A One-Vendor Multiple-Buyer Production-Distribution System: Vendor-Managed Inventory and Retailer-Managed Inventory

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1 A One-Vendor Multiple-Buyer Production-Distribution System: Vendor-Managed Inventory and Retailer-Managed Inventory Xiao Huang Department of Decision Sciences and MIS, Concordia University, Montreal, Quebec, Canada H3G 1M8, Yunfan Li China Academy of elecommunication Research, Ministry of Industry and Information echnology of China, Beijing, China , he paper considers a production/distribution system that involves one vendor and multiple buyers. he buyers belong to an entity that is operationally more cost efficient than the vendor. We study two settings that have been widely applied in practice: vendor-managed inventory (VMI) and retailer-managed inventory (RMI). Specifically for the two-buyer problem, we characterize equilibrium under RMI and optimal solution under VMI. Our VMI solution is shown to be more cost efficient than others in literature. he general multiple-buyer problem can be non-tractibal and a heuristic is provided to approach decisions under both settings. One surprising finding is that under VMI, the production decision need not involve operational information of the buyers (e.g., inventory/logistic costs) but only their market information (e.g., demand). Numerical results suggest that VMI and just-in-time production are substitutes in improving the efficiency of the distribution network. VMI can be more critical for companies with rigid capacity, seasonally low demand, or complex distribution networks. Key words : production; inventory; vendor managed inventory; supply chain; game theory History : his draft: Jul Introduction In recent decades, vendor-managed inventory (VMI), whereby manufacturers/vendors store unsold items at the buyers locations and take the responsibility of monitoring inventory levels and replenishing stocks, received great attention in both practice and academia. While there is a vast body of literature concerning VMI from many different aspects (Marquès et al. (010)), our paper is most 1

2 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System relevant to those modeling operational decisions in supply chains. his stream of literature begins with the study of a one-vendor one-buyer supply chain. Early papers focused on joint economic lot size. Goyal (1977) and Banerjee (1986) are among the firsts that examine an integrated production-inventory system with infinite and finite production rate respectively. Improved production/inventory schedules are proposed in many subsequent works such as Goyal (1988), Hill (1997) and Hill (1999). Later literature also takes into account of other realistic factors such quantity discount (Lal and Staelin (1984), Rosenblatt and Lee (1985), Munson and Rosenblatt (001)), stochastic demand and lead time (Ben-Daya and Hariga (004)), capacity constraint (Goyal (000)), shortage (Zhou and Wang (007)), pricing (Boyaci and Gallego (00)), negative echelon holding cost (Hill and Omar (006)). he value and implementation of VMI, however, were not highlighted until recently. Particularly for the one-vendor one-buyer system, Dong and Xu (00) analyze the channel benefit of VMI; Braglia and Zavanella (003) consider implementing VMI through consignment stocks and Valentini and Zavanella (003) provide more managerial insights on such contracts through case studies. Gümüs et al. (008) compare the benefit VMI where vendor makes all ordering decisions, versus consignment inventory where buyer places orders but the vendor bears a portion of the inventory holding costs. Nagarajan and Rajagopalan (008) investigate the value of holding-cost-subsidy contract in VMI where vendor makes ordering decisions, as opposed to retailer-managed inventory where buyer makes ordering decisions. In coping with today s complex supply chain, it is paramount to understand extended systems comprising multiple buyers. Compared with the single-vendor-single buyer system, the one-vendor multi-buyer problems have been known non-trivial. Banerjee and Burton (1994) show that the optimal policy in a multi-buyer scenario can be quite different than with single buyer. Lu (1995) considers a one-vendor-multi-buyer setting in which each buyer procures a distinct type of product, and provides heuristics to solve the integrated inventory problem. Zavanella and Zanoni (009) study a one-vendor-multi-buyer system with one common product, aiming to derive optimal solution for the integrated system as well as equilibrium solution in a decentralized system. It has been

3 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 3 noted that this analysis only applies to consecutive delivery patterns (Zavanella and Zanoni (010)) among multiple buyers, which is shown less superior than rotational delivery patterns in Ben-Daya et al. (01). Further, Goyal et al. (01) note that the general one-vendor-multi-buyer problem (particularly the characterization of total cost function) has not been fully explored in Zavanella and Zanoni (009) or Ben-Daya et al. (01), thus deserving more in-depth study. We are therefore motivated to continue this line of research by studying one-vendor-multi-buyer problems under a similar framework of Zavanella and Zanoni (009) with rotational delivery patterns. In this paper, we consider a vendor selling to multiple heterogeneous buyers. he buyers are owned by one entity, thus representing the distribution system of a large enterprise, e.g., Walmart, arget, Home Depot and many auto dealers. he vendor is a specialist (e.g., a just-in-time manufacturer) who incurs higher inventory holding cost than any of the buyers. It is then more expensive to keep the inventory at the vendor than at the buyers site, which gives incentive for the vendor to shift the inventories downstream. From this aspect, unlike many papers (e.g., Lu (1995), Hill (1997)) which assume that the buyers holding costs are higher than the vendor/manufacturer s (e.g., wholesalers are usually more cost efficient than retailers), the scenario of this problem is closer to those considered in Valentini and Zavanella (003), Hill and Omar (006), Nagarajan and Rajagopalan (008), Zavanella and Zanoni (009), etc. We next describe the model in and derive the total cost expression for the general multi-buyer problem in ts 3. hen we characterize solutions for both VMI and RMI when there are two buyers ( 4). Finally in 5, we provide heuristic solutions for the general Y -buyer problem. Managerial insights are summarized in 6.. Model Consider Y +1 firms a vendor (production rate P ) taking orders from Y buyers (demand rate d i for buyer i) on a continuous-time horizon. he production and demand rates are all deterministic. We index i = 0 for the vendor and i = 1,,..., Y for the buyers. Let denote the production cycle

