Coordinating a Constrained Channel with Linear Wholesale Price Contracts.

Transcription

1 Coordinating a Contrained Channel with Linear Wholeale Price Contract. Navid Sabbaghi, Yoi Sheffi, John N. Titikli Maachuett Intitute of Technology, abbaghi@mit.edu, heffi@mit.edu, jnt@mit.edu We how that when a upply channel i capacity-contrained and the contraint i tight, there i a et of linear wholeale price contract that coordinate the channel while allowing the upplier to make a profit. We prove thi for the one-upplier/one-newvendor upply channel a well a the many-upplier/one-newvendor channel configuration (with each upplier elling a unique product. We analyze how thi et of wholeale price change a we change the channel capacity contraint. We alo explore condition under which thee channel-efficient linear wholeale price contract reult from the equilibrium behavior of a newvendor procurement game. Our newvendor procurement game generalize the Stackelberg game introduced in Lariviere and Porteu (21 to allow for multiple upplier a well a a capacity contraint at the newvendor. Finally, we find the et of rik-haring contract (uch a buy-back and revenue-haring contract that coordinate a contrained upply channel and contrat that et with the et of rik-haring contract that coordinate an uncontrained channel. Key word : Wholeale Price Contract; Supply Contract; Channel Coordination; Capacity Contraint; Newvendor. 1. Introduction There i a wealth of upply contract available that coordinate a newvendor deciion for uncontrained upplier-retailer channel: buy-back contract, revenue-haring contract, etc. (Cachon 23 A contract coordinate the action of a newvendor for a upply channel if the contract caue the newvendor to take action when olving hi own deciion problem that are alo optimal for the channel. 1 Our paper how that impler contract, namely linear wholeale price contract, (which are thought to be unable to coordinate a newvendor deciion for uncontrained channel can, in fact, coordinate a newvendor procurement deciion for reource-contrained channel. Thi i relevant for upply channel in which capacity of ome reource i limited. For example, helf pace at retail tore, eat on airline, warehoue pace, procurement budget, time available for manufacturing, raw material, etc. (Corten 26 1 Sometime we alo ay a contract channel-coordinate a newvendor deciion. Therefore, achieving coordination for the channel equate to attaining channel optimality (and thu efficiency when the newvendor i allowed to decide for himelf. 1

2 2 LIDS Technical Report #2749, February 27. Toward the end of our paper we how how rik-haring contract uch a buy-back and revenueharing coordinate the procurement deciion of a reource-contrained newvendor thereby generalizing the treatment of thee contract. But the primary inight we how in thi paper, i that if newvendor capacity i a binding contraint, then a et of linear wholeale-price contract can coordinate the procurement deciion of a capacity-contrained newvendor. 2 Furthermore, thi et include wholeale price that allow both the upplier and the newvendor to profit. Wholeale price contract are commonplace ince they are traightforward and eay to implement. While rik-haring contract uch a revenue-haring agreement can coordinate a retailer deciion in a newvendor etting, Cachon and Lariviere (25 note that thee alternative contract impoe a heavier adminitrative burden. For example, thee alternative contract may require an invetment in information technology or a higher level of trut between the trading partner due to the additional procee involved. Our tylized capacity-contrained newvendor etting provide a laboratory for undertanding the et of wholeale price contract that lead the retailer to take coordinating action under variou channel configuration: one-upplier/one-retailer and multiple-upplier/one-retailer. In thi paper, we are concerned with the coordination capability of wholeale price contract for a upply channel in both a negotiation etting and an equilibrium etting. In our negotiation etting, we are concerned with the entire et of coordinating wholeale-price contract. The wholeale price in thi et are Pareto-optimal, a ueful property for getting win/win reult in negotiation etting. Thi i in contrat to an equilibrium etting, where chooing the wholeale price( i an initial tage of a game for the upplier(. In the equilibrium etting we explore condition for the game equilibria wholeale-price vector to coordinate the newvendor procurement deciion for the channel (i.e., neceary and ufficient condition o that the game equilibria are included in the et of coordinating wholeale price contract, and characterize the extent of the efficiency lo when thee condition are violated Organization of thi paper In Section 2, we provide an overview of the upply contract literature and emphaize the point that the literature ha underetimated and not conidered the coordination capability of wholeale 2 In addition to capacity being a binding contraint, the relative power of the partie and their competitive environment are alo important for the wholeale-price contract to coordinate the action of the newvendor in practice. For example, even if the et of wholeale price W(k that coordinate the retailer action i enlarged beyond the upplier marginal cot (due to the retailer capacity contraint k, the retailer and upplier till need to agree upon ome wholeale price in that et. Their outide-alternative and the power in the upply channel could determine if ome wholeale price in the et W(k i acceptable for the partie involved.

3 LIDS Technical Report #2749, February price contract for a contrained upply channel. In Section 3, we provide a tylized 1-upplier/1- retailer model and formally define what it mean for a wholeale price contract to coordinate the retailer ordering deciion for a upply channel. Thi model i analyzed in Section 4. In Section 5, we extend the model to include more upplier and generalize our earlier analyi. Section 6 how that the et of revenue-haring and buyback contract that coordinate a newvendor deciion for a contrained channel i a uperet of the et of coordinating contract in the uncontrained etting. Finally, we ummarize our finding and provide managerial inight in Section Literature Review The upply contract literature ha been baed on the obervation, pointed out, for example, by Lariviere and Porteu (21, that wholeale price contract are imple but do not coordinate the retailer order quantity deciion for a upplier-retailer upply chain in a newvendor etting. Thi obervation ha led to the tudy of an aortment of alternative contract. For example, buy back contract (Paternack 1985, quantity flexibility contract (Tay 1999, and many other. Cachon (23 provide an excellent urvey of the many contract and model that have been tudied in the upply contract literature. The mindet urrounding wholeale price contract inability to channel-coordinate i true under appropriate aumption which the upply contract literature ha been implicitly auming: that there are no capacity contraint (e.g., helf pace, budget, etc.. A mentioned before, we alo conider the cae of multiple upplier erving a ingle retailer. Thi exploration i motivated, in part, by Cachon (23 and Cachon and Lariviere (25, who emphaize that coordination for channel configuration with multiple upplier ha yet to be explored. The relevant literature on multi-product newvendor with ide contraint (which ha developed independently from the coordination literature include Lau and Lau (1995, Abdel-Malek and Montanari (25a,b. Conidering capacity contraint in a upply channel i not new to the upply contract literature. However, mot other paper in the literature conider chooing capacity a one tage of a game (before downtream demand i realized that alo involve a production deciion after demand i finally realized (Cachon and Lariviere 21, Gerchak and Wang 24, Wang and Gerchak 23, Tomlin 23. Our paper, although complementary to thi tream of literature, doe not involve an endogenou capacity choice for any party but rather analyze how an exogenou capacity contraint determine the et of wholeale price that can coordinate the retailer deciion for the channel. Paternack (21 conider an exogenou budget contraint, but not for the purpoe of tudying coordination. Rather, he analyze a retailer optimal procurement deciion when the retailer ha two available trategie: buying on conignment and outright purchae.

