CoordinationMechanisms for a Distribution System with One Supplier and Multiple Retailers
Fangruo Chen * Awi Federgruen * Yu-ShengZheng GraduateSchool of Business, Columbia University, New York,New York10027 GraduateSchool of Business, Columbia University, New York,New York10027 The WhartonSchool, University of Pennsylvania, Philadelphia,Pennsylvania 19104 [email protected] * [email protected]* [email protected]
W e address a fundamental two-echelon distributionsystem in which the sales volumes of the retailers are endogenously determined on the basis of known demand functions. Specifically, this paper studies a distribution channel where a supplier distributes a single product to retailers, who in turn sell the product to consumers. The demand in each retail market arrives continuously at a constant rate that is a general decreasing function of the retail price in the market. We have characterized an optimal strategy, maximizing total sys- temwide profits in a centralized system. We have also shown that the same optimum level of channelwide profits can be achieved in a decentralized system, but only if coordination is achieved via periodically charged, fixed fees, and a nontraditional discount pricing scheme under which the discount given to a retailer is the sum of three discount components based on the retailer's (i) annual sales volume, (ii) order quantity, and (iii) order frequency, respec- tively. Moreover, we show that no (traditional) discount scheme, based on order quantities only, suffices to optimize channelwide profits when there are multiple nonidentical retailers. The paper also considers a scenario where the channel members fail to coordinate their deci- sions and provides numerical examples that illustrate the value of coordination. We extend our results to settings in which the retailers' holding cost rates depend on the wholesale price. (Coordination;Pricing; Quantity Discounts; Supply Chain Management)
1. Introduction integration, supply chain partnerships such as vendor A production and distribution channel often encom- managed replenishments, contracts specifying deci- passes independent firms or decentralized divisions sion rules for all channel members, and profit-sharing of the same firm. The channel members typically schemes. See Jeuland and Shugan (1983), Johnston optimize their own performance based on locally and Lawrence (1988), Tirole (1988), and Buzzell and available information. Driven by competitive pres- Ortmeyer (1995) for discussions of these mechanisms sures and enabled by modern information technology, and their limitations. The preferred way to achieve many supply chains have come to realize that their coordination is often to maintain decentralizeddecision overall performance can be improved dramatically by making but to structure the costs and rewards of all employing novel mechanisms to coordinate decisions. members so as to align their objectives with the sys- Traditional mechanisms to optimize the overall temwide objective, i.e., to identify a coordinationmech- channel performance include vertical or horizontal anism. If the decentralized cost and reward structure
0025-1909/01/4705/0693$5.00 MANAGEMENTSCIENCE ? 2001 INFORMS 1526-5501electronic ISSN Vol. 47, No. 5, May 2001 pp. 693-708 CHEN, FEDERGRUEN,AND ZHENG SupplyChain Coordination Mechanisms results in channelwide profits equal to those under demand functions and cost parameters are stationary a centralized system, the coordination mechanism is and common knowledge among the channel mem- perfect. bers. Extensions to settings with asymmetric informa- We consider the following two-echelon system. A tion, as in Corbett and de Groote (2000), are worthy supplier distributes a single product to multiple retail- of study. ers who in turn sell to consumers. The retailers serve We first characterize the solution to the centralized geographically dispersed, heterogeneous markets, in system with a central planner who makes all pric- which demands occur continuously at a rate that ing and replenishment decisions so as to maximize depends on the price charged by the retailer according the channelwide profits. We then show that the cen- to a general demand function. The supplier replen- tralized solution can be realized in a decentralized ishes his inventory through orders (purchases, pro- system if the supplier offers a discount from a list duction runs) from a source with ample supply. If the price based on the sum of three discount components; system operates in a decentralized manner, the sup- each is based on a single retailer characteristic, i.e., plier charges a wholesale price (schedule) and makes (1) annual sales volume, (2) order quantity, and (3) order its own replenishment decisions, each of the retailers frequency. The first two discount components have determines its retail price as well as its replenishment been widely studied in the marketing, operations, and policy from the supplier, and all parties maximize industrial organization literature, but this appears to their own profit functions. (Our model applies equally be the first model in which order-frequency-based dis- to settings where the "supplier" and the "retailers" counts arise as an essential component in the coor- represent different divisions of the same firm, each dination mechanism. Under this pricing scheme, the operating as an independent profit center.) centralized solution emerges as a strong type of equi- The costs consist of holding costs for the invento- librium in the decentralized system. ries at the supplier and the retailers, and fixed and We also show, with simple counterexamples, that variable costs for their orders. For each retailer order, traditional order-quantity discount schemes alone the supplier incurs a fixed order-processing cost and (pricing rules that determine discounts on an order- the retailer incurs a setup cost. We initially assume, as by-order basis as a function of the absolute order size, in most standard inventory models, that the holding- such that the average price per unit decreases with cost rates are exogenously specified parameters. (The the order size) do not guarantee perfect coordina- results carry over to the case where the retailer tion. While virtually all of the operations management holding-cost rates are functions of the wholesale price, literature has focused on traditional order-quantity see ?8.) Our model permits an additional cost compo- discounts, in practice the vast majority of discount nent: The supplier may incur a specific annual cost for schemes are based on other criteria. In a recent field managing each retailer's needs and transactions. In study Munson and Rosenblatt (1998) document that the consumer electronics industry, for example, some no less than 76% of the study participants experi- suppliers establish