4 4 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System S 0 n 1 =3,d 1 =1, n =4,d =10, P =3. q 1 q O t S q O q 1 + q P t S 1 q 1 O q 1 P t Figure 1 Inventory levels of the vendor (S 0) and two buyers (S 1, S ): P = 3, d 1 = 1, d = 10, n 1 = 3, n = 4 and n i the number of deliveries to buyer i per cycle. In aligning with relevant literature, we assume that that the system requires there be at least one order per cycle, i.e., n i 1 for i = 1,,..., Y. Orders will be produced and delivered to each buyer at equal-size shipments. In particular, during each cycle the batches of the buyers will be delivered on a rotational basis (e.g., buyer 1, buyer, buyer 3, buyer 1, buyer, buyer 3,...). If the delivery schedule for a buyer is finished for the cycle, then simply skip it and serve the next unfulfilled buyer on the list. he rotational delivery pattern initially appeared in Zavanella and Zanoni (009), and Ben-Daya et al. (01) show its superiority to consecutive delivery pattern (e.g., buyer 1, buyer 1, buyer 1, buyer, buyer, buyer, buyer 3,... as in Zavanella and Zanoni (010)). Figure 1 illustrates how this works in a one-vendor-two-buyer system. he problem then calls for the following decisions to be made (the sequence and decision makers of these variables will be discussed later): Production cycle for the vendor;

5 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 5 Delivery frequency n = (n 1, n,..., n Y ) for the buyers; Delivery pattern ρ P : {1], [],..., [Y ]} {1,,..., Y } where ρ([i]) ρ([j]) if i j. ρ describes the rotational delivery sequence among the Y buyers. he set {[1], [],..., [Y ]} is used to denote the original index of the buyers. hus the rotation list always starts with buyer 1 and ends on Y, or equivalently, buyer [ρ 1 (1)] and [ρ 1 (Y )] respectively. Denote A i and h i the setup cost and holding cost per unit-time per item for firm i. As discussed before, we only consider the case when h 0 h i for i = 1,,..., Y. Let C i represent the total cost in a cycle incurred at firm i s site, and C ave,i = C i / the average total cost at firm i s site. We consider two kinds of systems: Vendor-Managed Inventory (VMI): Under VMI, inventories are under the ownership of the vendor regardless of the storage site. he vendor therefore bears all the costs at each site, and will determine the cycle and delivery frequency n i that minimize total average cost of the system C ave,s = Y i=0 C ave,i. he optimal decisions are denoted as (, n, ρ ). Retailer-Managed Inventory (RMI): Under RMI, the buyers initiate orders on their own and the inventory will be under the ownership of the buyers. Assume that all buyers belong to the same entity. hey will then agree on a delivery frequency (n 1,..., n Y ) and pattern P that minimizes C ave,b = Y C ave,i. On the other hand, the vendor will only focus on determining cycle that minimizes C ave,0. he equilibrium decision will be denoted by (, n, ρ ). All notations are summarized in able otal Cost for the Vendor and the Buyers In this section, we derive the total cost expression at each firm s site, which has not been explicitly characterized in previous literature. We start from a simple one-vendor one-buyer problem (Y = 1). Analysis and notations can then naturally be extended to the multiple-buyer case otal cost in a one-vendor one-buyer problem In this case, the vendor delivers n 1 batches at equal size q 1 = d 1 /n 1 to the solo buyer (indexed 1) during each cycle. he vendor manufactures at rate P and ships a batch to the buyer once it

