research article an optimization model for inventory system and...

Research ArticleAn Optimization Model for Inventory System andthe Algorithm for the Optimal Inventory Costs Based onSupply-Demand Balance

Qingsong Jiang,1 Wei Xing,1,2 Ruihuan Hou,1 and Baoping Zhou1

1College of Information Engineering, Tarim University, Alar, Xinjiang 843300, China2College of Computer Science and Technology, Zhejiang University, Hangzhou, Zhejiang 310027, China

Correspondence should be addressed to Baoping Zhou; [email protected]

Received 13 June 2015; Revised 18 November 2015; Accepted 23 November 2015

Academic Editor: Anna M. Gil-Lafuente

Copyright © 2015 Qingsong Jiang et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In order to investigate the inventory optimization of circulation enterprises, demand analysis was carried out firstly consideringsupply-demand balance. Then, it was assumed that the demand process complied with mutually independent compound Poissonprocess. Based on this assumption, an optimization model for inventory control of circulation enterprises was established withthe goal of minimizing the average total costs in unit time of inventory system. In addition, the optimal computing algorithm forinventory costs was presented.Meanwhile, taking the agricultural enterprises in Aksu, Xinjiang, China, for example, the researchersconducted numerical simulation and sensitivity analysis. Through constantly adjusting and modifying the parameters values inmodel, the optimal stock and the optimal inventory costs were obtained.Therein, the numerical results showed that the uncertaintyof lead time greatly influenced the optimal inventory strategy. Besides, it was demonstrated that the research results provided avaluable reference for the agricultural enterprises in terms of optimal management for inventory system.

1. Introduction

Warehousing is an important part of logistics system. Inven-tory control of warehousing has been widely focused on bycirculation enterprises and relevant scholars all the time.If the stock is high, smooth business process can be fullyguaranteed to improve service level and customer satisfac-tion, while if the stock is low, capital backlog of enterprisesand the corresponding management costs are able to bereduced (as in [1]), to optimize and control stock mattersto the service quality and economic benefits of circulationenterprises. In addition, the sustainable development of cir-culation enterprises is crucial.

Owing to the significance of inventory optimization andcontrol in circulation enterprises, there are many relevantscholars that begin to pay attention to this. In the lasttwo decades, the issue has attracted much attention frommany researchers. Among these researches, economic orderquantity (EOQ) model based on stock-dependent demandwas established (as in [2]). Besides, the production-inventory

model for perishable items with definite productivity andwith demand linearly depending on inventory level was con-sidered (as in [3]). Some scholars explored the inventory issuewith allowable shortages under inventory-level-dependentdemand; at the same time, they also took monetary value aswell as the expansion rate caused by external and internalcosts into consideration (as in [4]). In addition, EOQ modelfor perishable items was established, where the perishableitems were under the following conditions: the demand ratewas related to inventory level and some stock-outs could besupplemented later (as in [5]). Cardenas-Barron et al. (asin [6]) studied the optimal solution of multiproduct EOQmodel. Moreover, the optimal replenishment strategy forperishable items was investigated aiming at maximizingprofits (as in [7]), while the inventory optimization forperishable items under stock-dependent demandwas studiedas well (as in [8]). Wang et al. (as in [9]) discussed the inven-tory control model for fresh agricultural products onWeibulldistribution under the assumption that the inflation rate ishigher than the natural decay rate. Paul and Rajendran (as

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 508074, 11 pageshttp://dx.doi.org/10.1155/2015/508074

2 Mathematical Problems in Engineering

in [10]) studied the problem of rationing mechanisms andinventory control-policy parameters for a divergent supplychain operating with lost sales and costs of review. Krish-namoorthy andNarayanan (as in [11]) considered the stabilityand performance analysis of a production-inventory system.Yadavalli et al. (as in [12]) studied the problem of updatingservice facilities for inventory system to achieve productionand service synchronization. From the inventory cost andthe cost of order to determine the optimal order point andquantity, Dogru et al. (as in [13]) pointed out enterprisesadjust inventory quantity through a large number of bufferstocks and there is a serious bullwhip effect. Hua et al. (as in[14]) studied the carbon emissions in inventorymanagement.The corresponding model is established by the joint ofreplenishment strategy, and through genetic algorithm, Zhouet al. (as in [15]) gave the solution and simulation of themodel. Murray et al. (as in [16]) studied the multiproductpricing and inventory issues. Choi and Ruszczynski (as in[17]) established a multiproduct risk-averse newsvendor, andthey pointed out that the increase of risk aversion doesnot necessarily lead to the reduction of the order quantity.Schrijver et al. (as in [18]) studied the optimization modeland algorithm of multiproduct demand inventory networkdesign for stochastic demand and inventory decision. Basedon the theory of nonlinear integer programming, Yang et al.(as in [19]) studied the integrated multiproduct optimizationmodel. Liu et al. (as in [20]) studied the flexible servicepolicies for a Markov inventory system with two demandclasses. Zhao and Lian (as in [21]) studied the priorityservice rule of a queueing-inventory system with two classesof customers. Karimi-Nasab and Konstantaras (as in [22])studied an inventory control model with stochastic reviewinterval and special sale offer. By determining the level ofcustomer’s anchoring effect, Liu and Shum (as in [23]) studiedthe joint control of pricing and inventory allocation in twoperiods of retailers based on constructing the customer’sdisappointment aversion utility function.Mo et al. (as in [24])researched the inventory issue for the perishable multi-itemswith just-in-time (JIT) inventory-level-dependent demand.Ji and Jin (as in [25]) established an inventory optimizationmodel meeting the restrained conditions of being control-lable in lead time and service level. Li (as in [26]) studied thecontrol and optimization model for multiechelon inventoryin supply chain, while the optimization method for two-echelon inventory system based on stochastic lead time wasresearched by Dai et al. (as in [27]). Zhao (as in [28]) pre-sented an optimization study on multiechelon inventory insupply chain on the basis of time competition, while Wang(as in [29]) studied the optimization model for production-inventory under uncertain environments. Fu and Pan (asin [30]) mainly explored disposing the inventory manage-ment problem by using fuzzy theory under uncertainty toderive the fuzzy mathematical model for single inventorymanagement with multiple fuzzy parameters in the case ofallowingmoderate shortages. Besides, supply chain inventoryoptimization with controllable lead time under fuzzy envi-ronment was investigated by Li and Xu (as in [31]). Wangand Guo (as in [32]) analyzed the inventory risk loss ledby the EOQ and order cycles of classical inventory models

under fuzzy demand to deduce the economic risk functionin fuzzy situation. Kong and Jirimutu (as in [33]) researchedthe inventory optimization under stochastic demand basedon Monte Carlo simulation. Xu et al. (as in [34]) exploredthe inventory control model during random replenishmentinterval with inventory-level-dependent demand.

