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The inventory billboard effect on the lead-time decision
Int. J. Production Economics 170 (2015) 45–53
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
The inventory billboard effect on the lead-time decision Zhengping Wu a, Xin Zhai b,n, Zhongyi Liu c a b c
Whitman School of Management, Syracuse University, Syracuse, NY 13244-2450, United States Guanghua School of Management, Peking University, Beijing 100871, China School of Police Administration, People's Public Security University of China, Beijing 100038, China
art ic l e i nf o
a b s t r a c t
Article history: Received 4 July 2014 Accepted 25 August 2015 Available online 16 September 2015
This research brings together two important research streams: lead-time management and the inventory billboard effect. While traditional inventory theory, which assumes that demand is independent of the average on-hand inventory, recommends that lead-time be reduced to the lowest level possible, it is clearly not the case when inventory exhibits the billboard effect, which refers to the demand stimulating effect of inventory. We use analytical models to examine the firm's optimal lead-time decision in the presence of the inventory billboard effect in two scenarios: with and without inventory based competition. We begin with the single firm scenario, where a firm employs the base stock policy for the infinite horizon continuous-review inventory problem with a deterministic lead-time and inventory dependent demand. The firm's decisions are to choose the optimal lead-time and the corresponding base stock level to maximize its expected profit rate. We find that the inventory billboard effect favors a long lead-time, because a long lead-time results in a higher inventory level, which in turn induces more demand. We then consider the case where two firms compete on inventory for customer demand. We completely characterize the unique Nash equilibrium and provide closed-form solutions, which allow us to conclude that inventory based competition pressures both firms to increase the lead-time. Our numerical studies show that the extent to which demand depends on inventory amplify the impact of competition on the optimal decisions and the associated profit. Our findings suggest that lead-time reduction, although widely advocated by popular production philosophies such as Just-in-Time, has to be carefully evaluated when inventory exhibits billboard effect. & 2015 Elsevier B.V. All rights reserved.
Keywords: Lead-time decision Billboard effect Inventory management Competition Game theory
1. Introduction Lead-time has long been recognized as a crucial component of inventory management. Its importance has been well explained in operations management textbooks: it affects the degree of uncertainties in the demand during lead-time, which in turn determines the safety stock level and the associated total inventory costs. An immediate implication of the above logic is that lead-time reduction, if possible, leads to lower safety stock, hence lower average on-hand inventory, lower inventory holding cost, and higher profit. As a result, many companies regard the leadtime decision as an important strategy and strive to shorten it as much as possible to improve inventory efficiency, as evidenced by the widespread use of the Just-in-Time (JIT) system, which places a lot of emphasis on lead-time reduction, e.g., locating facilities in close proximity to suppliers. n
Corresponding author. Tel.: þ 86 10 62757460; fax: þ86 10 62753182. E-mail addresses: [email protected] (Z. Wu), [email protected] (X. Zhai), [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.ijpe.2015.09.008 0925-5273/& 2015 Elsevier B.V. All rights reserved.
A very short lead-time, however, may not necessarily be a sound strategy in some cases. In fact, Blackburn (2012) observes that over the past decades supply chains have gotten longer instead of shorter, and the flow of goods through the chains has become slower rather than faster, which implies a longer leadtime in effect. This can be explained by several reasons. First, leadtime reduction does not come free. Instead, it usually requires substantial investments. For instance, Marks & Spencer had to invest heavily in mobile radio frequency identification technology to achieve lead-time reduction by shortening the reading time for each tagged dolly (Delgardo and Wilding, 2004). Gerchak and Parlar (1991) and Ray et al. (2004) use analytical models to show that lead-time reduction has to be carefully evaluated against its investment requirements. Second, the characteristics of the product may favor a relatively long lead-time. In a well-known qualitative framework, Fisher (1997) categorizes products as functional or innovative depending on such factors as contribution margin, margin of forecast error, and product variety. He uses a number of real life examples to illustrate that supply chains featuring relatively long lead-times are more suitable for functional products. de Treville et al. (2014)
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use quantitative finance tools to provide a useful foundation for quantifying demand unpredictability. They demonstrate that leadtime reduction is not warranted unless the demand volatility is high and demand forecast evolves over time. Third, in traditional inventory theories, demand is commonly assumed to be independent of the amount of inventory carried. Therefore, the advantages of lead-time reduction are discussed without recognizing the inventory billboard effect, which refers to the positive impact of inventory on customer demand (Cachon and Olivares, 2012). This effect is also known as inventory (or shelf space) dependent demand, and has been observed in a wide range of industries (Whitin, 1957; Wolfe, 1968; Koschat, 2008), especially at the retail level in the fast moving consumer goods (FMCG) industry. It is clear that when demand is stimulated by inventory, the firm has an incentive to stock more, which in turn calls for a longer lead-time. Therefore, it is no longer optimal for the leadtime to be as short as possible. Our paper focuses on the inventory billboard effect, and investigates the following research questions: in the presence of the inventory billboard effect, how should the firm determine the cost minimizing lead-time together with the corresponding inventory decision? Further, how does the inventory based competition affect the optimal lead-time choice? To answer these questions, we set up a stylized model to conduct an economic analysis of the lead-time decision, taking into consideration the inventory billboard effect. We begin with the single firm scenario, where a firm employs the base stock policy for the infinite horizon continuous-review inventory problem with deterministic leadtime. The average demand rate depends on the firm's average on-hand inventory. Both the mean and the range (a measure of uncertainties) of the random demand during lead-time increase in lead-time and average demand rate. The firm's decision is to choose the optimal lead-time and the corresponding base stock level to maximize its expected profit rate. We then extend the single firm model to the scenario where two firms compete on inventory for customers. We study how the firms should set their equilibrium lead-times and base stock levels simultaneously. Our main results are summarized as follows. In the single firm case, we obtain closed-form solutions for the optimal lead-time and base stock strategies, which allow us to understand how various parameters affect the optimal decisions. We find that the inventory billboard effect favors a long lead-time, because a long lead-time results in a higher inventory level, which in turn induces more demand. The optimal decisions are driven by tradeoffs between the gain from the extra demand stimulated and the increased inventory holding cost. In the competition case, we completely characterize the equilibrium lead-time and base stock decisions and provide closed-form solutions for the unique Nash equilibrium. We find that the effect of the inventory based competition is to further prolong the lead time. Finally, we observe from numerical experiments that the extent to which demand depends on inventory amplifies the impact of competition on the optimal decisions and profit. Our paper makes contributions to the literature from both theoretical and managerial standpoints. From a theoretical perspective, we endogenize the lead-time decision while demands exhibit inventory billboard effects and provide closed-form solutions to this complex problem. To the best of our knowledge, we are among the first to bring together two important research streams: lead-time management and the inventory billboard effect. Moreover, we add to the knowledge base by using a rigorous analytical model to uncover the effect of inventory based competition in this context. From a managerial perspective, we offer new insights into lead-time related decisions to industry practitioners. We caution that although lead-time reduction strategy suggested by prior research is viable when demand is
independent of inventory, it deserves further investigation in the presence of the inventory billboard effect. Specifically, when demand is stimulated by inventory, firms cannot take for granted that they can always benefit from lead-time reduction. It is only rational to do so when the current lead-time is above a certain threshold. Furthermore, inventory based competition tends to be against lead-time reduction. The remainder of the paper is organized as follows. Section 2 outlines related literature. We study the inventory billboard effect on the lead-time reduction decision of a single firm in Section 3, and extend the model in Section 4 to consider the case where two firms compete for customers on inventory. Numerical examples are presented in Section 5. The paper concludes in Section 6 with a discussion of future research directions. All proofs are collected in the appendix.
