imperfect production in three-layer supply chain system...
TRANSCRIPT
Imperfect production in three-layer supply chain systemincorporating deterioration and screening effect
Sudip Adak, and G.S. Mahapatra∗
Department of Mathematics, National Institute of Technology Puducherry, Karaikal-609609, India.
E-mail: [email protected], and *E-mail: [email protected]
September 30, 2020
Abstract: This paper develops a three-layer supply chain for defective and non-defective types of pro-duced items by supplier and manufacturer. The condition of the chain for the supplier and manufacturerthat, after complition of screening defective items are sold to the supplier to produce other items usingthese defective items. In the subsequent stage, the retailer accepts the non-defective items produced andscreening by the manufacturer. Hence the retailer receives the perfect quality items for selling to the cus-tomers, but the retailer considers the effect of the deterioration of items. This model also considers theimpact of several business strategies such as; optimal order size of raw materials, production rate, unitproduction cost, idle time costs of the supplier, and manufacturer in a collaborative marketing system, etc.to determine the optimum average profit of the integrated model. This study discusses the selling priceof the retailer, demand rate of the customer, purchase cost of supplier and holding cost, which can be asignificant breakthrough in expanding the profit of the business in real terms. Numerical example andsensitivity analysis are presented to illustrate the phenomenon of theoretical study and demonstrate themanagerial implication of the model.
Keywords: Three-layer supply chain; Defective items; Deterioration; Production; Cost for Idle time;Screening.
1 Introduction
Supply chain management is a consistent progression in which an organization manages the flow of prod-ucts, services, money, etc. A supply chain model (SCM) is a network of suppliers, manufacturer, retailersand customers, to obtain the maximum profit with minimum costing, to fulfilling the customer’s demand.As per the facts, SCM includes the movement and storage of raw materials from nature, raw materialssupply, screening the raw materials, buyback/return policies, transportation of raw materials into a man-ufacturer warehouse, production of goods in the production centre, screening the finished goods, and dis-tribution of these finished products to retailers for sale to the customers. A single supplier and singlecustomer problems are developed by the researchers ([1] – [6]) for integrated production inventory prob-lem. However, it is highly imperative that a systematic network is created between suppliers, manufacturer,retailers and customers. Thus, in this case, we consider a three-layer supplier chain in our model.
One of the key assumptions of the classical economic order quantity (EOQ) model is that the items pro-duced and received are of perfect quality. However, in reality, production is not always accurate, and hencethere is a need to incorporate the defective items. This paper develops supply chain for supplier, manu-facturer and retailer considering the defective and non-defective types of produced items. This model alsoconsiders the impact of several business strategies such as optimal order size of raw materials, productionrate, unit production cost, idle costs of supplier and manufacturer in a collaborating marketing system anddetermines the optimum average profit of the integrated model. The supply chain considers the condition
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that defective items will be sent back to supplier after screening to repair/produce other items using thesedefective items on the manufacturer. The manufacturer produces items including perfect and defectiveitems, but only perfect items are supplied to the retailer after screening, in addition to that the defectiveitems are sent back to the supplier of the retailer. Thereupon, the retailer receives the perfect quality itemsfor selling to the customers but considers the effect of deterioration.
Deterioration means decay or spoilage in the item, and thereafter, the effect of the deterioration of theitems in the lot size must be taken into account. For items such as steel, hardware, glassware, and toys, therate of deterioration is too low; but some items such as alcohol, gasoline, radioactive chemical, medicine,and food items deteriorate remarkably over time, and hence arises the need for considering deteriorationconcept. In this paper, we propose a three-layer supply chain model with imperfect production qualityfor supplier and manufacturer. There are very few literatures that studied a three-layer supply chain withdeterioration. The proposed supply chain system considers constant deterioration of an item at the retailerend without any replacement of items of the deteriorated items.
This paper presents mathematical modelling and analysis of supply chain system without shortages,and cost of the idle time at the supplier and the manufacturer. The production rate of supplier and man-ufacturer is greater than the demand rate of manufacturer and retailer, respectively. The unit productioncost of the modelling is a function of production rate. Uniform distribution functions for production ofdefective item has been considered along with simultaneously carrying out the screening process with arate less than, or equal to production rate, but higher than or equal to the demand rate.
In this study, the production-based supply chain system has these advantages: (i) For the supplier, thedefective items are sent back to the supplier at one lot with the sales price. So, the faulty items can be usedby the supplier to prepare another product or can sell at the rate of a few percentages of the selling price.(ii) For the manufacturer, the defective items are sent back to the supplier and which can be sold at therate of a few portions of the selling price to recover some profit due to defective of items. (iii) The retailerreceives the perfect quality items for selling to the customer, which will help to improve their goodwill. (iv)In this SCM, it is easy to recognize the relationship between the demand rate of the retailer and the sellingprice of the retailer.
The organization of the rest of the paper is as follows: Section 2 describes the details literature reviewwhich motivate to study the proposed three-layer SCM. Section 3 describes the formulation of the sup-ply chain model and mathematical computation based on the proposed production inventory. Section 4presents the optimization of the proposed supply chain model. An example of the proposed three-layersupply chain model with the detail numerical analysis present in Section 5. Section 6 presents sensitivityanalysis for the various parameters to illustrate the vital aspect of the model, and further, the managerialimplications are also discussed. Concluding remarks with limitations of this study and scope for futureresearch on the three-layer SCM is given in Section 7.
2 Literature Review
This section focuses on reviewing the relevant literature of the three-layer supply chain with the productionrate of supplier and manufacturer, selling price of the retailer, demand rate of the customer, and purchasecost of the supplier. With rapid changes and enlargement of the supply chain system business management,it is very vital to construct a systematic network between suppliers, manufacturer, retailers, and customers.In accordance, in our model, we have considered a three-layer supplier chain. The literature of supplychain management has developed a lot during the last few decades ([7],[8]). Li et al. [9] constructed asingle retailer and two-supplier supply chain model.
AlDurgam et al. [10] developed a single-vendor and single-manufacturer integrated inventory modelwith stochastic demand and variable production rate. Pasandideh et al. [11] discussed vendor-managedinventory in the joint replenishment problem of a multi-product single-supplier multiple-retailer supplychain. Several researchers ([12] - [25]) constructed integrated production model consisting of multi-stagesupply chain system. Ben-Daya and Seliaman [26] developed inventory model in a three-layer SCM with
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raw material replenishment. Several researchers ([27] - [28]) developed fuzzy dynamic multi-objective,multi-item model by considering customer satisfaction in the supply chain. Taleizadeh and Noori-Daryan[29] studied the pricing, manufacturing and inventory policies for raw material in a three-level supplychain. Gamasaee and Zarandi [30] discussed incorporating demand, orders, lead time, and pricing de-cisions for reducing the bullwhip effect in supply chains. Taleizadeh et al. [31] developed a sustainableclosed-loop supply chain problem with pricing decisions and discounts on returned products.