Most of the above researches were conducted on thebasis of continuous normal population, which made theresearches convenient and operable to some extent. However,on the premise of uncertain supply and demand, there werea lot of uncertain factors for inventory optimization. In fact,most of the demand and supply in reality cannot distributecontinually but present in the form of discrete random vari-ables usually. As a result, on the assumption that the demandprocess of each subwarehouse submitted to the mutuallyindependent compound Poisson process, the authors carriedout the researches on some aspects, including the opti-mization and control of inventory system based on supply-demand balance as well as the algorithm design for theoptimal inventory costs. In addition, the related researcheshave a certain value on theoretical research.

Although there have been quite a few researches on inven-tory control, it is still necessary to takemany factors and vari-ables into account due to its systematicness and complexityof inventory problem. Besides, it is difficult to quantify anddefine the optimal inventory because the correlation degreesbetween each factor are fuzzy. In view of the above facts, theoptimization and control of inventory can be summarizedas a complex dynamic system containing multifactors, whilethe quantitative model about the optimization and control ofinventory is considered as a complex system with multiplevariables and multiple parameters. It is very difficult to solvethe problem once and for all by using a single model and aunified algorithm for the research work of the optimizationand control of inventory. Most of the existing researches werefocused on some specific fields of a particular region to onlywork out the specific issues under a certain environment. Inaddition, it is inevitable that the research process is influencedby the subjectivity of the researchers themselves, whichsuggests that the optimization and control of inventory undervarious situations cannot be solved. Hence, this issue willundoubtedly attract the persistent attention from the relevantexperts and scholars. Actually, it still plays a very realisticrole in carrying out pertinent researches on optimization andcontrol model for inventory and algorithm with respect tosome specific fields in different areas.

2. Inventory System Model

From a practical point of view of research object, a necessarysimplification for the research object was conducted duringthe research combining the actual conditions of regionalcirculation enterprises. For the underdeveloped regions, thetwo-echelon inventory system is more common. In orderto improve the practical significance of research results andenhance the operability, a typical two-echelon inventorysystem composed of a central warehouse and several sub-warehouses was emphatically studied.

Mathematical Problems in Engineering 3

3. Model Assumption and Symbol Description

3.1. Model Assumption

(1) The central warehouse of the two-echelon inventorysystem mentioned purchases products from materialsuppliers, while the subwarehouses order goods fromthe central warehouse.

(2) Both the central warehouse and subwarehouses ofthe system carry out the (𝑅, 𝑄) ordering strategyof continuous review inventory. In other words, theinventory levels are continuously observed by thesubwarehouses and central warehouse. When theinventory level reduces to order point 𝑅, the ware-houses will purchase with lot-size of 𝑄, where theinventory level refers to the result by subtracting thestock-outs from the total of on-hand inventory andthe goods of the orders in transit. In this way, theinventory level is within a range [𝑅, 𝑅 + 𝑄] afterdistribution centers and retailers ordering.

(3) The material suppliers can supply materials unlimit-edly and the delivery time for central warehouse is aconstant, while the transportation time from centralwarehouse to subwarehouses is a random variable.Then, the lead time of subwarehouses consists ofrandom delay and random transportation time.

(4) The product demand process of subwarehouses isa mutually independent compound Poisson process,that is to say, Poisson arrival of consumers. In addi-tion, the demand of each consumer is a randominteger.

(5) 𝑅𝑖≥ −𝑄𝑖(𝑖 = 0, 1, 2, . . . , 𝑁) are available for all order

points.(6) All stock-outs in the two-echelon inventory system

are waiting. Besides, the delayed order-to-deliveryfollows the principle of “first come first serve.”

3.2. Symbol Description. The meaning of the symbols in theresearch is as follows:

𝑁 is the number of retailers.𝐿0is the fixed delivery time from manufacturers to

distribution centers, namely, the lead time of distri-bution centers.𝑇𝑖is the random transportation time of goods from

distribution centers to retailers 𝑖.𝜏 is the random delay of retailer’s orders in distribu-tion centers.𝐿𝑖is the lead time of retailers 𝑖; 𝐿

𝑖= 𝑇𝑖+ 𝜏.

𝑄0is the order quantity of distribution centers.

𝑄𝑖is the order quantity of retailers 𝑖.

𝑅0is the order point of distribution centers.

𝑅𝑖is the order point of retailers 𝑖.

ℎ0is the storage costs of unit goods in unit time of

distribution center.

ℎ𝑖is the storage costs of unit goods in unit time of

retailers 𝑖.𝑃0is the stock-out losses of unit goods in unit time of

distribution center.𝑃𝑖is the stock-out losses of unit goods in unit time of

retailers 𝑖.𝐶0(𝑅0, 𝑄0) is the average holding costs and shortage

costs of distribution center in unit time.𝐶𝑖(𝑅𝑖, 𝑄𝑖) is the holding costs and shortage costs of

retailers 𝑖 in unit time.𝑇𝐶 is the expected gross costs of the inventory sys-tem.

