An integrated vendor-buyer model with stockdependent

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1 WORKING PAPER L Mohsen S. Sajadieh, Anders Thorstenson & Mohammad R. Akari Jokar An integrated endor-uyer model with stockdependent demand CORAL Centre for Operations Research Applications in Logistics

2 An integrated endor uyer model with stock dependent demand Mohsen S. Sajadieh a, *, Anders Thorstenson, Mohammad R. Akari Jokar a a Department of Industrial Engineering, Sharif Uniersity of Technology, Tehran, Iran CORAL Centre for OR Applications in Logistics, Department of Business Studies, Aarhus School of Business, Aarhus Uniersity, Denmark Astract We deelop an integrated endor-uyer model for a two-stage supply chain. The endor manufactures the product and deliers it in a numer of equal-sized atches to the uyer. The items deliered are presented to the end customers in a display area. Demand is assumed to e positiely dependent on the amount of items displayed. The ojectie is to maximize total supply chain profit. The numerical analysis shows that uyer-endor coordination is more profitale in situations when demand is more stock dependent. It also shows that the effect of doule marginalization proides a link etween the non-coordinated and the coordinated case. Keywords: Batch production; Doule marginalization; Inentory; Integrated endor-uyer model; Stock-dependent demand. * Corresponding author. msajadieh@mehr.sharif.edu 1

3 1 Introduction In order to satisfy customer demands in today s competitie markets, critical information needs to e shared along the supply chain. A high leel of coordination etween endors and uyers decision making is also required. The concept of joint economic lot sizing (JELS) has een introduced to refine traditional methods for independent inentory control. The purpose is to find a more profitale joint production and inentory policy, as compared to the policy resulting from independent decision making. The idea of optimizing the joint total cost in a single-endor, single-uyer model was considered early on y Goyal (1976). Banerjee (1986) deeloped the model y incorporating a finite production rate and following a lot-for-lot policy for the endor. By relaxing Banerjee s lot-for-lot assumption, Goyal (1988) proposed a more general joint economic lotsizing model. Lu (1995) specified the optimal production and shipment policies when the shipment sizes are equal. He relaxed the assumption of Goyal (1988) aout completing a whole atch efore starting shipments. Goyal (1995) then deeloped a model where successie shipment sizes increase y a ratio equal to the production rate diided y the demand rate. He found an expression for the optimal first shipment size as a function of the numer of shipments. Later, Hill (1997) took this idea one step further y considering the geometric growth factor as a decision ariale. He suggested a solution method ased on an exhaustie search for oth the growth factor and the numer of shipments within certain ranges. Finally, Hill (1999) determined the form of the oerall optimal policy. This turns out to e a comination of the policy suggested y Goyal (1995) used initially and an equal shipments policy used susequently. Howeer, ecause the policy with equal-sized shipments etween the endor and the uyer is straightforward to implement in practice, this shipment policy is usually employed in the JELS modeling literature.

4 The asic JELS model has een extended in seeral different directions. The literature on JELS may thus e diided into different categories treating issues such as quality (e.g., Affisco et al., 00), controllale lead time (e.g., Hoque and Goyal, 006), stochastic lead time (e.g., Sajadieh et al., 008), multiple uyers (e.g., Chan and Kingsman, 007), setup and order-cost reduction (e.g., Chang et al., 006), transportation (e.g., Ertogral et al., 007), deteriorating items (e.g., Yang and Wee, 000), fuzzy logic (e.g., Pan and Yang, 008), and three-leel supply chains (e.g., Khouja, 003). Some contriutions to the literature may elong to more than one of these categories. Moreoer, Hill and Omar (006) deried the optimal policy of an integrated production-inentory model where, contrary to most of the preious work, it is assumed that the unit stock-holding cost decreases as stock moes downstream in the supply chain. We refer to Ben-Daya et al. (008) for a comprehensie reiew of the JELS literature. It is eyond the scope of this paper to discuss all contriutions in detail. Demand in inentory control models is most commonly assumed to e exogenous, although models with partly endogenous demand exist. In the marketing literature there is empirical eidence showing that consumer demand may indeed ary with the inentory on display or on the shelf at a retailer. For example, an inestigation y Desmet and Renaudin (1998) supported the hypothesis that direct shelf-space elasticities are significantly non-zero for many product categories. In particular, they concluded that product categories typical of impulse uying hae higher space elasticities. Moreoer, Koschat (008) proided empirical eidence from magazine retailing and demonstrated that the demand for a specific rand decreases as the on-shelf inentory of that rand decreases. Gupta and Vrat (1986), and Baker and Uran (1988) were among the first to introduce a class of inentory models in which the demand rate is inentory dependent. They considered a single-period model, where the 3

