Robust inventory management with stock-out substitution

Accepted Manuscript Robust inventory management with stock-out substitution Zhaolin Li, Qi (Grace) Fu PII:

S0925-5273(17)30295-5

DOI:

10.1016/j.ijpe.2017.09.011

Reference:

PROECO 6824

To appear in:

International Journal of Production Economics

Received Date: 30 June 2016 Revised Date:

7 September 2017

Accepted Date: 15 September 2017

Please cite this article as: Li, Z., Fu, Q.(G.), Robust inventory management with stock-out substitution, International Journal of Production Economics (2017), doi: 10.1016/j.ijpe.2017.09.011. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Robust Inventory Management with Stock-Out Substitution Zhaolin Li

RI PT

School of Management, Ji Nan University, Guangzhou, China

Discipline of Business Analytics, The University of Sydney, Australia

Qi (Grace) Fu

SC

[email protected]

M AN U

Faculty of Business Administration, The University of Macau, Taipa, Macau [email protected]

AC C

EP

TE D

Please address all correspondence to Erick Li, Room 4159, Business School Building (H70), The University of Sydney, NSW 2006, Australia.

ACCEPTED MANUSCRIPT

Robust Inventory Management with Stock-Out Substitution

RI PT

ABSTRACT Stock-out substitution is a well-documented phenomenon that occurs when customers seek a different product as a substitute for their first-choice item if it runs out of stock. We consider a single-period inventory model with limited information regarding the external demands (i.e., mean, variance, and covariance) and

SC

focus on identifying the inventory levels that maximize the worst-case expected profit. We formulate a two-stage optimization model: the second stage characterizes the worst-case joint demand distribution by

M AN U

treating the inventory levels as input parameters, and the first stage identifies the optimal inventory levels based on the results of the second stage. The closed form solution of the second stage problem is analytically intractable except for two special cases. We develop closed form solutions for these two special cases and use them to develop a heuristic for the general case. An extensive numerical study over a wide range of parameters indicates that the performance of the heuristic that we develop is nearly optimal. We also provide insights into the effects of demand correlation, substitution rate and other parameters on the optimal

TE D

worst-case expected profit of the system.

AC C

EP

[Keywords: Robust optimization, Stock-out substitution, Inventory]

1

ACCEPTED MANUSCRIPT

1.

INTRODUCTION

Our research is motivated by some of the practical issues for Brightstar, the largest distributor of telecommunication devices in Australia. Brightstar procures various brands of telecommunication devices, such

RI PT

as iPhone and Galaxy devices, from the original equipment manufacturers (OEMs) and then allocates the available devices to each store in its supply chain, which includes more than 3, 000 stores. In the telecommunications industry, new models or generations of telecommunication devices are released each year with uncertain market reaction, making demand forecasting for various devices especially challenging. One ob-

SC

stacle that Brightstar faces is the difficulty in characterizing the joint distribution of external demands and determining the amount of inventory to be allocated to each store.

M AN U

In the extant literature on inventory management, the demand distribution is usually assumed to be known to the decision maker (Qiu et al. 2014). For example, in the classic newsvendor model, the optimal solution is a percentile of the known demand distribution. In practice, fully characterizing the demand distribution is notoriously difficult. Scarf (1958) argued that “the sample size of the past demands may be quite small” or “the future demand will come from a distribution which differs from that governing past history in an unpredictable way.” However, estimating the mean and standard deviation of demand

TE D

distributions may be easier. Therefore, Scarf (1958) proposed a max-min approach to the newsvendor problem when only the mean and variance of the demand are known without any further assumptions about the form of the demand distribution, aiming at constructing solutions that maximize the worst-case profit

EP

for all possible distributions with the given mean and variance. Because of various challenges to accurately characterizing the demand distributions, Brightstar’s headquarters may aim to utilize limited knowledge

rule.

AC C

about demands when allocating inventory. A simple decision rule suitable for such an aim is the max-min

An important phenomenon that Brightstar observes and considers in its supply chain planning process is stock-out substitution, which occurs when a customer finds his/her first choice out of stock. A certain proportion of customers are willing to take a second choice as a substitute (Anupindi et al. 1998). For instance, a customer whose initial choice is a Nokia phone may switch to buying a Samsung phone if the store runs out of Nokia phones. In this circumstance, the actual demand for any given product includes two parts: i) the demand from those who initially select this product as the first choice, which can be represented 2

ACCEPTED MANUSCRIPT

by a non-truncated random variable, and ii) the excess demand from those who initially selected some other product as the first choice but switch to this product after finding that the first choice is out of stock, which can be represented by the sum of several truncated random variables. Thus the actual demand for any

RI PT

given product involves a convolution of one non-truncated random variable and multiple truncated random variables.

An inventory model with stock-out substitution is generally difficult to solve because of the complexity in the actual demand distribution. For instance, Netessine and Rudi (2003) showed that the objective

SC

function may not be well behaved for some joint demand distributions. In addition, the existing literature on stock-out substitution assumes that the decision maker possesses complete knowledge regarding the demand distributions and attempts to maximize the expected profit. Given the aforementioned challenges in specify-

M AN U

ing the demand distributions, in this paper we study the stock-out substitution problem using the max-min approach. To the best of our knowledge, our research is the first attempt to address such a problem. We consider a retailer selling two substitutable products. The retailer has limited information on the joint demand distribution. Specifically, only the means, variances and the covariance are known. We formulate a two-stage robust optimization problem, which in general is intractable. However, we can solve two special

TE D

cases, no substitution and perfect substitution. For the first case without substitution, if the correlation coefficient between the two product demands is within a certain interval, the covariance constraint does not affect the worst-case expected profit and the problem can be decomposed into two single-product problems. For the second case with perfect stock-out substitution, the problem can be solved by pooling the two product

EP

demands. We then use the results of these two special cases to develop a heuristic for the general model. An extensive numerical study reveals the conditions, under which the heuristic does (or does not) perform well.

AC C

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 introduces the model and describes how to solve the model using an approximation method based on discrete demands. Section 4 explicitly solves two special cases and uses the relevant results to develop a heuristic solution for the general case. Section 5 describes the numerical experiments to characterize the worstcase demand distribution and to investigate the performance of the proposed heuristic solution. Section 6 concludes the study and discusses directions for future works. Proofs are provided in the Appendix.

3

ACCEPTED MANUSCRIPT

2.