4. Hypothesis Testing and Demonstration ofPoisson Distribution

In order to test whether the order demand is subject to Pois-son distribution, we give an empirical research by taking theagricultural warehousing company in Akesu area of Xinjiangas an example. Based on balance theory between supply anddemand, the fertilizers supply quantity of the agriculturalwarehousing company within a week is determined by thedemand quantity of farmers, while the supply quantity of theagricultural warehousing company decides its order quantity.Hence, the demand-based order quantity of the agriculturalwarehousing companywithin aweek depends on the demandquantity of the farmers in this area in this period. In order tosimplify the problem, the order quantity of each time withlittle fluctuation is taken as a constant. By doing so, the orderquantity can be determined by controlling the order timeswithin a single period.

In general, agricultural production presents seasonalityand periodicity. This epically can be obviously found inagriculture plantation of Aksu area in Xinjiang province.As the demand quantity of fertilizers for the farmers inAksu area is shown to be stable within some period or agiven time in a season, to some extent, the order times ofthe agricultural warehousing company for an arbitrary timeinterval within the demand period merely rely on the spanof time interval, instead of the end point of time interval.In addition, the order events of the agricultural warehous-ing company happen independently in the nonoverlappingintervals. Moreover, the probability for the occurrence of twoor more than two order times can be nearly neglected whenthe time interval is enough small. As above mentioned, it isindicated that the order times of the agricultural warehousingcompany conform to Poisson stream, indicating stability, noaftereffect stream, and ordinary (as in [35]). On this basis,it is supposed that the order demand of the agriculturalwarehousing company is subject to Poisson distribution.

In order to verify our supposing, we conducted on-siteinterviews to the large and middle size agricultural ware-housing company and the farmers in various corps. for sur-veying the demands of various fertilizers. With in-depthinvestigation, we have acquired large volume of informationand first-hand data: through the close communication andcontact with themanagers in each corp., the total arable areas


in Aksu and actual demand quantity of various fertilizers ofeach corp. were acquired. Afterwards, we verified the good-ness of fit for Pearson 𝜒2 of the sample data, to judge whetherthe sample data are subject to Poisson distribution or not. Itis noteworthy that the actual demand quantity of fertilizersis the order demand quantity of the agricultural warehousingcompany in fact. The method is described as follows.

By randomly selecting a set of sample data with eachsample size more than 50, the order quantity of each timeis assumed to be a constant when the order quantity of eachtime shows little fluctuation. In this case, the order demandcan be directly presented by controlling order times. Theanalysis of chosen data indicates that third- or fourth-ordertimes of the agricultural warehousing company are shown tobe reasonable.

According to statistical quantity,

𝜒2

𝑖(𝛼)=

𝑚

∑𝑗=1

[

[

(𝑓𝑖𝑗− 𝑝𝑖𝑗)2

𝑓𝑖

]

]

. (1)

𝑚 is the number of purchase groups, where𝑚 = 2.The empirical and theoretical frequencies of stochastic

events are 𝑝𝑖and𝑓𝑖, respectively, and they indicate significant

difference, with a given confidential level 𝛼; the distributionof 𝑝𝑖does not agree with that of 𝑓

𝑖; if there is no significant

difference between 𝑝𝑖and 𝑓

𝑖, the random variable distribu-

tion is subject to the theoretical scheme. To validate whetheror not the empirical scheme of the fertilizers demand in atime interval in some areas is subject to Poisson distribution,a significant difference between the theoretical and empiricalfrequencies needs to be checked. In the case of 𝛼 = 0.05,𝜒2

𝛼= 3.841 refers to the value of 𝜒2 when the degree of

freedom is 1. On this basis, 𝜒2 = 3.5011 is calculated through𝑅 software programming this is obviously obtained that 𝜒2 <𝜒2𝛼. As can be seen, the selected sample data have passed the

verification of Poisson distribution. This is to say, these dataare subject to the Poisson distribution; the detailed processesand methods can be seen in the researches (as in [35, 36]). Byfrequent sampling on the sample data, various sets of sampledata are obtained by repeating abovementionedmethods andprocesses. All chosen sample data have been validated inPoisson distribution.This further confirmed the feasibility ofour assumption.

Apart from verifying the goodness of fit for Pearson𝜒2, the repeated samplings have been conducted on the

sample data. Moreover, the density function was performedon the simulation of different samples.The simulating resultsindicate that the density function approximates to normaldensity with increasing quantity of sample size; consideringthat the order quantity or order number of a warehouse isstochastic and independent, the data obtained are discretedata, as the order number is relatively large. This is becausethe corps. in Aksu area mainly make a living by planting;the demand quantity of various fertilizers is relatively large:the quantities are usually more than 1,000 tons on average.The average value of real sample data investigated is 1,093tons. The total average value estimated is the average value

of the sample data. Based on these two points, the parameterlambda (total average value) of Poisson distribution is largeaccording to probability knowledge; as its limit distributionis normal, it is supposed that the demand data are subjectto Poisson distribution which is assumed to be reasonableaccording to the abovementioned analysis results.

As shown in Figure 1, 𝑁 = 16 represents the densityfunction image of all data investigated, where 𝑁 = 16 refersto 16 corps. in Aksu. The density functions are obtained bythe simulation when three sample sizes are 30, 60, and 300,respectively. The analysis and investigation into the simula-tion of density function images reveal that, in the averagevalue range of the samples [800, 1200], there is a good sim-ilarity between Poisson and normal distributions. The largethe sample size, the more favorable the approximation effect.

5. Modeling

On the assumption that the lead time of retailers is a randomvariable and the demand process is a compound Poissonprocess, the (𝑅, 𝑄) storage model was established with theaim of minimizing the mean total cost of two-echelon inven-tory system in unit time (as in [5]).