5 demand is a polynomial function of the inentory on hand. Seeral other contriutions extended this model to other inentory situations. Balakrishnan et al. (004) and Uran (005) conducted comprehensie reiews of inentory models for products exhiiting inentory-dependent demand. In particular, the latter studied two types of models in which the demand rate is either a function of the initial inentory leel, or it depends on the currently aailale inentory leel. Recently, Hariga et al. (007) proposed an inentory model to determine the product assortment, inentory replenishment policy, display area, and shelf-space allocation decisions that maximize the retailer s profit. Aott and Palekar (008) studied a single-store, multi-product inentory prolem in which product sales are a composite function of the shelf space. Warurton (009) also considered the stock-dependent demand prolem. Wang and Gerchak (001) deeloped models for coordinating decentralized two-stage supply chains when demand is shelf-space dependent. They characterized retailers Nash equilirium and explored whether the manufacturer can use incenties to coordinate such supply chains. Zhou et al. (008) also considered the coordination issues in a decentralized two-echelon supply chain, ut in cases where the manufacturer follows a lot-for-lot policy, and the demand is dependent on the inentory leel on display. The Stackelerg game structure was discussed. Their model proides the manufacturer with a quantity discount scheme to entice the retailer to increase the order quantity. Also recently, Goyal and Chang (009) proposed an inentory model with oth ordering and transfer lot sizes, where the demand rate depends on the stock leel displayed. Howeer, they determined the ordering and transfer schedules ased on the uyer s costs only. In this paper, we propose a joint economic lot-sizing model for coordination in a centralized supply chain when determining the optimal endor and uyer policies. The 4

6 endor manufactures the product in atches at a finite rate and deliers it in equal-sized transfer lots to the uyer. Some of the deliered items are displayed on the sheles in the uyer s retail store, while the rest of the items are kept in the uyer s warehouse. Final customer demand is positiely dependent on the amount of items shown on the shelf/in the display area. The ojectie is to maximize the total system profit when there is centralized coordination of the supply chain memers. In the sense emphasized y the italicized text, our analysis differs from and adds to the contriutions referred to in the preious paragraph. The result is then compared to the total profit otained in the corresponding non-coordinated supply chain. Finally, we also show and discuss how the so-called doule marginalization effect (Spengler, 1950; Jeuland and Shugan, 1983; Weng, 1995) impacts the performance of the supply chain in the non-coordinated case. This proides a linkage etween the coordinated and the non-coordinated supply chains. The rest of this paper is organized as follows. In Section, the modeling assumptions and notation are proided. In Section 3, we deelop the non-coordinated supply chain model and show how to find the independently optimal policies for the uyer and the endor. We introduce the coordinated supply chain model in Section 4 and deelop an algorithm to find the jointly optimal policy. Section 5 uses numerical examples to compare the two models. Conclusions and further research directions are presented in Section 6. Assumptions and notation The following assumptions are used throughout this paper to deelop the models proposed: 1. The supply chain consists of one endor supplying a single product to one uyer, i.e. it forms a ilateral monopoly. 5

7 . The uyer faces a deterministic consumer demand rate D(I) which is an increasing function of the stock on display I. It has the polynomial form (see e.g., Baker and Uran, 1988, Balakrishnan et al., 004, and Zhou et al., 008): D(I)=αI β, where α>0 and 0<β<1 are the scale and the shape parameters, respectiely. The shape parameter, β, reflects the elasticity of the demand rate with respect to the stock leel on display. 3. There is a limited capacity C d of the display area, i.e. I C d.. This limitation could e interpreted as a gien shelf space allocated to the product. 4. The endor has a finite production rate P which is greater than the maximum possile demand rate, i.e. P >. β αc d 5. At each setup the endor manufactures a production atch n Q, where n is an integer and the size of each shipment to the uyer is Q. 6. Inentory at the uyer s warehouse is continuously reiewed. The uyer orders a lot of size Q when the inentory leel reaches the reorder point. The items are transferred from the warehouse to the display area in n equal lots of size q until the inentory leel in the warehouse falls to zero and a new lot of size Q is deliered. Hence, Q=n q, where n is an integer. 7. Shortages are not permitted, and transfers to the display area occur as the inentory on display reaches the leel zero. This is the so-called run-out replenishment policy for stock-dependent demand models (see Balakrishnan et al., 004). It is not the optimal policy in general, ecause some demand may e lost when the inentory on display is low. The run-out replenishment policy is used for simplicity in this multi-stage model. 8. Both the lead time etween the endor and the uyer, and the lead time etween the uyer s warehouse and the display area are constant. Howeer, as demand is 6