2. Literature review Our work brings together the research stream that studies supply lead-time in inventory systems and the literature on the inventory billboard effect. 2.1. Lead-time in inventory systems Lead-time plays a critical role in supply chain management, especially in inventory management, as lead-time is a fundamental factor that affects key decisions and performance metrics such as system base stock level and total cost. Considerable research has been done to understand the importance of leadtime. Some papers consider deterministic lead-time for technical tractability. For instance, Kouki et al. (2015) study a perishable inventory system that operates under stochastic demand and a constant lead-time. They characterize the properties of the cost function and present an approximation procedure to find the policy parameters. Through simulations, they demonstrate their simple and efficient algorithm outperforms another approximation procedure in the literature. Das (1975) is among the first to discuss the effect of stochastic lead time on inventory management. So and Zheng (2003) use a two-echelon supply chain model to show how the supplier's variable lead-times can amplify the variability in the order quantities of the downstream member in the supply chain. Song (1994) investigates how the behavior of the optimal base stock level and long-run average costs are influenced by the lead-time. She shows that (1) a stochastically larger lead-time requires a higher optimal base stock level, and (2) while a more variable lead-time always leads to a higher optimal average cost, the effect of lead-time variability on optimal policies depends on the inventory cost structure. Arikan et al. (2014) use a serial inventory system consisting of a manufacturer who works with overseas suppliers to investigate the inter-relation between lead-time uncertainty and the economic and environmental performance of supply chains. They quantify the effects of lead-time variability on emissions and total cost for a retailer or manufacturer with high service level requirements, and find that a change in the optimal policy from cost to emission minimization has a low impact on cost, but can have a considerably high impact on emissions. Hoque (2013) also considers uncertain lead-times. He recognizes the limitations of exponential lead-times assumed in previous vendor–buyer integrated production–inventory models, and develops a model with normal distribution of lead-time. Extensive comparative studies are carried out to highlight the significance of his different modeling approach. Heydari et al. (2008) study a four-echelon supply chain using a structural model. They find that lead-time variance affects the inventory system through changing the order variances
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and increasing both holding and stockout amounts. Zipkin (2000) provides a general framework for examining the role of lead-time in inventory management. He reviews different settings and concludes that the impact of lead-time hinges on the structure and control of the supply system. Through proper lead-time management, firms could improve their inventory systems by lowering the safety stock level directly, and reducing loss caused by stockouts and thus delivering a higher customer service level indirectly. In a recent paper, Cobb et al. (2015) discuss ways to model leadtime demand in continuous review systems. To construct an accurate probability density function for lead-time demand in inventory management models, they use B-spline functions from empirical data on demand per unit time to estimate a mixture of polynomials distributions. Inventory policy can then be determined without knowledge of the underlying demand and/or leadtime distribution. They also present an improvement to a mixture distribution approach that models the lead-time demand distribution with a mixture of truncated exponentials distribution. They argue that both methods provide reasonable accuracy, but the former approach requires lower computational time to determine optimal inventory policies. Two papers explore the impact of lead-time reduction. Ryu and Lee (2003) consider a dual-sourcing model with both suppliers investing in lead-time reduction respectively. They compare the two suppliers' expected total cost per unit time in cases with and without lead-time reduction, and conclude that lead-time reduction results in significant savings. de Treville et al. (2004) explore the relationship between lead-time reduction and demand information transfer in demand chains. Their analysis shows that relative supply lead-times should be prioritized over demand information transfer. In contrast to the above studies, which assume an exogenous lead-time and focus on parametric analysis, some other papers investigate the endogenous lead-time decision. Early work along this line of research includes van Beek and van Putten (1987); BenDaya and Raouf (1994). They show how the deterministic leadtime duration can be optimized to minimize total system inventory costs. Gerchak and Parlar (1991) consider a stochastic leadtime and examine the optimal investment strategy to reduce leadtime. Ray et al. (2004) expand Gerchak and Parlar (1991) to give a better understanding of the impact of investing in lead-time reduction. They assume an exact (Q, r) policy and a stochastic replenishment lead-time, and show that the firm's inventory costs are jointly convex in inventory policy and lead-time. They find that the firm will undoubtedly reduce lead-time to zero if there is no cost involved in lead-time reduction. Lo (2009) considers leadtime as a decision variable, and develops an economic order quantity (EOQ) model with backorder price discount. The author derives the optimal lead-time to minimize the expected total annual cost. Two interesting papers, Lee and Schwarz (2007) and Lee (2009), use the principal-agent framework to study incentive compatible contracts for decentralized supply chains to undertake lead-time reduction. Our paper also considers endogenous leadtime, but differs from the aforementioned publications in that we are the first to integrate lead-time decision into the context of the inventory billboard effect. Zhu (2015) integrates capacity, pricing, and lead-time decisions in a decentralized supply chain consisting of a supplier and a retailer who faces price- and lead-time-sensitive demand. He characterizes the impact of capacity decision on the players' profit and demonstrates that the profit loss caused by double marginalization can be significantly reduced by optimizing capacity. Kaman et al. (2013) study the value of shop floor information for a manufacturer that operates under a make-to-stock queue scheme and quotes lead-times to arriving customers. They examine the cases with perfect and imperfect information about the shop floor
47
status, and find that the increased likelihood of observing the true status of the system does not necessarily lead to higher profit. Wu et al. (2012) investigate the optimal pricing, inventory, and leadtime quotation decisions when demand is stochastic and priceand lead-time-sensitive. Their model enables the firm to determine the optimal selling price, quoted lead-time, and order quantity simultaneously, and provides a new set of insights to managers. In these papers, the lead-time is defined as the delivery lead-time to customers, whereas our paper considers the supply lead-time. 2.2. Inventory billboard effect A second research stream that is closely related to our paper studies the inventory billboard effect. Classical inventory models assume that demand rate is independent of inventory levels. In practice, however, it is widely recognized that an increase in inventory (or shelf space) of an item induces more consumers to buy it. Some existing studies have analytically demonstrated the motivating effect that inventory levels have on demand rate of retail items. Interested readers may refer to Urban (2005) for a comprehensive review on the inventory level dependent demand literature up to year 2005. This review classifies the literature into two distinct streams according to how the dependence of demand rate on the inventory displayed is modeled. One is the initial inventory level dependent demand rate, and the other is instantaneous inventory level dependent demand rate. The author demonstrates the equivalence of the two types of models through a simple periodic-review model. We now turn our attention to the more up-to-date literature. A couple of recent papers have reported empirical evidence in support of the inventory billboard effect. Koschat (2008) find that in magazine retailing, demand can indeed vary with inventory. They suggest making inventory decision for retail categories rather than individual brands. Olivares and Cachon (2009) examine inventory held by competing retailers empirically. They discuss mechanisms of separating competing influence: a sales effect and a service level effect, and observe that when additional competition is in presence, increasing quality (e.g., through raising inventory level, etc.) is a better choice. The above empirical evidence has sparked revived research interests in this area. Arya et al. (2009) look at three inventory models for perishable items. They develop an optimization framework for the replenishment policy when demand depends on current stock level. Roy and Chaudhuri (2009) investigate two production inventory models, assuming that demand rate is a function of instantaneous inventory level. Chang et al. (2010) extend the EOQ model by assuming that demand rate depends on both stock-on-display and selling price. Petruzzi et al. (2009) explore a newsvendor model in which demand is controlled strategically. They develop new insights into classifying newsvendor models that incorporate demand management effects. Balakrishnan et al. (2008) generalize the newsvendor model by considering the initial-stock-level dependent demand. They study the inventory and pricing policies when the uncertain demand is stochastically increasing in stocking quantity. Consistent with previous research, they find that the consideration of demandstimulating effect of stock leads to higher initial stock levels and higher fill rates. Qin et al. (2011) provide a quick review of the newsvendor problem by specifically analyzing the impact of market price, market effort, and stock on customer demand. Sapra et al. (2010) study the opposite of the inventory billboard effect in a finite-horizon periodic-review two-echelon supply chain where demand is a decreasing function of the initial inventory level. They look at the retailer's inventory replenishment problem when demand can be manipulated by restricting supply. By drawing
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comparisons with the case where demand is independent of inventory, they conclude that understocking is optimal in various scenarios when inventory influences demand. Baron et al. (2011) investigate the joint shelf space allocation and inventory control problem for multiple products with demand that depends on both the inventory level and the shelf space allocated to their complement or supplement products. Their heuristics show that in some settings ignoring the demand dependency on the inventory level can lead to profit losses of more than 6%. None of the above papers consider the inventory billboard effect on the optimal lead-time choice, which is the main research question we attempt to address in this work.
3. The single firm model In this section, we analyze a stylized model where a single firm faces an infinite time horizon continuous-review inventory problem with a deterministic supply lead-time l, and employs the widely used base stock replenishment policy. In such a setting, the firm orders up to the base stock level R whenever its inventory position drops to R 1 (Zipkin, 2000; Wang and Gerchak, 2001). The firm's demand during lead-time (also called lead-time demand), X, follows a uniform distribution Uðμ δ; μ þ δÞ, where μ denotes the mean lead-time demand, and δ is half of the range, which measures the variability of the lead-time demand. Both μ and δ increase in the lead-time l, as commonly assumed in the literature. In particular, μ ¼ lλ, and δ ¼ βμ ¼ β lλ, where λ stands for the average demand rate (i.e., the average demand per unit time), and the constant β is a non-negative coefficient, which is restricted to the range ð0; 1 to ensure non-negativity of the demand. Furthermore, we draw on Wang and Gerchak (2001) and assume that the average demand rate λ is increasing in the average on-hand inventory I, recognizing the inventory billboard effect:
λ ¼ aI2b þ 1 :
ð1Þ
In the above equation, the constant b measures the degree to which demand depends on inventory – a larger b means stronger demand stimulating effect of inventory. It is assumed that 1=2 rb o 0, implying diminishing marginal return of each extra unit of average inventory. Note that b ¼ 1=2 corresponds to the case where demand is independent of inventory. The constant a 4 0 represents the base demand absent the inventory billboard effect, i.e., λ¼a when b ¼ 1=2. Given Eq. (1), the dependence of demand mean μ and variability δ on lead-time l and average inventory I can be described as follows.
μ ¼ lλ ¼ laI2b þ 1 ;
ð2Þ
δ ¼ βlλ ¼ lβaI2b þ 1 :
ð3Þ
In classical inventory theories (Zipkin, 2000), the average onhand inventory I is computed in the following fashion: x ¼ R Z R 1 1 1 I¼ ðR xÞϕðxÞ dx ¼ Rx x2 ¼ ðR μ þ δÞ2 ; ð4Þ 2 2 4δ δ μδ x ¼ μδ where ϕðxÞ denotes the probability density function (pdf) of the random demand X. Since each unit of demand results in either a sale of a lost sale, we have
μ ¼ Expected Sales þ B; where B stands for the average backorders. Moreover, every unit of inventory purchased is either sold or left over, so R ¼ Expected Sales þ I:
The above two equations, taken together, give rise to the following expression of the average backorders B: B ¼ μ þ I R: The firm decides on the optimal supply lead-time l together with the optimal base stock level R to maximize its long-run average profit per unit time. Note that R is non-negative, and thus Eq. (4) indicates a one-to-one correspondence between I and R, which can be used to transform the average inventory I into the decision variable in place of R. In other words, the decisions of the firm are turned into (l, I) from (l, R). This technical treatment greatly facilitates the tractability of the problem. Under the transformation, we express R as a function of I by substituting Eqs. (2) and (3) into Eq. (4): qffiffiffiffiffiffiffi pffiffiffiffiffi 2b þ 1 R ¼ μ δ þ 2 δI ¼ laI lβaI 2b þ 1 þ2 lβ aI b þ 1 : ð5Þ Likewise, backorders B can be expressed as a function of (l, I) as follows: qffiffiffiffiffiffiffi B ¼ μ þ I R ¼ I þlβaI 2b þ 1 2 lβ aI b þ 1 : ð6Þ It is worth noting that R is increasing in I, as one would intuit. This relationship between them implies that although the average demand rate λ is assumed to be increasing in the average on-hand inventory I for analytical convenience, it can be easily interpreted as an increasing function of the base stock level R, which is another common approach in the literature. Following standard techniques (Zipkin, 2000), we are now ready to formulate the firm's optimization problem as follows: max Π ðl; IÞ ¼ ðp cÞλ sB hI ¼ ðp cÞaI 2b þ 1 l;I qffiffiffiffiffiffiffi slβaI 2b þ 1 þ 2s lβ aI b þ 1 ðs þ hÞI;
ð7Þ
where the constants p, c, s, h are the unit selling price, variable cost, backorder cost per unit time, and holding cost per unit time, respectively. The first step in the above formulation shows the firm's expected profit per unit time (i.e., the long-run average profit rate). The first term ðp cÞλ is the riskless (i.e., when there is no demand uncertainty) profit per unit time, while the other two terms are backorder cost per unit time and inventory holding cost per unit time, respectively. 3.1. Solution procedure It is straightforward to verify that ∂Π ðl; IÞ ¼ ðp cÞað2b þ 1ÞI 2b slβað2b þ 1ÞI 2b ∂I qffiffiffiffiffiffiffi þ 2s lβaðbþ 1ÞI b ðh þ sÞ; qffiffiffiffiffiffiffi ∂2 Π ðl; IÞ ¼ 2ðp c slβÞabð2b þ 1ÞI 2b 1 þ 2s lβabðb þ 1ÞI b 1 ; 2 ∂I
ð8Þ
where the second order derivative in the second equation above is not guaranteed to be non-positive, which means that in general the profit function is not piecewise concave in the average inventory I, let alone jointly concave in both decisions I and l. This poses great technical challenge in finding the optimal solutions. Fortunately, we are able to take a sequential optimization approach as in Petruzzi and Dada (1999) to characterize the unique pair of I and l that maximizes the objective function, by optimizing l first for a given I. Taking I as given, we can show that the first and second order derivatives of the objective function with respect to l are as follows: rffiffiffiffiffiffi ∂Π ðl; IÞ βa b þ 1 ¼ sβaI 2b þ 1 þ s I ; ∂l l
Z. Wu et al. / Int. J. Production Economics 170 (2015) 45–53
∂2 Π ðl; IÞ 2
∂l
¼
s 2
qffiffiffiffiffiffi βal 3=2 Ib þ 1 o0:
Corollary 1. The inventory billboard effect increases the optimal lead-time and average on-hand inventory.
