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MANAGEMENT SCIENCE

informs

Vol. 51, No. 12, December 2005, pp. 1873–1891 issn 0025-1909 eissn 1526-5501 05 5112 1873

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doi 10.1287/mnsc.1050.0452 © 2005 INFORMS

Tailored Supply Chain Decision Making Under Price-Sensitive Stochastic Demand and Delivery Uncertainty Saibal Ray, Shanling Li, Yuyue Song

Desautels Faculty of Management, McGill University, 1001 W. Sherbrooke Street, Montréal, Quebec, Canada H3A IG5 {[email protected], [email protected], [email protected]}

I

n this paper, we study a serial two-echelon supply chain selling a procure-to-stock product in a price-sensitive market. Our analytical modelling framework incorporates optimal pricing and stocking decisions for both echelons in the presence of stochastic demand and random delivery times. We focus on understanding how these decisions for the chain are affected by its management paradigm (centralized or decentralized), and its business characteristics—price sensitivity, demand uncertainty, and delivery time variability. A novel combination of transformations enables us to analyze the framework and determine the unique optimal choices for centralized and wholesale price-based decentralized supply chains. More detailed investigation reveals that, in general, the business characteristics influence both the behavior and the optimal values of the decision variables, while the management paradigm primarily governs the optimal values. We illustrate the significance of these results in terms of how managers should tailor their decisions to align with their business requirements. Subsequently, comparison of the optimal profits between the channel partners and the management paradigms provides implications for decentralization strategy. A decentralized chain is most inefficient for moderately price-sensitive customers and uncertain environments, but is relatively more effective when dealing with mature products. We propose a contracting scheme that can improve the decentralized chain profit in reliable delivery time settings. The salient modelling insight of this paper is that ignoring the randomness of delivery time trivializes the interaction between pricing and stocking decisions. On the other hand, from a managerial viewpoint, we establish that optimal pricing policies provide the means to increase revenue and also act as strategic tools for tackling uncertainty. Key words: pricing; stochastic demand; random delivery times; multiechelon inventory; customized supply chains History: Accepted by Paul Zipkin, operations and supply chain management; received March 19, 2004. This paper was with the authors 3 months for 2 revisions.

1.

Motivation

an analytical framework, which can help managers to consistently and optimally tailor their decisions to the particular business situation that the chain is facing? To answer the above question, the first step is to understand the complexity of the managerial decisionmaking process in a supply chain. The uncertainty framework developed by Lee (2002) and Fisher (1997) suggests this complexity to be driven by what we term as “supply chain characteristics.” More precisely, complexity in a chain is generated by two sources: customer side and supply side. For Lee (2002) and Fisher (1997), customers create planning problems because of randomness in the number and the pattern of the demand process (demand uncertainty). We believe that the demand-side complexity is exacerbated by the propensity of end customers to either change firms or to totally drop the purchase decision if the product is too expensive (price sensitivity). Of these two customer-driven complexities, the mag-

Increased product variety, discerning customers and global markets have resulted in a fundamental shift in the business philosophy of most organizations. One particularly striking development is the growing reliance on outsourcing and resulting fragmentation of supply chains. The focus of competition has now shifted from the level of integrated organizations to that of decentralized and independent firms collaborating with each other. But the primary goal still remains the same: to satisfy the end customers because they provide the only source of real revenue in any chain. However, in the present economic scenario, each system faces its unique customer niche and distinct business environment. There is substantial anecdotal evidence attributing the profitability of chains to successful customization of strategic decisions for specific business requirements (Poirier and Quinn 2003). But is it possible to develop 1873

Ray et al.: Tailored Supply Chain Decision Making

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Management Science 51(12), pp. 1873–1891, © 2005 INFORMS

Figure 1

Examples of Supply Chains with Different Characteristic Trinities Customer (demand) complexity Low demand uncertainty Low price sensitivity

High demand uncertainty

High price sensitivity

Low price sensitivity

High price sensitivity

Supply complexity Low delivery time variability

Health care products

Low-end automobiles

Fashion apparel, laptops

DVD players, desktops

High delivery time variability

Healthy fast food (e.g., Subway)

Basic apparel outsourced to developing countries

High-end telecommunications products

Semiconductors

nitude of demand uncertainty depends on the type of product that the firm is selling and the amount and quality of demand information available to the firm. For example, functional products like groceries tend to have predictable demand, while innovative high-end telecommunications products face highly variable demand. Similarly, online firms with faster access to better demand information can be more accurate in their demand forecasting compared to their offline competitors (Magretta 1998). On the contrary, price sensitivity depends on the target customer niche. For example, the same cereal is now sold as branded products for customers willing to pay more, and as no-name brands for price-sensitive ones (www.pbs.org/moneymoves). Although functional products are normally associated with higher price sensitivities compared to innovative ones (Fisher 1997), there are instances when innovative products are also aimed at price-sensitive customers. A recent example is the DVD player market, where a number of firms have pursued bargain hunters (Bloom 2003). The complexity on the supply side is influenced by the nature of the supply source and the information flow between the parties in a chain. The consequence can be summarily represented by the variability of the delivery time between the parties (Lee 2002). A stable and flexible supply source with quick and accurate flow of information results in reliable delivery times between the parties. Lack of such attributes, on the other hand, might cause randomness in the delivery time. This issue has gained in importance recently because of the growth in offshore sourcing (from developing countries) by many North American firms. Such outsourcing gives rise to elongated chains and enhanced supply risks because of factors like security measures, transportation risks, diseases (e.g., SARS), and weather (e.g., tsunami). The ultimate implication is an increase in the variability of the replenishment time for the chain. In this paper, we define the business environment of a supply chain by the levels of the following

trinity of characteristics it is facing—demand uncertainty, price sensitivity, and delivery time variability. Some real-life examples of such characterizations are provided in Figure 1.1 This characterization not only applies to different supply chains at a particular point in time, but also to the same chain (i.e., for a particular product) over time. As indicated in Figure 2, most chains during their life cycles move from uncertain, low price-sensitive environments (incubation stage) to stable but highly price-sensitive settings (mature stage). Moreover, this classification is intricately related to whether a supply chain is managed centrally or in a decentralized manner. The decisions of a supply chain must then be in accordance with its specific management paradigm and business characteristics. Among the many decisions made by a manager, the two most generic and strategic ones are those of setting prices and planning for the amount of inventory to hold (Stern and El-Ansary 1992, Li and Atkins 2002). Hence managers are increasingly focusing on better integrating these two decisions into a single problem. However, optimal intra- and interorganizational supply chain management requires careful consideration of costs and benefits associated with all the relevant parties. In this paper, we develop models to support tailored pricing and inventory decisions for both centralized and decentralized supply chains by understanding the impact of the three business characteristics on such managerial decisions and channel performance.

2.

Model Framework

The basic investigative framework of this paper is a two-echelon (distributor-retailer) serial supply chain dealing with a single procure-to-stock product. We assume the perpetual end-customer demand, occurring one unit at a time at the retail site, to be price 1 Note that Lee (2002) provides examples of supply chains (products) categorized only on the basis of demand and supply uncertainties. We have incorporated the price-sensitivity categories into the framework.