The screening process has used to separate the perfect quality products from the whole batch of theproduced items. After screening process, the non-defective items are supplied to the supplierer. Recently,some exciting research articles deal with imperfect quality items, which also have practical importance.Thereafter, EOQ models ([32]–[44]) were developed for items with imperfect quality. Pal et al. [45] con-structed a three-layer supply chain EPQ model for price-and stock-dependent stochastic demand with theimperfect item under rework. Pal et al. [46] represented a three-layer supply chain model for reworkableitems. Jian et al. [47] presented a three-echelon reverse supply chain for economic incorporating coordina-tion strategies and social benefit. Pal et al. [48] developed a manufacturing-oriented supply chain modelfor imperfect quality with inspection errors, stochastic demand under rework, and shortages. Khan et al.[49] presented optimal vendor-buyer inventory policy by accounting for quality inspection errors of imper-fect items at the buyer’s end. Feng et al. [50] discussed pricing and lot-sizing policies for perishable goodswhen the demand depends on selling price, displayed stocks, and expiration date.
Deterioration of the items in the inventory model is inevitable. In reality, we cannot neglect the effect ofdeterioration. Dye [51] developed a finite horizon deteriorating inventory model of two-phase pricing withtime-dependent demand and cost under trade credit financing. Chung and Cardenas-Barron [52] presenteda simplified solution procedure for deteriorating items under stock-dependent demand and two-level tradecredit in the supply chain management. Gao et al. [53] presented a two-layer supply chain system withstochastic customer demand and delay-in-payment. Chan et al. [54] developed an integrated production-inventory model for deterioration during delivery. Tiwari et al. [55] constructed an optimal pricing andlot-sizing policy for supply chain system with deteriorating items under limited storage capacity. Liao etal. [56] developed a lot-size model with deterioration for two warehouses under order-size dependent tradecredit. Chung et al. [57] developed an inventory model with non-instantaneous and exponentially deteri-oration for an integrated three-layer supply chain under two levels of trade credit. Inventory model ([58],[59]) studied for deteriorating items with expiration dates and partial backlogging in two layers. Puga etal. [60] proposed a supply chain system for facility location, safety stock placement, and delivery strategywith time preferences for two customer classes. Modak and Kelle [61] presented a dual-channel supplychain under price and delivery-time dependent stochastic customer demand incorporating the price andorder quantity for both the retail. Tiwari et al. [62] discussed retailer’s optimal ordering policy for dete-riorating items under order-size dependent trade credit and complete backlogging. Adak and Mahapatra[63] developed an effect of reliability on multi-item inventory system with shortages and partial backlogincorporating time dependent demand and deterioration. Bhunia et al. [64] studied single deterioratingitem with variable demand dependent on marketing strategy and displayed stock level. Teimoury et al.[65] studied integrated pricing and inventory model for deteriorating products in a two-stage supply chainunder replacement and shortage.
3 Formulation of Three Layer Supply Chain Model
This paper develops a SCM of production inventory, considering that the production rate of supplier andproduction rate of (a decision variable) manufacturer is greater than the demand rate of manufacturer andretailer, respectively. The proposed study also considers the start time of production at the supplier, andthe manufacturer to be initiated in such a way that the shortages won’t occur. The supply chain modelconsiders unit production cost as a function of production rate. We also consider the cost of idle timesfor supplier and manufacturer. This production inventory-based supply chain system proposes that theproduced defective items of supplier and manufacturer follow uniform distribution, i.e., f (x) = 1
b−a , a < x < b.
3
Hence the screening process has been carried out simultaneously during the production period. It furtherconsiders that the screening rate is greater than or equal to the demand rate, but less than or equal to theproduction rate. The proposed supply chain system considers that deterioration of an item is constant dueto time or transit, and there is no repair or replenishment of the deteriorated items. The proposed study isvery much interesting due to the nature of the system and the real-life practical application of the concept.
3.1 Notation
The proposed supply chain model will be developed using the following notations:As, Am, Ar - Set up cost of the supplier, manufacturer and retailer, respectively.Hs, Hm, Hr - Unit holding cost per unit time at supplier, manufacturer and retailer, respectively.Ws,Wm,Wr - Selling price per unit of non-defective item at supplier, manufacturer and retailer, respec-
tively.APS, APM, APR - Average profit of the supplier, manufacturer and retailer, respectively.Ps, P - Production rate for the supplier and manufacturer (i.e. demand rate at supplier), respectively.Rs, Rm - Screening rate per unit time at supplier and manufacturer, respectively.Ss, Sm - Screening cost per unit item at supplier and manufacturer, respectively.Is, Im - Cost per unit idle time of supplier and manufacturer, respectively.Dr, Dc - Demand rate of the retailer and customers, respectively.α, β - Probability of defective items for supplier and manufacturer, respectively.R - Initial stock level of supplier.Cs - Purchasing cost per unit item of supplier.Sp - Selling price per unit defective item of supplier.Mp - Selling price per unit defective item of manufacturer.δm - Cost per unit finished product.θ - Deterioration rate of finished item.dc - Cost of each deteriorated item.L - Total labour /energy costs per unit time of a production system.γ - Tool costs per unit of the production rate.E(x) - Expectation of variable x.T - Cycle length of retailer.
3.2 Production inventory based three layer supply chain model
The proposed supply chain model consists of three stages of economic production lot-sizing model fora single supplier, manufacturer and retailer. Let the production rate of raw materials be Ps with non-defective as well as defective items. After completion of screening, the non-defective items are supplied tothe manufacturer at rate P up to time t1, and the defective items are sent back at one lot with sales price Spper unit item. The manufacturer produces the item at rate P and meets the demand of retailer at a rate Drup to time t2. The manufacturer produces both defective as well as non-defective items. After completionof screening, the defective items accumulated at time t1, are sold at the rate of Mp% of selling price perunit of the non-defective item at the manufacturer end. Thus, the retailer receives the perfect quality itemsfor selling to the customer at rate Dc up to time t2. In retailer warehouse, many items deteriorate to time.Figure 1 and Figure 2 represent the pictorial representation and graphical representation respectively inthree-layer supply chain model with deterioration.
