explore the impact of option contracts on thesupply chain through using wholesale price contractsmodel as the benchmark and discuss which kind ofcontracts is more suitable for supply chain membersunder inflation scenarios

(4) We design the bilateral coordination mechanismfrom the perspective of both the supplier and theretailer and achieve an efficient channel under infla-tion scenarios

The remaining part of this paper is structured as followsA summary of related literature is presented in Section 2Themodel formulation and assumptions are given in Section 3We establish supply chain models with three different con-tracts and explore the effect of inflation on the optimaldecision policies in Section 4Wediscuss the impact of optioncontracts on the supply chain decisions and performancein Section 5 Coordination conditions with option contractsunder inflation scenarios are considered in Section 6Wepro-vide a numerical example to illustrate the effect of inflationon the optimal decisions and the optimal expected profitsin Section 7 We conclude our findings in Section 8 andhighlight possible future work

2 Literature Review

We first review the literature on enterprise operation man-agement under inflation scenarios Dey et al [10] solve adeteriorating inventory problem with two warehouses con-sidering time value of money and interval-valued lead time

Discrete Dynamics in Nature and Society 3

under inflation Jaggi andKhanna [11] formulate an inventorymodel for deteriorating itemswith inflation-induced demandwhen trade credit policy and time discounting with allowableshortages are considered Yang et al [12] investigate theoptimal replenishment policy with stock-dependent con-sumption rate under inflation when partial backloggingis considered Sarkar et al [13] establish an EMQ modelwith time-dependent demand and an imperfect productionprocess under inflation and time value of money Tripathi[14] investigates the optimal pricing and ordering decisionsfor a deteriorating inventory when demand is a functionof price inflation and delay in payment is permissibleTaheri-Tolgari et al [15] study the production problem forimperfect items where inflation and inspection errors areconsideredMirzazadeh [16] deals with a deteriorating inven-tory model with inflation-dependent demand and partialbacklogging under stochastic inflationary conditions Guriaet al [17] present an inventory policy considering inflationand selling price-dependent demandunder deterministic andstochastic planning horizons Mousavi et al [18] use themixed binary integer mathematical programming approachto resolve multi-item multiperiod inventory control problemconsidering quantity discounts interest and inflation factorsGilding [19] proposes the optimal inventory replenishmentschedule with time-dependent demand and inflation in afinite time horizon Pal et al [20] formulate a productioninventory model with the effect of inflation under fuzzy envi-ronment These papers are from the viewpoint of inventorymanagement andmainly focus on the optimal replenishmentor production strategy for one single enterprise under infla-tionary conditionsTheydonot consider the effect of inflationon a supply chain In addition they do not also consideroption contracts

We now examine the literature on supply chain man-agement with option contracts Li et al [21] investigate thevalue of forward contracts and option contracts on a supplier-retailer system with asymmetric information in which bothmembers face price and demand uncertainty Zhao et al [22]adopt a cooperative game method to study the coordina-tion problem with option contracts They find that optioncontracts can coordinate the supply chain to achieve Paretoimprovement Xu [23] obtains the optimal procurementand production decisions in a supplier-manufacturer systemunder option contracts when the uncertainties such as thesupplierrsquos production yield the instant price and the marketdemand are considered Fu et al [24] concentrate on a single-period portfolio procurement problem and then extend thediscussion to a two-period setting when both the demandand the spot price are random Xia et al [25] analyze how toshare the supply disruption risk and the demand random riskunder two different contract mechanisms wholesale pricecontracts and option contracts They find that the benefit ofreliable supplier depends on the type of contracts and thebuyer prefers the reliable supplier under option contractsChen and Shen [26] describe that portfolio contracts bringmore benefit for the supply chain members than wholesaleprice contracts in the presence of a service requirementTheyalso consider the conditions for the channel coordinationwith a service requirement Liang et al [27] analyze the value

of option contracts on the relief material supply chain andfind that option contract mechanism can help reduce theimpact of disaster and maintain social stability Lee et al[28] study the multiple-supplier procurement problem withcapacity constraints and fixed ordering costs when optioncontracts and spot market are simultaneously used by thebuyer Liu et al [29] introduce option contracts into containerplanning mechanism and analyze the application strategiesof unilateral and bidirectional option contracts in differentpractical scenarios Chen et al [30] investigate how to applyoption contracts to coordinate a channel which contains onesupplier and one loss-averse retailer Hu et al [31] establishthe decision-makingmodel in amanufacturer-retailer systemwith option contracts and partial backordering when boththe production yield and the market demand are stochasticThese papers do not discuss the effect of inflation on a supplychain

We eventually review the literature on supply chain coor-dination with contracts Several contracts such as revenuesharing contracts and buyback contracts are proved to be anefficient approach to coordinate the supply chain under dif-ferent scenarios Linh andHong [32] study how to coordinatea supply chain through revenue sharing contracts in the two-period setting Xiao et al [33 34] design a mechanism tocoordinate a manufacturer-retailer system through buybackcontracts and investigate the effects of the consumer returnon the coordination strategy Chiu et al [35] describe howthe policy which contains wholesale price channel rebateand returns can achieve the supply chain coordination withboth additive and multiplicative price-dependent demandsA comprehensive analysis on supply chain coordination withcontracts can be found in Cachon [36] In these papers thediscussion on the sellerrsquos production decision is neglected andonly the buyerrsquos order quantity needs coordination Howeverthe seller can plan the production schedule that maximizesits own profit in the presence of option contracts To thebest of our knowledge only two papers [26 30] discuss howto coordinate both the buyerrsquos ordering quantity and thesellerrsquos production quantity All the other papers consideringoption contracts assume that the seller adopts the make-to-order production policy However these two papers abovedo not consider both option contracts and portfolio contractssimultaneously In addition all the papers do not discuss theeffect of inflation on the supply chain

3 Model Formulation and Assumptions

We consider a one-period two-stage supply chain in whichone supplier manufactures one type of seasonal productsand one retailer purchases from the upstream supplier andsells to the downstream consumers The retailer obtainsthe products through three different contracts respectivelywholesale price contracts option contracts and portfoliocontracts consisting of wholesale price contracts and optioncontracts Under wholesale price contracts the retailer placesa firm order denoted as 119876

1199030 at unit wholesale price 119908 before

the selling period Then the supplier receives the firm orderand manufactures the products up to 119876

1199040at unit production

cost 119888 When the selling period starts the retailer obtains


the products through the firm order Under option contractsthe retailer only purchases call options denoted as 119902

1199031 at unit

purchase price 119888119900before the selling periodThen the supplier

receives the options order and manufactures the productsup to 119876

1199041 During the selling period the retailer observes

the realized market demand and then determines how muchproducts to obtain through the options order at unit exerciseprice 119888

119890 Under portfolio contracts the retailer places a firm

order denoted as 1198761199032 and purchases call options denoted as

1199021199032 Then the supplier receives the orders of the two different

types and manufactures the products up to 1198761199042 When

the selling period starts the retailer obtains the productsthrough the firm order During the selling period the retailerobtains the additional products through the options orderThe retailer incurs a unit shortage cost 119892

119903for each unsatisfied

demand The supplier incurs a unit penalty cost 119892119904for each

exercised option that cannot be immediately filled Thus thesupplierrsquos unit penalty cost 119892

119904represents the cost to obtain

an additional unit of product by expediting production orbuying from an alternative source

In the seasonal product industry the length of sellingperiod is short but the length of production lead time islong [37] During the production lead time both the retailprice and the market demand vary with time owing to theeffect of inflation In reality sometimes the production leadtime is considered to be exogenous [33 34] The empiricalstudies [38 39] show that the length of production lead timeis uncertain We assume that the length of production leadtime denoted as 119905 is an exogenous random variable over(0 119879) with probability density function (PDF) 119892(sdot) Similarto Jaggi and Khanna [11] we assume that unit retail pricedenoted as 119901(119905) follows a continuous exponential growthduring the production lead time under inflation scenariosthat is 119901(119905) = 119901119890

120574119905 where 119901 is the initial retail price and120574 (120574 gt 0) is the price rising factor Similar to Xiao et al[33 34] we assume that the market demand denoted as119863(119905) is decomposed into a deterministic form 119889(119905) and anondeterministic error 120585 that is 119863(119905) = 119889(119905) + 120585 where 119889(119905)is a decreasing function of the production lead time owingto the effect of inflation and 120585 is a random variable over(0 +infin) with probability density function (PDF) 119891(sdot) andstrictly increasing cumulative distribution function (CDF)119865(sdot) 119865(0) = 0 119864(120585) = 120583 and 119865(sdot) = 1 minus 119865(sdot) denote thetail distribution As we know various time-varying functionscan be used to describe 119889(119905) such as linear Weibull andexponential distribution forms Similar to Tripathi [14] anexponentially decreasing pattern is used to describe themarket contraction caused by the effect of inflation for thepurpose of making the model analytically tractable that is119889(119905) = 120582119890

minus120572119905 where 120582 is the initial market scale and 120572 (120572 gt 0)is the demand contraction factor

Throughout this paper we use the parameters and vari-ables as shown in ldquoNotationsrdquo

We assume that the supply chain members are rationaland self-interested and all the information available is sym-metric between the supplier and the retailer Moreover weassume that the retailerrsquos initial inventory is zero and anyexcess product either owned by the retailer or by the supplier

can be ignored Furthermore we assume that 119901119890120574119905 gt 119888119900+ 119888119890gt

119908 gt 119888 and 119888119900+ 119888119890gt 119908 gt 119888

119900 The first condition can ensure

profits for two parties The second condition can ensure thatthe retailer places a firm order and purchases call optionssimultaneously

4 Supply Chain Models

In this section we plan to study the retailerrsquos optimalordering policy and the supplierrsquos optimal production policyconsidering the effect of inflation under three different typesof contracting arrangement wholesale price contracts optioncontracts and portfolio contracts

41 Wholesale Price Contracts Model

411 Optimal Ordering Policy under Wholesale Price Con-tracts Since wholesale price contracts are widely used inpractice we use wholesale price contracts model as thebenchmark and compare with option contracts model andportfolio contracts model developed in the remaining part

Under wholesale price contracts only products are pur-chased from the upstream supplier and the expected profit ofthe retailer denoted as Π

1199030(1198761199030) is given by

Π1199030(1198761199030) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199030] minus 119908119876

1199030

minus 119892119903[119863 (119905) minus 119876

1199030]+

119892 (119905) 119889119905

(1)

The first term is the sales revenue The second term is thecosts of purchasing products and the last term is the shortagecost Then the above equation can be simplified as

Π1199030(1198761199030)

= 1198761199030int

119879

0

(119901119890120574119905

+ 119892119903minus 119908)119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198761199030minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(2)

As to the retailerrsquos optimal ordering policy under whole-sale price contracts we can derive the following proposition

Proposition 1 Under wholesale price contracts the retailerrsquosoptimal firm order quantity 119876lowast

1199030is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

1199030minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119908 (3)

Proof From (2) we can derive that 119889Π1199030(1198761199030)1198891198761199030

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(1198761199030minus120582119890minus120572119905

)119892(119905)119889119905 and1198892

Π1199030(1198761199030)1198891198762

1199030= minusint119879

0

(119901119890120574119905

+119892119903)119891(1198761199030minus120582119890minus120572119905

)119892(119905)119889119905 lt 0so Π1199030(1198761199030) is concave in 119876

1199030 Let 119889Π

1199030(1198761199030)1198891198761199030

= 0we can obtain that the optimal solution to (2) is int

119879

0

(119901119890120574119905

+

119892119903)119865(119876lowast

1199030minus 120582119890minus120572119905

)119892(119905)119889119905 = 119908


This proposition shows that the effect of inflation has asignificant impact on the retailerrsquos optimal ordering policyunder wholesale price contracts We have the followingcorollary

Corollary 2 The retailerrsquos optimal firm order quantity underwholesale price contracts is decreasing in 120572 and increasing in 120574

Proof Let 1198710(119876lowast

1199030) = int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876lowast

1199030minus

120582119890minus120572119905

)119892(119905)119889119905 minus 119908 we can derive that 120597119876lowast

1199030120597120572 =

minus(1205971198710(119876lowast

1199030)120597120572)(120597119871

0(119876lowast

1199030)120597119876lowast

1199030) = minus int

119879

0

120582119905119890minus120572119905

(119901119890120574119905

+

119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876lowast

1199030120597120574 = minus(120597119871

0(119876lowast

1199030)120597120574)(120597119871

0(119876lowast

1199030)120597119876lowast

1199030) =

int119879

0

119901119905119890120574119905

119865(119876lowast

1199030minus 120582119890

minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903)119891(119876lowast

1199030minus

120582119890minus120572119905

)119892(119905)119889119905 gt 0 that is the retailerrsquos optimal firm orderquantity under wholesale price contracts is decreasing in 120572

and increasing in 120574

From Corollary 2 we can see that when the demandcontraction factor 120572 grows the retailer will reduce the sizeof the firm order When the price rising factor 120574 grows theretailer will enlarge the size of the firm order Since both priceand demand vary in two opposite directions due to the effectof inflation this poses a challenge for the retailer to decidewhether to increase or decrease the size of the firm orderAt this moment the retailer needs to seek the right balancebetween the rising price and the shrinking demand Whenthe increase in the retail price is more obvious the retailerwill increase the firm order quantity When the decrease inthemarket demand is more obvious the retailer will decreasethe firm order quantity

412 Optimal Production Policy under Wholesale Price Con-tracts Since the retailer obtains the products through thefirmorder at the beginning of the selling period the supplierrsquosoptimal production quantity is equivalent to the retailerrsquosoptimal firm order quantity under wholesale price contractsThat is 119876

lowast

1199040= 119876

lowast

1199030 Obviously the supplierrsquos optimal

production quantity under wholesale price contracts is alsodecreasing in 120572 and increasing in 120574

Under wholesale price contracts the optimal expectedprofit of the supplier denoted as Π

1199040(119876lowast

1199040) is given by

Π1199040(119876lowast

1199040) = (119908 minus 119888)119876

lowast

1199040= (119908 minus 119888)119876

lowast

1199030 (4)

42 Option Contracts Model

421 Optimal Ordering Policy under Option ContractsUnder option contracts only call options are purchased fromthe upstream supplier and the expected profit of the retailerdenoted as Π

1199031(1199021199031) is given by

Π1199031(1199021199031) = int

119879

0

119901 (119905)min [119863 (119905) 1199021199031] minus 1198881199001199021199031

minus 119888119890min [119863 (119905) 119902

1199031] minus 119892119903[119863 (119905) minus 119902

1199031]+

119892 (119905) 119889119905

(5)

The first term is the sales revenue The second term is thecosts of purchasing call optionsThe third term is the costs ofexercising call options and the last term is the shortage costThen the above equation can be simplified as

Π1199031(1199021199031)

= 1199021199031int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1199021199031minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(6)

As to the retailerrsquos optimal ordering policy under optioncontracts we can derive the following proposition

Proposition 3 Under option contracts the retailerrsquos optimaloptions order quantity 119902lowast

1199031is

int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119902

lowast

1199031minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (7)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1199021199031

minus

120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199031(1199021199031)1198891199022

1199031= minusint

119879

0

(119901119890120574119905

+ 119892119903minus

119888119890)119891(1199021199031

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 so Π1199031(1199021199031) is concave in 119902

1199031 Let

119889Π1199031(1199021199031)1198891199021199031

= 0 we can obtain that the optimal solutionto (6) is int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 = 119888119900

This proposition shows that the effect of inflation has asignificant impact on the retailerrsquos optimal ordering policyunder option contracts We have the following corollary

Corollary 4 The retailerrsquos optimal options order quantityunder option contracts is decreasing in 120572 and increasing in 120574