8 deterministic, we may assume (without loss of generality) that oth lead times are zero. 9. Inentory holding costs increase downstream in the supply chain, ut they are not related directly to the unit prices. 10. The time horizon is infinite. A graphical representation of the inentory leels at the endor, as well as at the two stocking points at the uyer is presented in Figure 1. The cost parameters are: A Vendor s setup cost Figure A Buyer s fixed ordering cost including any fixed deliery cost for each atch shipped to the uyer S c Buyer s fixed cost for transfer from the warehouse to the display area The net unit purchasing price (charged y the endor to the uyer, ut net of the endor s, i.e. the supply chain s, acquisition costs) δ The net unit selling price (charged y the uyer to the consumer, ut net of the endor s, i.e. the supply chain s, acquisition costs) h Inentory holding cost per unit per unit time at the endor stage h w Inentory holding cost per unit per unit time at the uyer s warehouse, h w >h. h d Inentory holding cost per unit per unit time at the uyer s display area, h d >h w. We note that the simple cost structure suggested here is also in accordance with the theoretical implications for inentory control concluded in Curşeu et al. (009) from their empirical study of handling operations in grocery retail stores. 7

9 3 Non-coordinated supply chain For comparatie purposes, we first otain the independently optimal policies for the endor and the uyer, respectiely. In this non-coordinated case, each supply chain memer tries to maximize its own profit. The result is then compared to the coordinated system (Section 4), where the two parties cooperate and/or are controlled centrally. 3.1 Buyer s optimal policy The ojectie of the uyer is to maximize its own profit. The elements of the uyer s profit are as follows: the reenue from selling the product, the fixed cost of ordering from the endor, the holding cost at the warehouse, the holding cost at the display area, the fixed cost of transfer from the warehouse to the display area, and the ariale purchasing cost. The demand rate at time t is equal to the decrease in the inentory leel at that time. Therefore, the inentory dynamics I(t) are descried y the differential equation di() t β = αit (), 0 t Td, dt where T d is the cycle time defined in Figure 1. Thus, the (on-hand) inentory at time t can e otained y soling I(t) -β di(t)=-αdt. By integrating oth sides, we hae t Hence, β It () dit () = αdt. 0 0 t It I t 1 β 1 β () (0) = α (1 β ). As I(0)=q, we get 1 1 β [ α (1 β ) t + ] β I ( t) = q 1. Sustituting I(T d )=0 into the aoe expression, we get T d ased on the transfer quantity q as T d =q 1-β /[α(1-β)]. 8

10 The uyer s total cost TC w at the warehouse is otained as TC w A hw n( n 1) qtd = +, T T w w where T w is the warehouse cycle time specified in Figure 1. Sustituting T w =n T d and T d =q 1- β /[α(1-β)] into the aoe expression and simplifying, the total cost per unit time at the warehouse is TC w A α (1 β ) h ( n 1) q = +. (1) w 1 nq β The uyer s total cost TC d at the display area consists of the fixed cost of transfer from the warehouse to the display area, and the holding cost at the display area. Therefore, TC d β S h T d d S h d q = + I() t d. 0 t Td T = + d Td Td α( β) Sustituting T d =q 1-β /[α(1-β)] into the aoe expression and simplifying, the uyer s total cost per unit time at the display area is TC d Sα(1 β ) h = + 1 q β d (1 β ) q. β Total net reenue per unit time is TR=(δ-c)q/T d =(δ-c)α(1-β)q β, and the uyer s total profit is TP =TR-TC w - TC d. Thus, () β α(1 β)( A n + S) hw( n 1) hd(1 β) TP( q, n) = ( δ c) α(1 β) q q. 1 β + q β (3) Taking the second partial deriatie of TP (q,n ) with respect to q, we get TP( q, n) = α β δ β + β + < q β 1 (1 ) q ( c) ( )( A n S) q 0. Hence, TP (q,n ) is concae in the transfer quantity q for a gien alue of n. Howeer, there is no closed-form solution for the optimal q. Therefore, we employ a one-dimensional search algorithm to find its optimal alue. Moreoer, assume (temporarily) that n is a continuous ariale. Taking the second 9

11 partial deriatie of TP (q,n ) with respect to n, we otain TP ( q, n ) α(1 β) A = < n q β n Thus, TP (q,n ) is also concae in n for a gien alue of q. Taking the first partial deriatie of TP (q,n ) with respect to n, and equalizing it to zero, we hae TP ( q, n ) α(1 β ) A h q = = 0, w 1 n q β n from which we otain n α (1 β ) A =. β hwq As expected, there is an inerse relation etween n and q: as the transfer quantity decreases, the numer of shipments increases. Although theoretically the transfer quantity is only assumed to e greater than zero, for practical purposes we can assume that it is not less than one. In other words, there is a finite smallest unit for the product which the transfer is ased upon. The upper ound for the optimal numer of transfers can then e otained y considering q=1 as the smallest possile quantity. Hence, max α(1 β) A n =, (4) hw where the upper part rackets indicate rounding up to the nearest integer. The lower ound for the optimal numer of transfers can e otained y considering q=c d as the largest possile transfer quantity. Hence, n min α(1 β) A = max,1, hc β w d where the lower part rackets indicate rounding down to the nearest integer. The ounds in (4) and (5) are used in the solution algorithm elow for finding the uyer s optimal inentory policy. (5) Solution algorithm for uyer s prolem 10