Insert Figure 1Insert Figure 2
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3.3 Formulation of Supplier Individual Cost
Let qs(t) is the inventory of non-defective items, and R is the initial stock (t = 0) for the supplier. Thesupplier starts the production at the rate Ps unit per unit time and supplies it to the manufacturer. At t = tssuppliers stop their production, and at t = t1 the inventory level becomes zero. The lot size R and the itemsproduced by the supplier for run time ts are screened with rate Rs at cost Ss per unit item. After completionof screening, the defective items are sold to the supplier at a price Sp per unit item. Then the differentialequations for the supplier as shown in Figure 3 during [0, T] is given by
dqs(t)dt
=
(1−α)Ps−P, for 0 ≤ t ≤ ts
−P, for ts≤ t ≤ t10, for t1≤ t ≤ T
(1)
with the boundary conditions are qs(0) = (1−α)R and qs(t1) = 0.
Insert Figure 3
The solution of the above differential equations using the boundary conditions qs(0) = (1−α)R andqs(t1) = 0 is
qs(t) ={{(1−α)Ps−P} t + (1−α)R, for 0 ≤ t ≤ ts
P (t1−t) , for ts≤ t ≤ t1(2)
Lemma 1 The suppliers stop their production at time ts=1Ps
(Pt11−α−R
)where the inventory level becomes zero at
time t1, and defective items at supplier be α.
Proof. The inventory of supplier for t is given by the equation (2). Satisfying the condition of qs(t) at timet = ts we obtain,
qs(t) ={(1−α)Ps−P
}ts+(1−α)R
Again the inventory level at t = ts is qs(t) = P (t1−ts)Therefore, by satisfying the condition of qs(t) at ts a relation is obtained as follows{(1−α)Ps−P
}ts+(1−α)R = P (t1−ts)
The time to stop the production by the supplier is given by
ts=1Ps
( Pt1
1−α−R
)(3)
The holding cost of non-defective items during (0, t1) is
Hs
∫ t1
0qs(t)dt =
Hs
2
[Pt2
1+1Ps
{2RPt1−
P2t21
1−α−R2 (1−α)
}](4)
The holding cost of defective items during (0, ts) is
Hs
∫ ts
0αPs (ts−t)dt +αR
RRs
+αPstsPs
Rs
= αHs
[1
2Ps
( Pt1
1−α−R
)2+
R2
Rs+
Ps
Rs
( Pt1
1−α−R
)](5)
The screening cost is
Ss (R + Psts)= SsPt1
1−α(6)
5
The income from selling the non-defective and defective items of the supplier is
Ws (1−α) (R + Psts)+Spα (R + Psts)
= Pt1
(Ws+
Spα
1−α
)(7)
The purchase cost of the supplier is
CsR + CsPsts= CsPt1
1−α(8)
The fixed set-up cost of the supplier is given by As.The idle cost for the supplier is
Is (T− t1) (9)
Therefore, the average profit of supplier (APS) is given by the following equation
APS =1T
[Pt1
(Ws+
Spα
1−α−Cs+Ss
1−α
)−As−
Hs
2
{Pt2
1+1Ps
(2RPt1−
P2t21
1−α−R2 (1−α)
)}−αHs
{1
2Ps
( Pt1
1−α−R
)2+
R2
Rs+
Ps
Rs
( Pt1
1−α−R
)}−Is (T− t1)
](10)
3.4 Formulation of Manufacturer Individual Cost
Let qm(t) is the inventory of non-defective items of the manufacturer who start their production at the rateP unit per unit time and supply it to the retailer. At t = t1 manufacturers stop their production, and at t = t2the inventory level becomes zero. The items produced by the manufacture up to the production run timet1 are screened with rate Rm at cost Sm per unit item. After completion of screening, the defective items areaccumulated at time t1, and those are sold at Mp% of selling price per unit good item at the supplier. Thedifferential equations as per Figure 4 for the manufacture in [0, T] is given by
dqm(t)dt
=
(1− β)P−Dr for 0 ≤ t ≤ t1−Dr for t1≤ t ≤ t20 for t2≤ t ≤ T
(11)
with the boundary conditions are qm(0) = 0 and qm(t2) = 0.
Insert Figure 4
The solution of the above differential equations using the boundary conditions qm(0) = 0 and qm(t2) = 0is as follows
qm(t) ={ {
(1− β)P−Dr
}t for 0 ≤ t ≤ t1
Dr (t2−t) for t1≤ t ≤ t2(12)
Lemma 2 The inventory level of manufacturer becomes zero at time t2= (1−β)Pt1Dr
,where at time t1 the manufacturestop their production and β is the probability of defective items at the manufacturer.
6
Proof. The inventory of the manufacturer for the time t is given by the equation (12).The inventory level at t1 is qm(t) =
{(1− β)P−Dr
}t1.
Again the inventory level at t = t1 is qm(t) = Dr (t2−t1) .Therefore, a relation satisfying the condition of qm(t) at t1 is obtained as follows{(1− β)P−Dr
}t1= Dr (t2−t1)
From the above expression, we get,
t2=(1− β)Pt1
Dr(13)
The holding cost of non-defective items during (0, t2) is
Hm
∫ t2
0qm(t)dt =
Hm(1− β)Pt21
2
{(1− β)P
Dr−1
}(14)
The holding cost of defective items during (0, t1) is
Hm
∫ t1
0βP (t1−t)dt + βPt1
PRm
= HmβPt1
(t1
2+
PRm
)(15)
The screening cost isSmPt1 (16)
The income from selling the non-defective and defective items is
Wm(1− β)Pt1+MpWmβPt1 (17)
= WmPt1
(1− β + Mpβ
)The total production cost is (
Ws+δm+LP
+γP)
Pt1 (18)
The set-up cost of the manufacturer is fixed which is Am.The idle cost for the manufacturer is
Im (T− t1) (19)
Therefore, the average profit of the manufacturer (APM) is given by
APM =1T
[WmPt1
(1− β + Mpβ
)−Am−SmPt1−HmPt1
{(1− β)2Pt1
2Dr+βPRm
+t1
(β−1
2
)}−(Ws+δm+
LP
+γP)Pt1−Im (T− t1)
](20)
7
3.5 Formulation of Retailer Individual Cost
After receiving the item from the manufacturer, qr(t) is an on-hand inventory for the retailer. The retailerstarts its production at the rate Dr unit per unit time and supplies it to the customer. At t = t2 retailers stoptheir production, and the inventory level becomes zero at t = T. The differential equations for the retaileras per Figure 5 in [0, T] is given by
dqr(t)dt
={
Dr−Dc−θqr(t), if 0 ≤ t ≤ t2−Dc−θqr(t), if t2≤ t ≤ T (21)
with the boundary conditions are qr(0) = 0 and qr(T) = 0.