a "management team" of logistics ence discount plans that "aggregate over time," i.e., managers and sales and production representatives where the discount is based on the annual sales vol- for each of their major retailer accounts to monitor ume. They also document the prevalence of schemes the retailers' needs, transactions, and forecasts, and to where a discount is given "for every extra week's negotiate and implement sales and logistical terms, worth of sales that is added to an order" (p. 360), etc. The same management team may be devoted as well as schemes based on multiple criteria, e.g., exclusively to a single retailer account or to multi- annual sales volumes as well as individual order ple accounts. We model the "management costs" by sizes. The authors conclude that new models are a concave function of the retailer's annual sales vol- needed to "suggest schedules proposing business vol- ume, reflecting economies of scale. ume discounts aggregated over products and time The retailers are nonidentical, i.e., they can have to help suppliers appropriately parameterize sched- different demand functions and cost parameters. All ules for these increasingly common quantity-discount
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aggregations" (p. 365), and "to address the increas- These mechanisms are designed to eliminate dou- ing practice of quantity-discount combinations, e.g., ble marginalization (Spengler 1950). Ingene and Parry per-purchase, per-item discounts combined with time (1995) generalize the above model to multiretailer The of discounts aggregation" (p. 365). prevalence settings. based on annual sales volumes has also been docu- The operations literature on channel coordination, mented in the economics literature. See, Brown e.g., on the other hand, has until recently been confined and Medoff (1990) and Lilien et al. (1992) for a gen- to replenishment decisions, assuming that all demand eral discussion of different types of discount schemes. processes are exogenously given. Except for Lal and The discount scheme usually needs to be comple- Staelin (1984), this literature restricts itself to channels mented with periodic fixed fees from each of the with a retailer (or multiple identical retailers). retailers to the supplier. The combined scheme is usu- single Various mechanisms have been to induce ally referred to as a block tariff, see, e.g., Oi (1971) proposed the retailer to the order and Schmalensee (1982). The fixed fees clearly do adopt globally optimal quan- not affect the total profits of the supply chain or the tity, effectively compensating the retailer for devi- .parties' operational policies. These fees are, however, ating from his locally optimal order quantity, see, essential to achieve a proper allocation of the channel e.g., Crowther (1964), Monahan (1984), and Lee and profits; see ?5 below. Rosenblatt (1986). Dealing with multiple noniden- It is of interest to compare the performance of tical retailers, Lal and Staelin (1984) and Joglekar chain under an centralized strat- the supply optimal and Tharthare (1990) propose different coordination in a decentralized with egy (or equivalently system mechanisms that are based on order-quantity dis- the above coordination mechanism) with that perfect counts and do not necessarily achieve the central- of more typical decentralized chains without proper ized optimum. See Dolan (1987), Boyaci and Gallego coordination. One important benchmark for the value (1997), Cachon (1998), Lariviere (1998), Munson and of coordination arises when one of the parties in Rosenblatt (1998), and Tsay et al. (1998) for further the chain, e.g., the supplier, is able to specify the reviews. terms (i.e., the wholesale price) unilaterally, so as to one of the first to maximize his own profits. We consider two Stackel- Weng (1995) represents attempts combine the above two streams of research. We berg games with the supplier as the leader and the gen- retailers as followers. In one, the supplier sets a con- eralize his model with a single retailer or multiple stant wholesale price, and in the other, the supplier identical retailers to allow for an arbitrary number offers an order-quantity discount scheme with one of nonidentical retailers. Although a scheme using an breakpoint. The differences between the channelwide order-quantity discount and a periodic franchise fee in profits these Stackelberg games and those in the suffices to achieve perfect coordination in the setting coordinated two measures system represent possible studied by Weng, we show that when the retailers of the value of coordination. Numerical examples sug- are not identical, such a scheme is not guaranteed to that the gest value of coordination can be significant. coordinate the channel. The marketing literature on channel coordination The remainder of the paper is organized as follows. focuses on decisions. Jeuland and pricing Shugan Section 2 introduces the model. Section 3 considers (1983) consider a channel with one supplier and one the centralized system. Section 4 shows that order- retailer, without inventory replenishment considera- discounts alone are insufficient to tions. There, a simple quantity discount based on quantity guarantee coordination when retailers are nonidentical. the annual sales volume results in perfect coordina- perfect 5 tion. Moorthy (1987) points out that perfect coordi- Section describes our coordination mechanism. Sec- nation can also be achieved with a simple two-part tion 6 analyzes the two Stackelberg games. Section 7 tariff, i.e., the supplier sells the goods to the retailer contains the numerical examples, and ?8 considers an at its marginal cost and charges a fixed franchise fee. alternative model of retailer holding costs.
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2. Model ci = transportation cost per unit shipped from the to retailer i A supplier distributes a single product to N retail- supplier = annual cost incurred the ers, who in turn serve geographically dispersed (thus T(di) by supplier for independent) retail markets. The consumer demand managing retailer i's account, with T(.) = in each retail market occurs continuously at a con- nondecreasing, concave and T(0) 0. stant rate, which is determined by the retail price Let pi(di) denote the inverse demand function charged in the market in accordance with a general (because di(.) is We assume that pi(di)di time-invariant (i.e., stationary) demand function. The decreasing). is concave in di. We assume that > 0 for all i, retail price in each market, once determined, remains hi constant over time. All demands must be satisfied which means the cost of carrying a unit at retailer i is at least as as the cost of it in without backlogging. The supplier and the retailers large carrying the warehouse. Without loss of are independent firms, and each firm has the objec- supplier's generality, we assume the cost is borne the tive to maximize its long-run average profits. transportation ci by retailer. The above costs do not include The supplier replenishes its inventory from a specified any transfer between the and the retail- source with no capacity limit. The retailers place payments supplier ers. For notational convenience, we assume that all orders with the supplier. All replenishment orders incur fixed and variable costs. The fixed cost asso- orders are received instantaneously upon placement. Positive but deterministic leadtimes can be handled ciated with a retailer order has two components: a shift in time of all desired one incurred by the retailer and the other by by simple replenishment the supplier (e.g., an order-processing cost). As epochs. with the consumer demand, retail orders cannot The supplier sets the wholesale price and deter- mines its own The retailers set be backlogged. Each firm incurs inventory-carrying replenishment policy. the retail the demand in costs which at any point in time are proportional price (or alternatively rate) their markets and determine their own to its inventory level. In addition, the supplier replenishment The of each firm is to maximize its incurs a specific annual cost for managing each policy. objective retailer's account. All cost parameters are stationary. long-run average profits. For i = 1,..., N, define
pi = retail price charged by retailer i 3. Centralized Solution di(pi) = annual consumer demand in the market Assume that there is a central planner who makes all served by retailer i, a decreasing function the pricing and replenishment decisions so as to max- of Pi imize the systemwide profit, which is equal to the Ko = fixed cost incurred by the supplier per order revenue (EN=pi(di)di) minus costs. The cost compo- placed by the supplier nents are: variable costs EN (co + c)di, account man- Ks = fixed cost incurred by the supplier per order agement costs EN=1 (di), and inventory-carrying and placed by retailer i setup costs. Kr = fixed cost incurred by retailer i per order Suppose, for a moment, that all the retail prices placed by retailer i (thus demands) are given. Therefore, maximizing Ki = Ki + Kr profit reduces to minimizing the inventory-carrying ho = annual holding cost per unit of inventory and setup costs. While a single-location model of this at the supplier type is easily solved by the well-known EOQ formula, hi = annual holding cost per unit of inventory the two-stage distribution system is much more dif- at retailer i ficult and was not well understood until the seminal = hi hi- ho, incremental or echelon holding work of Roundy (1985). Roundy shows that, while the cost at retailer i truly optimal replenishment strategy is intractable, a co = cost per unit ordered by the supplier near-optimal solution can be found in the class of
696 MANAGEMENTSCIENCE/Vol. 47, No. 5, May 2001 CHEN, FEDERGRUEN, AND ZHENG Supply Chain CoordinationMechanisms integer-ratio policies. (The optimal integer-ratio pol- consider the problem of maximizing the systemwide icy is proven to come within 6% of a lower bound on profit: the minimum achievable costs.) Let Ti be the replen- ishment interval for retailer i, i = 1,..., N, and To the - - H(d, T) = Z: (pi(di) -c c)di (di)- T replenishment interval for the supplier. An integer- di=l- a T hT KTiO ratio that either or be a policy requires To/Ti Ti/To - 1 - K of is used in hodimax{To,}-Ti h positive integer. (This type policy widely 2 To practice to synchronize replenishment activities.) To N=G(d, Ko synchronize, the system starts out empty (so that all =i (d,T T (2) locations place an order at Time 0), and it is easily i=l o) ? verified that it is for each to ensure optimal facility where d is the demand vector, T the vector of replen- that its is down to zero at all inventory subsequent ishment intervals, and replenishment epochs. This implies that each retailer i's order is in the amount of diTi units. Gi(di, Ti, To) = (pi(di) - c - ci)di - (di) - Roundy also shows that the replenishment policies Ti can be further restricted to the so-called power-of-two 1 1 - hodimax{To, T} - hidiTi. policies with virtually no loss of optimality. A power- 2 2 of-two is one with policy The problem is to determine a power-of-two vector T (satisfying (1)) and a demand vector d that maxi- Ti = 2miTb, mi integer, i = 0, 1,..., N, (1) mize l. Let (To*,T*,... , TN) and (dr,... , d) be the optimal solution, which we assume to be unique. Let where Tbis a given base time period a day). That (e.g., TI*be the maximum channel For an is, the reorder interval at each location is restricted to profits. algo- rithm that determines the optimal solution, see Chen be an integer power-of-two multiple of Tb.Obviously, et al. (1998). (Multiple optimal solutions may arise if a power-of-two policy is also an integer-ratio policy. the functions pi(d)d- T(d) fail to be strictly concave We refer the reader to Roundy (1985) for an algorithm or due to the rounding of replenishment intervals to that determines an optimal power-of-two policy. power-of-two values. On the other hand, if the func- Throughout this paper, we assume that the order tions pi(d)d - T(d) are strictly concave, the optimal intervals at the supplier and any of the retailers are solution is unique almost surely when assuming that all chosen from the discrete set of power-of-two val- the cost parameters are drawn from general continu- ues (2mTb:m = -oo,..., -1, 0,1,...}. Although the ous distributions.) power-of-two grid may appear to be sparse, it in fact limits the cost increase due to the grid restric- tion to a small percentage, while maximizing the ben- 4. Order-Quantity Discounts efits of coordination of replenishment intervals. For The existing literature on channel coordination has in the example, basic EOQ model, restriction to the repeatedly shown that a discount scheme based set of power-of-two values results in an increase of on the order quantity (i.e., an order-quantity dis- the cost value no than by more 6%; see Brown (1959) count scheme) coupled with fixed transfer payments and Roundy (1985). Moreover, power-of-two policies achieves perfect coordination. We show, however, that are appealing for a decentralized system because they an order-quantity discount scheme that is uniformly require less coordination than integer-ratio policies. A applied to all retailers does not guarantee perfect power-of-two policy requires each firm to satisfy the coordination for systems with nonidentical retailers. power-of-two constraint independently; they do not In the terminology employed by the economics liter- have to coordinate their reorder intervals explicitly. ature on vertical contracting, uniform order-quantity Having focused on the replenishment decisions discount schemes are an insufficient set of instru- given pricing/demand decisions, we now proceed to ments; see Mathewson and Winter (1984, 1986). While
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this is not surprising if the supplier's costs of serving in Roundy's model with exogenously prespecified the retailers are themselves nonidentical, it is more demand rates? The following proposition sheds light striking when this cost structure is uniform across all on this question. retailers. We therefore confine ourselves to the case THEOREM2. Assume all retailer demand rates {d*, i = with a uniform cost structure, i.e., KS = Ks for all 1, ... , N} are exogenouslygiven (or equivalently,all retail i=1,... ,N. prices are fixed at their centralized-optimalvalues). Then, THEOREM1. Even when the cost structure is uniform, incremental, as well as all-unit, order-quantitydiscounts a uniform order-quantitydiscount scheme cannot be guar- cannot be guaranteed to achieve perfect coordinationeven anteed to coordinatethe channel with multiple nonidentical when the cost structure is uniform. retailers. PROOF. Consider a system with N = 2 retailers. = 0 PROOF. We prove this via a counterexample. Let Assume KI for i= 1,2, and Ko is sufficiently in w(Q) be the average unit wholesale price when a large relationship to the other parameters that in the centralized retailer orders Q units from the supplier. (Thus, solution To > max{T*, T2}. It is eas- verified from Qw(Q) is the total price paid by the retailer for the ily Roundy (1985) that in this case, = lot.) For w(-) to be a discount scheme, we must have T* = V2Ki/(hid*) 2mi for i = 1,2 if Ki = 22mi-lhid* for some i = w(Q1) > w(Q2) for any Q1< Q2. Suppose pi(di) = ai - integer mi, 1, 2. (Assume the base period = Let = for i bidi for some positive constants ai and bi, i = 1, ... , N. Tb 1.) Qt d7Ti* = 1, 2 denote the central- ized order Without loss of Let Q* = diTi*, i.e., the (globally) optimal order quan- optimal quantities. gener- assume < or tity at retailer i. ality, Q7 Q,, induces the retailers to Suppose w(-) order the dhKl dhK2 not, it does not ' (3) optimal quantity Q*. (If certainly h1 h2 achieve maximum channel profits.) Let w* = w(Qt). Let denote the total the retailers The remaining problem for the retailers is to solve t(Q) price for an order of size Q under an incremen- max{(pi(di) - ci - wi)di - diKi/Q - hi = (ai - b,d, - pay - tal discount scheme. Thus, = Ci *)di- dKQ -hiQi di > 0}. It is easy to see order-quantity t(Q) minl=l . L{kl+ wlQ}, with < k2 < .. < and > that the optimal solution to the above problem is k, kL w, 2 > ... > Because there are two d? = (a, - ci - w - K /Ql)/2bi. To achieve maximum WL. only retailers, it suffices to have L = 2. Now consider the retailers' channel profits, we must have d?= d* or wZ = ai - ci - optimization (in the decentralized Kl/Q - 2bid[, i = 1,..., N. If there exists a pair i and problems system with an incremental order-quantity discount scheme). j such that Q* < Q1 and wi < wj, then w(.) is not a Suppose retailer i chooses k, + wlQ as its whole- discount scheme, i.e., there does not exist any tradi- sale cost function. Then its annual cost can be writ- tional quantity-discount scheme that achieves maxi- ten as diwl + + + where is his mum channel profits. This is the case when, for exam- (di*(Ki k))/Q hiQ, Q order quantity. This is the standard EOQ cost func- ple, N = 2, Ko = 100, K' = K = 0, K = K = 10, ho = tion, and thus the order for retailer h = h2 = 1, c0 = 10, c = 1, P(.) = 0, al = a2 = optimal quantity i is = 100, b1= 10, and b2 = 5. In this example, the glob- Q? /2d*(Ki +kl)/hi. To induce retailer i to use ally optimal solution is: d* = 4.3, d? = 8.6, To = 4, T* = his optimal order quantity in the centralized solution, T2 = 2. Therefore, QT= 8.6 and Q* = 17.2. However, i.e., Q? = Q*, we must have kl/ho = Ki/hi. Assume w* = 11.8372 and w2 = 12.4186. O Kl/h1 : K2/h2. Thus the intercepts k, that induce the retailers to Finally, is the insufficiency of (traditional) order- choose their optimal order quantities are different. < quantity discounts to achieve perfect coordination Because QT Q*, it must be the case that retailer i chooses as his entirely due to the fact that demands are price sen- ki + wiQ wholesale cost func- i = 2. sitive and that retail prices are endogenously deter- tion, 1, Therefore, mined? In other words, would, under a uniform cost k _=--KlhKh2h (. structure, an order-quantity discount scheme suffice 698 MANAGEMENT SCIENCE/Vol. 47, No. 5, May 2001 CHEN, FEDERGRUEN, AND ZHENG SupplyChain Coordination Mechanisms However, there exist examples where (3) and (4) can- impossible to find an all-unit order-quantity discount not be satisfied simultaneously. Here is one such scheme that achieves perfect coordination. O example: d* = 1 and d* = 8, ho = h = h2 = 1, K1 = 2 Note that because any concave increasing func- and K2 = 1. tion t(Q) can be approximated arbitrarily closely We proceed to consider all-unit order-quantity dis- by an incremental order-quantity discount scheme counts. Let 0 < ql < q2 < . < qL-1 be the breakpoints, minl=l,, L{kl+ wlQ), it is clear that any order-quantity and w1 > W2> ... > W the corresponding wholesale discount scheme based on a concave t(.) function is prices, for some positive integer L, i.e., if a retailer's still insufficient to achieve perfect coordination. Of order quantity Q is less than ql, he pays the whole- course, Theorem 2 leaves open the possibility of the sale price wl for every unit he buys, if Q is in existence of an even more general order-quantity dis- [ql, q2) then the wholesale price is w2, etc. Define count scheme that can achieve perfect coordination. ci(Q) = dK,/Q + hiQ, i = 1, 2, i.e., retailer i's aver- However, such a scheme, even if it existed, is likely age holding and setup costs if Q is his order quantity to have a most complex and unintuitive form, and (in the decentralized system). Let Q. be the (uncon- might therefore be impractical. REMARK. The version of our model strained) minimum point of ci(.), i.e., Q' = /2dlKi/hi. single-retailer (without account costs, i.e., T = 0) was Clearly, Q' < Qt. For Q E [ql1_,ql), retailer i's total management first introduced (1995). While it is stated cost function is wld? +ci(Q). Retailer i's objective is to by Weng in his that an discount scheme choose an order quantity that minimizes his total cost. paper order-quantity results in perfect coordination, and Because his objective function is piecewise convex, Boyaci Gallego (1997) question the validity of this result. Chen et al. retailer i's optimal order quantity can be either Q' or a (1997) formally establish that a scheme using an breakpoint. To induce retailer i to choose the central- order-quantity discount and a periodic franchise fee