The Effect of Supply Uncertainty on Dynamic Procurement and Pricing Strategies Under Lost Sales

Abstract

The uncertainty in the supplier’s material flows has become a norm rather than an exception in supply chains. The supply uncertainty can result in unexpected inventory shortfall, amplifying lost sales. However, the design of inventory replenishment and product pricing policy to mitigate both supply uncertainty and demand loss remains unexplored. This is because the resulting dynamic planning problem is highly nonconcave and thus intractable. To address this challenge, we propose an approach that focuses on a class of intuitively appealing and practically plausible policies. Specifically, as the level of on-hand inventory increases, we expect an increased amount of demand fulfillment and a decreased product price. Applying the notions of stochastic functions, we show that, under general conditions of the stochastic supply and demand functions, the dynamic planning problem becomes a concave optimization problem over the restricted policy class. We further reduce the set of candidate policies to a refined class by excluding the dominated policies. A refined policy can be easily computed, and appropriately selected refined policies produce close-to-optimal profits. These developments allow us to evaluate the consequences of demand loss. In particular, demand retention through backordering can be beneficial when overstocking is costly in relation to understocking, but the benefit is insensitive to supply and demand uncertainties. Moreover, inventory-based dynamic pricing is more valuable in mitigating supply risk under lost sales than under backordering.

1 Introduction

The impact of supply uncertainty, amplifying the mismatch between the demand and supply, has gained increasing attention recently. Firms are realizing that effective management of supply chains requires careful coordination between inventory control and demand shaping. On the one hand, inventory replenishment through procurement should aim at meeting the demand, avoiding excessive shortage and surplus. On the other hand, the price adjustment should account for the product availability, ensuring the appropriate profit margin and sales. Amihud and Mendelson (1983) discover the price-smoothing effect, suggesting that a carefully designed procurement policy can reduce the sensitivity of the material flow to the price. Feng (2010) demonstrates the complementarity between replenishment and pricing—When the procured amount is not guaranteed for delivery due to high supply uncertainties, frequent price adjustments can generate a significant benefit.

The combined inventory-pricing strategies have been extensively studied in the existing literature, while almost exclusively for the situation where the unmet demand can be backordered. In many situations, customers may not be willing to postpone the consumption and wait for product availability (Chen and Wu, 2019). Lost sales often happen in the event of stockouts. This is the situation frequently observed in retail, where either there are alternative options for the customers or there is a certain time frame for consumption (e.g., Christmas decorations or snow throwers).

During the COVID-19 pandemic, supply issues due to factory closures and clogged ports aggravated unmet consumer demands in many industries. Gap reported an estimated $300 million in lost sales in a quarter of 2021 (Scott, 2021). Bed Bath and Beyond posted a 7% sales decline because of empty shelves in a quarter of 2021 including a holiday season (Kapner and Grossman, 2022). Under Armour, a sports apparel brand, and Chemours, a chemical company, canceled a large number of orders in 2022 because of supply issues (Williams-Alvarez, 2022). The significant amounts of lost sales due to supply uncertainties have induced retailers to adjust prices to maintain profitability. For example, apparel sellers including Abercrombie and Fitch, Guess, and Gap significantly reduced price discounts to boost profit margins, while closely monitoring stock levels (Maurer and Trentmann, 2021).

In other situations, supply uncertainties can lead to excess inventories. As reported by Smith and O’Neal (2021), many companies, worried about instability in upstream material flows, ordered in large volumes for the end-of-year holidays in 2021 and ended up with stock pileups. For example, Target and Big Lots complained that they had to offer deep discounts to clear excess inventories at the beginning of 2022 (Broughton and Maurer, 2022).

The analysis of a dynamic planning system facing lost sales and supply uncertainty is nontrivial. Performance evaluation and policy design for the general system are unknown. We attempt to address these in this study. Specifically, we consider a periodic-review system involving both supply and demand uncertainties. The amount of supply is a stochastic function of the supply decision and the amount of demand is a stochastic function of the pricing decision. The decision maker determines the order and price for the current period, fulfills the demand with the available inventory, and receives the delivery from the supplier.

The profit function is neither concave nor convex in the decisions even in the special case where the supply is deterministic and the inventory cannot be stored (i.e., the planning horizon is one period). The optimal policy is complex and cannot have a characterization in general. Instead of looking for approximations of the profit function or asymptotic behaviors of the system, we examine potential solutions that are intuitive for implementation.

In the special case of a single-period setting, we show that, if the optimal supply decision is decreasing in the price, the profit function is concave in the pricing decision. This observation suggests we restrict the solution space to decreasing supply paths. Such solutions are practically plausible because firms often cut down the prices in the event of overstock, and firms often order bulk amounts in preparation for price discounts. The policy can be analyzed by adapting a similar approach developed by Feng et al. (2020). This approach applies to general supply functions (that are stochastically linear in midpoint) and to general demand functions (that are stochastically linear in midpoint and stochastically decreasing in the increasing concave order).

When the planning involves multiple periods, the evolution of the material flow is triggered by the dynamics of the supply and demand uncertainties. In this case, the argument used for the single-period setting does not apply anymore. Instead, we need to examine how the decisions are related to the on-hand inventory level. In view of the potential shortage of future supply, a portion of the on-hand inventory is allocated to meet the current demand, and the rest is reserved for future sales. Such inventory rationing provides decision flexibility to mitigate supply uncertainties. We show that, under general conditions of the stochastic supply and demand functions, the profit function is concave for a given inventory rationing decision. The optimal profit function, however, is generally not concave in the on-hand inventory level, making the analysis of the dynamic program intractable.

In practice, firms often allocate more inventories to meet the incoming demand and lower the product price when a larger amount of stock is available (Gimpl-Heersin et al., 2008). Theoretically, however, such a monotone policy may not be optimal. We prove that, if an optimal policy is indeed monotone, then the value function of the dynamic program is concave. This result entices us to construct a set of candidate policies that reveal the intuitive monotone behaviors. We show that the profit function is concave when any candidate policy is implemented. We further refine the candidate policy set by requiring the policy to be close enough to the truly optimal policy, and show that any refined policy can be computed efficiently. Certainly, the policy refinement reduces the feasible set of the decision space, which could result in a suboptimal profit. An extensive numerical experiment, however, suggests that the profit loss from implementing the refined policies is negligible.

To understand the impact of lost sales on system performance, we use a backorder system as a benchmark, under which the unmet demand can be fulfilled in the future at a cost. Our analysis suggests that the profit difference between a backorder system and a lost-sales system is larger when the inventory carrying cost gets higher and the shortfall penalty gets lower. In other words, retaining unsatisfied customers through backordering can be beneficial when overage is costly in relation to underage. Interestingly, the profit difference between the two models is not sensitive to either supply or demand uncertainty. Furthermore, inventory-based dynamic pricing, as opposed to static pricing, is more beneficial in the lost-sales system than in the backorder system.

The remainder of the article is organized as follows. In Section 2, we review the related literature and discuss the contribution of our study. Section 3 lays out the model formulation. The single-period version of the problem is treated in Section 4. In Section 5, we analyze the multi-period problem by discussing the properties of the profit function, constructing the potential policies, and evaluating the performance of the proposed policies. Section 6 concludes the study. The preliminary results of stochastic functions are relegated to the appendix, and the proof of formal results can be found in Online Appendix EC.1. The computational algorithm is described in Online Appendix EC.2 and supplemental numerical experiments are provided in Online Appendix EC.3. Throughout this article, we name monotonicity (i.e., increasing or decreasing) in the weak sense.

2 Literature Review

Coordinating inventory and pricing decisions is one of the most researched problems in the operations management literature. Whitin (1955) first formulates the price-setting newsvendor model, in which the demand distribution depends on the product price. Since then, many authors have analyzed this problem by assuming either an additive demand, in which the price affects only the location of the distribution (e.g., Whitin, 1955; Mills, 1959), or a multiplicative demand, in which the price affects only the scale of the distribution (e.g., Karlin and Carr, 1962; Monahan et al., 2004). Early developments on this problem are summarized by Petruzzi and Dada (1999).

The major difficulty in the single-period price-setting newsvendor problem lies in the fact that the profit function is, in general, not jointly concave in the ordering and pricing decisions. The focus is given to identifying the properties of the demand–price relationship that lead to the desired structure of the profit function so that an optimal solution can be computed efficiently. Kocabıyıkoǧlu and Popescu (2011) propose a measure, called the lost sales elasticity, which is essentially the price elasticity of the cumulative hazard rate of the demand distribution. They show that the profit function is jointly concave if this measure is above a certain threshold. Lu and Simchi-Levi (2013) derive a general set of conditions for the family of location-scale demands (i.e., the price affects only the location and the scale of the demand distribution), under which the profit function is log-concave and thus the optimal solution can be easily computed. Feng et al. (2020) demonstrate that the conditions obtained by Kocabıyıkoǧlu and Popescu (2011) are restrictive, as they fail under multiplicative demands. Feng et al. (2020) also generalize the results obtained by Lu and Simchi-Levi (2013) for the case of general stochastic demand functions beyond the location-scale family.

Most of the studies on the multi-period extensions of this problem assume that the unmet demand can be fully backordered and a penalty is charged for unfilled demand at the end of the planning horizon (e.g., Federgruen and Heching, 1999; Feng et al., 2014). The advantage of the backorder model lies in the fact that the amount of backorder can be represented as the negative of the inventory level. As a result, the profit function is concave in the realized inventory level under general supply and demand functions (Feng and Shanthikumar, 2018). The shortcoming of the backorder models comes from the assumption that the customers pay the current price, though the demands are filled in the future when the price may be revised. Such an assumption may not be realistic.

When the unmet demand is lost, a situation observed in many practices, the problem becomes analytically challenging because the single-period profit is not jointly concave and neither is the profit-to-go function due to the potential shortage. Zabel (1972) analyzes a two-period version of the model with a deterministic supply and an additive demand that involves a uniformly or exponentially distributed noise. While he is able to identify conditions that involve the mean demand function, holding cost and feasible prices, to ensure a unique solution for additive demands, his analysis does not go too far for multiplicative demands. Thowsen (1975) focuses on additive demands in a nonstationary system. The conditions characterized contain the profit-to-go function, which makes them difficult to verify. Pang (2011) attempts to extend the analysis by Kocabıyıkoǧlu and Popescu (2011) for multi-period settings with additive demands. In summary, the analysis of the optimal solution for the multi-period model is mostly limited to the additive demands.

In view of the difficulty of the multi-period problem, several authors restrict the attention to asymptotes of the model. For example, Polatoglu and Sahin (2000), Chen et al. (2006), Huh and Janakiraman (2008) analyze long-term stationary settings of the problem under a given class of policies. Bu et al. (2020) analyze the performance of the constant-ordering policies and establish the asymptotic optimality of such a policy when the procurement lead time becomes long.

Instead of looking for likely restrictive optimality conditions or focusing on limiting behaviors, Feng et al. (2020) propose to restrict the solution space within a class of intuitively appealing policies in order to address the analytical difficulty of the problem. By restricting to decreasing price paths, they show that the dynamic optimization problem is tractable for a wide class of stochastic demand functions. However, the constructed policy, relying heavily on monotonicity along a single dimension, cannot be applied to our problem with stochastic supply. To deal with supply uncertainty, we need to develop an appropriate approximation of the expected profit and construct policies along three dimensions.

In an essential contrast to the aforementioned studies, the consideration of supply uncertainty leads to a random post-order inventory position. Consequently, the usual trick of dimension reduction by transforming the ordering decision to the order-up-to level does not work any longer. The matching of supply and demand becomes subtle due to the uncertainties from both. As our analysis suggests, we need to modify the development made by Feng et al. (2020) to deal with the single-period version of our model. But such modification would not work for the multi-period setting. Instead, we need a new approach of profit approximation and policy construction.