Insert Figure 5The solution of the above differential equations, after applying the boundary conditions qr(0) = 0 and
qr(T) = 0 is
qr(t) =
1θ (Dr−Dc)
(1− e−θt
), if 0 ≤ t ≤ t2
Dcθ
(eθ(T−t)−1
), if t2≤ t ≤ T
(22)
Lemma 3 The inventory level of retailer becomes zero at time T = Drt2Dc, where at time t2 the inventory level be-
comes zero for the manufacturer.
Proof. The inventory of the manufacturer for the time t is given by the equation (22).The inventory level at t2 is qr(t) = 1
θ (Dr−Dc)(1− e−θt2
).
Again the inventory level at t = t2 is qr(t) = Dcθ
(eθ(T−t2)−1
).
Therefore, a relation satisfying the condition of qr(t) at t2 is obtained as follows1θ (Dr−Dc)
(1− e−θt2
)= Dcθ
(eθ(T−t2)−1
)The above expression gives the time of inventory level vanishes at the retailer level as follows:
T =Drt2
Dc(23)
The holding cost of items during (0, T) is
Hr
∫ T
0qr(t)dt =
Hr
θ(Drt2−DcT) (24)
The deterioration cost of the retailer is
dc (Drt2−DcT) (25)
The income from selling the items is
WrDcT (26)
The fixed set-up cost of the retailer is Ar.The purchase cost of the retailer is
WmDcT (27)
Therefore, the average profit of the retailer (APR) is given by
APR=1T
[(Wr−Wm+
Hr
θ+dc
)DcT−Ar−
(Hr
θ+dc
)(1− β)Pt1
](28)
8
3.6 Integrated expected average profit of the supply chain model
To find the integrated expected average profit of the proposed three-layer supply chain model, we have todetermine the expected average profit for the supplier, manufacturer and retailer.
From the equation (10), the expected average profit of the supplier is
EAPS =1T
[Pt1
{Ws+SpE
( α1−α
)− (Cs+Ss)E
( 11−α
)}−Hs
2
{Pt2
1+1Ps
(−P2t2
1E( 1
1−α
)−R2E (1−α)
+2RPt1)}−E (α)Hs
{1
2PsE( Pt1
1−α−R
)2+
R2
Rs+
Ps
Rs
(Pt1E
( 11−α
)−R
)}−As−Is (T− t1)
]
=1T
[Pt1
{Ws+SpE
( α1−α
)− (Cs+Ss)E
( 11−α
)−HsR
Ps−Hs
(Ps
Rs− R
Ps
)E (α)E
( 11−α
)}+Ist1−
Hs
2Pt2
1
+P2t21
Hs
2Ps
{E( 1
1−α
)−E (α)E
(1
(1−α)2
)}−{
As+IsT−HsR2
2PsE (1− 2α)+
HsRRs
(R−Ps)E (α)}]
(29)
From equation (20) the expected average profit of the manufacturer is
EAPM =1T
[Pt1
{Wm
(1−E (β)+MpE (β)
)−Sm−Ws−δm
}− (Am+ImT)−Hm
P2t21
2DrE(1− β)2
+Hm
(12−E (β)
)Pt2
1+(Im−L) t1−(γ+
E (β)Hm
Rm
)P2t1
](30)
From equation (28) the expected average profit of the retailer is
EAPR=1T
[(Wr−Wm+
Hr
θ+dc
)DcT−Ar−
(Hr
θ+dc
)(1−E (β))Pt1
](31)
Thus, the integrated expected average profit (EIAP) of the proposed three-layer supply chain is
EIAP = EAPS+EAPM+ EAPR
From equations (29), (30) and (31), the expected integrated average profit of the supply chain is
EIAP =1T
[Pt1
{SpE
( α1−α
)− (Cs+Ss)E
( 11−α
)−HsR
Ps−Hs
(Ps
Rs− R
Ps
)E (α)E
( 11−α
)−Sm−δm
+Wm
(1−
(1−Mp
)E (β)
)−(Hr
θ+dc
)(1−E (β))
}+{Hm
(12−E (β)
)−Hs
2
}Pt2
1−(γ+
E (β)Hm
Rm
)P2t1
+P2t21
{Hs
2Ps
(E( α
1−α
)−E (α)E
(1
(1−α)2
))− Hm
2DrE(1− β)2
}+(Im+Is−L) t1− (Ar+As+Am)
−(Is+Im)T+HsR2
2PsE (1− 2α)−HsR
Rs(R−Ps)E (α)+
(Wr−Wm+
Hr
θ+dc
)DcT
](32)
The total expected average profit for the system of three-layer supply chain model can be obtained fromthe above equation (32) for two variables P and t1.
9
4 Optimization of Proposed Supply Chain Model
The expected total profit EIAP of three-layer supply chain model is also a function of two variables P andt1. Now we have to calculate the optimality of two decision variables P, t1 and also for the optimum valueof EIAP, the EIAP equation (32) can be written as follows
EIAP =1T
[U1P2t2
1−U2P2t1+U3Pt21+U4Pt1+U5t1+U6
](33)
Where U1= Hs2Ps
(E(α
1−α
)−E (α)E
(1
(1−α)2
))− Hm
2DrE(1− β)2, U2= γ+ E(β)Hm
Rm, U3= Hm
(12−E (β)
)−Hs
2 ,
U4= SpE(α
1−α
)− (Cs+Ss)E
(1
1−α
)−HsR
Ps−Hs
(PsRs− R
Ps
)E (α)E
(1
1−α
)+Wm
(1−
(1−Mp
)E (β)
)−Sm−δm−
(Hrθ +dc
)(1−E (β)) ,U5= Im+Is−L,
U6= −(Is+Im)T+ HsR2
2PsE (1− 2α)−HsR
Rs(R−Ps)E (α)+
(Wr−Wm+ Hr
θ +dc
)DcT− (Ar+As+Am).
Proposition 4 To obtain the maximum average profit there exists at least one positive point(P∗, t∗1
)at which
∂2EIAP∂P2 and ∂2EIAP
∂t21
both are negative, and Hessian Matrix
∂2EIAP∂P2
∂2EIAP∂t1∂P
∂2EIAP∂P∂t1
∂2EIAP∂t2
1
or ∂2EIAP∂P2
∂2EIAP∂t2
1−(∂2EIAP∂P∂t1
)2= CD− E2
must be positive where ∂2EIAP∂P2 = C, ∂
2EIAP∂t2
1= D and ∂2EIAP
∂P∂t1= E.
It is not possible to show the existence of the optimal solution analytically. For this reason, the condi-tions are assessed by a numerical example; however, the concavity is shown by graphically. Figure 6 showsthat the integrated expected average profit (EIAP) equation (33) of the proposed three-layer supply chainmodel is followed by concavity property.
Insert Figure 6
Lemma 5 The maximum expected integrated average profit (EIAP) exits if C < 0, D < 0 and CD− E2> 0.
Proof. We know that for a stationary point (x∗, y∗) a function f (x,y) is called maximum if∂2f (x∗, y∗)
∂x2∂2f (x∗, y∗)
∂y2 −(∂2f (x∗, y∗)∂x∂y
)2> 0 and ∂2f (x∗, y∗)
∂x2 < 0, ∂2f (x∗, y∗)∂y2 < 0.
Now ∂2EIAP∂P2 = C > 0 as C > 0 and ∂2EIAP
∂t21
= D > 0 as D > 0 at(P∗, t∗1
).
Also ∂2EIAP∂P2
∂2EIAP∂t2
1−
(∂2EIAP∂P∂t1
)2> 0 since CD− E2
> 0 where ∂2EIAP∂P2 = C, ∂
2EIAP∂t2
1= D and ∂2EIAP
∂P∂t1= E.
Therefore all conditions of Proposition 1 are satisfied at point(P∗, t∗1
). Hence the maximum average profit (EIAP)
exits if C < 0, D < 0 and CD− E2> 0.
Lemma 6 If C < 0 and D > 0 then the maximum EIAP does not exit.Proof. Here C < 0 represents ∂2EIAP
∂P2 < 0 at(P∗, t∗1
)since ∂2EIAP
∂P2 = C, and
D > 0 represents ∂2EIAP∂t2
1> 0 at
(P∗, t∗1
)since ∂2EIAP
∂t21
= D.
Therefore one condition of Proposition 1 is not satisfied, and so there does not exist the maximum average profit.
Lemma 7 If CD− E2< 0 then the maximum expected integrated average profit does not exit.
Proof. Now if CD− E2< 0 then ∂2EIAP
∂P2∂2EIAP∂t2
1−
(∂2EIAP∂P∂t1
)2< 0, since ∂2EIAP
∂P2 = C, ∂2EIAP∂t2
1= D and ∂2EIAP
∂P∂t1= E at(
P∗, t∗1).
10
Thus all conditions of Proposition 1 are not satisfied at point(P∗, t∗1
).
So if CD− E2< 0 then there does not exist the maximum EIAP at
(P∗, t∗1
).
The expected integrated average profit (EIAP) equation (33) of the three-layer supply chain model willgive the optimal solution based on the following lemma. The conditions for the optimal total EIAP areanalysed.
Lemma 8 The condition of total expected integrated average profit is maximum as follows
i) HmPs
{2DrE (β)+Rmt1E(1− β)2
}> RmDr
[Hst1
{E(
11−α
)−E (α)E
(1
(1−α)2
)}−2γPs
]ii) HsDr
[P{E(
11−α
)−E (α)E
(1
(1−α)2
)}−Ps
]<HmPs
[PE(1− β)2−2Dr
{12−E (β)
}]Proof. We can write EIAP = 1
T
[U1P2t2
1−U2P2t1+U3Pt21+U4Pt1+U5t1+U6
]∂EIAP∂P
=1T
(2U1Pt2
1−2U2Pt1+U3t21+U4t1
)(34)
∂EIAP∂t1
=1T
(2U1P2t1−U2P2+2U3Pt1+U4P + U5
)(35)
∂2EIAP
∂P2 =2t1
T(U1t1−U2) (36)
∂2EIAP
∂t21
=2PT
(U1P + U3) (37)
∂2EIAP∂t1∂P
=1T
(4U1Pt1−2U2P + 2U3t1+U4) (38)
Based on the concept of optimality for several variables, from equation (34) and equation (35) we get
2U1Pt1−2U2P + U3t1+U4= 0
2U1P2t1−U2P2+2U3Pt1+U4P + U5= 0
and from equation (36) and equation (37) we get
U1t1−U2< 0 and U1P + U3< 0
Thus, the necessary conditions are obtained as follows
HmPs
{2DrE (β)+Rmt1E(1− β)2
}> RmDr
[Hst1
{E( 1
1−α
)−E (α)E
(1
(1−α)2
)}−2γPs
]
HsDr
[P{
E( 1
1−α
)−E (α)E
(1
(1−α)2
)}−Ps
]<HmPs
[PE(1− β)2−2Dr
{12−E (β)
}]Therefore, the optimal solution of the proposed three-layer supply chain exists.
11
5 Numerical Example
The theoretical demonstration of the proposed three-layer supply chain system is illustrated through nu-merical example to study the feasibility. The values of the parameters of the model for numerical examplesare not selected from any real-life case study; however, these values have seemed to be realistic. This modelcan be used in heavy industries like steel, chemicals, machinery, food, textile, etc.
We assume replenishment lot size of the raw material of the supplier is 300 units which also produces400 units per unit time. The supplier purchases the raw material at the cost of 30$ per unit item, and theholding cost of these raw materials is 3$ per unit item. Let the cost associated with setting up a piece ofproduction equipment for the supplier be 300$ per unit item. Again, the screening rate of supplier is 1800per unit time, and the cost of screening per unit item is 0.5$. The selling price of the non-defective itemis 54$ per unit item and for the defective item is 10$ per unit item. At the end of the supplier’s inventorycycle if it remains idle, then the costs for idle for supplier be 200$ per unit time.
The manufacturer purchases the raw material from the supplier for 54$ per unit item, and the holdingcost of these raw materials is 3$ per unit item. The setup cost for the manufacturer is 400$ per unit item,and the cost per unit finished product is 1$. The total labour/energy costs per unit time of a productionsystem are 1000$, and the cost for tool per unit of the production rate is 0.02$. The screening rate and thecost of screening of manufacturer are 1750$ per unit time and 0.5$ per unit item, respectively. Again, theselling price of the non-defective item is 130$ per unit item, and for the defective item is 20% of the sellingprice of non-defective item per unit item. At the end of the supplier’s inventory cycle, if it remains idle,then the costs for unit idle time is 150$ per unit time.
The retailer purchases the finished product from the manufacturer at the cost of 130$ per unit item,and the holding cost of these finished products is 5$ per unit item. The retailer cost associated with thesetting of infrastructure and to maintain the inventory is 350$ per unit item. Again, the selling price ofthe finished product item is 140$ per unit item. If the finished product is held for more time in retailerwarehouse, then the product will deteriorate with the rate of 0.08 per unit time and the cost of deteriorateditems is 8$ per unit item. The demand rate of the retailer is 170 units per unit time, and the demand rateof the customer is 130 units per unit time.
Numerical result and analysis of the above problem are solved using Mathematica and MATLAB soft-ware. For this model, we consider the data with the appropriate unit as below:
R = 300, As= 300, Hs= 3, Ws= 54, Sp= 10, Ps= 400, Rs= 1800, Ss= 0.5, Is= 200, Cs= 30, Im= 150, Hm= 2,Sm= 0.5, Rm= 1750, Wm= 130, Mp= 0.2, Am= 400, δm= 1, θ = 0.08, L = 1000, γ = 0.02, Ar= 350, Wr= 140,Hr= 5, Dr= 170, Dc= 130, dc= 8, T = 12.
Insert Table 1
From Table 1, the optimal value of t1 is t∗1 = 5.126, and the optimum production rate is P∗= 298.846,and the optimum total integrated expected average profit EIAP∗ for the three-layer supply chain model is10702$.
Insert Figure 7Insert Figure 8
Figure 7 shows the percentage of optimal cost at individual level of three layers supply chain model,where we observe that the total cost of retailer is the maximum share of the total cost. In figure 8, wealso compare the individual cost of supplier, manufacturer, and retailer, along with the optimum totalintegrated expected average profit.
12
6 Sensitivity Analysis and Managerial Implications
We discuss the effect of changes of different parameters in different percentage change for analysing theproposed three-layer supply chain model. Sensitivity analysis is performed by increasing and decreasingthe parameters by taking changes of one parameter at a time for analysis, and the other parameter remainsunchanged.
Insert Table 2Insert Figure 9
Insert Figure 10Insert Figure 11
From Table 2, Figure 9, 10, and 11, we have observed the following nature of the proposed three-layersupply chain model. The model is highly sensitive to the demand rate of customer (Dc). As the demandrate of customer increases the total profit of the system increases and vice versa. The total profit of theSCM rises with the demand of retailer (Dr), which managerial significance leads to more items producedby the manufacturer. And consequently, these increase the sales to the customer; therefore, the systemprofit will increase. The proposed supply chain model has responded to the screening rate of supplier(Rs) and screening rate of manufacturer (Rm). Therefore a proper managerial decision can be taken fromthis study of the supply chain model to increase the profit of the supply chain system. The optimal totalcost is very less sensitive to the initial stock level of the supplier (R) in reverse proportion; because initialstock level increases mean the supplier produces less items and so the profit decreases for the supply chainsystem. The proposed supply chain model interacts slowly with the production rate of supplier (Ps) inreverse proportion; because the increase of (Ps) would imply more items will be produced in a short time,and thus it will decrease the supplier’s total profit.
This study reflects the quality concept of the products for both the supplier and manufacturer due toscreening implementation. This study has developed a SCM of production model to convert the raw mate-rials into a finished product to satisfy the demand of customers in time, which is a common phenomenonof managerial insights. We have considered idle cost of supplier and manufacturer to time, which will helpto reduce the idle time of machines.
Insert Table 3Insert Figure 12Insert Figure 13
On the basis of sensitivity analysis which is presented in the above Figure 9, 10, 11, 12 and 13, andTable 3, the following observations are made for different cost parameters of the three-layer supply chainmodel. The model is highly sensitive to the selling price of retailer (Wr). If we increase (Wr), the profitof the retailer automatically increases and thus, the total profit increases. It is also seen that the decisionvariables P and t1 are constant because the variables are independent of (Wr). The model is sensitive to thescreening cost of supplier ( Ss) and manufacturer (Sm) in reverse proportion.
The expected total profit is moderately sensitive to the purchasing cost of supplier (Cs) in reverse pro-portion. The managerial impact of this change of supplier is real because an increase of purchasing costrestricts supplier to buy of items. Accordingly, fewer items are sold to the manufacturer. Hence the totalprofit of the proposed supply chain system decreases as an impact of fewer items produced by the man-ufacturer. The supply chain management would take decision about the selling price of defective item ofsupplier (Sp) to increase the profit of supplier, and hence the total profit of the supply chain model in-creases. This paper shows that the profit will decrease for a higher cost of screening at the level of supplierand manufacturer. The cost due to idle time of supplier (Is) and manufacturer (Im) has an impact on the
13
optimal expected total profit of the model. Cost for idle time increases mean the purchase cost increases,and hence, the total profit of the system decrease. The model is sensitive to the cost of deterioration item(dc), because the managerial decision would help the retailer to gain profit from deterioration items forhigher cost after discount. However, the cost of deterioration item should be decided to concentrate on thefact of the cost of non-defective items.
Insert Table 4
Based on the sensitivity analysis, shown in Table 4, Figure 10 and 13, few observations are noticed. Thetotal profit of supplier decreases due to increase in the average of the holding cost (Hs). It is also seenthat as holding cost of supplier increases, the demand rate of supplier P increases, thus more items aresold to the manufacturer. Also, the time t1 decreases, which means that the supplier has to hold for lesstime, hence the total profit of the model decreases. For supply chain system shows that if holding cost atmanufacturer (Hm) level increases then as a resulting in the total profit decreases of the manufacturer. It isalso observed from this proposed model that if holding cost increases, then the production rate of supplierdecreases, thus less quantity of items will be produced by the manufacturer. The rise of labour/energycosts (L) implies that the total profit decreases and vice versa. The total profit of this model decreases dueto increase of tool cost (γ) and cost per unit finished product (δm) due to the increase in (γ) and (δm). Thissupply chain system shows that as tool cost and cost per item of the finished product increases, the demandrate of supplier decreases and the time t1 decreases, which means that the manufacturer produces items inless quantity in less time. Therefore, the manufacturer’s profit as well as total profit decreases of SCM.