LITERATURE REVIEW

Our research is related to two streams of literature on inventory management. The first literature stream concerns robust inventory management and has grown since the ground-breaking work of Scarf (1958). Scarf

RI PT

applied the distribution-free approach to the classical newsvendor problem to derive a closed form solution for the order quantity that maximizes the worst-case expected profit using only information about the mean and variance of the demand. Gallego and Moon (1993) provided a simpler proof for Scarf’s ordering rule and studied several extensions. A variety of extensions to this model have been studied in the literature

SC

(e.g., Moon and Choi 1995, Alfares and Elmorra 2005, Liao et al. 2011) to incorporate various operational details such as the shortage cost, balking, and lost sales penalty. Perakis and Roels (2008) studied a ro-

M AN U

bust newsvendor problem with partial demand information (e.g., mean, variance, range, and unimodality) to minimize the maximum regret. More recently, Raza (2014) applied this distribution-free approach to the newsvendor problem with joint pricing and inventory decisions. Wagner (2015) applied the robust optimization approach to a supply chain with asymmetric information, where one firm has limited information about demand (i.e., the mean and variance), while the other firm knows the full distribution. Several other works extend the analysis to a multi-product setting. For example, Vairaktarakis (2000) investigated a multi-item

TE D

robust newsvendor problem to minimize the maximum regret when the support and the mean of the demands are known. Moon and Silver (2000) developed a heuristic for the multi-item robust newsvendor problem with a fixed setup cost. These studies assume that the demands are independent and there is no stock-out

EP

substitution. By contrast, we formally identify the impact of correlated demands on the worst-case profit when stock-out substitution exists. A few studies examined the robust newsvendor problem with different decision rules. For example, Jammernegg and Kischka (2009) considered a robust newsvendor model with

AC C

maximin and minimax regret decision rules, in which the exact demand distribution is unknown but there is a given set of possible demand distributions that are stochastically ordered. Qiu et al. (2014) investigated the robust newsvendor decisions using CVaR-based profit maximization with ellipsoid and box discrete distributions, and transformed the problem as a tractable convex program using a second-order cone. The second related stream of research considers stock-out substitution but aims to maximize the expected profit (rather than the worst-case expected profit). The demand distribution, however, is assumed to be known. The first subset of this stream of literature focuses on centralized inventory management with 4

ACCEPTED MANUSCRIPT

substitutable products. McGillivray and Silver (1978) considered a multi-product model with substitution in a multi-period setting and developed a heuristic approach for determining the order-up-to levels. Pasternack and Drezner (1991) studied a single-period two product system with perfect substitution and compared the

RI PT

optimal inventory decisions with and without substitution. Ernst and Kouvelis (1999) studied the optimal stocking policy for a two-product model, in which each product is sold independently or as a part of a package. The second subset of this literature stream investigates the substitution effect using a probabilistic demand model to capture the dynamics of substitution (e.g., Smith and Agrawal 2000). The third subset of

SC

this stream focuses on competition between retailers selling substitutable products by applying game theory. Parlar (1988) developed a single-period inventory competition model between two retailers with stock-out substitution and obtained the equilibrium ordering decisions. Wang and Parlar (1994) extended the results of

M AN U

Parlar (1988) to the case with three players. Netessine and Rudi (2003) further extended Parlar’s model to n players and examined the optimal stocking decisions under both centralized and decentralized environments. Netessine et al. (2006) studied an inventory competition model with demand substitution in a multi-period setting. Ye (2014) proposed a novel substitution structure with both horizontal inter-brand and vertical intra-brand substitution. The equilibrium inventory decisions in both centralized and decentralized settings

TE D

were derived and compared. Inventory competition with substitution has also been studied in the literature by incorporating, for example, reactive capacity (Li and Ha 2008), price competition (Zhao and Atkins 2008), and quick response (Caro and Mart´ınez-de-Alb´eniz 2010). Recently, Jiang et al. (2011) applied the absolute regret minimization criterion to a competitive newsvendor problem with stock-out substitution and

EP

asymmetric information in which the firm knows only the support of its own demand distribution but not that of its competitors. Our paper belongs to the first subset, focusing on centralized inventory management with

AC C

substitutable products, and studies centralized inventory control with limited information about the demand distribution. Our paper contributes to the literature by making the first attempt to combine max-min robust optimization and stock-out substitution.

3.

THE MODEL

Before introducing our inventory model, we summarize Scarf’s single-product model and show some preliminary results that facilitate the later analysis. 5

ACCEPTED MANUSCRIPT

3.1

Theoretical Background: Scarf’s Model

In the single-period model developed by Scarf (1958), a firm determines how many units of inventory (denoted by q) to be procured from the supplier. The procurement cost of the product is c per unit. The firm

RI PT

sells the inventory to external customers for r per unit. For expositional simplicity, we presume that any leftover inventory will be salvaged for zero value. For many telecommunication devices, the salvage value is high; however, a positive salvage value does not fundamentally alter the analysis. The model assumes that customers are impatient and will walk away if the inventory is insufficient. The external demand follows an

SC

unknown distribution F with mean µ and standard deviation σ. The firm knows the values of µ and σ but does not know the exact distribution F . We use a non-negative random variable D to represent the external

σ. The store’s expected profit equals

M AN U

demand. Let Ω be the collection of all the possible demand distributions with mean µ and standard deviation

v (q) = −cq + rE min (D, q) = rµ − cq − rE(D − q)+ ,

(3.1)

where the expectation is taken with respect to the demand distribution F and (x)+ = max(0, x). Because

TE D

the store does not precisely know the demand distribution F , it wishes to maximize the worst-case expected profit by solving the following model:

Z = max inf {v (q)} .

(3.2)

EP

q≥0 F ⊂Ω

We let vwst (q) = inf F ⊂Ω {v (q)} be the worst-case expected profit function, q ∗ be the optimal max-min solution, and Z ∗ be the optimal worst-case expected profit. Using a primal-dual approach (see the Appendix

AC C

for details), Scarf (1958) showed that the worst-case demand distribution is a two-point distribution and that the corresponding worst-case expected profit function equals

vwst (q) =

   −cq +   −cq +

rµ2 q 2 µh2 +σ√ i r µ− (µ−q)2 +σ 2 +q 2

6

q≤

µ2 +σ 2 2µ ,

q≥

µ2 +σ 2 2µ .

(3.3)

ACCEPTED MANUSCRIPT

We observe that the worst-case profit function vwst (q) in equation (3.3) is a continuous and concave function of q. By solving the first-order condition with respect to q, we obtain the max-min robust solution q ∗ , which is often referred to as “Scarf’s rule.”

max-min robust solution q ∗ is given by

∗

q =

   0, σ 2

q

r−c c

−

q

c r−c



,

r−c c

≤

σ2 , µ2

r−c c

>

σ2 . µ2

(3.4)

SC

  µ+

RI PT

Lemma 1 (Scarf 1958) If the external random demand D satisfies E(D) = µ and V ar (D) = σ 2 , then the

M AN U

Consequently, if q ∗ > 0, the optimal worst-case expected profit equals Z ∗ = vwst (q ∗ ) = (r − c) µ − σ

In equation (3.4), the fraction costs, whereas the fraction

σ2 µ2

r−c c

p c (r − c).

(3.5)

measures the relative magnitude of the under-stock and over-stock

measures the demand variability relative to the mean demand. Intuitively, a

max-min store orders a positive quantity of the inventory when the under-stock cost is high relative to the

TE D

over-stock cost or when the demand variability is low relative to the mean. Gallego and Moon (1993) derived the same result as that in Lemma 1. However, they used a different approach that is based on the Cauchy-Schwarz inequality. Because our model involves correlated demands and stock-out substitution, the Cauchy-Schwarz inequality becomes inapplicable in the context of our prob-

3.2

AC C

our analysis.

EP

lem. Therefore, applying the primal-dual methodology developed by Scarf (1958) is more convenient for

The Robust Stock-Out Substitution Model

We now describe the stock-out substitution model in detail. Consider a firm selling two substitutable products, which are labeled product 1 and product 2. We assume that the two products have the same selling price r. As most substitutable products usually differ only in some minor characteristics, such as color, favor, or size, they are often sold at the same price. The procurement costs of the products are denoted by ci for i = 1, 2. Without loss of generality, we assume c1 ≤ c2 . The demand for each product is random, 7

ACCEPTED MANUSCRIPT

denoted by a non-negative random variable Di for i = 1, 2. The two product demands may be correlated. Let F (·, ·) denote the joint distribution of (D1 , D2 ), which is unknown to the firm. The firm has only limited information about the demands, including the means E (Di ) = µi , variances V ar (Di ) = σi2 , and covari-

RI PT

ance Cov (D1 , D2 ) = ρσ1 σ2 , where −1 ≤ ρ ≤ 1 represents the correlation coefficient. In other words, the joint distribution may not be symmetric. If product i (i = 1, 2) is out of stock, a fraction 0 ≤ α ≤ 1 of the excess demands will switch to product j (j 6= i, j = 1, 2). Specifically, α (Dj − qj )+ represents the excess demand that switches to product i after experiencing stock-out in product j, where α is a known constant. We call α the stock-out substitution rate. Therefore, the aggregate demand for product i equals

SC

Di + α (Dj − qj )+ , where Di is a non-truncated random variable and α (Dj − qj )+ is a truncated random variable. The actual demand for product i is the convolution of these two random variables.

v (q1 , q2 ) =

2  X

M AN U

For any given joint demand distribution F , the store’s total expected profit equals

+


rE min qi , Di + α (Dj − qj )

i=1

For expositional simplicity, we define

2  X

− ci qi .

Di − min qi , Di + α (Dj − qj )+




(3.6)



TE D

L (D1 , D2 |q1 , q2 ) =



(3.7)

i=1

as the lost sales quantity, which represents the number of customers whose demands are not satisfied by either product. Table 1 summarizes the expression of the lost sales quantity for various cases.

EP

If the joint distribution F is known and the firm’s objective is to maximize the expected total profit, the

AC C

first-order conditions are as follows:

Pr (D1 < q1 ) − Pr(D1 < q1 < D1 + α (D2 − q2 )+ ) + α Pr(D2 + α (D1 − q1 )+ < q2 , D1 > q1 ) =

8

r − c1 r

(3.8)

ACCEPTED MANUSCRIPT

and Pr (D2 < q2 ) − Pr(D2 < q2 < D2 + α (D1 − q1 )+ ) r − c2 . r

(3.9)

RI PT

+ α Pr(D1 + α (D2 − q2 )+ < q1 , D2 > q2 ) =

Solving these two simultaneous equations, we can find the optimal inventory levels that maximize the total expected profit. However, these two equations remain difficult to solve because they involve nested integrals of high dimensions over six regions. In particular, the six regions can be found using the conditions of the

SC

six cases listed in Table 1.

Table 1: Lost Sales Quantity L (D1 , D2 |q1 , q2 )

C1 B2 C2 D

L (D1 D2 |q1 , q2 ) If 0 < α ≤ 1 If α = 0 D1 + D2 − q1 − q2 D1 + D2 − q1 − q2

M AN U

B1

Condition D1 ≥ q1 and D2 ≥ q2 D 1 ≥ q1 , D 2 ≤ q2 D2 + α (D1 − q1 ) ≥ q2 D 1 ≥ q1 , D 2 ≤ q2 D2 + α (D1 − q1 ) < q2 D 2 ≥ q2 , D 1 ≤ q1 D1 + α (D2 − q2 ) ≥ q1 D 2 ≥ q2 , D 1 ≤ q1 D1 + α (D2 − q2 ) < q1 D1 < q1 and D2 < q2

Total sales q1 + q2 q1 + q2

D 1 + D 2 − q1 − q2

D 1 − q1

D 2 + q1 +α (D1 − q1 )

(1 − α) (D1 − q1 )

D 1 − q1

q1 + q2

D 1 + D 2 − q1 − q2

D 2 − q2

(1 − α) (D2 − q2 )

D 2 − q2

0

0

TE D

Case A

D 1 + q2 +α (D2 − q2 ) D1 + D2

EP

Without stock-out substitution (i.e., α = 0), the second and third terms in the left-hand-side (LHS) of equation (3.8) will disappear, and the optimal solution is the standard newsvendor solution. With stock-out

AC C

substitution (i.e., α > 0), the second term in the LHS of equation (3.8) pertains to case B2 in Table 1, and the third term pertains to case C1 in Table 1. If case B2 in Table 1 occurs, the inventory of product 1 is sufficient to satisfy the local demand for product 1 (i.e., D1 < q1 ) but is insufficient to satisfy the sum of the local demand and the excess demand from product 2 (i.e., q1 < D1 + α (D2 − q2 )+ ). If case C1 in Table 1 occurs, the inventory of product 2 is sufficient to satisfy the sum of the local demand for product 2 and the excess demand from product 1. Proposition 1 of Netessine and Rudi (2003, page 332) also provided results similar to those from equations (3.8) and (3.9).