5.1. The Inventory Model of Central Warehouse. It was as-sumed that the demand process of subwarehouse 𝑖 is followedby the compound Poisson process with parameter of 𝜆

𝑖,

which is the mean unit time of purchasers arriving at sub-warehouse 𝑖. If 𝑗 is the demand of purchasers, 𝑓

𝑖,𝑗is the

probability of demand 𝑗 (𝑗 > 0) in retailer 𝑖, 𝜇𝑖is the average

demand of retailer 𝑖 in unit time, and 𝜎2𝑖is the demand

variance of retailer 𝑖 in unit time, then

𝜇𝑖= 𝜆𝑖

∞

∑𝑗=1

𝑗2𝑓𝑖,𝑗,

𝜎2

𝑖= 𝜆𝑖

∞

∑𝑗=1

𝐿0𝜎2

𝑖.

(2)

Thereby, the average demand and demand variance ofretailer 𝑖 in the lead time of distribution centers can beexpressed as

𝜇𝑖(𝐿0) = 𝐿0𝜇𝑖,

𝑉𝑖(𝐿0) = 𝐿0𝜎2

𝑖.

(3)

Because the demand discussed here was Poisson demand,it could be known according to probability knowledge thatif the sample capacity was great, Poisson distribution wouldbe approximate to normal distribution. Thus, it was assumedthat the demand of subwarehouse in the lead time of centralwarehouse was close to normal distribution (as in [5, 25]).

If the probability for placing 𝑘 orders by subwarehouse 𝑖 tocentral warehouse in lead time 𝐿

0was 𝑝

𝑖,𝑘(𝐿0), the first order

placed after demand was up to 𝑥, where 𝑥 obeyed uniformdistribution on (0, 𝑄

𝑖), so placing 𝑘 orders by subwarehouse


Density default (x = w3)Density default (x = w2)

Density default (x = w1)Density default (x = w)

1000 1200 1400 1600 1800800

0.0000

0.0010

0.0020

0.0030

Den

sity

0.0000

0.0010

0.0020

0.0030

Den

sity

1000 1200 1400 1600 1800800

1000 1200 1400 16008001000 1200 1400 1600800

0.000

0.001

0.002

0.003

0.004

Den

sity

0.000

0.001

0.002

0.003

0.004

0.005D

ensit

y

N = 300, bandwidth = 57.62N = 60, bandwidth = 80.04

N = 30, bandwidth = 92.73N = 16, bandwidth = 104.2

Figure 1: The simulation of density function images.

𝑖 within 𝐿0means that the demand is between 𝑥 + (𝑘 − 1)𝑄

𝑖

and 𝑥 + 𝑘𝑄𝑖; then,

𝑝𝑖,𝑘(𝐿0) =

1

𝑄𝑖

∫𝑄𝑖

0

[𝜙(𝑥 + 𝑘𝑄

𝑖− 𝜇𝑖(𝐿𝑜)

𝜎𝑖(𝐿0)

)

− 𝜙(𝑥 + (𝑘 − 1)𝑄𝑖 − 𝜇

𝑖(𝐿0)

𝜎𝑖(𝐿0)

)] 𝑑𝑥

=𝜎𝑖(𝐿0)

𝑄𝑖

[𝜙(1)

((𝑘 − 1)𝑄𝑖 − 𝜇

𝑖(𝐿0)

𝜎𝑖(𝐿0)

)

+ 𝜙(1)

((𝑘 + 1)𝑄𝑖 − 𝜇

𝑖(𝐿0)

𝜎𝑖(𝐿0)

)

− 2𝜙(1)

(𝑘𝑄𝑖− 𝜇𝑖(𝐿0)

𝜎𝑖(𝐿0)

)] ,

(4)

where 𝜙(𝑥) is the standard normal density function and𝜙(1)(𝑥) is first-order loss function; then, it can be obtainedthat

𝜙(1)

(𝑥) = ∫∞

𝑥

(𝑢 − 𝑥) 𝜛(𝑢)𝑑𝑢 = 𝜛(𝑥)

− 𝑥 (1 − 𝜙(𝑥)) . (5)

Thereby, the mean 𝜇0𝑖(𝐿0) and variance 𝑉0

𝑖(𝐿0) of the

demand of central warehouse obtained from subwarehouse𝑖 in lead time 𝐿

0can be calculated as

𝜇0

𝑖(𝐿0) = 𝜇 (𝐿

0) ,

𝑉0

𝑖(𝐿0) = ∑(𝑘𝑄

𝑖− 𝜇0

𝑖(𝐿0))2

𝑝𝑖,𝑘(𝐿0) .

(6)


To sum up 𝜇0𝑖(𝐿0) and 𝑉0

𝑖(𝐿0), the mean 𝜇0(𝐿

0) and

variance𝑉0(𝐿0) of demand of central warehouse in lead time

𝐿0can be obtained; namely,

𝜇0(𝐿0) =

𝑁

∑𝑖=1

𝜇0

𝑖(𝐿0) ,

𝑉0(𝐿0) =

𝑁

∑𝑖=1

𝑉0

𝑖(𝐿0) .

(7)

𝐼0and 𝐼

𝑖are set as the stochastic inventory levels of

central warehouse and subwarehouse 𝑖, respectively (𝑖 =

1, 2, 3, . . . , 𝑁). Besides, it has been known that the inventorylevel of central warehouse is independent of the random vari-able of demand in lead time and the stationary distribution atarbitrary time 𝑡 in the interval [𝑅

0+ 1, 𝑅

0+ 𝑄0] is uniform

distribution (as in [25]).The total demand of central warehouse is made up of the

random demand of each subwarehouse. In addition, basedon central limit theorem, when sample capacity 𝑁 is largeenough, it can be considered that the demand of centralwarehouse in lead time almost obeys normal distribution.Therefore, the average stock-out quantity of central ware-house can be acquired as

𝐸 (𝐼0)−=

1

𝑄0− 1

⋅ ∫𝑄0

1

∫∞

𝑅0+𝑦

(𝑥 − 𝑅0− 𝑦) 𝑑𝜙(

𝑥 − 𝜇0 (𝐿0)

𝜎0 (𝐿0)

) 𝑑𝑦

=(𝜎0 (𝐿

0))2

𝑄0− 1

[𝜙(2)

(𝑅0+ 1 − 𝜇0 (𝐿

0)

𝜎0 (𝐿0)

)

− 𝜙(2)

(𝑅0+ 𝑄0− 𝜇0 (𝐿

0)

𝜎0 (𝐿0)

)] ,

(8)

where 𝜙(2)(𝑥) is the quadratic loss function of standard nor-mal distribution, so

𝜑(2)

(𝑥) = ∫∞

𝑥

𝜑(1)

(𝑢) 𝑑𝑢

=1

2[(𝑥2+ 1) (1 − 𝜑 (𝑥)) − 𝑥𝜛 (𝑥)] .

