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...manifestation of the "double-marginalization" phenomenon: In a decentralized supply chain with two monopolists, two successive markups occur, causing the final price to be higher and the aggregate profits to be lower than if the firms were vertically integrated (Spengler 1950). When demand is stochastic and the retail price is fixed, double marginalization is reflected through reduced inventory levels.
The limited performance of price-only contracts was first investigated in two-stage supply chains by Lariviere and Porteus (2001) and Cachon and Lariviere (2001). It was then analyzed in more complex supply chains, such as assembly systems (Wang and Gerchak 2003, Gerchak and Wang 2004, Tomlin 2003), multitier assembly systems (Bernstein and DeCroix 2004), distribution systems when demand is stochastic (Anupindi and Bassok 1999, Cachon 2003, Bernstein and Federgruen 2005) or deterministic (Chen et al. 2001, Wang 2001). However, no formal analysis has accurately quantified the loss of efficiency associated with these contracts.
To improve coordination in supply chains, various alternative contracts have been proposed: buyback, revenue sharing, quantity flexibility, sales rebate, and quantity discount contracts (see Cachon 2003 and Lariviere 1999 for a review). However, these more elaborate contracts are typically more costly to negotiate, more complex to administrate, or might create additional moral hazard problems (e.g., see Krishnan et al. 2004).
Because of the prevalence of price-only contracts in practice and the additional cost of using more elaborate contracts, it is important to quantify the loss of efficiency associated with price-only contracts. Numerical examples in Cachon (2004) show that the relative efficiency of a two-stage decentralized supply chain could be as low as 70%-85% for push configurations and 75%-90% for pull configurations. But to the best of our knowledge, there has been no formal analysis to quantify the loss of efficiency in decentralized supply chains.
Quantifying the efficiency of a decentralized system relative to the performance of a centralized system has generated a lot of research interest over the past few years. The price of anarchy (PoA)--a concept introduced by Koutsoupias and Papadimitriou (1999) and dubbed by Papadimitriou (2001)--measures the ratio of the performance of the centralized system over the worst performance of the decentralized system (corresponding to the worst Nash equilibrium). The PoA has been used as a measure of performance for transportation networks (Roughgarden and Tardos 2000, 2002; Correa et al. 2004, 2005; Perakis 2005), in network resource allocation games (Johari and Tsitsiklis 2004), and network pricing games (Acemoglu and Ozdaglar 2007).
In contrast, there has been little research on quantifying the efficiency of supply chains. Chen et al. (2000) quantified the inefficiency due to the bullwhip effect by comparing the variance of orders with the variance of demand; because inventory costs are related to the variance of orders, their work relates to the literature on PoA. Martinez de Albeniz and Simchi-Levi (2003) computed the PoA of a procurement game with option contracts. There are two main differences between option contracts and price-only contracts. First, option contracts are two-dimensional (there is the reservation price and the execution price); as a result, competition is multidimensional and preserves the diversity of suppliers. Second, there is no moral hazard with option contracts, as supply capacity is assumed to be contractible. Finally, Chan et al. (2006) characterized the loss of efficiency in a distribution system, where the manufacturer chooses the wholesale price and the retailer chooses the retail price under a buy-back contract.
In this paper, we quantify the impact of double marginalization in supply chains that use price-only contracts. All our bounds are tight and distribution free. In our analysis, we assume that the demand distribution has an increasing generalized failure rate (IGFR). This assumption is commonly introduced in the supply chain literature (e.g., see Lariviere and Porteus 2001, Cachon 2004) because it guarantees a unique solution to the game. Moreover, many probability distributions satisfy this property (e.g., normal, uniform, exponential, gamma, Weibull). The IGFR characterization is nevertheless questionable as the set of IGFR distributions is not closed under convolution or shifting (Paul 2005). Our results critically depend on this assumption as our bounds are typically attained "at the boundary" of the set of IGFR distributions. We also show that non-IGFR distributions can lead to a worse performance than what our bounds suggest (see Remark 1).
We focus on price-only contracts, as their inability to coordinate the supply chain lies at the foundation of the large body of research on more elaborate contracts (buyback, quantity flexibility, etc.). We assume that the only decisions are the wholesale price and the inventory/capacity levels at each stage. In particular, we assume that the retail price is fixed and that no efforts can be made to improve forecast accuracy, increase sales, or reduce costs. We also ignore the effect of repeated interaction (Anupindi and Bassok 1999), nonzero reservation profits (Lariviere and Porteus 2001, Bernstein and Marx 2005), or renegotiation to a Pareto-improving situation (Ertogral and Wu 2001, Cachon 2004), as they diminish the impact of double marginalization.
We characterize the efficiency of different supply chain configurations: push or pull inventory positioning, two or more stages, serial or assembly systems, single or multiple competing suppliers, and single or multiple competing retailers. We also test the validity of our findings with a numerical study, measuring supply chain efficiency for commonly used demand distributions. Our analysis generates the following insights:
1. The loss of efficiency from double marginalization constitutes a major concern: Even in a two-stage supply chain, there might be a loss of efficiency of 42%; with more stages, the supply chain performance deteriorates further.
2. The efficiency of price-only contracts generally drops with the number of intermediaries but rallies when competition is introduced. There are however a few exceptions.
3. From a double-marginalization perspective, a pull inventory configuration generally outperforms a push configuration. That is, a make-to-order environment is generally more efficient than a make-to-stock environment, even in the absence of lost sales costs and salvage costs. In particular, significant savings can be realized if the supply chain adopts an assemble-to-order production policy, i.e., assembles components after observing the demand.
The paper is organized as follows. In [section]2, we introduce the model framework and characterize the solution of the integrated supply chain. In [section]3, we compute the PoA of a serial system, first limited to two stages, then consisting of an arbitrary number of stages, in both pull and push configurations. Section 4 is devoted to deriving the PoA of assembly systems, in which the components are procured from different suppliers. In [section][section]5 and 6, we quantify the efficiency of competition among suppliers and among retailers respectively. Finally, we summarize and discuss our results in [section]7 to provide insights into supply chain design. All proofs appear in the appendix.
2. Model Framework
2.1. Model Notations
Consider a supply chain facing the newsvendor problem. The supply chain has to build its inventory (or its capacity) Q before a selling season, without knowing the demand. We assume that costs are linear and denote by c the per-unit purchasing cost and by p the per-unit selling price. We suppose that the salvage value of the end products is zero. (A nonzero salvage value would attenuate the opportunity cost from not selling a unit, and it would then lower the PoA.) Therefore, the gross profit margin equals 1 - r, with r = c/p. Demand D is random and has a cumulative distribution function F(x) that is strictly increasing and continuous, with probability density function f(x). We denote by [bar.F](x) the complementary distribution function, i.e., [bar.F](x) = 1 - F(x).
In the sequel, we assume that the demand distribution has an IGFR (see Lariviere and Porteus 2001, Lariviere 2006, Paul 2005). Let h(x) = f(x)/[bar.F](x) be the hazard rate and let g(x) = xh(x) be the generalized failure rate, approximating the percentage decrease in the probability of a stockout from increasing the stocking quantity by 1%. The IGFR assumption is sufficient to guarantee a well-behaved (unimodal) problem for the contract initiator in a decentralized setting.
We also define l(x) = (h(x) [[integral].sub.0.sup.x][bar.F]([xi])d[xi])/[bar.F](x), roughly representing the percentage decrease in the probability of stockout from increasing the expected sales by 1%. The quantity l(x) is increasing if the distribution is IGFR (Cachon 2004).
2.2. Centralized Supply Chain
As a benchmark, we consider the centralized (or integrated) supply chain, as if there were a single decision maker operating the entire supply chain. The level of inventory is chosen to maximize the total supply chain expected profits:
[max.[Q[greater than or equal to]0]] p E[min{Q, D}] - cQ. (1)
The above problem--called the newsvendor problem--is concave and has the unique optimal solution [Q.sup.c] = [bar.F.sup.-1](r), where superscript c refers to the centralized (or integrated) supply chain. The optimal inventory level is such that the probability of stockout equals r = c/p.