There is an extensive study on the effect of supply uncertainty in inventory–pricing systems (e.g., Li and Zheng, 2006; Federgruen and Yang, 2008; Feng, 2010; Chen et al., 2013). All of the existing studies assume backordering. The reader is referred to a recent survey by Feng and Shanthikumar (2022) for references therein. Feng and Shanthikumar (2018) show that the problem with backordering is tractable under general supply and demand functions that are stochastically linear in midpoint. It is generally optimal to follow an almost threshold policy (Feng and Shi, 2012; Federgruen et al., 2022). To our knowledge, this study is the first to consider supply uncertainty in a dynamic planning problem facing lost sales.

In terms of methodology, we adopt the notions of stochastic functions (Feng and Shanthikumar, 2022). These notions have been applied to study inventory-based pricing strategies (Feng and Shanthikumar, 2018; Feng et al., 2020) and sourcing strategies under uncertain supplies (Feng and Shanthikumar, 2018; Feng et al., 2019, 2021; Federgruen et al., 2022). In these studies with demand backorder or deterministic supply, the properties of stochastic linearity and concavity are sufficient to ensure a well-behaved profit function and the almost threshold structure of the optimal policy. For our problem involving both supply uncertainty and lost sales, the profit function does not have a concave transformation and the optimal policy reveals a complex structure. In addition to utilizing the properties of the stochastic functions, we need to examine the interaction between supply uncertainty and inventory shortfall in order to design an efficient policy, leading to new developments of the solution approach.

3 Problem Description

We consider a dynamic planning problem over a planning horizon of $T$ periods. The firm replenishes stock of a certain product from a supplier. Both the demands and the supplies of the product are random. When an order of $q$ units is placed at the beginning of period $t\in \{1,2,\dots ,T\}$ , the supplier delivers a random amount of ${\tilde{S}}_t(q)$ at the end of that period. The demand occurs during period $t$ , denoted by ${\tilde{D}}_t(p)$ , is a random quantity depending on the price $p$ chosen from some feasible set $[\underset{\_}{p},\overline{p}]$ . The dynamics of the system are depicted in Figure 1.

Figure 1.

The dynamics of the system.

We assume that the stochastic functions $\{{\tilde{S}}_t(q),q\ge 0\}$ and $\{{\tilde{D}}_t(p),\underset{\_}{p}\le p\le \overline{p}\}$ are mutually independent. Though our analysis can be easily extended to deal with the situation where $\{{\tilde{S}}_t(\cdot ),t=1,2,\dots ,T\}$ and $\{{\tilde{D}}_t(\cdot ),t=1,2,\dots ,T\}$ are nonstationary or Markov modulated, we assume that both processes are independent and identical across periods for ease of exposition. We may thus drop the subscript $t$ when it does not create confusion.

The unmet demand is lost and the leftover inventory is carried over to the next period. Thus, the dynamics of the inventory follows:

$$X_{t+1}=(x_t-\tilde{D}(p){)}^{+}+\tilde{S}(q),$$

(1)

where $x_t\ge 0$ is the inventory available at the beginning of period $t$ and $x^{+}=max\{0,x\}$ .

For each unit of product delivered by the supplier, the firm pays $c$ . It costs $h$ to carry one unit of product for one period, and a penalty $b$ is incurred for each unit of unmet demand. We denote $H(x)=h{x}^{+}+b{x}^{-}$ , where $x^{-}=max\{0,-x\}$ . When the on-hand inventory is $x$ at the beginning of period $t$ , and the firm orders $q$ units from the supplier and sets a price $p$ for period $t$ , the expected profit is

$$\begin{array}{rl}{J}_t^0(x,q,p)& =p\mathsf{E}[\tilde{D}(p)\wedge x]-c\mathsf{E}[\tilde{S}(q)]-\mathsf{E}[H(x-\tilde{D}(p))]\\ & +\rho \mathsf{E}[V_{t+1}^0((x_t-\tilde{D}(p){)}^{+}+\tilde{S}(q))],\end{array}$$

(2)

where

$$V_t^0(x)=max\{J_t^0(x,q,p):q\ge 0,\underset{\_}{p}\le p\le \overline{p}\}$$

(3)

and $\rho \in [0,1]$ is the discount factor. We assume a concave terminal value $V_{T+1}^0(x),x\ge 0$ . We should note that the consideration of nonstationary parameters $(c,h,b,\rho )$ does not increase the complexity of the analysis.

The above problem formulation can be seen as an extension of the price-inventory model studied by many (e.g., Federgruen and Heching, 1999; Feng et al., 2014). Two features of this formulation, namely, lost sales and supply uncertainties, impose new challenges to this problem that are not addressed in the previous studies.

We make the following assumptions of the demand and supply functions:

• A1.

$\mathsf{E}[\tilde{S}(q)]$ is finite and increasing in $q\ge 0$ with $\tilde{S}(0)=0$ almost surely.

A2.

$\mathsf{E}[\tilde{D}(p)]$ is finite and decreasing in $p\in [\underset{\_}{p},\overline{p}]$ .

These are the minimal conditions required in all studies involving random supplies and/or random demands, and they do not imply stochastic monotonicity of the supply and demand functions. With these assumptions, we can define the inverses of the mean supply and mean demand, respectively, as $q({\mu }_S)=inf\{q\ge 0:\mathsf{E}[\tilde{S}(q)]\ge {\mu }_S\}$ and $p({\mu }_D)=sup\{p\in [\underset{\_}{p},\overline{p}]:\mathsf{E}[\tilde{D}(p)]\ge {\mu }_D\}$ . Also, denote ${\overline{\mu }}_S=\tilde{S}(\mathrm{\infty })$ , ${\underset{\_}{\mu }}_D=\mathsf{E}[\tilde{D}(\overline{p})]$ , and ${\overline{\mu }}_D=\mathsf{E}[\tilde{D}(\underset{\_}{p})]$ . Then, we may express the supply and demand functions as

$$\begin{array}{rl}S({\mu }_S)& =\tilde{S}(q({\mu }_S)),{\mu }_S\in [0,{\overline{\mu }}_S]\text{ and }\\ D({\mu }_D)& =\tilde{D}(p({\mu }_D)),{\mu }_D\in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D].\end{array}$$

(4)

It is straightforward to see that the expected supply and demand functions are linear in the transformed decisions. In our analysis below, we may transform the order quantity $q$ to ${\mu }_S$ , the supply decision, and the price $p$ to ${\mu }_D$ , the demand decision, whenever such transformations simplify the derivations. When the unmet demand is backordered, the development of Feng and Shanthikumar (2018) suggests that the above decision transformations are sufficient to make the dynamic decision-making problem concave, as long as the supply and demand functions are stochastically linear in midpoint. With lost sales, however, this is not true and the problem remains intractable. Thus, additional development is needed.

The analysis of the problem relies heavily on the notions of stochastic orders and stochastic functions. We introduce the concepts needed for our analysis below in Definition 1, while discussing their properties in the appendix. Throughout the article, we use the capital letter, say, $X$ , to denote a random variable and the corresponding lowercase letter, $x$ , for its realization. We denote $F_X$ and ${\overline{F}}_X$ as the cumulative distribution and the survival function, respectively, associated with random variable $X$ .

Definition 1 Stochastic Functions

A stochastic function $\{X(y),y\in \mathcal{Y}\}$ with $\mathcal{Y}\subseteq \mathbb{R}$ is

(i)

stochastically linear in midpoint, written as $SL(mp)$ , if for any $y_1,y_2\in \mathcal{Y}$ , there exist random variables $X_1$ and $X_2$ defined on the common probability space such that $X_i{=}^{d}X(y_i),i=1,2$ , and $(X_1+X_2)/2{\le }_{cv}X((y_1+y_2)/2)$ ;

(ii)

stochastically concave in midpoint, written as $SCV(mp)$ , if for any $y_1,y_2\in \mathcal{Y}$ , there exist random variables $X_1$ and $X_2$ defined on the common probability space such that $X_i{=}^{d}X(y_i),i=1,2$ , and $(X_1+X_2)/2{\le }_{icv}X((y_1+y_2)/2)$ ; and

(iii)

stochastically decreasing in the increasing concave order, written as $SD(icv)$ , if for any $y_1,y_2\in \mathcal{Y}$ with $y_1\le y_2$ , $X(y_1){\ge }_{icv}X(y_2)$ .

Some basic properties of stochastic functions are discussed in the appendix, which are essential for our subsequent derivations.

4 The Single-Period Problem

In this section, we examine the single-period version of the model. This version is interesting in its own right as a vast literature is devoted to analyzing the price-setting newsvendor problem, a special case of our model with unlimited deterministic supply. To deal with the analytical challenge of the single-period problem, we focus our attention on the class of monotone policies, under which the supply decision and the pricing decision move in opposite directions. Indeed, it is a common practice that firms reduce product prices when inventories are piling up (e.g., Byrnes, 2003; Gimpl-Heersin et al., 2008). In the analysis below, we first prove that the single-period profit function is well-behaved if the optimal policy is indeed monotone. When it is not, we construct a set of candidate monotone policies, with which the resulting profit function is well-behaved. Later in Section 5.3, we will demonstrate that, though the optimal policy may not always be monotone, the restriction to the constructed policies does not lead to a significant profit loss.

When there is only one random demand to be met, the objective is to choose a supply decision, represented by the mean supply ${\mu }_S$ (or, equivalently, an order $q$ ), and a price $p$ that maximizes the expected one-period profit. There are two ways to formulate a single-period model in the literature, depending on the assumption of decision flexibility. When the supply lead time is long and the pricing policy can be quickly implemented, one may assume that the product price is set after the supply is observed. Such a situation, often termed responsive pricing in the literature, can be analyzed using an approach that is similar to that of price-independent demand, a special case of the model treated by Feng and Shanthikumar (2018), because the profit function under any realized supply is concave in the responsive price. We, instead, focus on the other way of modeling, in which the pricing decision is made before the supply is observed. Such a situation arises when the implementation of a pricing policy requires significant effort (e.g., a uniform price across the retail channels to avoid price matching, or manual changes of price tags on individual items). Analysis of the model below also sheds light on the multi-period policy design.

Specifically, under a supply decision ${\mu }_S$ and a price decision $p$ , the expected one-period profit is

$$\begin{array}{rl}J({\mu }_S,p)& =p\mathsf{E}[\tilde{D}(p)\wedge S({\mu }_S)]-c{\mu }_S-h\mathsf{E}[(S({\mu }_S)-\tilde{D}(p){)}^{+}]\\ & =(p+h)\mathsf{E}[\tilde{D}(p)\wedge S({\mu }_S)]-(c+h){\mu }_S.\end{array}$$

(5)

Here $h$ is the per-unit disposal cost (if positive) or the per-unit salvage value (if negative) of the unsold inventory. For ease of exposition, we have skipped the penalty for the unmet demand, as the analysis with a positive lost-sales penalty can be carried out in the same way.

The profit function $J$ defined in (5) is not well-behaved even in the special case of deterministic supply, that is, $S({\mu }_S)={\mu }_S$ (see, e.g., Lu and Simchi-Levi, 2013). In particular, the expected revenue $\varphi (p,{\mu }_S)=min\{p\mathsf{E}[\tilde{D}(p)],p{\mu }_S\}$ is not concave because $p{\mu }_S$ is neither concave nor convex. However, if $\hat{p}(\cdot )$ is a decreasing function, then $\hat{p}({\mu }_S){\mu }_S$ is concave in ${\mu }_S$ , and thus $\varphi (\hat{p}({\mu }_S),{\mu }_S)$ is concave in ${\mu }_S$ . In other words, the revenue is concave in ${\mu }_S$ along the direction of any decreasing price path $\hat{p}(\cdot )$ . This observation, made by Feng et al. (2020), leads to a solution for the problem with deterministic supply. When the supply is uncertain, however, their analysis does not apply to general stochastic supply functions. Instead, we establish that the profit function is concave if the optimal supply decision is decreasing in the price, as stated in the next theorem.