ized optimal order quantity Qt(>Q)), we must have suffices to achieve perfect coordination in the single QT as a breakpoint. Because there are only two retail- retailer or multiple identical retailers settings. ers, it suffices to consider an all-unit order-quantity discount scheme with {Qt, Q*} as its only breakpoints. That is, L = 3, q1 = Q, q2 = Q*, and (wl, w2, w3) are 5. Coordination Mechanisms the corresponding wholesale prices. This section specifies a perfect coordination mecha- Assume Q* = Q'. (There exist examples that sat- nism for the model described in ?2. Every decentral- isfy this condition.) For both retailers to choose ized supply chain requires an upfront specification of their centralized optimal order the follow- quantities, a contract,i.e., a set of ground rules for the commercial ing conditions are w3d* + c2(Q*) < w2d* + necessary: interactions between the different parties involved. c2(Q2) = w2d* + and w2dt + < w3d + C2(QT) c1(QT) Such a contract may involve the specification of a where the first ensures that retailer cl(Q*), inequality pricing rule, the commitment to deliver in whole or 2 does not choose the smaller and the breakpoint, in part (possibly within a specified lead time), return second retailer 1 from a inequality prevents choosing policies, restrictions on the times at which orders may lower wholesale the two price. Combining inequali- be placed and delivered (e.g., daily, every Tuesday or ties, we have Friday, etc.), among others. As explained in Tirole for with information it c2(Q2*) - c2(QQ1) c< (Q2) - c1(QM) systems symmetric (1988, p. 173) < W2- W3< . (5) is immaterial whether the contract is one 2 d* 1 specified by of the channel members, a consortium representing = However, in an example with ho = 3 and hi h2 = 1, all, or a supply chain consultant, as long as all parties = = = d= 2 and d* 1, K1 1 and K2= 8, it can be eas- agree upon the terms. The channel members accept = - ily verified that QT Q' and (c2(Q) c2(Qt))/d* > a contract only if it permits them to achieve a profit (c1(Q*) - cl(Qt))/dl, in which case (5) cannot be sat- value at least equal to their outside opportunity or isfied. Therefore, there exist examples where it is status quo. MANAGEMENTSCIENCE/Vol. 47, No. 5, May 2001 699 CHEN, FEDERGRUEN, AND ZHENG SupplyChain Coordination Mechanisms In our model, the contract consists of the following i = 0,1,...,N, where V = I* - EN=0? > 0. provisions. The supplier commits himself to satisfy all (This corresponds with Fi = I - ? - V/(N + 1), i = retailer orders in their entirety and to deliver them 0, 1,... , N, and is in fact the Nash bargaining solu- with a fixed leadtime, which is without loss of gen- tion of an (N + 1)-person bargaining game. See, e.g., erality normalized to be zero. All channel members Myerson (1991) for a systematic discussion of vari- place their replenishment orders at epochs taken from ous allocation schemes and their properties.) Clearly, the discrete set 2mTb: m = ... , -2, -1,0, 1,2,... ; all fee vectors in 9 = {F : F, = 0, Fi 700 MANAGEMENTSCIENCE/Vo1. 47, No. 5, May 2001 CHEN, FEDERGRUEN,AND ZHENG SupplyChain Coordination Mechanisms inventory is down to zero. Adding Components (i)- management costs. Therefore, the supplier's profit (iii), the average wholesale price retailer i pays to the function is supplier is - K0 N1 + E hodT(A-min{A, T*}-max{To, T }+ T*). KP 'k(di) 1 1 To 2 w= +Co+ + (8) i=1 TidiT d, 22hA-2ho 2 min(A,Ti}. Collecting terms related to T0, the supplier maxi- The for retailer i is long-run average profit given by mizes -Ko/T o-E >N hodi*max{T, T*7}that is equal / 1 1 to n(d*, T) with T = (T0, T*, ... , T) plus a constant, ~(di) - pi(di)di- co+ci+ +di 2 hA 2 hmin{A,Ti}) di and it is maximized at a unique point To= To*.Hence, the is to use a 1- Ks+Kr supplier's optimal strategy power- Ti . of-two with interval -21 hidi7Ti- (9) strategy replenishment To. (The coordination mechanism has been designed to make Notice that this profit function is independent of each player's profit function equal to the total profit the other players' decisions. Therefore, any unique plus a constant, an application of the Groves mech- solution to the above problem represents a domi- anism; see Groves 1973.) In addition, observe that nant strategy for retailer i. Because A- mintA, T} = the centralized solution arises as an iterated domi- - max{A, Ti} Ti and A = To, (9) can be written as nant strategy equilibrium, considering any sequence 1 1 of the players in which the supplier is last. More- - (pi(di) co-ci)di - (di)- hodimax{T*, Ti} - -hidiTi over, the above analysis indicates that the centralized solution is a unique Nash equilibrium of the game. - - Gi(di, Ti, To*), follows from the fact that is a T, (The uniqueness (d*, T*) unique solution of the centralized problem.) Note that which is maximized at a unique point (d*, T*) (see the supplier makes a negative profit, -Ko/To, under (2)). Therefore, the dominant strategy for retailer i is the contract. Therefore, transfer payments from the to employ the demand rate d* and the power-of-two retailers are necessary for the supplier to accept the replenishment strategy with interval Ti*. contract. We conclude: Now suppose the retailers all follow their domi- THEOREM 3. Under the above described coordination nant strategies. Under the above mechanism, the sup- mechanism,in particular, the wholesale mechanism plier receives the following average payments from pricing the retailers: (i)-(iii) and any fee vector in ', the centralized solution (d*, T*) prevails as an iterated dominant strategy equilib- rium (and also a unique Nash equilibrium) in the decen- E + Co+c + hoA tralized channel, and the total channel profit is maximized. Our coordination mechanism combines three kinds - h min{A, T*})d* (10) 2 of discounts. Consider (8), the average wholesale retailer i. Note that diTi is the It is again easily verified that the supplier is best off price paid by (1) retailer's order The the order placing his orders when his inventory is down to zero. quantity. higher quan- the lower the unit This He then incurs the following average costs: tity, price (ceteris paribus). is an order-quantity discount. (2) The ratio I(di)/di decreases with di because T(.) is concave and + E hod max{ To, Ti*- -2 hod*T ) T ( T(0) = 0. Thus, the higher the annual sales volume di, the lower the wholesale price. This is a volume dis- (11) count. As the reorder interval the +K1++|i=+1+ cod* + (d) )j (11) (3) Ti increases, dis- count increases first and stays constant after it exceeds that include fixed costs, holding costs, order- A, a contract parameter. This is an order-frequency processing costs, variable purchase costs, and account discount. As mentioned in the introduction, all of the MANAGEMENTSCIENCE/Vol. 47, No. 5, May 2001 701 CHEN, FEDERGRUEN,AND