Retailer's sourcing strategy under consumer stockpiling in anticipation ...

International Journal of Production Research, 2017 https://doi.org/10.1080/00207543.2017.1401748

Retailer’s sourcing strategy under consumer stockpiling in anticipation of supply disruptions Jiho Yoona, Ram Narasimhanb and Myung Kyo Kima* a

Department of Management, College of Business Administration, Kansas State University, Manhattan, KS, USA; bDepartment of Supply Chain Management, The Eli Broad Graduate School of Management, Michigan State University, East Lansing, MI, USA (Received 2 January 2017; accepted 23 October 2017) We study a retailer’s sourcing strategy under consumers’ stockpiling behaviour and the factors associated with the selection of an optimal strategy in multi-tier supply chains in the presence of supply disruption risk. Stockpiling behaviour occurs when consumers attempt to mitigate the negative impact of a supply shortage. We prove that those behaviours become stronger if consumers have experienced similar problems before and weaker as more inventories are hoarded. Based on these findings, numerical analysis is carried out to compare the superiority of single sourcing versus dual sourcing from retailer’s perspective. Our results suggest that the superiority is highly dependent on factors such as supplier’s volume flexibility, retailer’s purchasing price, and supplier reliability. Keywords: sourcing strategy; consumer stockpiling; supply disruption risk; multi-tier supply chain; retail supply chain

1. Introduction As supply chains become extended and globalised, firms are more vulnerable to supply disruptions due to a variety of external factors such as supply shortages, adverse market conditions and transportation disruptions. The direct impacts of supply disruptions are catastrophic as they cascade down the supply chain (Sokolov et al. 2016; Ivanov et al. 2017). The tsunami in the northeast coast of Japan on 11 March 2011 created a severe shortage of Xirallic, a specialty pigment made in a single plant suspended until May 2011. This forced most of global automakers including BMW, Chrysler, Ford, General Motors, Toyota and Volkswagen to stop taking orders for certain colours using Xirallic. In July 2016, a sudden bankruptcy of a small, just-in-time vendor of acoustic insulation and interior trim parts nearly brought General Motors’ most profitable North American operations to a halt. The indirect effect of supply disruptions also should be noted. Upon detecting any event that can lead to upstream disruptions, consumers often stockpile supplies to mitigate the risk of future shortages. When rice production in Australia was reduced by 98% after a long drought period in 2008, for instance, consumers in Vietnam, India and Hong Kong began to stockpile rice due to the fear of insufficient future supply, which actually made the shortage problem worse. After the radioactive leak at Japan’s Fukushima nuclear plant in 2011, for instance, consumers went on panic buying of various products such as iodine tablets (in Bulgaria), salt (in China), red wine and seaweed (in Russia) that were rumoured to prevent radiation poisoning. This consumer stockpiling behaviour can immediately affect the decisions of the retailer (i.e. consumer’s supplier) with respect to: (i) order quantity and inventory level, and/or (ii) sourcing strategy (i.e. single versus dual/multiple sourcing) and order allocation (if dual/multiple sourcing strategy is selected). This could sequentially influence more upstream suppliers’ (i.e. wholesaler and manufacturer) decisions about pricing and volume flexibility. Despite this close association between supply chain partners, much of the existing literature on sourcing strategy has not jointly analysed the decisions of upstream and downstream firms taking into account consumers’ stockpiling behaviours. Consumer stockpiling has been studied in marketing and economics literature, with a common focus on the impact of price changes or price competition between suppliers. This important topic, however, has been largely neglected in the traditional approach to sourcing strategies, based on the assumption of rational expectations (e.g. Liu and Ryzin 2008; Tereyağoğlu and Veeraraghavan 2012). Further, these studies did not thoroughly consider how supplier’s volume flexibility (SVF) impacts the effectiveness of the retailer’s sourcing strategies in the presence of supply disruption risk. We thus attempt to fill the gaps in our knowledge about the optimal sourcing strategy of a retailer

*Corresponding author. Email: [email protected] © 2017 Informa UK Limited, trading as Taylor & Francis Group

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positioned between a supplier and consumers in a three-tier supply chain by considering consumers’ stockpiling behaviours and supplier’s volume flexibility. The objectives of this paper are twofold. First, we investigate the interdependent but conflicting requirements of the retailer that of mitigating supply disruption risk while maximising profit. As the industry matures, U.S. retailers have witnessed notable changes in competitive dynamics characterised by thinner margins, globally dispersed suppliers, etc. Although these trends have forced retailers to satisfy the two aforementioned requirements simultaneously, extant studies have mainly discussed the sources and/or consequences of supply disruptions (e.g. Craighead et al. 2007; Hendricks, Singhal, and Zhang 2009; Hendricks and Singhal 2014; Bode and Wagner 2015). By bridging this gap in the literature, we seek to better understand the decision-making behaviour of partnering firms. This will enable them to select optimal sourcing strategy that can deal with supply disruption risks more effectively. 2. Literature review In supply chain management literature, risks that focus on manufacturing processes and demand or lead-time uncertainties have been discussed at the tactical level. Naylor, Naim, and Berry (1999), for instance, showed the stock out risks under demand uncertainty can be mitigated by postponing the decoupling point. Gupta and Maranas (2003) proposed a stochastic programming-based bi-level optimisation model for manufacturing and distribution timing decisions to minimise cost under demand uncertainty. So and Zheng (2003) presented an analytical model that considers the relationship between order quantity variability and supplier’s lead time in a supply chain and suggest how to reduce the negative effect of supplier’s variable delivery lead times in a two-tier supply chain. Mitigating supply uncertainty has gained increased attention in recent years (Feng 2010). This topic has been mainly addressed via two approaches: random-yield model and supply disruption model. Random-yield model assumes that the supply level is a random function of the input level (Grosfeld-Nir and Gerchak 2004). Gurnani, Akella, and Lehoczky (2000) introduced a cost function and derived the function’s bound values with consideration of random yields in supply. He and Zhang (2008) also investigated the performance impact of random yields in a two-tier supply chain comprised of a supplier and a retailer. In the supply disruption model, in contrast, the supplier’s availability status is either ‘UP’ (i.e. when the orders are fulfilled in full and on time) or ‘DOWN’ (when no order can be fulfilled). Parlar and Berkin (1991) analysed this case with exponential UP and DOWN periods under the EOQ assumption. Parlar and Perry (1995) and Parlar (1997) considered random supply disruptions by applying a Markov Chain model with stochastic demand and lead-times under different inventory policies. Literature in this area has discussed sourcing strategies to mitigate supply disruption risks. Tomlin (2006) applied the Markovian approach to model supplier’s availability with the consideration of disruptions (i.e. frequent, but short versus rare, but long disruptions) and suggested contingent risk mitigation strategies when there are two suppliers, one that is unreliable and the other that is reliable but more expensive. Based on a similar setting, Chopra, Reinhardt, and Mohan (2007) analysed the effectiveness of backup sourcing by considering recurrent risks (i.e. random yield) as well as disruption risks. To date, however, few studies have fully addressed the impact of SVF (i.e. supplier’s ability to change output level of products produced) in the presence of supply disruption risks. Existing literature on volume flexibility has been predominantly focused on downstream uncertainties, but little attention has been paid to analysing and valuing its economic benefits under upstream uncertainty. For example, Eppen and Iyer (1997), Tsay and Lovejoy (1999), and Wu (2005) showed the importance and effectiveness of SVF, but they took into account demand uncertainty only, not upstream (i.e. supply) risks. Tomlin (2006) and Chopra, Reinhardt, and Mohan (2007) also considered SVF, but assumed that reliable supplier might possess infinite volume flexibility. Further, these studies did not address the effect of strategic consumer behaviour commonly observed in the retailing setting, which can affect the retailer’s ordering decision. Our paper is closely linked to the work of Dana and Petruzzi (2001) that assumes consumers observe the retailer’s price and inventory level and can compute the expected fill rate. Based on this, consumers learn whether they value the good and the value of their outside option, and then decide between visiting firm and consuming their outside option. However, they did not consider the consumer stockpiling behaviour. Several works in marketing literature have examined optimal consumer stockpiling behaviour in response to changing sales prices (e.g. Assunção and Meyer 1993; Ho, Tang, and Bell 1998; Guo and Villas-Boas 2007). Extending this to studying endogenous demand process has generated emerging research streams on the topics of strategic consumer behaviour and consumer-driven demand in operations management (e.g. Netessine and Tang 2009). These works also assumed the absence of supply disruption. To the best of the authors’ knowledge, Shou, Xiong, and Shen (2013) is the only available study that considered consumer behaviour (panic buying, in specific) and supply disruption simultaneously. The focus of their work was confined to the use of retailer’s internal control systems such as optimal safety