12 The optimal solution to the uyer s prolem can e otained y using the following algorithmic steps. Step 1. Initialize y computing n max and min n using Equations (4) and (5), respectiely. Set opt n = and TP = 0. min n Step. Find the alue of q, employing a one-dimensional search algorithm on Equation (3), so that TP (q,n ) is maximized for the gien alue of n. Step 3. If q>c d, then q=c d. Step 4. Compute TP (q,n ) using Equation (3). Step 5. If opt TP ( q, n ) > TP, then set ( opt TP = TP q, n ), n = n, and q opt =q. opt Step 6. Increment n y 1. If gloally optimal. max n n, then go to Step. Otherwise, the current solution is Eidently, this algorithm terminates with the optimal solution in a finite numer of iterations. 3. Vendor s optimal policy When the transfer quantity and the numer of transfers hae een decided y the uyer, orders are receied y the endor at known interals T w (see Figure 1). The endor s aerage inentory leel is then otained as 1 Q ( n 1) n Q TQ I = nq( + T) ( ( n 1)) T P n P n Q nq nq = ( n 1)(1 ) +, TP TP where the cycle time T is specified as in Figure 1. The endor s total profit per unit time can now e expressed as cnq A Q nq nq TP( n) = h ( n 1)(1 ) +. T T TP TP (6) 11

13 Sustituting T = n n T d, T d = q 1-β /[α(1-β)], and Q= n q into expression (6) and simplifying, we otain the total profit per unit time for the endor as β β α(1 β) A nq ( n) α(1 β) q TP( n) = cα(1 β) q h ( 1). 1 n β + nnq P (7) Taking the first and second deriaties of TP (n ) with respect to n (assumed temporarily to e continuous), we otain β dtp ( n ) α(1 β) A n q α(1 β) q = h (1 ), dn n q n P 1 β dtp( n) α(1 β) A 1 3 dn nq β n = < 0. Hence, TP (n ) is concae in n. Therefore, the following optimality conditions can e otained for n β α(1 β) APq n( n 1) n ( 1). n + β hnq ( P αq (1 β)) (8) * * In the non-coordinated supply chain, the uyer chooses its own optimal policy ( q, ), * and the endor then chooses its optimal numer of shipments n. Thus, total system profit per * * * * * * unit time is otained as TP ( q, n, n ) = TP ( q, n ) + TP ( n ). N n 4 Coordinated supply chain In this section, we consider the situation in which the two parties in the supply chain either cooperate fully or are centrally directed to follow the jointly optimal policy deried y maximizing the total profit TP C (q, n, n ). The optimal policy of this coordinated system is found y soling the following prolem: 1

14 [ A n n A n S] β α(1 β) ( ) + + Maximize TPC( q, n, n) = δα (1 β ) q qn,, 1 n β q hw( n 1) + hn( n 1) hd(1 β) hn( n) α(1 β) q + q β P st.. 0< q C n, n integer. d 1+ β (9) Taking the second partial deriatie of TP C (q,n,n ) with respect to q, we otain TPC( q, n, n) = δαβ (1 β ) q α (1 β ) ( β ) [ A ( n n ) + A n + S] q q hn ( n) αβ (1 β ) q P β β 3 β 1. It cannot e concluded that TP C (q, n, n )/ q is necessarily negatie. One of two cases may occur depending on the numer of shipments n : Case 1: n All three terms in TP C (q, n, n )/ q are negatie, and therefore total system profit is concae in q for known alues of n and n. Howeer, there is no closed-form solution for the transfer quantity q. We then employ a one-dimensional search algorithm to find its optimal alue. Case : n > The first two terms in TP C (q, n, n )/ q are negatie. Howeer, the third term is positie. Rewriting TP C (q, n, n )/ q, we hae TP ( q, n, n ) = q C β 3 ( ϑ3q ϑ1q ϑ ), q whereϑ δαβ ( 1 β =, ϑ = α(1 β ) ( )[ A ( n n ) + A n S] 1 ) ϑ = h n ( n ) αβ (1 β ) ( ). 3 P β +, and 13

15 As can e seen directly, ϑ 1, ϑ, and ϑ 3 are all positie. Setting the second partial deriatie of TP C (q, n, n ) with respect to q equal to zero, and soling, we otain the two saddle points ϑ1 ϑ1 + 4ϑϑ 3 q1 = ϑ 3, ϑ1+ ϑ1 + 4ϑϑ 3 q = ϑ 3. Therefore < 0 if q 1 < q < q, TPC( q, n, n) q > 0 otherwise. Thus, the total profit function is concae etween the two saddle points, and conex when q q 1 or q q. Moreoer, as ϑ 1 + 4ϑ ϑ3 > ϑ and ϑ > 1 3 0, it follows that q 1<0 and q >0. Therefore, the optimal transfer quantity for any gien alues of n and n is the smallest of the local optimum point, LO, and the maximum capacity of the display area, C d (see Figure ). As there is no closed form solution for the local optimum point, we employ a onedimensional search algorithm to find its alue Figure Taking the second partial deriaties of TP C (q, n, n ) with respect to n and n (relaxing temporarily the integrality rquirements), we otain TP ( q, n, n ) α(1 β) A = < 0, C 3 1 n nnq β [ A A n ] TP ( q, n, n ) α(1 β) + = < 0. C 3 1 n nq β Hence, TP C (q, n, n ) is concae in n for gien alues of q and n, and concae in n for 14