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Figure 2

Values of Characteristic Trinities over the Life Cycle of a Supply Chain Demand uncertainty

Price sensitivity

Delivery time variability

High Moderate Low

Low Moderate High

High Moderate Low

Life-cycle stage Incubation Growth Maturity

sensitive, as well as stochastic. The retailer has many replenishment opportunities to procure the finished product from a distributor following a continuous review local base-stock policy. A key modelling feature of this paper is the randomness of the delivery time between the channel partners. As indicated before, this particular feature is of growing concern for many global chains. The retailer incurs cost for holding excess inventory, and any unsatisfied demand can be backordered at some extra cost. The distributor also follows a local base-stock policy for procurement decisions from a manufacturer, bears the cost for holding excess stock, and is charged a penalty for backordering retail orders. For ease of analysis, suppose that the manufacturer has infinite capacity, i.e., the replenishment time for the distributor is a positive constant, and that the costs for any in-transit inventory can be ignored. Building on this framework, we study two alternative management (decision-making) paradigms for the chain prevalent in real life. In a centralized chain, both the retailer and the distributor are managed by a single decision maker who has to decide on the system-optimal static values of the retail price and the local base-stock levels for the two parties. In a decentralized setting, the two parties are independent profit-maximizing entities operating under a wholesale-price-only contract. In particular, the distributor charges a per unit wholesale price to the retailer, and the decision-making responsibilities are decentralized: the distributor sets the wholesale price and the local base-stock level, while the retailer decides on the retail price and its local base stock (again all decisions are static). The above stylized model captures all the features of a supply chain that we are interested in. Both parties make pricing and inventory decisions, and all of the three characteristics influence those decisions: demand uncertainty, price sensitivity by way of price-sensitive stochastic demand, and supply uncertainty through delivery time variability. Note that for analytical purposes, we explicitly assume the optimal pricing and base-stock policies to be static, i.e., they are chosen at the beginning of the time horizon (infinite) and remain fixed throughout. However, for

developing managerial insights, the implicit assumption is that the life cycle of a product can be segmented into a number of “independent” stages, e.g., incubation, growth, and maturity. The pricing and inventory decisions remain static within each stage, but can change from stage to stage depending on the corresponding values of the business characteristics indicated in Figure 2. That is, in our framework, managers can estimate the relevant system parameter values for a particular stage, and determine the resulting optimal decisions. These decisions then remain fixed all through that stage. Only when the business environment shows a definite movement into the next stage, e.g., substantial changes in the levels of the characteristics, managers would adjust the parameter values. They would then re-optimize and choose the corresponding “new” optimal decisions, independent of the other stages. These new decisions also remain fixed throughout the next stage. Such a process is repeated over the entire life cycle of the product/chain (the life cycle might have more than three stages). Hence the discussion in this paper with respect to tailored decision making over the life cycle, which involves changing optimal policies, needs to be interpreted in this particular context. We believe our insights are consistent with the static assumption as long as each stage is of sufficiently long duration. Evidently, the stocking and pricing decisions of the two channel partners are interrelated. Another point worth noting is the profit loss caused by “double marginalization” in a decentralized management paradigm (Spengler 1950). Our focus is on understanding how the three supply chain characteristics interact with each other to influence the optimal managerial decisions, as well as the extent of profit loss. Specifically, we will address the following issues in this paper: (1) What should be the optimal price and inventory decisions for the retailer and the distributor under a centralized or a wholesale-price-based decentralized arrangement? How do the decisions of the channel partners affect each other, and how are they affected by the system characteristics in the two arrangements? (2) How do the profit performances of the channel partners compare with each other in a decentralized setup? What is the degree of profit loss for the whole chain because of decentralization? What types of business environments would benefit most from a central decision maker? (3) What alternate contracting mechanism would allow a decentralized supply chain to attain the optimal (or near-optimal) centralized system (CS) profit in its existing configuration, i.e., “coordinate” the chain? How is the contract choice affected by the business environment?


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The analytical models developed in §4 enable us to determine the unique optimal price and basestock decisions for both centralized and decentralized chains. By analyzing the absolute and relative optimal values in §§5.1 and 5.2, we gain critical insights into how the three characteristics and the management paradigm act in unison to shape such decisions. For example, the optimal retail prices in both centralized and decentralized chains are almost insensitive to demand uncertainty, but decrease (increase) with price sensitivity (delivery time variability). On the other hand, the optimal base stocks are normally higher in centralized chains, although highly price sensitive and uncertain environments might reverse the result. Section 5.3 demonstrates the parallels and contrasts between the profit performance of the channel partners and the management paradigms. It is then possible to identify the most favorable conditions for either party in a decentralized setup. Moreover, we show that the argument for a central decision maker is most compelling during the growth phase of a product, while decentralized chains might perform reasonably well in mature phases. We also propose a combination of contracts from the literature that can improve the profit performance for decentralized chains. Our analysis suggests that many of the existing contracts in the literature, which are developed based on the assumption of reliable supply systems, need to be revisited because contract performances might be quite adversely affected by uncertainties in delivery times. We conclude this paper (§6) by discussing its managerial significance, and suggesting potential future research directions. One of our primary contributions is the clear recognition that ignoring the variability of delivery time might have strategic implications from both marketing and operations viewpoints. But before going into the details, we present a summary of the relevant literature in the following section.

3.

Related Literature

Most of the related research comes from the literature on coordination and multiechelon inventory models. These can be categorized into two main streams: (1) those that deal with stocking and pricing issues in a single-period, single-echelon framework without any delivery time considerations and (2) those that analyze multiechelon systems with perpetual demand and many replenishment opportunities, but have no pricing decisions involved. Our model essentially bridges the gap between the two disparate streams. However, rather than providing a comprehensive review (those interested should refer to Yano and Gilbert 2003), here we aim to position this paper in relation to the existing literature.

The first stream of research is based on the idea of a distributor selling to a retailer facing a standard newsvendor problem. Lariviere and Porteus (2001) investigate the performance of such a decentralized system (DS) under a wholesale-price-only contract. Coordination cannot be achieved with such a contract, and the profit inefficiency as well as the profit division depends largely on the demand rate variability. Coordination is realized only with parameters like buybacks or revenue sharing (Cachon 2003). The growing popularity of integrated operationsmarketing models led researchers to focus more on the price-dependent newsvendor problem, i.e., the retail demand is not only stochastic, but also price sensitive. Petruzzi and Dada (1999) analyze this problem in a centralized setting to show the effect of the nature of the stochastic demand function (additive or multiplicative) on the optimal price and order quantity decisions. Subsequently, researchers like Bernstein and Federgruen (2005), Emmons and Gilbert (1998), Wang et al. (2004), and Wang (2004) study the issue of coordination and/or allocation of profits for the above problem in a decentralized setup. However, in all these models, the distributor does not make any stocking decision, and the subject of delivery time is not relevant. On the contrary, for Li and Atkins (2002), price is decided by the retailer and quantity by the distributor. The authors propose a coordinating contract for such a scenario, although their single-period model does not involve retail base-stock decision or delivery time issues. The second stream of literature considers models with perpetual price-insensitive stochastic demand and many replenishment opportunities. Both channel partners follow base-stock policies for inventory control and it is possible to have backorders at both sites (delivery time is usually constant). In this classical cost-based, two-echelon inventory problem (Clark and Scarf 1960, Chen 1998, Axsäter 2000), both parties are only concerned about their holding and backordering costs. Coordination, in such cases, implies attaining the minimum cost associated with a CS. Cachon and Zipkin (1999) show that the excess cost because of decentralized decision making depends on the parameter values, and a linear transfer payment based on the distributor’s backorders can coordinate the chain. Alternative coordination policies for related problems have been proposed by Porteus (2000), Chen (1999), and Lee and Whang (1999). In another related research stream, Caldentey and Wein (2003) also analyze coordination issues for a two-echelon supply chain, but from a queueing theory perspective. Although some parts of the basic model seem to be well investigated, there are two distinctive features of our framework. There are only a handful of papers in the literature (notably Eliashberg and Steinberg 1987



and Zhao and Wang 2002, which are based on multiperiod deterministic frameworks) that allow both channel partners to make inventory and pricing decisions, as we do. More importantly, we are able to incorporate delivery time variability in a decentralized price-sensitive setting. Each of these two features has significant effects on managerial decisions and reveals interesting insights. In effect, integration of retail/wholesale pricing, supply/demand uncertainty, and multiechelon inventory issues into a single framework facilitates analysis of a wide variety of decision-making contexts.

4.

Analysis of the Optimal Policies for Centralized and Decentralized Supply Chains

The goal of this section is to establish the optimal price and inventory decisions for centralized and decentralized supply chain paradigms. We first focus on understanding the specific features of the endcustomer demand process and the lead-time demand (LTD) distributions for the two parties that drive those decisions. 4.1.