4 4 LIDS Technical Report #2749, February 27. Alo our paper i not the firt to reconider wholeale price contract and their benefit beyond implicity. Cachon (24 look at how inventory rik i allocated according to wholeale price contract and the reulting impact on upply chain efficiency. A far a we are aware, our paper i the firt to conider the coordination-capability of linear wholeale price contract under a imple capacity-contrained production/procurement newvendor model. 3. Model A rik-neutral retailer r face a newvendor problem in ordering from a rik-neutral upplier for a ingle good: there i a ingle ale eaon, the retailer decide on an order quantity q and order well in advance of the eaon, the entire order arrive before the tart of the eaon, and finally demand i realized, reulting in ale for the retailer (without an opportunity for replenihment. Without lo of generality, we aume that unit remaining at the end of the eaon have no alvage value and that there i no cot for tocking out. The model parameter are ummarized in Figure 1 with the arrow denoting the direction of product flow. In particular, the upplier ha a fixed marginal cot of c per unit upplied and charge the retailer a wholeale price w c per unit ordered. The retailer price p per unit to the market i fixed, and we aume that p > w. For that price, the demand D i random with probability denity function (p.d.f. f and cumulative ditribution function (c.d.f. F. We alo define F (x def = 1 F (x = P (D > x. We ay that a c.d.f. F ha the IGFR property (increaing generalized failure rate, if g(x def = x f(x F (x i weakly increaing on the et of all x for which F (x > (Lariviere and Porteu 21. Mot ditribution ued in practice (uch a the Normal, the Uniform, the Gamma, and the Weibull ditribution have the IGFR property. We aume that the retailer capacity i contrained by ome k > ; for example, the retailer can only hold k unit of inventory, or accept a hipment not larger than k. For a different interpretation, k could repreent a contraint on the capacity of the channel or a budget contraint. Figure 1 ingle upplier & ingle capacity contrained retailer model. r c w q p D F q k Note. Supplier with marginal cot c (per unit offer a product at wholeale price w (per unit to a capacitycontrained retailer r that face uncertain demand D downtream, when the price for the product i fixed at p (per unit. The retailer mut decide on a quantity q to order from the upplier.

5 LIDS Technical Report #2749, February Aumption 1. The p.d.f. f for the demand D ha upport [, l], with l > k, on which it i poitive and continuou. A a conequence, F ( = 1 and F i continuouly differentiable, trictly decreaing, and invertible on (, l Retailer problem Faced with uncertain ale S(q def = min{q, D} (when ordering q unit and a wholeale price w (from the upplier, the retailer decide on a quantity to order from the upplier in order to maximize expected profit π r (q def = E[pS(q] wq while keeping in mind the capacity contraint k. Namely, it olve the following convex program with linear contraint in the deciion variable, q: RETAILER(k,w maximize pe[s(q] wq (1 ubject to k q q. Becaue of our aumption on the c.d.f. F, it can be hown that RETAILER(k,w ha a unique olution which we denote by q r (w Channel problem Denote the channel expected profit by π (q def = E[pS(q cq]. Under capacity contraint k, the optimal order quantity q for the ytem/channel i the olution to convex program (2, CHANNEL(k. Note that CHANNEL(k ha identical linear contraint but a lightly altered objective function when compared to RETAILER(k,w: CHANNEL(k maximize pe[s(q] cq (2 ubject to k q q. Again becaue of our aumption on the c.d.f. F it can be hown that CHANNEL(k alo ha a unique olution which we denote by q. We denote the unique olution, arg max q< π (q, for the uncontrained channel problem by q. It i well known that q = F 1 (c/p (e.g., Cachon and Terwiech (26. Becaue of convexity, it i alo eaily een that q = min{q, k}.

6 6 LIDS Technical Report #2749, February Definition: Coordinating the retailer action A wholeale price contract w coordinate the retailer ordering deciion for the upply channel when it caue the retailer to order the channel-optimal amount, i.e., q r (w = q. In Section 4 we are intereted in the following quetion: For a fixed capacity k, what i the et of wholeale price W(k for which q r (w = q? What doe thi et W(k reemble geometrically? If there i no capacity contraint (or equivalently if k i very large, recall that double marginalization reult in the retailer not ordering enough (i.e., q r (w < q under any wholeale price contract, w > c. In the next ection, we will how that when the capacity contraint k i mall relative to demand, there exit a et of wholeale price contract w > c that can coordinate the retailer order quantity, i.e., q r (w = q. 4. Analyi Our firt reult decribe the et of coordinating wholeale price under a capacity contraint. Theorem 1. In a 1-upplier/1-retailer configuration where the retailer face a newvendor problem and ha a capacity contraint k, any wholeale price w W(k def = [ c, p F (min{q, k} ] will coordinate the retailer ordering deciion for the upply channel, i.e., q r (w = q. Furthermore, if q r (w = q and c w p, then w W(k. Proof. See Appendix A. Notice that if the capacity contraint k i larger than or equal to the uncontrained channel optimal order quantity, q, then p F (min{q, k} = p F (q = c, reducing to the claic reult in the upply contract literature. However, thi i true only when the capacity contraint i not binding for the channel (i.e., q k. When the capacity contraint k i binding for the channel (i.e., q > k, then any wholeale price w [c, p F (k] will coordinate the retailer action and only wholeale price in the range [c, p F (k] can coordinate the retailer action. Many factor uch a power in the channel, outide alternative, inventory rik expoure, and competitive environment ultimately influence the actual wholeale price (elected from the et [c, p] charged by the upplier. In the uncontrained etting, regardle of thee factor, coordination i not poible with a linear wholeale price contract (becaue the upplier preumably would not agree to price at cot. However, when the capacity contraint i binding for the channel, coordination become poible (becaue the et of coordinating wholeale price contract become