16 gien alues of q and n. Taking the first partial deriatie of TP C (q, n, n ) with respect to n, and setting it equal to zero, we otain n α(1 β ) A =, β β hnq 1 q α(1 β) P which shows that there is a negatie relation etween n and the optimal n. Thus, the maximal optimum alue of n for a gien q is otained when n =1. In order to find the maximal optimum alue of n, we also need to find the minimum of β ( 1 q α(1 β P) β γ = q ) saddle point q to e. Taking the second deriatie of γ with respect to q, we find the ( β ) P q = α 1 β, where γ is conex when q q, and concae when q>q. Taking the first deriatie of γ with respect to q, and equalizing it to zero, we otain the local minimum and maximum as q Local min = 0, q Local max ( β ) P = α(1 β) 1 β. β β Because P > αc d αq, γ is a positie function. The minimum of γ is otained at q=0 or q=c d. Howeer, the minimum transfer quantity is 1. Therefore, the minimum of γ will e β β { P Cd Cd P } γmin = min 1 α(1 β), (1 α(1 β) ). Consequently, we hae n max α(1 β) A =. hγ min (10) Similarly, taking the first partial deriatie of TP C (q,n,n ) with respect to n, and equalizing it to zero, we hae 15

17 n α(1 β )( A + A n) =. hq β 1 q β α(1 β) P n + ( hw h) q β + hα(1 β) q P β β As shown aoe, γ = q ( 1 q α(1 β ) P) is positie y assumption. Hence, there is also a negatie relation etween the optimal n and any gien n. The maximal optimum alue of n is then otained for n =1. For n =1, the equation aoe reduces to n α(1 β )( A + A ) = 1 =. hq h q P ( n ) w β + α(1 β) Note, that there is also a negatie relation etween the optimal n and any gien q. The maximal optimum alue of n is then otained at q=1. Consequently, we hae max α(1 β)( A + A) n =. hw + hα(1 β) P (11) Solution algorithm for joint optimum We hae specified the following algorithm for determining the jointly optimal alues of the three decision ariales n, n, and q in the coordinated model. The algorithm is similar in spirit to the algorithm specified in Section 3. opt Step 1. Initialize y setting n =1, n =1, and TP = 0. Compute n max and C max n using Equations (10) and (11). Step. Find the alue of q, employing a one-dimensional search algorithm (in Case 1 or Case aoe), such that TP C (q, n, n ) is maximized for the gien alues of n and n. Step 3. If q>c d, then set q=c d. Step 4. Calculate TP C (q, n, n ) using Equation (9). opt opt opt Step 5. If TP >, then set TP = TP q, n, n ), n = n, n = n, and q opt =q. opt C TP C Step 6. Increment n y 1. If C C ( max n n, then go to Step. 16

18 Step 7. Increment n y 1. If max n n, then set n =1, and go to Step. Otherwise, the current solution is gloally optimal. Oiously, after a finite numer of iterations, the algorithm terminates with the optimal solution. 5 Numerical study We first consider a ase case with the following data: α=100, β=0., P=4500/year, S=$5/transfer, A =$100/order, A =$400/setup, h d =$0/unit/year, h w =$5/unit/year, h =$4/unit/year, δ=$30/unit, c=$0/unit, and C d =500. Most of these parameter alues hae een collected from earlier studies of JELS prolems. In order to analyze the effect of stockdependent demand on the enefits of supply chain coordination, we also specify three and twele leels for the parameters α and β, respectiely. Specifically, α [75,100,15], and β [0.00,0.05,..., 0.55] are used. To represent gains from using coordinated ersus noncoordinated policies, we define the percentage gain as PG=100 (TP C -TP N )/TP N. In Tale 1, for different alues of the demand parameters α and β, we compare the optimal alues of the decision ariales in the coordinated and non-coordinated supply chain. The uyer s optimal order quantity Q=n q is always smaller in the non-coordinated case compared to the coordinated case. Howeer, there are no such general relations etween the optimal transfer quantities q, or etween the endor s optimal production atches n Q in the coordinated and non-coordinated cases. Moreoer, as β increases, the percentage gain PG in the last column of Tale 1 grows initially and susequently declines,. This is due to the fact that as demand ecomes more sensitie to the stock leel in the display area, coordination ecomes relatiely more profitale. Howeer, ecause of the limited capacity of the display area, the transfer quantity 17