6 6 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System P Production rate for the vendor d j Demand rate for buyer j, j = 1,,.., Y A 0 Setup cost for the vendor A j Setup cost for buyer j, j = 1,,.., Y h 0 Holding cost for the vendor h j Holding cost for buyer j, j = 1,,.., Y Production cycle for the vendor n j Number of orders that buyer j places in a cycle, j = 1,,.., Y C 0 otal cost for the vendor in a cycle C j otal cost for buyer j in a cycle, j = 1,,.., Y C b = Y j=1 j otal cost for the buyers in a cycle Y C C sys j=0 j otal cost for the entire system in a cycle, j = 1,,.., Y C ave,0 = C 0 / Average total cost for the vendor C ave,j = C j / Average otal cost for buyer j, j = 1,,.., Y C ave,b = Y j=1 j/ Average total cost for the buyers Y C C ave,s j=0 j/ Average total cost for the entire system, j = 1,,.., Y able 1 Notations is done. he no-stockout condition requires that P d 1. Figure illustrates the inventory levels of the vendor and the buyer in this setting. It can be observed that he k th batch arrives at a k = k q 1 P he k th batch is completely sold (and the (k + 1) st batch can start to sell) by t k = a 1 + k q 1 d 1 = q 1 P + k q 1 d 1 It can be observed that batches are generally delivered before the last one is completely sold, and the inventory of the buyer piles up during the course of the cycle. For example, the first batch is not completely gone until t 1, yet the second batch is delivered at a < t 1. hus the second batch is idle and untouched during [a, t 1 ]. Also the amount of idle time is extended for later batches (i.e., t 1 a < t a 3 < t 3 a 4 ). herefore for any batch, the inventory holding cost consists of two parts: 1. Holding Cost due to Economies of Scale: q 1 (t k t k 1 )h 1. Holding Cost due to Idle Batch: h 1 q 1 W 1,k where k > 1 and W 1,k = t k 1 a k = (k 1)( q 1 d 1 q 1 P ) is the idle time for buyer 1 s k th batch.

7 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 7 S 0 q 1 q 1 = d1 n 1 a i = i q1 P O a 1 a a 3 a 4 t S 1 t i = a 1 + qi d i q 1 O a 1 a t 1 a 3 t a 4 t 3 t 4 t Figure Inventory level with one vendor and one buyer he total cost for the buyer in one cycle is then the sum of the set-up cost as well as two kinds of holding costs: where C 1 = h 1d 1 n 1 C 1 = A 1 n 1 + cumulative batch idle time. k=1 q 1 (t k t k 1 )h 1 + = A 1 n 1 + C 1 + C 1W 1 n 1 n 1 n 1 k= n 1 n 1 and W 1 = W 1,k = (k 1) k= k= h 1 q 1 W 1,k = A 1 n 1 + h 1 q 1 + h 1q 1 W 1 ( q1 q ) 1 d 1 P he total cycle cost for the vendor can be readily derived as = n 1(n 1 1) n 1 ( 1 d ) 1 P (1) is the q 1 d 1 C 0 = A 0 + h 0 P = A 0 + h 0d 1 n 1 P () 3.. otal cost in the general one-vendor multi-buyer problem We now extend our analysis to the general Y -buyer case. Figure 3 illustrates the change of inventory levels where there are three buyers.

8 8 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System S 0 n 1 =3,n =4,n 3 =4. q 1 q 3 S 3 O t q 3 O q 1 + q + q 3 P t S q O q 1 + q P t S 1 q 1 O q 1 P t Figure 3 Inventory levels with one vendor and three buyers In general, the equal-batch size for buyer j is q j = d j /n j, for j = 1,,..., Y (3) he no-stockout condition in this case is a little bit different than that with one buyer, in which we only ask for the supply to cover demand, i.e., P d 1. For multiple buyers, the vendor produces and delivers batches of each buyer on a rotational basis. For any buyer j, after getting its k th order, it has to wait another round of production (that produces the a batch for buyer j +1,..., Y, 1,,...j 1, and j itself) before the (k + 1) th order is delivered. herefore, the batch q j should be able to cover buyer j s demand during this period of time ( Y q i/p ). hat is Y q i /P q j /d j j = 1,,..., Y

9 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 9 By (3) the no-stockout condition is practically Y P max {n d i j}. (4) j=1,,...,y n i Specifically when Y = 1, the condition is degenerated to P d 1. We therefore assume (4) for the general problem. Recall that batches of the buyers are delivered on a rotational mode in each cycle, and fulfilled buyers will be skipped. We can therefore derive the following for buyer j: he k th batch arrives at a j,k = j l=1 q l P min{n l, k} + Y j+1 q l P min{n l, k 1} he k th batch gets completely sold (and the (k + 1) th batch can start selling) at t j,k = a j,1 + k q j d j = j l=1 q l P + k q j d j. Similar to that of the one-buyer case, the total cost for buyer j s k th batch consists of two parts: 1. Holding Cost due to Economies of Scale: q j (t j,k t j,k 1 )h j. Holding Cost due to Idle Batch: h j q j W j,k where k > 1 and W j,k = t j,k 1 a j,k = j l=1 is the idle time for buyer j s k th batch. q l P + (k 1) q j d j he total cost for buyer j in a cycle is then n j C j = A j n j + k=1 = A j n j + C j + C j n j q j (t k t k 1 )h j + n j k= j l=1 q l P min{n l, k} Y j+1 q l P min{n l, k 1} (5) h j q j W j,k = A j n j + h j q j + h jq j W j W j n j (6) where C j = h j d j / and W j = n j k= W j,k is the cumulative batch idle time for buyer j. he total cycle cost for the vendor is C 0 = A 0 + h 0 Y j=1 q j d j P = A 0 + h 0 P Y j=1 d j n j (7)