End-Customer and Lead-Time Demand Processes for the Chain Let X1 = yp + be the random end-customer demand arriving at the retailer within one unit of time, where yp is a downward sloping, concave, deterministic function of the unit retail price p, and is a continuous random variable.2 Both the additive demand form and the assumptions about yp are common in the literature (Petruzzi and Dada 1999, Federgruen and Heching 1999). Examples of yp include A − Bpk , A > 0, B > 0, k ≥ 1, ≤ 1 (k = 1 and = 1 results in linear demand), A − B p , A > 0, B > 1 and ln A−Bp , > 0, B > 0. For specificity, we further assume that follows a normal distribution with mean d >0 and variance d2 , i.e., N d d2 . To eliminate unrealistic solutions, we constrain p ∈ c P u , where c is the replenishment cost per unit for the distributor and P u is an upper bound on the retail price such that yP u ≥ 0 (“null price”). Thus the demand rate X1 ∼ N p d2 , where p = yp + d . Obviously, p is decreasing concave. We also suppose that the demand process exhibits stationary and independent increment properties. As the retailer follows a local base-stock policy for inventory control and the end-customer demand is one unit at a time, the demand process at the distributor is exactly the same as that of the retailer. Based on the assumption that the delivery time between 2

A comprehensive list of notations is provided in the appendix.

1877 the manufacturer and the distributor is of a constant integer duration (say, L0 > 0), the random variable representing the LTD for the distributor (LT D0 ) is then N L0 L20 , where L0 = pL0 and L20 = d2 L0 (Zipkin 2000) (throughout this paper, decisions/ parameters with subscript 0 refer to the distributor and the subscript-free ones are for the retailer). We assume the distribution system to be exogenous and sequential. This implies that there is no crossing of retailer’s orders and the delivery times between the parties are independent of the number and sizes of outstanding retail orders (Zipkin 2000). Nevertheless, the LTD distribution for the retailer is quite involved. First, the delivery time between the distributor and the retailer is not constant, rather an integral positive random number L with a mean value of EL and variance VarL. Hence the precise mean L = pEL and variance L2 = p2 VarL + d2 EL of the random variable representing the LTD for this delivery time (nominal lead-time demand (NLTD)) is known, but not the exact distribution (Ross 1989, pp. 22–23; Axsäter 2000, p. 77). Secondly, the LTD for the retailer is also affected by the base-stock level at the distributor because there is a delay for an order if the on-hand inventory at the distributor is zero. In what follows, we characterize the distribution of the random variable corresponding to the number of these “delayed demands” () at the distributor location. Characterization of . It is obvious that = LT D0 − S + , where S is the local base-stock level at the distributor. Note that the events ≤ x and LT D0 ≤ S + x are equivalent for any given x ≥ 0. Thus Pr ≤ x = S + x − L0 /L0 , where and " are the distribution and the density functions for the standard normal, respectively. Let #0 = S − L0 /L0 denote the safety factor for the distributor, and the mean and the variance of are represented by m and v, respectively. Then, we can show that3 m = L0 "#0 − #0 1 − #0 and v = L20 #20 + 1 1 − #0 − #0 "#0 − m#0 2 & Lemma 1. The mean m and the variance v of are decreasing in the safety factor #0 ∈ − + . Consequently, m and v can be reduced by either increasing S and/or p, or decreasing L0 and/or d . It is intuitive that the magnitude and the randomness of the number of delayed demands can be alleviated by increasing the base stock at the distributor facility and/or by reducing the delivery time L0 or the 3 Proofs for all lemmas and theorems are provided in an electronic companion to this paper, available at the Management Science website (http://mansci.pubs.informs.org/ecompanion.html).


1878 demand uncertainty d (all result in high #0 ). Interestingly, higher p also results in higher #0 by reducing demand, and thus offers an alternate mechanism to deal with the problem. Based on means and variances of and NLTD, the distribution for the overall LTD of the retailer can be represented approximately by a normal distribution, N 2 , where #0 p = pEL + m#0 and (1) #0 p = p2 VarL + d2 EL + v#0 & It is worthwhile to discuss the rationale behind the above approximation. Because and NLTD are independent for any given realization of L (Graves 1985), we can show that the above mean and variance values for LTD are indeed exact. However, the distribution for LTD is quite complicated and not necessarily normal. Even the distribution for NLTD can be nonnormal, and is available only for specific cases. For analytical tractability, the most common approach in the inventory literature is to use a second-order approximation, whereby the LTD is assumed to be normally distributed with the precise mean and variance given by and 2 , respectively (Hadley and Whitin 1963; Silver et al. 1998; Zipkin 2000, Chapter 7; Axsäter 2000, Chapters 3 and 5).4 Tyworth and O’Neill (1997) have shown this approximation technique to be quite robust regarding inventory decisions in real-life settings (interested readers can also refer to Lau and Lau 2003 and references therein for a detailed discussion about the accuracy of the approximation technique). In fact, such approximations seem to be most frequently used in stochastic lead-time scenarios, as in this paper (Axsäter 2000, p. 85). Our research has an added layer of complexity in the form of pricing compared to the inventory models. Moreover, we aim to develop managerial insights rather than accurate decision support systems. Under these conditions, the above approximation approach seems to be reasonable for this paper. It is evident from (1) that supply chain decisions might be considerably affected by the three system characteristics. Note that in our additive demand framework, if the uncertainty of the delivery process is ignored, becomes independent of the pricing policy (for a given #0 ). On the other hand, a positive VarL provides the opportunity to influence the uncertainty perceived by the retailer through pricing. This expression will form the basis of the following analysis about determining the profit-maximizing decision variables for centralized and decentralized chains. 4 Note that in the inventory literature, this approximation is even used only for the NLTD component, which is assumed to be normally distributed with mean L and variance L2 .


4.2.

Optimal Pricing and Stocking Policies for the Centralized Chain In a centralized chain, both the retailer and the distributor are controlled by the same manager. It is well known that such a chain, if optimally managed, can generate higher profits than any decentralized structure (under similar operating conditions). Consequently, we will use the optimal centralized system profit as a benchmark for evaluating the profit performance of various decentralized paradigms. The manager of a centralized supply chain needs to decide on the optimal values for three decision variables: (1) the retail price p, (2) the retail local base stock R, and (3) the distributor local base stock S. However, we will subsequently use the transformed decision variables #0 =S − L0 /L0 in place of S, and # =R − /, the safety factor for the retailer, for R in our analysis. Let h >0 and b >0 be the constant holding cost rate and the backordering cost rate at the retail location, respectively, while h0 >0 is the constant holding cost rate for the distributor. Because the backordering assumption implies that the retailer eventually receives the revenue p, we can define b as the loss-ofgoodwill cost or the cost of handling complaints from backordered customers. Therefore, constant b indeed makes sense. Moreover, such assumptions are established in the literature (e.g., Federgruen and Heching 1999). The implications of alternative definitions for h and b (dependent on the replenishment cost and the retail price, respectively) on the modelling framework are discussed in §6. The objective of the centralized chain is to maximize the following profit function: *#0 p # = p − cp + #0 p · b# − h + b# # − h + b"# − L0 h0 #0 #0 + "#0 &

(2)

Clearly, the optimal #, #C , for any given p and #0 is a constant,5 i.e., #C = −1 b/b + h. Substituting this #C in *#0 p #, we get *#0 p = p − cp − K c1 p2 + c2 + v#0 − L0 h0 #0 #0 + "#0 where for expositional clarity c1 = VarL, c2 = d2 EL and K = h + b"#C . Based on the assumption that p is decreasing concave, we can then ascertain that the profit function for the CS is well behaved. 5 In this paper, the superscript C is used to denote the optimal values for the CS, and later on D is used to represent the optimal values for the DS.