Proof Let1198711(119902lowast

1199031)=int119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905minus 119888119900 we

can derive that 120597119902lowast1199031120597120572 = minus(120597119871

1(119902lowast

1199031)120597120572)(120597119871

1(119902lowast

1199031)120597119902lowast

1199031) =

minus int119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+

119892119903minus 119888119890)119891(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119902lowast

1199031120597120574 = minus(120597119871

1(119902lowast

1199031)

120597120574)(1205971198711(119902lowast

1199031)120597119902lowast

1199031) = int

119879

0

119901119905119890120574119905

119865(119902lowast

1199031minus 120582119890

minus120572119905

)119892(119905)119889119905

int119879

0

(119901119890120574119905

+119892119903minus119888119890)119891(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905 gt 0 that is the retailerrsquosoptimal options order quantity under option contracts isdecreasing in 120572 and increasing in 120574

FromCorollary 4 we can see that whenmore call optionsare purchased the retailer has a higher ability to be resistantto the price rising and a lower ability to be resistant to thedemand contractionWhen fewer call options are purchasedthe retailer has a lower ability to be resistant to the pricerising and a higher ability to be resistant to the demandcontraction Owing to the effect of inflation the retailermust make careful observations on the changes in price anddemand and then decide whether to increase or decrease theoptions order When the increase in the retail price is more


obvious the retailer will order more call options When thedecrease in the market demand is more obvious the retailerwill order fewer call options

422 Optimal Production Policy under Option ContractsSince the retailer obtains the products through the optionsorder during the selling period and the retailer may notexercise all the options purchased the supplierrsquos optimalproduction quantity cannot exceed the retailerrsquos optimaloptions order quantity under option contracts That is 119876

1199041le

119902lowast

1199031Under option contracts the optimal expected profit of the

supplier denoted as Π1199041(1198761199041) is given by

Π1199041(1198761199041) = int

119879

0

119888119900119902lowast

1199031+ 119888119890min [119863 (119905) 119902

lowast

1199031]

minus 119892119904[min (119863 (119905) 119902

lowast

1199031) minus 1198761199041]+

minus 1198881198761199041 119892 (119905) 119889119905

(8)

The first term is the revenue realized from options salesThe second term is the revenue realized from exercisedoptions The third term is the penalty cost and the last termis the production cost Then the above equation can besimplified as

Π1199041(1198761199041) = (119888119900+ 119888119890minus 119892119904) 119902lowast

1199031

+ (119892119904minus 119888119890) int

119879

0

int

119902lowast

1199031minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199041

minus 119892119904int

119879

0

int

1198761199041minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(9)

The supplierrsquos decision problem under option contracts isdescribed as

max1198761199041gt0

Π1199041(1198761199041)

st 1198761199041

le 119902lowast

1199031

(10)

As to the supplierrsquos optimal production policy underoption contracts we can derive the following proposition

Proposition 5 Under option contracts the supplierrsquos optimalproduction quantity 119876lowast

1199041satisfies

119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(11)

where int1198790

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199041(1198761199041)1198891198762

1199041=

minus119892119904int119879

0

119891(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 so Π1199041(1198761199041) is concave

in 1198761199041 Let 119889Π

1199041(1198761199041)1198891198761199041

= 0 we can derive that theoptimal solution to (9) is int119879

0

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)

119892119904

Considering the constraint in (10) the supplierrsquos optimalproduction quantity under option contracts satisfies

119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(12)

This proposition shows that owing to the productionconstraint condition the supplierrsquos optimal production quan-tity under option contracts is expressed as an interval If119876120593

1199041lt 119902lowast

1199031 the production constraint condition is inef-

fective If 119876120593

1199041ge 119902lowast

1199031 the production constraint condition

is effective At this point the supplier will try the best toraise the production quantity so as to improve the expectedprofit Obviously the production quantity equivalent to theoptions order quantity is the supplierrsquos best choice Moreoverthis proposition also shows that the effect of inflation alsohas a significant impact on the supplierrsquos optimal produc-tion policy under option contracts We have the followingcorollary

Corollary 6 The supplierrsquos optimal production quantity underoption contracts is decreasing in 120572 and nondecreasing in 120574

Proof Let 1198671(119876120593

1199041) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199041minus

120582119890minus120572119905

)119892(119905)119889119905 From Proposition 5 we see that if119876120593

1199041lt 119902

lowast

1199031 then 119876

lowast

1199041= 119876

120593

1199041 We can deduce that

120597119876120593

1199041120597120572 = minus(120597119867

1(119876120593

1199041)120597120572)(120597119867

1(119876120593

1199041)120597119876120593

1199041) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

and 120597119876120593

1199041120597120574 = minus(120597119867

1(119876120593

1199041)120597120574)(120597119867

1(119876120593

1199041)120597119876120593

1199041) = 0 so

in this case the supplierrsquos optimal production quantity isdecreasing in 120572 and constant in 120574 If119876120593

1199041ge 119902lowast

1199031 then119876

lowast

1199041= 119902lowast

1199031

The supplierrsquos optimal production quantity is decreasing in 120572

and increasing in 120574 Hence the supplierrsquos optimal productionquantity under option contracts is decreasing in 120572 andnondecreasing in 120574

From Corollary 6 we can see that when the increase inthe retail price is more obvious the supplier will observe theoperation status carefully and then decidewhether to increaseor maintain the production quantity When the decrease inthe market demand is more obvious the supplier will reducethe production quantity

43 Portfolio Contracts Model

431 Optimal Ordering Policy under Portfolio ContractsUnder portfolio contracts both products and call options arepurchased from the upstream supplier and the expected profitof the retailer denoted as Π

1199032(1198761199032 1199021199032) is given by

Π1199032(1198761199032 1199021199032) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199032

+ 1199021199032]

minus 1199081198761199032

minus 1198881199001199021199032

minus 119888119890min [(119863 (119905) minus 119876

1199032)+

1199021199032]

minus 119892119903[119863 (119905) minus (119876

1199032+ 1199021199032)]+

119892 (119905) 119889119905

(13)


The first term is the sales revenue The second term isthe costs of purchasing products The third term is the costsof purchasing call options The fourth term is the costs ofexercising call options and the last term is the shortagecost Set 119876

2= 1198761199032

+ 1199021199032 Note that determining (119876

1199032 1199021199032) is

equivalent to determining (1198761199032 1198762)Then the above function

can be rewritten as

Π1199032(1198761199032 1198762)

= (119888119900+ 119888119890minus 119908)119876

1199032

minus 119888119890int

119879

0

int

1198761199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ 1198762int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198762minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(14)

As to the retailerrsquos optimal ordering policy under portfo-lio contracts we can derive the following proposition

Proposition 7 Under portfolio contracts the retailerrsquos optimalfirm order quantity 119876lowast

1199032is

int

119879

0

119865 (119876lowast

1199032minus 120582119890minus120572119905

) 119892 (119905) 119889119905 =119888119900+ 119888119890minus 119908

119888119890

(15)

The retailerrsquos optimal total order quantity 119876lowast2is

int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119876

lowast

2minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (16)

Proof From (14) we can derive that 120597Π1199032(1198761199032 1198762)1205971198761199032

=

(119888119900+119888119890minus119908)minus119888

119890int119879

0

119865(1198761199032minus120582119890minus120572119905

)119892(119905)119889119905 1205972Π1199032(1198761199032 1198762)1205971198762

1199032=

minus119888119890int119879

0

119891(1198761199032

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 120597Π1199032(1198761199032 1198762)1205971198762

=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1198762minus

120582119890minus120572119905

)119892(119905)119889119905 1205972

Π1199032(1198761199032 1198762)1205971198762

2= minusint

119879

0

(119901119890120574119905

+ 119892119903

minus

119888119890)119891(1198762minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 1205972

Π1199032(1198761199032 1198762)12059711987611990321205971198762=

1205972

Π1199032(1198761199032 1198762)12059711987621205971198761199032

= 0 Hence

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972

Π1199032(1198761199032 1198762)

12059711987621199032

1205972

Π1199032(1198761199032 1198762)

12059711987611990321205971198762

1205972

Π1199032(1198761199032 1198762)

12059711987621205971198761199032

1205972

Π1199032(1198761199032 1198762)

12059711987622

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt 0 (17)

So Π1199032(1198761199032 1198762) is concave in 119876

1199032and 119876

2 Let 120597Π

1199032(1198761199032

1198762)1205971198761199032

= 0 and 120597Π1199032(1198761199032 1198762)1205971198762= 0 we can obtain that

the optimal solution to (14) is int1198790

119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+

119888119890minus119908)119888

119890and int

119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

2minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900

From Proposition 7 we can deduce that the retailerrsquosoptimal options order quantity is described as 119902lowast

1199032= 119876lowast

2minus119876lowast

1199032

which implies that the optimal total order quantity is alwayshigher than the optimal firm order quantity under portfoliocontracts Note that 119876lowast

2gt 119876lowast

1199032is equivalent to 119888

119900lt ((119901119890

120574119905

+

119892119903minus119888119890)(119901119890120574119905

+119892119903))119908This inequality shows that if the supplier

charges an exorbitant option price the retailer will refuse toorder any options Moreover this proposition shows that theeffect of inflation has a significant impact on the retailerrsquosoptimal ordering policy under portfolio contracts We havethe following corollary

Corollary 8 The retailerrsquos optimal firm order quantity underportfolio contracts is decreasing in 120572 and constant in 120574


1199032) = (119888

119900+ 119888119890

minus 119908) minus 119888119890int119879

0

119865(119876lowast

1199032minus

120582119890minus120572119905

)119892(119905)119889119905 we can derive that 120597119876lowast

1199032120597120572 =

minus(1205971198712(119876lowast

1199032)120597120572)(120597119871

2(119876lowast

1199032)120597119876lowast


119879

0

120582119905119890minus120572119905

119891(119876lowast

1199032minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119876lowast

1199032120597120574 =

minus(1205971198712(119876lowast

1199032)120597120574)(120597119871

2(119876lowast

1199032)120597119876lowast

1199032) = 0 that is the retailerrsquos

optimal firm order quantity under portfolio contracts isdecreasing in 120572 and constant in 120574

Corollary 9 The retailerrsquos optimal total order quantity underportfolio contracts is decreasing in 120572 and increasing in 120574


2) = int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 minus

119888119900 we can derive that 120597119876

lowast

2120597120572 = minus(120597119871

3(119876lowast

2)120597120572)

(1205971198713(119876lowast

2)120597119876lowast

2) = minus int

119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903

minus 119888119890)119891(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and120597119876lowast

2120597120574 = minus(120597119871

3(119876lowast

2)120597120574)(120597119871

3(119876lowast

2)120597119876lowast

2) = int119879

0

119901119905119890120574119905

119865(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 gt 0 thatis the retailerrsquos optimal total order quantity under portfoliocontracts is decreasing in 120572 and increasing in 120574

From Corollaries 8 and 9 we can see that when the risingdegree of price runs faster than the falling degree of demandthe retailer will raise the total order quantity through increas-ing the options order quantity It is worth noting that theretailer does not attempt to make any alteration in the firmorder quantity under this situation When the falling degreeof demand runs faster than the rising degree of price theretailer will reduce both the firm order quantity and the totalorder quantity

432 Optimal Production Policy under Portfolio ContractsSince the firm order is required to be delivered to thedownstream retailer at the beginning of the selling period thesupplierrsquos optimal production quantity cannot remain belowthe retailerrsquos optimal firm order quantity under portfoliocontracts Moreover since the retailer can obtain the addi-tional products through the options order during the sellingperiod and the options exercising quantity may not exceedthe options order quantity the supplierrsquos optimal productionquantity cannot surpass the retailerrsquos optimal total orderquantity under portfolio contracts That is 119876lowast

1199032le 1198761199042

le 119876lowast

2


Under portfolio contracts the optimal expected profit ofthe supplier denoted as Π

1199042(1198761199042) is given by

Π1199042(1198761199042) = int

119879

0

119908119876lowast

1199032+ 119888119900(119876lowast

2minus 119876lowast

1199032)

+ 119888119890min [(119863 (119905) minus 119876

lowast

1199032)+

119876lowast

2minus 119876lowast

1199032]

minus 119892119904[min (119863 (119905) 119876

lowast

2) minus 1198761199042]+

minus 1198881198761199042 119892 (119905) 119889119905

(18)

The first term is the revenue realized from firm ordersThe second term is the revenue realized from options salesThe third term is the revenue realized from exercised optionsThe fourth term is the penalty cost and the last term is theproduction cost Then the above equation can be simplifiedas

Π1199042(1198761199042) = (119888119900+ 119888119890minus 119892119904) 119876lowast

2

+ (119892119904minus 119888119890) int

119879

0

int

119876lowast

2minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119908 minus 119888119900minus 119888119890) 119876lowast

1199032

+ 119888119890int

119879

0

int

119876lowast

1199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199042

minus 119892119904int

119879

0

int

1198761199042minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(19)

The supplierrsquos decision problem under portfolio contractsis described as

max1198761199042gt0

Π1199042(1198761199042)

st 119876lowast

1199032le 1198761199042

le 119876lowast

2

(20)

As to the supplierrsquos optimal production policy underportfolio contracts we can derive the following proposition

Proposition 10 Under portfolio contracts the supplierrsquos opti-mal production quantity 119876lowast

1199042satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(21)

where int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199042(1198761199042)1198891198762

1199042=

minus119892119904int119879

0

119891(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 so Π1199042(1198761199042) is concave in

1198761199042 Let 119889Π

1199042(1198761199042)1198891198761199042

= 0 we can derive that the optimalsolution to (19) is int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904

Considering the constraint in (20) the supplierrsquos optimalproduction quantity under portfolio contracts satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(22)

This proposition shows that owing to the productionconstraint condition the supplierrsquos optimal production quan-tity under portfolio contracts is expressed as an interval If119876120593

1199042le 119876lowast

1199032 the constraint condition plays an important role

in the production decision At this point the supplier willmake every effort to reduce the production quantity so asto increase the expected profit Obviously the productionquantity equivalent to the firm order quantity is the supplierrsquosbest choice If119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2 the constraint condition plays

no role in the production decision If1198761205931199042

ge 119876lowast

2 the constraint

condition plays an important role in the production decisionAt this point the supplier will try the best to raise theproduction quantity so as to increase the expected profitObviously the production quantity equivalent to the totalorder quantity is the supplierrsquos best choice Moreover thisproposition also shows that the effect of inflation also hasa significant impact on the supplierrsquos optimal productionpolicy under portfolio contracts We have the followingcorollary

Corollary 11 The supplierrsquos optimal production quantityunder portfolio contracts is decreasing in 120572 and nondecreasingin 120574

Proof Let 1198672(119876120593

1199042) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905FromProposition 10 we see that if119876120593

1199041lt 119876lowast

1199032 then119876

lowast

1199042= 119876lowast

1199032


and constant in 120574 If119876lowast1199032

lt 119876120593

1199041lt 119876lowast

2 then119876

lowast

1199042= 119876120593

1199042 We can

deduce that 1205971198761205931199042120597120572 = minus(120597119867

2(119876120593

1199042)120597120572)(120597119867

2(119876120593

1199042)120597119876120593

1199042) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876120593

1199042120597120574 = minus(120597119867

2(119876120593

1199042)120597120574)(120597119867

2(119876120593

1199042)120597119876120593

1199042) = 0

so in this case the supplierrsquos optimal production quantity isdecreasing in 120572 and constant in 120574 If119876120593

1199041ge 119876lowast

2 then119876

lowast

1199041= 119876lowast

2


and increasing in 120574 Hence the supplierrsquos optimal productionquantity under portfolio contracts is decreasing in 120572 andnondecreasing in 120574

