تعیین کمیت بهره وری قراردادهای تک نرخی در فشار زنجیره های تامین در طول توزیع تقاضای پشتیبانی شناخته شده

In this paper, we quantify the efficiency of price-only contracts in supply chains with demand distributions by imposing prior knowledge only on the support, namely, those distributions with support [a,b][a,b] for 0 < a ≤ b < + ∞ 0 < a ≤ b < + ∞ . By characterizing the price of anarchy (PoA) under various push supply chain configurations, we enrich the application scope of the PoA concept in supply chain contracts along with complementary managerial insights. One of our major findings is that our quantitative analysis can identify scenarios where the price-only contract actually maintains its efficiency, namely, when the demand uncertainty, measured by the relative rangeb/ab/a, is relatively low, entailing the price-only contract to be more attractive in this regard.

Price of anarchy (PoA), a quantifier measuring the inefficiency of a multi-agent system due to selfish behavior of its agents, has been an extremely popular concept in computer science and operations research communities in the last decade Nisan et al. [17]. Perakis and Roels [19] pioneered its application in supply chain contracts and obtained the PoA for price-only contract for several configurations of the underlying supply chain when the demand distributions possess the (weakly) increasing generalized failure rate (IGFR) property. A non-negative random variable X with cumulative distribution function (cdf) F(x) and probability density function (pdf) f(x) is of IGFR property if View the MathML sourcexf(x)/F¯(x) is nondecreasing for all x such that View the MathML sourceF¯(x)>0, where View the MathML sourceF¯(x)=1−F(x). We will use FIGFRFIGFR to denote the class of all distributions with IGFR property. One of the most important managerial insights observed from their analysis is that the worst PoA under FIGFRFIGFR is at least 1.71 (a 71% loss of efficiency) even for the simple two-stage chain, and consequently price-only contract may not be a viable practical contract with certain demand distributions due to this large loss of efficiency. Nevertheless, the price-only contract has been widely adopted in many real-life practices. This popularity has been constantly attributed to its low administrative cost (cf. [1]). Therefore, the following important questions arise naturally: is it possible that the assumption of IGFR on demand distributions leads to the overwhelmingly negative image on the price-only contract? Can we identify and justify those situations where price-only contract is attractive not just because of its low administrative cost? These questions of significant practical consequences serve the main motivation of this work, which investigates another class F[a,b]F[a,b] (0<a≤b<+∞)0<a≤b<+∞) of all distributions with support of the form [a,b][a,b]. The adoption of this class is a distributionally robust (or distribution-free, semi-parameter or min–max) approach similar to those by Scarf (1965) (see also, e.g., [8], [9], [10], [11], [12], [13], [21] and [20], etc.), when only distribution parameters, such as support, mean or variance, rather than the full distribution itself, are assumed to be known. The main contribution of this work is to derive PoA bounds over all distributions in the class F[a,b]F[a,b] under various supply chain configurations, as compared with the work of Perakis and Roels [19] for distribution class FIGFRFIGFR. These two classes of distributions are overlapping but not inclusive: there are IGFR distributions with support [0,∞)[0,∞) and there are distributions of support [a,b][a,b] that are not IGFR. The bounds derived by Perakis and Roels [19] depend on the number of supply chain partners n and the profit margin, whereas the bounds derived here depend on the number of supply chain partners n and the relative range b/ab/a. Hence, different and complementary managerial insights are obtained, especially with regard to the degree of uncertainty of demand, measured by the parameter b/ab/a. We only present the results on the push mode [2], where the downstream partner(s) hold(s) the supply chain inventory. Interested readers are referred to our working paper [6] for results concerning the other mode, pull mode. Moreover, throughout this paper, we will only consider pure equilibria for all the (sub-)games involved, as mixed strategies are not well-accepted in supply chain management [3]. Finally, we only focus on the nontrivial cases where the upstream partner in the game is the leader and where full efficiency cannot be achieved, that is, PoA>1PoA>1. The paper is organized as follows. After this introduction, we first provide some preliminary results in Section 2, and then consider the serial supply chain system, the assembly system, and two distribution systems depending on two different customer behaviors in Section 3, Section 4, and Section 5, respectively. We conclude the paper by some important observations in Section 6. All technical proofs can be found in the Appendices.

We have extended the application of the PoA analysis in supply chain management. Our results have revealed some new performance behavior of the price-only contract in various supply chain systems and hence deepened our understanding of it. The following observations follow readily from our analysis: 1.The bounds derived in this paper are independent of costs, prices and the boundaries a and b of the demand distribution support. In particular: (a) The bounds in the present work do not depend on upstream supply costs. This property is an attractive feature in environments of fluctuating commodity prices. (b) It is also significant that the bounds are independent of retail prices, individual values of a and b. Instead, the bounds only depend on their ratio ρ=b/aρ=b/a. (c) Moreover, in the assembly setting where multiple equilibria can exist in the absence of the IGFR property, our PoA bounds serve the purpose to necessitate the need of coordinating contracts in settings where ambiguity surrounds the demand distribution, cost/price parameters, and/or the particular equilibrium reached. 2.One of the major contributions of this work is the identification of the relative rangeρ=b/aρ=b/a, a measure of uncertainty, as a pivotal parameter in that the PoA obtained usually improves with reduced fluctuation ratio, although the exact analytical formula for PoA is highly nontrivial. Intuitively, on the one hand, when there is demand certainty, namely ρ=1ρ=1, it should be clear that both the decentralized and centralized solution will be to order the exact demand, leading to perfect coordination with PoA=1. On the other hand, when there is uncertainty in the demand, namely, ρ>1ρ>1, the decentralized solution can be forced to order a while the centralized solution is to order b in the worst case. 3.Our analysis under F[a,b]F[a,b] in this work shows that the price-only contract actually maintains its efficiency when the demand uncertainty, measured by ratio ρ=b/aρ=b/a, is relatively low, entailing the price-only contract to be more attractive in this regard than those administratively more expensive contracts as those considered, e.g., by Chen et al. [4], Jörnsten et al. [14] and Palsule-Desai [18] and hence justifying the efforts in demand forecasting to reduce uncertainties. Moreover, it actually offers a deeper reason on the wide acceptance and popularity of the price-only contract in many real-life practices, besides its low administrative cost. This insight of efficiency improvement with decrease of ρρ is empirically and qualitatively intuitive, given that the main source of double marginalization is demand uncertainty [15]. However, to the best of our knowledge, our work here is the first one to theoretically quantify this effect, achieved by the introduction of an uncertainty measure, ρρ. 4.Our analysis also shows that worst-case PoAs under F[a,b]F[a,b] and FIGFRFIGFR are complementary in the following sense: the former is in general independent of the profit margin 1−r1−r, while the latter is in general increasing in the profit margin. Note that the aforementioned insights are obtained through worst-case analysis and for a given demand distribution other than the worse-case distribution. The PoA may not perform as described above (see Sections 3.3, 4.4, 5.1.3, and 5.2.3 for the exact PoA under the uniform distribution, for which the PoA behaves differently from the worst-case situation). Therefore, one should exercise caution and care when applying the observations based on worst-case analysis to a given demand distribution other than the worst-case distributions. An apparent open problem is to find a tight bound for the distribution system with herd behavior (Theorem 3). Moreover, due to the practical importance of the price-only contract, an investigation of other demand classes would be important for identifying the situations where the price-only contract is relatively efficient.