7 LIDS Technical Report #2749, February [c, p F (k] (rather than {c} and ultimately depend on thee other factor. Theorem 6 in Section 4.4 conider a equilibrium etting where the retailer take on all the inventory rik (akin to the Stackelberg game in Lariviere and Porteu (21 and puh mode in Cachon (24, and provide additional condition that mut be met o that the equilibrium wholeale price contract i a member of the et of coordinating wholeale price contract, [c, p F (k] Size of W(k. The geometry of the et of wholeale price W(k that coordinate the retailer deciion for the upply channel i depicted in Figure 2. Figure 2 The et of wholeale price that coordinate the action of a ingle retailer when procuring from a ingle upplier. p F(q p F(k c Note. Note that p F (q = c and W(k = [c, p F (k] (the interval denoted in bold when k q. p Note that the ize of W(k i increaing a k decreae. Corollary 1 formalize thi notion and follow directly from Theorem 1 becaue F (k i decreaing in k. Corollary 1. If k 1 k 2, then W(k 2 W(k 1 [c, p]. Thu, the more contrained the channel i with repect to the channel optimal order quantity, q, the larger the et of coordinating wholeale price contract W(k. Conider two upply channel elling the ame good with the ame retail price p and upplier cot c. Aume that the probability of exce demand in the firt channel i larger, in the ene F 1 (k F 2 (k. Let W i (k denote the et of coordinating wholeale price contract for channel i when the channel i contrained by k unit. The channel with the higher probability of exce demand ha a larger et of coordinating wholeale price. Corollary 2 to Theorem 1 make thi precie. Corollary 2. Given two demand ditribution F 1 and F 2, if F 1 (k F 2 (k >, then W 2 (k W 1 (k [c, p]. Proof. See Appendix B.

8 8 LIDS Technical Report #2749, February Revenue requirement implicit in W(k. By agreeing to focu on the et W(k in negotiating over a wholeale price for coordination purpoe, the upplier and retailer are implicitly agreeing to a minimum hare of expected revenue requirement for the retailer and thu a maximum hare of expected revenue retriction for the upplier. Thi notion i formalized in Theorem 2. Theorem 2. If the capacity contraint k i binding for the channel (i.e., q > k, then any coordinating linear wholeale price contract w W(k guarantee that the retailer receive at leat a fraction a fraction R k F (x dx k F (k R k F (x dx of the channel expected revenue, and that the upplier receive at mot k F (k R k F (x dx of the channel expected revenue. Furthermore, if F ha the IGFR property, then the upplier maximum revenue hare i weakly decreaing a k increae. Proof. See Appendix C. An important ditinction regarding the upplier and retailer hare of expected revenue guarantee formalized in Theorem 2 i that the upplier hare reult in a guaranteed income (i.e., no uncertainty wherea the retailer hare reult in an uncertain income. For example, from Theorem 2 there exit ome wholeale price w W(k, where the upplier receive a fraction k F (k R k F (x dx of the expected channel revenue, pe[s(k]. But the upplier income i certain, wk, wherea the retailer income i an uncertain amount, ps(k wk. A a numerical example, if k F (k R k F (x dx = 1/2, the upplier can receive up to fifty percent of the expected channel revenue and till keep the channel coordinated, wherea we require that the retailer receive at leat fifty percent of the revenue in order for the wholeale price to coordinate the action of the retailer. Recall that the et of coordinating wholeale price contract W(k increae with the probability F (k of exce demand, when k i held fixed (Corollary 2. Theorem 3 formalize a related idea: the larger the expected exce demand, the greater the maximum poible hare of revenue at the upplier without acrificing channel-coordination. Theorem 3. Conider two different demand D 1 and D 2, with each D i aociated with a c.d.f. F i, that have the ame mean and uch that F 1 (k F 2 (k. Suppoe that (a the capacity contraint k i binding for the channel under both ditribution (i.e., min{q 1, q 2} > k, and (b E[(D 1 k + ] E[(D 2 k + ] (i.e., the expected exce demand under D 1 i higher than that under D 2. Then, Proof. See Appendix D. k F 1 (k k F 1 (x dx k F2 (k k F 2 (x dx.

9 LIDS Technical Report #2749, February Wholeale price contract and flexibility in allocating channel-optimal profit The benefit of rik haring contract in the uncontrained etting include the ability to channelcoordinate the retailer deciion a well a flexibility (due to the extra contract parameter that allow for any allocation of the optimal channel profit between the upplier and retailer. Cachon (23 provide excellent example of the channel-profit allocation flexibility inherent in thee more complex contract. Theorem 4 demontrate that in a reource contrained etting, wholeale price contract alo have flexibility in allocating the channel-optimal profit. Namely, thee impler contract allow for a range of diviion of the optimal channel profit among the firm. The diviion allowed (without loing coordination depend on the channel capacity, k. Similar to our obervation in Section 4.2 for the implicit revenue requirement, the upplier hare reult in a guaranteed income (i.e., no uncertainty wherea the retailer hare reult in an uncertain income. Theorem 4. If the capacity contraint i binding for the channel (i.e., q > k, there exit a wholeale price contract w W(k that can allocate a fraction t of the channel-optimal profit to the upplier and a fraction 1 t to the retailer, if and only if t [, t max (k; F ], where = k ( F (k c/p t max (k; F def k ( F (x c/p dx. Furthermore, if F ha the IGFR property, then t max (k; F i weakly decreaing a k increae in the range [, q. Proof. See Appendix E. Let u interpret Theorem 4 at two extreme value for the capacity k. A k approache q, t max (k; F approache zero. Thu the upplier can not get any fraction of the channel-optimal profit with any wholeale price contract from W(k (thi wa to be expected becaue W(k = {c} when k q. At the other extreme, a k tend to zero, t max (k; F tend to one. Thu any allocation of the channel-optimal profit become poible with ome wholeale price contract from W(k (thi i natural, becaue a k tend to zero, the interval W(k become [c, p]. See Figure 3. Theorem 5 parallel Theorem 3 and make precie the idea that when we erve a larger market the flexibility in allocating the channel-optimal profit increae. Theorem 5. Under the ame aumption a in Theorem 3, we have Proof. See Appendix F. t max (k; F 1 t max (k; F 2. Theorem 5 ugget that a upplier (and retailer can find flexibility in profit allocation by joining a upply channel that erve a larger market.

10 1 LIDS Technical Report #2749, February 27. Figure 3 Flexibility in allocating channel-optimal profit a a function of the capacity contraint. Maximum fraction of allowable channel profit allocation to the upplier 1..9 fraction of channel optimal profit to the upplier Channel optimal Not channel optimal capacity, k Note. Demand i ditributed according to a Gamma ditribution with mean 1 and coefficient of variation 2 1/2.77. The retail price i p = 1, and the cot i c = 4. (thee are imilar to parameter ued in Cachon (24. Thu, q The haded region denote the fraction of profit to the upplier conitent with a channel-optimal outcome (i.e., the et [, t max (k; F ]. Or in other word, the haded region repreent the fractional allocation of channel-optimal profit to the upplier that are achievable with ome wholeale price contract w W(k Equilibrium etting. The equilibrium etting we analyze i a two-tage (Stackelberg game. In the firt tage, the upplier (the leader et a wholeale price w. In the econd tage, the retailer (the follower chooe an optimal repone q, given the wholeale price w. The upplier produce and deliver q unit before the ale eaon tart and offer no replenihment. Both the upplier and retailer aim to maximize their own profit. The upplier payoff function i π (w; q = (w cq and the retailer payoff function i π r (q; w = E[pS(q wq]. Lariviere and Porteu (21 analyze thi Stackelberg game, for an uncontrained channel with one upplier and one retailer. They find that when F ha the IGFR property, the game reult in a unique outcome (q e, w e defined implicitly in term of the equation p F (q e (1 g(q e c =, (3 p F (q e w e =, (4

11 LIDS Technical Report #2749, February where g i the generalized failure rate function g(y def = yf(y/ F (y. Furthermore, they how that the outcome i not channel optimal. In thi ection, and in Section 4.6, we explore the efficiency of the outcome when the channel ha a capacity contraint (i.e., q k. Theorem 6 provide neceary and ufficient condition on the channel capacity contraint k for the Stackelberg game to reult in a channel-optimal equilibrium. Theorem 6. Aume F ha the IGFR property. Conider the above decribed game, when the channel capacity i k unit. Thi game ha a unique equilibrium, given by q eq (k = min{k, q e } and w eq (k = max{p F (k, w e }, where q e and w e are defined by equation (3 and (4, repectively. Thi equilibrium i channel optimal if and only if k q e. (5 Under thi condition, we have q eq = k and w eq = p F (k. Proof. See Appendix G. The function p F (y (1 g(y c repreent the upplier marginal profit on the yth unit, when y < k. When F ha the IGFR property, the upplier marginal profit i decreaing in y, while the marginal profit i nonnegative. Thi fact and equation (3 imply that inequality (5 i equivalent to the inequality p F (k (1 g(k c, which can be interpreted a a tatement that the upplier marginal profit (when relaxing the capacity contraint on the kth unit i greater than zero. Therefore, inequality (5 ugget that when the capacity contraint i binding for the upplier problem (the leader in the Stackelberg game, then the outcome of the game i channel optimal and vice-vera. If the channel capacity k i large enough, o that inequality (5 i not atified, how inefficient i the channel? In Section 4.6, we provide a ditribution-free meauring tick for the efficiency lo in channel with a capacity contraint When can both partie be better off? The et of coordinating wholeale price contract W(k introduced in Theorem 1 ha many merit in a negotiation etting. For example, uch contract are Pareto optimal. In contrat, Theorem 7 examine the et of wholeale price contract D(k that have little merit in that they are Paretodominated by ome other wholeale price contract in [c, p]. A contract i Pareto-dominated if there exit an alternative linear wholeale price contract that make one party better off without making any other party wore off. Having a complete picture of the contract that are channel-optimal and the contract that are Pareto-dominated i helpful in a negotiation etting.

12 12 LIDS Technical Report #2749, February 27. Theorem 7. Aume F ha the IGFR property and that the quantity q e and wholeale price w e are defined implicitly in term of equation (3 and (4. If k q, then the et of Pareto-dominated wholeale price contract D(k i Proof. See Appendix H. D(k def = (max{w e, p F (k}, p] = (p F (min{q e, k}, p]. Note that W(k and D(k are dijoint. Corollary 3 to Theorem 7 formalize the idea that when k i mall enough, W(k and D(k partition the et [c, p]. Figure 4 illutrate thee idea when demand ha a Gamma ditribution. Figure 4 An example illutrating W(k and D(k. Two et of wholeale price a a function of capacity: W(k and D(k 1 9 D(k 8 wholeale price 7 6 W(k Neither capacity, k Note. We ue the ame parameter a in Figure 3, reulting in q 1.112, q e 4.784, and w e The et of coordinating wholeale price contract W(k lie under the olid curve. The et of Pareto-dominated wholeale price contract D(k lie above both the olid and dahed curve. The et of contract that lie between the olid and dahed curve are neither in W(k nor in D(k. Such contract do not coordinate the channel, but neverthele, are not Pareto dominated by coordinating wholeale contract. Corollary 3. Aume F ha the IGFR property. If k q e, then W(k D(k = [c, p], (6 W(k D(k =. (7

13 LIDS Technical Report #2749, February Corollary 3 i epecially intereting: it aert that when capacity i mall enough there are only two type of contract: good contract, W(k, and bad contract, D(k. Furthermore, both partie will alway have a reaon to avoid the bad contract becaue they are Pareto-dominated by ome channel-optimal contract in the et W(k Efficiency Lo. When the outcome of the Stackelberg game we decribed in Section 4.4 reult in a wholeale price contract that i not channel optimal, how much doe the channel loe a a reult? What i the price paid for the gaming between the upplier and retailer? To quantify the anwer we analyze the wort-cae efficiency. Our definition of efficiency i related to the concept of Price of Anarchy, PoA, a ued by Koutoupia and Papadimitriou (1999, and Papadimitriou (21. PoA ha been ued a a meauring tick in an aortment of gaming context: facility location (Vetta 22, traffic network (Schulz and Moe 23, reource allocation (Johari and Titikli 24. More recently Peraki and Roel (26 analyze the PoA for an aortment of upply channel configuration with the IGFR retriction, but not for reource contrained channel. Theorem 8 complement their reult, by providing an efficiency reult for the Stackelberg game of Section 4.4, in the preence of a capacity contraint k. For a channel with a capacity contraint k and probability F (k of exce demand, we define the parameter β def = max{ F (k,c/p} c/p. The parameter β depend on the probability F (k of exce demand and take value from the et [1, p/c]. It quantifie how contrained the channel i with repect to the channel optimal order quantity q, becaue β def = max{ F (k,c/p} = max{ F (k, F (q } c/p F (q. In the Stackelberg game with a capacity contraint k and parameter β, the efficiency, Eff(k, β, i defined according to equation (8 below. Eff(k, β = inf F F(k,β Channel profit under gaming Optimal channel profit E[pS(q eq (k cq eq (k] = inf F F(k,β E[pS(q (k cq (k] (8 The et F(k, β repreent the et of probability ditribution that atify Aumption 1, have the IGFR property, and uch that the probability F (k of exce demand atifie max{ F (k,c/p} c/p = β. Note that Eff(k, β i a ditribution-free method of quantifying the wort-cae efficiency. When Eff(k, β i low (much maller than one, there i ignificant efficiency lo due to gaming. Theorem 8. Define m def = (p c/p (the channel gro profit margin. For the Stackelberg game decribed in Section 4.4, we have ( (β 1 + m ( 1 Eff(k, β = m β 1 1/m ( 1 1. (9 1 m 1 m

14 14 LIDS Technical Report #2749, February 27. Proof. See Appendix I. Note that Eff(k, β i decreaing in the channel gro profit margin m and increaing in β. ( ( When β = 1, the channel i not contrained and Eff(k, β equal 1 1/m 1 1 m 1 1 m which, after ome algebraic manipulation, matche the reult in Peraki and Roel (26. On the other hand, when the channel i mot contrained (i.e., k, F (k 1, and β p/c, then Eff(k, β implifie to 1. In other word there i no efficiency lo becaue the equilibrium outcome involve the retailer ordering exactly k. Our reult i thu a more general verion of the two-tage puh-mode PoA reult in Peraki and Roel (26 in that we account for a capacity contraint. Alo our proof technique differ from and complement Peraki and Roel (26, in that we indirectly optimize over the pace of probability ditribution by optimizing over the pace of generalized failure rate. Figure 5 An example illutrating Eff(k, β when m = Efficiency beta Note. We fix the margin (p c/p =.35 and ee how Eff(k, β change a a function of β. Figure 5 provide an example of the Eff(k, β when the channel gro profit margin i 35 percent. Figure 5 illutrate that for channel with maller capacity (i.e., higher β, the wort-cae efficiency (a meaured by Eff(k, β i larger. 5. Extenion to the cae of Multiple Supplier We conider a retailer who order from multiple upplier (where each upplier offer one differentiated product, ubject to a contraint on the total amount of inventory that can be tocked. The

15 LIDS Technical Report #2749, February market price for each product i fixed. The retailer face a random demand for each one of the product (product ubtitution i not allowed, which i independent of the quantitie tocked. In thi context, the retailer mut make a portfolio deciion: which upplier to order from, and how much to order from each. For thi model, we explore quetion imiliar to thoe tudied for the ingle-product cae. Do there exit nontrivial (wholeale price different than the unit cot wholeale contract that coordinate the retailer portfolio deciion, reulting in an order quantity vector which i optimal from the channel point of view? How doe the et of coordinating wholeale price vector change a we change the retailer capacity contraint? I everyone better off or no wore off by picking a wholeale price vector in thi et? We will how that our main finding for the 1-upplier/1-retailer cae (Theorem 1 and 6 extend to thi more general etting with many upplier Many-upplier/1-retailer model A rik-neutral retailer r order from m 2 rik-neutral upplier, for m different good, differentiated by upplier. There i a ingle ale eaon, the retailer decide on an order quantity vector/portfolio (q 1, q 2,..., q m and order well in advance of the eaon, the entire order arrive before the tart of the eaon, and finally demand i realized, reulting in ale for the retailer (without an opportunity for replenihment. Without lo of generality, unit remaining at the end of the eaon are aumed to have no alvage value, and there i no cot for tocking out. Supplier i ha a fixed marginal cot of c i per unit upplied and charge the retailer a wholeale price w i c i per unit ordered. The retailer price p i per unit to the market for good i i fixed and, at that price, the demand for good i, D i, i random with p.d.f. f i and c.d.f. F i. We aume that the demand D i are independent random variable, in the ene that their ditribution doe not depend on the ordered quantitie (q 1, q 2,..., q m. The retailer total capacity i again contrained by ome k >. We aume that the capacity a well a the quantitie of the different product are meaured with a common et of unit (e.g., helf-pace, o that the capacity contraint can be expreed in the form q q m k. The model parameter are ummarized in Figure 6, with the arrow denoting the direction of product flow. A before, we aume that the p.d.f. f i for the demand D i ha upport [, l i ], with l i > k, on which it i poitive and continuou. A a conequence, Fi ( = 1 and F i i continuouly differentiable, trictly decreaing, and invertible on (, l i.