19 cannot increase further for alues of β aoe a certain leel. The decision ariales then remain the same. Consequently, the percentage gains start decreasing and eentually anish. Another conclusion that can e drawn from Tale 1 is that for most of the cases, the enefits of coordination decrease y α for a fixed alue of β. A few exceptions to this general effect are related to the discrete nature of the decision ariales n and n. Thus, as the demand rate α increases, the difference etween the coordinated and the non-coordinated supply chain profits generally ecomes smaller (assuming no changes in the demand sensitiity) Tale We also analyze the effect of the unit holding cost in the display area, h d, on the enefits of coordination. In order to concentrate on this effect, the capacity of the display area is then assumed not to e inding in the solution, i.e. C d > M, where M is a sufficiently large numer. From the results in Tale, we conclude that the percentage gain does not exhiit strong sensitiity with respect to changes in h d. Decreasing the unit holding cost to one fourth of its initial alue still leaes a percentage gain of more than 13% from coordination compared to the initial gains of approximately 17%. There is no monotone relationship, howeer, etween the percentage gain and the unit holding cost. The reason appears to e related to the simultaneous discrete changes in the integer multiples n and n Tale Finally, we examine the effect on the gains of coordination of the unit (net) purchasing price c paid y the uyer to the endor. For this purpose, we use different leels of the purchasing price c expressed y the ratio c/δ (for a gien alue of δ). Specifically, we consider c δ [ 40%, 35%,...,0%,...,100%]. Howeer, the purchasing price paid is only an 18

20 internal transfer from one supply chain memer (the uyer) to another (the endor). Hence, it is not a cost for the supply chain as a whole. Therefore, the total system profit does not change with the unit purchasing (or transfer) price c when the control of the supply chain is coordinated. On the other hand, as c/δ increases, the uyer s marginal profit from selling the product δ-c decreases. The optimal transfer quantity from the warehouse to the display area then ecomes smaller if the uyer tries to maximize its own profit. Therefore, the numer of products in the display area is reduced, and as a result, demand as well as total system profit in the non-coordinated case decreases. Tale 3 shows how the asolute difference D = TP C TP N and the relatie difference PG etween total system profits under coordinated and non-coordinated regimes ary y the ratio c/δ. The non-coordinated supply chain s performance is exactly the same as in the coordinated case when the ratio c/δ is approximately -5% (more precisely: -4.%). In this case, the uyer s optimal order quantity Q = n q in the non-coordinated supply chain equals the quantity in the coordinated case. Hence, the net unit transfer price c could e employed as a coordination tool in a supply chain dyad with stock-dependent demand. If the two parties set their production, shipment and ordering policies ased on a transfer price c < 0, then the supply chain without further sharing of information performs similarly to a coordinated supply chain. Howeer, coordination of decisions generally also requires a mechanism for sharing of the total profit improements, as discussed further elow. In principle, full coordination, if costless, can always e made eneficial for oth supply chain parties. For reasons such as costly information or strategic considerations, the parties may not e inclined to share detailed information with each other. Using the precise alue of the net unit purchasing price that achiees perfect coordination etween the supply chain memers is then not the first-est solution either. It requires information sharing aout costs 19

21 and other parameters similarly to what is required in order to otain the full coordination solution directly. Howeer, ecause oth PG and its sensitiity with respect to ariations of the ratio c/δ are quite low around zero, we suggest that y choosing c=0 as the transfer price, it will still e possile to capture most of the potential gains of supply chain coordination in a case such as this with stock-dependent demand. Recall that c is specified as a net unit purchasing price. Therefore, using c=0 corresponds to a unit transfer price that equals the endor s unit acquisition cost. It might e possile to erify this cost with reasonale accuracy and relatiely little effort. The adantage of this arrangement is that the uyer and the endor can otain some coordination without further costly sharing and erification of information. Tale 3 also shows that the distriution of profit etween the endor and the uyer depends strongly on the ratio c/δ. In particular, the profit for the uyer and for the endor is negatie for large and small unit purchasing prices, respectiely. Different alues of the transfer price c (normalized as c/δ) were used to otain different solutions and therefore also different total profit alues in the non-coordinated case. Howeer, changing the transfer price not only changes the total profit ut also transfers profit etween the supply chain memers. A relatiely higher c oiously transfers profit to the endor and ice ersa. In order to otain final profit leels that are acceptale to oth parties and that encourage cooperation, some kind of profit-sharing mechanism needs to e employed as a supplement. For example, a side payment might e agreed upon etween the uyer and the endor in order to share the total profits, in particular the gains otained y a more coordinated solution. Again, howeer, reaching a conceded profit sharing agreement may require a fair amount of information sharing to motiate the agreement. A simple way that has een proposed to allocate the joint total profit etween the uyer 0