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inventory level and fixed quota policy. Such mitigation strategies are convenient to implement but may not be able to fully address the disruptions caused by external sources under less control of the retailer. Our work attempts to contribute to both strategic sourcing and consumer behaviour literatures by analytically characterising consumer stockpiling behaviours in the presence of supply disruptions, in a multi-tier supply chain setting rather than a dyadic setting. In doing so, the present study introduces additional strategic options for retailers, supplier’s reliability (SR) and SVF, to mitigate the adverse effects of both upstream and downstream disruptions. 3. Characterising consumers’ stockpiling behaviours We initially consider a simple three-stage supply chain consisting of a single supplier who provides homogeneous products to a single retailer, who in turn, sells the homogeneous products to N consumers. We assume that each tier’s basic information is known to all supply chain members. In the literature, information comes with its associated search time (Mitra, Reiss, and Capella 1999) and costs (Lynch and Ariely 2000). However, information technology now allows them to gather information quickly and inexpensively (Cachon and Fisher 2000). We thus assume that information search time and costs are negligible in that the basic information can include (i) supplier’s reliability (SR), (ii) retailer’s purchasing volume in period 1, and (iii) consumer’s stockpiling behaviour. Based on the setting, we examine how the uncertainty of future product availability affects consumers’ purchasing and stockpiling behaviour and the retailer’s ordering decisions. The primary decision for the consumers is to set the order quantity and timing in order to maximise their utility and the primary decision for the retailer is to set order quantities (purchasing volumes) under predefined per unit purchasing/selling prices to maximise profit. In general, an individual consumer can purchase the product from the retailer in each period. However, due to supply shortage, the consumer’s order may not be filled in a given period and thus a loss of utility occurs to the consumer. To avoid this, the consumer may stockpile in anticipation of a supply shortage by purchasing more than needed when the supply is available. This stockpiling behaviour, however, will incur holding costs for unused products within that period. We will analytically characterise the individual consumer’s optimal purchasing decisions and examine how the decisions are affected by various factors such as the likelihood of supply disruption, the product’s retail price, holding cost, and the individual consumer’s risk attitude. We start with studying the individual customer problem under twoperiod setting using the following notations. v pi pw h H α β τ(v) Qi Di π

A consumer’s valuation of one unit of product Retail price per unit of product in period i, i = {1, 2} Wholesale price per unit of product (retailer’s purchasing price, RPP) A consumer’s inventory holding cost per unit of product Retailer’s inventory holding cost per unit of product A consumer’s anticipation of product availability in period 2 Likelihood that supplier will be UP in period 2 (supplier reliability, SR) A consumer’s risk attitude at valuation v Retailer’s ordering quantities in period i, i = {1,2} Consumer’s demand in period i, i = {1, 2} Retailer’s expected profit

3.1 Fixed price case First, we consider the fixed retail price case over two periods: Periods 1 and 2 (p = p1 = p2) and derive results that form the basis for extensions of the basic model. A consumer purchases a homogeneous product from a single retailer over two periods (assume no initial inventory and/or backorder). A consumer’s valuation for consuming one unit of the product follows a general distribution with a c.d.f. FðvÞ. The valuations of different consumers are assumed to be i.i.d. We also assume that the market size, denoted by N, is known to all consumers as well as other supply chain entities. The assumption on consumer’s knowledge on market size is considered reasonable since the scope of this study is the B2C (business to consumer) products whose suppliers need retailers to reach end consumers. Different from B2B (business to business) or intermediate products (e.g. raw materials, components, etc.), the end consumers of such products can easily gain some general but important knowledge about the products that are of their interests or they frequently purchase through various sources (e.g. past experience, media reporting, reference groups, etc.). Recent studies have provided empirical support for this view by showing that end consumers are aware of the secondary data such as market size (e.g. Soysal 2008; Chevalier and Goolsbee 2009).

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Each individual consumer can consume at most one unit of product in each period but may purchase multiple units within a period. Each unit of the unconsumed product incurs a holding cost. For simplicity, we assume that all consumers’ demands can be satisfied in period 1. In period 2, however, consumers anticipate that their demand could be satisfied with probability α due to the risk of supply shortage. As such, (1 – α) denotes an individual consumer’s assessment of the likelihood of purchasing failure due to supply shortage in period 2. Each consumer is assumed to have a power utility function, U ðxÞ ¼ xc ; where 0\c ¼ sðvÞ 1. This affects quantity and timing of consumer’s purchases which are derived from the assumption that retail price is predefined and it influences the demands over two periods. We consider the following cases to account for consumer’s utility: (i) not purchase the product, (ii) purchase one unit of product in each period, or (iii) purchase two units of products in period 1. Note that 0 < τ(v) < 1 corresponds to riskaversion and τ(v) = 1 corresponds to risk neutrality (Gupta, Su, and Walter 2004). We assume that τ(v) is monotone decreasing, twice differentiable and convex w.r.t. v, ðs0 ðvÞ\0; and s00 ðvÞ [ 0Þ. This assumption implies that high valuation consumer is more risk-averse (Bardey 2004). An individual consumer with a valuation, v ≤ p, obviously will not purchase any product in either period. If v > p, the consumer’s utility is as follows: 8 c if one unit of the product is purchased in each period; < ð2v 2pÞ ; (1) U ¼ ð2v 2p hÞc ; if two unitsare purchased in period 1; : if one unit is purchased in period1 only ðv pÞc ; Based on this utility function, we can derive threshold anticipation of product availability (purchasing success) in c ðvpÞc that satisfies: period 2, aT ðp; h; v; cÞ ¼ ð2v2phÞ ð2c 1ÞðvpÞc ð2v 2p hÞc ¼ aT ðp; h; v; cÞð2v 2pÞc þ ð1 aT ðp; h; v; cÞÞðv pÞc

(2)

LHS of equation (2) is individual consumer’s total utility over two periods when the consumer purchases two units in period 1, and RHS implies the consumer’s total expected utility over two periods when the consumer purchases only one unit in period 1. RHS is composed of two cases: (i) the case of the purchasing success (first term) and (ii) the case of the purchasing failure in period 2 (second term). Note that when 2v 2p h\0 , v p h2\0 , v\p þ h2, the consumer will not consider purchasing two units in period 1 because of negative utility, which means that the consumer will purchase only one unit in period 1 and another unit in period 2 if v > p and available in this case. Therefore, we can conjecture that consumer’s purchasing policy is to try purchase one unit each period if p\v\p þ h2. From the logic in Equation (2) and the threshold αT(p, h, v, γ) we can derive a consumer’s optimal purchasing policy. Proposition 1: For any given valuation v, where v > p and p is fixed over two periods, there exists a threshold anticipac ðvpÞc , such that an individual consumer can use the foltion of product availability in period 2, aT ðp; h; v; cÞ ¼ ð2v2phÞ ð2c 1ÞðvpÞc lowing optimal purchasing policy: (a) If α > αT(p, h, v, γ), then the individual consumer will purchase 1 unit in period 1 and another 1 unit in period 2, when the product is available in period 2. (b) If α ≤ αT(p, h, v, γ), then the individual consumer will purchase 2 units in period 1 and zero unit in period 2 regardless of the availability of the product in period 2. Proof 1: See Appendix 1. Proposition 1 is quite intuitive. It indicates that when an individual consumer’s anticipation of product availability in period 2 is low enough, α ≤ αT(p, h, v, γ), the consumer will purchase two units in period 1 under an expectation that this proactive purchase would improve his or her total utility over the two periods under high supply disruption risk. In the same vein, the consumer will not purchase more than needed in period 1 when he/she is highly confident about the product availability in period 2, α > αT(p, h, v, γ). Note that according to proposition 1, when v – p < h, αT < 0, i.e. αT < α, which implies that the consumer will not consider purchasing two units in period 2. This is sufficient condition for the conjecture made from Equation (1), since if p < v < p + h is satisfied, p\v\p þ h2 is always true. Proposition 2: The threshold αT(p, h, v, γ) is decreasing in h, p, but increasing in v. Proof 2: See Appendix 1. Proposition 2 also provides intuitive implications. A consumer is less likely to stockpile by purchasing two units in period 1 as costs increase (i.e. the retail price and/or the holding cost becomes higher) to reduce negative effects of the high costs on the utility, but as consumer’s valuation of the product increases, the consumer is more likely to stockpile by purchasing two units in period 1, since the high valuation increases the consumer’s utility and, by assumption, the consumer becomes more risk averse as the consumer values the product higher.


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3.2 Price change case Now we consider the price change model over two periods. We can derive the threshold anticipation of product availability in period 2, αT(p1, p2, h, v, γ), for price change case with the following utility function Up and the corresponding equation that determines threshold anticipation (subscript p denotes price change case). Similarly, boundary condition for this case is v – p1 > 0. If p1 + h < p2, the consumer will simply purchase two units in period 1. Moreover, if p1 + h > p2, the consumer will purchase one or two unit(s) in period 1. Similar to the fixed price case, we consider 2v – 2p1 – h > 0, which implies that 2v – p1 – p2 > 0. 8 c < ð2v p1 p2 Þ ; if one unit of the product is purchased in each period; (3) Up ¼ ð2v 2p1 hÞc ; if two units are purchased in period 1; : c ðv p1 Þ ; if one unit is purchased in period 1 only ð2v 2p1 hÞc [ aT ðp1 ; p2 ; h; v; cÞð2v p1 p2 Þc þ ð1 aT ðp1 ; p2 ; h; v; cÞÞðv p1 Þc

(4)

From the example of hard disc supply shortage due to floods in Thailand in 2011, we were able to observe that price often increases after a supply disruption occurs (Luo et al. 2015). Thus, it is reasonable to assume that the consumers who have experienced supply shortage in the past will believe that the price will increase (i.e. p1 = p ≤ p2) if the supply cannot satisfy the consumers’ future demands in the anticipation of supply shortage, α < 1). Based on our intuition, we conjecture that high valuation consumer anticipates higher price increase in the future when he/she predicts supply shortages (i.e. p2 = δ(v) + p1, where δ(v) > 0, and δ(v) is monotone increasing in v). This price change leads to the threshold anticipation change. In other words, the joint impact of supply uncertainty and the belief of price change will affect consumer’s purchasing/stockpiling behaviour. c c 1 hÞ ðvp1 Þ and proposition 1 can Similar to the fixed price case, the threshold is derived by, aT ðp1 ; p2 ; h; v; cÞ ¼ ð2v2p ð2vp1 p2 Þc ðvp1 Þc be revised by replacing αT(p, h, v, γ) with αT(p1, p2, h, v, γ) as follows: Proposition 3: For any given valuation v, where v > p1 and p1 < p2, there exists a threshold anticipation of product c c 1 hÞ ðvp1 Þ , such that the individual consumer can adopt the following availability in period 2, aT ðp1 ; p2 ; h; v; cÞ ¼ ð2v2p ð2vp1 p2 Þc ðvp1 Þc optimal purchasing policy: (a) If α > αT(p1, p2, h, v, γ), then the individual consumer will purchase 1 unit in period 1 and another 1 unit in period 2, when the product is available in period 2. (b) If α ≤ αT(p1, p2, h, v, γ), then the individual consumer will purchase 2 units in period 1 and zero unit in period 2 regardless of the availability of the product in period 2. Proof 3: See the proof of proposition 1 in Appendix 1. Moreover, αT(p1, p2, h, v, γ) maintains the properties derived in proposition 2, since the increment of p1 impacts more on the denominator of the threshold and h still affects only the numerator of the threshold. Part C in the first derivative of aT ðp; h; v; cÞ w:r:t: c in proof 2 (see Appendix 1) will decrease in the new threshold αT(p1, p2, h, v, γ), i.e. (2v – 2p)γ(v – p)γ > (2v – p1 – p2)γ(v – p)γ, which implies that the effect of v on the threshold will be amplified. αT(p1, p2, h, v, γ), however, can be greater than 1, which is inadmissible. Thus, we have an additional boundary condition for our analysis of aT ðp1 ; p2 ; h; v; cÞ; p1 þ h [ p2 . c ðvpÞc and In the price change case, αT(p1, p2, h, v, γ) > αT(p, h, v, γ), since αT(p, h, v, γ)can be rearranged by ð2v2phÞ ð2v2pÞc ðvpÞc p1 = p ≤ p2. This fact implies that experiencing supply shortage can lead to the belief that the price increases following supply shortage; this, in turn, will be more likely to lead consumers to stockpile. For simplicity, we assume that τ(v) follows survival function of Pareto distribution with shape parameter hs ; i:e:; sðvÞ ¼ ðv v1 Þhs , so that 0\sðvÞ\sðvÞ ¼ 1 and dðvÞ ¼ hd lnðv v þ 1Þ so thus dðvÞ 0; where v ¼ maxfvg and v = min{v}. The individual consumer’s behaviour analysed in this section enables us to estimate the demand of each period. We generalise the product availability in period 2 anticipated by the individual consumer (i.e. α) to N-heterogeneous consumers by using superscript n; i:e:; an ; n = 1 … N. Based on the demand estimated, we can study retailer’s profit over the two periods. 4. Retailer’s profit Slightly different from the consumer’s perspective, we assume that the retailer needs to make order quantity decision before the beginning of each period. As in Section 3 we assume that the retailer is sure that the order for period 1 will be filled at the beginning of period 1, but not sure that the order for period 2 will be filled at the beginning of period 2. Figure 1 depicts this assumption.

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Figure 1. Time horizon.

This assumption is well illustrated with the chronic congestion and labour slowdowns at U.S. ports. A labour dispute between dock workers and port owners triggered numerous, intermittent shutdowns of West Coast port operations over nine months until 20 February 2015. Under such unpredicted circumstances, even a supplier that perfectly satisfied the retailer’s order for period 1 can run into a supply shortage in the following period. The retailer’s inventory level at the end of period 1 can be greater than one. We assume that the supplier has infinite capacity in period 1 so inventory level in period 1 cannot be negative in the optimal solution, while inventory level at the end of period 2 can have negative value (i.e. supply shortage) due to supplier disruption. We assume that the retailer orders Q1 units of products for sale before the beginning of period 1. Any unsold products at the end of period 1 are carried over to period 2 with a holding cost of H per unit. The retailer orders Q2 units of products for sale before the beginning of period 2, but due to supplier disruption these products are delivered only with a probability β. Based on Section 3, we can derive the total demand of N (maximum market potential) consumers in period 1 and 2, denoted by D1 and D2, respectively. One may think that the total demand is simply the same as market size over two periods, i.e. D1 + D2 = N + N = 2 N, or the demand is strictly decreasing in the market price (retail price) p1 and p2 , i.e. Di ¼ qi ðpi Þ ¼ N Epi ; where qi ðpi Þ is the quantity ordered by consumers at price pi ; and E is the price sensitivity. However, an individual consumer’s strategic behaviour studied in Section 3 can practically incorporate another facet of consumer’s behaviour other than that induced by the simple theory of demand as a decreasing function of price, such as the belief that price increase in the future will lead to demand increase in current period. We can define the demand Di in period i; i ¼ f1; 2g as follows (superscript n denotes nth consumer): N P 1; if an [ aT ðp1 ; pn2 ; h; vn ; cÞ d1n 1fvn [ p1 g ; where d1n ¼ D1 ¼ 2; otherwise n¼1 (5) N P 0; if vn \pR2 n n d2 1fd1n 1fvn [ p1 g ¼1g ; where d2 ¼ D2 ¼ 1; otherwise n¼1 where pR2 is actual retail price in period 2 when retailer cannot fully satisfy consumer’s demand and pR2 p1 . Note that pn2 is anticipated price in period 2 by consumer n. 4.1 Single sourcing case Now, we can derive retailer’s expected profit under single sourcing setting (SS) as following. pSS ðb; pR2 ; Q1 ; Q2 ; D1 ; D2 ; D02 Þ ¼ p1 minfD1 ; Q1 g pw Q1 HðQ1 D1 Þþ þ bðp1 minfQ2 þ ðQ1 D1 Þþ ; D02 g pw Q2 Þ þ ð1 bÞuSS pR2 minfðQ1 D1 Þþ ; D2 g þ ð1 bÞð1 uSSÞp1 D02 ; 0; if ðQ1 D1 Þþ [ D02 where ðÞþ ¼ maxf ; 0g and uSS ¼ 1; otherwise

(6)


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We further define D02

¼

N X

ðd2n Þ0

1fd1n 1fvn [ p1 g ¼1g ; where

ðd2n Þ0

¼

n¼1

0; if vn \p1 1; otherwise

(7)

We assume that the retailer will decide product’s price depending on demand, i.e. the price will be increased if demand exceeds available quantity. Thus, D02 is demand in period 2 when available quantity is high enough, while D2 is demand in period 2 when available quantity is not high enough. The first term represents the revenue of the retailer in period 1 and the second and third terms represent procurement cost and holding cost, respectively. The fourth term stands for the retailer’s profit in period 2, when supplier is UP and the last two terms represent the retailer’s profit, when supplier is DOWN. The last two terms make the switching expected profit function; when the inventory level is high enough to cover demand in period 2 at the initial price offered in period 1 (p1), the retailer does not charge a higher price in period 2 (pR2 ), while the retailer charges pR2 in period 2, if the inventory level is not enough to satisfy the demand in period 2 at p1. The last two terms imply that only when the supplier is DOWN and the retailer does not carry enough inventories, pR2 becomes the price charged in period 2. We assume that all consumers have their independent estimate of retailer’s fill rates. Some assume that all customers have the same estimates, which is a consequence of the assumption that consumers have fully rational expectations. This assumption implies that the consumers can perfectly anticipate the firm’s fill rate (Liu and Ryzin 2008, 2011), i.e. 1 for 8n. However, under our consumers’ heterogeneity assumption, each consumer’s valuation an ¼ a ¼ b þ ð1 bÞQ1DD 2 is different, and we assume that the consumer has a false belief that all consumers have the same valuation. Thus, our assumption (the modification of Liu and Ryzin’s fill rate estimation of consumers) implies that SS 1fvn [ p1 g , where NSS = min{Q1, N}. By proposition 3 when αn > αT(p1, p2, h, v, γ), the consumer an ¼ b þ ð1 bÞðQ1QN1 N SS Þ1fvn p1 g þNSS will purchase one unit in period 1. Note that this assumption implies that an individual consumer will anticipate the product availability in period 2 if the purchase of one unit in each period. When a consumer n does not want to purchase ði:e: vn p1 ; an ¼ 1Þ, he/she does not care about the purchasing failure by reasoning that ‘I will not buy this product so others might not.’ On the other hand, when vn [ p1 ; 0\b\an 1, the consumer must be concerned about the failure for future purchase, since he/she reasons that ‘everyone wants to buy this product just like me’. We further assume that SVF can restrict retailer’s sourcing size (Tomlin 2006). Based on these assumptions, the retailer’s problem under SS is: max pSS ðb; pR2 ; Q1 ; Q2 ; D1 ; D2 ; D02 Þ

(8)

Q1 ;Q2

s.t. anSS ¼ b þ ð1 bÞ

Q1 NSS 1fvn [ p1 g ðQ1 NSS Þ 1fvn p1 g þ NSS

Q2 2 ½bð1 xÞQ1 c; dð1 þ xÞQ1 e

for 8n

(9) (10)

where x is SVF; 0\x 1. ω = 1 implies the supplier is perfectly flexible in volume, since Q2 \2Q1 and Q2 \D2 due to the assumptions of pR2 p1 and no back order so that D2 ≤ D1 and D1 Q1 , while ω = 0 presents that the supplier is not flexible at all (Q1 = Q2). Intuitively, in period 1, the retailer will not source more than total demand over two periods (Q1 D1 þ D02 ), and, to satisfy the aforementioned assumption that all consumers can always purchase two units in period 1, the retailer sources at least total consumer demand for period 1 (D1 ≤ Q1). Therefore, the optimal sourcing size in period 1 (Q1 ) is contained in the range (D1 Q1 D1 þ D02 ). Proposition 4: As Q1 increases, anSS will increase, thus D1 will decrease but D2 and D02 will increase. Proof 4: See Appendix 1. Proposition 4 indicates that consumers will be less likely to carry inventory when retailer stocks enough inventories, since from the consumer’s perspective, the chance of the purchasing failure in the following period is low (so that the consumers will be likely to try to save the holding cost for unconsumed product in period 1 by purchasing the additional unit when it needs to be purchased). We employ these intuitions in justifying the following arguments of algorithm building. When vn [ p1 and Q1 ¼ 2N (total demand over two periods cannot exceed 2 N), obviously αn = 1. So, our analysis for achieving optimal sourcing size of the retailer in period 1 can start with Q1 = 2 Nand reduce Q1 gradually until the expected profit of the retailer does not increase or Q1 = D1. The retailer will source at least D1 in period 1 by assumption. At the beginning (i.e. Q1 = 2 N), aTSS ¼ 1 aTSS ðp1 ; pn2 ; h; vn ; cÞ; d1n ¼ 1; d2n ¼ 1fvn [ pR2 g ; and ðd2n Þ0 ¼ 1fvn [ p1 g .

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Figure 2. Evolution of pSS ðÞ and its optimal value & solution under algorithm for SS at b ¼ 0:5; x ¼ 0:85; p1 ¼ 1:2; pw ¼ 0:6p1 ; pR2 ¼ 1:1p1 ; h ¼ 0:15p1 ; hs ¼ 1; and hd ¼ 0:3:

However, before reaching the point where Q1 ¼ D1 ; anSS can become lower than the threshold ði:e:anSS αT(p1, p2, h, vn, γ)). In this case, d1n ¼ 2 and d2n ¼ ðd2n Þ0 ¼ 0: For Q2 ; the following proposition can be derived: the optimal Q2 is given by Proposition 5: At a certain point of Q 2, Q2 ¼ D02 ðQ1 D1 Þþ ; if bð1 xÞQ1 c D02 ðQ1 D1 Þþ : Otherwise, Q2 ¼ bð1 xÞQ1 c. Proof 5: See Appendix 1. Based on these intuitions and proposition 3 above, we can derive the following algorithms to find optimal sourcing size over two periods:

•

Algorithm for SS

Begin 0; Q1 2N and generate vn0 s and pn0 pSS ðÞ 2s Apply sub - algorithm 1 Apply sub - algorithm 2 Q1 Q1 1 While Q1 D1 Apply sub - algorithm 1 End Apply sub - algorithm 2 •

Sub-algorithm 1: Generating D1 , D2 , and D2 0

Begin For each n s:t:vn [ p1 ; calculate aTSS and aTSS ðÞ If anSS aTSS ðÞ D1 þ 1; D02 D02 þ 1 D1 If pR2 \vn D2 ← D2 + 1 Else D1 ← D1 + 2 End •

Sub-algorithm 2: Finding optimal solution at given Q1

Begin If bð1 xÞQ1 c D2 ðQ1 D1 Þþ Q2 D02 ðQ1 D1 Þþ and new pSS ðÞ pSS ðÞ at Q2 Else bð1 xÞQ1 c and new pSS ðÞ pSS ðÞ at Q2 Q2 If new pSS ðÞ [ previous pSS ðÞ new pSS ðÞ pSS ðÞ End


9

Figure 2 shows that our algorithm can efficiently find the optimal solution under given parameters. The improvement starts from lower right corner to upper left corner in the Figure 2 until pSS ðb; pR2 ; Q1 ; Q2 ; D1 ; D2 ; D02 Þ does not increase. 4.2 Dual sourcing case From the retailer’s perspective, having another supplier might have potential to increase expected profit. To analyse the impact of having another supplier on the retailer’s expected profit, we modify SS strategy structure. This structure adds another supplier in supplier tier (the two suppliers are not necessarily identical so that all the parameters such as SVF, RPP, SR, etc. can be different for each supplier) but maintains all the assumptions made in SS strategy structure (i.e. information of both SRs can be accessed by retailer and consumers, and consumers’ stockpiling behaviour can be accessed by both supplier 1 and 2 and retailer). For DS, we can derive the expected profit as following: pDS ðbS ; pR2 ; QS1 ; QS2 ; D1 ; D2 ; D02 Þ ¼ p1 minfD1 ; Q11 þ Q21 g ðp1w Q11 þ p2w Q21 Þ HðQ11 þ Q21 D1 Þþ þ ðb1 b2 Þðp1 minfQ12 þ Q22 þ ðQ11 þ Q21 D1 Þþ ; D02 g ðp1w Q12 þ p2w Q22 ÞÞ þ b1 ð1 b2 Þu1DS ðpR2 minfQ12 þ ðQ11 þ Q21 D1 Þþ ; D2 g p1w Q12 Þ þ b1 ð1 b2 Þð1 u1DS Þðp1 D02 p1w Q12 Þ þ b2 ð1 b1 Þu2DS ðpR2 minfQ22 þ ðQ11 þ Q21 D1 Þþ ; D2 g p2w Q22 Þ b2 ð1 b1 Þð1 u2DS Þðp1 D02 p1w Q12 Þ þ ð1 b1 Þð1 b2 ÞuDS pR2 minfðQ11 þ Q21 D1 Þþ ; D2 g þ ð1 b1 Þð1 b2 Þð1 uDS Þp1 D02

(11)

superscript where S = {1, 2}. Similarly, we define three additional binary variables: S stands for supplier, þ 1 2 0 0; if ðQ1 þ Q1 D1 Þ [ D2 0; if Q12 þ ðQ11 þ Q21 D1 Þþ [ D02 uDS ¼ , u1DS ¼ , and 1; otherwise 1; otherwise 0; if Q22 þ ðQ11 þ Q21 D1 Þþ [ D02 . u2DS ¼ 1; otherwise The first term represents the revenue of the retailer in period 1, and the second and third term represent procurement cost and holding cost respectively. The fourth term stands for the retailer’s profit in period 2, when both supplier 1 and 2 are UP. The fifth (seventh) and sixth (eighth) term represent the retailer’s profit in period 2, when supplier is supplier 1(2) is UP but supplier 2(1) is DOWN. The last two terms represent the retailer’s profit in period 2, when both supplier 1 and 2 are DOWN. Similar to the case of SS, the retailer’s problem under DS is: max pDS ðbS ; pR2 ; QS1 ; QS2 ; D1 ; D2 ; D02 Þ

(12)

QS1 ;QS2

s:t:

anDS

¼ b1 b2 þ

P k2S

1 2 kn n 1 þQ2 NDS 1fv [ p1 g bk ð1 bl6¼k Þmin 1; ðQ1QþQ1 þQ 2 þQkn N Þ1 n DS fv p g þNDS 1

1

2

þð1 b Þð1 b Þ QS2 2

1

2

Q11 þQ21 NDS 1fvn [ p1 g ðQ11 þQ21 NDS Þ1fvn p1 g þNDS

for 8S ð1 xS ÞQS1 ; ð1 þ xS ÞQS1

1

for 8n

(13)

(14)

where QSn 2 is consumer n’s estimation of retailer’s sourcing allocation for supplier S and NDS = min{Q1 + Q2, N}. Consumers can access the basic information only but cannot access further information including (i) how much SVF do the suppliers have? (ii) how much will the retailer source? and/or (iii) how does the retailer allocate the purchasing volume in period 2? Thus, we assume that an individual consumer n estimates retailer’s purchasing volume and its alloS cation in period 2 as those of period 1, i:e:QSn 2 ¼ Q1 ; S ¼ f1; 2g: n n Proposition 6: It is always true that aDS aSS at a certain level of total sourcing size in period 1, i:e:Q11 þ Q21 ¼ Q1 , when the retailer adds another non-inferior supplier (supplier 2) on existing supply base, i:e:b2 b1 ¼b. But, not always when adding inferior supplier, i:e:b2 \b1 ¼b. Proof 6: See Appendix 1. is decreasing in β = β1 at Proposition 7: When b2 b1 ; jD1 D2 jSS jD1 D2 jDS Q11 þ Q21 ¼ Q1 ; where jD1 D2 jSS and jD1 D2 jDS are demand fluctuation in SS and DS respectively. Proof 7: See Appendix 1. Proposition 6 indicates that consumers’ perception of retailer’s ability to satisfy their demand in period 2 will always increase when the retailer adds non-inferior supplier to its supply base. This perception further implies that consumers

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demand over two periods will be more balanced (Proposition 7: the fluctuation of demand over two periods will decrease). This implication is intuitive since with higher perception of future demand satisfaction will reduce consumer’s fear of purchasing failure in period 2 so that the consumers will be less likely to stockpile. Proposition 7 further indicates that non-inferior supplier addition always positively impacts the balancing of consumers’ demand over two periods, but the effectiveness of the supplier addition for demand balancing will decrease as incumbent supplier is more reliable. However, when the retailer adds inferior supplier, if the sourcing size in period 1 is high enough, the perception still increases, but if the sourcing size is relatively low, we cannot guarantee that the perception will increase. This implies that the addition of inferior supplier can exacerbate the consumer’s fear of purchasing failure in period 2 so that the consumers could be more likely to stockpile. We recognise that the sourcing strategies can affect consumers’ purchasing and stockpiling behaviour. Intuitively, consumers’ behaviour also can affect retailer’s sourcing decisions. We will study how the consumers’ changed behaviour influences retailer’s sourcing decisions. Moreover, we will analyse how the SVF influences the decisions. Our underlying assumption is that the information about SVF is not basic information so that it cannot be accessible to consumers. This assumption implies that SVF does not directly affect consumers’ behaviour, but it can affect retailer’s purchasing and inventory carrying decisions so that the effectiveness of sourcing strategies can vary depending on supplier’s characteristics. Similar to SS analysis, even though it might contain some more redundant range, we can set Q11 þ Q21 ¼ 2N as the upper bound of sourcing size in period for DS analysis. We solve DS case by applying modified Sub-algorithm 1 used in SS case and branch and bound to ensure optimal solution in the following section. 5. Numerical experiment In this section, we report results from numerical experiments based on our models in Sections 3 and 4. In addition to examining the impact of simultaneous parameter changes on the retailer’s optimal purchasing and inventory carrying decisions under SS, we conducted an extensive numerical analysis to compare the profitability of two sourcing strategies: SS and DS. More specifically, we explore the following three areas of inquiry: • Impact of supplier’s offers on retailer’s profit in SS: The expected profit of the retailer is determined by supplier’s offers including RPP and SVF and consumer’s purchasing and stockpiling behaviours (i.e. D1 and D2 ). In this regard, the key questions are: ‘how do SVF and RPP influence the retailer’s risk mitigation approach?’ and ‘how does that approach influence consumer’s behaviour and retailer’s expected profit over two periods under SS strategy?’, which can be useful in evaluating the impact of supplier’s offer on retailer’s profit and explaining how the external parameters modify retailer’s approach to mitigating supply disruption risk. Another key question is ‘how does the relationship between SVF and RPP affect retailer’s risk mitigation approach and expected profit over two periods under SS strategy’, which can address the trade-off between the offer’s positive and negative effects. • Impact of SR on retailer’s profit under SS: SR affects consumer’s perception of supply availability in the following period. Consumer’s demand over two periods will thus vary depending on SR. One important question is ‘how does SR affect: (i) consumer’s behaviour, (ii) retailer’s risk mitigation approach, and (iii) retailer’s expected profit under SS strategy’. This leads to another interesting question about SVF versus SR under SS; in other words, ‘which supplier provides more benefits to the retailer under SS – more flexible one or more reliable one’ (SVF versus SR under SS). • SS versus DS: Based on the results from the first two numerical analyses, we are able to investigate the superiority of SS over DS or DS over SS; in other words, ‘which strategy is better and whether such superiority always holds’ We will start with two identical suppliers setting and then expand our consideration to multiple possible settings.

5.1 Impact of supplier’s offers on retailer’s profit in SS Since retailer’s expected profit function contains two supplier’s offers (SVF and RPP), we generate four plots in one figure. In Figure 3, we maintain all parameters used in Section 4.1. except SVF and RPP. We vary SVF in the range ½0:555; 0:995 and RPP in the range ½0:505; 0:825 (no significant change in quantity or trend change in profit was found outside the two ranges above over the entire range from 0 to 1). The two plots on the left column of Figure 3 illustrate the impact of SVF, and the two plots on the right column show the impact of RPP on retailer’s profit. Figure 3 shows that higher SVF does not harm retailer’s profit and lower RPP leads to higher retailer’s profit and inventory level as noted by prior studies (Greenleaf 1995; Lee, Padmanabhan, and Whang 1997). One interesting


11

Figure 3. Impact of SVF and RPP on retailer’s profit and inventory levels at β = 0.5.

observation is that a higher SVF incurs a higher inventory level at the retailer; which contradicts the existing view (e.g. Avittathur and Swamidass 2007) that high SVF will reduce inventory level of the retailer. This counter intuitive result can be explained by the positive impact of retailer’s inventory level on consumer’s perception. A high level inventory allays consumer’s fears of purchasing failure due to stock-out that leads consumers to stockpile less. High SVF enables a retailer to source higher volume in period 1 that will balance consumer’s demand over two periods (proposition 6). This can result in an increase in inventory level. In contrast, however, high SVF also can enable the retailer to dramatically reduce retailer’s purchasing volume in the following period to avoid redundant purchases for that period. The retailer may try to reduce sourcing volume in period 1 with the purpose of minimising its inventory holding costs, but the demand in period 1 will increase as the volume decreases where the unmet demand is lost. Given this situation, high SVF would prevent the retailer from losing its consumers by allowing a high discrepancy between sourcing volumes over two periods. Not surprisingly, higher SVF corresponds to more costs (Asche, Osmundsen, and Tveterås 2002). We thus consider three different ‘retailer’s Purchasing Price – supplier’s Volume Flexibility’ (PPVF) structures: (i) linear (three plots in the first column of Figure 4), (ii) concave (three plots in the second column of Figure 4) and (iii) convex structure (three plots in the last column of Figure 4). We use structure functions x ¼ minfxg þ a ppw ; x ¼ ln ppw þ hx minpfpw g lnðhx Þ þ minfxg and x ¼ exp ppw hc for linear, concave, and convex, respectively, where –1 minfpw g minfxg; minfp g ¼ 0:42; and minfxg ¼ 0:595. a ¼ 0:085 w 0:006, θω = (a ⋅ 0.7) , hc ¼ exp p Figure 4 shows that the negative impacts of RPP increments on retailer’s profit dominate the positive impacts of SVF increments on retailer’s profit in all the three structures. Intuitively, however, the positive correlation between RPP and SVF may partially make up for the retailer’s profit loss from RPP increments. By comparing the independent structure (RPP and SVF are independent) and our linear structure, for instance, we can observe that the linear structure can reduce retailer’s profit loss by 8:978:44 ¼ 5:91% and 17:9416:02 ¼ 10:7% at RPP 0:76 and 0:81, respectively (see the ‘profit’ columns in Table 1). As can 8:97 17:94 be seen from the corresponding inventory levels in the linear structure, higher inventory plays a moderating role, and the high SVF makes this possible. Because SVF is non-decreasing in RPP, similar results (with more or less compensation effects) will be derived in the other cost structures. Figure 4 also shows that higher SVF, beyond the point where inventory level meets its maximum, gradually lose its effectiveness. This collectively implies that, after a certain threshold is reached, a high inventory level has only marginal impact on positive perception about the product availability in period 2, i.e. anSS ¼ 1 for 8n s:t:vn [ p.

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Figure 4. Impact of SVF – RPP relationship on retailer’s profit and inventory levels. Table 1. Compensation effect of SVF – RPP correlation.* SVP RPP

Independent

Linear

Inventory

Independent –

Independent

0.85

–

0.85

47

–

47

0.85

–

0.90

(+6.0%)

79.56

(–8.97%)

80.43

(–8.44%)

47

–

64

(+36.17%)

0.81

(+12.0%)

0.85

–

0.95

(+12.0%)

71.72

(–17.94%)

73.40

(–16.02%)

47

–

76

(+61.70%)

87.40

–

Linear

(+6.0%)

87.40

–

Linear

0.76

0.72

–

Profit

–

*First row is reference.

Convex structure is superior to linear structure at relatively low (0.63) and mid (0.73) RPP, but the two structures become identical with respect to both profit and inventory at a high (0.83) RPP (see Table 2). Thus, the superiority of convex structure increases with increasing RPP until inventory level reaches maximum. From the peak, the superiority of convex structure gradually decreases and becomes identical to linear structure. These results collectively indicate that supplier’s offers are tightly connected to retailer’s profit. At a certain level of RPP, higher SVF does not deteriorate retailer’s profit. Similarly, at a certain SVF, lower RPP always increases retailer’s profit. Two different offers show one common notable feature: the higher the inventory level, the better for the retailer when the offers work in positive characteristic (i.e. high SVF and/or low RPP). However, those two offers often conflict – high SVF comes with high RPP. In general, the negative effect of high RPP dominates the positive effect of high SVF, but the degree of domination can be attenuated or exacerbated depending on the RPP-SVF structure. 5.2 Impact of SR on retailer’s profit in SS To illustrate the impact of SR, we start by fixing pw ¼ 0:6p and x ¼ 0:85 as used in Section 4.1. Figure 5 is generated by varying supplier reliability β in the range ½0:09; 0:97. We further analyse the impact of SR sensitivity in RPP by characterising the relationship SR and RPP. Figure 6 is generated by varying SR sensitivity (i.e. varying ppw and x simultaneously) while fixing all the other parameters. One counter intuitive observation in Figure 5 is that retailer’s expected profit decreases when SR is low (segment 1: about lower than 0.34 under given parameter set). In general, high SR positively impacts buyer’s profit (retailer’s profit in this study) under SS (Yu, Zeng, and Zhao 2009). When SR is very low, the retailer may want to maintain a very high level inventory to avoid the loss of consumer’s demand in period 2. The high level of inventory drives consumers to stockpile less, which implies that inventory holding costs will be very high. As SR increases in this segment, however, the likelihood of supplier disruption decreases. This implies that the retailer will be more likely to purchase redundant

0.75 0.87 0.98

0.63 0.73 0.83

0.70 0.77 0.83

(–6.67%) (–11.49%) (–15.31%)

Concave

*Linear columns are reference.

Linear

RPP

SVP

0.77 0.92 1.07

Table 2. Comparison of three PPVF structures.*

(+2.67%) (+5.75%) (+9.18%)

Convex 98.56 84.38 69.31

Linear 97.01 81.51 66.29

(–1.57%) (–3.40%) (–4.36%)

Concave

Profit

99.25 86.01 69.31

(+0.70%) (+1.93%) (+0.00%)

Convex

18 53 76

Linear 7 28 43

(–61.11%) (–47.17%) (–43.42%)

Concave

Inventory

28 68 76

(+55.56%) (+28.30%) (+ 0.00%)

Convex

International Journal of Production Research 13

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J. Yoon et al.

Figure 5. Impact of SR on retailer’s profit and inventory levels.

Figure 6. Impact of SR sensitivity in RPP on retailer’s profit.

units of product in period 2 due to SVF constraint. Moreover, the increased SR leads more consumers not to stockpile. The increased SR, therefore, can negatively affect retailer’s profit in this segment. We can observe that retailer’s profit increases in SR when SR is medium (segment 2). Relatively lower purchasing volume in period 1 than that of segment 1 can cover supplier disruption risk. At the beginning of this segment, selling more units in period 1 is more likely to give higher profit since the retailer is still under pressure of supplier disruption risk. As SR increases, however, the retailer will have more chance to utilise the volume that will be purchased in period 2. The retailer might reduce purchasing volume in period 1 while increasing purchasing volume in period 2 to save inventory holding costs, but SR is still not high enough to convince the retailer that the increased purchasing volume scheduled in period 2 can be utilised. Therefore, although SR increases, the retailer will not reduce purchasing volume in period 1 to achieve the benefit from higher inventory level in this segment. When SR is high (segment 3), we can observe the inventory level drops to zero so that the retailer could increase profit by removing inventory holding cost since a high SR itself can alleviate consumer’s fear of purchasing failure in


15

period 2. The high SR even ensures that the purchasing volume scheduled in period 2 can be utilised. Maintaining high level inventory is thus not a good strategy for the retailer in this segment. Similar to our analysis used in Section 5.1, we characterise the relationship between SR and RPP in a linear fashion (Yu, Zeng, and Zhao 2009). As shown in Figure 6, although the shape of retailer’s profit is not identical, the basic structure of retailer’s profit is maintained under a positive relationship between SR and RPP. Different from Figure 5, the mid SR (segment 2) exhibits less of a positive trend for profit as SR is more sensitive in RPP. We thus can conclude

Figure 7. Comparison between SVF impact and SR impact on retailer’s profit.

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J. Yoon et al.

Table 3. Speed of SVF impact on retailer’s profit. Profit ISS = 0.5 SVF 0.08 point 1 0.45 point 2 +

pw = 0.6p 51.36 70.08 18.72

ISS = 0.6

pw = 0.7p pw = 0.8p 25.92 47.76 21.84

0.48 25.44 24.96

ISS = 0.7

pw = 0.6p pw = 0.7p pw = 0.8p point 1 point 2 +

51.89 73.49 21.60

26.14 51.34 25.2

0.39 29.18 28.79

pw = 0.6p pw = 0.7p pw = 0.8p point 1 point 2 +

56.69 80.38 23.69

31.17 58.12 26.94

5.66 37.25 31.59

that the retailer’s profit is negatively related to SR when SR is very low. This association weakens as SR increases, and the retailer’s profit slope becomes positive when SR is high enough. In sum, our results indicate that higher SR cannot always promise better outcome. Increment of SR can negatively affect retailer’s profit when it is very low and cannot be increased enough. This phenomenon can be alleviated as given SR increases, and the increase in SR positively affects retailer’s profit when SR becomes high enough. When SR is very low and there is no way to increase SR high enough, therefore, increasing SVF can be a better approach to increase retailer’s profit. The aforementioned findings naturally raise the question of ‘which supplier can provide more benefit to the retailer under SS: more reliable supplier or more flexible supplier?’ To answer this question, we initially assume SVF and SR are equal and maintain all the other parameters, and then vary SVF and SR. The underlying assumption here is that the marginal cost of increasing SVF and SR are the same so that we can rule out the RPP effects in this analysis. The initial procedure enables us to start with fair condition, and the assumption made above maintains the fairness in comparison of the impact of SVF and SR on retailer’s profit. In general, we observe the impact of SR is larger than SVF on the right side of the dashed line (R), while the impact of SVF is larger than SR on the left side of the dashed line (L). Note that the origin of R and L is the x-axis value of the dashed line. Based on Figure 7, as SR increases, the effect of SVF increment can positively affect retailer’s profit faster (see Table 3). As found in the previous section, however, the increment of SVF loses its positive impact at a certain point and Figure 7 shows that the SVF increment loses earlier as SR increases. Therefore, when SR is relatively low (Figure 7(a)), SVF impact is slightly higher than SR impact at the beginning of R, whereas when SR is relatively high (Figure 7(c)), SR impact is greater than SVF impact at the beginning of L. This suggests that it would be better off for the supplier to focus on increasing its SVF if it cannot make a significant increase in its SR. 5.3 Retailer’s maximum profit vs. supplier’s offers We start with the parameters used on Section 4.1 and apply high-sensitive SR in RPP used in Section 5.2. The only difference in this section is that the retailer has two identical suppliers. Figure 8 shows that DS strategy is not always better than SS with respect to retailer’s profit. Specifically, DS is better on the left side of the reference point whereas SS is better on the right side of the reference point. At the reference point ðSR ¼ b ¼ b1 ¼ b2 ffi 0:50Þ, DS strategy is almost the same as SS strategy (DS is only slightly better) in retailer’s profit under given parameters. We further analyse the effect of non-identical supplier addition to the existing supply base. To illustrate this effect, we compare three potential risk mitigation strategies: (i) SS with incumbent supplier (SS-I), (ii) SS with new supplier (SS-N), and (iii) DS of incumbent and new supplier. Note that incumbent supplier is supplier 1 and new supplier is supplier 2. We fix all SS-I parameters used in Section 4.1, and maintain the parameters for SS-N and DS except new supplier’s SR. New supplier’s SR varies in the range [0.01, 0.99] with high-sensitive SR in RPP used in Section 5.2. Figure 9 shows that DS and SS-N are superior to SS-I when new supplier’s SR is very low and/or very high (segment 1), and those two strategies provide almost same outcome in terms of retailer’s profit. When new supplier’s SR increases from very low to relatively low (segment 2), however, DS is the best strategy and SS-I and SS-N become the best strategy sequentially as SR increases (segments 3 and 4). For segment 2, DS is the best strategy by alleviating consumers’ fear with stocking maximum inventories. From segment 3 ðnew SR 0:61 and 0:72Þ, in contrast, maintaining the very high level of inventory burdens the retailer with relatively higher RPP. Thus, the increased SVF from DS loses its attractiveness.


17

Figure 8. SS vs. DS of two identical suppliers.

Figure 9. SS-I vs. SS-N vs. DS.

The feature observed in segment 3 continues in higher new SR than segment 3. In this range (segment 4), however, the high enough, new SR itself provides the same effect of high level inventory in low new SR (i.e. balancing consumer’s demand without increasing inventory level). SS-N would thus be better strategy within this range. DS primarily uses new supplier (supplier 2) when supplier 2 can provide the retailer with higher profit, while uses incumbent supplier (supplier 1) when supplier 1 is more profitable for the retailer. These results collectively indicate that the superiority of SS or DS strategy over the other is contingent. SS (SS-N) and DS are indifferent to retailer’s profit when new SR is very low and/or very high. As the superiority of SS-N in retailer’s profit decreases (compared to SS-I), however, DS becomes more likely to be the best strategy by taking advantage of high SR from SS-I and low RPP from SS-N as well as taking advantage of higher SVF without sacrificing cost. On the contrary, the retailer cannot take advantage of low RPP from supplier 2 when new SR becomes relatively high and

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SS-I is superior to SS-N. Since the new SR is not very significantly higher than supplier 1’s SR, the retailer cannot take advantage of high SR from supplier 2 either. Moreover, high level inventory loses the attractiveness in this range, and the retailer cannot enjoy the free SVF effect of DS. SS (SS-I) is therefore the best strategy when new SR is relatively high. Later, when new SR becomes high enough, SS (SS-N) becomes the best strategy by replacing the positive effect of high inventory with high SR (i.e. the high enough new SR eliminates the need for inventory). When the new SR becomes very high, finally, SS (SS-N) and DS are indifferent since DS will use supplier 2 as the main supplier. 6. Conclusions and implications Our study bridged the sourcing strategy and stockpiling behaviour literature under supply disruption risks and analysed the interactions among consumer’s stockpiling behaviour, SVF, SR and RPP. In doing so, we developed a conceptually unifying, mathematically rigorous and practically relevant framework to address gaps in our knowledge about the impacts of upstream as well as downstream disruption risks on supply chain decision-making in a multi-tier setting (rather than a simple dyadic setting). Our analytical investigation established convincing rationale that consumers are more likely to stockpile when they anticipate future supply shortages. It was shown that this behaviour becomes more pronounced when the consumer has experienced supply shortages in the past. We also proved that the demand fluctuation from the consumer’s stockpiling behaviour is highly affected by retailer’s inventory level in period 1. More specifically, a higher inventory level leads to more balanced demands over two periods. This result affects retailer’s decisions that will subsequently influence supplier’s decision. Further, we also conducted a comprehensive numerical study to investigate how the above sequential reactions influence retailer’s profit, which resulted in the following counterintuitive and interesting findings: (1) High SVF enables the retailer to decrease inventory level so that it can reduce inventory holding cost. However, our numerical analysis shows that the high SVF allows the retailer to stock more inventories to create more profit, when SR is low. This result indicates that from the supplier’s perspective, the supplier needs to improve its SVF at lower cost when it recognises that the retailer anticipates that its SR is low. (2) Higher SR is always better if RPP is constant. This intuition can be supported only when the SR significantly increases. Based on our analysis, the marginal increment of SR deteriorates retailer’s profit when the initial SR is very low. This result suggests that the retailer should find cheaper but more unreliable supplier or very highreliable but expensive supplier for better profit performance. For suppliers, this result recommends them to focus on increasing SVF rather than SR if they are incapable or unsure of making a significant improvement of their reliability. On the basis of the above findings, it can be concluded that the superiority of SS versus DS is contingent. More specifically, SS (SS-N) and DS are recommended to the retailer when ‘new SR’ is very low and/or very high. As ‘new SR’ increases (but still lower than the SR of the incumbent supplier), DS becomes more likely to be the best strategy for the retailer. SS (SS-I) would more likely be the best strategy when ‘new SR’ becomes higher than incumbent supplier’s SR. Later, SS (SS-N) becomes the best strategy when new SR becomes high enough, and finally, SS (SS-N) and DS becomes the best strategy again for the retailer when the new SR becomes very high. There are several interesting extensions for future research. First, it will be worthwhile to incorporate multiple suppliers’ competitions into our scenario. The competition in the supplier tier might affect supplier’s offers (i.e. RPP and SVF) that can significantly influence retailer’s decisions. This extension may provide more implications on how the competitive dynamics of a supply chain interplay with consumer stockpiling behaviour and corresponding sourcing strategy. A second avenue of future research is extending the time-frame of this research to more number of periods. Under this setting, consumers could consider not only the supply disruption but also preservation rate of unused units of products. This opens the possibility that the consumers can stockpile more than two units. It will also be worthwhile to investigate the supplier’s strategy to consumer stockpiling. Especially interesting will be the interrelationships among the supplier’s control over its reliability, retailer’s perception, and the corresponding sourcing strategies. Finally, the empirical examination of retailer’s behaviours in anticipation of supply disruption will be useful in refining our analytical model to better capture optimal decision makings in a multi-tier supply chain setting. Acknowledgment The corresponding author was supported by the Summer Research Grant from College of Business Administration at Kansas State University.


19

Disclosure statement No potential conflict of interest was reported by the authors.

ORCID Myung Kyo Kim

http://orcid.org/0000-0003-4840-5958

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Appendix 1. Proofs of Propositions Proof of Proposition 1: Based on the logic in equation (2) a consumer will purchase two units only if the utility of purchasing two units in period 1 is greater than the expected utility of the other case. So, we can express this argument as the following inequality. ð2v 2p hÞc [ að2v 2pÞc þ ð1 aÞðv pÞc ¼ að2c 1Þðv pÞc þ ðv pÞc ðvpÞ which implies that when a ð [ Þð2v2phÞ ¼ aT ðp; h; v; cÞ; purchasing two (one) units in period 1 will be better since 2γ – 1 > 0 ð2 1ÞðvpÞ ■ (because 0 < γ) and v > p (because the case of v > p need not be considered) so that (2γ – 1)(v – p)γ > 0. Proof of Proposition 2: As h increases, the numerator of αT(p, h, v, γ) decreases. Therefore, it is clear that αT(p, h, v, γ) is decreasing in h. Moreover, first derivative of αT(p, h, v, γ) w.r.t. p is negative, which implies that αT(p, h,v, γ) is decreasing in p. In addition, if cð2v2phÞ @ T 2 1 we treat v and γ as independent, since by assumption, @va ðp; h; v; cÞ ¼ 2v2ph vp [ 0, ðvpÞ c

c

c

c

c

c

v p [ h ) v p [ h=2;

1 h vp ¼ ð2v2phÞðvpÞ [ 0. 2v2ph @ T ðA CÞ lnð2Þ\0, Moreover, vp @ca ðp; h; v; cÞ ¼ ðA BÞ ln c c c c B ¼ ð2v 2p hÞ ðv pÞ ; and C ¼ ð2v 2pÞ ðv pÞ , c

cð2v2phÞ ðvpÞc

[ 0 and

2 2v2ph

cð1cÞð2v2pÞc 2c lnð2Þþcð1cÞð2v2pÞc ð12c Þ ln

þð2c 1Þ2 ð2v2pÞc

where

A = (2v – 2p – h)γ(2v – 2p)γ, since

ð Þ aT ðp; h; v; cÞ ¼ [ 0 but @c@ aT ðp; h; v; cÞ ¼ 0 at h = 0 (lower bound of h) and ð2v2phÞ T h = v – p(upper bound of h, since if h > v – p, thenα (p, h, v, γ) = 0. However, by assumption, γ decreases as v increases. Therefore, ■ αT(p, h, v, γ) increases in v. Proof of Proposition 4: As Q1 increases, the second term in equation (9) increases and b\1 so, anSS increases. Consequently, by proposition 2, the increased anSS leads d1n to more likely be 1. This implies that D1 will be likely to decrease, ■ i:e: Pr d1n 1fvn [ p1 g ¼ 1 increases, and sequentially, D2 ðD02 Þ will be likely to increase by definitions in equations (5) and (7). Proof of Proposition 5: When we assume that Q1 and Q2 are continuous, πSS is piecewise linear w.r.t. Q2 at a certain point of Q1. Regardless of ðQ1 D1 Þþ can satisfy D02 or not, Q2 [ 0 because of SVF, but min (Q2) = (1 – ω)Q1. If for increasing pSS . However, when ð1 xÞQ1 D02 ðQ1 D1 Þþ ; Q2 will increase until Q2 ¼ D02 ðQ1 D1 Þþ D02 ðQ1 D1 Þþ \Q2 ð ð1 þ xÞQ1 Þ; πSS starts decreasing and never bouncing up because Q2 ðD02 ðQ1 D1 Þþ Þ [ 0 means @2 @2h

@ @c

2c

2v2ph 2v2p


21

redundant amount so that this amount will be never sold. Therefore, the optimal Q2 for the best πSSat a certain point of Q1 should exist before it starts showing decreasing trend as Q2 increases, i:e: Q2 ¼ D02 ðQ1 D1 Þþ . This will still hold when Q1 and Q2 are discrete. In that sense, if bð1 xÞQ1 c [ D2 ðQ1 D1 Þþ ; πSS will never increase as Q2 increases. Thus, when πSS shows decreasing trend in Q2 from the beginning, Q2 ¼ bð1 xÞQ1 c: ■ Proof of Proposition 6: Assume that Q11 þ Q21 ¼ Q1 . If vn p1 ; anDS ¼ anSS ¼ 1: But if vn [ p1 ; constraint (12) can be rearranged by anDS ¼ b1 b2 þ ð1 b1 b2 ÞQ NN þ D for each consumer n by simple algebra with substituting QS1 for QS1n ; 8 0; if Q11 þ Q21 ¼ 2N I > > > k l6¼k > < if Q11 þ Q21 \2N \Q11 þ Q21 þ minðQS2 Þ II / + / [ 0; where D ¼ bk ð1 bl6¼k ÞNQ + /l6¼k [ 0; if Q11 þ Q21 þ Qk2 \2N \Q11 þ Q21 þ Ql62¼k III > > > > P bk ð1 bl6¼k Þ Q [ 0; otherwise, i.e:Q1 þ Q2 þ maxðQS Þ\2N IV : 1

SS

SS

k 1

SS

k 1

1

NSS

k2S

1

2

and /k ¼ bk ð1 bl6¼k Þ 1 Q NN [ 0 when Q1 ¼ Q11 þ Q21 \2N . anDS anSS ¼ D þ e; where e ¼ b1 ð1 b2 Þ 1 Q NN 0. For I, ε=0 so that anDS anSS ¼ 0 regardless of β2 value. For II, Q N Q N Q N 2 1 2 2 1 2 1 n n þ b ð1 b Þ 1 N so that aDS aSS ¼ b ð1 b Þ 1 N [ 0 regardless of β value. For III, D ¼ b ð1 b Þ 1 N Q N 1 2 2 1 Q 1 2 þ b ð1 b Þ 1 N ; if Q1 [ Q1 ; which is greater than –ε regardless of β2 value. However, when D ¼ b ð1 b Þ 1 N Q11 \Q21 , D ¼ b2 ð1 b1 Þ 1 Q NN þ b1 ð1 b2 Þ 1 NQ [ e can be guaranteed only if β2 ≥ β1. Similarly, for IV, when k 1

SS

1

SS

1

SS

1

SS

2 1

SS

1

SS

SS

SS

1

SS

SS

SS

SS

SS

1

1 1

SS

SS

SS

b2 b1 ; D ¼ b1 ð1 b2 ÞN þ b2 ð1 b1 ÞN [ b1 ð1 b2 Þ; since Q11 þ Q21 ¼ Q1 NSS , which implies that D [ e , anDS anSS [ 0. However, we cannot guarantee anDS anSS [ 0 when β2 < β1. Therefore, if β2 ≥ β1, anDS anSS is always true, otherwise anDS anSS cannot always be guaranteed. ■ Proof of Proposition 7: From proposition 2, as anSS decreases, the likelihood of d1n ¼ 2 increases, i.e. jD1 D2 j increases (unbalanced). So, to prove this proposition, we only need to see if the consumer who decides to purchase two units in period 1 under SS will change its mind to purchase one unit in each period under DS. For those who perceive that anSS ¼ b1 þ ð1 b1 ÞQ NN ; i:e:; vn [ p1 , when anSS aT ðp1 ; p2 ; h; v; cÞ, they will purchase two units in period 1. However, they antici1

Q11

Q21

SS

SS

SS

SS

pate the demand satisfaction under DS as anDS ¼ b1 b2 þ ð1 b1 b2 ÞQ NN þ D. Clearly, by proposition 4, anDS anSS when b2 b1 ;

which implies that Pr anDS [ aT ðp1 ; p2 ; h; v; cÞ anSS [ aT ðp1 ; p2 ; h; v; cÞ i:e:; anDS anSS ¼ D þ e 0, @ða a Þ 1 [ 0 so that jD1 D2 jSS jD1 D2 jDS 0. But, @b 0 in all of ①, ②, ③, and ④. Therefore, anDS anSS is decreasing in β , ■ which implies jD1 D2 jSS jD1 D2 jDS is nonnegative but is decreasing, as β1 increases. 1

SS

SS

n DS

n SS

1