16 16 LIDS Technical Report #2749, February 27. Figure 6 m upplier & 1 capacity contrained retailer model with independent downtream demand. 1 c 1 w 1 r p q1 1 D 1 F m q m D m F m p m c m w m q q m k Note. There are m upplier. Supplier i with marginal cot c i (per unit offer good i at wholeale price w i (per unit to a capacity-contrained retailer r who face uncertain demand D i downtream with c.d.f. F i (for good i when the price for the good i fixed at p i (per unit. The retailer decide on a portfolio of good to order from the upplier Retailer problem For product i {1,..., m}, let S i (q def = min{q, D i } denote the (uncertain amount of ale for product i given that the retailer order q i unit of product i. The retailer decide on a quantity vector q r (w = (q r 1, q r 2,..., q r m to order (for a given wholeale price vector w that maximize the expected profit π r (q def = E[ m p i min{q i, D i } w i q i ], ubject to the capacity contraint k. In particular, it olve the following convex program with linear contraint in the deciion vector, q: RETAILER(k,w maximize ubject to m ( pi E[S i (q i ] w i q i k m q i q i, i = 1,..., m. (1 Becaue of our aumption on the ditribution of the demand D i for each product, it can be hown that RETAILER(k,w ha a unique olution (vector, which we denote by q r (w Channel problem Given the channel expected profit π (q def = E[ m p i min{q i, D i } c i q i ] and capacity contraint k, the optimal order quantity vector q for the ytem/channel i the olution to the following convex program, CHANNEL(k, with the ame linear contraint on the deciion vector, q, but a lightly altered objective function: CHANNEL(k maximize m ( pi E[S i (q i ] c i q i (11

17 LIDS Technical Report #2749, February ubject to m k q i q i, i = 1,..., m. Again, becaue of our aumption on the demand ditribution, it can be hown that CHANNEL(k alo ha a unique olution (vector which we denote by q. Finally, we denote the unique olution for the uncontrained channel problem, max q< π (q, by q The et W(k. In thi ubection, (cf. Theorem 9 below, we derive condition under which the vector w = (w 1,..., w m belong to the et W(k of wholeale price vector that coordinate the retailer order quantity vector, i.e., q r (w = q. Throughout thi ubection, we aume that the capacity contraint i binding for the channel, that i, m q i k or equivalently m ( F 1 ci i k. p i Otherwie, the problem degenerate into m tandard 1-upplier/1-retailer problem in which the only way to coordinate the retailer action for the upply channel i with a wholeale price contract w = (c 1,.., c m. Theorem 9. Let Z = {i q i = } M = {1,..., m} be the et of product that are not ordered in the channel portfolio deciion problem, and define λ m+1 implicitly by the equation: ( F 1 pj c j λ m+1 j = k. p j j M\Z For any wholeale price vector w = (w 1, w 2,..., w m, the following two condition are equivalent. (a The vector w coordinate the retailer portfolio deciion, i.e., q r (w = q. (b There exit ome α that atifie Proof. See Appendix J. α [, λ m+1 ], (12 w j = c j + α, j M \ Z, (13 w j p j λ m+1 + α, j Z. (14 Let W(k be the et of all w for which q r (w = q. If Z = (o that every product i in the channel optimal portfolio, W(k can be repreented geometrically by a line egment that tart at the point (c 1, c 2,..., c m, ha unit partial derivative, and end at the interection of the line with the et of vector w that atify m et decribed by the condition (12 through (14. 1 F i ( w i p i = k. More generally, if Z, then W (k i the

18 18 LIDS Technical Report #2749, February The Stackelberg game with multiple upplier. We now conider a generalization of the Stackelberg game analyzed in Section 4.4. In the firt tage, all the upplier (the leader imultaneouly chooe their wholeale price w i. In the econd tage, the retailer (the follower chooe an order quantity vector q. When doe an equilibrium wholeale price vector of thi game belong to the et W(k? A full exploration of thi and other quetion related to thi intereting game i beyond the cope of thi paper and i reported elewhere (Sabbaghi et al. 27. We only provide here one reult that connect to and generalize Theorem 6. Theorem 1. Aume the game i ymmetric for the upplier, that i, c i = c, p i = p, and F i = F, for every upplier i. Furthermore, aume that F ha the IGFR property. Recall the definition of q e given in equation (3. If k m q e, then there exit a ymmetric equilibrium that belong to W(k. Proof. See Appendix K. 6. Rik haring contract We have provided neceary and ufficient condition o that linear wholeale price contract coordinate a newvendor procurement deciion and allow both the upplier( and the newvendor to profit. A natural related quetion i whether more complicated contract uch a buy-back contract and revenue-haring contract alo coordinate a newvendor procurement deciion when the newvendor i capacity-contrained. In thi ection, we prove that revenue-haring contract and buy-back contract continue to coordinate a newvendor ordering deciion even when the newvendor ha a contrained reource. Furthermore, we examine the advantage of thee more complex contract over a linear wholeale price contract for a contrained newvendor Buyback and revenue-haring contract for uncontrained newvendor till coordinate In Theorem 11, we how that buyback contract, which are known to coordinate an uncontrained newvendor procurement deciion, continue to coordinate a contrained newvendor procurement deciion. Theorem 11. Conider a 1-upplier/1-retailer configuration in the preence of a capacity contraint k. Buyback and revenue haring contract coordinate the retailer ordering deciion for the channel, and allow for any profit allocation. In particular, the buyback and revenue haring contract that coordinate an uncontrained retailer (in the correponding uncontrained channel continue to coordinate the contrained retailer order deciion and allow for any profit allocation.

19 LIDS Technical Report #2749, February Proof. See Appendix L. Figure 7 illutrate the et of buyback contract (w, b that channel-coordinate a capacitycontrained newvendor (a well a uncontrained retailer a decribed in Theorem 11. The buyback contract in Figure 7 are the only buyback contract that can coordinate an uncontrained newvendor. However, the buyback contract in Figure 7 are not the only buyback contract that can coordinate a contrained newvendor. There are more. In Subection 6.2 we find neceary and ufficient condition for a buyback contract (w, b to coordinate a capacity-contrained newvendor. Figure 7 Some buyback contract (w, b that channel-coordinate a contrained newvendor. buyback w parameter p F(kp c p buyback b parameter Note. The buyback contract (w, b that channel-coordinate an uncontrained newvendor ordering deciion (the one graphed in thi figure till coordinate a capacity-contrained newvendor. F (kp i labelled on the y-axi purely for comparion with Figure Neceary and ufficient condition for coordination In Theorem 12, we how that the et of buyback contract that coordinate an contrained newvendor procurement deciion i a uperet of the et of buyback contract that coordinate an uncontrained newvendor procurement deciion. Theorem 12. Conider a 1-upplier/1-retailer configuration in the preence of a capacity contraint k, and aume that F (k > c/p. A buy-back contract (w, b {(u, v c u p, v u} coordinate a newvendor procurement deciion for the channel if and only if (w, b {(u, v u = (1 λv + λp, λ [ c/p, F (k ] }. Proof. See Appendix M.