FromCorollary 11 we can see that when the rising degreeof price runs faster than the falling degree of demandthe supplier will observe the operation status carefully andthen decide whether to increase or maintain the productionquantity When the falling degree of demand runs fasterthan the rising degree of price the supplier will reduce theproduction quantity


5 The Impact of Option Contracts

In this section we plan to explore the impact of optioncontracts on supply chain through comparingwith the resultsunder different contracting arrangements

51The Impact of Option Contracts on Supply ChainDecisionsComparing the retailerrsquos optimal order quantity among thesethree different contracts we consider the impact of optioncontracts on the retailerrsquos ordering decision

Proposition 12 The retailerrsquos optimal order quantity underthree different types of procurement contracts is as follows themaximum is the optimal total order quantity under portfoliocontracts and the optimal options order quantity under optioncontracts the medium is the optimal firm order quantity underwholesale price contracts and the minimum is the optimal firmorder quantity under portfolio contracts That is 119876lowast

1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2

Proof From (7) and (16) we can deduce that 119902lowast1199031

= 119876lowast

2 From

(3) and (15) we can deduce that (119889Π1199030(1198761199030)1198891198761199030)|1198761199030=119876lowast

1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt

(119908(119908 minus 119888119900))[(119888119900+ 119888119890minus 119908) minus 119888

119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876

1199030 it follows that119876lowast

1199030gt

119876lowast

1199032From (3) and (7) we can deduce that

(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905

)119892(119905)119889119905 gt 0 Recalling that Π1199031(1199021199031) is concave in 119902

1199031 it

follows that 119902lowast1199031

gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2

Proposition 12 explains the change that occurs in theretailerrsquos optimal ordering policy after introducing optioncontracts under inflation scenarios Since the same marketenvironment is faced the retailerrsquos optimal options orderquantity under option contracts is equivalent to the retailerrsquosoptimal total order quantity under portfolio contracts More-over since call options give only the right but not theobligation to obtain the products the retailer can chooseto exercise call options or not use them at all Obviouslythe presence of option contracts provides more flexibility forthe retailer than wholesale price contracts Thus the optimalaggregate order quantity under portfolio contracts is alwayshigher than the optimal firm order quantity under whole-sale price contracts In addition the optimal options orderquantity under option contracts is always higher than theoptimal firm order quantity under wholesale price contractsIn other words comparing with the case of wholesale pricecontracts the retailer will increase the order quantity so asto reduce the shortage cost caused by unsatisfied demand inthe presence of option contracts Furthermore call optionsgive the retailer the right to adjust the order quantity upwardsObviously the presence of option contracts will suppress theretailerrsquos purchasing behavior and restrains increasing thefirm order quantity Thus the optimal firm order quantity

under wholesale price contracts is higher than that underportfolio contracts

Contrasting the supplierrsquos optimal production quantityamong these three different contracts we consider the impactof option contracts on the supplierrsquos production decision

Proposition 13 If 119876120593119904

isin (0 119876lowast

1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593

119904isin (119876lowast

1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593

1199042 From Propositions 5 and 10 we

can derive the following conclusions If 119876120593119904


1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593

119904 We can obtain

119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593

119904 We can obtain 119876

lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast

2 We can obtain119876

lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proposition 13 explains the change that occurs in thesupplierrsquos optimal production policy after introducing optioncontracts under inflation scenarios Owing to the produc-tion constraint condition the supplierrsquos production decisionbecomes more complicated but more flexible after intro-ducing option contracts Moreover the supplierrsquos optimalproduction quantity under portfolio contracts is not lessthan that under option contracts The main reason is thatthe supplier must deliver the firm order at the beginningof the selling period under portfolio contracts In contrastthe supplier does not need to deliver a certain quantity ofproducts at the beginning of the selling period under optioncontracts

Nowwe summarize the conclusions regarding the impactof option contracts on the supply chain membersrsquo optimaldecision policies under inflation scenarios in Table 1

Table 1 shows that the introduction of option contractswill suppress the urge to buy more products and make theretailer decrease the size of the firm order under inflationscenariosMoreover the introduction of option contracts willprompt increasing the size of the total order and help theretailer reduce the shortage risk under inflation scenariosFurthermore the introduction of option contracts will con-tribute to adjusting the production quantity and make thesupplier more flexible under inflation scenarios

52 The Impact of Option Contracts on Supply Chain Per-formance Comparing the retailerrsquos optimal expected profitamong these three different contracts we examine the impactof option contracts on the retailerrsquos performance

Proposition 14 The retailerrsquos optimal expected profit underthree different types of procurement contracts is as follows themaximum is that under portfolio contracts the medium isthat under option contracts and the minimum is that underwholesale price contracts That is Π

1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)

Proof First we compare Π1199032(119876lowast

1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast

1199031) we can derive that

Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905


Table 1 The impact of option contracts on supply chain decisions

Wholesale price contracts Option contracts Portfolio contractsOptimal firm order quantity Maximum mdash MinimumOptimal options order quantity mdash Maximum MinimumOptimal total order quantity Minimum MaximumOptimal production quantity Fixed value Interval value Interval value

Because Δ1(0) = 0 and (119889Δ

1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0

gt 0we can obtain that Π

1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we

compare Π1199031(119902lowast

1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast

1199030) we derive that Δ

2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905

)119892(119905)119889119905] gt 0 we obtain that Π1199031(119876lowast

1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast

1199030) we obtain thatΠ

1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)

Proposition 14 explains the change that occurs in theretailerrsquos optimal expected profit after introducing optioncontracts under inflationary conditions Since call optionsgive the right to obtain the products based on the realizedmarket demand the retailer can adjust the order quantityupwards in the presence of option contracts Obviously theintroduction of option contracts brings more benefits to theretailer than wholesale price contracts Thus the retailerrsquosoptimal expected profit under wholesale price contracts islower than that under the other two contracts Moreover wecan see that wholesale price contracts are characterized byonly one parameter unit wholesale price of products Optioncontracts are characterized by two parameters one is unitpurchase price of call options and the other is unit exerciseprice of call options Since 119888

119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the

retailerrsquos procurement cost under portfolio contracts is alwayslower than that under option contracts Thus the retailerrsquosoptimal expected profit under portfolio contracts is higherthan that under option contracts In conclusion portfoliocontracts are the most preferable for the retailer among thesethree different contracts under inflation scenarios

Comparing the supplierrsquos optimal expected profit amongthese three different contracts we examine the impact ofoption contracts on the supplierrsquos performance

Proposition 15 The supplierrsquos optimal expected profit underthree different types of procurement contracts is as follows themaximum profit is that under option contracts the medium isthat under portfolio contracts and the minimum is that underwholesale price contracts That is Π

1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast

1199042minus 119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905

119865(119909)119892(119905)119889119909 119889119905 Fromthe previous analysis we can see that (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905

119865(119909)119892(119905)119889119909 119889119905 = 0 We can obtainthat Δ

3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905

119865(119909)119892(119905)119889119909 119889119905 gt 0 Wecan obtain that Δ

3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)

Then we compare Π1199042(119876lowast

1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast

1199040) we can derive that Δ

4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905minus(119908minus119888)119876lowast

1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast

1199030 We can see that Δ

4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From

(3) we can see that 119876lowast1199030is decreasing in 119908 If 119876120593

1199042gt 119876lowast

1199032 then

119889119876lowast

1199042119889119908 = 0 We can obtain that (119889Δ

4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)

Proposition 15 explains the change that occurs in thesupplierrsquos optimal expected profit after introducing optioncontracts under inflation scenarios When wholesale pricecontracts are employed the supplier determines how manyproducts to manufacture based on the retailerrsquos firm orderWhen option contracts or portfolio contracts are employedthe supplier can determine the production quantity thatmaximizes its own interest based on the retailerrsquos flexiblepurchasing Obviously the introduction of option contractsbrings more benefits to the supplier than wholesale pricecontracts Moreover since there is no constraint for theminimumproduction quantity the supplier can plan the pro-duction schedule withmore flexibility under option contractsthrough comparing with the case of portfolio contractsThusthe supplierrsquos optimal expected profit under option contractsis higher than that under portfolio contracts In conclusionoption contracts are the most preferable for the supplieramong the three different contracts under inflation scenar-ios Furthermore owing to the supplierrsquos market dominantposition the supply chain is willing to implement optioncontracts ultimately

Nowwe summarize the conclusions regarding the impactof option contracts on the supply chain membersrsquo optimalexpected profit under inflation scenarios in Table 2


Table 2 The impact of option contracts on supply chain performance

Wholesale price contracts Option contracts Portfolio contractsRetailerrsquos optimal expected profit Minimum Medium MaximumSupplierrsquos optimal expected profit Minimum Maximum Medium

Table 2 shows that the introduction of option contractsbenefits both the supplier and the retailer Obviously thepresence of option contracts is a good thing for the sup-ply chain members under inflation scenarios Moreoverthe retailer prefers ordering products and purchasing calloptions but the supplier is only inclined to provide calloptions under inflation scenarios Furthermore the supplychain will implement option contracts ultimately underinflation scenarios

6 Supply Chain Bilateral Coordination

In this section we plan to study how to design optioncontract mechanism to achieve supply chain coordinationunder inflation scenarios

To derive the optimal expected profit of the supply chainwe take the channel as a centralized entity and consider acentral controller which decides the production quantity 119876

119868

The expected profit of the supply chain denoted asΠ119868(119876119868) is

given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)

The first term is the sales revenue The second term is theshortage cost and the last term is the production cost Thenthe above equation can be simplified as

Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)

As to the optimal production policy of the supply chainsystem under centralized decision-making we can derive thefollowing proposition

Proposition 16 The central controllerrsquos optimal productionquantity 119876lowast

119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can

obtain that the optimal solution to (24) is int1198790

(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888

From the analysis above we see that the supply chain willimplement option contracts ultimately under inflation sce-narios In the remaining part we plan to discuss how optioncontracts should be set to attain supply chain coordinationunder inflation scenarios

As we know the supplier is always assumed to committo manufacturing the products up to the retailerrsquos order inthe tradition approach [36] In this case the issue on how tomake a noncentralized supply chain achieve coordination canbe simplified to the unilateral coordination from the retailerrsquosperspective only However the supplier does not complywith the make-to-order policy and decides the productionquantity according to the profit maximization rule underoption contracts The supply chain coordination cannot beachieved according to the tradition unilateral coordinationmechanism Similar to Chen et al [26 30] we design thebilateral coordinationmechanism fromboth the retailerrsquos andthe supplierrsquos perspectives

Proposition 17 When 119892119904gt 119901119890120574119905

+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+119888119890minus119888)119888119890are satisfied the supply chainwith

option contracts can be coordinated under inflation scenarios

Proof Since the supplier decides the optimal productionquantity that maximizes its own profit under option con-tracts both the retailerrsquos order quantity and the supplierrsquosproduction quantity need coordinate Both Propositions 5and 16 provide a condition tomotivate the supplier to producethe same quantity as that in the coordinated supply chainThen 119876

120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From

Propositions 3 and 16 we can see that to ensure the retailerrsquosorder is coordinated the retailerrsquos order quantity must satisfyint119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can

derive that int1198790

119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two

aforementioned conditions are the supply chain coordinationconditions

Proposition 17 shows the sufficient conditions for the sup-ply chain coordination Let = Π

119868(119876lowast

119868) minusΠ1199031(119902lowast

1199031) then we can

obtain thatΔ120587 = (119888119900+119888119890minus119888)119876lowast

119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast

119868) (0 lt 120578 lt 1) We can deduce

that the expected profits of the supplier and the retailerafter coordinating are Π

1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868

respectively This implies that the supply chain coordinationcan be achieved through option contracts and the expectedprofit of the channel can be allocated arbitrarily between thesupplier and the retailer under inflationary conditions


Table 3 The impact of 120574 on the optimal decisions

120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667

This proposition also shows that the supply chain coordi-nation conditions are determined by the unit purchase priceof call option 119888

119900 unit exercise price of call option 119888

119890 and unit

production cost 119888 are not related to unit wholesale price 119908This indicates that the unit wholesale price cannot be usedto control the division of profit between the two membersinvolved under option contracts and cannot influence theexpected profit of the supply chain Moreover we can seethat unit purchase price of call option is negatively relatedto unit exercise price of call option in the coordinatingcontracts The main reason is that if there is an increase inboth unit purchase price and unit exercise price this presentsan advantage to the supplier and a disadvantage to theretailer Only when there is an inverse relationship betweenunit purchase price and unit exercise price it is feasible toreconcile the conflicting interests between the retailer andthe supplier under option contracts Furthermore we canobserve that with the coordinating contracts the expectedprofits of each party do not decrease and at least one of them isstrictly better offComparedwith the case of noncoordinatingcontracts there always exists a Pareto contract

7 Numerical Example

In this section a numerical example is provided to illustratethe impact of inflation on the optimal decisions and theoptimal expected profits

We assume that the default values of parameters are usedas 119901 = 10 119908 = 4 119888 = 12 119888

119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)

The above values of parameters satisfy the basic assumptionsof this paper

71 The Impact of Inflation on the Optimal Decisions Theimpact of price rising factor (120574) on the optimal decisions isshown in Table 3

FromTable 3 the following observations can be obtainedOn the one hand the scale of market demand will increase

when there is an obvious increase in 120574 which will cause thefollowing (1)The retailer will increase the optimal firm orderquantity under wholesale price contracts (2)The retailer willincrease the optimal options order quantity under optioncontracts (3) The retailer will increase the optimal optionsorder quantity and leave the optimal firm order quantityunchanged under portfolio contracts On the other handthe retailerrsquos order requirement will increase when there isan obvious increase in 120574 which will cause the supplier toincrease the optimal production quantity with and withoutoption contracts

The impact of demand contraction factor (120572) on theoptimal decisions is shown in Table 4

FromTable 4 the following observations can be obtainedOn the one hand the scale of market demand will declinewhen there is an obvious increase in 120572 which will cause thefollowing (1)The retailer will decrease the optimal firm orderquantity under wholesale price contracts (2)The retailer willdecrease the optimal options order quantity under optioncontracts (3) The retailer will decrease both the optimaloptions order quantity and the optimal firm order quantityunder portfolio contracts On the other hand the retailerrsquosorder requirement will decrease when there is an obviousincrease in 120572 which will cause the supplier to decreasethe optimal production quantity with and without optioncontracts

From Tables 3 and 4 we can deduce the followingconclusions (1) Comparing with the case of wholesale pricecontracts the introduction of option contracts will make theretailer raise the total ordering quantity and reduce the firmorder quantity under inflation scenarios (2) Comparing withthe case of wholesale price contracts the introduction ofoption contracts will make the supplier raise the productionquantity under inflation scenarios

72 The Impact of Inflation on the Optimal Expected ProfitsThe impact of price rising factor (120574) on the optimal expectedprofits is shown in Table 5



120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182

Table 5 The impact of 120574 on the optimal expected profits

120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018

FromTable 5 the following observations can be obtainedOn the one hand when there is an obvious increase in 120574 thescale ofmarket demand increases whichwill cause the retailerto obtain more profit with and without option contractsOn the other hand when there is an obvious increase in 120574the retailerrsquos order requirement increases which will causethe supplier to obtain more profit with and without optioncontracts

The impact of demand contraction factor (120572) on theoptimal expected profits is shown in Table 6

FromTable 6 the following observations can be obtainedOn the one hand when there is an obvious increase in 120572 thescale of market demand declines which will cause the retailerto obtain less profit with and without option contracts Onthe other hand when there is an obvious increase in 120572the retailerrsquos order requirement decreases which will causethe supplier to obtain less profit with and without optioncontracts