22 and the endor is the following (see Ouyang et al., 004; Wu and Ouyang, 003; and Goyal, 1976) TP ( n ) TP C = TP ( q, n, n N TP ) TP ( q, n ) TP C = TPC TP ( q, n, n ) N C ( q, n ( q, n, n, n ) ) where TP C and TP C are the resulting final profits for the endor and the uyer under a coordinated supply chain, respectiely. With this arrangement the endor and the uyer presere their relatie shares of the total profits. Howeer, there are also seeral other possile arrangements, for example ased on game-theoretic approaches. To conclude this discussion, it should e osered that if β=0, then TC N and therefore D and PG do not change with c. Thus, if there is no relation etween the stock in the display area and the consumer demand, then the net unit purchasing price c will hae no effect on coordination in the supply-chain dyad. This demonstrates that it is the stock-dependent demand that creates the doule marginalization effect discussed aoe. Spengler (1950) was one of the first to identify the prolem of doule marginalization; there is coordination failure in a supply chain dyad if each memer of the dyad only considers its own net profit margin when making decisions aout prices and quantities that affect total profits. Stock-dependent demand and its effect on the uyer s choice of inentory policy correspond to what Jeuland and Shugan (1983) referred to as a point-of-sale effort other than price adjustments in their generic analysis of the doule marginalization effect for a ilateral monopoly. In the model treated in this paper, the doule marginalization effect comines with the coordination prolem caused y the production- and inentory-related setup, ordering, transfer, and holding costs also included in the model. In such a case, as shown y Weng (1995), a simple 1

23 discount scheme is generally not sufficient to coordinate the supply chain dyad. As confirmed aoe in the numerical analysis with our models, a profit sharing mechanism, such as a side payment mechanism, is then required for coordination of independent parties to e otained. We refer to Chapters 6-7 in Graes and de Kok (003) for further discussions aout this issue Tale Conclusions This paper deals with deeloping an integrated production-inentory model for a twostage supply chain. The contriution of the paper to the joint economic lot-sizing literature is to add the stock-dependency of demand to existing integrated endor-uyer models. Hence, a more general model is estalished in which we assume that demand is not constant ut sensitie to the amount of inentory displayed on the retailer s sheles. Stock-dependency also incurs a doule marginalization effect in a non-coordinated supply chain dyad. Therey, our study proides a link etween the literature on distriution channel coordination and the joint economic lot-sizing literature. The main findings can e summarized as follows. Taking the relationship etween demand and stock into consideration, we find a further adantage of supply chain coordination in the form of an increase in the total selling amount (the demand). When oth parties cooperate or are coordinated, the optimal amount of the product in the display area is higher than in the non-coordinated case when the parties optimize indiidually. As a result, end customer demand, and consequently total system profit, is higher for the coordinated supply chain. Assuming that the capacity limitation of the display area is not inding, the numerical results show a strong relation etween the improements otained with coordination and the stock-sensitiity of demand. This implies

24 that it is particularly eneficial for supply chain parties to cooperate in cases where consumer demand is increased as more of the product is displayed on the retailer s sheles. The transfer price agreed etween the supply chain parties is also found to e a useful tool for partially coordinating the supply chain with only limited sharing of information. This is due to the presence of a doule marginalization prolem in the supply chain model. The endor and the uyer can improe the coordination of their production and inentory decision ariales y appropriately choosing the transfer price. Howeer, in addition they also need to agree on a suitale profit sharing mechanism in order to allocate the net enefits that can e otained from the improed coordination. It may require additional information sharing in order to reach such an agreement. A future research direction might e to extend this study to other product shipment policies, e.g., the geometric shipment policy. It might also e useful to compare coordinated and non-coordinated supply chains under other types of functions for the stock-dependent demand, as well as under stochastic demand settings. In our model we assume that the inentory on the shelf/in the display area is zero efore it is replenished. Relaxing this run-out replenishment policy is also an interesting suggestion for future research. Acknowledgement The first two authors work has een partly supported y grant no from the Danish Social Science Research Council. The authors would like to thank two anonymous referees for their aluale comments and helpful suggestions on improing the quality of the paper. References Aott, H. and Palekar, U.S., 008. Retail replenishment models with display-space elastic demand. European Journal of Operational Research, 186,

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28 Tale Captions Tale 1. Decision ariales of non-coordinated ersus coordinated optimization Tale. Models sensitiity to the unit holding cost at the display area Tale 3. Effect of net purchasing unit price on non-coordinated and coordinated optimization Figure Captions Figure 1. Inentory leels at the three stocking points of the supply chain Figure. Total profit function for different alues of q if n > 7

29 Tale 1 Tale 1. Decision ariales of non-coordinated ersus coordinated optimization Parameters Non-coordinated optimization Coordinated optimization α β q n n TP N q n n TP C TP C TP N PG

30 Tale Tale. Models sensitiity to the unit holding cost at the display area Non-coordinated optimization Coordinated optimization PG h d q n n TP TP TP N q n n TP C TP C TP C (%)