20 2 LIDS Technical Report #2749, February 27. Notice that if capacity become large enough (o that k q, then the et of coordinating buyback contract implied by Theorem 12 and Figure 8 implifie to the claical et of coordinating buyback contract implied by Theorem 11 and Figure 7. Figure 8 Neceary and ufficient condition for a buyback contract (w, b to channel-coordinate a contrained newvendor. buyback w parameter p F(kp c 1 p buyback b parameter Note. The haded area repreent all the buyback contract (w, b that channel-coordinate a capacity-contrained newvendor when k q c. Compare with Figure Dicuion One of the reaon that revenue-haring, buyback, and an aortment of other contract are able to coordinate the retailer in an uncontrained etting i becaue thoe contract have two or more parameter. Intuitively, the flexibility of thoe parameter create contract where the retailer ha an incentive to order the ytem-optimal amount and that allow the upplier to earn a profit. Interetingly, our model alo introduce another parameter, capacity. But capacity i not part of the contract. Rather it i part of the ytem. So intead of introducing complexity into the contract (with another contract parameter one hould check if an inherent reource parameter (uch a capacity can lead to the ue of impler contract. Demand and capacity are both lever in the ytem. If demand i large relative to capacity for the channel problem, then wholeale price contract that coordinate the channel and allow both the upplier and retailer to profit exit. Conequently the potential to reach a channel optimal outcome in a negotiation etting exit. Furthermore, in the Stackelberg game (Section 4.4 where the upplier act a the leader, if the capacity contraint i tight, the equilibrium outcome i channel optimal. Otherwie, when the

21 LIDS Technical Report #2749, February equilibrium i not efficient (becaue the capacity k i not mall enough, we provide a ditributionfree wort-cae characterization of the efficiency lo, a meaured by Eff(k, β (ee Section 4.6. Another leon for contrained channel i that buyback and revenue haring contract till coordinate the channel (ee Section 6.1. And thoe contract coordinate the contrained channel for a larger et of parameter than for the uncontrained cae. Cachon (23 mention that coordination in multiple upplier etting ha not been explored. Theorem 9 and 1 are initial tep in that direction. Reource contraint are a part of many upply channel. Thi paper how that taking them into conideration in the analyi i important in aeing the actual efficiency of contract for contrained channel. Acknowledgment Thi reearch wa upported by the MIT-Zaragoza Project funded by the government of Aragón. Appendix In order to not dirupt the flow of preentation, the proof for our reult are contained here. A. Proof: 1-upplier/1-retailer, Set of wholeale price W(k Proof of Theorem 1. We tart by proving that if w W(k, then q r (w = q. Suppoe firt that q k. We then have p F (min{q, k} = p F (q = c. Therefore, W(k = {c}. Thu, for any w W(k, the problem RETAILER(k,w and CHANNEL(k are the ame and q r (w = q. Suppoe now that q > k. We then have q = k and, furthermore, p F (min{q, k} = p F (k > p F (q = c. (The trict inequality i obtained becaue F i trictly decreaing. Therefore, W(k = [c, p F (k]. Solving ( x E[pS(x] p F (kx = for x [, l] and noting S(x = F (x, we obtain q r (p F (k = k. Since q r (w i x nondecreaing a we decreae w, we ee that for all w W(k, q r (w = k = q. Suppoe now that q r (w = q and c w p. We have hown that { {c}, if q k; W(k = [c, p F (k], if q > k. When q k, the firt order condition imply that p F (q r (w w = = p F (q c for any w c, which implie w mut equal c. When q > k, we know that q = k. Aume w > p F (k when q r (w = q. Due to invertibility around k, q r (w < k. Thi i a contradiction becaue q = q r (w < k. B. Proof: Impact of ize of Market on ize of W(k Proof of Corollary 2. Let q i = demand ditribution F i. 1 F i (c/p be the order quantity (for an uncontrained channel under the If k q 2, then c/p F 2 (k F 1 (k, which implie that k q 1. Thu, W i (k = [c, p F i (k] for i 1, 2. Since F 2 (k F 1 (k, we can conclude that W 2 (k W 1 (k [c, p]. Similarly, if q 2 < k, then W 2 (k = {c}. Thu, W 2 (k W 1 (k.

22 22 LIDS Technical Report #2749, February 27. C. Proof: Revenue requirement implicit in W(k Proof of Theorem 2. If the capacity contraint k i binding for the channel (i.e., q > k, then W(k = [ c, p F (k ]. For any wholeale price, the upplier fraction of expected revenue i r (w def = wq(w/e[ps(q(w] where q(w i the retailer order quantity for a wholeale price w. Thu for any coordinating linear wholeale price contract w W(k, r (w = wk E[pS(k] = wk p k F (x dx. The maximum poible value for r (w, when w W(k, i ( r max (k; F p F (k k = pe[s(k] = k F (k k F (x dx. Accordingly, the expected revenue that the retailer receive with any linear wholeale price contract w W(k i at leat a fraction of the total. 1 k F k (k k F (x dx = F (x dx k F (k F (x dx Next we how that if F ha the IGFR property, then r max (k; F i weakly decreaing a k increae. We firt note that where g(x def = xf(x F (x lim k r max r max (k; F = k k k F (k F (x dx (1 g(k r max (k; F, (15 i the generalized failure rate function. From L Hôpital rule, we alo have (k; F = 1. Furthermore, the function r max (k; F i bounded above by 1 and goe to zero a k. If thi function i not weakly decreaing, there mut exit ome value t uch that the derivative of r max (k; F at t i zero, and poitive for value lightly larger than t. We then have ince the derivative of r max r max (t; F = 1 g(t (16 (k; F at t i zero. For k lightly larger than t, the function r max (k; F increae, and g(k i nondecreaing, by the IGFR aumption. But then, equation (15 implie that the derivative of r max (k; F i negative, which i a contradiction. D. Proof: Revenue requirement a we vary F Proof of Theorem 3. Note that k F i (x dx = ( F i (x dx ( F k i (x dx = E[D i ] E[(D i k 1 {Di >k}]. Thu, k F 1 (x dx = E[D 1 ] E[(D 1 k 1 {D1 >k}] = E[D 2 ] E[(D 1 k 1 {D1 >k}] E[D 2 ] E[(D 2 k 1 {D2 >k}] = k The inequalitie (17 and F 1 (k F 2 (k imply that F 2 (x dx. (17 F 1 (k R k F 1 (x dx F 2 (k R k F 2 (x dx.