9

ACCEPTED MANUSCRIPT

We are interested in the case in which the joint demand distribution can not be fully characterized, but the firm has partial information regarding the means, variances and covariance of the demands. Let Ω be the collection of all possible joint demand distributions with means µi , standard deviations σi and correlation

RI PT

coefficient ρ. The firm’s objective is to maximize the worst-case expected profit as follows: X  2 Z = max inf {v (q1 , q2 )} = max (rµi − ci qi ) − r max EF [L (D1 , D2 |q1 , q2 )] . qi ≥0 F ⊂Ω

qi ≥0

F ⊂Ω

i=1

(3.10)

The above robust optimization problem is a two-stage model. In the second stage, we regard the inventory

SC

levels (q1 , q2 ) as exogenously given and try to identify the worst-case bound for the total lost sales quantity by maximizing

L = max EF [L (D1 , D2 |q1 , q2 )] ,

M AN U

F ⊂Ω

(3.11)

subject to the moment constraints on the joint demand distribution: Z

dF (x, y) dxdy = 1

TE D

E (Di ) = µi , for i = 1, 2.
E Di2 = µ2i + σi2 , for i = 1, 2. E (D1 D2 ) = µ1 µ2 + ρσ1 σ2 .

(3.12) (3.13) (3.14) (3.15)

Constraint (3.12) states that the total probabilities equal 1, constraint (3.13) pertains to the mean of the

EP

demand for each product, constraint (3.14) pertains to the variance of the demand for each product, and constraint (3.15) pertains to the covariance of the two product demands. Let vα (q1 , q2 ) = inf F ⊂Ω {v (q1 , q2 )}

AC C

be the worst-case expected profit function with substitution rate α. The second stage problem is an infinite-dimensional linear programming problem where the variable is the cumulative distribution function F (·, ·). Because this second stage problem involves optimization over an infinite number of variables (the densities of the continuous joint distribution with given means, variances, and covariance), it is extremely difficult to solve. Therefore, we take the dual problem for the

10

ACCEPTED MANUSCRIPT

second stage problem as follows:   

,

subject to the constraint

(3.16)

 

RI PT


  y0 + y1 µ1 + y2 µ2 + y3 µ2 + σ 2 1 1 L = min
yi   +y4 µ22 + σ22 + y5 (µ1 µ2 + ρσ1 σ2 )

y0 + y1 D1 + y2 D2 + y3 D12 + y4 D22 + y5 D1 D2 ≥ L (D1 , D2 |q1 , q2 ) , ∀D1 , D2 ≥ 0,

(3.17)

SC

where the signs of the dual decision variables {yi }, for i = 0, 1, ..., 5, are unrestricted.

Anderson and Nash (1987) proved a strong duality theorem for the general continuous semi-infinite

M AN U

program, which implies the following strong duality result in Lemma 2.

Lemma 2 (Strong duality) If the dual problem (3.16) has a strictly or interior feasible solution and its optimal value L is finite, then the primal problem (3.11) is feasible and has the same optimal value. It is easily shown that the dual problem (3.16) has interior feasible solutions and that its optimal objective value is finite (see the proof of Proposition 1); therefore there is no duality gap and we can solve the

TE D

dual problem (3.16) instead. Substituting the dual problem into the original problem (3.10), the first stage problem is to identify the inventory levels that maximize the worst-case expected profit

(3.18)

AC C

EP

      −c1 q1 − c2 q2 + 2rµ        
2 2 Z = max ,  y0 + y1 µ1 + y2 µ2 + y3 µ1 + σ1   qi ≥0    −r minyi    
    +y4 µ22 + σ22 + y5 (µ1 µ2 + ρσ1 σ2 ) subject to constraint (3.17).

The dual problem of (3.16) is still difficult to solve because constraint (3.17) has a fairly complicated structure. First, because both yi and Di are variables, the LHS of constraint (3.17) is a quadratic function of variables. Second, the right-hand side (RHS) of constraint (3.17) is complicated. For instance, if we fix D1 and vary D2 , the RHS of constraint (3.17) is piece-wise linear with respect to D2 and has three pieces. Each of these three pieces (or three lines) corresponds to one of the six cases listed in Table 1 and has different start and end points. Suppose that D1 < q1 . When we increase D2 , case D of Table 1 occurs first, 11

ACCEPTED MANUSCRIPT

then case C2 occurs, and finally case B2 occurs. The first line starts from D2 = 0 and ends at D2 = q2 (with a slope of 0), the second line starts from D2 = q2 and ends at D2 = slope of α), and the third line starts from D2 =

1 α

1 α

(q1 − D1 ) + q2 (with a

(q1 − D1 ) + q2 and ends at infinity (with a slope of 1).

RI PT

Conversely, if D1 > q1 , a different path arises. First, case C1 occurs, then case B1 occurs, and finally case A occurs. The first line starts from D2 = 0 and ends at D2 = q2 − α (q1 − D1 ), the second line starts from D2 = q2 − α (q1 − D1 ) and ends at D2 = q2 , and the third line starts from D2 = q2 and ends at infinity. The complexities in constraint (3.17) make an exact analysis of the maximum lost sales quantity intractable

SC

except for two special cases that we will discuss in Section 4.

To solve the worst case distribution, we need to identify the values of (D1 , D2 ) such that constraint (3.17) is active. From Table 1, the values of the RHS lost sales function form four facets, (A, B1, B2),

M AN U

C1, C2, and D. The LHS is a quadratic function with two variables. Thus, the binding points of constraint (3.17) are at most four, which correspond to the tangent points of the four-facet surface and the quadratic surface. Supposing that there are four tangent points, we need to solve 18 variables for the dual problem 3.16-3.17, including, 8 variables for the two-dimensional coordinates of the 4 tangent points, 4 variables for the probability densities at these four points, and 6 dual variables. There are also 18 active constraints: 6

TE D

for the moments constraints, and 3 constraints for each of the four tangent points. We refer readers to the Appendix for the details of the 18 binding constraints. Solving these eighteen variables with the eighteen equations is analytically intractable. In the section on the numerical experiment, we provide an illustrative

3.3

EP

example of the worst-case distribution.

Discrete Demand Scenario Approximation

AC C

Despite the challenges in solving equation (3.18) in closed form, we can apply a discrete demand scenario approximation method and derive a structural result of the objective function. Specifically, we presume that for each product i, demand Di is a discrete random variable that takes one of the following values: {0, d1 , d2 , ..., dn−1 }, where di = (i − 1)δ and δ > 0 is a constant. Suppose that k ∈ {0, 1, ..., n − 1} and l ∈ {0, 1, ..., n − 1}. When (D1 , D2 ) = (dk , dl ) are fixed, the LHS of constraint (3.17) is linear with respect to the decision variables {yi }. There are a total of n2 different pairs of (dk , dl ). In other words, after using the discrete demand scenario approximation method, we can re-write the dual problem of (3.16) as a linear

12

ACCEPTED MANUSCRIPT

programming (LP) model with n2 constraints and six decision variables. Conversely, the decision variables of the primal model in equations (3.11) to (3.15) include θkl = Pr(D1 = dk , D2 = dl ). We observe that the primal model is also an LP model, so the strong duality holds.

RI PT

The dual objective function of (3.16) remains unchanged, but the constraint (3.17) changes to y0 + y1 dk + y2 dl + y3 d2k + y4 d2l + y5 dk dl ≥ L (dk , dl |q1 , q2 ) ,

(3.19)

for any k ∈ {0, 1, ..., n − 1} and l ∈ {0, 1, ..., n − 1}. The accuracy of the approximation depends on the

SC

values of n and δ, whereas the computational effort depends on the value of n. For appropriately chosen values of n and δ (i.e., n and nδ are sufficiently large while δ is sufficiently small), we are able to cover a

M AN U

wide range of actual demand and obtain a high-quality solution. Using the well-known theory of LP models, we find that the objective function in (3.16) is well behaved. By letting n → ∞, δ → 0, and nδ → ∞, we obtain a limiting case result in Proposition 1 and provide the details in the Appendix. Proposition 1 The objective function in equation (3.18) is concave with respect to the inventory levels

4.

TE D

(q1 , q2 ).

SPECIAL CASES AND HEURISTIC

In this section, we solve two special cases and then use the relevant results to develop a heuristic solution to

i

EP

the general problem in equation (3.18). For ease of exposition, we presume hereafter that the cost parameters (σ12 +σ22 +2ρσ1 σ2 ) σi2 r−c2 i > , which also implies that such that the optimal inventory levels > satisfy r−c 2 ci c2 (µ1 +µ2 )2 µ

4.1

AC C

are positive in both special cases.

Without Stock-Out Substitution

First, we consider the case of α = 0, which implies no stock-out substitution. The constraint (3.17) is reduced to

y0 + y1 D1 + y2 D2 + y3 D12 + y4 D22 + y5 D1 D2 ≥ (D1 − q1 )+ + (D2 − q2 )+ ,

13

ACCEPTED MANUSCRIPT

for any Di ≥ 0. In the above inequality, the LHS is a quadratic surface, whereas the RHS consists of two components. Each of these two components has two linear pieces related to only one of the two random variables, D1 and D2 . If we omit the covariance constraint (3.15), the optimization problem (3.11) can be

RI PT

decomposed into two single-product models. Each can be solved using Scarf’s model. After this decomposition step, we must construct a worst-case joint distribution of D1 and D2 satisfying all the moment constraints (3.12) to (3.15).

We now analyze the conditions on the existence of such a joint distribution and how to construct it.

SC

Without the covariance constraint, the worst-case distribution for each individual product is a two-point distribution. Let li and hi denote the low and high realizations of demand Di , i = 1, 2, in the worstcase distribution given by Scarf’s model, respectively. In addition, we define the corresponding marginal

M AN U

probabilities associated with the two realizations as Pr (Di = li ) = βi and Pr (Di = hi ) = 1 − βi . We define the following notations for the joint distribution:

Pr(Di = di and Dj = dj ) = θij ,

TE D

where di ∈ {li , hi } , dj ∈ {lj , hj } and θij ≥ 0. Note that constraint (3.12) is equivalent to

θll + θlh + θhl + θhh = 1.

(4.1)

EP

The constraints (3.13) and (3.14) imply that the marginal probabilities satisfy

(4.2)

AC C

θll + θlh = βi and θll + θhl = βj .

The remaining task is to find a joint distribution (θij ) that satisfies the covariance constraint (3.15). Using the properties of covariance, we obtain

Cov (Di , Dj ) = Cov (Di − li , Dj − lj )   Di − li Dj − lj , . = (hi − li ) (hj − lj )Cov hi − li hj − lj

Let xi =

Di −li hi −li

and xj =

Dj −lj hj −lj

be the transformed variables of Di and Dj . The joint distribution of (x, y) 14

ACCEPTED MANUSCRIPT

satisfies

θhh = Pr(xi = xj = 1),

θlh = Pr(xi = 0 and xj = 1), θll = Pr(xi = xj = 0).

RI PT

θhl = Pr(xi = 1 and xj = 0),

SC

We have the following relationship between the variance of Di (Dj ) and the transformed variable xi (xj ), which is a Bernoulli distribution:

M AN U

V ar(Di ) = (li − hi )2 V ar(xi ) ⇒ σi2 = (li − hi )2 βi (1 − βi ).

It is readily verified that

Cov (x, y) = E (xy) − E (x) E (y) = θhh − (θhh + θhl ) (θhh + θlh )

TE D

= θhh (1 − θhh − θhl − θlh ) − θhl θlh = θhh θll − θhl θlh .

Hence, the covariance constraint (3.15) implies that

q ρσi σj Cov (Di , Dj ) = = ρ βi βj (1 − βi )(1 − βj ), (hi − li )(hj − lj ) (hi − li ) (hj − lj )

(4.3)

EP

θhh θll − θhl θlh =

which is the fourth equation pertaining to {θij }. Solving these four equations (4.1) to (4.3) yields a unique

AC C

joint distribution of (Di , Dj ) such that all the moment constraints (3.12) to (3.15) are satisfied. We present the result in Proposition 2.