From Tables 5 and 6 we can deduce the followingconclusions (1) Comparing with the case of wholesale pricecontracts the introduction of option contracts will makeboth the retailer and the supplier obtain more profits underinflation scenarios (2) Among the three different contractsthe retailer prefers portfolio contracts and the supplier prefersoption contracts under inflation scenarios

8 Conclusion

Most countries in the world have suffered from differentdegrees of inflation for years The effect of inflation exertsan important influence on supply chain management Inthis context the studies on how to use various contractsto protect against the effect of inflation and achieve acoordinated channel are very meaningful To the best ofour knowledge most papers focus on how to use variouscontracts to manage demand uncertainty So far the effect of



120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945

inflation has not been addressed in supply chainmanagementapplications Given this we explore the effect of inflationon supply chain decisions and performance Moreover theeffect of inflation has been studied in financial applicationsand options are deemed to be effective to protect against theeffect of inflation However the studies which incorporateoption contracts and the effect of inflation are rare in supplychain management applications Given this we explore theimpact of option contracts on supply chain decisions andperformance under inflation scenarios Furthermore thesupplierrsquos optimal production quantity also needs coordinatein the presence of option contracts Given this we designthe bilateral coordination mechanism from the perspectiveof both the supplier and the retailer Our objective is tocontribute to providing insights into the effect of inflation andoption contracts on supply chain

In this paper we study the procurement production andcoordination strategies with and without option contractsunder inflation scenarios We take wholesale price contractsmodel as the benchmark and give insights into the impact ofoption contracts on supply chain decisions and performanceunder inflation scenarios By comparing results under threedifferent contracts we deduce that the retailerrsquos total orderquantity is higher with option contracts than without optioncontracts while the retailerrsquos firm order quantity is lowerwith option contracts than without option contracts Theseconclusions provide guidance for the companies on howto make appropriate procurement decisions under inflationscenarios We also deduce that the supplierrsquos productiondecision is more complex with option contracts than withoutoption contracts These conclusions provide guidance for thecompanies on how tomake appropriate production decisionsunder inflation scenarios Moreover we deduce that theintroduction of option contracts can benefit both the supplierand the retailer under inflation scenariosThe retailer prefersadopting portfolio contracts and the supplier is only inclinedto provide option contracts These conclusions provideguidance for the companies on how to choose appropriate

contracting arrangement under inflation scenarios Further-more we deduce that option contracts are ultimately carriedout by the supply chain under inflation scenarios Thuswe design the bilateral coordination mechanism from theperspective of both the supplier and the retailer and deducethe coordination conditions with option contracts underinflation scenariosThese conclusions provide a good way onhow to align the supply chain membersrsquo incentives to drivethe optimal action under inflation scenarios Our objectiveis to help the members better optimize their decisions andincrease their revenues when they face inflationary pressureand option games

There still exist some limitations in this paper We onlyconsider a one-period setting and we plan to extend ourresearch to a multiperiod setting Besides both the retailerand the supplier are supposed to be risk-neutral In reality thedecision-making behaviors of most enterprises are consistentwith loss-aversion [30] In addition we assume that themarket structure is supplier Stackelberg in this paper and weshould consider other market power such as retailer Stack-elberg and vertical Nash [40 41] in the future Furthermoreoption contracts are classified into three different categoriesIn this paper we focus our discussion on call option contractsPut or bidirectional option contracts [42] will be taken intoconsideration in the future

Notations

119905 The length of production lead timefollowing probability density function(PDF) 119892(sdot) within the interval (0 119879)

119889(119905) Deterministic part of the market demandfollowing an exponentially decreasingfunction of the production lead time dueto the effect of inflation that is119889(119905) = 120582119890

minus120572119905 where 120582 is the initial marketscale and 120572 (120572 gt 0) is the demandcontraction factor


120585 Stochastic part of the marketdemand following probabilitydensity function (PDF) 119891(sdot) andstrictly increasing cumulativedistribution function (CDF) 119865(sdot)within the interval (0 +infin)

119901(119905) Unit retail price of product followingan exponentially increasing functionof the production lead time due to theeffect of inflation that is 119901(119905) = 119901119890

120574119905where 119901 is the initial retail price and120574 (120574 gt 0) is the price rising factor

119908 Unit wholesale price of product119888119900 Unit purchase price of call option

119888119890 Unit exercise price of call option

119888 Unit production cost of product119892119903 Retailerrsquos unit shortage cost for

unsatisfied demand119892119904 Supplierrsquos unit penalty cost for each

exercised option that cannot beimmediately filled

119876119903119894 Retailerrsquos firm order quantity

(119894 = 0 1 2)

119902119903119894 Retailerrsquos options order quantity

(119894 = 0 1 2)

119876119894 Retailerrsquos total order quantity

(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894

119876119904119894 Supplierrsquos production quantity

(119894 = 0 1 2)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the National NaturalScience Foundation of China (nos 71432003 and 71272128)Program forNewCentury Excellent Talents inUniversity (noNCET-12-0087) Specialized Research Fund for the DoctoralProgram of Higher Education of China (no 20130185110006)and Youth Foundation for Humanities and Social Sciences ofMinistry of Education of China (no 11YJC630022)

References

[1] Statistics Global Inflation Rate from 2004 to 2014 2015httpwwwstatistacomstatistics256598

[2] Z Bodie ldquoInflation insurancerdquo The Journal of Risk and Insur-ance vol 57 no 4 pp 634ndash645 1990

[3] Q Wang and D-B Tsao ldquoSupply contract with bidirectionaloptions the buyerrsquos perspectiverdquo International Journal of Pro-duction Economics vol 101 no 1 pp 30ndash52 2006

[4] Y Zhao L Ma G Xie and T C E Cheng ldquoCoordination ofsupply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[5] V Nagali J Hwang D Sanghera et al ldquoProcurement riskmanagement (PRM) at Hewlett-Packard companyrdquo Interfacesvol 38 no 1 pp 51ndash60 2008

[6] Z J Ren M A Cohen T H Ho and C Terwiesch ldquoInforma-tion sharing in a long-term supply chain relationship the roleof customer review strategyrdquoOperations Research vol 58 no 1pp 81ndash93 2010

[7] C Haksoz and K D Simsek ldquoModeling breach of contract riskthrough bundled optionsrdquoThe Journal of Operational Risk vol5 no 3 pp 3ndash20 2010

[8] J J Spengler ldquoVertical integration and antitrust policyrdquo TheJournal of Political Economy vol 58 no 4 pp 347ndash352 1950

[9] G J Gutierrez and X He ldquoLife-cycle channel coordinationissues in launching an innovative durable productrdquo Productionand Operations Management vol 20 no 2 pp 268ndash279 2011

[10] J K Dey S K Mondal and M Maiti ldquoTwo storage inventoryproblem with dynamic demand and interval valued lead-timeover finite time horizon under inflation and time-value ofmoneyrdquo European Journal of Operational Research vol 185 no1 pp 170ndash194 2008

[11] C K Jaggi and A Khanna ldquoRetailerrsquos ordering policy fordeteriorating items with inflation-induced demand under tradecredit policyrdquo International Journal of Operational Research vol6 no 3 pp 360ndash379 2009

[12] H-L Yang J-T Teng and M-S Chern ldquoAn inventory modelunder inflation for deteriorating items with stock-dependentconsumption rate and partial backlogging shortagesrdquo Interna-tional Journal of Production Economics vol 123 no 1 pp 8ndash192010

[13] B Sarkar S S Sana and K Chaudhuri ldquoAn imperfect pro-duction process for time varying demand with inflation andtime value of moneymdashan EMQ modelrdquo Expert Systems withApplications vol 38 no 11 pp 13543ndash13548 2011

[14] R P Tripathi ldquoOptimal pricing and ordering policy for inflationdependent demand rate under permissible delay in paymentsrdquoInternational Journal of Business Management and Social Sci-ences vol 2 no 4 pp 35ndash43 2011

[15] J Taheri-Tolgari A Mirzazadeh and F Jolai ldquoAn inventorymodel for imperfect items under inflationary conditions withconsidering inspection errorsrdquo Computers amp Mathematics withApplications vol 63 no 6 pp 1007ndash1019 2012

[16] A Mirzazadeh ldquoOptimal inventory control problem withinflation-dependent demand rate under stochastic conditionsrdquoResearch Journal of Applied Sciences Engineering and Technol-ogy vol 4 no 4 pp 306ndash315 2012

[17] A Guria B Das S Mondal and M Maiti ldquoInventory policyfor an item with inflation induced purchasing price sellingprice and demand with immediate part paymentrdquo AppliedMathematical Modelling vol 37 no 1-2 pp 240ndash257 2013

[18] S M Mousavi V Hajipour S T A Niaki and N AlikarldquoOptimizing multi-item multi-period inventory control systemwith discounted cash flow and inflation two calibrated meta-heuristic algorithmsrdquo Applied Mathematical Modelling vol 37no 4 pp 2241ndash2256 2013

[19] B H Gilding ldquoInflation and the optimal inventory replen-ishment schedule within a finite planning horizonrdquo EuropeanJournal of Operational Research vol 234 no 3 pp 683ndash6932014

[20] S Pal G Mahapatra and G Samanta ldquoA production inventorymodel for deteriorating item with ramp type demand allowinginflation and shortages under fuzzinessrdquo Economic Modellingvol 46 pp 334ndash345 2015


[21] H Li P Ritchken and Y Wang ldquoOption and forward contract-ing with asymmetric information valuation issues in supplychainsrdquo European Journal of Operational Research vol 197 no1 pp 134ndash148 2009

[22] Y Zhao S Wang T C E Cheng X Yang and Z HuangldquoCoordination of supply chains by option contracts a cooper-ative game theory approachrdquo European Journal of OperationalResearch vol 207 no 2 pp 668ndash675 2010

[23] H Xu ldquoManaging production and procurement through optioncontracts in supply chains with random yieldrdquo InternationalJournal of Production Economics vol 126 no 2 pp 306ndash3132010

[24] Q Fu C-Y Lee and C-P Teo ldquoProcurement managementusing option contracts Random spot price and the portfolioeffectrdquo IIE Transactions vol 42 no 11 pp 793ndash811 2010

[25] Y Xia K Ramachandran and H Gurnani ldquoSharing demandand supply risk in a supply chainrdquo IIE Transactions vol 43 no6 pp 451ndash469 2011

[26] X Chen and Z-J Shen ldquoAn analysis of a supply chain withoptions contracts and service requirementsrdquo IIE Transactionsvol 44 no 10 pp 805ndash819 2012

[27] L Liang X Wang and J Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[28] C-Y Lee X Li and Y Xie ldquoProcurement risk managementusing capacitated option contracts with fixed ordering costsrdquoIIE Transactions vol 45 no 8 pp 845ndash864 2013

[29] C Liu Z Jiang L Liu and N Geng ldquoSolutions for flexible con-tainer leasing contracts with options under capacity and orderconstraintsrdquo International Journal of Production Economics vol141 no 1 pp 403ndash413 2013

[30] X Chen G Hao and L Li ldquoChannel coordination with a loss-averse retailer and option contractsrdquo International Journal ofProduction Economics vol 150 pp 52ndash57 2014

[31] F Hu C-C Lim and Z Lu ldquoOptimal production and procure-ment decisions in a supply chain with an option contract andpartial backordering under uncertaintiesrdquoAppliedMathematicsand Computation vol 232 pp 1225ndash1234 2014

[32] C T Linh and Y Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[33] T Xiao J Jin G Chen J Shi and M Xie ldquoOrdering wholesalepricing and lead-time decisions in a three-stage supply chainunder demand uncertaintyrdquo Computers amp Industrial Engineer-ing vol 59 no 4 pp 840ndash852 2010

[34] T Xiao K Shi and D Yang ldquoCoordination of a supply chainwith consumer return under demand uncertaintyrdquo Interna-tional Journal of Production Economics vol 124 no 1 pp 171ndash180 2010

[35] C-H Chiu T-M Choi and C S Tang ldquoPrice rebate andreturns supply contracts for coordinating supply chains withprice-dependent demandsrdquo Production and Operations Man-agement vol 20 no 1 pp 81ndash91 2011

[36] G P Cachon ldquoSupply chain coordination with contractsrdquo inHandbooks in Operations Research and Management Sciencevol 11 chapter 6 pp 227ndash339 University of PennsylvaniaPhiladelphia Pa USA 2003

[37] A V Iyer and M E Bergen ldquoQuick response in manufacturer-retailer channelsrdquo Management Science vol 43 no 4 pp 559ndash570 1997

[38] J G Shanthikumar and U Sumita ldquoApproximations for thetime spent in a dynamic job shop with applications to due-dateassignmentrdquo International Journal of Production Research vol26 no 8 pp 1329ndash1352 1988

[39] U S Karmarkar ldquoManufacturing lead times order release andcapacity loadingrdquo in Handbooks in Operations Research andManagement Science vol 4 pp 287ndash329 Elsevier 1993

[40] X Chen XWang and X Jiang ldquoThe impact of power structureon the retail service supply chain with an O2Omixed channelrdquoJournal of the Operational Research Society 2015

[41] X Chen and X Wang ldquoFree or bundled channel selectiondecisions under different power structuresrdquo Omega vol 53 pp11ndash20 2015

[42] N Wan and X Chen ldquoBilateral coordination strategy of supplychain with bidirectional option contracts under inflationrdquoMathematical Problems in Engineering vol 2015 Article ID369132 16 pages 2015

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


demand contraction caused by inflation It is worth notingthat options in supply chain management applications dif-ferent from options in financial applications are consideredas real options In addition option contracts are alwaysclassified into three categories [3 4] and we limit ourdiscussion to call option contracts in this paper Optioncontracts can help the demand-side obtain the goods after themarket demand is realized and provide the demand-side withmore flexibility to accommodate changing market Optioncontracts can also help the supply-side make a productionplan that maximizes its own profit and provide the supply-side with more flexibility to reduce the production cost Inthe real world many famous companies such as HP [5] Sun[6] and IBM [7] adapt derivatives of option contracts topurchase various inputs such as memory chips and scannerassemblies Since then extensive attentions from scholars arepaid to option contracts So far there aremany papers relatingto option contacts under various scenarios in supply chainmanagement applications However all these papers do notconsider the effect of inflation Motivated by this we plan tostudy whether option contracts are an efficient tool to resolvechannel conflicts and achieve channel coordination underinflation scenarios

On the other hand the coordination problem has alwaysbeen an important part in the study on supply chain man-agement As we know there exists double marginalizationphenomenon under the decentralized decision-making [8]The primary purpose of channel coordination is to promptthe members involved to carry out the actions which areconsistent with the supply chainrsquos objective [9] To the bestof our knowledge except for some scenarios such as randomyield the supply-side is always assumed to adopt the make-to-order production policy and commit tomanufacturing theproducts up to the order requirement of the demand-sideunder various contract types such as buyback contracts andrevenue sharing contracts In this case supply chain coordi-nation can be achieved only when the demand-side decisionunder the decentralized situation is the same as the integrateddecision under the centralized situation At this momenthow to make a nonintegrated supply chain coordinate issimplified to the supply chain unilateral coordination fromthe demand-side perspective However the supply-side has astrong incentive to decide the production quantity accordingto the profit maximization rule in the presence of optioncontracts The major reason is because option contracts givethe right not the obligation to obtain the items and so thedemand-side may not exercise all the options purchasedObviously this poses a challenge on the implementationof supply chain unilateral coordination mechanism in thepresence of option contracts Motivated by this we designthe supply chain bilateral coordination mechanism from theperspective of both the demand-side and the supply-side inthe presence of option contracts

In this paper we consider one-period two-stage supplychain which contains one supplier and one retailer Consid-ering price increase and demand contraction due to the effectof inflation we introduce option contracts into supply chaindecision-making in order to hedge against these risks justmentionedThis paper mainly solves the following problems