10 10 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 4. VMI and RMI in the wo-buyer Problem We now consider the problem with exactly two buyers, i.e., Y =. he goal is to derive optimal solution under VMI and the equilibrium for RMI. By (7), the vendor s average cost is C ave,0 = C 0 = A 0 + h ( ) 0 d 1 + d P n 1 n he buyers cost when Y = are given in Goyal et al. (01). We provide a brief proof in the appendix, and conduct optimum/equilibrium analysis based on these cost functions. Lemma 1. otal average cost for the buyers when Y =. (Goyal, Huang and Li 01) C ave,1 = A 1 n 1 A 1 n 1 + h 1d 1 + h 1d 1 h 1d 1 P (n 1 1)( d 1 n 1 + d n ) if n 1 n h 1d 1 n 1 P [(d 1 + d )(n 1 1) d (n n 1 )] if n 1 > n C ave, A n = A n + h d + h d h d n P [ ( d 1 n 1 + d n )n (n 1) d 1 n 1 (n n 1 )(n n 1 + 1) ] if n 1 n h d P (n 1)( d 1 n 1 + d n ) if n 1 > n 4.1. Equilibrium solution in RMI Consider that the vendor chooses the optimal cycle that minimizes its cost C ave,0, and buyers jointly determine order frequencies (n 1, n ) that minimize the total cost C ave,b = C ave,1 + C ave,. It is not hard to derive that the vendor will set the cycle as A 0 P = ( ) d (8) h d n 1 n

11 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 11 For the buyers, first assume that the delivery pattern ρ is chosen. he first-order condition (FOC) of buyers cost follows immediately after Lemma 1: C ave,b ρ = n 1 A 1 [ d 1 h 1 + d 1d (h 1 h ) + d ] 1d h P n 1 n n 1 A 1 [ ] d 1 h 1 + d 1 d (h 1 + h + h 1 n h n ) P n 1 if n 1 n if n 1 > n C ave,b ρ = n A [ ] d h d 1 d (h 1 h )(n 1 1) P n A [ d h + d ] 1d (h h 1 ) P n 1 n if n 1 n if n 1 > n Solving min n1,n C ave,b would involve two sub-problems for the regions of n 1 n and n 1 > n, and they may not be convex for general parameters. However, in practice buyers ordering the same kind of products from the vendor are highly likely to incur the same storage spaces, electricity usages, insurances, etc., which yield the same holding costs. herefore we focus on examining the scenario when h 1 = h = h. In this case, the FOCs are identical for both regions (n 1 n ): C ave,b ρ = A 1 n 1 d 1 (d 1 + d )h P n 1 C ave,b ρ = A n d h P n We can therefore solve the optimal decision as n d 1 (d 1 + d )h 1(ρ) = A 1 P n d (ρ) = h A P (9) and the minimum buyers cost is Cave,b(ρ) = h h P (d 1 + d )(P d 1 d ) + P ( A 1 d1 (d 1 + d ) + A d ) he optimal delivery pattern can thus be figured out as ρ = arg min ρ P C ave,b(ρ).

12 1 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System Particularly, ρ ([i]) i if A [1] d[1] (d [1] + d [] ) + A [] d [] A [] d[] (d [] + d [1] ) + A [1] d [1] = j o/w (10) where i, j {1, } and i j. Under delivery pattern ρ, solving (8) and (9) together we have the following result. Proposition 1. (Retailer-managed inventory in a one-vendor-two-buyer system) In a two-buyer problem with retailer-managed inventory, assume that the buyers incur identical holding cost, i.e., h 1 = h = h. he the vendor and the set of buyers make competing decisions, and the equilibrium is as follows: P h A = 0 h 0 d1 d 1 A1 + d A d 1 + d he delivery pattern ρ follows (10). n 1 = h d1 (d 1 + d )A 0 h 0 d1 d 1 A 1 + d A1 A d 1 + d n = h d A 0 h 0 d1 d 1 A1 A + d A d 1 + d (11) One may observe from (10) that the optimal delivery pattern matters with A i and d i for i = 1, only. It does not depend upon the production rate P or cost structure (A 0, h 0 ) of the vendor. Corollary 1. Under retailer-managed inventory, the delivery pattern ρ is an internal decision that is not affect by upstream operational efficiencies. 4.. Optimal solution in VMI For vendor-managed inventory, we enforce the same condition that h 1 = h = h. he goal is to find (, n 1, n ) that minimize the system s total cost C ave,s = i=0 C ave,i. Given ρ, the FOCs are given by C ave,s ρ = A 0 + A 1 n 1 + A n + h P [ ( 1 1)d 1 + ( 1 1)d + ( 1 ] 1)d 1 d + (d 1 + d )P n 1 n n 1