For ease of exposition, we define the following thresholds of the correlation coefficient:

ρl

ρh

s ) βi βj (1 − βi )(1 − βj ) = max − ,− , (1 − βi )(1 − βj ) βi βj s (s ) βi (1 − βj ) (1 − βi )βj = min , , (1 − βi )βj βi (1 − βj ) ( s

15

(4.4)

(4.5)

ACCEPTED MANUSCRIPT

where

q

r−ci ci

−

q

ci

r−ci 1 1 βi = + q q ci 2 2 r−ci + ci r−ci

is the marginal probability associated with the low demand realization in the single product case based on

RI PT

equation (A-8). The thresholds ρl and ρh are within the interval [−1, 1] and do not depend on the values of µi and σi .

Proposition 2 For the case of no demand substitution, i.e., α = 0, we obtain the following:

SC

(a) If ρl ≤ ρ ≤ ρh , the worst-case expected profit function v0 (qi , qj ) = v0i (qi ) + v0j (qj ), where v0i (qi ), i = 1, 2 satisfies

=

 

r 2 µi

qi ≤

µ2i +σi2 2µi ,

(qi − µi )2 + σi2 qi ≥

µ2i +σi2 2µi ,

rµ2i qi µ2i +σi2

M AN U

v0i (qi )

   −ci qi +

+ ( 2r − ci )qi −

r 2

q

(4.6)

and the optimal worst-case expected profit, Z0∗ , equals Z0∗

=

2  X

TE D

i=1

(r − ci ) µi − σi

p

+ ci (r − ci ) .

p

+ ci (r − ci ) .

(4.7)

(b) If ρ ≤ ρl or ρ ≥ ρh , then we have

Z0∗ >

2  X

(r − ci ) µi − σi

EP

i=1

The subscript 0 in v0i (qi ) and Z0∗ indicates that the substitution rate is α = 0. We emphasize that Propo-

AC C

sition 2 was not proven in the existing literature. For instance, in equation (15) of Gallego and Moon (page 833, 1993), the product demands are assumed to be independent, and therefore, the worst-case expected profit function equals the sum of all the product-specific worst-case expected profit functions. By contrast, we formally prove that when the demand correlation is larger than a negative threshold ρl and less than a positive threshold ρh , the worst-case expected profit function equals the sum of the product-specific worst-case expected profit functions. Our proof is performed by first relaxing constraint (3.15) on the covariance and solving the dual problem using a decomposing method. The worst-case distribution of each product for this

16

ACCEPTED MANUSCRIPT

relaxed problem is a two-point distribution. Based on the product-specific marginal probabilities as well as the values of the low and high realizations of the demand, we then construct the joint worst-case distribution that satisfies the covariance constraint (3.15). We then show that for any correlation coefficient ρl ≤ ρ ≤ ρh ,

RI PT

we can identify a joint distribution that satisfies the omitted constraint on covariance. Therefore, demand correlation does not affect the worst-case expected profit when stock-out substitution is absent. We provide the details in the Appendix.

However, when ρ ≤ ρl or ρ ≥ ρh , the relaxed solution is no longer feasible for the covariance constraint

SC

(3.15). The infeasibility of the relaxed solution implies that the covariance constraint (3.15) must be binding. In the prime model of equation (3.11), a binding covariance constraint (3.15) reduces the maximal total lost sales quantity because the maximal objective value decreases as the number of binding constraints increases.

M AN U

Note that a smaller value of the maximal total lost sales quantity actually improves the worst-case expected profit. In other words, if the correlation coefficient is outside the interval between ρl and ρh , the demand correlation increases (rather than decreases) the worst-case profit. Such a counter-intuitive result has not been reported in the extant literature. Unfortunately, we are unable to solve the worst-case distribution in closed form for the case in which the correlation coefficient is either ρ ≤ ρl or ρ ≥ ρh . The worst-case

TE D

distribution in this case is very complex. In Section 5, we provide some numerical examples to illustrate the characteristics of the worst-case demand distribution.

4.2

With Perfect Stock-Out Substitution

EP

We now consider that α = 1, which implies perfect stock-out substitution. If α = 1,

AC C

L (D1 , D2 |q1 , q2 ) = (D1 + D2 − q1 − q2 )+ .

The store’s total expected profit becomes  2  X +
v (q1 , q2 ) = rE min qi , Di + (Dj − qj ) − ci qi i=1

= rE min (D1 + D2 , q1 + q2 ) − (c1 q1 + c2 q2 ) = v1 (q1 , q2 ).

17

ACCEPTED MANUSCRIPT

We can aggregate the total demand as D = D1 + D2 and the total supply as q = q1 + q2 . We observe that E (D) = µ1 + µ2 and V ar (D) = σ12 + σ22 + 2ρσ1 σ2 . By applying Lemma 1, we can derive the max-min

Proposition 3 With α = 1, the worst-case expected profit function equals

(µ1 +µ2 )2 +σ12 +σ22 +2ρσ1 σ2 , 2(µ1 +µ2 )

The optimal worst-case expected profit equals

(µ1 +µ2 )2 +σ12 +σ22 +2ρσ1 σ2 . 2(µ1 +µ2 )

q (σ12 + σ22 + 2ρσ1 σ2 )c1 (r − c1 ).

M AN U

Z1∗ = (r − c1 )(µ1 + µ2 ) −

(4.8)

SC

 r(µ1 +µ2 )2 (q1 +q2 )   , q 1 + q2 ≤ −(c1 q1 + c2 q2 ) + (µ +µ  2 2 2  1 2 ) +σ1 +σ2 +2ρσ1 σ2  r r r v1 (q1 , q2 ) = 2 (µ1 + µ2 ) + ( 2 − c1 )q1 + ( 2 − c2 )q2  √   2 2   −r (q1 +q2 −µ1 −µ2 )2 +σ1 +σ2 +2ρσ1 σ2 , q1 + q2 ≥ 2

RI PT

solution.

(4.9)

Before developing the heuristic solution, we compare the two special cases and discuss a useful property. Using the definition of L (D1 , D2 |q1 , q2 ), it is straightforward to show that for any given (q1 , q2 ), the lost sales quantity L (D1 , D2 |q1 , q2 ) decreases with respect to α. Because the lost sales quantity L (D1 , D2 |q1 , q2 )

TE D

appears in the RHS of constraint (3.17), a smaller value of L (D1 , D2 |q1 , q2 ) must weaken constraint (3.17) and reduce the maximum expected lost sales quantity.

es with increasing α.