(1) What are the optimal ordering and production poli-cies for the supply chain members in the presence ofoption contracts under inflation scenarios

(2) What are the optimal expected profits for the supplychain members in the presence of option contractsunder inflation scenarios

(3) What effect does inflation have on the optimal order-ing and production policies

(4) What effect do option contracts have on supply chainunder inflation scenarios

(5) How should option contracts be set to achieve supplychain coordination under inflation scenarios

The main contributions of our work are as follows

(1) To the best of our knowledge there are no publishedpapers that study the use of option contracts to protectagainst the effect of inflation in supply chain manage-ment applications We develop supply chain modelsthat incorporate the effect of inflation and optioncontracts in this paper Our objective is to providemanagement insights into the effect of inflation andoption contracts on the supply chain

(2) We explore the effect of inflation on the retailerrsquosoptimal ordering policy and the supplierrsquos optimalproduction policy and gain many management inter-esting results

(3) We explore the impact of option contracts on thesupply chain through using wholesale price contractsmodel as the benchmark and discuss which kind ofcontracts is more suitable for supply chain membersunder inflation scenarios

(4) We design the bilateral coordination mechanismfrom the perspective of both the supplier and theretailer and achieve an efficient channel under infla-tion scenarios

The remaining part of this paper is structured as followsA summary of related literature is presented in Section 2Themodel formulation and assumptions are given in Section 3We establish supply chain models with three different con-tracts and explore the effect of inflation on the optimaldecision policies in Section 4Wediscuss the impact of optioncontracts on the supply chain decisions and performancein Section 5 Coordination conditions with option contractsunder inflation scenarios are considered in Section 6Wepro-vide a numerical example to illustrate the effect of inflationon the optimal decisions and the optimal expected profitsin Section 7 We conclude our findings in Section 8 andhighlight possible future work

2 Literature Review

We first review the literature on enterprise operation man-agement under inflation scenarios Dey et al [10] solve adeteriorating inventory problem with two warehouses con-sidering time value of money and interval-valued lead time














1199031 at unit





















119908 gt 119888 and 119888119900+ 119888119890gt 119908 gt 119888








1199030(1198761199030) is given by

Π1199030(1198761199030) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199030] minus 119908119876

1199030

minus 119892119903[119863 (119905) minus 119876

1199030]+

119892 (119905) 119889119905

(1)


Π1199030(1198761199030)

= 1198761199030int

119879

0

(119901119890120574119905

+ 119892119903minus 119908)119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198761199030minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(2)



1199030is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

1199030minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119908 (3)


=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(1198761199030minus120582119890minus120572119905

)119892(119905)119889119905 and1198892

Π1199030(1198761199030)1198891198762

1199030= minusint119879

0

(119901119890120574119905

+119892119903)119891(1198761199030minus120582119890minus120572119905


1199030 Let 119889Π

1199030(1198761199030)1198891198761199030


119879

0

(119901119890120574119905

+

119892119903)119865(119876lowast

1199030minus 120582119890minus120572119905

)119892(119905)119889119905 = 119908





1199030) = int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876lowast

1199030minus

120582119890minus120572119905


1199030120597120572 =

minus(1205971198710(119876lowast

1199030)120597120572)(120597119871

0(119876lowast

1199030)120597119876lowast


119879

0

120582119905119890minus120572119905

(119901119890120574119905

+

119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876lowast

1199030120597120574 = minus(120597119871

0(119876lowast

1199030)120597120574)(120597119871

0(119876lowast

1199030)120597119876lowast

1199030) =

int119879

0

119901119905119890120574119905

119865(119876lowast

1199030minus 120582119890

minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903)119891(119876lowast

1199030minus

120582119890minus120572119905





lowast

1199040= 119876

lowast




1199040(119876lowast


Π1199040(119876lowast

1199040) = (119908 minus 119888)119876

lowast

1199040= (119908 minus 119888)119876

lowast

1199030 (4)



1199031(1199021199031) is given by

Π1199031(1199021199031) = int

119879

0

119901 (119905)min [119863 (119905) 1199021199031] minus 1198881199001199021199031

minus 119888119890min [119863 (119905) 119902

1199031] minus 119892119903[119863 (119905) minus 119902

1199031]+

119892 (119905) 119889119905

(5)


Π1199031(1199021199031)

= 1199021199031int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1199021199031minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(6)



1199031is

int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119902

lowast

1199031minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (7)


=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1199021199031

minus

120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199031(1199021199031)1198891199022

1199031= minusint

119879

0

(119901119890120574119905

+ 119892119903minus

119888119890)119891(1199021199031

minus 120582119890minus120572119905


1199031 Let

119889Π1199031(1199021199031)1198891199021199031


0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 = 119888119900




1199031)=int119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905minus 119888119900 we


1(119902lowast

1199031)120597120572)(120597119871

1(119902lowast

1199031)120597119902lowast

1199031) =

minus int119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+

119892119903minus 119888119890)119891(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119902lowast

1199031120597120574 = minus(120597119871

1(119902lowast

1199031)

120597120574)(1205971198711(119902lowast

1199031)120597119902lowast

1199031) = int

119879

0

119901119905119890120574119905

119865(119902lowast

1199031minus 120582119890

minus120572119905

)119892(119905)119889119905

int119879

0

(119901119890120574119905

+119892119903minus119888119890)119891(119902lowast

1199031minus120582119890minus120572119905






1199041le

119902lowast



Π1199041(1198761199041) = int

119879

0

119888119900119902lowast

1199031+ 119888119890min [119863 (119905) 119902

lowast

1199031]

minus 119892119904[min (119863 (119905) 119902

lowast

1199031) minus 1198761199041]+

minus 1198881198761199041 119892 (119905) 119889119905

(8)


Π1199041(1198761199041) = (119888119900+ 119888119890minus 119892119904) 119902lowast

1199031

+ (119892119904minus 119888119890) int

119879

0

int

119902lowast

1199031minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199041

minus 119892119904int

119879

0

int

1198761199041minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(9)


max1198761199041gt0

Π1199041(1198761199041)

st 1198761199041

le 119902lowast

1199031

(10)



1199041satisfies

119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(11)

where int1198790

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199041(1198761199041)1198891198762

1199041=

minus119892119904int119879

0

119891(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 so Π1199041(1198761199041) is concave

in 1198761199041 Let 119889Π

1199041(1198761199041)1198891198761199041


0

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)

119892119904


119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(12)


1199041lt 119902lowast


fective If 119876120593

1199041ge 119902lowast




Proof Let 1198671(119876120593

1199041) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199041minus

120582119890minus120572119905


1199041lt 119902

lowast

1199031 then 119876

lowast

1199041= 119876

120593


120597119876120593

1199041120597120572 = minus(120597119867

1(119876120593

1199041)120597120572)(120597119867

1(119876120593

1199041)120597119876120593

1199041) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

and 120597119876120593

1199041120597120574 = minus(120597119867

1(119876120593

1199041)120597120574)(120597119867

1(119876120593

1199041)120597119876120593

1199041) = 0 so


1199041ge 119902lowast

1199031 then119876

lowast

1199041= 119902lowast

1199031






1199032(1198761199032 1199021199032) is given by

Π1199032(1198761199032 1199021199032) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199032

+ 1199021199032]

minus 1199081198761199032

minus 1198881199001199021199032

minus 119888119890min [(119863 (119905) minus 119876

1199032)+

1199021199032]

minus 119892119903[119863 (119905) minus (119876

1199032+ 1199021199032)]+

119892 (119905) 119889119905

(13)



2= 1198761199032


1199032 1199021199032) is


can be rewritten as

Π1199032(1198761199032 1198762)

= (119888119900+ 119888119890minus 119908)119876

1199032

minus 119888119890int

119879

0

int

1198761199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ 1198762int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198762minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(14)



1199032is

int

119879

0

119865 (119876lowast

1199032minus 120582119890minus120572119905

) 119892 (119905) 119889119905 =119888119900+ 119888119890minus 119908

119888119890

(15)


int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119876

lowast

2minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (16)


=

(119888119900+119888119890minus119908)minus119888

119890int119879

0

119865(1198761199032minus120582119890minus120572119905

)119892(119905)119889119905 1205972Π1199032(1198761199032 1198762)1205971198762

1199032=

minus119888119890int119879

0

119891(1198761199032

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 120597Π1199032(1198761199032 1198762)1205971198762

=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1198762minus

120582119890minus120572119905

)119892(119905)119889119905 1205972

Π1199032(1198761199032 1198762)1205971198762

2= minusint

119879

0

(119901119890120574119905

+ 119892119903

minus

119888119890)119891(1198762minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 1205972

Π1199032(1198761199032 1198762)12059711987611990321205971198762=

1205972

Π1199032(1198761199032 1198762)12059711987621205971198761199032

= 0 Hence

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972

Π1199032(1198761199032 1198762)

12059711987621199032

1205972

Π1199032(1198761199032 1198762)

12059711987611990321205971198762

1205972

Π1199032(1198761199032 1198762)

12059711987621205971198761199032

1205972

Π1199032(1198761199032 1198762)

12059711987622

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt 0 (17)

So Π1199032(1198761199032 1198762) is concave in 119876

1199032and 119876

2 Let 120597Π

1199032(1198761199032

1198762)1205971198761199032



119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+

119888119890minus119908)119888

119890and int

119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

2minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900


1199032= 119876lowast

2minus119876lowast

1199032


2gt 119876lowast


119900lt ((119901119890

120574119905

+

119892119903minus119888119890)(119901119890120574119905





1199032) = (119888

119900+ 119888119890

minus 119908) minus 119888119890int119879

0

119865(119876lowast

1199032minus

120582119890minus120572119905


1199032120597120572 =

minus(1205971198712(119876lowast

1199032)120597120572)(120597119871

2(119876lowast

1199032)120597119876lowast


119879

0

120582119905119890minus120572119905

119891(119876lowast

1199032minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119876lowast

1199032120597120574 =

minus(1205971198712(119876lowast

1199032)120597120574)(120597119871

2(119876lowast

1199032)120597119876lowast





2) = int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 minus


lowast

2120597120572 = minus(120597119871

3(119876lowast

2)120597120572)

(1205971198713(119876lowast

2)120597119876lowast

2) = minus int

119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903

minus 119888119890)119891(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and120597119876lowast

2120597120574 = minus(120597119871

3(119876lowast

2)120597120574)(120597119871

3(119876lowast

2)120597119876lowast

2) = int119879

0

119901119905119890120574119905

119865(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905




1199032le 1198761199042

le 119876lowast

2



1199042(1198761199042) is given by

Π1199042(1198761199042) = int

119879

0

119908119876lowast

1199032+ 119888119900(119876lowast

2minus 119876lowast

1199032)

+ 119888119890min [(119863 (119905) minus 119876

lowast

1199032)+

119876lowast

2minus 119876lowast

1199032]

minus 119892119904[min (119863 (119905) 119876

lowast

2) minus 1198761199042]+

minus 1198881198761199042 119892 (119905) 119889119905

(18)


Π1199042(1198761199042) = (119888119900+ 119888119890minus 119892119904) 119876lowast

2

+ (119892119904minus 119888119890) int

119879

0

int

119876lowast

2minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905


1199032

+ 119888119890int

119879

0

int

119876lowast

1199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199042

minus 119892119904int

119879

0

int

1198761199042minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(19)


max1198761199042gt0

Π1199042(1198761199042)

st 119876lowast

1199032le 1198761199042

le 119876lowast

2

(20)



1199042satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(21)

where int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199042(1198761199042)1198891198762

1199042=

minus119892119904int119879

0

119891(1198761199042

minus 120582119890minus120572119905


1198761199042 Let 119889Π

1199042(1198761199042)1198891198761199042


0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(22)


1199042le 119876lowast



1199032lt 119876120593

1199042lt 119876lowast



ge 119876lowast

2 the constraint



Proof Let 1198672(119876120593

1199042) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905


1199041lt 119876lowast

1199032 then119876

lowast

1199042= 119876lowast

1199032



lt 119876120593

1199041lt 119876lowast

2 then119876

lowast

1199042= 119876120593

1199042 We can


2(119876120593

1199042)120597120572)(120597119867

2(119876120593

1199042)120597119876120593

1199042) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876120593

1199042120597120574 = minus(120597119867

2(119876120593

1199042)120597120574)(120597119867

2(119876120593

1199042)120597119876120593

1199042) = 0


1199041ge 119876lowast

2 then119876

lowast

1199041= 119876lowast

2









1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2


= 119876lowast

2 From


1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt


119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876


1199030gt

119876lowast


(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199031 it


gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2




Proposition 13 If 119876120593119904


1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593




1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593


119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040






1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)


1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast


Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905





1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















1199031 at unit





















119908 gt 119888 and 119888119900+ 119888119890gt 119908 gt 119888








1199030(1198761199030) is given by

Π1199030(1198761199030) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199030] minus 119908119876

1199030

minus 119892119903[119863 (119905) minus 119876

1199030]+

119892 (119905) 119889119905

(1)


Π1199030(1198761199030)

= 1198761199030int

119879

0

(119901119890120574119905

+ 119892119903minus 119908)119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198761199030minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(2)



1199030is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

1199030minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119908 (3)


=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(1198761199030minus120582119890minus120572119905

)119892(119905)119889119905 and1198892

Π1199030(1198761199030)1198891198762

1199030= minusint119879

0

(119901119890120574119905

+119892119903)119891(1198761199030minus120582119890minus120572119905


1199030 Let 119889Π

1199030(1198761199030)1198891198761199030


119879

0

(119901119890120574119905

+

119892119903)119865(119876lowast

1199030minus 120582119890minus120572119905

)119892(119905)119889119905 = 119908





1199030) = int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876lowast

1199030minus

120582119890minus120572119905


1199030120597120572 =

minus(1205971198710(119876lowast

1199030)120597120572)(120597119871

0(119876lowast

1199030)120597119876lowast


119879

0

120582119905119890minus120572119905

(119901119890120574119905

+

119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876lowast

1199030120597120574 = minus(120597119871

0(119876lowast

1199030)120597120574)(120597119871

0(119876lowast

1199030)120597119876lowast

1199030) =

int119879

0

119901119905119890120574119905

119865(119876lowast

1199030minus 120582119890

minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903)119891(119876lowast

1199030minus

120582119890minus120572119905





lowast

1199040= 119876

lowast




1199040(119876lowast


Π1199040(119876lowast

1199040) = (119908 minus 119888)119876

lowast

1199040= (119908 minus 119888)119876

lowast

1199030 (4)



1199031(1199021199031) is given by

Π1199031(1199021199031) = int

119879

0

119901 (119905)min [119863 (119905) 1199021199031] minus 1198881199001199021199031

minus 119888119890min [119863 (119905) 119902

1199031] minus 119892119903[119863 (119905) minus 119902

1199031]+

119892 (119905) 119889119905

(5)


Π1199031(1199021199031)

= 1199021199031int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1199021199031minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(6)



1199031is

int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119902

lowast

1199031minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (7)


=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1199021199031

minus

120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199031(1199021199031)1198891199022

1199031= minusint

119879

0

(119901119890120574119905

+ 119892119903minus

119888119890)119891(1199021199031

minus 120582119890minus120572119905


1199031 Let

119889Π1199031(1199021199031)1198891199021199031


0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 = 119888119900




1199031)=int119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905minus 119888119900 we


1(119902lowast

1199031)120597120572)(120597119871

1(119902lowast

1199031)120597119902lowast

1199031) =

minus int119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+

119892119903minus 119888119890)119891(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119902lowast

1199031120597120574 = minus(120597119871

1(119902lowast

1199031)