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Theorem 1. For any given retail price p, the profit function *#0 p is unimodal for #0 ∈ − + . Hence there exists a unique optimal distributor safety factor #0 p for the system. Furthermore, the net profit function *#0 p p itself is unimodal for p ∈ c P u . We can also show that #0 p is increasing in p. That is, the higher the retail price set by the chain, the higher safety factor it can afford in terms of distributor stocking policy. A more important implication of Theorem 1 is that the overall optimal decisions can be uniquely determined by the first-order conditions. Corollary 1. The profit-maximizing *C managerial policies for the centralized chain are as follows: (1) Set the optimal retail price pC and the optimal distributor safety factor #C0 from the unique simultaneous solutions of -*#0 p/-p = 0 and -*#0 p/-#0 = 0. (2) Set the optimal local retail base stock at RC = #C0 pC #C + pC #C0 , where #C = −1 b/b + h and the optimal local distributor base stock at S C = #C0 L0 + L0 pC . As indicated before, a large family of demand functions is of a decreasing concave nature. It is almost a standard assumption in related economics/ marketing/operations literature. The uniqueness of the optimal values for the centralized chain is then quite a general result. However, large-scale outsourcing means that most current systems have independent managers for the channel partners. Investigation of such DSs is considerably more intricate. 4.3.

Optimal Pricing and Stocking Policies for the Decentralized Chain The basis for analysis of the DS is a wholesale-priceonly contract, and a sequential (Stackelberg) framework with the distributor as the leader of the chain. We initially assume the distributor’s wholesale price and safety factor as given and determine the profitmaximizing end-customer price and the safety factor for the retailer. Provided all relevant parameters are common knowledge, the distributor can decide on the optimal wholesale price and safety factor by substituting the optimal retail decisions into its profit function. Inferring the overall optimal prices and base-stock decisions is then relatively straightforward. 4.3.1. Optimal Policies for the Retailer. For a given wholesale price w ≥c and safety factor #0 ∈ − + of the distributor, the decision variables for the retailer are the retail price pw #0 and the safety factor #w #0 . We assume that the retailer charges some industry standard explicit backordering cost at

the rate of b0 (normally less than b) per unit of unsatisfied retail demand per unit time because of stockouts at the distributor facility.6 This amount accrues as revenue for the retailer. The expected profit function to be maximized for the retailer per unit time can be written as (after some simplification) /# p = p − wp + #0 p · b# − h + b# # − h + b"# + L0 b0 "#0 − #0 1 − #0

(3)

where L0 b0 "#0 − #0 1 − #0 is the backordering penalty from the distributor per unit time. For a given p, the optimal safety factor for the retailer will be the same constant as the CS. That is, #D = −1 b/b + h = #C . Substituting #D into /# p results in /p = p − wp − K c1 p2 + c2 + v#0 + L0 b0 "#0 − #0 1 − #0 &

(4)

Analysis of the above equation defines the optimal decisions for the retailer. Theorem 2. For any given wholesale price w and safety factor #0 of the distributor, there is a unique retail price pw #0 such that the expected profit for the retailer /p is maximized. The optimal value of the retail price pw #0 can be deduced from the unique solution of d/p p p =0 ⇒ p−w+ − Kc1 = 0& dp p

(5)

The optimal local retail base stock is given by Rw #0 = #D #0 pw #0 + #0 pw #0 . We can then analyze the effects of the distributor’s actions on the optimal retail decisions. Corollary 2. The optimal retail price pw #0 is increasing in the wholesale price w and the distributor safety factor #0 , while the optimal local retail base stock Rw #0 is decreasing in w and #0 . The effects of wholesale price on the optimal retail price and the optimal base-stock level are expected. However, note how the distributor’s base stock affects the retail behavior. When the distributor increases its local base stock (hence higher #0 ), the number of delayed demands decrease. The resulting more reliable supply enables the retailer to reduce its base stock R. That is, there is less inventory for the retailer to sell and it has less risk of leftovers. It can then 6 This b0 might be the imputed backordering cost rate from some industry standard service level (Boyaci and Gallego 2001). In §5.3, we investigate the scenario with b0 as a contract parameter.


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increase the retail price p to maximize its profit. This increase further reduces the LTD and, hence, the base stock for the retailer. With knowledge about the retailer’s optimal strategy for a given action by the distributor, we can shift our attention to characterizing the optimal decisions for the distributor. 4.3.2. Optimal Policies for the Distributor. The analysis of the distributor’s profit function is quite complicated. The distributor’s revenue function is given by w − cp. However, analytically it is more convenient to work with p rather than w. Based on (5), we can express w explicitly in terms of p, for a given #0 , as (note that p ≥ w) wp #0 = p +

p p − Kc1 & p #0 p

(6)

Hence, the distributor’s profit can be transformed as a function of p and #0 . /0 #0 p p p = p+ − Kc1 − c p p #0 p + L0 b0 #0 − h0 + b0 #0 #0 − h0 + b0 "#0 & (7) It is then possible to generate structural insights regarding the behavior of the distributor’s profit. Before that, we present some bounds regarding the distributor’s optimal decisions—pD and #D0 . Lemma 2. Let #¯ 0 = −1 b0 /b0 + h0 . If EL ≥ L0 , then #D0 ≤ #¯ 0 and pD > pC . 2

Theorem 3. If EL ≥ L0 , VarLpC − 2d2 EL + v#¯ 0 ≤ 0, and + ≥ 0 ∀ p ∈ pC P u , for any given #0 ∈ − #¯ 0 , then there exists a unique p#0 ∈ pC P u such that /0 #0 p is maximized. Additionally, if b0 /h0 ≤ 105, then /0 #0 p#0 itself is unimodal for #0 ∈ − #¯ 0 . Note that when VarL = 0, the optimal safety factor for the distributor #D0 is a constant =#¯ 0 . This basically implies that the retailer’s does not depend on the decision variables (see (1)), whereas w =p + p/ p depends only on p. As a result, /0 turns out to be solely a function of p. We can exploit this particular property to develop an even stronger result regarding the behavior of the profit function for the distributor. Corollary 3. If VarL = 0, the optimal safety factor for the distributor is a constant, #D0 = −1 b0 /b0 + h0 = #¯ 0 , and then if + ≥ 0 ∀ p ∈ pC P u , /0 #D0 p is concave in retail price p ∈ pC P u .

Three remarks are in order here. First, the con2 ditions (i) EL ≥ L0 , (ii) VarLpC − 2d2 EL + v#¯ 0 ≤ 0, and (iii) b0 /h0 ≤ 105 only involve system parameters, not decision variables (pC is a function of the system parameters). The first condition EL ≥ L0 is self-explanatory. The second one requires that the delivery time between the parties should not be too random and/or the demand rate should have fairly high levels of uncertainty. The last condition means that the penalty cost rate charged to the distributor should not be too large compared to its holding cost rate (or equivalently, fill rate expectation from the distributor should be less than approximately 99% (Boyaci and Gallego 2001)). Second, although + ≥ 0 involves the retail price, it is primarily related to the structure of the demand function.7 Specifically, this condition is more general than linear demand, i.e., yp = A − Bp, a common assumption in the related literature (Eliashberg and Steinberg 1987, Petruzzi and Dada 1999, Li and Atkins 2002). For example, if yp = A − Bpk , k ≥ 1, then the condition is satisfied for 1 ≤ k ≤ 2 by any p ∈ 0 P u ; for k > 2, p ≥ A+d /2B1/k is sufficient. If yp = A−B p , B > 1, then the condition is valid for p ≥ lnA + d /2/ ln B. Our numerical experiments suggest that + ≥ 0 normally holds true in the range of our interest. Third, note that all of the conditions in the theorem are sufficient ones. Based on our numerical study, the profit function is usually unimodal even when they are not satisfied. The above theorem facilitates determination of the optimal prices and local basestock levels for both parties in a decentralized chain. Corollary 4. For a decentralized supply chain that satisfies the conditions of Theorem 3: (1) The optimal retail price to be charged pD and the optimal distributor safety factor #D0 can be uniquely obtained from simultaneously solving -/0 #0 p/-p = 0 and -/0 #0 p/-#0 = 0. The retail manager should then maintain a local base stock RD = #D #D0 pD + #D0 pD , where #D = −1 b/b + h. (2) The distributor should optimally charge a wholesale price w D = wpD #D0 (obtained from (6)) and maintain a local base stock of S D = #D0 L0 + L0 pD . The managerial relevance of the above results lies in the guidance it provides regarding optimal decision making. However, the analysis in this paper has modelling contribution as well. In particular, the simultaneous use of two transformations, stocking variables to safety factors, and wholesale price to retail price, is rather effective in obtaining analytical results. To the best of our knowledge, to date these transformations have only been used separately in the literature (Petruzzi and Dada 1999, Lariviere and Porteus 7 For Corollary 3, we only require 2 + ≥ 0. To be consistent, we use + ≥ 0.


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2001, respectively). The effectiveness of this approach is most evident when VarL = 0. In such settings, our method results in decoupling of the safety factor and pricing decisions for additive demand (Corollary 3), an issue not captured in the literature before. We have already seen how the decisions of the two parties affect each other. However, from a managerial perspective, it is important to provide more specific insights regarding optimal tailored decision making under varying business environments. The next section accomplishes this objective through a combination of analytical and numerical approaches.

5.

Effects of the Characteristics and the Management Paradigms on the Decisions and Performance of Supply Chains

In this section, we first study how the three business characteristics affect the optimal prices and local basestock levels for centralized and decentralized supply chains. Subsequently, we compare the optimal profit performance for the two management paradigms and the channel partners, and how their relative values are also affected by the business conditions. In the last part of this section, we propose a simple contracting scheme that can improve the profit level of the decentralized setup. To gain clear insights for the rest of this paper, the focus is on a particular form of the expected demand rate p = A − Bp + d , p ∈ c A/B , where A represents the potential market size and B denotes the price sensitivity of the customers. This standard assumption satisfies the demand condition for the profit function to be unimodal in §4. Henceforth the set of values B VarL d2 will characterize a particular supply chain—based either on the business environment (Figure 1) or on the life-cycle stage (Figure 2). But even with a specific p, complicated first-order conditions sometimes render analytical comparative statics intractable. In those instances, we had to rely on a numerical study to generate relevant insights. Let us first explain the numerical experiment setting. Numerical Experiment Setting. The basic data set for the numerical experiments is as follows: A = 100, B = 10, c = 4, h = 0&03, b = 2, h0 = 0&01, b0 = 0&5, d = 15, d2 = 1000, EL = 20, VarL = 20, and L0 = 15. This data set satisfies all the unimodality conditions in Theorem 3. Varying the values of B, VarL, and d2 , while keeping other parameters fixed at the base level, we have been able to portray a whole spectrum of supply chains. The interesting effects of changes in the three system characteristics on the optimal policies and profits are shown in the figures (captions are self-explanatory). The horizontal axis

(h axis) in all the figures refers to the relative parameter values, not absolute ones. That is, we have represented the range of the characteristics (5.8–17.5 for B, 1.0–40.0 for VarL, and 80–1,640 for d2 , unless otherwise stated) by 1–40 on the h axis, with 1 being the lowest value and 40 the highest value (equal stepsizes). For example, in Figure 3(a), 1 refers to B = 5&8 and 40 to B = 17&5, with the step size being 0.3. In certain cases, we have changed two characteristics simultaneously (over the same ranges as when they are varied individually) to better understand their joint effects. For example, 1 in Figure 3(d) stands for B = 5&8 and d2 = 80, 2 for B = 6&1 and d2 = 120, and so on until 40, which represents B = 17&5 and d2 = 1640. Note that most of the data set, even when the parameters are changed from the base level, comply with the conditions in Theorem 3. 5.1. Effects on the Optimal Prices In this section, we examine the impact of changes in B VarL d2 on the optimal prices for both management paradigms. The centralized supply chain only needs to decide on the optimal retail price, while in a DS, the distributor sets the wholesale price and the retailer determines the retail price. Based on our analytical and numerical study, we claim that: Claim 1. Any increase in B, VarL, or d2 affects the optimal prices as follows:8

Retail price (pC or pD ) Wholesale price w D

B

VarL

d2

Decrease

Increase

Decrease

Decrease

Might increase or decrease Increase

While the optimal retail prices in both systems behave in a similar fashion, their values might be quite different. One would presume that because a decentralized retailer pays a unit purchase cost of w compared to c in the centralized setup, its retail price would be higher (“double marginalization”). As shown in Lemma 2, this indeed holds true in our setting and can be analytically verified. Specifically, if EL ≥ L0 , then pD > pC (note that when VarL = 0 the condition is not required). Based on our numerical experiments, we also observe that the optimal wholesale price w D is lower than the optimal centralized retail price pC , i.e., pD > pC > w D (see Figure 3). For decentralized chains with low B (e.g., fashion apparel), the distributor’s margin (w D − c) is higher than that of the retailer pD − w D , 8 The behavior of the optimal retail prices under the conditions of Theorem 3 is analytically proved in the online companion.


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Figure 3

Effects of the Characteristics on the Optimal Prices (b) Effects of σd2

(a) Effects of B 16

(c) Effects of Var(L )

9.6

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but the retailer earns a higher margin for high B scenarios (e.g., basic apparel). Finally, while pD − pC remains almost constant with changes in d2 (Figure 3(b)), this difference tends to decrease as the chain faces more price sensitive or uncertain supply conditions (Figures 3(a) and 3(c)). As expected, the major

10

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part of the difference is accounted for by the price sensitivity. The optimal retail prices are evidently most expensive for chains that serve low price-sensitive customers, but encounter considerable delivery variability, and vice versa. But note that the optimal retail

Effects of the Characteristics (Individually) on the Optimal Stocks (a) Effects of B on base stocks

(b) Effects of Var(L) on base stocks

1,450

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Figure 4

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Figure 5

Effects of the Characteristics (Simultaneously) on Optimal Stocks (a) Simultaneous effects of B and Var(L) on base stocks

1,400

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( b) Simultaneous effects of B and Var(L) on total stocks

5.2. Effects on the Optimal Local Base Stocks As Corollaries 1 and 4 indicate, the optimal local basestock levels depend on the optimal retail prices and optimal safety factors. When VarL = 0 and yp = A − Bp, we can obtain closed-form expressions for all the decision variables (e.g., #D = −1 b/b + h, #D0 = −1 b0 /b0 + h0 , and pD = 3A + Bc/4B). It is then possible to prove the overall effects of the characteristics on RC , RD , S C , and S D (as specified in Claim 2). However, a broad generalization is analytically intractable. Nevertheless, our numerical studies illustrate that: Claim 2. Any increase in the characteristic value affects the optimal stocks as follows.

2,000

B

1,800

C

R +S

1,400 D

1,000

R +S

0

10

d2

C

1,600

1,200

VarL

20

D

30

40

prices are virtually insensitive to the management paradigm (i.e., pD ≈ pC ) for highly price-sensitive customers and high delivery time variabilities (Figure 3(e)). Furthermore, the uncertainty related to delivery time seems to have a greater impact on retail prices than demand rate variability (Figures 3(b) and 3(c)). Hence, managers need to be specific about the underlying cause of system uncertainty—demand or supply—while deciding on the optimal retail price. The reason for this is intricately related to the effects of the characteristics on the optimal base stocks, which we discuss in §5.2. In summary, the above results generate implications in terms of tailored pricing strategy. For example, retail prices need to be highest for products in the incubation phase of the life cycle, and lowest for mature products like low-end automobiles or detergents (see Figure 1). Additionally, retailers like Wal-Mart that procure basic apparel from developing countries (high B, high VarL), would not only have higher margins compared to their distributors, but also do not need much adjustment of their retail prices irrespective of whether the chain is centralized or decentralized. On the contrary, a fashion boutique buying from nearby distributors (low B, low VarL) would be squeezed for margins and has to charge quite different retail prices, depending on whether it owns the distributor or not.