ZHENG SupplyChain Coordination Mechanisms above three types of quantity discounts have been that the challenged offer, while entailing a different used in practice, and many schemes use combina- marginal price, was made available to all competing tions of the three types of discounts (see Munson and buyers." Depending upon what allocation mechanism Rosenblatt 1998). is used to allocate systemwide profits, the franchise The coordination mechanism is appealing because fees paid by the retailers may either be guaranteed to the differentials in the wholesale prices paid by differ- be identical or not. It is our understanding that U.S. ent retailers are based on the underlying economics fair trade regulations donot require that identical fran- in the system. (1) The discount based on the order chise fee be charged to all retailers or that differences size is due to the order-processing costs KP the sup- between them need to be justified as a sample func- plier incurs. (2) The discount based on the annual tion of one or more of the retailer characteristics. We sales volume reflects the economies of scale in man- refer to Stein and El-Ansary (1992) and Handler et al. aging a retailer account. (3) The discount based on the (1990) for a more detailed discussion of the Robinson- order frequency reflects the savings in holding costs Patman Act. the supplier enjoys when the retailer orders less fre- The above described coordination mechanism is by quently (i. e., the retailer keeps larger inventories). To no means unique. For example, in lieu of the annual see this, first suppose that the retailer's reorder inter- franchise fee F, one can use a profit-sharing plan val is very small. In this case, the supplier has to under which the supplier receives a predetermined hold all the inventory for the retailer. Because the sup- fraction 0i of retailer i's profit, i = 1,..., N. This mod- plier's replenishment cycle is T*, the supplier incurs ified coordination mechanism induces the same equi- an average holding cost of h0oT/2 for every unit librium strategies and is therefore perfect as well. sold by the retailer. This expense becomes part of the (To verify the latter, note that the profit function for basic wholesale price in (6). Now suppose the retailer retailer i is now given by (1 - Oi)Gi(di, ,,A). The replenishes once every Ti units of time. If Ti < To, then supplier maintains the same cost function as in (11) the supplier on average saves hoTj/2 of holding costs while receiving profit shares ENOiGi(d, Ti*,A) in for every unit sold by retailer i. This explains the dis- addition to revenues specified in (10), both of which count given in (7). Now if Ti exceeds To, it is easy to are independent of the supplier's decision variable see that the supplier does not have to hold any inven- To.) More generally, the coordination mechanism may tory for the retailer, and therefore the discount in (7) is use a combination of this type of profit sharing as designed to cancel the holding-cost expense charged well as annual franchise fee F or for that matter, as part of the basic wholesale price. any profit-sharing rule that is a fixed, predetermined Our coordination mechanism (including the fixed and increasing function of the retailer's profits. In the fees) ensures compliance with fair trade laws, in par- remainder of this paper, we confine ourselves to the ticular the Federal Robinson-Patman Act. Section 2(a) original coordination mechanism with franchise fees of this act specifies that it is unlawful "either directly as the sole instrument to reallocate profits. Our choice or indirectly to discriminate in price between pur- is based on the simplicity and transparency of the chasers of commodities of like grade and quality." scheme as well as the fact that most of the economics Quantity discounts are permitted under this section literature on vertical contracting confines itself to this but only to the extent that they are fully justified instrument; see, e.g., Tirole (1988). cost by savings and, hence, identically applied to all One possible limitation of the coordinating pricing retailers. This is precisely the case with our coor- scheme arises when the setup costs I{K, i = 1, ..., N} dination mechanism because any discount awarded are retailer-specific. In this case, the Pricing Scheme to a retailer is directly based on the costs incurred (8) has a component that fails to be uniform across all by the supplier to fill the retailer's order. McAfee retailers, and it is therefore possible that a retailer i and Schwartz (1994, p. 217) report that "courts and with a larger sales volume di and a longer replenish- enforcement agencies (primarily the FTC) have gen- ment interval Ti than some retailer j incurs a higher erally been willing to consider as a defense the fact wholesale price, nevertheless. 702 MANAGEMENTSCIENCE/Vol. 47, No. 5, May 2001 CHEN, FEDERGRUEN, AND ZHENG SupplyChain Coordination Mechanisms REMARK. A critical assumption in this paper is that T(To) = 7r(T) + maxm, TmGm(dm, Tm, To) < 7T(T*) + all demand functions are deterministic, i.e., demands maxdm,T Gm(dm, Tm, To) = ri(To), where the inequal- are deterministically known once the retailer prices ity follows because per definition 7r(To)< Ir(To), and are selected. In practice, demands may be subject Gm(dm,T,, To) < Gm(dm,Tm, To) due to max{T0, Tm}> to a considerable amount of uncertainty, i.e., the max{ T, Tm}.Therefore, the To that maximizes r(To) demand functions are often stochastic. It is interest- must be less than or equal to To. O ing to observe that the identified discount pricing scheme can be implemented in a stochastic setting. The supplier and retailers can continue to agree to 6. An Uncoordinated Channel place orders with replenishment intervals of constant As mentioned, the franchise fees are necessary to length, possibly again chosen to be power-of-two ensure that the contract is attractive to all parties. To multiples of a given base period. The three-part cost this end, all channel members must derive a profit structure, specified in ?5, can continue to be imple- value equal to or in excess of their status quo profit. mented, except for the term in (6). The latter may be This section considers a status quo, where the chan- replaced by its expectation over the relevant stochas- nel members fail to coordinate their decisions. tic demand process; alternatively, the cost component Consider the following scenario. The supplier may be paid periodically on the basis of the real- charges a constant wholesale price and chooses a ized demand volume. Most importantly the three- replenishment strategy with the objective of maximiz- part cost structure continues to adequately reflect the ing its own profits. The retailers take the wholesale economics of the system even when demands are price as given and maximize their individual profits stochastic. While we do not surmise that the coordi- by choosing an optimal retail price and replenishment nation mechanism is perfect in a stochastic setting, we strategy. In other words, the channel members play conjecture that in expectation it is close to being per- a Stackelberg game with the supplier as the leader fect. This needs to be explored in future work. and the retailers as followers. This modus operandi We close this section with the following result, represents many traditional distribution channels in which shows how the coordination mechanism which the channel members fail to coordinate their changes as new retailers enter the system. decisions. Once again, we assume that all replenishment inter- THEOREM 4. Assume that a new retailer, called retailer vals have to be chosen from the discrete set of power- m (m Z 1,... , N), joins the system with demandfunction of-two values {2mTb: m = .. ,-2, -1,0,1,2,... . It dm(Pm)and parametersKm, Cm and hm. Assume the con- is then easily verified that all players are best tract is modifiedto accountfor the new retailer,i.e., a new off employing power-of-two strategies: Each player wholesale pricing scheme and a new set of franchise fees. places an order when his inventory is down to zero The entry of this retailer results in a lower wholesaleprice and adopts a constant, player-specific reorder inter- schedulefor all the existing retailers. val. Thus, in the Stackelberg game, the supplier's PROOF. Note that (8) can be rewritten as w= decision variables are the wholesale price, w, and its K/T,id, + Co+ T(di)/di + homax{A - T,, 01. Thus, the reorder interval To.Given these decisions, the retailers wholesale price schedule shifts down if A (=To) then determine their own retail prices Pi (or demand decreases. Let To (resp., To**) denote the opti- rates di) and reorder intervals Ti to maximize their mal reorder interval at the supplier obtained in own profits. The supplier incurs a fixed cost (K0) the centralized solution before (respectively, after) and a per unit variable cost (Co)for every order he retailer m enters the system. It thus suffices to places, an order-processing cost (Ks) for each retailer show that To**< T. Let Ir(To) (respectively, 7r(To)) order, holding costs at rate ho for his own invento- denote the optimal channel profit in the absence ries, as well as the account management costs (t(di)). (respectively, presence) of retailer m given the The retailers incur a fixed cost (Kr) and a per unit supplier's reorder interval To. For any To > To, transportation cost (ci) for each order they place with MANAGEMENTSCIENCE/Vol. 47, No. 5, May 2001 703 CHEN, FEDERGRUEN,AND ZHENG SupplyChain Coordination Mechanisms the supplier, and holding costs at rate hi for the inven- the former sets, N = 10 and the base case has tories they carry at their sites. Recall that the retail- the following parameters: K0 = 100; c0 = 10; Ks = 0, ers' orders are always shipped immediately. There- Kr = 10 and ci =1 for i = 1,... ,N; hi =1 for i= fore, retailer i's problem is 0, 1,..., N; pi(di) = ai -bidi with ai = 100 and bi = 20 for i = Kr 1,... , N; (d) = f + ed with f = 10 and e= 1. def= max 7ri(di, Ti w) (Pi(di)- ci- w)di- The other examples in this set are obtained by mod- di, Ti Ti ifying the base case one parameter at a time. The set --hidiT i= 1, .,N. (12) with nonidentical retailers has a base case with the same parameters as the previous base case except for that the function Note objective depends only on a ci and bi, now varying among the retailers as follows: the single parameter specified by supplier, i.e., the ci = 1 + i6c/N and bi = 10 + iSb/N, i = 1,... , N, with wholesale w. is of Let price (It independent To.) di(w) 8c = 10 and 8b = 10. The other examples in this set and be the solution to i= 1,..., N. Ti(w) optimal (12), are again obtained by modifying the base case one On the other hand, the is supplier's problem parameter at a time. (w The results are summarized in Figures 1 and 2. max 7r(w, To) f (w- o)di(w)- (di(w))- The channel under the two w, To T profits Stackelberg games are around 70% of the maximum, suggesting that the 1 - value of coordination can be significant. This sug- 2 hodi(w)[To T(w)]+ }- (13) T gests that a perfect coordination scheme can result in We have developed an efficient algorithm for solving major performance improvements for the channel as the above Stackelberg game; see Chen et al. (1998) for a whole, thus encouraging companies and their sup- details. ply chains to engage in coordination mechanisms of A more general Stackelberg game arises when the the type discussed here. Note that when comparing supplier offers a wholesale price schedule, i.e., the the two Stackelberg games, it is clear that the supplier average unit wholesale price w(Q) is a nonincreas- is always better off with a quantity-discount scheme ing function of the order quantity Q. One can, for because the latter provides him with a larger set of example, restrict w(.) to the class of incremental order- feasible strategies, but the same cannot be said of the quantity discounts with a finite number of break- channel profits. The is much harder points. resulting game, however, The double-marginalization phenomenon (Spengler to solve. If there is one only breakpoint, it is possible 1950 and Tirole 1988) recurs in our model, i.e., coor- to solve the via a resulting Stackelberg game complete dination leads to lower (average) wholesale prices, search over all combinations of the basic wholesale (i) lower retail prices, and larger demand rates. For the discounted wholesale and the price, (ii) price, (iii) the base case with identical retailers, the coordina- breakpoint. tion mechanism requires retailer i to pay a wholesale price 12 + (10 + di)/di -0.5 min{4, Ti per unit ordered. 