The second equation above establishes the concavity. Therefore, the optimal lead-time as a function of I, denoted by l(I), can be found from the first order condition, as summarized in the following lemma. Lemma 1. For a fixed I, the optimal lead-time is characterized by the following equation: lðIÞ ¼
1
βaI2b
:
ð9Þ
Substituting l(I) into Eq. (5) yields R ¼ μ þ I, which can then be substituted into Eq. (4) to get I ¼δ. Therefore, R ¼ μ þ δ. In other words, the optimal lead-time l(I) is set in such a way that the base stock level R attains the upper support of the demand distribution. As a result, there will be no backorders, i.e., B ¼0, as can be verified from Eq. (6). The managerial implication of this observation is that the positive impact of inventory on the top line (revenue) outweighes any potential increase in the inventory holding cost. Therefore, the firm should set a lead-time commensurate with the base stock level R that enables the firm to capture all demands without worrying about the inventory holding cost. Substituting l(I) back into the objective function in Eq. (7) turns it into an optimization problem of the single variable I only: max Π ðlðIÞ; IÞ ¼ ðp cÞaI 2b þ 1 hI: I
ð10Þ
It is worth noting that the above objective function has a very appealing interpretation: for any given I, the optimal lead-time l(I) eliminates the backorders, and the remaining decision I is to strike a balance between the riskless profit and the inventory holding cost. We next proceed to find the optimal decisions. 3.2. Optimal decisions Let a decision with the superscripts “N” and “C” denote its optimal solution in the single firm case and the competition case, respectively. Proposition 1. The firm's optimal decisions and the associated optimal profit are as follows: (a) In the absence of the inventory billboard effect, i.e., when N b ¼ 1=2, the optimal lead-time l lðI N Þ ¼ 0, the corresponding optimal average inventory IN ¼0, and the associated optimal N profit is Π ¼ aðp cÞ; (b) In the presence of the inventory billboard effect, i.e., when 1=2 ob o 0, we have N
l lðI N Þ ¼
ð2b þ1Þðp cÞ ; βh
að2bþ 1Þðp cÞ I ¼ h N
ð11Þ
1=ð 2bÞ ;
ð12Þ
Corollary 1 highlights the inventory billboard effect on the firm's optimal lead-time decision, which is one of our key findings. To elaborate on the intuition, let us now take a closer look at the drivers of the optimal decision making process. In the absence of the inventory billboard effect, i.e., when b ¼ 1=2, our problem reduces to the traditional newsvendor problem where demand is independent of inventory. In this case, as can be seen from Eq. (7), the riskless profit remains constant at aðp cÞ, and the firm's optimal decisions are to minimize inventory mismatch costs (including both backordering cost and holding cost). Clearly, it is in the firm's best interest to set both lead-time and average inventory to 0 to completely eliminate inventory mismatch costs. Not surprisingly, our conclusion accords well with existing findings in the lead-time reduction literature (e.g., Ray et al., 2004), and suggests that the firm should strive to eliminate the supply lead-time and achieve instantaneous replenishment. When the inventory billboard effect comes into the picture (i.e., when 1=2 o b o0), however, a downside risk of a short leadtime emerges and has to be taken into account: a short lead-time will result in lower uncertainty in the lead-time demand, thus less safety stock, and consequently, less average inventory. As a result, the firm faces a low average customer demand rate, which is detrimental to the bottom line. To optimize its profit, the firm has to carefully strike a balance between the two opposite forces. It turns out that it pays off for the firm to maintain a positive leadtime and carry more inventory to stimulate more customer demand, although the benefit of the enlarged demand will be partially offset by the increased inventory holding cost. There have been strong advocates in the literature to shorten supply lead-time as much as possible, so that inventory can be reduced. Our findings, however, are in sharp contrast to this recommendation. We show that the lead-time decision has to be carefully evaluated when inventory exhibits billboard effect. If the current lead-time of the firm is too long (above the optimal level lN), it pays off to take actions to reduce the lead-time, as long as investments required for those actions do not outweigh the benefits. Nevertheless, lead-time reduction may hurt the firm's bottom line if the current lead-time is already low. We caution that firms cannot take for granted that they can always benefit from lead-time reduction, even if such reduction do not consume a lot of resources. 3.3. Comparative statics In this section, we examine comparative statics to understand how various parameters affect the optimal decisions and the associated profit. Since we have obtained closed-form solutions for the optimal decisions and profit, it is straightforward to check the first order derivatives to see how they vary with parameters. The results are summarized in Table 1 without proofs, as all derivations are easy to Table 1 Comparative statics. Parameters
Optimal lead-time Optimal average inventory optimal profit (lN) (IN) (ΠN)
p c h s β a b
↑ ↓ ↓ – ↓ – ↑
and the associated optimal profit is
ΠN ¼
1=ð 2bÞ 2bh að2b þ 1Þðp cÞ : 2b þ 1 h
49
ð13Þ
A comparison of the two parts of Proposition 1 leads to the following corollary.
↑ ↓ ↓ – – ↑ ↑
↑ ↓ ↓ – – ↑ ↑
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verify. In the table, up arrows, down arrows, and flat lines mean increase, decrease, and no effect, respectively. The results in Table 1 can be interpreted as follows. Note that the impact of p, c, and h on the optimal decisions and the associated optimal profit is similar to that in the classical newsvendor problem. To elaborate, let us take the parameter p for example. An increase in the unit price p not only increases the newsvendor critical fractile to drive up inventory, it also raises the revenue and thus profit from the extra demand stimulated by the increased inventory. Therefore, the optimal average inventory increases in p. Correspondingly, the optimal lead-time is longer to justify the higher inventory. The logic for the impact of the unit cost c is similar. As for the unit holding cost h, it has no bearing on the profit from the extra demand stimulated by inventory, and the only role it plays is to decrease the critical fractile. Therefore it drives down both inventory and lead-time. The backorder cost s does not play any role, and the potential reason is more intricate: As s increases, the critical fractile also increases, and the traditional newsvendor logic calls for a higher inventory to minimize the inventory mismatch costs. This clearly favors a longer lead-time. On the other hand, increased inventory stimulates more demand, which may aggravate the shortages and thus increase the total backordering cost. In this sense, it is better to shorten the lead-time. As indicated by our results, it turns out that these two opposite effects of s always exactly cancel each other out, so the optimal decisions and profit are invariant with s. This is consistent with our previous observation that the optimal lead-time is set in such a way that backorders are eliminated. As for the parameter β, note that both β and the lead-time decision l affect the uncertainty of the lead-time demand. If β is large, it makes sense to set a short l to counteract the negative effect of β on the lead-time demand uncertainty, and vice versa. This explains why the optimal lead-time decreases in β. Our results also show that the lead-time decision can totally balance the effect of β, therefore both the average inventory and profit are independent of β. The effect of the parameter a is somewhat opposite that of β. It does not affect the optimal lead-time, but drives up the optimal average inventory and profit. The latter is not surprising, because the average demand rate λ increases in a. For the same reason, both decisions and the profit are increasing in b. Before concluding this section, we would like to remark on one of our modeling assumptions. Our model does not include a cost of lead-time reduction actions. This is because we intend to demonstrate that even if it is free to achieve a shorter lead-time, it may not be economically viable to do so. In the event that there is a cost associated with shortening lead-time, it is straightforward to incorporate this cost into our model. For example, we can augment the objective function by k(l), which is convex decreasing in the lead-time l, to reflect this cost. Obviously, such a modification does not alter any of our qualitative insights at all.
4. Two competing firms In this section, we extend the single firm problem to the case with two firms competing on inventory for customer demand. Following the common practice in the literature, we limit our attention to the symmetric case in which all cost parameters of the two firms are identical. Following Wang and Gerchak (2001), we model the customer demand and the firms' competition processes in the following way: the aggregate demand rate λ ¼ λ1 þ λ2 now depends on the total average on-hand inventory of the two firms. Specifically,
Eq. (1) is modified as
λ ¼ aðI 1 þI 2 Þ2b þ 1 : Each firm's share of the aggregate demand rate is proportional to its relative average on-hand inventory, i.e., we use an allocation function wi ¼ I i =ðI i þ I 3 i Þ, for i ¼ 1; 2. As a result, firm i's demand rate is
λi ¼ wi λ ¼ aIi ðIi þ I3 i Þ2b : In addition,
μi ¼ li λi ¼ ali Ii ðIi þ I3 i Þ2b ; and
δi ¼ βli λi ¼ βali Ii ðIi þ I3 i Þ2b : The two firms make a simultaneous move and play a Nash game to determine their lead-times and base stock levels. Similar to the single firm case, we choose to treat the average on-hand inventory Ii as a decision variable in place of the base stock level Ri to facilitate technical developments. Following that, we can rewrite Ri and Bi as functions of ðli ; I i Þ as follows: qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Ri ¼ μi δi þ 2 δi I i ¼ ð1 β Þali I i ðI i þ I 3 i Þ2b þ 2 li β aI i ðI i þ I 3 i Þb ; qffiffiffiffiffiffiffiffiffi Bi ¼ μi þ I i Ri ¼ I i þ β ali I i ðI i þI 3 i Þ2b 2 li β aI i ðI i þ I 3 i Þb : We can now formulate the firms' decision problems below: for i¼1, 2, max Π i ðli ; l3 i ; I i ; I 3 i Þ ¼ ðp cÞλi sBi hI i ¼ ðp c sβli ÞaI i ðI i þI 3 i Þ2b li ;I i
þ 2s
qffiffiffiffiffiffiffiffiffi βali I i ðI i þ I3 i Þb ðs þ hÞIi :
ð14Þ
As the first step toward finding the equilibrium outcome of the game, we characterize below firm i's best response function, given the other firm's lead-time and average inventory decisions. In this process, we use the same sequential optimization approach as in the single firm case. That is, we first fix Ii to find the optimal li ðI i Þ as a function of Ii, then substitute li ðI i Þ back into the objective function of firm i to transform it into a function of the single variable Ii, and finally identify the optimal Ii. Lemma 1 below summarizes the results. Lemma 2. Firm i's best response functions are as follows:
βaI2b li ¼ 1;
ð15Þ
aðp cÞI 2b þ 2abðp cÞI i I 2b 1 h ¼ 0:
ð16Þ
Clearly, firm 3 i's best response function can be written in a similar fashion. Solving the four best response functions simultaneously yields the Nash equilibrium (NE) presented in the following proposition. Proposition 2. In the presence of the inventory billboard effect, i.e., when 1=2 o b o 0, we have the following optimal solutions: l1 ¼ l2 ¼
ðbþ 1Þðp cÞ ; βh
ð17Þ
I C1 ¼ I C2 ¼
1=ð 2bÞ 1 aðbþ 1Þðp cÞ ; 2 h
ð18Þ
C
C
and the associated optimal profits are
Z. Wu et al. / Int. J. Production Economics 170 (2015) 45–53
Π C1 ¼ Π C2 ¼
1=ð 2bÞ bh aðb þ 1Þðp cÞ : 2ðb þ 1Þ h
51
ð19Þ
Proposition 2 largely resembles Proposition 1, the counterpart in the single firm case. Consequently, the comparative statics are also similar. Furthermore, a quick comparison between the two propositions uncovers that, in the presence of inventory based competition, the firms set longer lead-times. In other words, the inventory based competition tends to be against lead-time reduction. This is because, in an attempt to win customers (or to avoid losing customers), both firms are induced to carry more inventory, which in turn encourages longer lead-times. As a side note, when the inventory billboard effect is absent, i.e., when b ¼ 1=2, there is no allocation function per se, because the two firms do not compete on inventory. In effect, they behave just like two separate firms without any interaction at all. Therefore, their optimal decisions are exactly the same as those studied in the single firm case, i.e., they set lead-time to zero and carry no inventory.