2.3. The Price of Anarchy
The inventory level in a decentralized supply chain, denoted by [Q.sup.d], is in general not equal to [Q.sup.c], as each partner optimizes her own profit locally.
To measure the loss of efficiency, we derive a worst-case bound, computed over all IGFR distributions. We restrict our analysis to the class of IGFR distributions to ensure that the decentralized problem has a well-defined solution. This bound is a proxy for the magnitude of the loss of efficiency, without requiring to estimate the demand distribution. In fact, supply chain design, which is essentially strategic, must often be done without knowing the demand distribution (especially if the same supply chain is used for several generations of products), in contrast to inventory decisions, more tactical. All of our bounds are tight; that is, there exists an IGFR demand distribution for which the supply chain efficiency is characterized by the worst-case ratio.
DEFINITION 1. The PoA is the worst-case ratio of the profit of the centralized supply chain to the profit of the decentralized supply chain, that is,
PoA = [sup.[F[member of]F]] [[-c[Q.sup.c] + p E[min{[Q.sup.c], D}]]/[-c[Q.sup.d] + p E[min{[Q.sup.d], D}]]], (2)
where F is the set of nonnegative demand distributions that have the IGFR property.
2.4. Decentralized Game Framework
In the following, we analyze the performance of the decentralized supply chain depicted in Figure 1. The supply chain is divided into three parts: the procurement stage, the bill of materials of an assembly, and the distribution stage. A square represents a product and a circle represents a decision maker. A solid arrow symbolizes a "goes-into" relationship while a dashed arrow represents a supply/distribution channel.
In the center of the figure, the bill of materials represents an assembly structure, where [N.sub.C] components are assembled into one end product. At the procurement stage, in the left part of the figure, each component can be procured from any of [N.sub.S] competing suppliers. For simplicity, we assume the same number of suppliers corresponding to each component. Finally, at the distribution stage, the end-product can be sold to the end market through any of the [N.sub.R] retailers. Accordingly, the structure of the supply chain is parameterized by the triplet ([N.sub.C], [N.sub.S], [N.sub.R]).
To highlight the impact of double marginalization of supplier interdependence through a bill of material, supplier competition, and retailer competition, we analyze several special cases of the general supply chain depicted in Figure 1, corresponding to specific values of parameters [N.sub.C], [N.sub.S], and [N.sub.R]. In general, the structure of the game depends on whether the parameter values equal one or many (i.e., n, where n > 1). Table 1 summarizes the different supply networks that we analyze in the next sections.
[FIGURE 1 OMITTED]
For each supply chain structure presented in Table 1, we consider two inventory configurations, according to the push-pull classification introduced by Cachon (2004): When the downstream (respectively upstream) partner holds the supply chain inventory, the supply chain is said to be operated in a pull (respectively push) mode.
Consistently with the literature (Lariviere and Porteus 2001, Cachon and Lariviere 2001), we model the problem as a Stackelberg game where a leader proposes a "take-it-or-leave-it" contract to a follower, and we assume perfect information. The timing of the game for a two-stage supply chain is outlined below. Note that the order of the last two steps of the game depends on the push-pull configuration of the supply chain.
1. The leader offers the follower a contract specifying the per-unit wholesale price w.
2. The follower accepts the contract if his expected profit is above his reservation profit, assumed to be zero. Otherwise, there is no transaction between the parties.
3. The manufacturer chooses his/her level of inventory at a per-unit cost c.
Pull:
4. Demand D is realized.
5. The retailer orders what is needed to meet demand at a per-unit cost w. Each unit of satisfied demand generates a revenue p. If the manufacturer does not have enough inventory, the excess demand is lost at no cost.
Push:
4. The retailer places an order at a per-unit cost w.
5. Demand D is realized. Each unit of satisfied demand generates a revenue p. If the retailer does not have enough inventory, the excess demand is lost at no cost.
3. Serial Supply Chain
We first characterize the efficiency of a two-stage supply chain, under a push or a pull configuration, constituted of a manufacturer and a retailer. We then extend our results to multistage serial structures.
3.1. Push Serial Supply Chain
In a push supply chain, the inventory is held at the retailer's site, i.e., the retailer makes to stock. We consider two...
NOTE: All illustrations and photos
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