Retail stock (RD or RC ) Distributor stock (S D or S C ) Total stock (RD + S D or RC + S C )

Decrease

Increase

Increase

Decrease

Decrease

Increase

Decrease

Might increase or decrease

Increase

Since higher price sensitivities result in lower demands, it is intuitive that the optimal local base stocks will decrease with B (Figure 4(a)). However, note the following: (1) Any increase in VarL reduces the optimal wholesale price w D , while increasing the optimal retail price, pC or pD (Figure 3(c)). The system thereby ensures a lower demand rate without affecting p − w considerably. The lower demand rate provides two benefits. First, since #0 p contains the term p2 VarL (refer to (1)), it restrains the increasing effect of VarL on the retail base stocks, RC or RD . Second, lower demand rate also results in lower L0 =pL0 , allowing the distributor to reduce its base stock, S C or S D (Figure 4(b)). Note that the reduction in w D makes sense for the distributor, because any increase in w D would induce even higher levels of pD . The resulting lower demand rate would then adversely affect the distributor’s revenue. (2) On the other hand, the retail price (and so the demand rate) is not affected much by an increase in demand uncertainty d2 . As a result, any increase in d2 increases both and L0 . This implies higher local base stocks for both parties (see Claim 2). The distributor must then raise its optimal wholesale price w D with d2 to counterbalance the increase in inventory costs (Figure 3(b)). Note that the two forms of uncertainty might have dissimilar effects on w D and base stocks.

1884 (3) The higher retail price, and resulting lower demand rate, implies that the retail base stock does not increase substantially (or might even decrease) with VarL, while the distributor base stock decreases with VarL. Under certain scenarios, the total system stock might then indeed be nonmonotone/decreasing in VarL (although mostly it is increasing). For the parameter set in our numerical experiments, the retail base stocks and the total stocks are both increasing (Figures 4(b) and 4(c)). But if the holding cost rates for both channel partners are high, then the retail base stocks (especially RD ) and the total system stocks might be decreasing or nonmonotone. In Figure 4(d), we present such an example for total system stocks, where the only change from the original data set is that h = 0&1 and h0 = 0&033 (conditions of Theorem 3 are still satisfied). As we observed for the pricing policy, the values for the optimal base stocks are also different, depending on whether a supply chain is centralized or decentralized. Specifically: Claim 3. S D < S C and RD < RC , except for chains with highly price-sensitive customers (high B) in uncertain environments (high d2 , VarL), in which case it is possible to have S D > S C and S D + RD > S C + RC (Figures 4(a), 5(a), and 5(b)).9 Usually, the stocking quantity for a CS is more than that of a DS (e.g., in price-dependent newsvendor or cost-based multiechelon settings). Hence, at first glance, the above result seems to be counterintuitive. But the underlying reason for this behavior can be explained as follows. When facing very price-sensitive customers, a centralized decision maker can be proactive in reducing both RC and S C . The reduction would be more for the former because price sensitivity primarily affects the retail stock. Even for a decentralized chain, the decrease in RD is more pronounced than S D . However, in the DS, although both parties recognize the need for reducing local base stocks, the independent decisionmaking framework limits any such action for fear of increased backorders. Consequently, the optimal retail and distributor base stocks come closer as the system faces more price-sensitive customers. For highly price-sensitive customers, the distributor and the total 9 In Figure 5(b), the horizontal axis denotes B = 7–18&7 (equal stepsizes), and not 5.8–17.5, to exhibit the crossing property. We have not encountered RD > RC , although the possibility cannot be discarded. However, note that the retailer directly faces the end customers and is exposed to a higher degree of possible profit loss because of backordering, compared to the distributor. Hence, a centralized system would be more cautious about reducing the base stock at the retail location rather than the base stock at the distributor, even for high B. Intuitively, RD > RC would then be rarer than SD > SC .



system stock might be lower in a centralized setup than a decentralized setup, especially for uncertain environments. On the whole, the behavior of the optimal base stocks generates insights as to how managers should adapt their stocking policy with changes in the business environment and decision-making responsibilities. For both CSs and DSs, the system stock is the lowest for mature products like low-end automobiles (high B, low VarL, d2 ), and (generally) highest for products like fashion apparel with uncertain delivery times (low B, medium-high VarL, high d2 ). On the other hand, firms selling products with high B, VarL, d2 (e.g., offshore distributors of cheap DVD players) should be aware that if they decide to sell the products themselves in the North American market, they might get the benefit of holding lower stocks in the system than when they sell through decentralized retailers. The above discussion demonstrates how managerial decisions are influenced by the interaction among the three supply chain characteristics. This interaction is most evident in the uncertainty associated with the LTD for the retailer #0 p = p2 VarL+d2 EL+ v#0 1/2 . For high-volume products, i.e., large p, the delivery uncertainty component p2 VarL is the driving force of #0 p. Note that in cost-based inventory models with price-independent stochastic demand rate (mean ), the variability of the LTD for the retailer can be represented by #0 = 2 VarL + d2 EL + v#0 1/2 . Since 2 VarL is a constant, the behavior of #0 does not change irrespective of whether VarL is zero or positive, although the values do. Any increase in VarL is qualitatively equivalent to increasing d2 . That is, the behavior of the results in such cost-based models are consistent irrespective of whether VarL > 0 or VarL = 0 (Song and Zipkin 1996, Zipkin 2000). On the contrary, when pricing issues are taken into consideration, assuming VarL = 0 can result in dual distortion of the modelling framework. First, it again considerably reduces the magnitude of the uncertainty. More importantly, it alters the behavior of the system compared to when VarL > 0. In an additive demand framework, VarL = 0 assumption decouples the pricing and safety factor decisions limiting the flexibility of managers to mitigate the effects of uncertainty through pricing. Besides, positive VarL implies that the significance of price sensitivity is not solely in terms of its effect on the revenue function. Note that the intensity of the effects of pricing decisions on uncertainty levels is also driven by the price sensitivity of the target customer niche. This recognition of the crucial role that delivery time uncertainty plays in any pricesensitive multiechelon framework is one of the primary modelling insights of this paper.


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Figure 6

Effects of the Characteristics on the Optimal Absolute and Relative Profits

(a) Effects on the relative optimal profits (ΠD/ΠC)

0.79

B

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0.82 0.78

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While comparing the optimal policy values is interesting, a more exact comparison between the management paradigms should be based on a profit viewpoint (Lariviere and Porteus 2001, Wang 2004). We present this in the following section. 5.3. Effects on the Optimal Profit Performance The optimal profit of a centralized chain (*C ) is obviously more than that of a decentralized one (*D = /0D + / D , where /0D and / D are the optimal profits for the distributor and the retailer, respectively, in a decentralized setup) (Figure 6(a)). The primary interest here is in understanding the effects of varying business conditions on (1) the relative profit penalty from decentralization, *D /*C and (2) the profit allocation among the channel partners in an optimal decentralized system, /0D /*D . Moreover, while it is obvious that the total profits of both centralized and decentralized chains will decrease with an increase in the characteristics values, the effects on the optimal profits of the individual channel partners are not so clear. We also study this particular issue in this section. Note from Figure 6 that: Claim 4. The ratio *D /*C or CS total stock

procurement from Asia makes sense. Similarly, Dell has been successful with decentralization by targeting high-end customers and reducing system uncertainties (Magretta 1998). Detailed analysis of the DS shows that although the distributor normally garners the higher share, either of the channel partners might be more profitable.11 For products like laptops (medium B, high d2 , medium VarL), the DS as a whole performs poorly compared to a CS, but the profit allocation is skewed toward the distributor. Firms like Wal-Mart (high B, low d2 , medium-high VarL), on the other hand, have relatively more efficient DS, and the profit allocation between the partners is almost equal or slightly larger for the retailer. As far as the absolute profits for the channel partners are concerned, the noteworthy effect is that of demand uncertainty. This can be explained as follows: from previous sections, we know that any increase in d2 does not significantly affect the retail price, and hence the demand rate, but increases the retail base stock. This results in a reduction of the retailer’s optimal profit. But for the distributor, both its revenue and stocking costs increase with d2 (the demand rate remains almost constant, while w D and S D increase). Depending on which element is stronger, the distributor’s optimal profit might be increasing/decreasing/ nonmonotone. Although the discussion in this section is based on a linear demand form, our numerical experiments have indicated that most of the results remain valid for more general nonlinear structures. In Figure 7, we present a summary of the significant effects of the supply chain characteristics and the management paradigms on the optimal decisions (CS and DS, /0D /*D < 0&5 is not shown. This can happen for high B, high VarL, and low d2 scenarios, especially when the holding and backordering costs for the channel partners are not too different.