31 Tale 3 Tale 3. Effect of net purchasing unit price on non-coordinated and coordinated optimization Coordinated optimization q n n TP C Parameters Non-coordinated optimization δ c/δ q n n TP TP TP N TP C TP N PG 30-40% % % % % % % % % % % % % % % % % % % % % % % % % % % % %

32 Figure 1 Inentory leel P Vendor s inentory Q T =n T w =n n T d Time Buyer s warehouse Q=n q T w =n T d Time Buyer s display area q T d Time Figure 1. Inentory leels at the three stocking points of the supply chain 31

33 Figure Figure. Total profit function for different alues of q if n > 3

34 Working Papers from CORAL Centre for Operations Research Applications in Logistics L L L L L L Mohsen S. Sajadieh, Anders Thorstenson & Mohammad R. Akari Jokar: An integrated endor-uyer model with stock-dependent demand. Ellis L. Johnson & Sanne Wøhlk: Soling the Capacitated Arc Routing Prolem with Time Windows using Column Generation. Søren Bloch & Christian H. Christiansen: Simultaneously Optimizing Storage Location Assignment at Forward Area and Resere Area a Decomposition Based Heuristic. Bisheng Du & Christian Larsen: Base stock policies with degraded serice to larger orders. Karina H. Kjeldsen: Classification of routing and scheduling prolems in liner shipping. Daniele Pretolani, Lars Relund Nielsen, Kim Allan Andersen & Matthias Ehrgott: Time-adaptie ersus history-adaptie strategies for multicriterion routing in stochastic time-dependent networks. L Christian Larsen: Comments to Özkaya et al. (006). L L L Christian Larsen: Deriation of confidence interals of serice measures in a ase-stock inentory control system with low-frequent demand. Jens Lysgaard: The Pyramidal Capacitated Vehicle Routing Prolem. Jens Lysgaard & Janni Løer: Scheduling participants of Assessment Centres. L Christian Larsen: Note: Comments on the paper y Rosling (00). L L L L Christian Larsen, Claus Hoe Seiding, Christian Teller & Anders Thorstenson: An inentory control project in a major Danish company using compound renewal demand models. Christian Larsen: The Q(s,S) control policy for the joint replenishment prolem extended to the case of correlation among item-demands. Daniele Pretolani, Lars Relund Nielsen & Kim Allan Andersen: A note on Multicriteria adaptie paths in stochastic, time-arying networks. Lars Relund Nielsen, Kim Allan Andersen & Daniele Pretolani: Bicriterion a priori route choice in stochastic time-dependent networks.

35 L L L L L L L L L L L L L L Christian Larsen & Gudrun P. Kiesmüller: Deeloping a closed-form cost expression for an (R,s,nQ) policy where the demand process is compound generalized Erlang. Eduardo Uchoa, Ricardo Fukasawa, Jens Lysgaard, Artur Pessoa, Marcus Poggi de Aragão, Diogo Andrade: Roust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Prolem oer a Large Extended Formulation. Geir Brønmo, Bjørn Nygreen & Jens Lysgaard: Column generation approaches to ship scheduling with flexile cargo sizes. Adam N. Letchford, Jens Lysgaard & Richard W. Eglese: A Branch-and- Cut Algorithm for the Capacitated Open Vehicle Routing Prolem. Ole Mortensen & Olga W. Lemoine: Business integration etween manufacturing and transport-logistics firms. Christian H. Christiansen & Jens Lysgaard: A column generation approach to the capacitated ehicle routing prolem with stochastic demands. Christian Larsen: Computation of order and olume fill rates for a ase stock inentory control system with heterogeneous demand to inestigate which customer class gets the est serice. Søren Glud Johansen & Anders Thorstenson: Note: Optimal ase-stock policy for the inentory system with periodic reiew, ackorders and sequential lead times. Christian Larsen & Anders Thorstenson: A comparison etween the order and the olume fill rates for a ase-stock inentory control system under a compound renewal demand process. Michael M. Sørensen: Polyhedral computations for the simple graph partitioning prolem. Ole Mortensen: Transportkoncepter og IT-støtte: et undersøgelsesoplæg og nogle foreløige resultater. Lars Relund Nielsen, Daniele Pretolani & Kim Allan Andersen: K shortest paths in stochastic time-dependent networks. Lars Relund Nielsen, Daniele Pretolani & Kim Allan Andersen: Finding the K shortest hyperpaths using reoptimization. Søren Glud Johansen & Anders Thorstenson: The (r,q) policy for the lostsales inentory system when more than one order may e outstanding.

36 L L Erland Hejn Nielsen: Streams of eents and performance of queuing systems: The asic anatomy of arrial/departure processes, when focus is set on autocorrelation. Jens Lysgaard: Reachaility cuts for the ehicle routing prolem with time windows.

37 ISBN Department of Business Studies Aarhus School of Business Aarhus Uniersity Fuglesangs Allé 4 DK-810 Aarhus V - Denmark Tel Fax