Theorem 1

Suppose that

• (C1)

the demand satisfies one of the following conditions:

• (C1-a)

$\{(p+h)\tilde{D}(p),p\in [\underset{\_}{p},\overline{p}]\}\in SCV(mp)$ , or

(C1-b)

$\mathsf{E}[\tilde{D}(p)]$ is concave and $\{D({\mu }_D):{\mu }_D\in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D]\}\in SL(mp)\cup SD(icv)$ , and

(C2)

the supply satisfies $\{S({\mu }_S),{\mu }_S\in [0,{\overline{\mu }}_S]\}\in SL(mp)$ .

If ${\hat{\mu }}_S(p)\in \arg max\{J({\mu }_S,p),{\mu }_S\in [0,{\overline{\mu }}_S]\}$ is decreasing in $p$ , then $J({\hat{\mu }}_S(p),p)$ is concave in $p$ .

Feng et al. (2020) proved that condition (C1-b) implies condition (C1-a). Each condition is satisfied by a wide class of demand functions. Feng and Shanthikumar (2018, 2022) showed that condition (C2) covers almost all the supply functions analyzed so far in the existing literature. Thus, the result in Theorem 1 is established for general demand and supply functions, when it is optimal to set a high mean supply for a low price.

The condition of a decreasing ${\hat{\mu }}_S(\cdot )$ in Theorem 1 may or may not hold in general. Our next task is to develop an efficient approach that verifies this condition, and produces a solution when this condition fails. For any supply path $\mu :[\underset{\_}{p},\overline{p}]\to [0,{\overline{\mu }}_S]$ , which may or may not be monotone, define

$$\begin{array}{rl}\mathcal{D}[\mu ]& =\{p\in [\underset{\_}{p},\overline{p}]:\mu (p-\delta )>\mu (p+\delta ),\\ & \mathrm{\forall }\delta \in (0,min\{p-\underset{\_}{p},\overline{p}-p\})\},\text{ and }\\ \mathcal{C}[\mu ]& =\{p\in [\underset{\_}{p},\overline{p}]:\mu (p-\delta )=\mu (p+\delta ),\\ & \mathrm{\forall }\delta \in (0,\mathrm{\Delta })\text{ for some }\mathrm{\Delta }>0\}.\end{array}$$

The set $\mathcal{D}[\mu ]$ contains all the price points at which the supply path $\mu$ is strictly decreasing, and the set $\mathcal{C}[\mu ]$ contains all the price points at which the supply path $\mu$ is constant. Clearly, the two sets are disjoint, that is, $\mathcal{D}[\mu ]\cap \mathcal{C}[\mu ]=\mathrm{\varnothing }$ . If the two sets together contain all feasible prices, that is, $\mathcal{D}[\mu ]\cup \mathcal{C}[\mu ]=[\underset{\_}{p},\overline{p}]$ , then the supply path $\mu$ must be decreasing. We are interested in decreasing supply paths. However, the set of all decreasing supply paths is large, and, ideally, we would like to focus on those that are “close” to the optimal supply path and are easily identifiable. In particular, we define the following class of decreasing supply paths:

$$\begin{array}{rl}{\mathcal{U}}_0& =\{\mu :[\underset{\_}{p},\overline{p}]\to [0,{\overline{\mu }}_S]|\mathcal{D}[\mu ]\cup \mathcal{C}[\mu ]=[\underset{\_}{p},\overline{p}];\\ & \mu (p)={\hat{\mu }}_S(p)\text{ for }p\in \mathcal{D}[\mu ]\}.\end{array}$$

A supply path $\mu$ in the set ${\mathcal{U}}_0$ is certainly decreasing. Whenever $\mu$ is strictly decreasing at some $p$ , its value must coincide with the optimal supply path ${\hat{\mu }}_S(p)$ (as defined in Theorem 1). Otherwise, $\mu$ stays constant. Thus, the paths in ${\mathcal{U}}_0$ are closely related to the optimal supply path because each path in ${\mathcal{U}}_0$ , whenever strictly decreasing, overlaps with the optimal supply path.

However, the set ${\mathcal{U}}_0$ contains some obviously undesired supply paths. For example, a decision that sets a zero supply regardless of the price (i.e., $\mu (p)=0,p\in [\underset{\_}{p},\overline{p}]$ ) belongs to ${\mathcal{U}}_0$ . We would like to remove such elements from ${\mathcal{U}}_0$ and focus only on the candidate supply paths that capture the overall trend of the optimal path. For that, we define ${\mathcal{U}}^L=\{\mu \in {\mathcal{U}}_0:\mu (p)\le {\hat{\mu }}_S(p),p\in [\underset{\_}{p},\overline{p}]\}$ and ${\mathcal{U}}^U=\{\mu \in {\mathcal{U}}_0:\mu (p)\ge {\hat{\mu }}_S(p),p\in [\underset{\_}{p},\overline{p}]\}$ , respectively, as decreasing supply paths that are always below and above the optimal one. The lower and upper boundary paths are then defined, respectively, as for each $p\in [\underset{\_}{p},\overline{p}]$ ,

$$\begin{array}{rl}{\mu }^L(p)& =sup\{\mu (p):\mu \in {\mathcal{U}}^L\}\text{ and }\\ {\mu }^U(p)& =inf\{\mu (p):\mu \in {\mathcal{U}}^U\}.\end{array}$$

Now we can focus on the supply paths bounded by ${\mu }^L$ and ${\mu }^U$ by defining the candidate set

$$\mathcal{U}=\{\mu \in {\mathcal{U}}_0:{\mu }^L(p)\le \mu (p)\le {\mu }^U(p),p\in [\underset{\_}{p},\overline{p}]\}.$$

The next theorem characterizes important properties of the set $\mathcal{U}$ , which ensure not only analytical tractability but also profit performance within the class of decreasing supply paths.

Theorem 2

The set $\mathcal{U}$ of candidate supply paths satisfies the following properties.

• (i)

For any ${\mu }_S\in \mathcal{U}$ , $J({\mu }_S(p),p)$ is concave in $p$ when conditions (C1) and (C2) in Theorem 1 hold.

(ii)

If ${\hat{\mu }}_S$ is decreasing, that is, $\mathcal{D}[{\hat{\mu }}_S]\cup \mathcal{C}[{\hat{\mu }}_S]=[\underset{\_}{p},\overline{p}]$ , then set $\mathcal{U}$ contains only ${\hat{\mu }}_S$ , that is, $\mathcal{U}=\{{\hat{\mu }}_S\}$ .

(iii)

Suppose ${\mu }_0:[\underset{\_}{p},\overline{p}]\to [0,{\overline{\mu }}_S]$ is a decreasing function and ${\mu }_0\notin \mathcal{U}$ . There exists a ${\mu }_S\in \mathcal{U}$ such that $J({\mu }_0(p),p)\le J({\mu }_S(p),p)$ for any $p\in [\underset{\_}{p},\overline{p}]$ .

According to Theorem 2(i), the profit as a function of the price is concave along any candidate supply path in set $\mathcal{U}$ . Thus, the optimal price along each supply path is essentially unique and can be computed. In general there can be infinitely many supply paths in $\mathcal{U}$ . However, when the optimal supply path ${\hat{\mu }}_S(p),p\in [\underset{\_}{p},\overline{p}],$ is indeed decreasing, Theorem 2(ii) ensures that set $\mathcal{U}$ contains only the optimal supply path, and thus restricting the decision space to $\mathcal{U}$ does not hurt profit performance. When the optimal supply path is not decreasing, the set $\mathcal{U}$ contains a collection of decreasing supply paths that dominates those outside of the set, as implied by Theorem 2(iii).

The result in Theorem 2 is in the similar spirit to that developed by Feng et al. (2020) for the special case of our model with a deterministic supply. This result, however, does not extend to the multi-period setting. As our discussion unfolds in the next section, it becomes clear that the presence of supply uncertainty prevents the dimension reduction used by Feng et al. (2020). Thus, a new development is needed to treat the multi-dimensional decision space for the multi-period model. To facilitate the understanding of our later development, we take an alternative view of the candidate supply path in $\mathcal{U}$ . Specifically, define, for $\check{p}\in [\underset{\_}{p},\overline{p}]$ ,

$${\mu }_S^{\check{p}}(p)=\{\begin{array}{cc}\arg max\{J({\mu }_S,p)|{\mu }_S\in [0,{\overline{\mu }}_S],& \\ {\mu }_S\ge \underset{\tilde{p}\in (p,\check{p}]}{sup}{\mu }_S^{\check{p}}(\tilde{p})\},& \underset{\_}{p}\le p<\check{p},\\ {\hat{\mu }}_S(p),& p=\check{p},\\ \arg max\{J({\mu }_S,p)|{\mu }_S\in [0,{\overline{\mu }}_S],& \\ {\mu }_S\le \underset{\tilde{p}\in [\check{p},p)}{inf}{\mu }_S^{\check{p}}(\tilde{p})\},& \check{p}<p\le \overline{p}.\end{array}$$

Thus, a path ${\mu }_S^{\check{p}}(\cdot )$ is constructed by first finding the optimal supply decision at the chosen price, $\check{p}$ . Then, we increase (decrease) the price, while maintaining the local monotonicity of the supply decision. As stated in the next lemma, each ${\mu }_S^{\check{p}}(\cdot )$ belongs to the candidate set $\mathcal{U}$ because whenever ${\mu }_S^{\check{p}}(\cdot )$ is strictly decreasing it coincides with the optimal path. By construction, the path ${\mu }_S^{\check{p}}(\cdot )$ attempts to trace the optimal path as much as possible in view of the local monotonicity constraint. This alternative view of the candidate policies will be expanded for the multi-period problem.

Lemma 1 Policy Construction

$\{{\mu }_S^{\check{p}}(\cdot ),\check{p}\in [\underset{\_}{p},\overline{p}]\}\subseteq \mathcal{U}$ .

The advantage of the policy ${\mu }_S^{\check{p}}(p)$ lies in the fact that it can be computed efficiently through a single-dimensional search over the range $[\underset{\_}{p},\overline{p}]$ , as it is monotone. In particular, we start by computing the supply decision ${\mu }_{S0}=\arg max\{J({\mu }_S,p_0),0\le {\mu }_S\le {\overline{\mu }}_S\}$ for the price $p_0=\check{p}$ . Then, we compute the supply decisions over $[\underset{\_}{p},\check{p}]$ and $[\check{p},\overline{p}]$ by progressively moving toward the left and right, respectively. Specifically, over $[\underset{\_}{p},\check{p}]$ , we set $p_j=p_j-\delta$ and ${\mu }_{Sj}=\arg max\{J({\mu }_S,p_j),{\mu }_S^{\check{p}}(p_j+\delta )\le {\mu }_S\le {\overline{\mu }}_S\}$ in step $j$ , where $\delta >0$ is the computational accuracy. We proceed to the next step until $p_j$ reaches $\underset{\_}{p}$ . Over $[\check{p},\overline{p}]$ , we set $p_j=p_j+\delta$ and ${\mu }_{Sj}=\arg max\{J({\mu }_S,p_j),0\le {\mu }_S\le {\mu }_S^{\check{p}}(p_j-\delta )\}$ in step $j$ until we reach $\overline{p}$ . It is easy to see that by the time we complete the computation of ${\mu }_S^{\check{p}}(p),\underset{\_}{p}\le p\le \overline{p}$ , the range of $[0,{\overline{\mu }}_S]$ is traversed exactly once during the entire process. Thus, the computational complexity does not exceed that of a single-dimensional search. Furthermore, it is easy to see that the boundary supply paths ${\mu }^L(p)$ and ${\mu }^U(p)$ coincide with ${\mu }_S^{\underset{\_}{p}}(p)$ and ${\mu }_S^{\overline{p}}(p)$ . Thus, they can also be computed efficiently.

5 The Multi-Period Problem

In the context of multi-period planning, the problem formulated in equations (2) and (3) is known to be challenging because neither the instantaneous profit nor the profit-to-go function is well behaved. For the special case of our model with unlimited and deterministic supply, Feng et al. (2020) address the second challenge by deriving an upper bound of the marginal profit. Specifically, the saving of carrying inventory to the next period is capped by the discounted purchase cost and thus by any feasible price. With this property, the profit-to-go function can be transformed into a jointly concave function by applying the notions of stochastic functions. This approach, however, does not work in general with random supply. To see that, we take a one-period example and plot the value function $V_1(x)$ in the left panel of Figure 2. We observe that the marginal value ${\mathrm{\partial }}^{+}{V}_1(0)=\underset{\mathrm{\Delta }\downarrow 0}{lim}(V_1(\mathrm{\Delta })-V_1(0))/\mathrm{\Delta }$ of inventory is 2, which is higher than the procurement cost $c=1$ . If we were to apply the trick of Feng et al. (2020), we must restrict the feasible price to be above ${\mathrm{\partial }}^{+}{V}_1(0)=2$ . In other words, the gross margin ( $(p-c)/p$ ) of the product must be above 50%, which makes analysis inapplicable for most low-margin products. Therefore, we need to develop a new approach.

Figure 2.

An example with non-concave value function and non-monotone optimal policy. Notes. $T=1$ , $\rho =0.9$ , $c=1$ , $h=b=V_2(\cdot )=0$ , $\tilde{D}(p)=Z_D+(z_0-Z_D{)}^{+}\cdot \frac{\overline{p}-p}{\overline{p}-\underset{\_}{p}}$ for $p\in [\underset{\_}{p},\overline{p}]$ with $\underset{\_}{p}=1$ and $\overline{p}=2$ and $\tilde{S}(q)=q\wedge Z_S$ , where both $Z_D$ and $Z_S$ follow Bernoulli distribution with mean $0.5$ . The value function satisfies ${\mathrm{\partial }}^{+}{V}_1(0)=\underset{\mathrm{\Delta }\downarrow 0}{lim}(V_1(\mathrm{\Delta })-V_1(0))/\mathrm{\Delta }=2$ .

5.1 A Generalized Formulation

When there is no guarantee of receiving the ordered amount from the supplier, the decision maker would face the risk of potential shortage and revenue loss if the inventory level becomes low. It thus makes sense to think of rationing the available stock between meeting the immediate demand and hedging for future shortage. Such a ration would allow the decision maker to best leverage the dynamics of the inventory and pricing strategy to cope with the supply uncertainty. Therefore, we introduce a lever, $z_0$ , as the amount of stock to be reserved for future demand. Then, the available stock satisfies $x=z+z_0$ , where $z$ is the amount of stock allocated to meet the current demand. With this consideration, we can reformulate the multi-period problem as

$$\begin{array}{rl}{V}_t(x)& =max\{J_t(x,z,{\mu }_S,{\mu }_D):z\in [0,x],{\mu }_S\in [0,{\overline{\mu }}_S],\\ & {\mu }_D\in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D]\},\end{array}$$

(6)

where

$$\begin{array}{rl}{J}_t(x,z,{\mu }_S,{\mu }_D)& =p({\mu }_D)\mathsf{E}[z\wedge D({\mu }_D)]-c{\mu }_S\\ & -h\mathsf{E}[x-z\wedge D({\mu }_D)]\\ & -b\mathsf{E}[D({\mu }_D)-D({\mu }_D)\wedge z]\\ & +\rho \mathsf{E}[V_{t+1}(x-z\wedge D({\mu }_D)+S({\mu }_S))].\end{array}$$

(7)

It is straightforward to see that this model generalizes that formulated in (2) and (3), because the former reduces to the latter by setting $z=x$ . Moreover, because of inventory rationing across periods, the optimal profit obtained from (6) and (7) should be higher than that from (2) and (3). An extensive numerical analysis suggests that the difference in profit performance between the two formulations is generally negligible; see the details in Online Appendix EC.3.1.

Lemma 2 Componentwise Concavity

Suppose that

(C1)

$\mathsf{E}[\tilde{D}(p)]$ is decreasing and concave, and $\{D({\mu }_D):{\mu }_D\in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D]\}\in SL(mp)$ , and

(C2)

$\{S({\mu }_S),{\mu }_S\in [0,{\overline{\mu }}_S]\}\in SL(mp)$ .

If $V_{t+1}$ is concave, then the following results hold.

• (i)

  For given $z$ , $J_t(x,z,{\mu }_S,{\mu }_D)$ is jointly concave in $(x,{\mu }_S,{\mu }_D)$ .

• (ii)

  Suppose for every fixed $(x,{\mu }_S)$ , $\hat{z}(\mu ;x,{\mu }_S)\in \arg max\{J_t(x,z,{\mu }_S,\mu ):\mu \in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D]\}$ is increasing in $\mu$ . Then $J_t(x,\hat{z}({\mu }_D;x,{\mu }_S),{\mu }_S,{\mu }_D)$ is concave in ${\mu }_D$ .

Lemma 2(i) underscores the complexity of the problem due to the interaction between the pricing decision ${\mu }_D$ and the inventory decision $z$ through the term $p\mathsf{E}[\tilde{D}(p)\wedge z]$ (equivalently, by (4), $p({\mu }_D)\mathsf{E}[D({\mu }_D)\wedge z]$ ). The profit function is jointly concave in $(x,{\mu }_S,{\mu }_D)$ , but is not jointly concave once $z$ is not fixed. Lemma 2(ii) further suggests that, if the optimal inventory decision $\hat{z}$ is increasing in the mean demand $\mu$ , then along the direction of the optimal $\hat{z}$ , the profit function is jointly concave in $(x,{\mu }_S,{\mu }_D)$ . These observations provide a useful guide to designing monotone policies.

5.2 Policy Design

Given that we are facing a dynamic decision-making process, the needed structure of the value function (i.e., the concavity of $V_t$ ) must be preserved over time to ensure the tractability of the problem. With the observations from Lemma 2, we can characterize the properties of the optimal solution, under which the value function preserves concavity in the dynamic programing iterations.

Theorem 3 Monotone Optimal Policy

Suppose that

(C1)

$\mathsf{E}[\tilde{D}(p)]$ is decreasing and concave, and $\{D({\mu }_D):{\mu }_D\in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D]\}\in SL(mp)$ , and

(C2)

$\{S({\mu }_S),{\mu }_S\in [0,{\overline{\mu }}_S]\}\in SL(mp)$ .

If the optimal inventory decision $z^{*}(x)$ and demand decision ${\mu }_D^{*}(x)$ are increasing in $x$ , then $V_t$ is concave whenever $V_{t+1}$ is concave, where $(z^{*}(x),{\mu }_S^{*}(x),{\mu }_D^{*}(x))\in \arg max\{J_t(x,z,{\mu }_S,{\mu }_D):z\in [0,x],{\mu }_S\in [0,{\overline{\mu }}_S],{\mu }_D\in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D]\}$ .

According to Theorem 3, the value function of the dynamic programing equation is concave for general stochastic demand and supply functions if the optimal inventory decision and the optimal demand decision are monotone in the on-hand inventory level. This monotone property, however, is difficult to verify as it is endogenous, and may not hold in general. Any easy-to-check sufficient condition for the optimal solution to exhibit the monotone property appears restrictive even in the special case when the supply is deterministic (see the discussions by Feng et al., 2020). Thus, we needs to develop an approach to verify whether or not the optimal solution is indeed monotone to apply Theorem 3. In general, the optimal policy may not be monotone (evident from the example in Figure 2). In such a situation, we need to derive an easy-to-implement solution that produces good performance. Our analysis below devices a solution approach that verifies the optimality of a monotone policy, and generates an efficient solution when the optimal policy fails to be monotone.

To enable the analysis of the dynamic program, the value function needs to be concave in the on-hand inventory level so that decisions can be computed efficiently and the desired property of the objective function is retained across periods. Theorem 3 suggests that when a candidate policy exhibits a certain monotone structure, the resulting profit function can preserve concavity to enable tractability.

Specifically, we would like to focus on a class of policies, say $\{{\mathcal{P}}_C^t,t=1,2,\dots ,T\}$ , so that implementing policies within this class in every period would lead to a concave profit function. Our goal in the analysis below is to characterize the class ${\mathcal{P}}_C^t$ . In light of Theorem 3, the identified class ${\mathcal{P}}_C^t$ should be within the class of monotone policies:

$$\begin{array}{rl}{\mathcal{P}}_{inc}& =\{(z,{\mu }_S,{\mu }_D)|z:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+},\\ & {\mu }_S:{\mathbb{R}}_{+}\to [0,{\overline{\mu }}_S],{\mu }_D:{\mathbb{R}}_{+}\to [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D];\\ & z(x)\le x,z(x)\le z(x+\delta ),{\mu }_D(x)\le {\mu }_D(x+\delta ),\\ & x\in [0,\mathrm{\infty }),\mathrm{\forall }x\ge 0\text{ and }\mathrm{\forall }\delta >0\}.\end{array}$$

(8)

Suppose we will follow some policies $(z^j,{\mu }_S^j,{\mu }_D^j)$ within some ${\mathcal{P}}_C^j$ from period $j=t+1$ to period $j=T$ , and will obtain the expected profit

$$\begin{array}{rl}{\hat{V}}_{t+1}(x_{t+1})& =\mathsf{E}[\sum _{j=t+1}^T(p(D({\mu }_D^j(X_j))\wedge z^j(X_j))-c{\mu }_S^j(X_j)\\ & -h(X_j-z^j(X_j)\wedge D({\mu }_D^j(X_j)))-b(D({\mu }_D^j(X_j))\\ & -D({\mu }_D^j(X_j))\wedge z^j(X_j)))|X_{t+1}=x_{t+1}],\end{array}$$

with the inventory dynamics $X_{j+1}=X_j-z^j(X_j)\wedge D({\mu }_D^j(X_j))+S({\mu }_S^j(X_j)),j=t+1,t+2,\dots ,T$ . When implementing a policy $(z,{\mu }_S,{\mu }_D)$ in period $t$ with the observed state $x$ , we obtain a profit of ${\hat{J}}_t(x,z(x),{\mu }_S(x),{\mu }_D(x))$ . Note that ${\hat{J}}_t$ and ${\hat{V}}_{t+1}$ follow the same relation of $J_t$ and $V_{t+1}$ as in the dynamic programing equation. That is,

$$\begin{array}{rl}{\hat{J}}_t(x,z,{\mu }_S,{\mu }_D)& =p({\mu }_D)\mathsf{E}[D({\mu }_D)\wedge z]-c{\mu }_S\\ & -h\mathsf{E}[x-z\wedge D({\mu }_D)]\\ & -b\mathsf{E}[D({\mu }_D)-D({\mu }_D)\wedge z]\\ & +\rho \mathsf{E}[{\hat{V}}_{t+1}(x-z\wedge D({\mu }_D)+S({\mu }_S))].\end{array}$$

(9)

Note that we do not impose any assumptions on ${\hat{V}}_{t+1}$ . Instead, we try to characterize the policy class ${\mathcal{P}}_C^t$ so that ${\hat{J}}_t$ and thus ${\hat{V}}_t$ are concave in each period $t$ . Let $({\hat{z}}^{*}(x),{\hat{\mu }}_S^{*}(x),{\hat{\mu }}_D^{*}(x))$ denote a currently optimal policy, that is, a maximizer of ${\hat{J}}_t(x,z,{\mu }_S,{\mu }_D)$ , $x\in [0,\mathrm{\infty })$ .

Unfortunately, not every element $(z,{\mu }_S,{\mu }_D)$ in ${\mathcal{P}}_{inc}$ would ensure the concavity of the resulting profit function ${\hat{J}}_t(x,z(x),{\mu }_S(x),{\mu }_D(x))$ . We need to identify an appropriate subclass within ${\mathcal{P}}_{inc}$ . Moreover, we would like the identified policy class to be close enough to the currently optimal policy $({\hat{z}}^{*}(x),{\hat{\mu }}_S^{*}(x),{\hat{\mu }}_D^{*}(x))$ so that the difference between ${\hat{J}}_t(x,z(x),{\mu }_S(x),{\mu }_D(x))$ and the currently optimal profit ${\hat{J}}_t(x,{\hat{z}}^{*}(x),{\hat{\mu }}_S^{*}(x),{\hat{\mu }}_D^{*}(x))$ is not too large. Based on the property of the policy identified in Theorem 3, we define, for $(z,{\mu }_S,{\mu }_D)\in {\mathcal{P}}_{inc}$ ,

$$\begin{array}{rl}{\text{I}}_c[z,{\mu }_D]& =\{x:z(x+\delta )>z(x-\delta )\text{ and }{\mu }_D(x+\delta )\\ & >{\mu }_D(x-\delta )\text{ for any }\delta >0\},\\ {\text{I}}_z[z,{\mu }_D]& =\{x:z(x+\delta )>z(x-\delta )\text{ for any }\delta >0,\\ & \text{ and }{\mu }_D(x+\delta )={\mu }_D(x-\delta )\text{ for some }\delta >0\},\\ {\text{I}}_{{\mu }_D}[z,{\mu }_D]& =\{x:z(x+\delta )=z(x-\delta )\text{ for some }\delta >0,\\ & \text{ and }{\mu }_D(x+\delta )>{\mu }_D(x-\delta )\text{ for any }\delta >0\}.\end{array}$$

Thus, ${\text{I}}_c[z,{\mu }_D]$ is the set of state $x$ , over which both the inventory decision $z$ and the demand decision ${\mu }_D$ are strictly increasing. The set ${\text{I}}_z[z,{\mu }_D]$ or ${\text{I}}_{{\mu }_D}[z,{\mu }_D]$ is, respectively, the collections of states over which the inventory decision or the demand decision is strictly increasing, but not both. We also define ${\text{I}}_w[z,{\mu }_D]={\text{I}}_c[z,\mu ]\cup {\text{I}}_z[z,{\mu }_D]\cup {\text{I}}_{{\mu }_D}[z,{\mu }_D]$ as the set of the states over which at least one of the inventory and demand decisions is strictly increasing. For a policy in ${\mathcal{P}}_{inc}$ , the decisions $(z,{\mu }_D)$ must stay constant for the remainder states, that is, $\text{C}[z,{\mu }_D]=[0,\mathrm{\infty })\mathrm{\setminus }{\text{I}}_w[z,{\mu }_D]$ .

Now define the class of candidate policies ${\mathcal{P}}_C^t\subset {\mathcal{P}}_{inc}$ , such that $(z,{\mu }_S,{\mu }_D)\in {\mathcal{P}}_C^t$ satisfies the following conditions:

• (P-1)

For $x\in {\text{I}}_c[z,{\mu }_D]$ , we have $(z(x),{\mu }_S(x),{\mu }_D(x))=({\hat{z}}^{*}(x),{\hat{\mu }}_S^{*}(x),{\hat{\mu }}_D^{*}(x))$ . That is, whenever both $z$ and ${\mu }_D$ are increasing, the candidate policy coincides with the currently optimal policy.

(P-2)

For $x\in \text{C}[z,{\mu }_D]$ , we have

• (2-a)

$z(x-\delta )=z(x+\delta )$ and ${\mu }_D(x-\delta )={\mu }_D(x+\delta )$ for some $\delta >0$ (i.e., both $z$ and ${\mu }_D$ are constant around the neighborhood of $x$ ), and

(2-b)

${\mu }_S(x)\in \arg max\{{\hat{J}}_t(x,z(x),{\mu }_S,{\mu }_D(x)):{\mu }_S\in [0,{\overline{\mu }}_S]\}$ .

(P-3)

For $x\in {\text{I}}_{{\mu }_D}[z,{\mu }_D]$ , we have

• (3-a)

$z(x-\delta )=z(x+\delta )$ for some $\delta >0$ (i.e., $z$ is constant around the neighborhood of $x$ ), and

(3-b)

$({\mu }_S(x),{\mu }_D(x))\in \arg max\{{\hat{J}}_t(x,z(x),{\mu }_S,{\mu }_D):{\mu }_S\in [0,{\overline{\mu }}_S],{\mu }_D\in [{\underset{\_}{\mu }}_D,{\overline{\mu }}_D]\}$ .

(P-4)

For $x\in {\text{I}}_z[z,{\mu }_D]$ ,

• (4-a)

${\mu }_D(x-\delta )={\mu }_D(x+\delta )$ for some $\delta >0$ (i.e., ${\mu }_D$ is constant around the neighborhood of $x$ ), and

(4-b)

$(z(x),{\mu }_S(x))\in \arg max\{{\hat{J}}_t(x,z,{\mu }_S,{\mu }_D(x)):z\in [0,x],{\mu }_S\in [0,{\overline{\mu }}_S]\}$ .

Thus, a candidate policy in ${\mathcal{P}}_C^t$ must have weakly increasing inventory and demand decisions (with respect to the state $x$ ), while maintaining local optimality of the decisions.

Theorem 4 Concavity Along Candidate Policies

Suppose that

(C1)

$\mathsf{E}[\tilde{D}(p)]$ is decreasing and concave, $\{D(\mu ),\mu \in [\underset{\_}{\mu },\overline{\mu }]\}\in SL(mp)$ , and

(C2)

$\{S({\mu }_S),{\mu }_S\in [0,{\overline{\mu }}_S]\}\in SL(mp)$ .

For any $(z(\cdot ),{\mu }_S(\cdot ),{\mu }_D(\cdot ))\in {\mathcal{P}}_C^t$ , ${\hat{V}}_t(x)={\hat{J}}_t(x,z(x),{\mu }_S(x),{\mu }_D(x)),x\in [0,\mathrm{\infty }),$ is concave if the policy implemented for period $j$ is within ${\mathcal{P}}_C^j,j\in \{t+1,t+2,\dots ,T\}$ .

According to Theorem 4, implementing any policy within the constructed set ${\mathcal{P}}_C^t$ leads to a concave profit function. The set ${\mathcal{P}}_C^t$ , however, can be large, and some policies within ${\mathcal{P}}_C^t$ maybe far away from the currently optimal policy $({\hat{z}}^{*},{\hat{\mu }}_S^{*},{\hat{\mu }}_D^{*})$ . For example, a policy with ${\mu }_D(x)=\underset{\_}{\mu }$ or ${\mu }_S(x)={\overline{\mu }}_S$ for all $x$ can satisfy the conditions for ${\mathcal{P}}_C^t$ . But such a policy can perform very poorly when ${\overline{\mu }}_S$ is much larger than $\underset{\_}{\mu }$ . Thus, we would like to further refine the candidate policy set. Inspired by Lemma 1, we would like the candidate policy to coincide with the currently optimal policy at some inventory level. At other inventory levels, the policy deviation is limited as local optimality is maintained by the candidate policy. Specifically, define a refined $\check{x}$ -policy as the following triple of functions over $x\in [0,\mathrm{\infty })$ :

$$\begin{array}{rl}& (z^{\check{x}}(x),{\mu }_S^{\check{x}}(x),{\mu }_D^{\check{x}}(x))\\ & =\{\begin{array}{ll}\arg max\{{\hat{J}}_t(x,z,{\mu }_S,{\mu }_D)|{\mu }_S\in [0,{\overline{\mu }}_S],& \\ z\le \underset{\tilde{x}\in (x,\check{x}]}{inf}{z}^{\check{x}}(\tilde{x}),{\mu }_D\le \underset{\tilde{x}\in (x,\check{x}]}{inf}{\mu }_D^{\check{x}}(\tilde{x})\},& x<\check{x},\\ ({\hat{z}}^{*}(x),{\hat{\mu }}_S^{*}(x),{\hat{\mu }}_D^{*}(x))& x=\check{x},\\ \arg max\{{\hat{J}}_t(x,z,{\mu }_S,{\mu }_D)|{\mu }_S\in [0,{\overline{\mu }}_S],& \\ z\ge \underset{\tilde{x}\in [\check{x},x)}{sup}{z}^{\check{x}}(\tilde{x}),{\mu }_D\ge \underset{\tilde{x}\in [\check{x},x)}{sup}{\mu }_D^{\check{x}}(\tilde{x})\},& x>\check{x}.\end{array}\end{array}$$

(10)

When there are multiple maximizers, we arbitrarily choose one. According to its definition in (10), the $\check{x}$ -policy coincides with the currently optimal policy at the inventory level $x=\check{x}$ . When the inventory level is below $\check{x}$ (i.e., $x<\check{x}$ ), the $\check{x}$ -policy maximizes the profit under the restriction that the inventory decision $z^{\check{x}}(x)$ and the demand decision ${\mu }_D^{\check{x}}(x)$ is not higher than their counterparts for any inventory level above $x$ . When the inventory level is above $\check{x}$ (i.e., $x>\check{x}$ ), the $\check{x}$ -policy achieves the maximum profit under the condition that $z^{\check{x}}(x)$ and ${\mu }_D^{\check{x}}(x)$ are the highest for any inventory level below $x$ . To compute $\check{x}$ -policy, we first find the currently optimal solution for inventory level $x=\check{x}$ as a starting point. Then, we gradually decrease (increase) the inventory level from $\check{x}$ by searching $(z,{\mu }_D)$ to ensure that they are below (above) the lowest (highest) values obtained so far. We use ${\mathcal{P}}_R^t$ to denote the collection of the refined policies, that is,

$${\mathcal{P}}_R^t=\{(z^{\check{x}},{\mu }_S^{\check{x}},{\mu }_D^{\check{x}}):\check{x}\ge 0\}.$$

To demonstrate the construction, we present an example in Figure 3. In the figure, we plot the inventory decisions and the demand decisions. The supply decisions are optimized for the corresponding inventory and demand decisions. The solid line corresponds to the currently optimal solutions ${\hat{z}}^{*}(x)$ and ${\hat{\mu }}_D^{*}(x)$ . At $x=\check{x}=0.6$ , the constructed policy (in thickened dash lines) coincides with the currently optimal policy, that is, $z^{0.6}(0.6)={\hat{z}}^{*}(0.6)$ and ${\mu }_D^{0.6}(0.6)={\hat{\mu }}_D^{*}(0.6)$ . This is the starting point for constructing the $0.6$ -policy. Then we extend the policy based on the first and the third cases in (10) by gradually decreasing and increasing the inventory, respectively. Specifically, we reduce the inventory level by moving toward the left of $\check{x}=0.6$ . Because the currently optimal ${\hat{\mu }}_D^{*}(x)$ is decreasing around this neighborhood, the constructed solution would set ${\mu }_D^{0.6}(x)={\mu }_D^{0.6}(0.6)$ for $x$ slightly below $\check{x}=0.6$ . Thus we see a deviation of ${\mu }_D^{0.6}(x)$ from the currently optimal solution, until $x$ further decreases so that the currently optimal ${\hat{\mu }}_D^{*}(x)$ coincides with ${\mu }_D^{0.6}(0.6)$ , when we can set ${\mu }_D^{0.6}(x)={\hat{\mu }}_D^{*}(x)$ again. Correspondingly, we see a deviation of $z^{0.6}(x)$ from ${\hat{z}}^{*}(x)$ when we move from $\check{x}=0.6$ to the left, even though ${\hat{z}}^{*}(x)$ is increasing. This is the nature of multiple dimensional optimization. The policy to the right of $\check{x}=0.6$ is constructed similarly.

Figure 3.

Demonstration of the constructed $\check{x}$ -policy. Notes: $t=1$ , $T=2$ , $\rho =1$ , $c=1$ , $h=b=V_2(\cdot )=0$ , $\tilde{D}(p)=Z_D+(z_0-Z_D{)}^{+}\cdot \frac{\overline{p}-p}{\overline{p}-\underset{\_}{p}}$ for $p\in [\underset{\_}{p},\overline{p}]$ with $\underset{\_}{p}=1$ , $\overline{p}=2$ , and $\tilde{S}(q)=q\wedge Z_S$ , where $Z_D$ and $Z_S$ follow Bernoulli distribution with mean $0.5$ and $0.01$ , respectively.

Figure 3 also demonstrates two other $\check{x}$ -policies (the light dash lines) with $\check{x}=0$ and $\check{x}=2$ . We note that both policies do not cross the currently optimal policy, and they constitute lower and upper boundaries of any $\check{x}$ -policy for $\check{x}\in [0,2]$ .

Theorem 5 Policy Refinement

A refined policy is a candidate policy, that is, ${\mathcal{P}}_R^t\subset {\mathcal{P}}_C^t\subset {\mathcal{P}}_{inc}$ . Moreover, if $({\hat{z}}^{*},{\hat{\mu }}_S^{*},{\hat{\mu }}_D^{*})\in {\mathcal{P}}_{inc}$ , then ${\mathcal{P}}_R^t=\{({\hat{z}}^{*},{\hat{\mu }}_S^{*},{\hat{\mu }}_D^{*})\}$ .

Theorem 5 confirms that, indeed, the set ${\mathcal{P}}_R^t$ is a refinement of the candidate policy set ${\mathcal{P}}_C^t$ . It is important to note that when the currently optimal solution, $({\hat{z}}^{*},{\hat{\mu }}_S^{*},{\hat{\mu }}_D^{*})$ , satisfies the desired monotone property (i.e., belongs to ${\mathcal{P}}_{inc}$ ), then the refined set ${\mathcal{P}}_R^t$ reduces to a singleton, which is the currently optimal policy.

It is well-known that even in the special case of deterministic supply, the profit function may not be concave and the optimal policy may not be monotone; see the counterexample given by Feng et al. (2020). Several authors have derived conditions for an optimal monotone policy in this setting. The most recent development is given by Pang (2011), who shows the result by assuming that the demand is additive (i.e., $D({\mu }_D)={\mu }_D+Z$ for some zero-mean random variable $Z$ ), the expected revenue $p({\mu }_D){\mu }_D$ is concave, and the cumulative hazard rate of the demand has an elasticity above one (i.e., $(p-\rho c)/p^{\mathrm{\prime }}({\mu }_D)\cdot \mathrm{\partial }\mathrm{\Lambda }(y,{\mu }_D)/\mathrm{\partial }{\mu }_D\ge 1$ , where $\mathrm{\Lambda }(y,{\mu }_D)=-\log {\overline{F}}_D(y,{\mu }_D)$ ). As Feng et al. (2020) point out, the elasticity condition essentially requires that the feasible price must be significantly higher than the ordering cost. The analysis for the setting of deterministic supply heavily depends on the transformation from the ordering decision to the post-order inventory position. Such a transformation is not possible when the supply becomes uncertain. For our problem with stochastic supply functions, the analysis conducted by Pang (2011) and Feng et al. (2020) does not apply. One would expect even more restrictive conditions for the profit function in our model to be concave.

In view of this, the result in Theorem 5 provides theoretical justification of our policy construction. Given any problem instance, our approach identifies the optimal policy when it is indeed monotone. In this case, the refined policy becomes independent of $\check{x}$ , and can be computed efficiently because of the monotonicity. Thus, we can easily judge the optimality of the monotone policy without having to rely on restrictive sufficient conditions.

5.3 Policy Performance

We first evaluate the performance of the policy in the refined set ${\mathcal{P}}_R^t$ . For any given value of $\check{x}$ , the policy $(z^{\check{x}},{\mu }_S^{\check{x}},{\mu }_D^{\check{x}})$ can be computed efficiently in view of (10). This is because the monotonicity of the policy significantly reduces the search space, and the profit function is concave in that space (recall Lemma 2). To demonstrate the result, we report the performance of the two boundary policies, $(z^0,{\mu }_S^0,{\mu }_D^0)$ and $(z^{\overline{x}},{\mu }_S^{\overline{x}},{\mu }_D^{\overline{x}})$ , named as, respectively, policy $L$ and policy $R$ . The value $\overline{x}$ is an upper bound of the inventory level. A natural upper bound can be determined by the range of the supply and demand realizations (if either has a finite bound), or by excluding low probability sample paths (if both demand and supply can be unlimited). By definition, policy $L$ coincides with the optimal solution at the left boundary $\check{x}=0$ , while policy $R$ coincides with the optimal solution at the right boundary $\check{x}=\overline{x}$ . Algorithm 1 in Online Appendix EC.2 outlines the computational details.

We use Beta distribution to capture the random noises in the supply and demand functions. This distribution is flexible enough to capture various concentrations of the realizations. In particular, we test Beta distributions with shape parameters $(1.5,1.5)$ , $(1,1)$ , $(0.6,0.6)$ , $(1.5,5)$ , $(5,1.5)$ , $(1.5,0.5)$ , and $(0.5,1.5)$ , which capture, respectively, unimodal, uniform, bimodal, left skewed, right skewed, increasing, and decreasing densities. We choose demand functions that are polynomial in the price, as all smooth functions can be approximated by polynomial functions.

We randomly generate problem instances over a wide range of parameters (see the details described in Remark 1 in Online Appendix EC.2). For each instance, we compute the optimal profit under the truly optimal solution, $V^{*}$ , starting from an empty system. We also compute the profit ${\hat{V}}^L$ under policy $L$ and ${\hat{V}}^R$ under policy $R$ . Because of the one-period replenishment lead time, we initiate the system by placing an initial order before the first period, that is, $V^{*}=\underset{{\mu }_S\in [0,{\overline{\mu }}_S]}{max}\{\mathsf{E}[V_1^{*}(S({\mu }_S))]-c{\mu }_S\}$ and ${\hat{V}}^j=\underset{{\mu }_S\in [0,{\overline{\mu }}_S]}{max}\{\mathsf{E}[{\hat{V}}_1^j(S({\mu }_S))]-c{\mu }_S\},j\in \{L,R\}$ . We evaluate the profit gap against the optimal profit as

$$\lambda =\frac{V^{*}-max\{{\hat{V}}^L,{\hat{V}}^R\}}{V^{*}}\times 100\mathrm{\%}.$$

Out of 19,596 randomly generated problem instances, the optimal policy satisfies the monotone property (i.e., coincides with policies $L$ and $R$ ) in 14,845 instances. For the remaining 4751 instances, the average profit gap is $0.03$ % and the highest is $3.77\mathrm{\%}$ . Thus, the optimality loss due to restricting the feasible solution within the refined set ${\mathcal{P}}_R^t$ is negligible. We should note that in reporting the policy performance above, we have only evaluated the two boundary policies (i.e., for $\check{x}=0$ and $\check{x}=\overline{x}$ ) in the refined set ${\mathcal{P}}_R^t$ . The performance can be further improved when we evaluate additional solutions within ${\mathcal{P}}_R^t$ (i.e., by choosing several $\check{x}\in (0,\overline{x})$ and selecting the best performer).

To understand how the performance of policies $L$ and $R$ depends on the model parameters, we report the individual profit gaps of ${\hat{V}}^L$ and ${\hat{V}}^R$ under the linear demand form used by Feng et al. (2020), with which policy $L$ and policy $R$ are both suboptimal. A snapshot of the experiment is reported in Table 1 (a comprehensive experiment is reported in Tables EC.5 to EC.11 in Online Appendix EC.3.2, which confirms the robustness of the result). From Table 1, it is clear that both policy $L$ and policy $R$ generate close-to-optimal profits. We observe that the performance gaps generally reduce as the planning horizon $T$ increases. In most of the instances, the profit gaps of both policies are below 2.5%. The exceptions occur when the demand uncertainty is very high (e.g., ${\alpha }_{Z_D}={\beta }_{Z_D}=0.2$ and the distribution is bimodal). However, the performance is not sensitive to the mean supply capacity (determined by the parameter ${\mu }_{Z_S}$ ) or the supply uncertainty (determined by the parameters $({\alpha }_{Z_S},{\beta }_{Z_S})$ ).

Table 1.

Performance of policies $L$ and $R$ against the optimal policy.

Parameter	$V^{*}$	${\lambda }^L$	${\lambda }^R$		Parameter	$V^{*}$	${\lambda }^L$	${\lambda }^R$
$T=\mathbf{5}$	0.994	1.974%	0.314%		$z_0=0.4$	0.991	2.617%	0.016%
$10$	1.774	1.953%	0.103%		$0.5$	0.991	2.494%	0.083%
$15$	2.236	1.951%	0.048%		$\mathbf{0.6}$	0.994	1.974%	0.314%
$20$	2.509	1.951%	0.026%		$0.7$	0.999	1.290%	0.442%
$25$	2.671	1.950%	0.015%		$0.8$	1.009	0.780%	0.055%
$c=\mathbf{1.0}$	0.994	1.974%	0.314%		${\mu }_{Z_S}=0.5$	0.926	1.654%	0.347%
$1.1$	0.822	2.063%	0.419%		$0.6$	0.967	1.772%	0.326%
$1.2$	0.658	2.284%	0.550%		$\mathbf{0.7}$	0.994	1.974%	0.314%
$1.3$	0.503	2.418%	0.715%		$0.8$	1.013	2.122%	0.306%
$1.4$	0.359	2.407%	0.944%		$0.9$	1.027	2.155%	0.301%
$h=\mathbf{0}$	0.994	1.974%	0.314%		${\alpha }_{Z_S}={\beta }_{Z_S}=2.0$	1.082	2.220%	0.291%
$0.1$	0.887	2.117%	0.519%		$1.5$	1.065	2.148%	0.297%
$0.2$	0.799	2.266%	0.793%		$1.0$	1.036	2.062%	0.305%
$0.3$	0.727	2.241%	1.155%		$\mathbf{0.6}$	0.994	1.974%	0.314%
$0.4$	0.666	2.008%	1.642%		$0.2$	0.903	1.723%	0.317%
$b=\mathbf{0}$	0.994	1.974%	0.314%		${\alpha }_{Z_D}={\beta }_{Z_D}=2.0$	1.142	-%	-%
$0.1$	0.897	0.549%	0.293%		$1.5$	1.109	-%	-%
$0.2$	0.803	0.530%	0.278%		$1.0$	1.059	0.119%	0.005%
$0.3$	0.710	0.624%	0.268%		$\mathbf{0.6}$	0.994	1.974%	0.314%
$0.4$	0.618	0.722%	0.264%		$0.2$	0.884	7.090%	2.434%

Notes: $T=5$ , $\rho =0.9$ , $h=0$ , $c=1$ , $b=0$ , and $V_{T+1}(x)=0$ . The demand is $\tilde{D}(p)=Z_D+(z_0-Z_D{)}^{+}\cdot \frac{\overline{p}-p}{\overline{p}-\underset{\_}{p}}$ for $p\in [\underset{\_}{p},\overline{p}]$ , where $\underset{\_}{p}=1$ , $\overline{p}=2$ , $z_0=0.6$ , and $Z_D\sim Beta({\alpha }_{Z_D},{\beta }_{Z_D})$ with ${\alpha }_{Z_D}={\beta }_{Z_D}=0.6$ . The supply is $\tilde{S}(q)=q\wedge Z_S$ , where $Z_S\sim 2{\mu }_{Z_S}\times Beta({\alpha }_{Z_S},{\beta }_{Z_S})$ with ${\mu }_{Z_S}=0.7$ and ${\alpha }_{Z_S}={\beta }_{Z_S}=0.6$ . ${\lambda }^j=(V^{*}-{\hat{V}}^j)/{\hat{V}}^j,j=\{L,R\}$ .

In Table 2, we demonstrate the result for non-stationary cost parameters. In particular, the procurement cost $c_t$ and the holding cost $h_t$ are increasing with increments of ${\delta }_c$ and ${\delta }_h$ , respectively. Compared with their counterparts in Table 1, the profit gaps are slightly larger, though the difference is very small. Also, as the costs increase more rapidly (i.e., ${\delta }_c$ and ${\delta }_h$ get larger), the profit gaps become larger in most cases and become smaller occasionally. However, the changes are small. These observations suggest that the performance of the proposed policy is robust under non-stationary cost structures.

Table 2.

Performance of policies $L$ and $R$ against the optimal policy under increasing $c_t$ and $h_t$ .

Parameter	$V^{*}$	${\lambda }^L$	${\lambda }^R$		Parameter	$V^{*}$	${\lambda }^L$	${\lambda }^R$
${\delta }_c=0.03$	0.805	2.042%	0.860%		${\delta }_h=0.03$	0.814	1.990%	0.642%
$0.04$	0.787	2.070%	0.865%		$0.04$	0.791	1.992%	0.748%
$\mathbf{0.05}$	0.769	2.098%	0.864%		$\mathbf{0.05}$	0.769	2.098%	0.864%
$0.06$	0.752	2.070%	0.857%		$0.06$	0.749	2.042%	0.990%
$0.07$	0.736	2.079%	0.844%		$0.07$	0.730	2.093%	1.124%
$T=\mathbf{5}$	0.769	2.098%	0.864%		$z_0=0.4$	0.763	2.963%	0.110%
$10$	1.104	2.632%	0.378%		$0.5$	0.765	2.685%	0.373%
$15$	1.287	2.839%	0.194%		$\mathbf{0.6}$	0.769	2.098%	0.864%
$20$	1.395	2.936%	0.108%		$0.7$	0.776	1.395%	0.228%
$25$	1.459	2.987%	0.063%		$0.8$	0.786	0.880%	0.028%
$c_0=\mathbf{1.0}$	0.769	2.098%	0.864%		${\mu }_S=0.5$	0.721	1.774%	0.927%
$1.1$	0.621	2.249%	1.004%		$0.6$	0.750	1.922%	0.889%
$1.2$	0.481	2.546%	1.147%		$\mathbf{0.7}$	0.769	2.098%	0.864%
$1.3$	0.352	2.928%	1.274%		$0.8$	0.784	2.132%	0.847%
$1.4$	0.235	3.164%	1.336%		$0.9$	0.795	2.182%	0.835%
$h_0=\mathbf{0}$	0.769	2.098%	0.864%		${\alpha }_{Z_S}={\beta }_{Z_S}=2.0$	0.842	2.304%	0.799%
$0.1$	0.689	2.207%	1.207%		$1.5$	0.828	2.270%	0.815%
$0.2$	0.623	2.337%	1.631%		$1.0$	0.805	2.101%	0.838%
$0.3$	0.568	2.194%	1.473%		$\mathbf{0.6}$	0.769	2.098%	0.864%
$0.4$	0.521	1.939%	1.374%		$0.2$	0.692	1.866%	0.892%
$b=\mathbf{0}$	0.769	2.098%	0.864%		${\alpha }_{Z_D}={\beta }_{Z_D}=2.0$	0.932	-%	-%
$0.1$	0.660	0.456%	0.938%		$1.5$	0.896	-%	-%
$0.2$	0.553	0.286%	1.041%		$1.0$	0.839	0.072%	0.089%
$0.3$	0.449	0.409%	1.193%		$\mathbf{0.6}$	0.769	2.098%	0.864%
$0.4$	0.347	0.590%	1.434%		$0.2$	0.655	7.508%	2.456%

Notes: $T=5$ , $\rho =0.9$ , $b=0$ , $h_t=h_0+{\delta }_h\times (t\wedge 5)$ with $h_0=0$ and ${\delta }_h=0.05$ , $c_t=c_0+{\delta }_c\times (t\wedge 5)$ with $c_0=1$ and ${\delta }_c=0.05$ , and $V_{T+1}(x)=0$ . The demand is $\tilde{D}(p)=Z_D+(z_0-Z_D{)}^{+}\cdot \frac{\overline{p}-p}{\overline{p}-\underset{\_}{p}}$ for $p\in [\underset{\_}{p},\overline{p}]$ , where $\underset{\_}{p}=1$ , $\overline{p}=2$ , $z_0=0.6$ , and $Z_D\sim Beta({\alpha }_{Z_D},{\beta }_{Z_D})$ with ${\alpha }_{Z_D}={\beta }_{Z_D}=0.6$ . The supply is $\tilde{S}(q)=q\wedge Z_S$ , where $Z_S\sim 2{\mu }_{Z_S}\times Beta({\alpha }_{Z_S},{\beta }_{Z_S})$ with ${\mu }_{Z_S}=0.7$ and ${\alpha }_{Z_S}={\beta }_{Z_S}=0.6$ . ${\lambda }^j=(V^{*}-{\hat{V}}^j)/{\hat{V}}^j,j=\{L,R\}$ .

5.4 The Effect of Lost Sales

A natural question in managing lost demand is: Whether strategies to retain customers through backordering can improve profitability? Our analysis, enabling tractability of the lost-sales model, can be applied to explore the answer. Specifically, we compare the profit under our model and that under demand backorder (Feng and Shanthikumar, 2018). A common assumption imposed in the backorder models, is that a large penalty is charged for any ending backorder. That is, $V_{T+1}(x)=-b_{T+1}{x}^{-}$ with a sufficiently large $b_{T+1}$ . This assumption mitigates the potential issue of a large backorder at the end of the planning horizon, as customers have paid for the products but their demands are never filled. In our experiments below, we set $b_{T+1}=\overline{p}$ . To understand the difference in the material flows under the lost-sales system and the backorder system, we also compute the expected inventory, supply, demand, shortfall and surplus in each period, and compute the corresponding per-period average values; see Tables 3 and 4.

Table 3.

Lost-sales versus backorder systems: the effect of cost structure.

	$V^{*}$		$\overline{inventory}$		$\overline{supply}$		$\overline{demand}$		$\overline{shortfall}$		$\overline{surplus}$
Parameter	LS	BO	LS	BO	LS	BO	LS	BO	LS	BO	LS	BO
$T=5$	133	147	32.8	28.0	18.2	19.2	16.4	17.4	0.59	3.83	14.6	11.5
$\mathbf{10}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$15$	279	309	35.7	27.5	18.0	19.0	17.6	18.4	0.41	4.97	17.7	12.1
$20$	311	344	36.1	27.4	18.0	19.0	17.8	18.6	0.38	5.14	18.2	12.1
$25$	330	365	36.4	27.3	18.0	19.0	17.9	18.6	0.37	5.24	18.4	12.1
$c=\mathbf{1.0}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$1.5$	167	189	29.2	22.7	15.4	16.7	15.1	16.1	0.49	4.57	13.7	9.3
$2.0$	118	137	23.6	18.0	12.8	14.2	12.8	13.9	0.50	4.37	10.8	7.1
$2.5$	78	94	18.4	13.8	10.2	11.7	10.4	11.5	0.50	4.03	8.2	5.2
$3.0$	46	59	13.6	10.0	7.7	9.2	8.0	9.2	0.47	3.55	5.9	3.6
$h=0.2$	246	265	39.5	32.9	19.0	19.9	18.0	18.7	0.33	3.58	20.4	15.6
$\mathbf{0.4}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$0.6$	207	236	31.7	24.0	17.1	18.5	16.6	17.7	0.59	5.68	14.6	9.6
$0.8$	191	226	29.3	21.3	16.3	17.9	16.1	17.3	0.75	6.60	12.9	8.1
$1.0$	176	218	27.2	19.1	15.6	17.4	15.6	16.9	0.91	7.48	11.6	6.9
$b=0.6$	226	264	34.7	23.2	18.0	19.4	17.5	18.6	0.61	8.16	16.6	9.3
$0.8$	225	255	34.8	25.8	18.0	19.2	17.4	18.4	0.52	6.06	16.8	10.8
$\mathbf{1.0}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$1.2$	224	244	35.1	29.2	18.0	19.0	17.2	18.0	0.40	3.64	17.1	12.9
$1.4$	224	240	35.2	30.3	18.0	18.9	17.2	17.8	0.35	2.90	17.2	13.7

Notes: $T=10$ , $\rho =0.9$ , $h=0.4$ , $c=1$ , $b=1$ , and $V_{T+1}(x)=0$ under lost sales (LS) and $V_{T+1}(x)=-\overline{p}x$ under backordering (BO). The demand is $\tilde{D}(p)=(120-\theta p)Z$ for $p\in [\underset{\_}{p},\overline{p}]$ , where $\theta =20$ , $\underset{\_}{p}=1$ and $\overline{p}=120/\theta$ , and $Z_D\sim Beta({\alpha }_{Z_D},{\beta }_{Z_D})$ with ${\alpha }_{Z_D}={\beta }_{Z_D}=0.6$ . The supply is $\tilde{S}(q)=q\wedge Z_S$ , where $Z_S\sim 2{\mu }_{Z_S}\times Beta({\alpha }_{Z_S},{\beta }_{Z_S})$ with ${\mu }_{Z_S}=25$ and ${\alpha }_{Z_S}={\beta }_{Z_S}=0.6$ .

Table 4.

Lost-sales versus backorder systems: the effect of uncertainty.

	$V^{*}$		$\overline{inventory}$		$\overline{supply}$		$\overline{demand}$		$\overline{shortfall}$		$\overline{surplus}$
Parameter	LS	BO	LS	BO	LS	BO	LS	BO	LS	BO	LS	BO
$\theta =10$	632	702	44.8	32.8	21.3	22.2	19.9	20.9	0.45	7.31	23.5	15.3
$\mathbf{20}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$30$	101	111	26.7	22.6	14.6	15.6	14.2	15.0	0.45	3.04	12.2	9.4
$40$	46	51	20.1	17.7	11.2	12.1	11.2	11.7	0.43	1.89	8.8	7.2
$50$	20	22	14.1	12.9	8.0	8.7	8.1	8.5	0.35	1.11	6.1	5.1
${\mu }_{Z_S}=10.0$	163	176	17.4	12.1	10.0	10.0	9.8	9.7	0.54	3.65	7.5	4.6
$17.5$	210	230	29.8	22.8	15.8	16.2	15.2	15.5	0.49	4.44	14.1	9.7
$\mathbf{25.0}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$32.5$	232	257	36.7	29.3	18.9	20.3	18.1	19.2	0.44	4.69	17.9	12.5
$40.0$	236	261	37.7	29.9	19.4	20.9	18.5	19.8	0.44	4.65	18.3	12.7
${\alpha }_{Z_S}={\beta }_{Z_S}=2.0$	241	263	36.6	28.4	19.3	20.1	18.5	19.1	0.51	3.93	17.4	11.7
$1.5$	238	260	36.2	28.2	19.0	19.9	18.2	18.9	0.50	4.05	17.2	11.7
$1.0$	233	256	35.7	27.9	18.6	19.6	17.8	18.6	0.48	4.28	17.1	11.8
$\mathbf{0.6}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$0.2$	207	234	33.9	27.4	17.0	18.3	16.3	17.5	0.37	5.61	16.9	12.4
${\alpha }_{Z_D}={\beta }_{Z_D}=2.0$	248	269	30.7	24.9	18.5	19.5	18.4	19.0	0.55	4.06	12.2	8.4
$1.5$	243	265	31.8	25.6	18.4	19.5	18.2	18.8	0.57	4.21	13.3	9.3
$1.0$	234	258	33.2	26.5	18.3	19.3	17.8	18.5	0.55	4.43	15.0	10.5
$\mathbf{0.6}$	225	249	35.0	27.7	18.0	19.1	17.3	18.2	0.45	4.65	16.9	12.0
$0.2$	211	234	37.3	29.7	17.7	18.7	16.3	17.5	0.11	4.83	19.6	14.5

Notes: $T=10$ , $\rho =0.9$ , $h=0.4$ , $c=1$ , $b=1$ , and $V_{T+1}(x)=0$ under lost sales (LS) and $V_{T+1}(x)=-\overline{p}x$ under backordering (BO). The demand is $\tilde{D}(p)=(120-\theta p)Z$ for $p\in [\underset{\_}{p},\overline{p}]$ , where $\theta =20$ , $\underset{\_}{p}=1$ and $\overline{p}=120/\theta$ , and $Z_D\sim Beta({\alpha }_{Z_D},{\beta }_{Z_D})$ with ${\alpha }_{Z_D}={\beta }_{Z_D}=0.6$ . The supply is $\tilde{S}(q)=q\wedge Z_S$ , where $Z_S\sim 2{\mu }_{Z_S}\times Beta({\alpha }_{Z_S},{\beta }_{Z_S})$ with ${\mu }_{Z_S}=25$ and ${\alpha }_{Z_S}={\beta }_{Z_S}=0.6$ .

5.4.1 Effect of Cost Structure.

Certainly, the comparison between the lost-sales model and the backorder model depends on the costs of shortfall. To facilitate the comparison, we set the backorder cost to be the same as the lost-sales penalty, though one may expect the former to be smaller than the latter as the goodwill loss would be less if the demand is eventually met.

When it becomes costlier to operate the system (due to an increased procurement cost, holding cost, or shortfall penalty), the optimal policy calls for increased prices or reduced demands under both models. The reduced demands are accompanied with decreased demand uncertainties (as the demand function is decreasing in the dispersive order in the price). These reductions, however, do not necessarily lead to improved supply and demand matching by decreasing the amounts of shortfall or surplus because of supply uncertainties, leading to subtle differences among various cost parameters.

We observe from Table 3 that, when the procurement cost $c$ increases, the profits always decrease. An increased $c$ always leads to decreases in both supply and demand, driving down the variabilities in both. The supply–demand matching exhibits, however, different patterns under the two models. Under backordering, both the average amounts of shortfall and surplus are reduced due to reduced uncertainties in the material flow. Under lost sales, however, the average shortfall level is not very sensitive to the change of the procurement cost. This is because the optimal policy attempts to avoid lost sales by carefully rationing the inventories. When a highly uncertain supply is induced by a low cost $c$ , a larger amount of inventory is reserved to meet the current demand, leading to a smaller shortfall. If, however, the supply is very low due to a high cost, the demand is also low, making the magnitude of the shortfall small.

A higher holding cost $h$ induces smaller order quantities and higher prices. The tradeoff in demand and supply matching shifts toward preventing overage, leading to an increase in the average shortfall and a decrease in the average surplus, as suggested in Table 3. Moreover, we observe that the profit difference between the lost-sales model and the backorder model becomes bigger as the holding cost gets larger, suggesting a higher value of demand retention.

Intuitively, when the cost of shortfall (i.e., the lost-sales penalty or the backorder cost) is the same across the two models, the ability to fulfill the demand with future inventories allows the decision maker to reduce the inventory level, leading to an increased shortfall and a decreased surplus. This intuition is confirmed in all the experiments conducted. Moreover, the average supplies and demands under the lost-sales model are less sensitive to the shortfall penalty than those under the backorder model, while they are more sensitive to the holding cost under the lost-sales model than under the backorder model.

5.4.2 Effect of Uncertainties.

The results in Table 4 suggest that both the lost-sales model and the backorder model exhibit similar responses to the changes of the mean supply (via ${\mu }_{Z_S}$ ) and the supply or demand uncertainty. An exception is the level of shortfall. Under the backorder model, an increased mean supply capacity, supply uncertainty or demand uncertainty induces an increased level of shortfall. This is because of the increased risk of supply–demand mismatching. Under the lost-sales model, however, the shortfall level decreases. We note that when the mean supply capacity increases or demand uncertainty decreases, the optimal policy calls for increased average inventory and demand. However, in view of the potential loss of unmet demand, the increase of the inventory is more rapid than that of the demand, leading to a reduced shortfall. When the supply uncertainty increases, both the average inventory level and the average demand reduce. To avoid the loss of demand, the reduction in inventory is smaller than that in demand, making a lower shortfall level.

Interestingly, the profit difference between the lost-sales model and the backorder model is insensitive to the level of supply or demand uncertainty (the lower panels of Table 4). This suggests that customer retention through backordering generates a certain magnitude of benefit from smoothing the demand and supply matching.

5.5 The Value of Inventory-Based Pricing

In a stochastic inventory planning system, price adjustments can improve supply–demand matching. When the unmet demand is backordered, Feng (2010) demonstrates that the benefit of dynamic pricing, as opposed to static pricing, can be significant when the supplies are restricted. One would expect a similar behavior in the lost-sales system. To confirm this, we evaluate the value of dynamic pricing as the percentage profit improvement from the optimal profit $V^S$ under static pricing, that is,

$${\lambda }^S=\frac{V^{*}-V^S}{V^S}\times 100\mathrm{\%}.$$

In computing $V^S$ , we solve the $T$ -period dynamic procurement decisions under a fixed price across all periods, and then we optimize the pricing decision that maximizes the $T$ -period expected profit.

From Table 5, we find that the benefit of dynamic pricing becomes larger when the holding cost or the shortfall penalty increases. This suggests that dynamic pricing is more useful when the supply–demand mismatch is costlier. We also observe that the limited supply (induced by a low ${\mu }_{Z_S}$ ) amplifies the value of dynamic pricing, confirming the complementary relationship between pricing and procurement—when the ability of procurement is restricted, the ability to revise price can mitigate the potential shortage in the material flow. Moreover, the benefit of dynamic pricing is more significant when the supply or demand uncertainty becomes higher. It is worth noting that the benefit of dynamic pricing stabilizes quickly (i.e., after 20 periods) in both models. All these observations echo the message from Feng (2010) in her study of backorder systems.

Table 5.

The value of dynamic pricing, as opposed to static pricing.

	$V^S$		${\lambda }^S$				$V^S$		${\lambda }^S$
Parameter	LS	BO	LS	BO		Parameter	LS	BO	LS	BO
$T=5$	115	131	15.8%	12.7%		$\theta =10$	586	660	7.8%	6.4%
$\mathbf{10}$	198	228	13.5%	9.3%		$\mathbf{20}$	198	228	13.5%	9.3%
$15$	247	286	12.8%	8.2%		$30$	82	98	22.7%	13.8%
$20$	276	320	12.5%	7.6%		$40$	33	42	38.9%	22.2%
$25$	293	340	12.4%	7.3%		$50$	11	16	73.6%	40.9%
$c=\mathbf{1.0}$	198	228	13.5%	9.3%		${\mu }_{Z_S}=10.0$	137	148	19.0%	18.7%
$1.1$	146	174	14.9%	8.6%		$17.5$	180	201	16.6%	14.4%
$1.2$	101	127	17.2%	8.3%		$\mathbf{25.0}$	198	228	13.5%	9.3%
$1.3$	65	87	21.1%	8.2%		$32.5$	208	241	11.6%	6.6%
$1.4$	36	54	27.8%	8.8%		$40.0$	214	248	10.3%	5.2%
$h=\mathbf{0}$	220	243	11.9%	8.9%		${\alpha }_{Z_S}={\beta }_{Z_S}=2.0$	221	246	9.3%	7.0%
$0.1$	198	228	13.5%	9.3%		$1.5$	217	242	10.0%	7.5%
$0.2$	180	216	15.0%	9.7%		$1.0$	209	236	11.4%	8.2%
$0.3$	164	206	16.4%	9.8%		$\mathbf{0.6}$	198	228	13.5%	9.3%
$0.4$	150	198	17.6%	9.8%		$0.2$	174	209	18.8%	11.7%
$b=\mathbf{0}$	205	247	10.0%	6.7%		${\alpha }_{Z_D}={\beta }_{Z_D}=2.0$	223	248	11.5%	8.4%
$0.1$	202	237	11.7%	7.9%		$1.5$	217	244	11.7%	8.5%
$0.2$	198	228	13.5%	9.3%		$1.0$	209	237	12.2%	8.8%
$0.3$	195	220	15.3%	10.9%		$\mathbf{0.6}$	198	228	13.5%	9.3%
$0.4$	191	213	17.0%	12.7%		$0.2$	178	211	18.4%	10.9%

Notes: $T=10$ , $\rho =0.9$ , $h=0.4$ , $c=1$ , $b=1$ , and $V_{T+1}(x)=0$ under lost sales (LS) and $V_{T+1}(x)=-\overline{p}x$ under backordering (BO). The demand is $\tilde{D}(p)=(120-\theta p)Z$ for $p\in [\underset{\_}{p},\overline{p}]$ , where $\theta =20$ , $\underset{\_}{p}=1$ and $\overline{p}=120/\theta$ , and $Z_D\sim Beta({\alpha }_{Z_D},{\beta }_{Z_D})$ with ${\alpha }_{Z_D}={\beta }_{Z_D}=0.6$ . The supply is $\tilde{S}(q)=q\wedge Z_S$ , where $Z_S\sim 2{\mu }_{Z_S}\times Beta({\alpha }_{Z_S},{\beta }_{Z_S})$ with ${\mu }_{Z_S}=25$ and ${\alpha }_{Z_S}={\beta }_{Z_S}=0.6$ .

Interestingly, the effect of procurement cost $c$ reveals different behavior under the lost-sales model and the backorder model, as shown in Table 5. In the lost-sales model, the value of dynamic pricing increases as the procurement cost increases, while in the backorder model, the value is not monotone. We also highlight that, with the same shortfall penalty, dynamic pricing is more beneficial when unmet demand is lost than when inventory shortage is backordered.

6 Concluding Remarks

We study the inventory and price planning problem involving lost sales and supply uncertainties, a realistic situation that is underresearched. The challenge of the problem lies in the highly non-concave profit function, and the existing approaches developed for the variations of this problem (i.e., under a deterministic supply or under backordering) do not apply to this problem. We propose to restrict the solution space to a class of monotone policies. This policy class is intuitively appealing in practice, as one would expect to allocate more inventory to fulfill the demand and price lower when the on-hand inventory level is higher. We further refine the policy class by defining a subclass of easy-to-compute policies and demonstrate its close-to-optimal profit performance. These developments allow us to examine the effect of lost sales on profit generation and material flow dynamics, as well as the value of inventory-based dynamic pricing.

To our knowledge, this study is the first to examine the coordinated inventory and pricing decisions under both lost sales and supply uncertainties. Our analysis provides a new approach to overcome the technical difficulties in such problems and opens up several avenues for future research. For example, though our extensive numerical experiment suggests efficient performance of the proposed policy and, like in the case of most stochastic dynamic problems, evaluating theoretical performance bounds for the general model is challenging, it is worth exploring the worst-case performance for subclasses of supply and demand functions that are relevant to specific application contexts. Diversifying supplier base through multi-sourcing to mitigate supply risks has been underscored in practice in view of the recent challenges in supply chains. Though an almost threshold policy is known to be optimal in backorder systems (see, e.g., Federgruen et al., 2022), the policy structure for lost-sales systems remains unknown. In many procurement processes, there may be a significant fixed ordering cost. The policy design that accounts for economies of scale under uncertain supplies remains an open problem. Consideration of procurement lead time under a lost-sales system (Chen et al., 2018) is yet another challenging but important issue to be addressed.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478231224916 - Supplemental material for The Effect of Supply Uncertainty on Dynamic Procurement and Pricing Strategies Under Lost Sales

Supplemental material, sj-pdf-1-pao-10.1177_10591478231224916 for The Effect of Supply Uncertainty on Dynamic Procurement and Pricing Strategies Under Lost Sales by Qi Annabelle Feng, Lei Li and J George Shanthikumar in Production and Operations Management