Heuristic

AC C

4.3

EP

Corollary 1 For any given (q1 , q2 ), the firm’s optimal worst-case expected profit vα (q1 , q2 ) weakly increas-

We then develop a heuristic for the general case with 0 < α < 1. Recall that vα (q1 , q2 ) is the worst-case expected profit function with substitution rate α. Corollary 1 indicates that v0 (q1 , q2 ) ≤ vα (q1 , q2 ) ≤ v1 (q1 , q2 ). Define

vαh (q1 , q2 ) = (1 − α)v0 (q1 , q2 ) + αv1 (q1 , q2 ) .

We use vαh (q1 , q2 ) as an approximation of vα (q1 , q2 ). Because both v0 (q1 , q2 ) and v1 (q1 , q2 ) functions are piece wise, the vαh (q1 , q2 ) function is also piece-wise. Instead of showing all of the pieces, we provide the 18

ACCEPTED MANUSCRIPT

most relevant one as follows. When qi ≥

µ2i +σi2 2µi , i

= 1, 2 and q1 + q2 ≥

(µ1 +µ2 )2 +σ12 +σ22 +2ρσ1 σ2 , 2(µ1 +µ2 )

it holds

that

= (1 − α)

2  X r i=1

r r µi + ( − ci )qi − 2 2 2

 q 2 2 (qi − µi ) + σi

RI PT

vαh (q1 , q2 )

p  (q1 + q2 − µ1 − µ2 )2 + σ12 + σ22 + 2ρσ1 σ2 r r r +α (µ1 + µ2 ) + ( − c1 )q1 + ( − c2 )q2 − r 2 2 2 2   q 2 X r (µ1 + µ2 + q1 + q2 ) − (1 − α) ci qi + r (qi − µi )2 + σi2 = 2 i=1 p   (q1 + q2 − µ1 − µ2 )2 + σ12 + σ22 + 2ρσ1 σ2 . (4.10) −α c1 q1 + c2 q2 + r 2

SC



M AN U

When optimizing vαh (q1 , q2 ), equation (4.10) is the most relevant. Let q1h (α) and q2h (α) be the non-negative optimizers of vαh (q1 , q2 ). We call q1h (α) and q2h (α) the heuristic solution for the original problem in (3.18).

5.

NUMERICAL EXPERIMENT

In this section, we conduct numerical experiments to compare the heuristic solution with the optimal solution

TE D

and investigate how various parameters could affect the performance of the heuristic. We denote the optimal value of equation (3.18) by Z ∗ , which is solved using the discrete demand scenario approximation method described in Section 3.3. Because this method involves an LP model, we refer to it hereafter as the LP method. Let Z h be the worst-case expected profit when the inventory levels are q1h (α) and q2h (α) obtained

5.1

Z ∗ −Z h Z∗

× 100% in the literature.

AC C

as ∆Z =

EP

by the heuristic. To quantify the gap between Z ∗ and Z h , we use the percentage difference, which is defined

The Impact of Cost and Demand Asymmetry

We first examine how the cost difference between the two products could affect the performance of the heuristic. The parameter values for the experiment are µ1 = 0.35, σ1 = 0.1, µ2 = 0.5, σ2 = 0.2, ρ = 0.4, α = 0.5, r = 5. We keep c1 = 1, but change c2 in the range of [1, 2]. As shown in Figure 1, the performance gap between the heuristic solution and the optimal solution increases as the cost difference increases. Specifically, when the cost of product 2 is twice as that of product 1 (c2 = 2c1 ), the gap of

19

ACCEPTED MANUSCRIPT

the heuristic is 0.58%. If the cost of product 2 triples (c2 = 3c1 ), the gap becomes 3.82%, which is still acceptable. Hence, our heuristic performs the best when substitution occurs between products with similar cost structure. 2.9

Z* (LP)

h

2.7

2.6

2.5

3

2.5

2

1.5

2.3

1

1.2

1.4

1.6

1.8

c2

M AN U

2.4

SC

Worst−Case Expected Profit

Worst−Case Expected Profit

Zh (Heuristic)

Z (Heuristic)

2.8

RI PT

3.5

Z* (LP)

1 0.1

2

0.2

0.3

0.4

0.5

µ2

0.6

0.7

0.8

0.9

1

Figure 2: Effect of Demand Mean Asymmetry on Worst-Case Performance

Figure 1: Effect of Cost Asymmetry on Worst-Case Performance

TE D

Figure 2 plots the effect of mean demand asymmetry on the performance comparison. In this experiment, we change the mean demand of product 2 such that µ2 is in the range of [0.2, 0.9]. The other parameters are fixed at µ1 = 0.3, σ1 = 0.1, σ2 = 0.2, ρ = 0.4, α = 0.5, r = 5, c1 = 1, and c2 = 2. We observe that the worst-case profit increases in µ2 . The performance of the heuristic is very good and stable. We

EP

conclude that the asymmetry of the average demand does not affect the performance of the heuristic. Figure 3 presents the effect of asymmetric demand variance on the performance gap. We change the variance of

AC C

product 2, σ2 , in the range of [0.1, 0.6]. Other parameters include: µ1 = 0.4, σ1 = 0.15, µ2 = 0.6, ρ = 0.3, α = 0.5, r = 5, c1 = 1, and c2 = 1.5. We observe that the worst-case expected profit decreases along with σ2 and that the performance gap increases slightly along with σ2 . In summary, the performance of the heuristic solution deteriorates when the variances become more asymmetric.

20

ACCEPTED MANUSCRIPT

2.85

*

Z* (LP)

Zh (Heuristic)

Zh (Heuristic)

3

2.8

2.6

2.4

2.8

2.75

RI PT

Worst−Case Expected Profit

Worst−Case Expected Profit

3.2

Z (LP)

2.7

2.65

0.1

0.2

0.3

σ2

0.4

0.5

2.6 −0.6

0.6

−0.4

−0.2

0

0.2

ρ

0.4

0.6

0.8

1

Figure 4: Effect of Demand Correlation on WorstCase Performance

M AN U

Figure 3: Effect of Demand Variance on Worst-Case Performance

5.2

SC

2.2

The Impact of Demand Correlation and Substitution Rate

Figure 4 presents the performance gap along with the change in demand correlation ρ. The parameters are µ1 = 0.4, σ1 = 0.15, µ2 = 0.5, σ2 = 0.2, α = 0.4, r = 5, c1 = 1, and c2 = 1.5. Proposition 2 indicates that

TE D

the correlation should be within the upper and lower thresholds, ρh = 0.76 and ρl = −0.33. We can observe that the worst-case profit decreases in the demand correlation because the pooling effect from substitution decreases when the demand correlation increases. In addition, when the correlation coefficient increases, the

EP

gap between Z ∗ and Z h widens. This observation is attributable to the binding covariance constraint (3.15). As aforementioned, a binding covariance constraint (3.15) increases the worst-case profit. Our heuristic is based on two special cases that ignore the covariance constraint (3.15). Therefore, our approximated profit

AC C

function understates the total profit and yields a bigger performance gap when the correlation coefficient is outside the interval specified in Proposition 2. In Figure 4, the biggest gap is 0.85% when the correlation coefficient ρ = 0.9.