120597120574)(1205971198711(119902lowast

1199031)120597119902lowast

1199031) = int

119879

0

119901119905119890120574119905

119865(119902lowast

1199031minus 120582119890

minus120572119905

)119892(119905)119889119905

int119879

0

(119901119890120574119905

+119892119903minus119888119890)119891(119902lowast

1199031minus120582119890minus120572119905






1199041le

119902lowast



Π1199041(1198761199041) = int

119879

0

119888119900119902lowast

1199031+ 119888119890min [119863 (119905) 119902

lowast

1199031]

minus 119892119904[min (119863 (119905) 119902

lowast

1199031) minus 1198761199041]+

minus 1198881198761199041 119892 (119905) 119889119905

(8)


Π1199041(1198761199041) = (119888119900+ 119888119890minus 119892119904) 119902lowast

1199031

+ (119892119904minus 119888119890) int

119879

0

int

119902lowast

1199031minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199041

minus 119892119904int

119879

0

int

1198761199041minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(9)


max1198761199041gt0

Π1199041(1198761199041)

st 1198761199041

le 119902lowast

1199031

(10)



1199041satisfies

119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(11)

where int1198790

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199041(1198761199041)1198891198762

1199041=

minus119892119904int119879

0

119891(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 so Π1199041(1198761199041) is concave

in 1198761199041 Let 119889Π

1199041(1198761199041)1198891198761199041


0

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)

119892119904


119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(12)


1199041lt 119902lowast


fective If 119876120593

1199041ge 119902lowast




Proof Let 1198671(119876120593

1199041) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199041minus

120582119890minus120572119905


1199041lt 119902

lowast

1199031 then 119876

lowast

1199041= 119876

120593


120597119876120593

1199041120597120572 = minus(120597119867

1(119876120593

1199041)120597120572)(120597119867

1(119876120593

1199041)120597119876120593

1199041) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

and 120597119876120593

1199041120597120574 = minus(120597119867

1(119876120593

1199041)120597120574)(120597119867

1(119876120593

1199041)120597119876120593

1199041) = 0 so


1199041ge 119902lowast

1199031 then119876

lowast

1199041= 119902lowast

1199031






1199032(1198761199032 1199021199032) is given by

Π1199032(1198761199032 1199021199032) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199032

+ 1199021199032]

minus 1199081198761199032

minus 1198881199001199021199032

minus 119888119890min [(119863 (119905) minus 119876

1199032)+

1199021199032]

minus 119892119903[119863 (119905) minus (119876

1199032+ 1199021199032)]+

119892 (119905) 119889119905

(13)



2= 1198761199032


1199032 1199021199032) is


can be rewritten as

Π1199032(1198761199032 1198762)

= (119888119900+ 119888119890minus 119908)119876

1199032

minus 119888119890int

119879

0

int

1198761199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ 1198762int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198762minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(14)



1199032is

int

119879

0

119865 (119876lowast

1199032minus 120582119890minus120572119905

) 119892 (119905) 119889119905 =119888119900+ 119888119890minus 119908

119888119890

(15)


int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119876

lowast

2minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (16)


=

(119888119900+119888119890minus119908)minus119888

119890int119879

0

119865(1198761199032minus120582119890minus120572119905

)119892(119905)119889119905 1205972Π1199032(1198761199032 1198762)1205971198762

1199032=

minus119888119890int119879

0

119891(1198761199032

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 120597Π1199032(1198761199032 1198762)1205971198762

=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1198762minus

120582119890minus120572119905

)119892(119905)119889119905 1205972

Π1199032(1198761199032 1198762)1205971198762

2= minusint

119879

0

(119901119890120574119905

+ 119892119903

minus

119888119890)119891(1198762minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 1205972

Π1199032(1198761199032 1198762)12059711987611990321205971198762=

1205972

Π1199032(1198761199032 1198762)12059711987621205971198761199032

= 0 Hence

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972

Π1199032(1198761199032 1198762)

12059711987621199032

1205972

Π1199032(1198761199032 1198762)

12059711987611990321205971198762

1205972

Π1199032(1198761199032 1198762)

12059711987621205971198761199032

1205972

Π1199032(1198761199032 1198762)

12059711987622

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt 0 (17)

So Π1199032(1198761199032 1198762) is concave in 119876

1199032and 119876

2 Let 120597Π

1199032(1198761199032

1198762)1205971198761199032



119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+

119888119890minus119908)119888

119890and int

119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

2minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900


1199032= 119876lowast

2minus119876lowast

1199032


2gt 119876lowast


119900lt ((119901119890

120574119905

+

119892119903minus119888119890)(119901119890120574119905





1199032) = (119888

119900+ 119888119890

minus 119908) minus 119888119890int119879

0

119865(119876lowast

1199032minus

120582119890minus120572119905


1199032120597120572 =

minus(1205971198712(119876lowast

1199032)120597120572)(120597119871

2(119876lowast

1199032)120597119876lowast


119879

0

120582119905119890minus120572119905

119891(119876lowast

1199032minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119876lowast

1199032120597120574 =

minus(1205971198712(119876lowast

1199032)120597120574)(120597119871

2(119876lowast

1199032)120597119876lowast





2) = int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 minus


lowast

2120597120572 = minus(120597119871

3(119876lowast

2)120597120572)

(1205971198713(119876lowast

2)120597119876lowast

2) = minus int

119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903

minus 119888119890)119891(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and120597119876lowast

2120597120574 = minus(120597119871

3(119876lowast

2)120597120574)(120597119871

3(119876lowast

2)120597119876lowast

2) = int119879

0

119901119905119890120574119905

119865(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905




1199032le 1198761199042

le 119876lowast

2



1199042(1198761199042) is given by

Π1199042(1198761199042) = int

119879

0

119908119876lowast

1199032+ 119888119900(119876lowast

2minus 119876lowast

1199032)

+ 119888119890min [(119863 (119905) minus 119876

lowast

1199032)+

119876lowast

2minus 119876lowast

1199032]

minus 119892119904[min (119863 (119905) 119876

lowast

2) minus 1198761199042]+

minus 1198881198761199042 119892 (119905) 119889119905

(18)


Π1199042(1198761199042) = (119888119900+ 119888119890minus 119892119904) 119876lowast

2

+ (119892119904minus 119888119890) int

119879

0

int

119876lowast

2minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905


1199032

+ 119888119890int

119879

0

int

119876lowast

1199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199042

minus 119892119904int

119879

0

int

1198761199042minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(19)


max1198761199042gt0

Π1199042(1198761199042)

st 119876lowast

1199032le 1198761199042

le 119876lowast

2

(20)



1199042satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(21)

where int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199042(1198761199042)1198891198762

1199042=

minus119892119904int119879

0

119891(1198761199042

minus 120582119890minus120572119905


1198761199042 Let 119889Π

1199042(1198761199042)1198891198761199042


0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(22)


1199042le 119876lowast



1199032lt 119876120593

1199042lt 119876lowast



ge 119876lowast

2 the constraint



Proof Let 1198672(119876120593

1199042) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905


1199041lt 119876lowast

1199032 then119876

lowast

1199042= 119876lowast

1199032



lt 119876120593

1199041lt 119876lowast

2 then119876

lowast

1199042= 119876120593

1199042 We can


2(119876120593

1199042)120597120572)(120597119867

2(119876120593

1199042)120597119876120593

1199042) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876120593

1199042120597120574 = minus(120597119867

2(119876120593

1199042)120597120574)(120597119867

2(119876120593

1199042)120597119876120593

1199042) = 0


1199041ge 119876lowast

2 then119876

lowast

1199041= 119876lowast

2









1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2


= 119876lowast

2 From


1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt


119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876


1199030gt

119876lowast


(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199031 it


gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2




Proposition 13 If 119876120593119904


1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593




1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593


119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040






1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)


1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast


Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905





1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1199030) = int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876lowast

1199030minus

120582119890minus120572119905


1199030120597120572 =

minus(1205971198710(119876lowast

1199030)120597120572)(120597119871

0(119876lowast

1199030)120597119876lowast


119879

0

120582119905119890minus120572119905

(119901119890120574119905

+

119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+119892119903)119891(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876lowast

1199030120597120574 = minus(120597119871

0(119876lowast

1199030)120597120574)(120597119871

0(119876lowast

1199030)120597119876lowast

1199030) =

int119879

0

119901119905119890120574119905

119865(119876lowast

1199030minus 120582119890

minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903)119891(119876lowast

1199030minus

120582119890minus120572119905





lowast

1199040= 119876

lowast




1199040(119876lowast


Π1199040(119876lowast

1199040) = (119908 minus 119888)119876

lowast

1199040= (119908 minus 119888)119876

lowast

1199030 (4)



1199031(1199021199031) is given by

Π1199031(1199021199031) = int

119879

0

119901 (119905)min [119863 (119905) 1199021199031] minus 1198881199001199021199031

minus 119888119890min [119863 (119905) 119902

1199031] minus 119892119903[119863 (119905) minus 119902

1199031]+

119892 (119905) 119889119905

(5)


Π1199031(1199021199031)

= 1199021199031int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1199021199031minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(6)



1199031is

int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119902

lowast

1199031minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (7)


=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1199021199031

minus

120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199031(1199021199031)1198891199022

1199031= minusint

119879

0

(119901119890120574119905

+ 119892119903minus

119888119890)119891(1199021199031

minus 120582119890minus120572119905


1199031 Let

119889Π1199031(1199021199031)1198891199021199031


0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 = 119888119900




1199031)=int119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905minus 119888119900 we


1(119902lowast

1199031)120597120572)(120597119871

1(119902lowast

1199031)120597119902lowast

1199031) =

minus int119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119902lowast

1199031minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+

119892119903minus 119888119890)119891(119902lowast

1199031minus120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119902lowast

1199031120597120574 = minus(120597119871

1(119902lowast

1199031)

120597120574)(1205971198711(119902lowast

1199031)120597119902lowast

1199031) = int

119879

0

119901119905119890120574119905

119865(119902lowast

1199031minus 120582119890

minus120572119905

)119892(119905)119889119905

int119879

0

(119901119890120574119905

+119892119903minus119888119890)119891(119902lowast

1199031minus120582119890minus120572119905






1199041le

119902lowast



Π1199041(1198761199041) = int

119879

0

119888119900119902lowast

1199031+ 119888119890min [119863 (119905) 119902

lowast

1199031]

minus 119892119904[min (119863 (119905) 119902

lowast

1199031) minus 1198761199041]+

minus 1198881198761199041 119892 (119905) 119889119905

(8)


Π1199041(1198761199041) = (119888119900+ 119888119890minus 119892119904) 119902lowast

1199031

+ (119892119904minus 119888119890) int

119879

0

int

119902lowast

1199031minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199041

minus 119892119904int

119879

0

int

1198761199041minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(9)


max1198761199041gt0

Π1199041(1198761199041)

st 1198761199041

le 119902lowast

1199031

(10)



1199041satisfies

119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(11)

where int1198790

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199041(1198761199041)1198891198762

1199041=

minus119892119904int119879

0

119891(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 so Π1199041(1198761199041) is concave

in 1198761199041 Let 119889Π

1199041(1198761199041)1198891198761199041


0

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)

119892119904


119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(12)


1199041lt 119902lowast


fective If 119876120593

1199041ge 119902lowast




Proof Let 1198671(119876120593

1199041) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199041minus

120582119890minus120572119905


1199041lt 119902

lowast

1199031 then 119876

lowast

1199041= 119876

120593


120597119876120593

1199041120597120572 = minus(120597119867

1(119876120593

1199041)120597120572)(120597119867

1(119876120593

1199041)120597119876120593

1199041) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

and 120597119876120593

1199041120597120574 = minus(120597119867

1(119876120593

1199041)120597120574)(120597119867

1(119876120593

1199041)120597119876120593

1199041) = 0 so


1199041ge 119902lowast

1199031 then119876

lowast

1199041= 119902lowast

1199031






1199032(1198761199032 1199021199032) is given by

Π1199032(1198761199032 1199021199032) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199032

+ 1199021199032]

minus 1199081198761199032

minus 1198881199001199021199032

minus 119888119890min [(119863 (119905) minus 119876

1199032)+

1199021199032]

minus 119892119903[119863 (119905) minus (119876

1199032+ 1199021199032)]+

119892 (119905) 119889119905

(13)



2= 1198761199032


1199032 1199021199032) is


can be rewritten as

Π1199032(1198761199032 1198762)

= (119888119900+ 119888119890minus 119908)119876

1199032

minus 119888119890int

119879

0

int

1198761199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ 1198762int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198762minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(14)



1199032is

int

119879

0

119865 (119876lowast

1199032minus 120582119890minus120572119905

) 119892 (119905) 119889119905 =119888119900+ 119888119890minus 119908

119888119890

(15)


int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119876

lowast

2minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (16)


=

(119888119900+119888119890minus119908)minus119888

119890int119879

0

119865(1198761199032minus120582119890minus120572119905

)119892(119905)119889119905 1205972Π1199032(1198761199032 1198762)1205971198762

1199032=

minus119888119890int119879

0

119891(1198761199032

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 120597Π1199032(1198761199032 1198762)1205971198762

=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1198762minus

120582119890minus120572119905

)119892(119905)119889119905 1205972

Π1199032(1198761199032 1198762)1205971198762

2= minusint

119879

0

(119901119890120574119905

+ 119892119903

minus

119888119890)119891(1198762minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 1205972

Π1199032(1198761199032 1198762)12059711987611990321205971198762=

1205972

Π1199032(1198761199032 1198762)12059711987621205971198761199032

= 0 Hence

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972

Π1199032(1198761199032 1198762)

12059711987621199032

1205972

Π1199032(1198761199032 1198762)

12059711987611990321205971198762

1205972

Π1199032(1198761199032 1198762)

12059711987621205971198761199032

1205972

Π1199032(1198761199032 1198762)

12059711987622

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt 0 (17)

So Π1199032(1198761199032 1198762) is concave in 119876

1199032and 119876

2 Let 120597Π

1199032(1198761199032

1198762)1205971198761199032



119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+

119888119890minus119908)119888

119890and int

119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

2minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900


1199032= 119876lowast

2minus119876lowast

1199032


2gt 119876lowast


119900lt ((119901119890

120574119905

+

119892119903minus119888119890)(119901119890120574119905





1199032) = (119888

119900+ 119888119890

minus 119908) minus 119888119890int119879

0

119865(119876lowast

1199032minus

120582119890minus120572119905


1199032120597120572 =

minus(1205971198712(119876lowast

1199032)120597120572)(120597119871

2(119876lowast

1199032)120597119876lowast


119879

0

120582119905119890minus120572119905

119891(119876lowast

1199032minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119876lowast

1199032120597120574 =

minus(1205971198712(119876lowast

1199032)120597120574)(120597119871

2(119876lowast

1199032)120597119876lowast





2) = int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 minus


lowast

2120597120572 = minus(120597119871

3(119876lowast

2)120597120572)

(1205971198713(119876lowast

2)120597119876lowast

2) = minus int

119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903

minus 119888119890)119891(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and120597119876lowast

2120597120574 = minus(120597119871

3(119876lowast

2)120597120574)(120597119871

3(119876lowast

2)120597119876lowast

2) = int119879

0

119901119905119890120574119905

119865(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905




1199032le 1198761199042

le 119876lowast

2



1199042(1198761199042) is given by

Π1199042(1198761199042) = int

119879

0

119908119876lowast

1199032+ 119888119900(119876lowast

2minus 119876lowast

1199032)

+ 119888119890min [(119863 (119905) minus 119876

lowast

1199032)+

119876lowast

2minus 119876lowast

1199032]

minus 119892119904[min (119863 (119905) 119876

lowast

2) minus 1198761199042]+

minus 1198881198761199042 119892 (119905) 119889119905

(18)


Π1199042(1198761199042) = (119888119900+ 119888119890minus 119892119904) 119876lowast