(9)

Then, the average existing inventory of central warehouseis

𝐸 (𝐼0)+= 𝐸 (𝐼

0) + 𝐸 (𝐼

0)−

= 𝑅0+𝑄0+ 1

2− 𝜇0(𝐿0) + 𝐸 (𝐼

0)−.

(10)

Finally, according to the practical significance, make ℎ0

and 𝑃0the weight value of 𝐸(𝐼

0)+ and 𝐸(𝐼

0)−, respectively;

combine (8) and (10); the average holding cost and shortagecost of distribution center can be expressed as

𝐶0(𝑅0, 𝑄0) = ℎ0𝐸 (𝐼0)++ 𝑃0𝐸 (𝐼0)−. (11)

5.2. The Inventory Model of Subwarehouse. The randomnessof the lead time of subwarehouse is caused by the randomdelay of central warehouse and the uncertainty of transporta-tion time. Random delay means the random waiting time ledby the stock-outs of central warehouse when subwarehousesplace orders to central warehouse.

𝜏 was set as the random delay of the orders from sub-warehouses in central warehouse. For the sake of simplicity,assuming that it is the same to each subwarehouse, the meanand variance of random delay are 𝐸[𝜏] and𝑉[𝜏], respectively.The time for central warehouse demanding unit goods is 𝑡

1,

while the arrival time of the goods to central warehouse is 𝑡2.

If 𝑡1≥ 𝑡2, there is no delay, while if 𝑡

1< 𝑡2, there is a delay,

where 𝜏 = max(𝛿, 0) (𝛿 = 𝑡2− 𝑡1). According to calculation,

the following relationship can be obtained:

𝐸 [𝜏] =𝐸 (𝐼0)−

∑𝑁

𝑖=1𝜇𝑖

. (12)

If 𝛿 obeys normal distribution, then𝑃 (𝛿 ≤ 0) = (−𝜇

𝛿𝜎𝛿) = 𝑃 (𝜏 = 0) , (13)

where

𝑃 (𝜏 = 0) =1

𝑄0− 1

∫𝑄0

1

𝜑(𝑅0+ 𝑦 − 𝜇0 (𝐿

0)

𝜎0(𝐿0)

) 𝑑𝑦,

𝐸 [𝜏] = ∫∞

0

[1 − 𝜑(𝑥 − 𝜇𝛿

𝜎𝛿

)]𝑑𝑥

= 𝜎𝛿𝜙(1)

(−𝜇𝛿𝜎𝛿) .

(14)

So it can be obtained that

𝜎𝛿=

𝐸 [𝜏]

𝜙(1) (−𝜇𝛿𝜎𝛿),

𝜇𝛿= −𝜎−1

𝛿(𝑃 (𝜏 = 0)) ,

𝑉 [𝜏] = 𝐸 (𝜏2) − (𝐸 [𝜏])

2

= ∫∞

0

𝑥2

𝜎𝛿

𝜛(𝑥 − 𝜇𝛿

𝜎𝛿

)𝑑𝑥 − (𝐸 [𝜏])2

= 𝜎2

𝛿(1 − 𝑃 (𝜏 = 0)) + 𝐸 [𝜏] 𝜇𝛿 − (𝐸 [𝜏])

2.

(15)

If the transportation time 𝑇𝑖of subwarehouse 𝑖 obeys the

Gamma distribution with 𝛼𝑖and 𝛽

𝑖as parameters and the

mean and variance of random lead time of subwarehouse 𝑖are 𝐸𝑖(𝐿𝑇) and 𝑉

𝑖(𝐿𝑇), then

𝐸𝑖 (𝐿𝑇) = 𝐸 (𝑇

𝑖) + 𝐸 [𝜏] =

𝛼𝑖

𝛽𝑖

+ 𝐸 [𝜏] ,

𝑉𝑖 (𝐿𝑇) = 𝑉 [𝑇

𝑖] + 𝑉 [𝜏] =

𝛼2𝑖

𝑉 [𝜏].

(16)

Through further analysis, it can be known that the mean𝐸𝑖(𝐿𝑇𝐷) and variance 𝑉

𝑖(𝐿𝑇𝐷) of demand of subwarehouse

𝑖 in random lead time 𝐿𝑇 are𝐸𝑖 (𝐿𝑇𝐷) = 𝜇

𝑖𝐸𝑖 (𝐿𝑇) ,

𝑉𝑖 (𝐿𝑇𝐷) = 𝜎

2

𝑖𝐸𝑖 (𝐿𝑇) + (𝜇

𝑖)2𝑉𝑖 (𝐿𝑇) .

(17)


In general inventory model, if the demand process isPoisson process and the lead time obeys gamma distribution,most of the demand during lead time is approximated byusing Poisson distribution. However, if there are many sub-warehouses and themean demand of some subwarehouses inlead time is small, using normal distribution for approxima-tion will produce quite a number of negative values, whichis unreasonable. Because its mean and variance values arethe same, the change of demand during lead time cannot bewell reflected by approximating throughPoisson distribution.Besides, the characteristics of negative binomial distributioncan preferably conform to the practical situation of thedemand of retailers during lead time. In addition, it takesPoisson distribution as limiting distribution. As a result, itis more reasonable to utilize negative binomial distributionfor approximating the demanddistribution of retailers duringlead time (as in [26]).