13 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System + h ( ) 0 d 1 + d P n 1 n 13 C ave,s ρ = A 1 n 1 d 1 [d 1 (h + h 0 ) + d h] P n 1 C ave,s ρ = A n d (h + h 0 ) P n through which we can identify the optimum: A 0 P (ρ) = h(d 1 + d )(P d 1 d ) n A 0 d 1 [d 1 (h + h 0 ) + d h] 1 (ρ) = A 1 h(d 1 + d )(P d 1 d ) n A 0 d (ρ) = (h + h 0 ) A 1 h(d 1 + d )(P d 1 d ) (1) and the minimal total cost of the system ave,s(ρ) = [ A0 hd(p D) + A 1 d 1 (d 1 (h 0 + h) + hd ) + ] A d P (h 0 + h). he optimal delivery pattern then involves finding ρ = arg min ave,s(ρ) = arg min A1 d 1 (d 1 (h 0 + h) + hd ) + A d (h 0 + h) (13) ρ P ρ P We can now summarize our finding on VMI as follows. Proposition. (Vendor-managed inventory in a one-vendor-two-buyer system) In a two-buyer problem with vendor-managed inventory, assume that the buyers incur identical holding cost, i.e., h 1 = h = h. he cycle time and order frequencies n that minimize the system s total cost is characterized by (1) where the delivery pattern ρ follows (13). Compared with Corollary 1, one can tell that under vendor-managed inventory, the delivery pattern will involve operational performance on both the vendor side (h 0 ) and the buyer side (A i>0, d i, h). However, the decision of production cycle only relies on market information (d i ) of the buyers. he operational efficiency of the buyers (A i>0, h) would not affect how frequent the vendor sets up its production. hese give rise to the following corollary.

14 14 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System Corollary. Under vendor-managed inventory, the vendor only needs to know buyers market condition in determining its production schedule. Buyers operational efficiency would not affect the optimal production cycle Numerical examples We use the same parameters as in Goyal (1988), Zavanella and Zanoni (009) and Ben-Daya et al. (01) to illustrate our results on VMI and RMI: P = 300, d [1] = 500, d [] = 1000, A 0 = 400, A [1] = 75, A [] = 5, h 0 = 5, h = 4. ρ([1], []) n 1 n C ave,0 C ave,1 C ave, C ave,b C ave,sys RMI (, 1) VMI (, 1) For vendor-managed inventory, our solution that ρ([1], []) = (, 1), n = (5, ) and = 0.5 yields average total cost 437 for the system. he average total cost is 6.1% lower than that in Zavanella and Zanoni (009) (which suggests that the optimal solution be ρ([1], []) = (1, ), n = (1, 3) and = 0.45 with total cost verified by Goyal et al. (01)), and 14.8% lower than in Ben-Daya et al. (01) (which assumes common delivery frequency of all buyers, yielding optimal solution n = (, ) and = 0.49 with total cost 797.9). 5. Heuristics for the General One-Vendor-Multiple-Buyer Problem In the previous section, we have fully characterized the equilibrium and solutions of the two-buyer problem in both VMI and RMI. As the number of buyers grows, the general total cost function (6) and (5) imply that the problem will become more much complex and even intractable. Nevertheless, the analysis in 4 does provide us with a good start point to tackle the general problem. We analyze problem with Y > in the same spirit, and propose heuristics for calculating the total cost. Equilibrium and optimal solutions are derived thereafter. We first derive the total cost C j with given ρ under the assumption that n 1 n... n Y. For buyer 1, by (5) and (6) the total cost is

15 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System C 1 = A 1 n 1 + C 1 n 1 + C 1 n 1 W 1 = A 1n 1 + C 1 [ 1 (n 1 1) Y d i n i P ]. 15 For buyer, ] Y d i C = A n + C [1 (n 1) n i P + (n n 1 )(n n 1 + 1) d 1, n n 1 P and similarly, the total cost for buyer 3 is C 3 = A 3 n 3 + C 3 [1 (n 3 1) Y d i n i P + (n 3 n i )(n 3 n i + 1) n 3 d i n i P ]. One can observe a pattern from above that C j = A j n j + C j [1 (n j 1) Y j 1 d i n i P + (n j n i )(n j n i + 1) n j d i n i P ] (14) and as far as we can verify, the same pattern holds for higher indexed buyers. We therefor define (14) the heuristics total cost of buyer j in the one-vendor-multi-buyer system. In general, we should enforce all ranking assumptions on (n 1,..., n Y ), solve the sub-optimal total cost under each of them, and compare for global optimum. Yet it has been shown in the two-buyer problem that if holding costs are the same across buyers (h j = h), then C j is irrelevant to the ranking assumption of (n 1,..., n Y ). hat is, (14) is without loss of generality. We therefore study the system with identical holding cost among the buyers (h 1 =... = h Y = h) based on the heuristic in (14) Equilibrium under RMI Consider buyers jointly determine ρ and (n 1,..., n Y ) and the vendor makes its own decision on the cycle. Solving the FOCs of the average total cost (1/ of (7) and (14)) on given ρ, the optimal response functions for the vendor and the buyers are A 0 P (ρ) =, n Y d j (d j + D j )h j(ρ) =, d A j P i h 0 n i j = 1...Y. (15) where D j = Y i=j+1 d i and D = D 0.