2

+ (119892119904minus 119888119890) int

119879

0

int

119876lowast

2minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905


1199032

+ 119888119890int

119879

0

int

119876lowast

1199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199042

minus 119892119904int

119879

0

int

1198761199042minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(19)


max1198761199042gt0

Π1199042(1198761199042)

st 119876lowast

1199032le 1198761199042

le 119876lowast

2

(20)



1199042satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(21)

where int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199042(1198761199042)1198891198762

1199042=

minus119892119904int119879

0

119891(1198761199042

minus 120582119890minus120572119905


1198761199042 Let 119889Π

1199042(1198761199042)1198891198761199042


0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(22)


1199042le 119876lowast



1199032lt 119876120593

1199042lt 119876lowast



ge 119876lowast

2 the constraint



Proof Let 1198672(119876120593

1199042) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905


1199041lt 119876lowast

1199032 then119876

lowast

1199042= 119876lowast

1199032



lt 119876120593

1199041lt 119876lowast

2 then119876

lowast

1199042= 119876120593

1199042 We can


2(119876120593

1199042)120597120572)(120597119867

2(119876120593

1199042)120597119876120593

1199042) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876120593

1199042120597120574 = minus(120597119867

2(119876120593

1199042)120597120574)(120597119867

2(119876120593

1199042)120597119876120593

1199042) = 0


1199041ge 119876lowast

2 then119876

lowast

1199041= 119876lowast

2









1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2


= 119876lowast

2 From


1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt


119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876


1199030gt

119876lowast


(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199031 it


gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2




Proposition 13 If 119876120593119904


1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593




1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593


119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040






1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)


1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast


Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905





1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199041le

119902lowast



Π1199041(1198761199041) = int

119879

0

119888119900119902lowast

1199031+ 119888119890min [119863 (119905) 119902

lowast

1199031]

minus 119892119904[min (119863 (119905) 119902

lowast

1199031) minus 1198761199041]+

minus 1198881198761199041 119892 (119905) 119889119905

(8)


Π1199041(1198761199041) = (119888119900+ 119888119890minus 119892119904) 119902lowast

1199031

+ (119892119904minus 119888119890) int

119879

0

int

119902lowast

1199031minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199041

minus 119892119904int

119879

0

int

1198761199041minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(9)


max1198761199041gt0

Π1199041(1198761199041)

st 1198761199041

le 119902lowast

1199031

(10)



1199041satisfies

119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(11)

where int1198790

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199041(1198761199041)1198891198762

1199041=

minus119892119904int119879

0

119891(1198761199041

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 so Π1199041(1198761199041) is concave

in 1198761199041 Let 119889Π

1199041(1198761199041)1198891198761199041


0

119865(119876120593

1199041minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)

119892119904


119876lowast

1199041=

119876120593

1199041119876120593

1199041lt 119902lowast

1199031

119902lowast

1199031119876120593

1199041ge 119902lowast

1199031

(12)


1199041lt 119902lowast


fective If 119876120593

1199041ge 119902lowast




Proof Let 1198671(119876120593

1199041) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199041minus

120582119890minus120572119905


1199041lt 119902

lowast

1199031 then 119876

lowast

1199041= 119876

120593


120597119876120593

1199041120597120572 = minus(120597119867

1(119876120593

1199041)120597120572)(120597119867

1(119876120593

1199041)120597119876120593

1199041) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199041minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

and 120597119876120593

1199041120597120574 = minus(120597119867

1(119876120593

1199041)120597120574)(120597119867

1(119876120593

1199041)120597119876120593

1199041) = 0 so


1199041ge 119902lowast

1199031 then119876

lowast

1199041= 119902lowast

1199031






1199032(1198761199032 1199021199032) is given by

Π1199032(1198761199032 1199021199032) = int

119879

0

119901 (119905)min [119863 (119905) 1198761199032

+ 1199021199032]

minus 1199081198761199032

minus 1198881199001199021199032

minus 119888119890min [(119863 (119905) minus 119876

1199032)+

1199021199032]

minus 119892119903[119863 (119905) minus (119876

1199032+ 1199021199032)]+

119892 (119905) 119889119905

(13)



2= 1198761199032


1199032 1199021199032) is


can be rewritten as

Π1199032(1198761199032 1198762)

= (119888119900+ 119888119890minus 119908)119876

1199032

minus 119888119890int

119879

0

int

1198761199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ 1198762int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198762minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(14)



1199032is

int

119879

0

119865 (119876lowast

1199032minus 120582119890minus120572119905

) 119892 (119905) 119889119905 =119888119900+ 119888119890minus 119908

119888119890

(15)


int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119876

lowast

2minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (16)


=

(119888119900+119888119890minus119908)minus119888

119890int119879

0

119865(1198761199032minus120582119890minus120572119905

)119892(119905)119889119905 1205972Π1199032(1198761199032 1198762)1205971198762

1199032=

minus119888119890int119879

0

119891(1198761199032

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 120597Π1199032(1198761199032 1198762)1205971198762

=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1198762minus

120582119890minus120572119905

)119892(119905)119889119905 1205972

Π1199032(1198761199032 1198762)1205971198762

2= minusint

119879

0

(119901119890120574119905

+ 119892119903

minus

119888119890)119891(1198762minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 1205972

Π1199032(1198761199032 1198762)12059711987611990321205971198762=

1205972

Π1199032(1198761199032 1198762)12059711987621205971198761199032

= 0 Hence

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972

Π1199032(1198761199032 1198762)

12059711987621199032

1205972

Π1199032(1198761199032 1198762)

12059711987611990321205971198762

1205972

Π1199032(1198761199032 1198762)

12059711987621205971198761199032

1205972

Π1199032(1198761199032 1198762)

12059711987622

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt 0 (17)

So Π1199032(1198761199032 1198762) is concave in 119876

1199032and 119876

2 Let 120597Π

1199032(1198761199032

1198762)1205971198761199032



119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+

119888119890minus119908)119888

119890and int

119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

2minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900


1199032= 119876lowast

2minus119876lowast

1199032


2gt 119876lowast


119900lt ((119901119890

120574119905

+

119892119903minus119888119890)(119901119890120574119905





1199032) = (119888

119900+ 119888119890

minus 119908) minus 119888119890int119879

0

119865(119876lowast

1199032minus

120582119890minus120572119905


1199032120597120572 =

minus(1205971198712(119876lowast

1199032)120597120572)(120597119871

2(119876lowast

1199032)120597119876lowast


119879

0

120582119905119890minus120572119905

119891(119876lowast

1199032minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119876lowast

1199032120597120574 =

minus(1205971198712(119876lowast

1199032)120597120574)(120597119871

2(119876lowast

1199032)120597119876lowast





2) = int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 minus


lowast

2120597120572 = minus(120597119871

3(119876lowast

2)120597120572)

(1205971198713(119876lowast

2)120597119876lowast

2) = minus int

119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903

minus 119888119890)119891(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and120597119876lowast

2120597120574 = minus(120597119871

3(119876lowast

2)120597120574)(120597119871

3(119876lowast

2)120597119876lowast

2) = int119879

0

119901119905119890120574119905

119865(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905




1199032le 1198761199042

le 119876lowast

2



1199042(1198761199042) is given by

Π1199042(1198761199042) = int

119879

0

119908119876lowast

1199032+ 119888119900(119876lowast

2minus 119876lowast

1199032)

+ 119888119890min [(119863 (119905) minus 119876

lowast

1199032)+

119876lowast

2minus 119876lowast

1199032]

minus 119892119904[min (119863 (119905) 119876

lowast

2) minus 1198761199042]+

minus 1198881198761199042 119892 (119905) 119889119905

(18)


Π1199042(1198761199042) = (119888119900+ 119888119890minus 119892119904) 119876lowast

2

+ (119892119904minus 119888119890) int

119879

0

int

119876lowast

2minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905


1199032

+ 119888119890int

119879

0

int

119876lowast

1199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199042

minus 119892119904int

119879

0

int

1198761199042minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(19)


max1198761199042gt0

Π1199042(1198761199042)

st 119876lowast

1199032le 1198761199042

le 119876lowast

2

(20)



1199042satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(21)

where int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199042(1198761199042)1198891198762

1199042=

minus119892119904int119879

0

119891(1198761199042

minus 120582119890minus120572119905


1198761199042 Let 119889Π

1199042(1198761199042)1198891198761199042


0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(22)


1199042le 119876lowast



1199032lt 119876120593

1199042lt 119876lowast



ge 119876lowast

2 the constraint



Proof Let 1198672(119876120593

1199042) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905


1199041lt 119876lowast

1199032 then119876

lowast

1199042= 119876lowast

1199032



lt 119876120593

1199041lt 119876lowast

2 then119876

lowast

1199042= 119876120593

1199042 We can


2(119876120593

1199042)120597120572)(120597119867

2(119876120593

1199042)120597119876120593

1199042) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876120593

1199042120597120574 = minus(120597119867

2(119876120593

1199042)120597120574)(120597119867

2(119876120593

1199042)120597119876120593

1199042) = 0


1199041ge 119876lowast

2 then119876

lowast

1199041= 119876lowast

2









1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2


= 119876lowast

2 From


1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt


119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876


1199030gt

119876lowast


(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199031 it


gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2




Proposition 13 If 119876120593119904


1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593




1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593


119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040






1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)


1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast


Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905





1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2= 1198761199032


1199032 1199021199032) is


can be rewritten as

Π1199032(1198761199032 1198762)

= (119888119900+ 119888119890minus 119908)119876

1199032

minus 119888119890int

119879

0

int

1198761199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ 1198762int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119900minus 119888119890) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

1198762minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119909) 119892 (119905) 119889119909 119889119905

(14)



1199032is

int

119879

0

119865 (119876lowast

1199032minus 120582119890minus120572119905

) 119892 (119905) 119889119905 =119888119900+ 119888119890minus 119908

119888119890

(15)


int

119879

0

(119901119890120574119905

+ 119892119903minus 119888119890) 119865 (119876

lowast

2minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888119900 (16)


=

(119888119900+119888119890minus119908)minus119888

119890int119879

0

119865(1198761199032minus120582119890minus120572119905

)119892(119905)119889119905 1205972Π1199032(1198761199032 1198762)1205971198762

1199032=

minus119888119890int119879

0

119891(1198761199032

minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 120597Π1199032(1198761199032 1198762)1205971198762

=

int119879

0

(119901119890120574119905


119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(1198762minus

120582119890minus120572119905

)119892(119905)119889119905 1205972

Π1199032(1198761199032 1198762)1205971198762

2= minusint

119879

0

(119901119890120574119905

+ 119892119903

minus

119888119890)119891(1198762minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 1205972

Π1199032(1198761199032 1198762)12059711987611990321205971198762=

1205972

Π1199032(1198761199032 1198762)12059711987621205971198761199032

= 0 Hence

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205972

Π1199032(1198761199032 1198762)

12059711987621199032

1205972

Π1199032(1198761199032 1198762)

12059711987611990321205971198762

1205972

Π1199032(1198761199032 1198762)

12059711987621205971198761199032

1205972

Π1199032(1198761199032 1198762)

12059711987622

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

gt 0 (17)

So Π1199032(1198761199032 1198762) is concave in 119876

1199032and 119876

2 Let 120597Π

1199032(1198761199032

1198762)1205971198761199032



119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+

119888119890minus119908)119888

119890and int

119879

0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

2minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900


1199032= 119876lowast

2minus119876lowast

1199032


2gt 119876lowast


119900lt ((119901119890

120574119905

+

119892119903minus119888119890)(119901119890120574119905





1199032) = (119888

119900+ 119888119890

minus 119908) minus 119888119890int119879

0

119865(119876lowast

1199032minus

120582119890minus120572119905


1199032120597120572 =

minus(1205971198712(119876lowast

1199032)120597120572)(120597119871

2(119876lowast

1199032)120597119876lowast


119879

0

120582119905119890minus120572119905

119891(119876lowast

1199032minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and 120597119876lowast

1199032120597120574 =

minus(1205971198712(119876lowast

1199032)120597120574)(120597119871

2(119876lowast

1199032)120597119876lowast





2) = int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119865(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 minus


lowast

2120597120572 = minus(120597119871

3(119876lowast

2)120597120572)

(1205971198713(119876lowast

2)120597119876lowast

2) = minus int

119879

0

120582119905119890minus120572119905

(119901119890120574119905

+ 119892119903

minus 119888119890)119891(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905

)119892(119905)119889119905 lt 0 and120597119876lowast

2120597120574 = minus(120597119871

3(119876lowast

2)120597120574)(120597119871

3(119876lowast

2)120597119876lowast

2) = int119879

0

119901119905119890120574119905

119865(119876lowast

2minus

120582119890minus120572119905

)119892(119905)119889119905 int119879

0

(119901119890120574119905

+ 119892119903minus 119888119890)119891(119876lowast

2minus 120582119890minus120572119905




1199032le 1198761199042

le 119876lowast

2



1199042(1198761199042) is given by

Π1199042(1198761199042) = int

119879

0

119908119876lowast

1199032+ 119888119900(119876lowast

2minus 119876lowast

1199032)

+ 119888119890min [(119863 (119905) minus 119876

lowast

1199032)+

119876lowast

2minus 119876lowast

1199032]

minus 119892119904[min (119863 (119905) 119876

lowast

2) minus 1198761199042]+

minus 1198881198761199042 119892 (119905) 119889119905

(18)


Π1199042(1198761199042) = (119888119900+ 119888119890minus 119892119904) 119876lowast

2

+ (119892119904minus 119888119890) int

119879

0

int

119876lowast

2minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905


1199032

+ 119888119890int

119879

0

int

119876lowast

1199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199042

minus 119892119904int

119879

0

int

1198761199042minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(19)


max1198761199042gt0

Π1199042(1198761199042)

st 119876lowast

1199032le 1198761199042

le 119876lowast

2

(20)



1199042satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(21)

where int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199042(1198761199042)1198891198762

1199042=

minus119892119904int119879

0

119891(1198761199042

minus 120582119890minus120572119905


1198761199042 Let 119889Π

1199042(1198761199042)1198891198761199042


0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(22)


1199042le 119876lowast



1199032lt 119876120593

1199042lt 119876lowast



ge 119876lowast

2 the constraint



Proof Let 1198672(119876120593

1199042) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905


1199041lt 119876lowast

1199032 then119876

lowast

1199042= 119876lowast

1199032



lt 119876120593

1199041lt 119876lowast

2 then119876

lowast

1199042= 119876120593

1199042 We can


2(119876120593

1199042)120597120572)(120597119867

2(119876120593

1199042)120597119876120593

1199042) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876120593

1199042120597120574 = minus(120597119867

2(119876120593

1199042)120597120574)(120597119867

2(119876120593

1199042)120597119876120593

1199042) = 0


1199041ge 119876lowast

2 then119876

lowast

1199041= 119876lowast

2









1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2


= 119876lowast

2 From


1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt


119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876


1199030gt

119876lowast


(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199031 it


gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2




Proposition 13 If 119876120593119904


1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593




1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593


119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040






1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)


1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast


Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905





1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199042(1198761199042) is given by

Π1199042(1198761199042) = int

119879

0

119908119876lowast

1199032+ 119888119900(119876lowast

2minus 119876lowast

1199032)

+ 119888119890min [(119863 (119905) minus 119876

lowast

1199032)+

119876lowast

2minus 119876lowast

1199032]

minus 119892119904[min (119863 (119905) 119876

lowast

2) minus 1198761199042]+

minus 1198881198761199042 119892 (119905) 119889119905

(18)


Π1199042(1198761199042) = (119888119900+ 119888119890minus 119892119904) 119876lowast

