key: cord-0055702-4oa66o3i
authors: Biswas, Indranil; Adhikari, Arnab; Biswas, Baidyanath
title: Channel coordination of a risk-averse supply chain: a mean–variance approach
date: 2021-01-27
journal: Decision
DOI: 10.1007/s40622-020-00267-1
sha: 30dfdf097b0e30c7cd15fd48a7e5f90408e1dee8
doc_id: 55702
cord_uid: 4oa66o3i

In the age of rapid technological advancement and digitization, coordination strategy remains an important issue for the supply chain. Additionally, the uncertainty caused by the disruption often induces the risk aversion in the supply chain members. Motivated by this issue, here we propose a coordination mechanism for a risk-averse supply chain using mean–variance approach. Here, we consider both centralized and decentralized cases and show that our analysis holds good for a central planner as well as for a decentralized supply chain under channel coordinating contracts such as buyback and revenue-sharing schemes. With the help of theoretical and numerical analysis, we exhibit how an individual supply chain agent’s risk aversion behavior can impact the contracts selection mechanism - from the profitability perspective. We extend our analysis to a dyadic setting to a single-supplier multiple-retailer network and confirm that pure strategy Nash equilibrium exists when all the retailers are risk-averse with varying risk attitude.

Global supply chains are complex multi-echelon systems consisting of numerous agents. Due to rapid technological advancements, digitization, and a strive to improve operational efficiency, supply chains are increasingly becoming decentralized in nature. Firms are often relying on decentralization-enabling technologies such as blockchains to manage their decentralized supply chains (Catalini and Gans 2016; Alzahrani and Bulusu 2018) . However, these global supply chains often become vulnerable as they face different kinds of uncertainties and risks. For example, if a supplier firm employs a direct sales channel with traditional retailers, it exposes her to risk due to conflict from vertical as well as horizontal competition (Tsay and Agrawal 2004; Yao and Liu 2005) . The political instability of a region (Peck 2005) , natural disasters (Tomlin 2006) are other primary sources of supply chain risk because of their disruptive effects. As the uncertainties and the risks associated with supply chain decisions are increasing day by day, the risk-neutrality assumption seems to be inadequate (Wu et al. 2009; Katariya et al. 2013; Zhuo et al. 2018) from the perspective of supply chain coordination. In this context application of predictive analytics and associated models become crucial for devising optimal results for a firm. Predictive analytics enables supply chain managers to arrive at optimal decisions based on external market conditions. During the recent outbreak of the COVID-19 pandemic, supply chains are facing liquidity crunch 1 coupled with reduced demand in the markets 2 . As a result, businesses are increasingly turning risk-averse 3 . Modelling the channel coordination mechanism for a risk-averse firm using predictive analytics is typically based on gametheoretical techniques (Souza 2014) . In this paper, we focus on such a modeling technique for a risk-averse dyadic supply chain.

In supply chain management, the classical newsvendor problem-solving approach is employed to design channel coordinating supply contracts. Buyback contracts (Pasternack 2008; Cachon 2003) ; revenue-sharing contract (Pasternack 2005; Cachon and Lariviere 2005; Giri and Bardhan 2012) , and quantity discount contract (Huang et al. 2011 ) are some instances of channel coordination mechanisms. However, these mechanisms do not consider the risk attitude of the agent and focus on the objective of either maximizing the expected profit or minimizing the expected cost.

Scholars have adopted a multitude of approaches to incorporate the risk-averse nature of the members of a supply chain. Examples include mean-variance (MV) analysis (Lau 1980; Choi et al. 2008a, b; Wu et al. 2009; Wei and Choi 2010; Katariya et al. 2013) , conditional value-at-risk (CVaR) minimization (Gotoh and Takano 2007; Chen et al. 2009; Li et al. 2014; Soleimani et al. 2014) , and expected-utility (EU) maximization (Horowitz 1970; Eeckhoudt et al. 1995; Keren and Pliskin 2006) . The MV framework enables investors to analyze risk diversification of assets and helps them to design an optimal portfolio (Markowitz 1959) . This framework has been explored extensively by scholars within the realm of supply chain management to address supply chain risks, particularly those arising from uncertain market demand (Chiu and Choi 2016; Choi and Chiu 2012; Liu et al. 2016) . We further observe that in supply chain management both MV approach and von Neumann-Morgenstern utility (VNMU) approach are employed by scholars for studying optimal supply chain decisions under risk. In spite of being a precise approach, usage of VNMU approach is limited due to the difficulty in estimating an individual's utility function in practice (Choi et al. 2008a, b) . On the other hand, the MV approach aims at providing an implementable, useful, and approximate solution (Van Mieghem 2003; Buzacott et al. 2003) .