16 16 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System he buyers total cost is given by Cave,b(ρ) = h h D(P D) + P P Y Ai d i (d i + D i ). and the optimal delivery pattern can be found through ρ = arg min Cave,b(ρ) = arg min ρ P ρ P Solving (15) gives the equilibrium decision as follows: Y Ai d i (d i + D i ) (16) Proposition 3. (Retailer-managed inventory in a general one-vendor-multi-buyer system) Under retailer-managed inventory with heuristic total cost function (14), consider buyers incur identical holding cost, i.e., h j = h, j = 1,,..., Y. he vendor and the buyers will make the following decision in equilibrium: where B j = (16). = A 0P h 0 n j = A 0P h 0 ( Y B i d i B i ( Y ) 1 d i B i d j (d j + D j )h, D j = Y i=j+1 A j P d i, D = D 0, and the optimal delivery pattern ρ follows ) Optimal solution under VMI Now consider VMI. In minimizing the total cost of the system we have to make sure the following conditions hold under given ρ C ave,s ρ = 1 (A 0 + Y A i n i ) + h 0 P Y n j C ave,s ρ = A j d j n jp [(d j + D j )h + d j h 0 ] = 0 where D j = Y i=j+1 d i and D = D 0. [ Y d i + h d i (P D) + n i P Y j=1 d j n j (d j + D j ) ] = 0 Solving the above we have the optimal solution: P A 0 (ρ) = ha i D(P D) (17)

17 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System n j (ρ) = A 0 d j [(h + h 0 )d j + hd j ] A j hd(p D) 17 and ave,s(ρ) = [ A0 hd(p D) + P ] Y Ai d i (d i (h 0 + h) + hd i ) (18) where D j = Y i=j+1 d i and D = D 0 he optimal delivery pattern is therefore ρ = arg min Cave,s(ρ) = arg min ρ P ρ P Y Ai d i (d i (h 0 + h) + hd i ) (19) he findings on optimal decisions under VMI are summarized in the following proposition. Proposition 4. (Vendor-managed inventory in a general one-vendor-multi-buyer system) Under vendor-managed inventory with heuristic total cost function (14), consider buyers incur identical holding cost, i.e., h j = h, j = 1,,..., Y. he optimal production cycle, delivery frequency and delivery pattern follow (17) and (19) respectively. Specifically when the buyers share identical cost structure (A i = A, h i = h for i = 1,,..., Y ) yet heterogeneous demand rates, we can explicitly derive the optimal delivery pattern ρ for the system as follows. Proposition 5. (Optimal delivery pattern for identical buyers under VMI) For buyers with identical cost structure (A i = A, h i = h for i > 0), it is optimal to put buyers with higher demand on top of the rotation list. hat is, assume that d [1] d []... d [Y ], then ρ ([i]) = Y + 1 i. It practically suggests that in a distribution network where each location incurs similar ordering as well as holding cost, the rotational delivery should start from fulfilling buyers with the highest demand to those with lowest demand. herefore Proposition 4 and 5 together provides a handy VMI prescription for large distribution systems with identical buyers.

18 18 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System # of Buyers Prod. Rate RMI VMI Eff. Loss Y P n C ave,s n C ave,s δ 300 (14, 8) (7, 5) % (14, 1, 9, 5) (5, 4, 4, 3) % (14, 13, 11, (4, 4, 3, % 9, 7, 4) 3,, ) (14, 13, 1, 11, (3, 3, 3, 3, % 9, 8, 6, 4),,, 1) able RMI vs. VMI as the number of buyers increases, A 0 = 400, h 0 = 5, d i = 100, A i = 5, h i = Numerical examples We next conduct numerical experiments to illustrate how the outcomes are impacted by the number of buyers. Similar parameters as in 4.3 are adopted: A 0 = 400, h 0 = 5 and d i = 100, A i = 5, h i = 4 for i = 1,,..., Y. o examine the impact of the number of buyers, we increase the production rate P in accordance with Y, such that the system s utilization rate remains the same as the system grows, i.e., Y d i/p = 75%. he results are summarized in able. Specifically, the central columns are the equilibrium/optimal solution under RMI and VMI settings. he last column shows the efficiency loss in RMI compared with VMI. hat is, δ = ( C RMI ave,s C V MI ave,s )/ C V MI ave,s. It can be observed that as the number of buyers Y increases, the loss of efficiency also increases. his highlights the importance of vendor-managed inventory in large-scale distribution system. We also conduct similar analysis with different system utilities (Figure 4). It suggests that the efficiency gap can be significantly reduced if the system utilization is high. In other words, the impact of vendor-managed system is less obvious under flexible production which can be coped with the demand. hus far the numerical examples suggest two ways to enhance system performance. One is vendor-managed inventory (VMI), and the other is just-in-time (JI) production. Most interestingly, these two measures are substitutes to each other and VMI can be more critical under rigid production system with high capacity, or when demand is low (P >> D).