Fig. 2. The impact of parameter a on the optimal profit.
5. Numerical studies In this section, we complement our analytical results with numerical studies to investigate the magnitude of the impact of parameters a and b, which are the main determinants of the extent to which demand depends on inventory, as well as to understand the effect of competition on the optimal decisions and profit. All our studies exhibit a consistent pattern, and representative results are illustrated below. Fig. 1 shows that, regardless of competition, the optimal leadtime decision is independent of a, and the optimal average onhand inventory increases in a, which is consistent with our discussion in comparative statics. In addition, this figure also demonstrates the impact of inventory based competition. Compared to the single firm case, in the presence of competition, each firm faces additional pressure to have more on-hand inventory to avoid losing customers to the other. Consequently, both firms end up with a longer lead-time and higher average on-hand inventory, as we can see from the figure. Fig. 2 depicts how the optimal profit is affected by a. As one would expect, a has a positive effect on the optimal profit, and the effect of competition is to erode the bottom line of both firms. We also observe that the gap between the profits in the competition and no-competition cases widens as a increases. A potential reason is that, as a increases, the average demand rate λ also increases, which lures the two firms to fight more fiercely to win customers. Therefore, the larger the parameter a, the more the
Fig. 3. The impact of parameter b on optimal decisions: (a) optimal lead-time and (b) optimal average inventory.
Fig. 1. The impact of parameter a on optimal decisions: (a) Optimal lead-time and (b) optimal average inventory.
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Z. Wu et al. / Int. J. Production Economics 170 (2015) 45–53
optimal decisions are inflated in the competition case as compared to the no-competition case, leading to the increasing gap between the two scenarios. In other words, a amplifies the effect of competition on the optimal decisions and the associated profit. Figs. 3 and 4 portray the impact of parameter b on the optimal decisions and profit in both the single firm and competition cases. It can be observed that the effect of b is similar to that of a. Note from Eq. (1) that the role of b on the average demand rate λ is twofold: in general (i.e., when the average inventory is above unity, which is the typical case in real world settings), it not only increases λ, but it also increases its gradient (i.e., ∂λ=∂I) to magnify the billboard effect. With this understanding, it is no surprise that b is a much stronger catalyst than a to drive up the optimal average on-hand inventory and profit.
6. Conclusion and future research In this research, we study a firm's choice of the lead-time decision in the presence of the inventory billboard effect, and how it is affected by the inventory based competition. While traditional inventory theory, which assumes that demand is independent of the average on-hand inventory, recommends that lead-time be reduced to the lowest level possible, we examine if the firm will still be better off reducing their lead-time when inventory exhibits the billboard effect. We begin with the single firm scenario, where a firm employs the base stock policy for the infinite horizon continuous-review inventory problem with a deterministic lead-time and inventory dependent demand. The firm's decisions are to choose the optimal lead-time and the corresponding base stock level to maximize its expected profit rate. We obtain closed-form solutions, and find that the inventory billboard effect favors a long lead-time, because a long lead-time results in a higher inventory level, which, in turn, induces more demand. We then extend the single firm model to the case with two competing firms. We study how the firms set their equilibrium leadtimes and base stock levels simultaneously. We completely characterize the unique Nash equilibrium and provide closed-form solutions, which allow us to conclude that inventory based competition tends to be against lead-time reduction. Our numerical studies show that the parameters that define the extent to which demand depends on inventory amplify the impact of competition on the optimal decisions and the associated profit. Our findings suggest that leadtime reduction, although widely advocated by popular production philosophies such as JIT, has to be carefully evaluated when inventory exhibits billboard effect. If the firm's current lead-time is too long, it pays off to reduce it. Nevertheless, lead-time reduction may hurt the firm's bottom line if its current lead-time is already low. Furthermore, inventory based competition should also be taken into account to weigh on lead-time reduction.
Our paper contributes to the operations literature from both theoretical and managerial standpoints. From a theoretical perspective, we endogenize the lead-time decision in the presence of inventory billboard effect and provide closed-form solutions to this complex problem. To the best of our knowledge, we are among the first to bridge the literature on lead-time management and inventory billboard effect. Moreover, we use an analytical model to uncover the impact of inventory based competition. From a managerial perspective, we offer new insights into leadtime reduction to industry practitioners. We caution that although the recommendation to reduce lead-time suggested by prior research is viable when demand is independent of inventory, it deserves further investigation in the presence of the inventory billboard effect. Specifically, when demand is stimulated by inventory, firms cannot take for granted that they can always benefit from lead-time reduction. It is only rational to do so when the current lead-time is above a certain threshold. Furthermore, our results suggest that inventory based competition tends to be against lead-time reduction. This research can be extended along different dimensions. First, our model assumes a deterministic supply lead-time, which applies to settings with stable deliveries. Under circumstances where the leadtime is volatile, it will be more reasonable to consider a stochastic lead-time. Conceivably, such an extension will add considerable technical complexity to the problem. Second, our analysis adopts a specific demand distribution and functional form to model the dependence of demand on inventory. Other distributions and functional forms are worthy of investigation. Finally, we leave empirical validation of our results, which is beyond the scope of this paper, open for future research.
Acknowledgements The authors would like to thank Professor Edwin Cheng, the editor, and the anonymous referees for their constructive comments, which have led to significant improvements of this paper. Financial support from Beijing Planning Office of Philosophy and Social Science (No. 13JGB049, 15JGB128) and the Guanghua Leadership Institute (No. 1216R) is gratefully acknowledged.
Appendix Proof of Proposition 1. (a) It is evident that when b ¼ 1=2, the objective function in Eq. (10) reduces to Π ðlðIÞ; IÞ ¼ ðp cÞa hI, which is linearly decreasing in I. Hence IN ¼0 and the associated optimal profit reduces to aðp cÞ. (b) When 1=2 o b o 0, we can take the first and second order derivatives of the objective function in Eq. (10) with respect to I to obtain the following: ∂Π ðlðIÞ; IÞ ¼ ðp cÞað2b þ1ÞI 2b h; ∂I ∂2 Π ðlðIÞ; IÞ ∂I 2
Fig. 4. The impact of parameter b on the optimal profit.
¼ 2ðp cÞabð2b þ 1ÞI 2b 1 r 0;
where the inequality follows from the fact that 1=2 r b o 0. Therefore, the concavity of the objective function in I is established. The optimal average inventory characterized by Eq. (12) follows from the first order condition directly. The optimal leadtime can then be obtained from Eq. (9), and the associated expected profit is derived by plugging the optimal solutions into the objective function.□
Z. Wu et al. / Int. J. Production Economics 170 (2015) 45–53
Proof of Lemma 2. It is worth pointing out that when characterizing firm i's best response function, we take firm 3 i's decisions l3 i and I 3 i as given, i.e., they are treated as parameters. Therefore, we can use the same sequential optimization as that in the single firm case to solve the problem (note that arguments of functions are omitted for brevity whenever confusion does not arise). Let I ¼ I 1 þ I 2 for notational convenience. For a fixed Ii, we have the following: sffiffiffiffiffiffi ∂Π i βa b 2b II ; ¼ sβaI i I þ s li i ∂li qffiffiffiffiffiffi 2 ∂ Πi s ¼ βali 3=2 Ii Ib r 0; 2 2 ∂l i
which establishes the concavity, and thus the first order condition leads to Eq. (15). Substituting this result into the objective function Eq. (14) yields
Π i ¼ ðp cÞaIi I2b hIi ; whose first and second order derivatives with respect to Ii are given as follows: ∂Π i ¼ ðp cÞaI 2b þ 2ðp cÞabI i I 2b 1 h; ∂I i ∂2 Π i 2b 2 ¼ 2ðp cÞabI 2I þ ð2b 1ÞI i ∂I 2i 2b 2 ð2b þ 1ÞI i þ 2I 3 i r 0: ¼ 2ðp cÞabI Therefore the objective function is concave in Ii, and the first order condition leads to Eq. (16).□ Proof of Proposition 2. Similar to Eq. (16), we can write its counterpart as follows: aðp cÞI 2b þ 2abðp cÞI 3 i I 2b 1 h ¼ 0; which is then added up to Eq. (16) to obtain 2aðp cÞI 2b þ 2abðp cÞI 2b 2h ¼ 0: The above result can be simplified to get aðb þ 1Þðp cÞI 2b ¼ h, which is then substituted into Eq. (16) to get h 2bh Ii þ h ¼ 0: b þ 1 b þ 1 Ii þ I3 i Further simplifications yield that I i ¼ I 3 i ¼ I=2. Note that aðb þ 1Þðp cÞI 2b ¼ h also implies 1=ð 2bÞ aðbþ 1Þðp cÞ : I¼ h Hence the optimal average inventory result in Eq. (18). The optimal lead-time then follows directly from Eq. (15). The associated profit in Eq. (19) can be derived by substituting the optimal l and I into Eq. (14).□
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Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
The inventory billboard effect on the lead-time decision Zhengping Wu a, Xin Zhai b,n, Zhongyi Liu c a b c
Whitman School of Management, Syracuse University, Syracuse, NY 13244-2450, United States Guanghua School of Management, Peking University, Beijing 100871, China School of Police Administration, People's Public Security University of China, Beijing 100038, China
art ic l e i nf o
a b s t r a c t
Article history: Received 4 July 2014 Accepted 25 August 2015 Available online 16 September 2015
This research brings together two important research streams: lead-time management and the inventory billboard effect. While traditional inventory theory, which assumes that demand is independent of the average on-hand inventory, recommends that lead-time be reduced to the lowest level possible, it is clearly not the case when inventory exhibits the billboard effect, which refers to the demand stimulating effect of inventory. We use analytical models to examine the firm's optimal lead-time decision in the presence of the inventory billboard effect in two scenarios: with and without inventory based competition. We begin with the single firm scenario, where a firm employs the base stock policy for the infinite horizon continuous-review inventory problem with a deterministic lead-time and inventory dependent demand. The firm's decisions are to choose the optimal lead-time and the corresponding base stock level to maximize its expected profit rate. We find that the inventory billboard effect favors a long lead-time, because a long lead-time results in a higher inventory level, which in turn induces more demand. We then consider the case where two firms compete on inventory for customer demand. We completely characterize the unique Nash equilibrium and provide closed-form solutions, which allow us to conclude that inventory based competition pressures both firms to increase the lead-time. Our numerical studies show that the extent to which demand depends on inventory amplify the impact of competition on the optimal decisions and the associated profit. Our findings suggest that lead-time reduction, although widely advocated by popular production philosophies such as Just-in-Time, has to be carefully evaluated when inventory exhibits billboard effect. & 2015 Elsevier B.V. All rights reserved.