respectively). All these effects can be derived from our analysis in §§5.1 and 5.2. When seen in conjunction with Figures 1 and 2 of §1, the table offers managers a snapshot aid for custom decision making depending on the business environments of their chains and/or the life-cycle stages. For example, the statement “low wholesale price” follows from Claim 1. It implies that among the eight possible supply chain scenarios in the table (based on low/high values of B, VarL, and d2 ), the optimal wholesale prices are likely to be relatively cheaper for products facing high price sensitivity, high delivery time variability, and low demand uncertainty. The opposite business environment (low B, VarL, high d2 ), on the other hand, should charge comparatively premium wholesale prices among all scenarios. Although we do not show them explicitly, the other six scenarios should result in optimal wholesale prices of somewhat intermediate values. We believe that for highly uncertain environments of new product introductions, the focus of a decentralized chain should be on uncertainty reduction strategies. This can be accomplished by information coordination techniques like collaborative forecasting (Lee 2002). Contractual coordination,12 on the other hand, might be advisable in the growth period of a product (when customers are somewhat price sensitive, but there is still sufficient uncertainty), but not so much for a mature supply chain. From Figure 6(a), it is evident that the profit loss from using price-only contracts in a DS, compared to a CS, might be quite severe under certain situations. However, it is plausible that some alternate contracting scheme might be able to improve the profit performance of the decentralized chain. Below, we propose a contract that is better suited for coordinating the DS.

11

12

Contracts that allow a firm to attain the profit of a CS in a decentralized setup (Cachon 2003).


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Role of b0 and Possible Contractual Coordination Schemes. We will utilize b0 (the penalty charged by the retailer to the distributor per unit backordered per unit time) as one of our contractual levers. However, until now, b0 has been treated as an exogenous parameter. It might be useful to discuss the impact of b0 on the optimal decisions and performance of the channel partners in a decentralized setup. These are presented in Figure 8. The horizontal axis in the figure represents b0 = 0&001–0&196 (equal stepsizes).13 Note that from the distributor’s perspective, it would be preferable to set a high wholesale price and see the retailer charge a low price to the end consumers (so that the demand rate remains high). From Corollary 2, we know that the retail price increases with #0 and w. On the other hand, our numerical study illustrates that a lower b0 allows the distributor to charge a higher wholesale price w D and maintain a lower base-stock level S D , i.e., lower #0 (Figures 8(a) and 8(b), respectively). Hence, as b0 decreases, the retail price pD faces two contrasting effects: (1) higher w D that tends to increase pD and (2) lower S D that tends to decrease pD . Depending on their relative intensities, the retail price might increase or decrease or be nonmonotone as b0 decreases (Figure 8(a) shows pD decreasing as b0 decreases; but we have also noted other scenarios in our study). Furthermore, our numerical experiments also suggest that when there is a reduction in b0 , the retailer should increase the local base stock RD (Figure 8(b)). This makes intuitive sense since the base stock maintained by the distributor, S D , decreases under those conditions. As expected, the distributor’s optimal profit decreases with b0 , while that for the retailer increases (Figure 8(c)). However, recall that b0 is a transfer payment between the parties. That is, unlike b, it is not a loss to the system; it is more akin to the wholesale price although the financial flow is in the opposite direction. Consequently, the total channel profit is nonmonotone in b0 —it increases for low values, reaches a maximum, and subsequently decreases. The value of b0 , which maximizes the total channel profit (“profit-maximizing” b0 ), depends on the system parameters. For our base parameter set, h = 0&03 is very low compared to b = 2, but not too large when compared to h0 = 0&01. So from the chain’s perspective, it is most desirable to maintain a high base stock RD at the retailer to reduce the backordering cost, without substantially increasing the holding cost. For low b0 , the distributor’s stock level S D will also 13

We use the base parameter set of page 16 for this figure, however, the b0 values presented are deliberately small to clearly illustrate the “interesting” properties. For higher values of b0 , all the trends in Figures 8(a)–(c) persist. For example, w D in Figure 8(a) and *D in Figure 8(c) continue to decrease, while S D in Figure 8(b) continues to increase.

Effects of b0 on the Optimal Prices, Base Stocks, and Profits

Figure 8

(a) Effects on the optimal prices wD

pD 9.702

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40

be low. The resulting unreliable supply system will then force the retailer to have a high RD . By contrast, if b0 is high, then S D will be high, and the retailer can afford to maintain a low RD . Hence the low profitmaximizing b0 for Figure 8(c) (≈0&03, 1.5% of b) makes sense. Our extensive numerical study suggests that the profit-maximizing b0 for the chain increases with h and b (h seems to have a stronger effect than b) and decreases with h0 , but is relatively insensitive to B, VarL, or d2 . As h0 decreases, it is prudent from the chain’s viewpoint to maintain a higher level of stock at the distributor facility (S D ) to save on the holding costs. Therefore the profit-maximizing b0 increases with a decrease in h0 . Note that a higher level of S D allows the retailer to reduce RD . Thus the chain would also prefer to increase S D as h increases. It is then intuitive that the profit-maximizing b0 increases with h. As b increases, the retailer’s aim should be to increase RD to reduce the backordering costs. Since such an action


1888 will increase the retail holding cost, the chain ought to be interested in counterbalancing by reducing S D . However, note that the unit holding cost at the distributor level is lower than at the retailer. Thus it is also sensible for the chain as a whole to increase S D by increasing the profit-maximizing b0 . The higher supply reliability will then enable the chain to somewhat reduce the “expensive” stocking level RD at the retailer. From our numerical experiments, it seems that the profit-maximizing b0 increases with b (especially for large h − h0 ), so that the additional retail stock requirement for higher b is not too substantial. In fact, we are able to generate scenarios (e.g., when b is not too high compared to h, but h − h0 is large) with the profit-maximizing b0 in the range of 0&3b or higher. We should ideally treat b0 as a contractual parameter for coordination with its value being determined endogenously, in the spirit of Shang and Song (2003). We develop such a contract in the following analysis. We can show that a combination of two simple contracts: (1) revenue sharing (Cachon 2003, Bernstein and Federgruen 2005) and (2) backordering (distributor) penalty (Shang and Song 2003), which might be able to considerably reduce the profit loss for a wide variety of decentralized supply chains. Specifically, rather than a wholesale price-only contract (W contract), suppose that the decentralized channel operates under the following contract specifications (which we term the RWP contract): Contract. A fraction 0 ≤ 2 ≤ 1 of the retailer’s revenue pp is shared with the distributor, a wholesale price w = 1 − 2c is charged (by the distributor) per unit sold to the retailer, and a penalty cost of b0∗ (=h0 #C0 /1− #C0 , for more details see Shang and Song 2003) is charged (by the retailer) for each unit backordered at the distributor per unit time. Lemma 3. The RWP contract defined above can coordinate a decentralized chain, i.e., /0D + / D = *C , under the assumption VarL = 0. If the system variability originates only from demand uncertainty, then a RWP contract can indeed be an efficient one, i.e., /0D + / D = *C . However, note that the distributor would have to select a 2, which will maximize its own profit, subject to the constraints that the reservation profits of both parties are individually satisfied. Suppose that the reservation profits for the channel partners represent what they can earn under a W contract. Figures 9(a) and 9(b) illustrate the performance of a RWP contract when it satisfies the reservation profit constraints. For VarL > 0, a RWP contract does not result in coordination. However, for systems that are not too uncertain in terms of delivery


Figure 9

Profit Improvements from RWP Contract for Different Characteristics

(a) Effects of B on the optimal profit ratios (Π D/ΠC ) 1.00 RWP(0)

0.99

RWP (20)

0.98

RWP (40)

0.97

0.96 0

10

20

30

40

(b) Effects of σd2 on the optimal profit ratios D

C

(Π / Π ) 1.000 RWP (0)

0.999 RWP (20)

0.998

0.997 RWP (40)

0.996

0.995 0

10

20

30

40

Note. The numbers in parentheses refer to values of VarL.

time and/or not highly price sensitive, such a contract can substantially reduce the loss of decentralized supply chain profits compared to a W contract. It is well known that contracts that would be required for coordinating complex systems (e.g., with high VarL) have a hidden cost disadvantage in terms of administration (Cachon 2003). We believe that for decentralized chains dealing with low B, low VarL, low/high d2 products (e.g., laptops or healthy fast food that are procured locally), our proposed contract might be “enough” for improving the profit performance. The need for designing more sophisticated contracts arises primarily for products with relatively uncertain deliveries and medium-high price sensitivity. A prime example would be semiconductors used in North America, most of which are procured from Asia. Another noteworthy issue is that many of the existing contracts in the literature, which work well under the assumption of VarL = 0 (Cachon 2003), might be quite ineffective in the presence of supply uncertainty.