7. Numerical Examples Because d* = 2.1, and T* = 4, each retailer pays a wholesale of In this section, we investigate how large the value price 15.76. This is much lower than the of coordination, i.e., the increase in channel profits wholesale price of 53 in the Stackelberg game (with a resulting from perfect coordination, can be. We use constant wholesale price). With this Stackelberg game as our status quo scenario the Stackelberg games as the status quo, Table 1 shows for the base case a described in ?6. Their channel profits are compared franchise fee of 36.1 based on equal division of the with the maximum channel profits to yield a value of gains from coordination. coordination. The same phenomenon is observed in the sec- We consider two sets of examples, one with ond set with nonidentical retailers. For its base case, identical and one with nonidentical retailers. For the coordination mechanism uses the wholesale price 704 MANAGEMENT SCIENCE/Vol. 47, No. 5, May 2001 CHEN, FEDERGRUEN, AND ZHENG Supply Chain CoordinationMechanisms Figure1 UncoordinatedChannel Profit As a PercentageofMaximum Channel Profit in Centralized Solution, when Retailers Are Identical 80% 75% 75% 70% 70% 65% 65% 60%~I55% 50% I I 2 4 6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 9 10 Number of retailers (N) Unit Transportation Cost (ci) INConstantWholesale Price E QuantityDiscountI ISConstantWholesale Price U QuantityDiscount 80% 80% 75% 75% 70% 70%/ 65% 60% 60% 55% 55%-uu-uuu-.50%I 5 I0 I5 20 25 30 35 10 15 20 25 30 Markettype (bi) Economies of scale (K JAr) IUNConstantWhotesale Price EQuantity DiscountI IE0ConstantWholesale Price EQuantity DiscountjI Figure2 UncoordinatedChannel Profit As a PercentageofMaximum Channel Profit in Centralized Solution when Retailers Are Nonidentical 80% 80% 70% 7~0%1u ii 2 4 6 8 10 12 14 16 18 20 5 6 7 8 9 10 11 12 13 14 15 Number of retailters(N) Delta-c m ostn Wholesale Price E QuantityDiscount EConstant Wholesale Price EQuantity Discount 80% 80% 75% 75% - 70%---h-h----h-i 70% 65%-E------E-E-U--U------65% 60% ------60% 50% 50% 5 6 7 8 9 10 11 12 13 14 15 5 10 15 20 25 30 Delta-b Economiesof scale (KJAr) EoConstantWholesale Price EQuantity Discount!I InMConstantWholesale Price EQuantityDiscountI Table1 ProfitTable, Identical Retailers (Constant Wholesale Price) 11 + (10 + di)/di - 0.5 min{2, Ti}. In the coordina- Stackelberg Coordinated Bargaining Fixed tion system, we have, e.g., (d,, T1)= (3.86,2) and Game System Solution Payments (d10,T10) = (1.9, 2), resulting in a wholesale price of Supplier 316.00 -25.00 336.05 -361.05 13.59 for Retailer 1 and 16.26 for Retailer 10. Thus, OneRetailer 19.55 75.70 39.60 36.10 Retailer 1 receives a substantial volume discount. 511.50 732.00 732.00 Systemwide In the Stackelberg game (with a constant wholesale MANAGEMENTSCIENCE/Vol. 47, No. 5, May 200170 705 CHEN, FEDERGRUEN, AND ZHENG SupplyChain Coordination Mechanisms Table2 Profit NonidenticalRetailers Table, (ConstantWholesale assumed to be nonnegative. The average systemwide Price) inventory and setup costs can be expressed as Stackelberg Coordinated Bargaining Fixed Game System Solution Payments T+ -E +hodimax{To, Ti}+ hidiTi)? Supplier 405.00 -50.00 427.35 -477.35 O i=1i 2 Retailer1 43.09 149.20 65.44 83.77 Retailer2 37.19 132.00 59.53 72.47 The channel profit is thus I'(d, T) = EN1GG'(di, Ti, Retailer3 32.23 117.48 54.58 62.91 To)- Ko/To, where G;(di, i, T,) = (pi(di) - o - ci)d - Retailer4 28.02 - 105.07 50.36 54.71 (di) - Ki/Ti- hd max{T, T} hdiTi. Again, let Retailer5 24.40 94.35 46.74 47.61 T* = (T*, and d* = be the Retailer6 21.27 85.00 43.61 41.39 Ti,..., T) (d,..., d*) reorder intervals and the Retailer7 18.74 76.78 41.08 35.70 optimal optimal demand Retailer8 16.51 69.50 38.86 30.64 rates, respectively. This optimal solution can be Retailer9 14.55 63.01 36.90 26.12 obtained by using the same algorithm outlined in ?3. Retailer10 12.81 57.20 35.16 22.04 Our objective is to identify a coordination mechanism Total 653.81 899.60 899.60 0.00 that induces the decentralized system to achieve the maximum channel profit II'(d*, T*). Consider the decentralized system and suppose we price), the retailers pay a price of 50. The franchise still use the wholesale price mechanism (i)-(iii) pro- fees for this base case are in Table 2, the using posed in ?5. Then retailer i incurs an interest cost of Stackelberg game with a constant wholesale price as I(ci + w), where w is the unit wholesale price. How- the status quo and equal division of froms gains ever, from the system's the interest cost at coordination. perspective, the retail level should be I(ci + Co).Because w > c0, the price mechanism inflates the holding costs at the retail level. As a it can no 8. An Alternative result, longer induce the decen- Holding-Cost tralized channel to achieve the maximum total profit. Model The system again suffers from double marginaliza- So far we have assumed that the inventory holding tion. There are two ways to fix this problem. costs incurred by the retailers are independent of the The first remedy is easy. Instead of paying the fol- wholesale price paid. Because the holding costs typi- lowing full wholesale price at delivery, cally include both capital costs and physical holding such as () warehousing costs, this assumption may be co + + 1hoA- hmin{A, Ti}, (14) di 2 2 restrictive. In this section, we explicitly model hold- costs as of two ing consisting components: physical the retailers pay only c0 at delivery and the rest when costs and holding capital costs. We show that our the goods are sold. As a result, the retailers face an coordination mechanism can be modified to easily interest cost of I(co + ci). It is easy to verify that with accommodate this more model. general this partial consignment, the wholesale price mecha- Let I be the interest which is identical rate, across nism (i)-(iii) still makes the retailer's profit function the firms. Redefine ho and hi to be the physical coincide with G(di, Ti, T ), and therefore achieves carrying-cost rates at the supplier and retailer i, perfect coordination. respectively. The second approach does not use consignment but From the = system's perspective, it costs ho ho + Ico revises the wholesale price formula. Here we assume to hold a unit at the and h' = supplier, hi + Ico + Ic, that the retailers pay the wholesale price upon deliv- at retailer i. more accurate models based (For on the ery. (In practice, there are of course many other pos- net value of cash present flows, see Porteus 1985.) sible arrangements. One common arrangement is to the Hence, incremental, holding-cost rate at retailer i pay the wholesale price after a fixed grace period, but - = is h'=defh h hi -ho + Ici = hi + Ici, which is also this can easily be accommodated. To see this, let t be 706 MANAGEMENT SCIENCE/Vol. 47, No. 5, May 2001 CHEN, FEDERGRUEN,AND ZHENG SupplyChain Coordination Mechanisms the grace period. Set w' = w(1 + It), where w is the constant. Thus, the supplier chooses To. The central- desired unit wholesale price with zero grace period. ized solution again prevails as an iterated dominant Paying w' after the grace period is equivalent to pay- strategy equilibrium in the decentralized channel, and ing w upon delivery. Interested readers are referred the channel profits continue to be maximized. to Boyaci and Gallego 1997 for further discussions.) Change the wholesale price mechanism (i)-(iii) to the following. Retailer i, i = 1, ... , N, References (i)' is charged KS by the supplier for every order it Boyaci,T., G. Gallego. 1997.Coordination issues in a simple supply places; chain. Workingpaper, Columbia University, New York. Medoff. 1990. the dozen. (ii)' is charged by the supplier a basic per unit cost Brown,C., J. 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