Keywords: Lead-time decision Billboard effect Inventory management Competition Game theory
1. Introduction Lead-time has long been recognized as a crucial component of inventory management. Its importance has been well explained in operations management textbooks: it affects the degree of uncertainties in the demand during lead-time, which in turn determines the safety stock level and the associated total inventory costs. An immediate implication of the above logic is that lead-time reduction, if possible, leads to lower safety stock, hence lower average on-hand inventory, lower inventory holding cost, and higher profit. As a result, many companies regard the leadtime decision as an important strategy and strive to shorten it as much as possible to improve inventory efficiency, as evidenced by the widespread use of the Just-in-Time (JIT) system, which places a lot of emphasis on lead-time reduction, e.g., locating facilities in close proximity to suppliers. n
Corresponding author. Tel.: þ 86 10 62757460; fax: þ86 10 62753182. E-mail addresses: [email protected] (Z. Wu), [email protected] (X. Zhai), [email protected] (Z. Liu). http://dx.doi.org/10.1016/j.ijpe.2015.09.008 0925-5273/& 2015 Elsevier B.V. All rights reserved.
A very short lead-time, however, may not necessarily be a sound strategy in some cases. In fact, Blackburn (2012) observes that over the past decades supply chains have gotten longer instead of shorter, and the flow of goods through the chains has become slower rather than faster, which implies a longer leadtime in effect. This can be explained by several reasons. First, leadtime reduction does not come free. Instead, it usually requires substantial investments. For instance, Marks & Spencer had to invest heavily in mobile radio frequency identification technology to achieve lead-time reduction by shortening the reading time for each tagged dolly (Delgardo and Wilding, 2004). Gerchak and Parlar (1991) and Ray et al. (2004) use analytical models to show that lead-time reduction has to be carefully evaluated against its investment requirements. Second, the characteristics of the product may favor a relatively long lead-time. In a well-known qualitative framework, Fisher (1997) categorizes products as functional or innovative depending on such factors as contribution margin, margin of forecast error, and product variety. He uses a number of real life examples to illustrate that supply chains featuring relatively long lead-times are more suitable for functional products. de Treville et al. (2014)
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Z. Wu et al. / Int. J. Production Economics 170 (2015) 45–53
use quantitative finance tools to provide a useful foundation for quantifying demand unpredictability. They demonstrate that leadtime reduction is not warranted unless the demand volatility is high and demand forecast evolves over time. Third, in traditional inventory theories, demand is commonly assumed to be independent of the amount of inventory carried. Therefore, the advantages of lead-time reduction are discussed without recognizing the inventory billboard effect, which refers to the positive impact of inventory on customer demand (Cachon and Olivares, 2012). This effect is also known as inventory (or shelf space) dependent demand, and has been observed in a wide range of industries (Whitin, 1957; Wolfe, 1968; Koschat, 2008), especially at the retail level in the fast moving consumer goods (FMCG) industry. It is clear that when demand is stimulated by inventory, the firm has an incentive to stock more, which in turn calls for a longer lead-time. Therefore, it is no longer optimal for the leadtime to be as short as possible. Our paper focuses on the inventory billboard effect, and investigates the following research questions: in the presence of the inventory billboard effect, how should the firm determine the cost minimizing lead-time together with the corresponding inventory decision? Further, how does the inventory based competition affect the optimal lead-time choice? To answer these questions, we set up a stylized model to conduct an economic analysis of the lead-time decision, taking into consideration the inventory billboard effect. We begin with the single firm scenario, where a firm employs the base stock policy for the infinite horizon continuous-review inventory problem with deterministic leadtime. The average demand rate depends on the firm's average on-hand inventory. Both the mean and the range (a measure of uncertainties) of the random demand during lead-time increase in lead-time and average demand rate. The firm's decision is to choose the optimal lead-time and the corresponding base stock level to maximize its expected profit rate. We then extend the single firm model to the scenario where two firms compete on inventory for customers. We study how the firms should set their equilibrium lead-times and base stock levels simultaneously. Our main results are summarized as follows. In the single firm case, we obtain closed-form solutions for the optimal lead-time and base stock strategies, which allow us to understand how various parameters affect the optimal decisions. We find that the inventory billboard effect favors a long lead-time, because a long lead-time results in a higher inventory level, which in turn induces more demand. The optimal decisions are driven by tradeoffs between the gain from the extra demand stimulated and the increased inventory holding cost. In the competition case, we completely characterize the equilibrium lead-time and base stock decisions and provide closed-form solutions for the unique Nash equilibrium. We find that the effect of the inventory based competition is to further prolong the lead time. Finally, we observe from numerical experiments that the extent to which demand depends on inventory amplifies the impact of competition on the optimal decisions and profit. Our paper makes contributions to the literature from both theoretical and managerial standpoints. From a theoretical perspective, we endogenize the lead-time decision while demands exhibit inventory billboard effects and provide closed-form solutions to this complex problem. To the best of our knowledge, we are among the first to bring together two important research streams: lead-time management and the inventory billboard effect. Moreover, we add to the knowledge base by using a rigorous analytical model to uncover the effect of inventory based competition in this context. From a managerial perspective, we offer new insights into lead-time related decisions to industry practitioners. We caution that although lead-time reduction strategy suggested by prior research is viable when demand is
independent of inventory, it deserves further investigation in the presence of the inventory billboard effect. Specifically, when demand is stimulated by inventory, firms cannot take for granted that they can always benefit from lead-time reduction. It is only rational to do so when the current lead-time is above a certain threshold. Furthermore, inventory based competition tends to be against lead-time reduction. The remainder of the paper is organized as follows. Section 2 outlines related literature. We study the inventory billboard effect on the lead-time reduction decision of a single firm in Section 3, and extend the model in Section 4 to consider the case where two firms compete for customers on inventory. Numerical examples are presented in Section 5. The paper concludes in Section 6 with a discussion of future research directions. All proofs are collected in the appendix.
2. Literature review Our work brings together the research stream that studies supply lead-time in inventory systems and the literature on the inventory billboard effect. 2.1. Lead-time in inventory systems Lead-time plays a critical role in supply chain management, especially in inventory management, as lead-time is a fundamental factor that affects key decisions and performance metrics such as system base stock level and total cost. Considerable research has been done to understand the importance of leadtime. Some papers consider deterministic lead-time for technical tractability. For instance, Kouki et al. (2015) study a perishable inventory system that operates under stochastic demand and a constant lead-time. They characterize the properties of the cost function and present an approximation procedure to find the policy parameters. Through simulations, they demonstrate their simple and efficient algorithm outperforms another approximation procedure in the literature. Das (1975) is among the first to discuss the effect of stochastic lead time on inventory management. So and Zheng (2003) use a two-echelon supply chain model to show how the supplier's variable lead-times can amplify the variability in the order quantities of the downstream member in the supply chain. Song (1994) investigates how the behavior of the optimal base stock level and long-run average costs are influenced by the lead-time. She shows that (1) a stochastically larger lead-time requires a higher optimal base stock level, and (2) while a more variable lead-time always leads to a higher optimal average cost, the effect of lead-time variability on optimal policies depends on the inventory cost structure. Arikan et al. (2014) use a serial inventory system consisting of a manufacturer who works with overseas suppliers to investigate the inter-relation between lead-time uncertainty and the economic and environmental performance of supply chains. They quantify the effects of lead-time variability on emissions and total cost for a retailer or manufacturer with high service level requirements, and find that a change in the optimal policy from cost to emission minimization has a low impact on cost, but can have a considerably high impact on emissions. Hoque (2013) also considers uncertain lead-times. He recognizes the limitations of exponential lead-times assumed in previous vendor–buyer integrated production–inventory models, and develops a model with normal distribution of lead-time. Extensive comparative studies are carried out to highlight the significance of his different modeling approach. Heydari et al. (2008) study a four-echelon supply chain using a structural model. They find that lead-time variance affects the inventory system through changing the order variances
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and increasing both holding and stockout amounts. Zipkin (2000) provides a general framework for examining the role of lead-time in inventory management. He reviews different settings and concludes that the impact of lead-time hinges on the structure and control of the supply system. Through proper lead-time management, firms could improve their inventory systems by lowering the safety stock level directly, and reducing loss caused by stockouts and thus delivering a higher customer service level indirectly. In a recent paper, Cobb et al. (2015) discuss ways to model leadtime demand in continuous review systems. To construct an accurate probability density function for lead-time demand in inventory management models, they use B-spline functions from empirical data on demand per unit time to estimate a mixture of polynomials distributions. Inventory policy can then be determined without knowledge of the underlying demand and/or leadtime distribution. They also present an improvement to a mixture distribution approach that models the lead-time demand distribution with a mixture of truncated exponentials distribution. They argue that both methods provide reasonable accuracy, but the former approach requires lower computational time to determine optimal inventory policies. Two papers explore the impact of lead-time reduction. Ryu and Lee (2003) consider a dual-sourcing model with both suppliers investing in lead-time reduction respectively. They compare the two suppliers' expected total cost per unit time in cases with and without lead-time reduction, and conclude that lead-time reduction results in significant savings. de Treville et al. (2004) explore the relationship between lead-time reduction and demand information transfer in demand chains. Their analysis shows that relative supply lead-times should be prioritized over demand information transfer. In contrast to the above studies, which assume an exogenous lead-time and focus on parametric analysis, some other papers investigate the endogenous lead-time decision. Early work along this line of research includes van Beek and van Putten (1987); BenDaya and Raouf (1994). They show how the deterministic leadtime duration can be optimized to minimize total system inventory costs. Gerchak and Parlar (1991) consider a stochastic leadtime and examine the optimal investment strategy to reduce leadtime. Ray et al. (2004) expand Gerchak and Parlar (1991) to give a better understanding of the impact of investing in lead-time reduction. They assume an exact (Q, r) policy and a stochastic replenishment lead-time, and show that the firm's inventory costs are jointly convex in inventory policy and lead-time. They find that the firm will undoubtedly reduce lead-time to zero if there is no cost involved in lead-time reduction. Lo (2009) considers leadtime as a decision variable, and develops an economic order quantity (EOQ) model with backorder price discount. The author derives the optimal lead-time to minimize the expected total annual cost. Two interesting papers, Lee and Schwarz (2007) and Lee (2009), use the principal-agent framework to study incentive compatible contracts for decentralized supply chains to undertake lead-time reduction. Our paper also considers endogenous leadtime, but differs from the aforementioned publications in that we are the first to integrate lead-time decision into the context of the inventory billboard effect. Zhu (2015) integrates capacity, pricing, and lead-time decisions in a decentralized supply chain consisting of a supplier and a retailer who faces price- and lead-time-sensitive demand. He characterizes the impact of capacity decision on the players' profit and demonstrates that the profit loss caused by double marginalization can be significantly reduced by optimizing capacity. Kaman et al. (2013) study the value of shop floor information for a manufacturer that operates under a make-to-stock queue scheme and quotes lead-times to arriving customers. They examine the cases with perfect and imperfect information about the shop floor
47
status, and find that the increased likelihood of observing the true status of the system does not necessarily lead to higher profit. Wu et al. (2012) investigate the optimal pricing, inventory, and leadtime quotation decisions when demand is stochastic and priceand lead-time-sensitive. Their model enables the firm to determine the optimal selling price, quoted lead-time, and order quantity simultaneously, and provides a new set of insights to managers. In these papers, the lead-time is defined as the delivery lead-time to customers, whereas our paper considers the supply lead-time. 2.2. Inventory billboard effect A second research stream that is closely related to our paper studies the inventory billboard effect. Classical inventory models assume that demand rate is independent of inventory levels. In practice, however, it is widely recognized that an increase in inventory (or shelf space) of an item induces more consumers to buy it. Some existing studies have analytically demonstrated the motivating effect that inventory levels have on demand rate of retail items. Interested readers may refer to Urban (2005) for a comprehensive review on the inventory level dependent demand literature up to year 2005. This review classifies the literature into two distinct streams according to how the dependence of demand rate on the inventory displayed is modeled. One is the initial inventory level dependent demand rate, and the other is instantaneous inventory level dependent demand rate. The author demonstrates the equivalence of the two types of models through a simple periodic-review model. We now turn our attention to the more up-to-date literature. A couple of recent papers have reported empirical evidence in support of the inventory billboard effect. Koschat (2008) find that in magazine retailing, demand can indeed vary with inventory. They suggest making inventory decision for retail categories rather than individual brands. Olivares and Cachon (2009) examine inventory held by competing retailers empirically. They discuss mechanisms of separating competing influence: a sales effect and a service level effect, and observe that when additional competition is in presence, increasing quality (e.g., through raising inventory level, etc.) is a better choice. The above empirical evidence has sparked revived research interests in this area. Arya et al. (2009) look at three inventory models for perishable items. They develop an optimization framework for the replenishment policy when demand depends on current stock level. Roy and Chaudhuri (2009) investigate two production inventory models, assuming that demand rate is a function of instantaneous inventory level. Chang et al. (2010) extend the EOQ model by assuming that demand rate depends on both stock-on-display and selling price. Petruzzi et al. (2009) explore a newsvendor model in which demand is controlled strategically. They develop new insights into classifying newsvendor models that incorporate demand management effects. Balakrishnan et al. (2008) generalize the newsvendor model by considering the initial-stock-level dependent demand. They study the inventory and pricing policies when the uncertain demand is stochastically increasing in stocking quantity. Consistent with previous research, they find that the consideration of demandstimulating effect of stock leads to higher initial stock levels and higher fill rates. Qin et al. (2011) provide a quick review of the newsvendor problem by specifically analyzing the impact of market price, market effort, and stock on customer demand. Sapra et al. (2010) study the opposite of the inventory billboard effect in a finite-horizon periodic-review two-echelon supply chain where demand is a decreasing function of the initial inventory level. They look at the retailer's inventory replenishment problem when demand can be manipulated by restricting supply. By drawing
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comparisons with the case where demand is independent of inventory, they conclude that understocking is optimal in various scenarios when inventory influences demand. Baron et al. (2011) investigate the joint shelf space allocation and inventory control problem for multiple products with demand that depends on both the inventory level and the shelf space allocated to their complement or supplement products. Their heuristics show that in some settings ignoring the demand dependency on the inventory level can lead to profit losses of more than 6%. None of the above papers consider the inventory billboard effect on the optimal lead-time choice, which is the main research question we attempt to address in this work.
3. The single firm model In this section, we analyze a stylized model where a single firm faces an infinite time horizon continuous-review inventory problem with a deterministic supply lead-time l, and employs the widely used base stock replenishment policy. In such a setting, the firm orders up to the base stock level R whenever its inventory position drops to R 1 (Zipkin, 2000; Wang and Gerchak, 2001). The firm's demand during lead-time (also called lead-time demand), X, follows a uniform distribution Uðμ δ; μ þ δÞ, where μ denotes the mean lead-time demand, and δ is half of the range, which measures the variability of the lead-time demand. Both μ and δ increase in the lead-time l, as commonly assumed in the literature. In particular, μ ¼ lλ, and δ ¼ βμ ¼ β lλ, where λ stands for the average demand rate (i.e., the average demand per unit time), and the constant β is a non-negative coefficient, which is restricted to the range ð0; 1 to ensure non-negativity of the demand. Furthermore, we draw on Wang and Gerchak (2001) and assume that the average demand rate λ is increasing in the average on-hand inventory I, recognizing the inventory billboard effect:
λ ¼ aI2b þ 1 :
ð1Þ
In the above equation, the constant b measures the degree to which demand depends on inventory – a larger b means stronger demand stimulating effect of inventory. It is assumed that 1=2 rb o 0, implying diminishing marginal return of each extra unit of average inventory. Note that b ¼ 1=2 corresponds to the case where demand is independent of inventory. The constant a 4 0 represents the base demand absent the inventory billboard effect, i.e., λ¼a when b ¼ 1=2. Given Eq. (1), the dependence of demand mean μ and variability δ on lead-time l and average inventory I can be described as follows.
μ ¼ lλ ¼ laI2b þ 1 ;
ð2Þ
δ ¼ βlλ ¼ lβaI2b þ 1 :
ð3Þ
In classical inventory theories (Zipkin, 2000), the average onhand inventory I is computed in the following fashion: x ¼ R Z R 1 1 1 I¼ ðR xÞϕðxÞ dx ¼ Rx x2 ¼ ðR μ þ δÞ2 ; ð4Þ 2 2 4δ δ μδ x ¼ μδ where ϕðxÞ denotes the probability density function (pdf) of the random demand X. Since each unit of demand results in either a sale of a lost sale, we have
μ ¼ Expected Sales þ B; where B stands for the average backorders. Moreover, every unit of inventory purchased is either sold or left over, so R ¼ Expected Sales þ I:
The above two equations, taken together, give rise to the following expression of the average backorders B: B ¼ μ þ I R: The firm decides on the optimal supply lead-time l together with the optimal base stock level R to maximize its long-run average profit per unit time. Note that R is non-negative, and thus Eq. (4) indicates a one-to-one correspondence between I and R, which can be used to transform the average inventory I into the decision variable in place of R. In other words, the decisions of the firm are turned into (l, I) from (l, R). This technical treatment greatly facilitates the tractability of the problem. Under the transformation, we express R as a function of I by substituting Eqs. (2) and (3) into Eq. (4): qffiffiffiffiffiffiffi pffiffiffiffiffi 2b þ 1 R ¼ μ δ þ 2 δI ¼ laI lβaI 2b þ 1 þ2 lβ aI b þ 1 : ð5Þ Likewise, backorders B can be expressed as a function of (l, I) as follows: qffiffiffiffiffiffiffi B ¼ μ þ I R ¼ I þlβaI 2b þ 1 2 lβ aI b þ 1 : ð6Þ It is worth noting that R is increasing in I, as one would intuit. This relationship between them implies that although the average demand rate λ is assumed to be increasing in the average on-hand inventory I for analytical convenience, it can be easily interpreted as an increasing function of the base stock level R, which is another common approach in the literature. Following standard techniques (Zipkin, 2000), we are now ready to formulate the firm's optimization problem as follows: max Π ðl; IÞ ¼ ðp cÞλ sB hI ¼ ðp cÞaI 2b þ 1 l;I qffiffiffiffiffiffiffi slβaI 2b þ 1 þ 2s lβ aI b þ 1 ðs þ hÞI;
ð7Þ
where the constants p, c, s, h are the unit selling price, variable cost, backorder cost per unit time, and holding cost per unit time, respectively. The first step in the above formulation shows the firm's expected profit per unit time (i.e., the long-run average profit rate). The first term ðp cÞλ is the riskless (i.e., when there is no demand uncertainty) profit per unit time, while the other two terms are backorder cost per unit time and inventory holding cost per unit time, respectively. 3.1. Solution procedure It is straightforward to verify that ∂Π ðl; IÞ ¼ ðp cÞað2b þ 1ÞI 2b slβað2b þ 1ÞI 2b ∂I qffiffiffiffiffiffiffi þ 2s lβaðbþ 1ÞI b ðh þ sÞ; qffiffiffiffiffiffiffi ∂2 Π ðl; IÞ ¼ 2ðp c slβÞabð2b þ 1ÞI 2b 1 þ 2s lβabðb þ 1ÞI b 1 ; 2 ∂I
ð8Þ
where the second order derivative in the second equation above is not guaranteed to be non-positive, which means that in general the profit function is not piecewise concave in the average inventory I, let alone jointly concave in both decisions I and l. This poses great technical challenge in finding the optimal solutions. Fortunately, we are able to take a sequential optimization approach as in Petruzzi and Dada (1999) to characterize the unique pair of I and l that maximizes the objective function, by optimizing l first for a given I. Taking I as given, we can show that the first and second order derivatives of the objective function with respect to l are as follows: rffiffiffiffiffiffi ∂Π ðl; IÞ βa b þ 1 ¼ sβaI 2b þ 1 þ s I ; ∂l l
Z. Wu et al. / Int. J. Production Economics 170 (2015) 45–53
∂2 Π ðl; IÞ 2
∂l
¼
s 2
qffiffiffiffiffiffi βal 3=2 Ib þ 1 o0:
Corollary 1. The inventory billboard effect increases the optimal lead-time and average on-hand inventory.
The second equation above establishes the concavity. Therefore, the optimal lead-time as a function of I, denoted by l(I), can be found from the first order condition, as summarized in the following lemma. Lemma 1. For a fixed I, the optimal lead-time is characterized by the following equation: lðIÞ ¼
1
βaI2b
:
ð9Þ
Substituting l(I) into Eq. (5) yields R ¼ μ þ I, which can then be substituted into Eq. (4) to get I ¼δ. Therefore, R ¼ μ þ δ. In other words, the optimal lead-time l(I) is set in such a way that the base stock level R attains the upper support of the demand distribution. As a result, there will be no backorders, i.e., B ¼0, as can be verified from Eq. (6). The managerial implication of this observation is that the positive impact of inventory on the top line (revenue) outweighes any potential increase in the inventory holding cost. Therefore, the firm should set a lead-time commensurate with the base stock level R that enables the firm to capture all demands without worrying about the inventory holding cost. Substituting l(I) back into the objective function in Eq. (7) turns it into an optimization problem of the single variable I only: max Π ðlðIÞ; IÞ ¼ ðp cÞaI 2b þ 1 hI: I
ð10Þ
It is worth noting that the above objective function has a very appealing interpretation: for any given I, the optimal lead-time l(I) eliminates the backorders, and the remaining decision I is to strike a balance between the riskless profit and the inventory holding cost. We next proceed to find the optimal decisions. 3.2. Optimal decisions Let a decision with the superscripts “N” and “C” denote its optimal solution in the single firm case and the competition case, respectively. Proposition 1. The firm's optimal decisions and the associated optimal profit are as follows: (a) In the absence of the inventory billboard effect, i.e., when N b ¼ 1=2, the optimal lead-time l lðI N Þ ¼ 0, the corresponding optimal average inventory IN ¼0, and the associated optimal N profit is Π ¼ aðp cÞ; (b) In the presence of the inventory billboard effect, i.e., when 1=2 ob o 0, we have N
l lðI N Þ ¼
ð2b þ1Þðp cÞ ; βh
að2bþ 1Þðp cÞ I ¼ h N
ð11Þ
1=ð 2bÞ ;
ð12Þ
Corollary 1 highlights the inventory billboard effect on the firm's optimal lead-time decision, which is one of our key findings. To elaborate on the intuition, let us now take a closer look at the drivers of the optimal decision making process. In the absence of the inventory billboard effect, i.e., when b ¼ 1=2, our problem reduces to the traditional newsvendor problem where demand is independent of inventory. In this case, as can be seen from Eq. (7), the riskless profit remains constant at aðp cÞ, and the firm's optimal decisions are to minimize inventory mismatch costs (including both backordering cost and holding cost). Clearly, it is in the firm's best interest to set both lead-time and average inventory to 0 to completely eliminate inventory mismatch costs. Not surprisingly, our conclusion accords well with existing findings in the lead-time reduction literature (e.g., Ray et al., 2004), and suggests that the firm should strive to eliminate the supply lead-time and achieve instantaneous replenishment. When the inventory billboard effect comes into the picture (i.e., when 1=2 o b o0), however, a downside risk of a short leadtime emerges and has to be taken into account: a short lead-time will result in lower uncertainty in the lead-time demand, thus less safety stock, and consequently, less average inventory. As a result, the firm faces a low average customer demand rate, which is detrimental to the bottom line. To optimize its profit, the firm has to carefully strike a balance between the two opposite forces. It turns out that it pays off for the firm to maintain a positive leadtime and carry more inventory to stimulate more customer demand, although the benefit of the enlarged demand will be partially offset by the increased inventory holding cost. There have been strong advocates in the literature to shorten supply lead-time as much as possible, so that inventory can be reduced. Our findings, however, are in sharp contrast to this recommendation. We show that the lead-time decision has to be carefully evaluated when inventory exhibits billboard effect. If the current lead-time of the firm is too long (above the optimal level lN), it pays off to take actions to reduce the lead-time, as long as investments required for those actions do not outweigh the benefits. Nevertheless, lead-time reduction may hurt the firm's bottom line if the current lead-time is already low. We caution that firms cannot take for granted that they can always benefit from lead-time reduction, even if such reduction do not consume a lot of resources. 3.3. Comparative statics In this section, we examine comparative statics to understand how various parameters affect the optimal decisions and the associated profit. Since we have obtained closed-form solutions for the optimal decisions and profit, it is straightforward to check the first order derivatives to see how they vary with parameters. The results are summarized in Table 1 without proofs, as all derivations are easy to Table 1 Comparative statics. Parameters
Optimal lead-time Optimal average inventory optimal profit (lN) (IN) (ΠN)
p c h s β a b
↑ ↓ ↓ – ↓ – ↑
and the associated optimal profit is
ΠN ¼
1=ð 2bÞ 2bh að2b þ 1Þðp cÞ : 2b þ 1 h
49
ð13Þ
A comparison of the two parts of Proposition 1 leads to the following corollary.
↑ ↓ ↓ – – ↑ ↑
↑ ↓ ↓ – – ↑ ↑
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verify. In the table, up arrows, down arrows, and flat lines mean increase, decrease, and no effect, respectively. The results in Table 1 can be interpreted as follows. Note that the impact of p, c, and h on the optimal decisions and the associated optimal profit is similar to that in the classical newsvendor problem. To elaborate, let us take the parameter p for example. An increase in the unit price p not only increases the newsvendor critical fractile to drive up inventory, it also raises the revenue and thus profit from the extra demand stimulated by the increased inventory. Therefore, the optimal average inventory increases in p. Correspondingly, the optimal lead-time is longer to justify the higher inventory. The logic for the impact of the unit cost c is similar. As for the unit holding cost h, it has no bearing on the profit from the extra demand stimulated by inventory, and the only role it plays is to decrease the critical fractile. Therefore it drives down both inventory and lead-time. The backorder cost s does not play any role, and the potential reason is more intricate: As s increases, the critical fractile also increases, and the traditional newsvendor logic calls for a higher inventory to minimize the inventory mismatch costs. This clearly favors a longer lead-time. On the other hand, increased inventory stimulates more demand, which may aggravate the shortages and thus increase the total backordering cost. In this sense, it is better to shorten the lead-time. As indicated by our results, it turns out that these two opposite effects of s always exactly cancel each other out, so the optimal decisions and profit are invariant with s. This is consistent with our previous observation that the optimal lead-time is set in such a way that backorders are eliminated. As for the parameter β, note that both β and the lead-time decision l affect the uncertainty of the lead-time demand. If β is large, it makes sense to set a short l to counteract the negative effect of β on the lead-time demand uncertainty, and vice versa. This explains why the optimal lead-time decreases in β. Our results also show that the lead-time decision can totally balance the effect of β, therefore both the average inventory and profit are independent of β. The effect of the parameter a is somewhat opposite that of β. It does not affect the optimal lead-time, but drives up the optimal average inventory and profit. The latter is not surprising, because the average demand rate λ increases in a. For the same reason, both decisions and the profit are increasing in b. Before concluding this section, we would like to remark on one of our modeling assumptions. Our model does not include a cost of lead-time reduction actions. This is because we intend to demonstrate that even if it is free to achieve a shorter lead-time, it may not be economically viable to do so. In the event that there is a cost associated with shortening lead-time, it is straightforward to incorporate this cost into our model. For example, we can augment the objective function by k(l), which is convex decreasing in the lead-time l, to reflect this cost. Obviously, such a modification does not alter any of our qualitative insights at all.
4. Two competing firms In this section, we extend the single firm problem to the case with two firms competing on inventory for customer demand. Following the common practice in the literature, we limit our attention to the symmetric case in which all cost parameters of the two firms are identical. Following Wang and Gerchak (2001), we model the customer demand and the firms' competition processes in the following way: the aggregate demand rate λ ¼ λ1 þ λ2 now depends on the total average on-hand inventory of the two firms. Specifically,
Eq. (1) is modified as
λ ¼ aðI 1 þI 2 Þ2b þ 1 : Each firm's share of the aggregate demand rate is proportional to its relative average on-hand inventory, i.e., we use an allocation function wi ¼ I i =ðI i þ I 3 i Þ, for i ¼ 1; 2. As a result, firm i's demand rate is
λi ¼ wi λ ¼ aIi ðIi þ I3 i Þ2b : In addition,
μi ¼ li λi ¼ ali Ii ðIi þ I3 i Þ2b ; and
δi ¼ βli λi ¼ βali Ii ðIi þ I3 i Þ2b : The two firms make a simultaneous move and play a Nash game to determine their lead-times and base stock levels. Similar to the single firm case, we choose to treat the average on-hand inventory Ii as a decision variable in place of the base stock level Ri to facilitate technical developments. Following that, we can rewrite Ri and Bi as functions of ðli ; I i Þ as follows: qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Ri ¼ μi δi þ 2 δi I i ¼ ð1 β Þali I i ðI i þ I 3 i Þ2b þ 2 li β aI i ðI i þ I 3 i Þb ; qffiffiffiffiffiffiffiffiffi Bi ¼ μi þ I i Ri ¼ I i þ β ali I i ðI i þI 3 i Þ2b 2 li β aI i ðI i þ I 3 i Þb : We can now formulate the firms' decision problems below: for i¼1, 2, max Π i ðli ; l3 i ; I i ; I 3 i Þ ¼ ðp cÞλi sBi hI i ¼ ðp c sβli ÞaI i ðI i þI 3 i Þ2b li ;I i
þ 2s
qffiffiffiffiffiffiffiffiffi βali I i ðI i þ I3 i Þb ðs þ hÞIi :
ð14Þ
As the first step toward finding the equilibrium outcome of the game, we characterize below firm i's best response function, given the other firm's lead-time and average inventory decisions. In this process, we use the same sequential optimization approach as in the single firm case. That is, we first fix Ii to find the optimal li ðI i Þ as a function of Ii, then substitute li ðI i Þ back into the objective function of firm i to transform it into a function of the single variable Ii, and finally identify the optimal Ii. Lemma 1 below summarizes the results. Lemma 2. Firm i's best response functions are as follows:
βaI2b li ¼ 1;
ð15Þ
aðp cÞI 2b þ 2abðp cÞI i I 2b 1 h ¼ 0:
ð16Þ
Clearly, firm 3 i's best response function can be written in a similar fashion. Solving the four best response functions simultaneously yields the Nash equilibrium (NE) presented in the following proposition. Proposition 2. In the presence of the inventory billboard effect, i.e., when 1=2 o b o 0, we have the following optimal solutions: l1 ¼ l2 ¼
ðbþ 1Þðp cÞ ; βh
ð17Þ
I C1 ¼ I C2 ¼
1=ð 2bÞ 1 aðbþ 1Þðp cÞ ; 2 h
ð18Þ
C
C
and the associated optimal profits are
Z. Wu et al. / Int. J. Production Economics 170 (2015) 45–53
Π C1 ¼ Π C2 ¼
1=ð 2bÞ bh aðb þ 1Þðp cÞ : 2ðb þ 1Þ h
51
ð19Þ
Proposition 2 largely resembles Proposition 1, the counterpart in the single firm case. Consequently, the comparative statics are also similar. Furthermore, a quick comparison between the two propositions uncovers that, in the presence of inventory based competition, the firms set longer lead-times. In other words, the inventory based competition tends to be against lead-time reduction. This is because, in an attempt to win customers (or to avoid losing customers), both firms are induced to carry more inventory, which in turn encourages longer lead-times. As a side note, when the inventory billboard effect is absent, i.e., when b ¼ 1=2, there is no allocation function per se, because the two firms do not compete on inventory. In effect, they behave just like two separate firms without any interaction at all. Therefore, their optimal decisions are exactly the same as those studied in the single firm case, i.e., they set lead-time to zero and carry no inventory.
Fig. 2. The impact of parameter a on the optimal profit.
5. Numerical studies In this section, we complement our analytical results with numerical studies to investigate the magnitude of the impact of parameters a and b, which are the main determinants of the extent to which demand depends on inventory, as well as to understand the effect of competition on the optimal decisions and profit. All our studies exhibit a consistent pattern, and representative results are illustrated below. Fig. 1 shows that, regardless of competition, the optimal leadtime decision is independent of a, and the optimal average onhand inventory increases in a, which is consistent with our discussion in comparative statics. In addition, this figure also demonstrates the impact of inventory based competition. Compared to the single firm case, in the presence of competition, each firm faces additional pressure to have more on-hand inventory to avoid losing customers to the other. Consequently, both firms end up with a longer lead-time and higher average on-hand inventory, as we can see from the figure. Fig. 2 depicts how the optimal profit is affected by a. As one would expect, a has a positive effect on the optimal profit, and the effect of competition is to erode the bottom line of both firms. We also observe that the gap between the profits in the competition and no-competition cases widens as a increases. A potential reason is that, as a increases, the average demand rate λ also increases, which lures the two firms to fight more fiercely to win customers. Therefore, the larger the parameter a, the more the
Fig. 3. The impact of parameter b on optimal decisions: (a) optimal lead-time and (b) optimal average inventory.
Fig. 1. The impact of parameter a on optimal decisions: (a) Optimal lead-time and (b) optimal average inventory.
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optimal decisions are inflated in the competition case as compared to the no-competition case, leading to the increasing gap between the two scenarios. In other words, a amplifies the effect of competition on the optimal decisions and the associated profit. Figs. 3 and 4 portray the impact of parameter b on the optimal decisions and profit in both the single firm and competition cases. It can be observed that the effect of b is similar to that of a. Note from Eq. (1) that the role of b on the average demand rate λ is twofold: in general (i.e., when the average inventory is above unity, which is the typical case in real world settings), it not only increases λ, but it also increases its gradient (i.e., ∂λ=∂I) to magnify the billboard effect. With this understanding, it is no surprise that b is a much stronger catalyst than a to drive up the optimal average on-hand inventory and profit.
6. Conclusion and future research In this research, we study a firm's choice of the lead-time decision in the presence of the inventory billboard effect, and how it is affected by the inventory based competition. While traditional inventory theory, which assumes that demand is independent of the average on-hand inventory, recommends that lead-time be reduced to the lowest level possible, we examine if the firm will still be better off reducing their lead-time when inventory exhibits the billboard effect. We begin with the single firm scenario, where a firm employs the base stock policy for the infinite horizon continuous-review inventory problem with a deterministic lead-time and inventory dependent demand. The firm's decisions are to choose the optimal lead-time and the corresponding base stock level to maximize its expected profit rate. We obtain closed-form solutions, and find that the inventory billboard effect favors a long lead-time, because a long lead-time results in a higher inventory level, which, in turn, induces more demand. We then extend the single firm model to the case with two competing firms. We study how the firms set their equilibrium leadtimes and base stock levels simultaneously. We completely characterize the unique Nash equilibrium and provide closed-form solutions, which allow us to conclude that inventory based competition tends to be against lead-time reduction. Our numerical studies show that the parameters that define the extent to which demand depends on inventory amplify the impact of competition on the optimal decisions and the associated profit. Our findings suggest that leadtime reduction, although widely advocated by popular production philosophies such as JIT, has to be carefully evaluated when inventory exhibits billboard effect. If the firm's current lead-time is too long, it pays off to reduce it. Nevertheless, lead-time reduction may hurt the firm's bottom line if its current lead-time is already low. Furthermore, inventory based competition should also be taken into account to weigh on lead-time reduction.
Our paper contributes to the operations literature from both theoretical and managerial standpoints. From a theoretical perspective, we endogenize the lead-time decision in the presence of inventory billboard effect and provide closed-form solutions to this complex problem. To the best of our knowledge, we are among the first to bridge the literature on lead-time management and inventory billboard effect. Moreover, we use an analytical model to uncover the impact of inventory based competition. From a managerial perspective, we offer new insights into leadtime reduction to industry practitioners. We caution that although the recommendation to reduce lead-time suggested by prior research is viable when demand is independent of inventory, it deserves further investigation in the presence of the inventory billboard effect. Specifically, when demand is stimulated by inventory, firms cannot take for granted that they can always benefit from lead-time reduction. It is only rational to do so when the current lead-time is above a certain threshold. Furthermore, our results suggest that inventory based competition tends to be against lead-time reduction. This research can be extended along different dimensions. First, our model assumes a deterministic supply lead-time, which applies to settings with stable deliveries. Under circumstances where the leadtime is volatile, it will be more reasonable to consider a stochastic lead-time. Conceivably, such an extension will add considerable technical complexity to the problem. Second, our analysis adopts a specific demand distribution and functional form to model the dependence of demand on inventory. Other distributions and functional forms are worthy of investigation. Finally, we leave empirical validation of our results, which is beyond the scope of this paper, open for future research.
Acknowledgements The authors would like to thank Professor Edwin Cheng, the editor, and the anonymous referees for their constructive comments, which have led to significant improvements of this paper. Financial support from Beijing Planning Office of Philosophy and Social Science (No. 13JGB049, 15JGB128) and the Guanghua Leadership Institute (No. 1216R) is gratefully acknowledged.
Appendix Proof of Proposition 1. (a) It is evident that when b ¼ 1=2, the objective function in Eq. (10) reduces to Π ðlðIÞ; IÞ ¼ ðp cÞa hI, which is linearly decreasing in I. Hence IN ¼0 and the associated optimal profit reduces to aðp cÞ. (b) When 1=2 o b o 0, we can take the first and second order derivatives of the objective function in Eq. (10) with respect to I to obtain the following: ∂Π ðlðIÞ; IÞ ¼ ðp cÞað2b þ1ÞI 2b h; ∂I ∂2 Π ðlðIÞ; IÞ ∂I 2
Fig. 4. The impact of parameter b on the optimal profit.
¼ 2ðp cÞabð2b þ 1ÞI 2b 1 r 0;
where the inequality follows from the fact that 1=2 r b o 0. Therefore, the concavity of the objective function in I is established. The optimal average inventory characterized by Eq. (12) follows from the first order condition directly. The optimal leadtime can then be obtained from Eq. (9), and the associated expected profit is derived by plugging the optimal solutions into the objective function.□
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Proof of Lemma 2. It is worth pointing out that when characterizing firm i's best response function, we take firm 3 i's decisions l3 i and I 3 i as given, i.e., they are treated as parameters. Therefore, we can use the same sequential optimization as that in the single firm case to solve the problem (note that arguments of functions are omitted for brevity whenever confusion does not arise). Let I ¼ I 1 þ I 2 for notational convenience. For a fixed Ii, we have the following: sffiffiffiffiffiffi ∂Π i βa b 2b II ; ¼ sβaI i I þ s li i ∂li qffiffiffiffiffiffi 2 ∂ Πi s ¼ βali 3=2 Ii Ib r 0; 2 2 ∂l i
which establishes the concavity, and thus the first order condition leads to Eq. (15). Substituting this result into the objective function Eq. (14) yields
Π i ¼ ðp cÞaIi I2b hIi ; whose first and second order derivatives with respect to Ii are given as follows: ∂Π i ¼ ðp cÞaI 2b þ 2ðp cÞabI i I 2b 1 h; ∂I i ∂2 Π i 2b 2 ¼ 2ðp cÞabI 2I þ ð2b 1ÞI i ∂I 2i 2b 2 ð2b þ 1ÞI i þ 2I 3 i r 0: ¼ 2ðp cÞabI Therefore the objective function is concave in Ii, and the first order condition leads to Eq. (16).□ Proof of Proposition 2. Similar to Eq. (16), we can write its counterpart as follows: aðp cÞI 2b þ 2abðp cÞI 3 i I 2b 1 h ¼ 0; which is then added up to Eq. (16) to obtain 2aðp cÞI 2b þ 2abðp cÞI 2b 2h ¼ 0: The above result can be simplified to get aðb þ 1Þðp cÞI 2b ¼ h, which is then substituted into Eq. (16) to get h 2bh Ii þ h ¼ 0: b þ 1 b þ 1 Ii þ I3 i Further simplifications yield that I i ¼ I 3 i ¼ I=2. Note that aðb þ 1Þðp cÞI 2b ¼ h also implies 1=ð 2bÞ aðbþ 1Þðp cÞ : I¼ h Hence the optimal average inventory result in Eq. (18). The optimal lead-time then follows directly from Eq. (15). The associated profit in Eq. (19) can be derived by substituting the optimal l and I into Eq. (14).□
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