19 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System D/P=6.5% D/P=75% D/P=93.75% Efficiency Loss under Fixed System Usage Efficiency Loss Number of Buyers Y Figure 4 Efficiency Loss under fixed system utilization (6.5%, 75% and 93.75%) 6. Endnotes his paper examines the application of vendor-managed inventory (VMI) and retailer-managed inventory (RMI) in a one-vendor-multiple-buyer production-distribution system. We develop general total cost function for the vendor and the buyers, which adapts to a much wider set of scenarios than existing literature. For distribution systems consisting of two buyers, we explicitly derive the equilibrium and optimal solution (production, order, and shipping schedule) under both RMI and VMI settings. For general distribution systems, we provide good heuristic to approach solutions when buyers share similar cost structures. he results demonstrate significant improvement than existing solutions in literature. he model particularly suits for supply chains in which the downstream firm operates many distribution centres of similar sizes, and has greater logistics advantage than the upstream producer or vendor. his include industries such as retailing (e.g., Walmart, Home Depot) and automobile (dealers order from a single manufacturer). he findings also generate several interesting implications for managerial decision making. he first insight relates to the information requirements in reaching equilibrium/optimal outcomes (able 3, 4, and 5 summarize the comparison between RMI and VMI settings). Basically,

20 0 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System Production Cycle Buyers Information ( ) Operations Market RMI VMI able 3 Information required for the decision of production cycle : RMI vs. VMI Delivery Frequency (n) Vendor s Information Operations RMI VMI able 4 Information required for the decision of delivery frequency n: RMI vs. VMI Delivery Pattern (ρ) Vendor s Information Operations RMI VMI able 5 Information required for the decision of delivery pattern ρ: RMI vs. VMI there is less difference between RMI and VMI in making delivery frequency decisions (able 4) vendor s operational information is a must under both settings. For the production cycle (able 3), while the vendor will always need full information from the buyers under RMI, less will be demanded under VMI. In particular, under VMI the vendor would only need the market information in determining the production cycle whether the buyer is cost efficient in logistics matters little. However, such is reversed when buyers determine delivery patterns (able 5). Decisions on delivery pattern can be irrelevant to vendor s cost structure under RMI, while this information is required for VMI. he second insight is on understanding the value of vendor-managed inventory. In general, VMI

21 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 1 generates more savings as the distribution system involves more buyers. his impact is magnified when it is expensive to adjust production according to total demand rate. Such can happen in many industries such as automobile, where short-run capacity abandonment is unusual, or when an industry is highly unionized. Acknowledgments he research of Xiao Huang is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). he authors thank the research assistance of Hehui Wu. References Banerjee, A A Joint Economic-Lot-Size Model for Purchaser and Vendor. Decision Sciences 17(3) Banerjee, A., J. S. Burton Coordinated vs. independent inventory replenishment policies for a vendor and multiple buyers. International Journal of Production Economics Ben-Daya, M., M. Hariga Integrated single vendor single buyer model with stochastic demand and variable lead time. International Journal of Production Economics 9(1) Ben-Daya, M., E. Hassini, M. Hariga, M. M. AlDurgam. 01. Consignment and vendor managed inventory in single-vendor multiple buyers supply chains. International Journal of Production Research Forthcoming Boyaci,., G. Gallego. 00. Coordinating pricing and inventory replenishment policies for one wholesaler and one or more geographically dispersed retailers. International Journal of Production Economics Braglia, M., L. Zavanella Modelling an industrial strategy for inventory management in supply chains: he Consignment Stock case. International Journal of Production Research 41(16) Dong, Y., K. Xu. 00. A supply chain model of vendor managed inventory. ransportation Research Part E Goyal, S. K An integrated inventory model for a single supplier - single customer problem. International Journal of Production Research 15(1)

