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J. Serv. Sci. & Management, 2008, 1: 123-127 Published Online August 2008 in SciRes (www.SRPublishing.org/journal/jssm) Copyright © 2008 SciRes JSSM BBImpact of Supply Chain Coordination for Deteriorating Goods with Stock-Dependent Demand Rate Chuanxu Wang PSchool of Economy and Management ， Shanghai Maritime University Shanghai ， China Email: cxwang@shmtu.edu.cn P ABSTRACT To analyze effects of supply chain coordination for deteriorating goods with stock-dependent demand rate, this paper presents decision models for order quantity and ordering cycle under two scenarios( decentralized supply chain, cen- tralized supply chain). Numerical study is carried out to demonstrate the effectiveness of the proposed models, and to analyze the impact of supply chain coordination on supply chain profit. Sensitivity analysis is performed to study the impact of different parameters a ssociated with the model, su ch as the rate o f deterioration, the reta iler’s purcha se cost, the manufacturer’s production cost, the retailer’s and manufacturer’s holding cost on the supply chain profit increase percentages generated by t he sup pl y c h a i n c oordinati on. Keywords: supply chain coordination, deteriorating goods, ordering cycle, stock-dependent demand 1. Introduction In real life, many inventory goods, such as agricultural products, fashion goods, drugs and high-tech products, are subject to depletion through spoilage, shrinkage, de- cay and obsolescene [1]. The deterioration is quite preva- lent and should not be disregarded. Inventory manage- ment for deterioration goods has received many attentions from researchers and practitioners. Most of the existing researches focus on the EOQ-based inventory decision models. Ghare and Schrader [2] presented the EOQ model by considering the combined effects of demand, usage and linear decay. Covert and Philip [3]used the variable deterioration rate of the two-parameter Weibull distribution, to formulate the inventory decision model under the assumptions of a constant demand rate, with no shortages allowed. Philip [4] modified this model by us- ing the deterioration rate of the three-parameter Weibull distribution. Tadikamalla [5] adopted gamma distributed deterioration under constant demand over time, without shortages. Moon and Lee [6] presented the EOQ model with a normally distributed deterioration rate. Other dete- rioration inventory models have extended prior research by considering a time-varying demand function, with or without shortages. Dave and Patel [7] proposed an EOQ model under time-proportional demand, with no shortages allowed. Sachan [8] extended their model by considering shortages. Bahari-Kashani [9] generalized the problem by permitting variations in both replenishment cycle length and order quantity. Bose et al [10]developed an EOQ model for deterioration items incorporating the effects of inflation, time value of money, a linearly time-dependent demand rate and shortages. Replenishment decision mod- els under time-proportional demand and exponentially decaying deterioration rate was developed in [1]. It is observed that large quantities of consumer goods displayed in a supermarket generate higher demands. Sil- ver and Peterson [11] noted that the sales at the retail level tend to be proportional to the inventory displayed. Gupta and Vrat [12], Mandal and Phaujdar [13], Baker and Urban [14], Datta and Pal [15], etc developed the EOQ models with stock-dependent demand rate. Mandal and Phaujdar [16], Pal et al. [17] developed the inventory models for deteriorating items with stock-dependent de- mand rate. In this paper, we have extended these works, on deteriorating inventory research, by considering dete- riorating goods with stock-dependent demand in a two-echelon supply chain consisting one manufacturer and one retailer, the objective is to investigate the effects of supply chain coordination on profit increase in the supply ch ai n , an d study the impact of different parameters associated with the model, such as the rate of deteriora- tion, the retailer’s purchase cost, the manufacturer’s pro- duction cost, the retailer’s and manufacturer’s holding cost on the supply chain profit increase percentages gen- erated by the supply chain coordination. 2. Assumptions and Notations 2.1. Assumptions (1) The retailer replenishes the stocks from the exclusive source on an EOQ basis. Replenishments are instantane- ous. (2) Lead time is assumed to be zero for the sake of sim- plicity. 