In the context of MV analysis, there are three distinct ways to compute the objective(s) of a supply chain agent: (i) she tries to maximize the difference between the expected profit and a product multiplier of the variance of the profit (Wu et al. 2009; Katariya et al. 2013 ), (ii) she tries to maximize the expected profit while restricting its variance within a predefined level (Wei and Choi 2010) , (iii) she attempts to minimize the variance of the profit, thereby ensuring her expected profit to exceed a pre-defined minimum threshold (Choi et al. 2008a, b; Choi and Chiu 2012) . Choi et al. (2008a, b) show that the channel coordination in a two-tier supply-chain structure is a function of the net difference between the risk preferences of the supplier and the retailer. They conclude that coordination in the supply chain is not achievable in the presence of a highly risk-averse retailer. Wei and Choi (2010) propose a wholesale price and a profitsharing mechanism that coordinates the supply chain, depending on the risk-aversion threshold of the retailer.

While MV analysis follows the maximization of the difference between expected profit and risk-attitude multiplier of the variance of profit, Lau (1980) have established that the optimal order quantity for a riskaverse agent would be less than that for a risk-neutral one without considering the shortage cost. Wu et al. (2009) have demonstrated that a risk-averse newsvendor can order more than a risk-neutral one by 1 Economic Times Retail Report (June 10, 2020), ''Driving demand to drive economy'', Retrieved from: https://retail. economictimes.indiatimes.com/news/industry/driving-demandto-drive-economy/76298362, Accessed on: June, 10, 2020. 2 Ibid.

incorporating stock-out costs in their analysis. Katariya et al. (2013) show that under the MV criterion, a comparison between the optimal order quantities of the risk-neutral and risk-averse newsvendors depends solely on the model parameters and the nature of demand distribution chosen. As a result, channel coordination exercises under the MV criterion becomes parameter-dependent exclusively.

In this paper, we attempt to answer the following questions:

1. Is channel coordination in a risk-averse supply chain dependent on parameters (or preferences) of individual supply chain agents? 2. Is it possible for a supplier to coordinate a supply chain network consisting of a single supplier and multiple retailers where the retailers are all riskaverse?

In this paper, we derive the general condition for MV objective-maximization of a risk-averse supply chain. We consider both centralized and decentralized cases and show that our analysis holds good for a central planner as well as for a decentralized supply chain under channel coordinating contracts such as buyback and revenue-sharing schemes. Existing studies by Choi et al. (2008a, b) and Wei and Choi (2010) demonstrate that specially-designed contracts can achieve coordination in a risk-averse supply chain. We show that conventional contract forms, such as buyback and revenue-sharing schemes, can also lead to channel coordination. In the context of a two-tier supply chain structure, we first analyze the optimality conditions of a centralized risk-averse supply chain, which subsequently serves as a benchmark in our study. We further establish that an agent's MV optimization is dependent solely on the prior demand distribution. We prove that the maximizing condition is independent of the model parameters-a finding hitherto unreported in the extant literature. As a consequence of this novel finding, we successfully propose a relatively simplified technique to calculate the optimal order quantity compared to the existing mechanisms. Supported by extensive numerical analysis, we report interesting insights on how the risk aversion behavior of an individual supply chain agent can impact the contract selection mechanism, especially from the profitability perspective. We extend our analysis to a dyadic setting with a single-supplier multiple-retailer network and confirm that a pure-strategy Nash equilibrium exists when all the retailers are risk-averse with varying risk attitudes. We also establish the range of values for the risk-aversion parameter within which this pure Nash solution holds. Such analysis can assist supply-chain managers in designing their optimal supply contract forms in the context of a supply chain network.

The rest of the paper is organized as follows. In Sect. 2, we analyze the centralized supply chain under mean-variance approach with three contract forms, compare between risk shares, and develop the criterion to achieve channel coordination. In Sect. 3, we extend our analysis to a single-supplier multiple-retailers supply chain network when all the retailers are riskaverse and possess different risk attitude. We analytically investigate the condition(s) under which a purestrategy Nash equilibrium may exist for such a game. In Sect. 4, we conclude our study by discussing our key findings and future research avenues.

We consider a dyadic supply chain comprising one supplier and one retailer. The retailer experiences stochastic demand x during the selling season. Let F represent the cumulative distribution of that demand and f is its density function. We assume that F is differentiable and strictly increasing. Let the retail price be p, and at this price, the retailer's order quantity be q. The supplier's unit production cost is s and the retailer's marginal cost per unit is c. The retailer sells her season-end left-over inventory at per unit salvage price v 

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Þ, as from Theorem 1 we know, the order quantity decision is decreasing in aÖ .hProof of Theorem 2 The MV objective of retailer i is: maxis jointly quasi-concave in (p i , Y i ) when conditions (A) and (B) hold (for detailed proof refer to Zhao and Atkins 2008; Zhao 2008) . Therefore if ÀVar p i p i ; y i ð Þ ½ f g is concave in (p i , Y i ), then from the properties of concavity we can conclude thatg , which is a bivariate function of (p i , Y i ), as follows: We define w i as a random variable that corresponds to the truncated distribution of demand x i over [0, y i ].Using the properties of the order of integration and truncated distribution (A18) can be written as follows:where n i ¼ W i =y i and n i 2 0; 1 ½ . If retailer i's safety stock decision follows (35), the N retailers' game becomes a concave one and therefore pure-strategy Nash equilibrium exists for such a game, and the best response of an individual player is given by her firstorder condition.Proof of Proposition 3 From the expression of p d i p ð Þ, we obtain: