Supply process improvement decisions for a newsvendor with random capacity
Introduction
An increasing concern in global supply chains is vulnerability to supply risks, such as component shortages, late deliveries, machine breakdowns, supplier bankruptcies, and so forth. Supply risks can undermine operational performance and profits—with severe adverse consequences for a geographically or organizationally dispersed supply chain. Empirical studies by Hendricks and Singhal, 2005, Hendricks and Singhal, 2008, Hendricks and Singhal, 2014 document that various supply chain glitches, including production disruptions and capacity problems, degrade firms' financial and operational performance. For example, production glitches at Foxconn, Apple's contract manufacturer, led to extended supply shortfalls and shipping delays of the iPhone X in 2017 (Kubota, Mickle, & Mochizuki, 2017); also, Intel's stock price fell in response to the firm's inability to supply promised processor chips (Kim, 2018). Given the importance of abundant and stable supplies, companies are strongly inclined to participate in supply process improvements. For example, Tesla Inc. cooperated with its battery supplier Panasonic to resolve its “production hell” (i.e., not meeting planned volume targets) by adding more production lines (Ma, Masatsugu, & Furukawa, 2018). In this article, we focus on the firm's inventory and process improvement decisions under random capacity, a typical supply risk under which the number of items the firm can produce or procure is constrained by a stochastic quantity.
Given its critical role in achieving supply chain performance excellence, supply risk management has received considerable attention from operations management scholars (Heckmann et al., 2015, Ho et al., 2015, Kouvelis et al., 2011, Sodhi et al., 2012, Tang, 2006). Several forms of supply risk—including supply disruptions, random yield, random capacity, and uncertain lead times—are extensively examined in the literature. Our paper is most closely related to the stream of research that studies operations-related decisions under random capacity. Readers are referred to Snyder et al. (2016) for a recent review on supply disruptions; for reviews on random yield models, see Yano and Lee, 1995, Grosfeld-Nir and Gerchak, 2004.
Numerous studies have been conducted on inventory models under random capacity. Ciarallo, Akella, and Morton (1994) is a seminal paper on optimal inventory control with random capacity. With dynamic programming, the authors discover that the optimal order quantity for the single-period model is the same as that for the newsvendor model, and that a base stock policy is optimal for the multi-period setting with both finite and infinite periods. Since then, many extensions of this problem have been studied. For a supply system that features both random capacity and random yield, Wang and Gerchak (1996) demonstrate the optimality of a threshold-type policy under which the order-up-to levels are state-dependent. Dada, Petruzzi, and Schwarz (2007) consider the effect of supply diversification in a single-period problem where the newsvendor can procure from multiple unreliable suppliers. These authors propose a cost-based selection rule whereby unit procurement cost is the primary factor that determines whether a supplier is chosen. Feng (2010) analyzes multi-period joint inventory and pricing decisions under random capacity. She finds that an order is issued only when the inventory level is less than a reorder point, and explain how dynamic pricing mitigates supply risk. Li et al., 2013, Li et al., 2017 consider a newsvendor-like firm that sources from two suppliers with uncertain supply and examine the influence of responsive pricing, which—depending on particular model scenarios—can either complement or substitute for a supply diversification strategy. Other important model features can be found in studies of inventory control with random capacity. Examples include the firm's attitude toward risk (Liu et al., 2015, Wu et al., 2013), more complex supply structures such as assembly and serial systems (Bollapragada et al., 2004, Ji et al., 2016, Masih-Tehrani et al., 2011), and continuous-time settings (Erdem et al., 2006, Wang, 2010). Inventory management is also jointly applied with other strategies under random supply capacity, such as financial subsidy (Babich, 2010) and incentive design (Qin, Rao, Gurnani, & Bollapragada, 2014).
The tactics just described for managing supply uncertainty are implemented indirectly; that is, they treat random capacity as exogenous and depend on solutions that involve pricing and/or multi-sourcing to mitigate its adverse effect. However, few studies have addressed the direct and proactive improvement of capacity process and its benefit for firm performance, which is the focal problem of our study. Wang, Gilland, and Tomlin (2010) consider a firm that sources from one or two supplier(s) with risk (random capacity or yield), where the buyer firm can exert efforts to stochastically increase supplier reliability. These authors find that, under random capacity, process improvement is preferred over dual sourcing when suppliers exhibit greater cost differences and are less heterogeneous with respect to reliability. Using controlled lab experiments, Kalkanci (2017) tests the theoretical predictions in Wang et al. (2010) and reports both consistent and biased results. In the aforementioned papers, improvements in the magnitude of capacity are considered but other capacity characteristics, such as its variability, are not analyzed. Li and Arreola-Risa (2017) help fill this gap with their research on process improvements of random capacity while accounting for firm value. The authors establish that increasing (resp., decreasing) the mean (resp., standard deviation) of the firm's capacity can boost its value on the financial market. However, Li and Arreola-Risa do not consider the optimal investment in capacity process improvement, and they adopt the somewhat restrictive assumption that demand follows a normal distribution.
Significant efforts are devoted to investigating process improvement in the presence of other forms of supply risk such as random yield and disruptions. Liu, So, and Zhang (2010) consider a newsvendor that exerts marketing effort to affect demand when its supply has random yield; they find that, when the yield rate increases in the increasing concave order, the firm earns a higher expected profit. Similar findings are presented by Cheong and Song (2013) for the case of exogenous market demand. The strategic interactions among different firms have also been studied. For instance, Wang, Xiao, and Yang (2014) consider the case of two downstream manufacturers that can exert improvement efforts to increase the reliability of their common supply; they document the existence of spillover effects from such improvements. Lee and Lu (2015) focus on two competing firms that can increase their respective supply yields (in a stochastic sense) when facing stock-out substitution. These authors report that competition may discourage firms' reliability improvement efforts. For the case of supply disruptions, contracts are used by buyers in decentralized supply chains to incentivize suppliers' process improvement activities (Tang et al., 2014, Xia et al., 2011). A number of papers have studied suppliers’ development of or investment in improving product/process quality (Agrawal et al., 2016, Duarte et al., 2016, Quigley et al., 2018, Wang et al., 2013)—and reducing costs (Bernstein and Kök, 2009, Jin et al., 2019, Li and Wan, 2016)—both within and across firms. However, the models adopted in these papers are mostly stylized and do not involve procurement and inventory decisions.
Table 1 presents leading model features from the representative literature on supply process improvement and this paper. We follow the theoretical framework, presented by Li and Arreola-Risa (2017), of an expected profit--maximizing monopolistic newsvendor under random capacity. In this undertaking we explore the effects of improvements in the magnitude, variability, particular stochastic orders of capacity, and the improvements of parameters when the capacity follows different distribution profiles, which distinguishes our work from the extant literature.
As shown by our literature review, most studies on supply process improvement are about random yield and disruptions; models seldom incorporate random capacity, and reported results are usually based on a general capacity distribution. Hence we are motivated to address gaps in the related research by (i) considering a newsvendor that sells a product with random capacity, (ii) assessing how various capacity process improvements are related to corporate profit under different capacity distribution profiles, and (iii) analyzing the firm's optimal improvement-related decisions. More specifically, we evaluate the effects of two key aspects of capacity improvement (viz., increased magnitude and reduced variability) and find that each factor increases the firm's expected profit but with diminishing returns. We show also that the increasing concave order between capacities, which is closely associated with changes in mean and variance of capacity, is sufficient to guarantee the monotonicity of profit. In addition, we identify the positive moderating effects on the profit--capacity relationship of demand magnitude and the critical fractile. We then analyze, for the case when the random capacity follows a distribution (e.g., gamma, log-normal, uniform) whose mean and variance are jointly determined by model parameters (e.g., the gamma distribution's shape and scale parameters), how those parameters affect expected profit; the results are consistent with those derived in the general model. For some special cases where demand is uniformly distributed, we derive in closed form the optimal improvement decisions by selecting appropriate parameters.
Because of the difficulties in the analytical presentation of optimal process improvement decisions, we proceed by studying the patterns of those decisions numerically under several capacity distributions when system factors such as the critical fractile and demand variability vary. We find that the firm is more likely to implement process improvements for products with higher unit profit margins. Moreover, the firm is reluctant to reduce capacity variability when demand is more unpredictable, although whether the firm maintains a high average capacity level in response to changes in demand variability depends on the procurement cost.
We contribute to the field of supply chain risk management by articulating how firms benefit from process improvements in both the magnitude and variability of random capacity. We achieve this by first identifying the interconnections between corporate profit and specific parameters when capacity follows various distributions and then deriving the characteristics of optimal improvement decisions both analytically and numerically, which enables us to address many gaps in the literature. We complement the research framework of Liu et al., 2010, Cheong and Song, 2013 by applying the increasing concave order when comparing random capacities and by investigating its effect on profits. Also, we extend the analyses of Wang et al. (2010) by discussing how a reduction in variability improves the supply process. We augment the results reported by Paknejad et al., 2015, Aslani et al., 2017 with second-order effects of supply improvements, or the changing rates of a firm's profit as a function of improvement levels. Our study differs from Li and Arreola-Risa (2017) because it incorporates (a) a demand model with general distribution, (b) analyses for capacity with specific distributions whose parameters affect both the mean and the variance, and (c) a closed-form derivation of optimal improvement decisions.
The rest of this paper is organized as follows. Section 2 introduces a general model and delineates the effects of process improvement on profits. In Section 3, we discuss several cases where random capacity follows particular distributions and then assess analytically how expected profit varies with model parameters. Section 4 presents detailed numerical investigations into the relationships just described and then discusses how optimal process improvement decisions change environmental factors. We conclude in Section 5 with a summary of academic findings and managerial insights as well as suggestions for future research.
Section snippets
The general model
We consider a firm's decision in a single-period newsvendor setting. The firm acquires a product either by internal production or by external procurement, and then sells it in the market. Regardless of whether the firm makes the product itself, we refer to the source of this product as the supplier. The supplier has a random capacity in delivering the product, where is a nonnegative and continuous stochastic variable. Let and be, respectively, the cumulative distribution function
Effects of parameter changes under various random capacity distributions
In this section we focus on several distributions of random capacity, whereby the parameters (e.g., shape and scale parameters) may determine both the mean and variance of the capacity. We examine the effects of these parameters on optimal profit. The mean and variance are linked and so (as in Section 2) their effects cannot be observed separately. Nevertheless, our analysis yields insights that are largely consistent with Theorem 1 and Theorem 2.
Numerical studies
We have analytically studied the effects on expected profit of some model parameters when random capacity follows several commonly observed distributions. But unlike the analysis presented in Section 2, in many cases we can identify only the first-order effects of parameters i.e., the monotonicity of profit in the parameters. Thus in this section we investigate numerically their second-order effects, which deepens our understanding of the parameters' influence and verifies the conjectures
Academic findings
In this paper, we have studied a newsvendor-like firm's decisions under random supply capacity. Focusing on how process improvement affects the firm's expected profit, we find that both increasing the mean value of capacity and reducing its variance can increase profits, although these positive effects have diminishing returns. We also posit that an increasing concave order among random capacities would guarantee monotonic increases in profit. Moreover, we identify positive moderating effects
CRediT authorship contribution statement
Zhenyang Shi: Writing - original draft, Methodology, Investigation, Software. Bo Li: Conceptualization, Writing - review & editing, Funding acquisition. Shaoxuan Liu: Supervision, Validation.
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The research of Bo Li is supported by Ningbo Natural Science Foundation (2018A610120, 2019A610047), Zhejiang Qian- jiang Talent Program (QJC1803003), and the Pan-3315 Program of Ningbo (the Ningbo International Air Transport Research and Innovation Team).