124 Chuanx u Wang Copyright © 2008 SciRes JSSM (3) No backorders are allowed. (4) Demand rate is dependent on the instantaneous in- ventory level. The demand rate ()dI of the item, when the inventory is I , is considered in the form ()dI I β α =, where 0 α > and 01 β << are scale and shape pa- rameters ( Baker an d Urban 1988). (5) The manufacturer’s production rate is greater than or equal to the demand rate facing the retailer. (6)The manufacturer is a make-to-order producer; it has a lot-for-lot production policy in response to the retailer’s demand. In this particular case, the length of the manu- facturer’s production cycle is equal to the length of the retailer’s replenishment cycle. 2.2. Notations p the sale price for the retailer; S the order cost per order for the retailer; M the setup cost p er lot for the manufacturer; () rm cc procurement (manufacturing ) cost per unit for retailer (manufacturer); d the demand rate of the item in the marketplace; q the production rate; () rm hh the inventory holding cost as a fraction of the inventory cost for the retailer(manufacturer); ()t θ the deterioration rate facing both the retailer and manufacturer，01 θ ≤≤； (), () rm I tI t inventory level at timet for the retailer ( manufacturer)； Q the order quantity for the retailer; T the replenishment cycle ( or production cycle ) for the retailer( manufacturer); 3. Basic Model In this section, we first derive the profit model for the decentralized supply chain. Later, we present the profit model derived by considering the centralized supply chain. 3.1. The Decentralized Supply Chain In the decentralized supply chain, each entity within the supply chain aims to maximize its own profit functions, with no consideration given to its counterpart’s reaction or profit. The retailer makes a replenishment decision based on an EOQ policy that includes inventory holding cost and ordering cost. During the replenishment cycle, the change in retailer’s inventory level depends on demand and deterioration and is given by [1]: () ()[ ()] rrr dI t I tIt dt β θα +=− 0tT≤≤ , (1) As shown in Pal et al. (1993) [17], equation (1) can be rewritten as dttdI tI tI tI r r r r−= ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + −− −)( )]([ )]([ )( 11 1 2 β β αθ α θ (2) By integrating, we get 1 )1( 1 1lnln )]([ )]([ ln Ce tI tI t r r+= + − − − ϑβ β β αθ , (3) where 1 C is integration constant. It can be rearranged as t reC tI θβ β θα )1( 1 1 )]([ 1− −= + (4) By using the boundary condition on inventory (0) r I Q = ，we can get β θα − + =1 11 Q C (5) Substituting (5) in (4), the retailer’s inventory level at time t(0tT ≤ ≤) can be expressed as 1(1)1/(1) 1/(1) () [()]/ rt ItQ e β βθβ β αθα θ −−−− − =+ − (6) The inventory holding cost in a cycle for the retailer is 0() T rrr r H Chc Itdt=∫ (7) The retailer’s total number of deteriorated goods in a cycle is given by , 0 ()[ ()] T Dr rr QQITItdt β α =−− ∫ 1(1)1/(1) 1/(1) [()] / T QQe β βθβ β αθα θ −−−− − =− +− 1 (1)/(1) 0/(1) [( )] Tt Qe dt ββθββ ββ αα θα θ −−− − − +− −∫ (8) The profit per unit time for the retailer can be ex- pressed as 1 (1)/(1)/(1) 10 1(1)1/(1)1/(1), 0 ()[()] / [()]/ T rrt T rr trDr S pcQ edtTT hcQ edtcQ TT ββθββ ββ ββθββ αα θαθ αα θαθ −−− −− −−−− − =− +−− +− −− ∏∫ ∫ (9) The profit function (9) is highly nonlinear and cannot be solved by analytical methods. We solve it by using optimization technology on computer and get the follow- ing optimal values: Since the manufacturer is a make –to-order producer, it has a lot-for-lot production policy in response to the re- tailer’s demand. During the production cycle, the change in manufacturer’s inventory level is due to the combined effect of production and deterior ation: Impact of Supply Chain Coordination for Deteriorating Goods 125 with Stock-Dependent Demand Rate Copyright © 2008 SciRes JSSM () () mm dI tqIt dt θ =− m ttT≤≤ , (10) where m t and Tare the starting and stopping produc- tion times, respectively. With a make-to-order policy, the production quantity of the manufacturer is equal to the demand quantity of the retailer. Therefore, we get: 0() () m TT rm t I tdtItdt= ∫∫ (11) Solving the equation (6) by using the method proposed by Spiegel (1960)[18] , we can get: tm eC q tI θ θ − += 2 )( , (12) where 2 C is constant. By using the boundary condition on inventory () 0 mm It=，we can obtain: m t e q C θ θ −= 2 Therefore, the manufacturer’s inventory level at time t(m ttT≤≤ ) can be expressed as () ()() () m mm m tt t tt tt m t qqe It eeqdt θ θθ θθ −− −− − ==− ∫(13) The inventory holding cost in a cycle for the manufac- turer is () () 2 () [1( )] m mm m TT tt mmm mmm tt Tt mm m qq H Chc Itdthcedt q hceT t θ θ θθ θ θ −− −− ⎛⎞ ==− ⎜⎟ ⎝⎠ =−++− ∫∫ (14) The manufacturer’s total number of deteriorated goods in a cycle is given by ,()() Dm m m QqTtITQ=−− − (15) The profit per unit time for the manufacturer is , 1 () rm mmDm mccQ MHCcQ TTTT − =−−− ∏ () 2 [1( )] m Tt mm rm hcqeT t cQ M TT T θ θ θ −− −++ − =−− () () m mTt mm m cqT tcqcqe TT T θ θθ −− − −+− (16) Based on the optimal order quantity and ordering cycle as well as equation (11), we can obtain optimal profit for the manufacturer usi n g (16). 3.2. The Centralized Supply Chain In the centralized supply chain, the order quantity and replenishment cycle are determined by considering the total profit incurred by both the retailer and the manufac- turer, so that the overall profit is maximized. Th e central- ized supply chain requires information sharing between manufacture and retailer. Sequential or concurrent engi- neering will be beneficial to the information sharing. In this case, the total profit per unit time for the supply chain is the sum of Equation (9) and (16). 211 scr m = += ∏∏∏ 1 (1)/(1) 0/(1 ) [( )] Tt pQe dt S TT ββθββ ββ αα θα θ −−−− − +− − ∫ 1(1) 1/(1) 01/(1 ) [( )] T rr t hcQ edt T ββθβ β αα θα θ −−− − − +− −∫ 1(1) 1/(1) 1/(1) [( )] rT cQe T β βθ β β αθ α θ −−− − − +− + () 2 [1( )] m Tt mm m hcqeT t M TT θ θ θ −− −++ − −− () () m mTt mm m cqT tcq cqe TT T θ θθ −− − −+− (17) Similarly, since supply chain profit function (17) is highly nonlinear, we solve it by using optimization tech- nology on computer and get the optimal values. 4. Numerical Study To illustrate the effect of the models, we give the follow- ing numerical example: 0.5 α = units per time period, 0.4 β = , 0. 1 θ = , $10S = , $20M=, $20p = per unit, 200q = units, 0.35 r h=, 0.25 m h=, 3.5 r c=, 2.0 m c=. The models were implemented on a personal computer using Mathematica version 5.2 . The results are outlined in Table 1, revealing that the profit for centralized supply chain is greater than that for decentralized supply chain. Based on the numerical example considered above, we now perform the sensitivity analysis on the effects of changes in the model parameters such as the rate of dete- rioration, the retailer’s purchase cost, the manufacturer’s production cost, the retailer’s and manufacturer’s holding cost on the optimal order quantity for the retailer, the op- timal replenishment cycle ( or production cycle ) for the retailer( manufacturer), the optimal supply chain profit, and supply chain profit increase percentages generated by the centralized policy. The effects of changes in the pa rameters values are shown in Table 2. The sensitivity analysis is performed by changing each of the parameters by –50%, -20%, +30% and +50% and keeping the other parameters unchanged. The results are demonstrated in Table 1. The Sollution Results Supply chain * Q * T r Π m Πsc Π Decentralized 12.15 2.36 5.03 3.428.45 Centralized 20.63 2.75 4.76 4.569.32 126 Chuanx u Wang Copyright © 2008 SciRes JSSM Table 2. Effect of Changes in the Model Parameters Results of Optimization Procedure Decentralized Supply Chain Centralized Supply Chain Parameters Change (%) * Q * T * 1 SC ∏* Q * T * 2 SC ∏ Profit Increase (%) ＋50 10.434 2.342 6.345 18.675 2.735 7.243 14.153 ＋20 11.876 2.351 7.421 19.538 2.747 8.256 11.252 –20 13.216 2.375 9.236 21.562 2.769 9.768 5.760 θ –50 13.987 2.387 10.167 24.356 2.778 10.245 0.767 ＋50 11.579 1.987 17.948 17.917 2.294 18.831 4.920 ＋20 11.682 2.173 14.656 19.267 2.473 15.857 8.195 –20 15.876 2.563 7.063 30.013 2.931 7.771 10.024 r c –50 16.786 3.126 4.754 38.681 3.477 5.335 12.221 ＋50 10.342 2.031 7.346 18.625 2.334 7.658 4.247 ＋20 11.769 2.215 8.085 19.890 2.513 8.779 8.584 –20 16.987 2.685 11.886 31.325 2.973 13.256 11.526 m c –50 17.769 3.264 15.538 39.706 3.517 17.543 12.904 ＋50 10.876 2.046 7.258 17.018 2.305 7.589 4.560 ＋20 11.986 2.263 7.982 18.845 2.433 8.648 8.344 –20 16.765 2.765 12.189 32.098 3.384 13.767 12.946 r h –50 17.875 3.078 15.876 40.022 3.441 18.089 13.939 ＋50 11.054 2.178 7.458 17.357 2.338 7.915 6.128 ＋20 12.035 2.268 8.212 18.930 2.485 8.971 9.243 –20 16.497 2.589 11.687 32.021 3.264 12.893 10.319 m h –50 17.568 2.987 14.789 38.357 3.377 16.606 12.286 table 2. The following observations can be made from (1) Whether