7 Conclusion and Future Research
Several researchers have discussed integrated vendor–buyer inventory models and the joint optimization ofinventory policies, to maximize total profit for the supply chain. In this paper, we have developed a three-layer supply chain system for a production inventory model with supplier, manufacturer, and retailer as themembers of the chain. We have considered the idle cost of supplier and manufacturer, and also consideredthe constant demand rate of the supplier, manufacturer, and retailer. The defective items at suppliersand manufacturers are considered with the proportion factors of defective items that follow a Uniformdistribution function. It is also assumed that the screening rate is less than or equal to the production rateas well as greater than the demand rate. The deterioration is allowed for produced items at the retailer end.
This paper expands, upgrades, and complements many existing articles. The major contribution is finitereplenishment with supplier production rates, idle times, and the effect of imperfect items on the proposedsupply chain model compare to the existing literature. In the numerical example, the optimal total costalong with the optimal value of two decision variables P (production rate of the manufacturer) and t1(manufacturers stop their production) are evaluated by incorporating all the expenses of each three layers.The numerical example with sensitivity analysis has been given to study the feasibility of the proposedmodel for the effecting of changes in the various parameters involved in this study. From the sensitivityanalysis, we have observed that the model is highly sensitive to the change in demand rate of the customer,selling price of the retailer, and is less sensitive to the change of the purchasing cost of the supplier. Thisstudy observes that not only the production rate of supplier and manufacturer can be a decisive factorin optimizing the total profit of inventory, but also the selling price of the retailer, demand rate of thecustomer, purchase cost of supplier and holding cost can be a breakthrough in expanding the profit of thebusiness in real terms. This paper has established the coordination between production, demand rate, andselling price across the supply chain models for the optimum integrated average profit of the supply chain.
This study can be extended considering the machine breakdown, repairing costs of corrective and pre-ventive maintenance for the manufacturer. The model can be studied in the stock out situations in eachstage of the chain which occur due to uncertainties of the delivery, production, and demand of customers.The researchers can extend this work by considering the effect of inspection error and reworked by the
14
manufacturer. This model can further be extended about aspects such as defective items of supplier andmanufacturer, the production rate of supplier and manufacturer is time and reliability dependent, proba-bilistic demand rate of the retailer, variable deterioration rate, deterioration in each stage, shortages withfull or partial backlogging in every stage, quantity discounts, multiple suppliers, multiple manufacturers,etc. These are some crucial outlines for further topics of future research on the supply chain system.
Acknowledgement: We are grateful to Editor-in-Chief, Editor, and anonymous referees for their valu-able comments and helpful suggestions, which have helped us to improve this work significantly.
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Figure 1: Pictorial representation Three-layer supply chain ModelFigure 2: Graphical representation of the three-layer supply chainFigure 3: Sketch for inventory level of supplierFigure 4: Sketch for inventory level of manufacturerFigure 5: Diagram of inventory level of retailerFigure 6: Graph of total profit vs production rate of manufacturer vs timeFigure 7: Percentage of optimal cost at individual levelFigure 8: Compare between total cost and individual costFigure 9: Percentage change of total profit vs change of parameterFigure 10: Percentage change of total profit vs change of parameterFigure 11: Percentage change of total profit vs change of parameter Dc, WcFigure 12: Percentage change of total profit vs change of parameterFigure 13: Percentage change of total profit vs change of parameter
Table 1: Optimal solution of SCM for defective items at supplier and manufacturerTable 2: Sensitivity analysis of proposed model for rate parametersTable 3: Sensitivity analysis of the proposed model for cost parametersTable 4: Sensitivity analysis of the proposed SCM for other inventory parameters
Sudip Adak is an Assistant Teacher in Harit High School, Hooghly, West Bengal, India. He is currentlya Ph.D. scholar in NIT Puducherry, India. He obtained MSc degree in Applied Mathematics from JadavpurUniversity, India. His research interests include inventory control theory, inventory management, intervaloptimization, operations research, optimization theory, fuzzy systems, etc. He has published six researchpapers in international/national journals and one chapter in a book.
Dr. G.S. Mahapatra is an Associate Professor at National Institute of Technology Puducherry, India.He obtained M.Sc. and Ph.D. in Applied Mathematics from Bengal Engineering and Science University,Shibpur, presently IIEST Shibpur, India. Dr. Mahapatra has been involved in teaching and research formore than seventeen years and has published more than eighty research articles in various International,National journals and proceedings to his credit. His research interest include Inventory Management,Reliability, Optimization, Fuzzy Set theory, Soft Computing, and Mathematical Biology.