23 LIDS Technical Report #2749, February E. Proof: W(k flexibility in allocating the channel-optimal profit Proof of Theorem 4. We firt recall that given our aumption k < q, the et of coordinating wholeale price contract i W(k = [ c, p F (k ]. Firt we prove that t [, t max (k; F ], if and only if there exit a wholeale price contract w W(k uch that w allocate a fraction t of the channel-optimal profit to the upplier (and thu a fraction 1 t to the retailer. For any wholeale price w, the upplier fraction of the channel expected profit i t (w def = (w cq(w E[pS(q(w cq(w] where q(w i the retailer order quantity for a wholeale price w. For any coordinating linear wholeale price contract w W(k, the retailer order k unit; thu we can implify t (w: t (w = (w ck E[pS(k] ck = k k(w/p c/p (. (18 F (x c/p dx Oberve that t (c =, t (p F (k = t max (k; F, and t (w i trictly increaing and continuou in w for w [ ] c, p F (k. Thu, t (w i a one-to-one and onto map from the domain [ c, p F (k ] to the range [, t max (k; F ]. Next we how that if F ha the IGFR property, then t max (k; F def = k ( F (k c/p R k ( F (x c/p dx i weakly decreaing a k increae. Define H(x = F (x c/p. Since F (q = c/p, H(x 1 c/p retricted to the domain [, q i equal to 1 H(x, where H i a c.d.f. with upport [, q. The generalized failure rate function g H (x for H, defined in equation (19 below, can be rewritten in term of the generalized failure rate function g F (x for F, a follow: g H (x def = x H(x x (19 H(x = xf(x F (x c/p F (x = F (x c/p xf(x F (x F (x = F (x c/p g F (x. (2 Furthermore, ( x F (x = F (x c/p f(x c/p ( F, (21 (x c/p 2 which implie that Since F (x F (x c/p i weakly increaing (over the domain [, q. F (x i poitive and weakly increaing and F ha the IGFR property, we can deduce that H alo F (x c/p ha the IGFR property when retricted to the domain [, q (becaue of equation (2. Then, Theorem 2 (applied to H implie that k H(k R k H(x dx retricted to the domain [, q. But t max (k; F = k H(k a k increae (and k < q. R k H(x dx i weakly decreaing a k increae (while k i, which prove that tmax (k; F i weakly decreaing

24 24 LIDS Technical Report #2749, February 27. F. Proof: Flexibility in allocating the channel-optimal profit a we vary F Proof of Theorem 5. Given the definition of t max (k; F (cf. Theorem 4, we need to prove that F 1 (k c/p ( F1 (x c/p dx F2 (k c/p ( F2 (x c/p dx. (22 k We know that F 1 (k F 2 (k and that the capacity contraint i binding for the channel problem under both ditribution. Thu, k F 1 (k c/p F 2 (k c/p >. (23 From inequality (17 in the proof of Theorem 3, we alo know that k F 1 (x dx k F 2 (x dx. Thu, we can deduce that k ( < F1 (x c/p k ( dx F2 (x c/p dx. (24 Inequalitie (23 and (24 imply that inequality (22 hold. G. Proof: When i the equilibrium of the Stackelberg game channel optimal? Proof of Theorem 6. The retailer profit function π r (q; w under a wholeale price contract w i defined a π r (q; w def = E[pS(q wq]. Since π r (q; w i concave, in q, we can ue the firt order condition and conclude that for a wholeale price w [c, p], the contrained retailer order quantity q r (w i given by q r (w = min{k, F 1 (w/p}. (25 The upplier profit function π (w; q under a wholeale price contract w i defined a π (w; q def = (w cq. Since q r (w i the retailer bet repone in the econd tage to a wholeale price w by the upplier in the firt tage, equation (25 allow u to expre the upplier objective function a follow: { (w c k, if c w max{c, p π (w = F (k}; ( p F (qr (w c q r (w, if max{c, p F (k} < w p. For w > max{c, p F (k}, note that π(w w (26 = ( p F (q r (w (1 g(q r (w c qr (w. Since the function w p F (y (1 g(y c i trictly decreaing in y when it i nonnegative and equal zero at q e (ee equation (3, we can deduce that ( p F (q r (w (1 g(q r (w c > for w > w e (becaue q r (w < q e. Furthermore, qr (w w < for w > p F (k. Therefore, we can conclude that π(w w < for w > max{we, p F (k}. Either the inequality p F (k < w e hold or the inequality w e p F (k hold. Firt aume the inequality p F (k < w e hold. Equation (26 implie that π (w i increaing linearly between c and max{c, p F (k}. Furthermore, ince ( p F (q r (w (1 g(q r (w c < for w < w e (becaue q r (w > q e, we can deduce that π (w w = ( p F (q r (w (1 g(q r (w c qr (w w for w > max{w e, p F (k} = w e. Therefore, w eq (k = w e > for w (max{c, p F (k}, w e. And we know π(w w < and equation (25 and (4 imply q eq (k = q e. The inequality p F (k < w e i equivalent to the inequality q e < k (ee equation (4. Therefore, when q e < k hold, the inequality w eq (k = w e > max{c, p F (k} = p F (min{q, k} hold and we can deduce that w eq (k / W(k (uing Theorem 1. Next aume w e p F (k hold. Since π(w w < for w > max{we, p F (k} = max{c, p F (k}, equation (26 implie w eq (k = p F (k and equation (25 implie q eq (k = k. The inequality w e p F (k i equivalent to the inequality k q e (ee equation (4. Therefore, when k q e hold, the equality w eq (k = p F (k = max{c, p F (k} = p F (min{q, k} hold and we can deduce that w eq (k W(k (again uing Theorem 1.