2

+ (119892119904minus 119888119890) int

119879

0

int

119876lowast

2minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905


1199032

+ 119888119890int

119879

0

int

119876lowast

1199032minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

+ (119892119904minus 119888)119876

1199042

minus 119892119904int

119879

0

int

1198761199042minus120582119890minus120572119905

0

119865 (119909) 119892 (119905) 119889119909 119889119905

(19)


max1198761199042gt0

Π1199042(1198761199042)

st 119876lowast

1199032le 1198761199042

le 119876lowast

2

(20)



1199042satisfies

119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(21)

where int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


= (119892119904minus

119888) minus 119892119904int119879

0

119865(1198761199042

minus 120582119890minus120572119905

)119892(119905)119889119905 and 1198892

Π1199042(1198761199042)1198891198762

1199042=

minus119892119904int119879

0

119891(1198761199042

minus 120582119890minus120572119905


1198761199042 Let 119889Π

1199042(1198761199042)1198891198761199042


0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 = (119892119904minus 119888)119892

119904


119876lowast

1199042=

119876lowast

1199032119876120593

1199042le 119876lowast

1199032

119876120593

1199042119876lowast

1199032lt 119876120593

1199042lt 119876lowast

2

119876lowast

2119876120593

1199042ge 119876lowast

2

(22)


1199042le 119876lowast



1199032lt 119876120593

1199042lt 119876lowast



ge 119876lowast

2 the constraint



Proof Let 1198672(119876120593

1199042) = (119892

119904minus 119888) minus 119892

119904int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905


1199041lt 119876lowast

1199032 then119876

lowast

1199042= 119876lowast

1199032



lt 119876120593

1199041lt 119876lowast

2 then119876

lowast

1199042= 119876120593

1199042 We can


2(119876120593

1199042)120597120572)(120597119867

2(119876120593

1199042)120597119876120593

1199042) =

minus int119879

0

120582119905119890minus120572119905

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 int119879

0

119891(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 lt

0 and 120597119876120593

1199042120597120574 = minus(120597119867

2(119876120593

1199042)120597120574)(120597119867

2(119876120593

1199042)120597119876120593

1199042) = 0


1199041ge 119876lowast

2 then119876

lowast

1199041= 119876lowast

2









1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2


= 119876lowast

2 From


1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt


119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876


1199030gt

119876lowast


(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199031 it


gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2




Proposition 13 If 119876120593119904


1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593




1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593


119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040






1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)


1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast


Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905





1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1199032lt 119876lowast

1199030lt

119902lowast

1199031= 119876lowast

2


= 119876lowast

2 From


1199032

=

int119879

0

(119901119890120574119905

+119892119903minus119908)119892(119905)119889119905minusint

119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199032minus120582119890minus120572119905

)119892(119905)119889119905 gt


119890int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905] = 0Recalling thatΠ

1199030(1198761199030) is concave in119876


1199030gt

119876lowast


(119889Π1199031(1199021199031)1198891199021199031)|1199021199031=119876lowast

1199030

= int119879

0

(119901119890120574119905

+ 119892119903

minus 119908)119892(119905)119889119905 minus

int119879

0

(119901119890120574119905

+119892119903)119865(119876lowast

1199030minus120582119890minus120572119905

)119892(119905)119889119905minus(119888119900+119888119890minus119908)+119888

119890int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199031 it


gt 119876lowast

1199030 So 119876lowast

1199032lt 119876lowast

1199030lt 119902lowast

1199031= 119876lowast

2




Proposition 13 If 119876120593119904


1199032) then 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then 119876

lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 +infin)

then 119876lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040

Proof Let 119876120593119904= 119876120593

1199041= 119876120593




1199032) then

119876lowast

1199041= 119876120593

119904and 119876

lowast

1199042= 119876lowast

1199032 We can obtain 119876

lowast

1199041lt 119876lowast

1199042lt 119876lowast

1199040 If

119876120593


1199032 119876lowast

1199030) then119876

lowast

1199041= 119876120593

119904and119876

lowast

1199042= 119876120593


119876lowast

1199041= 119876lowast

1199042lt 119876lowast

1199040 If 119876120593119904isin (119876lowast

1199030 119876lowast

2) then 119876

lowast

1199041= 119876120593

119904and 119876

lowast

1199042=

119876120593


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040 If 119876120593119904isin (119876lowast

2 +infin) then

119876lowast

1199041= 119902lowast

1199031and119876

lowast

1199042= 119876lowast


lowast

1199041= 119876lowast

1199042gt 119876lowast

1199040






1199030(119876lowast

1199030) lt Π

1199031(119902lowast

1199031) lt

Π1199032(119876lowast

1199032 119876lowast

2)


1199032 119876lowast

2) with Π

1199031(119902lowast

1199031) Let

Δ1(119876lowast

1199032) = Π

1199032(119876lowast

1199032 119876lowast

2) minus Π

1199031(119902lowast


Δ1(119876lowast

1199032) = (119888

119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905





1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1(119876lowast

1199032)119889119876lowast

1199032)|119876lowast

1199032=0


1199032(119876lowast

1199032 119876lowast

2) gt Π

1199031(119902lowast

1199031) Then we


1199031) with Π

1199030(119876lowast

1199030) Let Δ

2(119876lowast

1199030) = Π

1199031(119876lowast

1199030) minus

Π1199030(119876lowast


2(119876lowast

1199030) = (119908 minus 119888

119900minus 119888119890)119876lowast

1199030+

119888119890int119879

0

int119876lowast

1199030minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 Because Δ2(0) = 0 and

119889Δ2(119876lowast

1199030)119889119876lowast

1199030= minus119888119890[int119879

0

119865(119876lowast

1199032minus 120582119890minus120572119905

)119892(119905)119889119905 minus int119879

0

119865(119876lowast

1199030minus

120582119890minus120572119905


1199030) gt Π

1199030(119876lowast

1199030) Since

Π1199031(119902lowast

1199031) gt Π1199031(119876lowast


1199031(119902lowast

1199031) gt Π1199030(119876lowast

1199030)


119900+ 119888119890gt 119908 and 119902

lowast

1199031= 119876lowast

2 the




1199040(119876lowast

1199040) lt Π

1199042(119876lowast

1199042) lt

Π1199041(119876lowast

1199041)


1199041) with Π

1199042(119876lowast

1199042) Let

Δ3(119876lowast

1199032) = Π

1199041(119876lowast

1199041) minus Π

1199042(119876lowast


Δ3(119876lowast

1199032) = (119888119900+ 119888119890minus 119908)119876

lowast

1199032minus 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 minus

(119892119904minus 119888)(119876

lowast


1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


119900+ 119888119890minus 119908)119876

lowast

1199032minus

119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 gt 0 Now let 119876120593119904= 119876120593

1199041= 119876120593

1199042

If 119876120593119904

isin (119876lowast

1199032 +infin) then 119876

lowast

1199041= 119876lowast

1199042and minus(119892

119904minus 119888)(119876

lowast

1199042minus

119876lowast

1199041) + 119892

119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 If 119876120593


1199032 +infin) then 119876

lowast

1199041lt 119876lowast

1199042and

minus(119892119904minus 119888)(119876

lowast


1199041) + 119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

119876lowast

1199041minus120582119890minus120572119905


3(119876lowast

1199032) gt 0 Hence Π

1199041(119876lowast

1199041) gt Π1199042(119876lowast

1199042)


1199042) with Π

1199040(119876lowast

1199040) Let Δ

4(119908) =

Π1199042(119876lowast

1199042) minus Π

1199040(119876lowast


4(119908) = (119908 minus

119888119900

minus 119888119890)119876lowast

1199032+ 119888119890int119879

0

int119876lowast

1199032minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119888119900

+ 119888119890minus

119892119904)119876lowast

2+ (119892119904minus 119888119890) int119879

0

int119876lowast

2minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905 + (119892119904minus 119888)119876

lowast

1199042minus

119892119904int119879

0

int119876lowast

1199042minus120582119890minus120572119905

0


1199030 Let1199080 = 119888

119900(119901119890120574119905

+

119892119903)(119901119890120574119905

+ 119892119903minus 119888119890) If 119908 = 119908

0 then 119902lowast

1199032= 0 119876lowast

1199032=

119876lowast

2= 119876lowast

1199030 and 119876

lowast

1199042= 119876lowast


4(119908) = 0

Notice (119889Δ4(119908)119889119908)|

119908=1199080 = 119892

119904[int119879

0

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 minus

int119879

0

119865(119876lowast

1199042minus120582119890minus120572119905

)119892(119905)119889119905](119889119876lowast

1199042119889119908)minus(119908

0

minus119888)(119889119876lowast

1199030119889119908) From


1199042gt 119876lowast

1199032 then

119889119876lowast


4(119908)119889119908)|

119908=1199080 gt 0

If 1198761205931199042

le 119876lowast

1199032 then 119876

lowast

1199042= 119876lowast

1199032 int1198790

119865(119876120593

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 le

int119879

0

119865(119876lowast

1199042minus 120582119890minus120572119905

)119892(119905)119889119905 and 119889119876lowast

1199042119889119908 lt 0 We can obtain

that (119889Δ4(119908)119889119908)|

119908=1199080 gt 0 HenceΠ

1199042(119876lowast

1199042) gt Π1199040(119876lowast

1199040)










119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119868


given by

Π119868(119876119868) = int

119879

0

119901 (119905)min [119863 (119905) 119876119868]

minus 119892119903[119863 (119905) minus 119876

119868]+

minus 119888119876119868 119892 (119905) 119889119905

(23)


Π119868(119876119868)

= 119876119868int

119879

0

(119901119890120574119905

+ 119892119903minus 119888) 119892 (119905) 119889119905

minus 119892119903int

119879

0

120582119890minus120572119905

119892 (119905) 119889119905 minus 119892119903120583

minus int

119879

0

int

119876119868minus120582119890minus120572119905

0

(119901119890120574119905

+ 119892119903) 119865 (119909) 119892 (119905) 119889119909 119889119905

(24)



119868is

int

119879

0

(119901119890120574119905

+ 119892119903) 119865 (119876

lowast

119868minus 120582119890minus120572119905

) 119892 (119905) 119889119905 = 119888 (25)


=

int119879

0

(119901119890120574119905

+ 119892119903minus 119888)119892(119905)119889119905 minus int

119879

0

(119901119890120574119905

+ 119892119903)119865(119876119868minus 120582119890minus120572119905

)119892(119905)119889119905

and 1198892

Π119868(119876119868)1198891198762

119868=minusint119879

0

(119901119890120574119905

+119892119903)119891(119876119868minus120582119890minus120572119905

)119892(119905)119889119905 lt 0

so Π119868(119876119868) is concave in 119876

119868 Let 119889Π

119868(119876119868)119889119876119868= 0 we can


(119901119890120574119905

+119892119903)119865(119876lowast

119868minus

120582119890minus120572119905

)119892(119905)119889119905 = 119888




+ 119892119903minus 119888119890and int

119879

0

119865(119876lowast

119868minus

120582119890minus120572119905




120593

1199041ge 119902lowast

1199031 that is 119892

119904gt 119901119890

120574119905

+ 119892119903minus 119888119890 From


0

(119901119890120574119905

+119892119903minus119888119890)119865(119876lowast

119868minus120582119890minus120572119905

)119892(119905)119889119905 = 119888119900 From (25) we can


119865(119876lowast

119868minus 120582119890minus120572119905

)119892(119905)119889119905 = (119888119900+ 119888119890minus 119888)119888

119890 The two



119868(119876lowast

119868) minusΠ1199031(119902lowast



119868minus119888119890int119879

0

int119876lowast

119868minus120582119890minus120572119905

0

119865(119909)119892(119905)119889119909 119889119905Now let 120578 = Δ120587Π

119868(119876lowast



1199041= 120578Π

119868and Π

1199031= (1 minus 120578)Π

119868




120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00005 15604 1657 9942 6628 15604 1657 16570001 15619 16578 9942 6636 15619 16578 1657800015 15635 16586 9942 6644 15635 16586 165860002 1565 16593 9942 6651 1565 16593 1659300025 15665 166 9942 6658 15665 166 1660003 1568 16608 9942 6666 1568 16608 1660800035 15695 16615 9942 6673 15695 16615 166150004 1571 16622 9942 668 1571 16622 1662200045 15725 16629 9942 6687 15725 16629 166290005 1574 16636 9942 6694 1574 16636 1663600055 15754 16643 9942 6701 15754 16643 166430006 15769 1665 9942 6708 15769 1665 166500065 15784 16657 9942 6715 15784 16657 166570007 15799 16664 9942 6722 15799 16664 1666400075 15813 1667 9942 6728 15813 1667 1667



119890 and unit


7 Numerical Example



119900119888= 2 119888

119890119888= 25 119892

119904= 20

119892119903= 7 120582 = 100 119879 = 60 119905 sim 119880(0 60) and 120585 sim 119880(0 100)











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572 119876lowast

1199030119902lowast

1199031119876lowast

1199032119902lowast

1199032119876lowast

1199040119876lowast

1199041119876lowast

1199042

00085 15715 16563 9833 6729 15715 16563 165630009 15604 16451 9727 6724 15604 16451 1645100095 15495 16342 9622 6719 15495 16342 16342001 15388 16234 952 6715 15388 16234 1623400105 15283 16129 9419 671 15283 16129 161290011 15181 16026 932 6705 15181 16026 1602600115 1508 15925 9224 6701 1508 15925 159250012 14981 15825 9128 6697 14981 15825 1582500125 14885 15728 9035 6693 14885 15728 157280013 1479 15633 8944 6689 1479 15633 1563300135 14696 15539 8854 6685 14696 15539 155390014 14605 15447 8765 6682 14605 15447 1544700145 14515 15357 8679 6678 14515 15357 153570015 14427 15268 8594 6675 14427 15268 1526800155 14341 15182 851 6671 14341 15182 15182


120574 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00005 63169 63541 67861 28088 45225 409050001 65043 65432 69753 28115 45234 4091400015 66955 67362 71682 28142 45243 409230002 68904 69329 73650 28169 45251 4093100025 70894 71337 75657 28196 4526 409390003 72923 73384 77704 28223 45268 4094800035 74994 75473 79793 2825 45276 409560004 77107 77603 81924 28277 45284 4096400045 79263 79777 84097 28304 45292 409720005 81464 81995 86315 28331 45300 409800055 83709 84258 88578 28358 45308 409880006 86000 86566 90887 28385 45316 4099600065 88339 88922 93242 28411 45323 410030007 90726 91326 95647 28437 45331 4101100075 93162 93780 98100 28464 45338 41018





8 Conclusion




120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572 Π1199030(119876lowast

1199030) Π

1199031(119902lowast

1199031) Π

1199032(119876lowast

1199032 119876lowast

2) Π

1199040(119876lowast

1199040) Π

1199041(119876lowast

1199041) Π

1199042(119876lowast

1199042)

00085 94502 95198 99449 28286 44966 407150009 93374 94131 98313 28086 44594 4041100095 92265 93082 97197 27891 44229 40113001 91175 92051 96100 27699 43870 3982100105 90104 91039 95022 27510 43517 395340011 89051 90044 93963 27325 43171 3925200115 88017 89067 92922 27144 42831 389760012 87000 88108 91900 26966 42497 3870500125 86002 87166 90896 26792 42169 384390013 85022 86241 8991 26621 41848 3817800135 84060 85333 88943 26453 41532 379220014 83115 84442 87992 26289 41221 3767100145 82188 83568 87060 26127 40917 374240015 81277 82709 86144 25969 40617 3718300155 80384 81867 85246 25813 40324 36945





Notations














(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















(119894 = 0 1 2)


(119894 = 0 1 2)


(119894 = 0 1 2) Note that 119876119894= 119876119903119894+ 119902119903119894


(119894 = 0 1 2)



Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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