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Table 1: Optimal solution of SCM for defective items at supplier and manufacturer
α β P∗ t∗1 EIAP∗($) 1b−a
(0.04,0.2) (0.05,0.2) 298.846 5.126 10702.0 10.16
(0.09,0.3) (0.1,0.3) 261.994 3.132 10252.1 10.21
(0.04,0.3) (0.05,0.3) 276.577 3.808 10370.3 10.26
(0.09,0.4) (0.1,0.4) 218.928 1.472 10082.4 10.31
Table 2: Sensitivity analysis of proposed model for rate parameters
Parameter Change (%) P ts t1 t2 EIAP($) Change EIAP(%)−20 298.846 3.614 5.126 7.885 8609.0 −19.557
Dc −10 298.846 3.614 5.126 7.885 9655.5 −9.77910 298.846 3.614 5.126 7.885 11748.5 9.77920 298.846 3.614 5.126 7.885 12795.0 19.557−20 271.941 2.438 4.116 7.201 10541.4 −1.501
Dr −10 284.840 2.988 4.607 7.505 10618.0 −0.78510 314.354 4.342 5.686 8.364 10795.5 0.87420 332.006 5.214 6.306 8.980 10902.0 1.869−20 298.432 3.607 5.125 7.872 10700.6 −0.013
Rm −10 298.662 3.611 5.126 7.880 10701.4 −0.00610 298.997 3.616 5.126 7.889 10702.5 0.00520 299.122 3.619 5.127 7.894 10702.9 0.008−20 298.604 3.603 5.117 7.864 10699.2 −0.026
Rs −10 298.738 3.609 5.122 7.876 10700.8 −0.01210 298.934 3.618 5.129 7.892 10703.0 0.00920 299.007 3.621 5.132 7.898 10703.9 0.017−10 311.773 4.686 5.594 8.975 10767.1 0.608
Ps −5 304.486 4.078 5.331 8.355 10730.4 0.2655 294.336 3.247 4.961 7.516 10679.3 −0.212
10 290.638 2.949 4.824 7.216 10660.7 −0.385−20 302.963 3.954 5.276 8.227 10745.1 0.403
R −10 300.905 3.783 5.201 8.055 10723.1 0.19710 296.785 3.445 5.051 7.716 10681.8 −0.18820 294.722 3.277 4.975 7.547 10662.6 −0.368
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Table 3: Sensitivity analysis of the proposed model for cost parameters
Parameter Change (%) P ts t1 t2 EIAP($) Change EIAP(%)−20 298.846 3.614 5.126 7.885 7062.0 −34.012
Wr −10 298.846 3.614 5.126 7.885 8882.0 −17.00610 298.846 3.614 5.126 7.885 12522.0 17.00620 298.846 3.614 5.126 7.885 14342.0 34.012−20 295.884 3.480 5.018 7.642 10666.9 −0.328
Sp −10 297.366 3.547 5.072 7.763 10684.3 −0.16510 300.325 3.682 5.180 8.007 10719.9 0.16820 301.804 3.750 5.234 8.131 10738.2 0.338−20 299.906 3.663 5.165 7.973 10714.8 0.120
Sm −10 299.376 3.638 5.145 7.928 10708.4 0.06010 298.315 3.590 5.107 7.842 10695.6 −0.06020 297.785 3.565 5.087 7.797 10689.3 −0.119−20 300.054 3.669 5.170 7.985 10716.6 0.137
Ss −10 299.450 3.642 5.148 7.935 10709.3 0.06810 298.241 3.586 5.104 7.835 10694.7 −0.06820 297.637 3.559 5.082 7.785 10687.5 −0.135−20 300.606 3.581 5.057 7.824 10719.3 0.161
Im −10 299.728 3.598 5.092 7.856 10710.6 0.08010 297.961 3.631 5.161 7.915 10693.4 −0.08020 397.072 3.647 5.196 7.945 10684.9 −0.160−20 301.190 3.570 5.035 7.805 10725.1 0.216
Is −10 300.021 3.592 5.080 7.845 10713.5 0.10710 297.665 3.636 5.172 7.924 10690.6 −0.10720 296.477 3.658 5.219 7.964 10679.2 −0.213−10 334.797 5.357 6.403 11.034 11224.2 4.879
Cs −5 316.903 4.464 5.776 9.421 10941.3 2.2365 280.650 2.807 4.449 6.427 10504.2 −1.849
10 262.348 2.047 3.742 5.053 10345.5 −3.331−20 313.647 4.037 5.357 8.648 10684.4 −0.165
dc −10 306.259 3.957 5.395 8.504 10690.9 −0.10410 291.409 3.278 4.852 7.278 10720.1 0.16920 283.951 2.950 4.574 6.685 10745.0 0.402
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Table 4: Sensitivity analysis of the proposed SCM for other inventory parameters
Parameter Change (%) P ts t1 t2 EIAP($) Change EIAP(%)−20 264.635 4.065 6.387 8.700 10822.5 1.126
Hs −10 282.800 3.812 5.663 8.243 10756.4 0.50810 313.364 3.454 4.709 7.595 10655.9 −0.43120 326.739 3.320 4.373 7.354 10615.9 −0.804−20 341.495 4.617 5.517 9.697 10806.7 0.978
Hm −10 317.968 4.033 5.280 8.641 10747.1 0.42210 282.535 3.298 5.029 7.313 10666.5 −0.33220 268.145 3.050 4.975 6.866 10637.8 −0.600−20 300.966 3.712 5.204 8.061 10727.8 0.241
δm −10 299.906 3.663 5.165 7.973 10714.8 0.12010 297.785 3.565 5.087 7.797 10689.3 −0.11920 296.723 3.517 5.048 7.710 10676.7 −0.236−20 286.736 3.831 5.608 8.277 10791.4 0.835
L −10 292.876 3.724 5.362 8.083 10745.7 0.40810 304.661 3.503 4.900 7.684 10660.2 −0.39020 310.334 3.389 4.682 7.479 10620.3 −0.763−20 355.039 4.510 5.201 9.504 10884.7 1.707
γ −10 324.254 4.027 5.171 8.630 10785.2 0.77710 277.492 3.257 5.069 7.240 10631.5 −0.65820 259.276 2.946 5.004 6.678 10571.1 −1.223
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Figure 1: Pictorial representation Three-layer supply chain Model
Figure 2: Graphical representation of the three-layer supply chain
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Figure 3: Sketch for inventory level of supplier
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Figure 4: Sketch for inventory level of manufacturer
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Figure 5: Diagram of inventory level of retailer
Figure 6: Graph of total profit vs production rate of manufacturer vs time
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22%EAPS
38%EAPM
40%
EAPR
Figure 7: Percentage of optimal cost at individual level
0
5000
10 000
EAPS
EAPM
EAPR
EIAP
Figure 8: Compare between total cost and individual cost
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-20 -15 -10 -5 0 5 10 15 20Parameters change (in %)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
To
tal p
rofi
t ch
ang
e (i
n %
)
Dr
Rm
Sm
R
Figure 9: Percentage change of total profit vs change of parameter
-20 -15 -10 -5 0 5 10 15 20Parameters change (in %)
-1
-0.5
0
0.5
1
1.5
To
tal p
rofi
t ch
ang
e (i
n %
)
Ss
Rs
Hs
Im
m
Figure 10: Percentage change of total profit vs change of parameter
28
-20 -15 -10 -5 0 5 10 15 20Parameters change (in %)
-40
-30
-20
-10
0
10
20
30
40T
ota
l pro
fit
chan
ge
(in
%)
Dc
Wr
Figure 11: Percentage change of total profit vs change of parameter Dc, Wc
-10 -8 -6 -4 -2 0 2 4 6 8 10Total cost change (in %)
-4
-2
0
2
4
6
Par
amet
ers
chan
ge
(in
%)
Ps
Cs
Figure 12: Percentage change of total profit vs change of parameter
29
-20 -15 -10 -5 0 5 10 15 20Total cost change (in %)
-1.5
-1
-0.5
0
0.5
1
1.5
2
Par
amet
er c
han
ge
(in
%)
Sp
dc
Is
Hm
L
Figure 13: Percentage change of total profit vs change of parameter
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