the supply chain is centralized or decen- tralized, the optimal order quantity, the optimal replen- ishment cycle (or production cycle) , the optimal supply chain profit are decreasing in the deterioration rate. Meanwhile the percentages of supply chain profit in- crease realized by employing the centralized policy are increasing in the deterioration rates. (2) Whether the supply chain is centralized or decen- tralized, the optimal order quantity and replenishment cycle (or production cycle) decrease, while the optimal supply chain profit increases in retailer’s unit procure- ment cost. (3) Whether the supply chain is centralized or decen- tralized, the optimal order quantity, the optimal replen- ishment cycle (or production cycle) and supply chain profit are decreasing in manufacturer’s unit manufactur- ing cost. (4) Whether the supply chain is centralized or decen- tralized, the optimal order quantity, the optimal replen- ishment cycle (or production cycle) and supply chain profit are decreasing in retailer’s and manufacturer’s in- ventory holding cost rate. (5) The percentages of supply chain profit increase re- alized by employing the centralized policy are increasing in the deterioration rates, but are decreasing in the re- tailer’s purchase cost, the manufacturer’s production cost, the retailer’s and manufacturer’s holding cost. 5. Conclusions This paper has investigated the effect of supply chain co- ordination for deteriorating goods with stock-dependent demand rate. Two profit models are developed with some assumptions based on exponentially decaying deteriora- tion rates. The numerical study is conducted to demon- strate the effectiveness of the proposed models, and to analyze the impact of supply chain coordination on supply chain profit. Sensitiv ity analysis is performed to stud y the impact of different parameters associated with the model, such as the rate of deterioration, the retailer’s purchase cost, the manufacturer’s production cost, the retailer’s and manufacturer’s h olding cost on the optimal ord er quantity for the retailer, the optimal replenishment cycle ( or pro- Impact of Supply Chain Coordination for Deteriorating Goods 127 with Stock-Dependent Demand Rate Copyright © 2008 SciRes JSSM duction cycle ) for the retailer( manufacturer), the optimal supply chain profit, and supply chain profit increase per- centages generated by the supply chain coordination. The following observations can be obtained from nu- merical analysis: (1) A centralized policy is found to be always superior to a decentralized policy in terms of profit increase, espe- cially when the deterioration rates are high. (2)Whether the supply chain is centralized or decen- tralized, the optimal order quantity , the optimal replen- ishment cycle (or production cycle) , the optimal supply chain profit are decreasing in the deterioration rate, re- tailer’s unit procurement cost, manufacturer’s unit manu- facturing cost, as well as retailer’s and manufacturer’s inventory holding cost rate. The optimal supply chain profit are decreasing in the deterioration rate, manufac- turer’s unit manufacturing cost, as well as retailer’s and manufacturer’s inventory holding cost rate. Meanwhile it is increasing in retailer’s unit procurement cost. (3)The percentages of supply chain profit increase re- alized by employing the centralized policy are increasing in the deterioration rates. The proposed models can be used to analyze gricultural products, fashion goods, drugs and high-tech products supply chain. It is observed that large quantities of these deteriorationg goods displayed in a supermarket tend to generate higher demands. However, the models consid- ered in this paper are somewhat idealized. In reality, when supply chain is coordinated, there are likely to be some costs incurred from information sharing scheme. The information sharing could also affect production, inventory, and other operations. The future research will further consider these factors. These models can be ex- tended in the future research to consider more general deterioration rates. Another extension possibility would be to use other replenishment policies. In addition to profit models, the cost models can also be applied to evaluate the extent of coordination. 6. Acknowledgement This paper is partially supported by Shanghai Shuguang Program under grant no. 06SG48. REFERENCES [1] Chen, J. M., Chen T. H. 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