6.

Conclusions and Future Research Opportunities

This paper presents the study of a two-echelon, serial supply chain framework selling a procure-to-stock product. Our objective is to analytically investigate how managers in such systems can optimally tailor their decisions to align with their business environments. We explored the effects of three defining characteristics for the chain—price sensitivity, demand uncertainty, and delivery time variability—on the optimal pricing and stocking decisions, as well as on the profit performance. Using a combination of transformation techniques, we were able to establish the unique profit-maximizing policies for both centralized and wholesale price-based decentralized chains. One of our primary contributions is the particular modelling framework—multiechelon inventory model under price-sensitive stochastic demand and random delivery times—and the solution technique. A more in-depth investigation leads to important insights regarding the main drivers of supply chain decision making. First of all, the behavior of the optimal decisions with changes in the business characteristics is not significantly affected by whether a chain is centralized or decentralized. For example, chains facing unreliable deliveries (resp., price-sensitive customers) should increase (resp., decrease) their retail price, irrespective of their management paradigms. By contrast, optimal decision values are affected both by the characteristics levels and the decision-making structure of the chain (centralized or decentralized). Specifically, while the double marginalization effect for retail prices persists in our framework, quite counterintuitively, for highly price-sensitive customers and uncertain environments, a centralized chain should set its total stock level lower than a decentralized one. In general, optimal decisions for centralized and decentralized chains tend to be similar in uncertain, highly price-sensitive environments, but quite different for stable, low price-sensitive settings. With respect to the optimal profit, we show that the efficiency of the DS and the profit allocation among the channel partners again depends crucially on the characteristic trinity. The distributor is substantially more profitable for innovative products aimed at medium price-sensitive customers. On the other hand, the retailer might be better off for mature products. The profit loss as a whole for the decentralized chain is significant for moderately price-sensitive customers in uncertain environments, but comparatively lower if the customers are very price sensitive and the demand/delivery time is relatively steady. To improve the profit performance of DS, we propose a contract that involves revenue sharing between the parties, in lieu of the distributor paying a backordering penalty and charging a low wholesale price.

1889 Our analysis yields insights not only in terms of tailored decision making, but also has implications concerning decentralization/contracting strategy and how supply chain decisions should change over the life cycle. It is clear that as a product matures, the primary mode of competition shifts to price, and managers should start focusing more on cost reduction by trimming down inventory levels. We argue that information coordination techniques to reduce uncertainties should be the priority for any chain at the beginning of its life cycle. Such strategies cannot only be effective in terms of profit improvement, but can also help chains use relatively simple contracting schemes to coordinate. We would also not encourage decentralization during the growth phase of a product. Decentralization (possibly in the form of outsourcing) seems to be more justifiable once products reach their mature stages. While the effects of demand uncertainty and price sensitivity have been researched before, the key modelling feature of this paper is the uncertainty related to delivery times. We would like to reiterate that largescale offshore outsourcing has indeed brought this aspect of uncertainty to the forefront of the business world. From a modelling viewpoint, incorporation of delivery time randomness highlights the interaction between the optimal inventory and pricing decisions. From a managerial perspective, delivery time randomness provides the option of using pricing both as a revenue-generating mechanism and as a strategic tool for handling uncertainty. This latter role of pricing has been ignored in traditional operations management models, which have suggested information, time, and inventory as substitutable tools to cope with an uncertain landscape (Lovejoy and Whang 1995). We also show that managers need to be precise about the cause of uncertainty—demand or supply— because decisions might need to be tailored differently depending on the underlying reason. Most of the managerial insights appear to be quite robust, and, we believe, will hold true for more general modelling frameworks. Nevertheless, this paper provides a number of opportunities for future research extensions. It is tempting to further generalize the framework in terms of nonlinear demand and/or a broader class of randomness distributions. Preliminary analysis has shown that in both cases, most of our modelling and managerial insights may remain valid. Incorporation of exact definitions for holding and/or backordering costs would run into the problem of analytical tractability. For example, it might be argued that ideally h = h + iw, where h is the physical holding cost and i is the cost of capital, and b = b + 4p, where b is some base backordering cost and 4 is a percentage of

1890 the retail price offered as discount to backordered customers. In that case, some of the modelling insights might be lost. However, we numerically noted the following. Suppose we start with some arbitrary holding and backordering costs; say, h1 = h and b1 = b + 4c. We can then use our model to determine the optimal pD h1 b1 and w D h1 b1 , and update the inventory costs h2 = h + iw D h1 b1 and b2 = b + 4pD h1 b1 . These h2 and b2 can then be used to determine the new pD h2 b2 and w D h2 b2 , and further update the holding and backordering costs. Such iterative use of the model quickly converges to the overall optimal retail and wholesale prices (i.e., when pD and w D barely change based on updated h and b). While the values of these “exact” optimal decisions are different from those in our framework, their behavior and the managerial insights are quite similar. We believe a more interesting extension would be to relax the static assumption and have dynamic inventory control and pricing for the channel partners (see Chen et al. 2004, for example). Such a model can shed further light on our claims about tailored decision making over the life cycle of a chain, especially for very short life cycle products. Another worthwhile research direction is to study the implication of having a nonlinear, multiplicative demand function with lost sales (Petruzzi and Dada 1999). Such a framework would be quite involved in terms of analytical treatment, and would lose some of our modelling properties. For example, one has to forego the independence property of pricing and the optimal distributor safety factor even if VarL = 0. However, Wang et al. (2004) show that the managerial insights are sometimes similar irrespective of whether the demand function is additive or multiplicative for price-sensitive newsvendor models. This needs to be investigated in our setting. Similarly, a more realistic model should perhaps involve multiple retailers competing on price (Bernstein and Federgruen 2005). An online companion to this paper is available on the Management Science website (http://mansci.pubs. informs.org/ecompanion.html). Acknowledgments

The authors thank the anonymous referees, the associate editor and the editor-in-chief for their valuable support, which significantly improved the relevance, positioning, and presentation of this paper. This research is supported by grants from the Natural Sciences and Engineering Research Council of Canada and Fonds de Recherche sur la Société et la Culture of Québec.

Appendix A: Glossary of Notation

p = mean price-dependent demand rate d2 = variance of demand per unit time L0 = constant delivery time from the manufacturer to the distributor



L = random delivery time from the distributor to the retailer, mean EL and variance VarL h h0 = holding cost rates for the retailer and the distributor, resp. b b0 = backordering cost rates for the retailer and the distributor, resp. # #0 = safety factors for the retailer and the distributor, resp. #C #D = retailer’s optimal safety factors in centralized and decentralized chains, resp. #C0 #D 0 = distributor’s optimal safety factors in centralized and decentralized chains, resp. c p w = replenishment cost/unit for the distributor, retail price/unit, and wholesale price/unit, resp. w D = distributor’s optimal wholesale price in the decentralized chain pC pD = optimal retail prices in centralized and decentralized chains, resp. R S = local base stocks for the retailer and the distributor, resp. RC RD = retailer’s optimal local base stocks in centralized and decentralized chains, resp. S C S D = distributor’s optimal local base stocks in centralized and decentralized chains, resp. * *C *D = profit function, and the optimal profits for centralized and decentralized chains, resp. /0 /0D = profit function and the optimal profit, resp. for the distributor in a decentralized chain / / D = profit function and the optimal profit, resp. for the retailer in a decentralized chain B = price sensitivity of the customers for linear yp = A − Bp

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