If the demand of retailer 𝑖 in lead time can obey thenegative binomial distribution with 𝑛

𝑖and 𝑃

𝑖as parameters,

distribution density function and distribution function are,respectively, 𝑔

𝑖(𝑥) and 𝐺

𝑖(𝑥) (𝑖 = 1, 2, . . . , 𝑁); then,

𝑃𝑖=𝑉𝑖 (𝐿𝑇𝐷) − 𝐸

𝑖 (𝐿𝑇𝐷)

𝑉𝑖 (𝐿𝑇𝐷)

,

𝑛𝑖=

𝐸2𝑖(𝐿𝑇𝐷)

𝑉𝑖 (𝐿𝑇𝐷) − 𝐸

𝑖 (𝐿𝑇𝐷).

(18)

Setting𝐺(0)𝑖(𝑥) = 1−𝐺

𝑖(𝑥), because 𝑛 is not an integer, it is

impossible to calculate accurately 𝑔𝑖(𝑥) and 𝐺

𝑖(𝑥). Thus, the

first-order loss function 𝐺(1)(𝑥) and quadratic loss function𝐺(2)(𝑥) (𝑥 ≥ 0) for 𝐿𝑇𝐷 are as follows:

𝐺(1)

(𝑥) = 𝐸 {[𝑋 − 𝑥]+} = ∑𝑦≥𝑥

𝐺(0)

𝑖(𝑦) = [

𝑛𝑖𝑃𝑖

1 − 𝑃𝑖

− 𝑥]

⋅ 𝐺(0)

𝑖(𝑥) + (𝑥 + 𝑛

𝑖)

𝑃𝑖

1 − 𝑃𝑖

𝑔𝑖 (𝑥) ,

𝐺(2)

(𝑥) =1

2𝐸 {[𝑋 − 𝑥]

+[𝑋 − 𝑥 − 1]

+} =

1

2

⋅ 𝐸 {𝑋 [𝑋 − 1]} − ∑0<𝑦≤𝑥

𝐺(1)

(𝑦)

=1

2{𝑛𝑖(𝑛𝑖+ 1) [

𝑃𝑖

1 − 𝑃2

]

2

− 2𝑛𝑖𝑃𝑖

1 − 𝑃𝑖

𝑥

+ 𝑥 (𝑥 + 1) 𝐺(1)

(𝑥)

+ [(𝑛𝑖+ 1) 𝑃

𝑖

1 − 𝑃𝑖

− 𝑥] (𝑥 + 𝑛𝑖)

𝑃𝑖

1 − 𝑃𝑖

𝑔𝑖 (𝑥)} .

(19)

In the same way of central warehouse, it can be obtainedthat subwarehouses 𝑖 are equally distributed on [𝑅

𝑖+1, 𝑅𝑖+𝑄𝑖]

at arbitrary time. Hence, it can be acquired that the averagestock-out quantity of subwarehouse 𝑖 is as follows:

𝐸 (𝐼𝑖)−=

1

𝑄𝑖− 1

∫𝑄

1

∫∞

𝑅𝑖+𝑦

(𝑥 − 𝑅𝑖− 𝑦) 𝑔

𝑖 (𝑥) 𝑑𝑥 𝑑𝑦

=1

𝑄𝑖− 1

[𝐺(2)

(𝑅𝑖+ 1) − 𝐺

(2)(𝑅𝑖+ 𝑄𝑖)] .

(20)

Then, corresponding to (10), the average existing inven-tory of subwarehouse 𝑖 is

𝐸 (𝐼𝑖)+= 𝐸 (𝐼

𝑖) + 𝐸 (𝐼

𝑖)−

= 𝑅𝑖+𝑄𝑖+ 1

2− 𝐸𝑖 (𝐿𝑇𝐷) + 𝐸 (𝐼

𝑖)−.

(21)

Therefore, according to the practical significance, makeℎ𝑖and 𝑃

𝑖the weight value of 𝐸(𝐼

𝑖)+ and 𝐸(𝐼

𝑖)−, respectively;

combine (20) and (21); the average holding cost and shortagecost of subwarehouse 𝑖 can be obtained as

𝐶𝑖(𝑅𝑖, 𝑄𝑖) = ℎ𝑖𝐸 (𝐼𝑖)++ 𝑃𝑖𝐸 (𝐼𝑖)−. (22)

As the total inventory cost is equal to the inventory costof the central warehouse and the warehouse, combine (11)and (22); through calculation and derivation, the expectedtotal cost function of two-echelon inventory system can beexpressed as

𝑇𝐶 = 𝐶0(𝑅0, 𝑄0) +

𝑁

∑𝑖=1

𝐶𝑖(𝑅𝑖, 𝑄𝑖) . (23)

5.3. Optimizing and Control Inventory Cost. Through analyz-ing the cost structure of two-echelon inventory system, theinventory control model based on cost optimization can beobtained as follows:

min𝑇𝐶 = min[𝐶0(𝑅0, 𝑄0) +

𝑁

∑𝑖=1

𝐶𝑖(𝑅𝑖, 𝑄𝑖)] . (24)

The key to solving the model is to select appropriate orderpoint 𝑅∗

𝑖and order quantity 𝑄∗

𝑖(𝑖 = 0, 1, . . . , 𝑁) to make the

objective function achieve the optimum value.Firstly, 𝑅

0and 𝑄

0are given to determine the order point

𝑅∗𝑖(𝑅0, 𝑄0) and order quantity 𝑄∗

𝑖(𝑅0, 𝑄0) of subwarehouse

𝑖. According to the analysis, it can be known that 𝐸(𝐼𝑖)− is a

convex function of𝑅𝑖and𝑄

𝑖, and𝐸(𝐼

𝑖)+ is a linear equation of

(𝑅𝑖, 𝑄𝑖).The nonlinear combination𝐶

𝑖(𝑅𝑖, 𝑄𝑖) is still a convex

function based on (𝑅𝑖, 𝑄𝑖). It can be known that theremust be

optimal 𝑅𝑖, 𝑄𝑖to minimize 𝐶

𝑖(𝑅𝑖, 𝑄𝑖) through the properties

of convex function.After further simplifying the objective function, it can be

obtained that

𝑇𝐶 (𝑅0, 𝑅𝑖) = 𝐶0(𝑅0) +

𝑁

∑𝑖=1

𝐶𝑖(𝑅0, 𝑅𝑖) . (25)


Table 1: Ordering costs of each cycle (104 Yuan).

Actual value Simulation value𝑁 = 6 𝑁 = 5 𝑁 = 7 𝑁 = 8 𝑁 = 9 𝑁 = 10 𝑁 = 50

Nitrogen 957.03 1109.0625 848.44 742.8 659.9 633.28 633.28Phosphate 783.025 783.02 114.6019 112.3126 110.5313 872.9776 872.9776Potassium 34.829 27.8632 39.80624 43.53625 46.43737 48.7606 48.7606Other 34.829 27.8632 39.80624 43.53625 46.43737 48.7606 48.7606Total 1914.0625 2060.1268 1102.778 996.515 913.0875 1696.259 1696.259

5.4. The Analysis and Calculation of the Optimal InventoryCost. The inventory model established in the research is atwo-echelon inventory system model containing a centralwarehouse and several subwarehouses. If the total cost of thesystem is the sum of the cost of the central warehouse and allthe subwarehouses, then it can be acquired by determiningthe optimal cost of each warehouse.

The optimal inventory cost can be computed by the fol-lowing two steps.

First step is to find the optimal order point 𝑅∗𝑖(𝑅0) of

retailer 𝑖 by fixing 𝑅0and minimizing 𝐶

𝑖(𝑅0, 𝑅𝑖). Through

the above theorem, it can be known that 𝐶𝑖(𝑅𝑖, 𝑄𝑖) is a

convex function of (𝑅𝑖, 𝑄𝑖); then, the necessary condition for

obtaining extreme value of 𝐶𝑖(𝑅0, 𝑅𝑖) is that

𝜕𝐶𝑖(𝑅0, 𝑅𝑖)

𝜕𝑅𝑖

= 0. (26)

The root of equation 𝑅∗𝑖(𝑅0) can be solved by using one-

dimensional search method.Second step is to solve 𝑅

∗

0by minimizing 𝑇𝐶. Because

𝑇𝐶 for 𝑅0may not be a convex function, a simple search

programwas designed to find out the locally optimal solution𝑅∗

0.

Step 0. Consider 𝑅0

= −𝑄0, step = ⌈𝜇0(𝐿

0)⌉, and 𝐶∗ =

𝑇𝐶[𝑅0, 𝑅∗𝑖(𝑅0)].

Step 1. Consider 𝑅0= 𝑅0+ step, marking the corresponding

order points as ��0.

Step 2. If step = 1, 𝑅∗0= ��0; otherwise, let 𝑅

0= max(𝑅

0, ��0−

step), step = ⌈step/10⌉, go back to Step 1.

The complexity of two search algorithms utilizedfor solving the optimum values of the order points ofcentral warehouse and subwarehouses was added up as𝑂(log𝑅∗

0∑𝑁

𝑖=1log𝑅∗𝑖), which is the linear function of 𝑁.

According to the solving steps above, the correspondingmathematical model is as follows:

min (𝐶0(𝑅0) +

𝑁

∑𝑖=1

𝐶𝑖(𝑅0, 𝑅𝑖))

s.t.𝑁

∑𝑖=1

𝐶𝑄𝑖≤ 𝑊

𝑄𝑖∈ (0,∞) , 𝑖 = 1, 2, . . . , 𝑁.

(27)

6. Model Solving and Main Results

6.1. The Empirical Analysis and Numerical Simulation Test.Through on-site survey to the 16 corps. in Aksu area ofSouthXinjiang, and the agricultural warehousing company inand surrounding this areas, a large logistics storage companywhose business covers whole Aksu area is taken as researchobject; the company has a developed warehouse systemconsisting of six bases which are one headquarter warehousebase and five subwarehouse bases. This company suppliesfertilizers to the 16 corps. around Aksu areas. We thereforeacquired the demand quantities of various fertilizers for eachcorp. and the order cycles of the logistics warehousing com-pany. Moreover, we constructed an optimization model forthe optimal order cost and inventory cost in each circle andprovided an optimal inventory cost algorithm. By combiningMATLAB programming, the optimal order cost in singlecircle and the optimal inventory cost for various fertilizers,as well as the corresponding total cost of single circle orderand the optimal total inventory cost, are solved. Besides, astatistical simulation method was used to simulate the caseswhen the warehouse quantities are 5, 7, 8, 9, 10, and 50,respectively. The simulated results of the optimal order costin single period, and corresponding total order cost in singleperiod, of various fertilizers are compared with empiricalresults, as shown in Table 1. Moreover, the empirical andsimulated results of the optimal inventory cost of variousfertilizers and corresponding optimal total inventory cost arecompared also, as indicated in Table 2.

Figure 2 indicates the simulated tendency of the ordercost and the total order cost in single period for various kindsof fertilizers including nitrogen, phosphate, and potassiumfertilizers.

6.2.The Analysis of Empirical and Simulated Results. Accord-ing to the statistics analysis of the original data obtained, theoptimal order cost in single period of nitrogen, phosphate,and potassium fertilizers aswell as other fertilizers is obtainedwhen the planting areas of the Aksu in Xinjiang provinceand 6 warehouses are given. Besides, statistics simulationmethod was used to conduct simulation when the number ofwarehouses is 5, 7, 8, 9, 10, and 50, respectively.The simulatedresults show that as the number of warehouses increases to 9from 5, the order costs and total order cost of single period,for nitrogen and phosphate fertilizers, decrease obviously.This is because the demands of nitrogen and phosphatefertilizers are large, which occupy higher proportion in


Table 2: The optimal inventory costs (104 Yuan).

Actual value Simulation value𝑁 = 6 𝑁 = 5 𝑁 = 7 𝑁 = 8 𝑁 = 9 𝑁 = 10 𝑁 = 50

Nitrogen 5742.1875 5545.3125 5939.08 5942.4 5939.1 6332.8 6332.8Phosphate 4698.1534 3915.1 802.2133 898.5008 994.7817 8729.776 8729.776Potassium 208.98 139.316 278.6437 348.29 417.9363 487.606 487.606Other 208.98 139.316 278.6437 348.29 417.9363 487.606 487.606Total 11484.375 10300.634 7719.446 7972.12 8217.788 16962.59 16962.59

20 30 40 5010N

1000

2000

3000To

tal

20 30 40 5010N

100

300

500

Pota

ssiu

m

20 30 40 5010N

200

400

600

800

Phos

phat

e

20 30 40 5010N

600

1000

1400

Nitr

ogen

Figure 2: The simulated tendency of the order cost and the total order cost in single period for various kinds of fertilizers.

order cost; the total order cost is mainly determined bynitrogen and phosphate fertilizers. In contrast, as the numberof warehouses rises, the order costs of single period forpotassium and other fertilizers increase. This is because thedemands of potassium fertilizer and other microelementfertilizers are small; the small number of warehouses cansufficiently store minor amount of potassium fertilizer andother fertilizers, while a large number of warehouses tend tocause waste resources and unnecessary cost. As the numberof warehouses increases to or greater than 10, the totalorder cost of single period for various fertilizers tend to bestable, as indicated in Figure 1. From the perspective of theorder cost of single period for a fertilizer, the demand ofnitrogen fertilizer is large and stable.With increasing numberof warehouses, the quantity of nitrogen fertilizers has beenwell supplied in each warehouse so as to satisfy the realtime demands of customers. This makes the order cost ofsingle period for nitrogen fertilizer reduce. Phosphate andpotassium fertilizers, as well as other fertilizers, all show anapparent fluctuationwhen the number of warehouses is equalto or greater than 10, and their order costs increase obviously.This is because the fact that as the number of warehousesincreases, the order quantities of these fertilizers are likelyto rise greatly so as to fully utilize warehouse. In addition,these fertilizers have high price; consequently, the order costsof single period increase. Regardless of construction andmanagement costs, the optimal order cost of single period is

proven to be most ideal when the number of warehouses is9. Based on the abovementioned results, the empirical resultsare consistent with simulated results.

Table 2 presents the calculated and simulated results ofthe optimal inventory cost.When the warehouse number is 7,the optimal total inventory cost of all fertilizers is minimum.For the perspective of nitrogen fertilizer, its optimal inventorycost is minimum when the warehouse number is 5; for phos-phate fertilizer, its optimal inventory cost is the smallest whenthe warehouse number is 7; however, the optimal inventorycosts of potassium and other fertilizers are minimum whenthe warehouse number is 5. When the warehouse number isequal to or greater than 10, the inventory costs and optimalinventory costs of different fertilizers tend to be stable. SinceXinjiang belongs to an arid and alpine area, the corps.demand great quantity of phosphate fertilizer. Consideringthe phosphate fertilizer is relatively expensive, it is suggestedto mainly concern a minimum optimal inventory cost ofphosphate fertilizer in our optimization decision with theoptimal warehouse number of 7.

7. Conclusion

In the process of numerical simulation, the optimal stock andthe optimal inventory costs of various agricultural productswere obtained by continuously modifying the values ofparameters in model. At the same time, the stock and the


ordering costs of all products in each cycle were acquired.Through constantly adjusting the initial values of the vari-ables in model, it was found that all models with reasonableinitial values can converge to a stable equilibrium solution,namely, the optimal solution in finite steps, so as to verify thereliability of the model. In order to investigate the stabilityof parameters, the authors performed sensitivity analysis forthe parameters in the model. As indicated by the numericalsimulation results, there only appeared some fluctuations byslightly varying and modifying the values of parameters inthe model during numerical simulation test. It was verifiedthat the stability of the parameters inmodel was preferable. Inthe simulation testmentioned, the authors discovered that theconvergence efficiency of algorithmwas greatly influenced byselecting different initial values for variables. Moreover, theconvergence efficiency of algorithms was high by choosinginitial values close to equilibrium solution.

Considering supply-demand balance, the research startedfrom demand analysis. On the assumption that the demandprocess obeyed mutually independent compound Poissonprocess, a two-echelon inventory system consisting of a cen-tral warehouse and several subwarehouses was constructedthrough simplifying the research object. Besides, aiming atminimizing the mean total cost of inventory system in unittime, the optimization and control model for inventory of thesystem was established. In addition, the optimal algorithmfor computing the inventory cost was provided. Meanwhile,a numerical simulation experiment was conducted, andthe numerical results illustrated that the optimal inventorystrategy wasmuch influenced by the uncertainty of lead time.In this paper, the researchers explored the issue about theoptimization and control of inventory system and establishedthe corresponding optimization and controlmodel for inven-tory to optimize the allocation of resources and utilize thelimited inventory resources effectively by managing themquantitatively. The related research results offered a certainpractical value and reference for the optimal managementof agricultural materials inventory of agricultural materialenterprises in Xinjiang. The modeling idea and methodpresented in this paper are excepted to be further developedto the higher-level multiechelon inventory structure.

Conflict of Interests

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

Acknowledgments

This research is supported by youth special funds for scientificand technological innovation of the Xinjiang Productionand Construction Corps. (2011CB001), the Doctor Founda-tion of the Xinjiang Production and Construction Corps.(2013BB013), the Doctor Foundation of Tarim University(TDZKBS201205), and the National Spark Program Project(2011GA891010). Here, thanks are due to the relevant com-pany and organizations.

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