22 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System Goyal, S. K A Joint Economic-Lot-Size Model for Purchaser and Vendor : A Comment. Decision Sciences 19(1) Goyal, S. K On improving the single-vendor single-buyer integrated production inventory model with a generalized policy. European Journal of Operational Research (15) Goyal, S. K., X. Huang, Y. Li. 01. Note on A one-vendor multi-buyer integrated production-inventory model: he Consignment Stock case. Working Ppaer 1 6. Gümüs, M., E. M. Jewkes, J. H. Bookbinder Impact of consignment inventory and vendor-managed inventory for a two-party supply chain. International Journal of Production Economics 113() Hill, R. M., M. Omar Another look at the single-vendor single-buyer integrated production-inventory problem. International Journal of Production Research 44(4) Hill, R. M he single-vendor single-buyer integrated production-inventory model with a generalised policy. European Journal of Operational Research Hill, R. M he optimal production and shipment policy for the single-vendor singlebuyer integrated production-inventory problem. International Journal of Production Research 37(11) Lal, R., R. Staelin An Approach for Developing an Optimal Discount Pricing Policy. Management Science 30(1) Lu, L A one-vendor multi-buyer integrated inventory model. European Journal of Operational Research Marquès, G., C. hierry, J. Lamothe, D. Gourc A review of Vendor Managed Inventory (VMI): from concept to processes. Production Planning & Control 1(6) Munson, C. L., M. J. Rosenblatt Coordinating a three-level supply chain with quantity discounts. IIE ransactions Nagarajan, M., S. Rajagopalan Contracting Under Vendor Managed Inventory Systems Using Holding Cost Subsidies. Production and Operations Management 17() Rosenblatt, M. J., H. L. Lee Improving profitability with quantity discounts under fixed demand. IIE ransactions 17(4) Valentini, G., L. Zavanella he consignment stock of inventories: industrial case and performance analysis. International Journal of Production Economics

23 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 3 Zavanella, L., S. Zanoni A one-vendor multi-buyer integrated production-inventory model: he Consignment Stock case. International Journal of Production Economics 118(1) 5 3. Zavanella, L., S. Zanoni Erratum to A one-vendor multi-buyer integrated production-inventory model: he Consignment Stock case. International Journal of Production Economics 15(1) Zhou, Y.-W., S.-D. Wang Optimal production and shipment models for a single-vendor single-buyer integrated system. European Journal of Operational Research 180(1)

24 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System 1 Appendix Proof of Lemma 1 We show how to get the average total cost for buyer 1 when n 1 n. Others can be proved in a similar fashion. ( q1 By (5), W 1,k = (k 1) q 1 d 1 P q ) hence P W 1 = n ( 1 (n q1 1 1) q 1 d 1 P q ) = n ( 1 P (n 1 1) d 1 n 1 n 1 P d ) n P Substitute this and (3) into (6), buyer 1 s average cost is then C ave,1 = A 1n 1 + h 1d 1 h 1d 1 ( d1 (n 1 1) n 1 P + d ). n P Proof of Proposition 5 We first propose the following lemma which will be used in the main proof. Lemma A1. Suppose a, b, c, d 0 satisfying 1. a + b = c + d, and. a b c d, then there is a + b c + d. Proof. It is sufficient to show that ( a + b) ( c + d), which by the first condition, can be reduced to ab cd. By the second condition, there is a b c d, hence (a+b) 4ab (c+d) 4cd. Applying the first condition, we can verify immediately that ab cd. For λ = h 0 + h h 1, it is sufficient to show that Y di (d i λ + D i ) reaches the minimum value when d 1 d Y. Suppose this is not true, then at the minimum there exists some 1 t Y 1 such that d t < d t+1. Define a new sequence {d i} Y where d t+1, if i = t; d i = d t, if i = t + 1; d i, otherwise.

25 Huang and Li: One-Vendor Multiple-Buyer Production-Inventory System Also denote D j = Y i=t+1 d i. hen D i = D i when i t (A1) and D i = D i+1 + d i+1 (A) Since {d i } Y yields the minimal total cost, there should be Y di (d i λ + D i ) Y d i (d iλ + D i) By (A1), the above is equivalent to dt (d t λ + D t ) + d t+1 (d t+1 λ + D t+1 ) d t(d tλ + D t) + d t+1(d t+1λ + D t+1) which by (A) can be further reduced to dt (d t λ + D t+1 + d t+1 ) + d t+1 (d t+1 λ + D t+1 ) d t+1 (d t+1 λ + D t+1 + d t ) + d t (d t λ + D t+1 ). Let a = d t (d t λ + D t+1 + d t+1 ), b = d t+1 (d t+1 λ + D t+1 ), c = d t+1 (d t+1 λ + D t+1 + d t ), d = d t (d t λ + D t+1 ). It is easy to verify that a + b = c + d. Also since d t < d t+1, we have a b = d t d t+1 (d t+1 d t )λ (d t+1 d t )D t+1 d t d t+1 + (d t+1 d t )λ + (d t+1 d t )D t+1 = c d. We can thus apply Lemma A1 and claim a + b c + d. hat is, dt (d t λ + D t+1 + d t+1 ) + d t+1 (d t+1 λ + D t+1 ) d t+1 (d t+1 λ + D t+1 + d t ) + d t (d t λ + D t+1 ). or equivalently Y di (d i λ + D i ) Y d i (d iλ + D i). However, this contradicts to our assumption of the minimality of Y di (d i λ + D i ). herefore when Y di (d i λ + D i ) gets the minimum value, there should be d 1 d Y.