We also investigate the effect of the substitution rate α on the performance gap by varying α in the range of [0, 1]. Figure 5 plots the performance gap under symmetric product costs, with the following parameters: µ1 = 0.4, σ12 = 0.15, µ2 = 0.5, σ22 = 0.2, ρ = 0.3, r = 5, c1 = 1, and c2 = 1. Figure 5 shows that the worst-case profit increases with the substitution rate. The performance of the heuristic is nearly optimal. We 21

ACCEPTED MANUSCRIPT

3.06

3.2

Z* (LP)

Z* (LP)

Zh (Heuristic)

Zh (Heuristic)

3

3.02

3

2.98

2.96

2.94

2.92

2.8

2.6

RI PT

Worst−Case Expected Profit

Worst−Case Expected Profit

3.04

2.4

2.2

2

2.88

0

0.1

0.2

0.3

0.4

0.5

α

0.6

0.7

0.8

0.9

1.8

1

0.1

0.2

0.3

0.4

0.5

α

0.6

0.7

0.8

0.9

1

Figure 6: Effect of the Substitution Rate on WorstCase Performance under Asymmetric Product Costs

M AN U

Figure 5: Effect of the Substitution Rate on WorstCase Performance under Symmetric Product Costs

0

SC

2.9

then evaluate the heuristic performance under asymmetric costs c1 = 1 and c2 = 3 with the other parameters the same with those of Figure 5. We plot the results in Figure 6. In this case, we can observe that when the substitution rate α is small or large, the heuristic performs very well. When α is moderate, the performance

TE D

gap widens with the maximal gap 3.88% occurring at α = 0.5 (The maximal gap for c2 = 2.5 is 1.81% occurring at α = 0.6). Therefore, the heuristic performance decreases when the product cost difference gets larger. However, we know that in practice large cost difference for substitutable product is rare. Another observation from Figure 5 and Figure 6 is that the profit curve changes from concave to convex when the

EP

product cost difference gets larger. The main reason is that with asymmetric cost (c1 < c2), the optimal solution of q2∗ could be zero for bigger α value. This is the bait-and-switch phenomenon mentioned by Smith

AC C

and Agrawal (2000).

One drawback of the LP method is that it involves n2 constraints, where n determines the accuracy of the approximation. The LP method suffers from the curse of dimensionality because the computation time increases exponentially as n increases. For instance, when we used n = 100, the computation time was approximately 16 seconds on our surface notebook equipped with 4GB memory, 2.5GHz CPU, and 64-bit Windows 10 Operating System; when we used n = 200, the computation increased to 4.5 minutes. By contrast, our heuristic solution can be solved instantaneously using EXCEL. The numerical examples in

22

ACCEPTED MANUSCRIPT

this subsection indicate that our heuristic performs remarkably well over a wide range of parameters except when the cost parameters are symmetric and the substitution rate is moderate. In summary, our heuristic method provides a good approximation of the original objective function, which is generally intractable, and

5.3

RI PT

yields an excellent solution that can be solved instantaneously.

Max-Min and Risk-Neutral

To compare the performance of the max-min solution with that of the risk-neutral solution, we consider a

When Di ∼ U [0, 1], we have µi = 21 , σi2 =

1 12 ,

SC

symmetric case with demand Di , i = 1, 2 independently and uniformly distributed over the interval [0, 1]. and ρ = 0. We let ri = 5, ci = 1, and αi = 0.5. In this

case, let i = 1, 2, j = 3 − i, and q1 = q2 = q. We can explicitly obtain the probabilities of the random

M AN U

events in the optimality Equations (3.8) and (3.9) as below:

Pr(Di < qi < Di + α(Dj − qj )+ ) =

   q − (1 +

1 2 2α )q ,

α 2

+

α 1+α

if q >

α 1+α .

if q ≤ αq 2 ),

q>

α 1+α

α 1+α .

TE D

2   α(1−q) , 2    q2 , 2α Pr(Dj + α(Di − qi )+ < q, Di > q) =   (1 − q)(q −

if q ≤

In addition, Pr(Di < q) = q. These expressions allow us to determine that the optimal inventory levels are q1 = q2 = 0.7287. The optimal expected profit can then be calculated in closed-form by taking expectation

EP

of the profit function (3.6) over the six regions given in Table 1. We find that the optimal expected profit equals 3.4595. Conversely, the optimal max-min solution is q1∗ = q2∗ = q ∗ = 0.68, based on the LP method. If we use the robust solution q1∗ = q2∗ = 0.68 in this risk-neutral setting, we find that the expected profit

AC C

equals 3.3875, which is 2.08% less than the optimal expected profit. Next we consider the same set of parameters but assume that the demand Di is normally distributed with mean µi = 0.5 and variance σi2 =

1 12 .

By keeping the mean and variance of the demand the same as those

of the uniform distribution, we can examine how a change in the demand distribution affects various solutions. With normal distribution, equations (3.8) and (3.9) cannot be solved in closed-form. We numerically compute the retailers optimal expected profit, which is 3.573. When the retailer uses the aforementioned max-min solution, the expected profit is 3.531, which is approximately 1.2% lower than 3.573. 23

ACCEPTED MANUSCRIPT

We conclude that the max-min solution also performs relatively well under the risk-neutral objective.

5.4

Characteristics of Worst-Case Distribution

RI PT

In Scarf’s model, the worst-case demand distribution is a two-point distribution. To determine the worstcase joint demand distribution when stock-out substitution exists, we provide a representative example to illustrate the characteristics of the optimal solution for equation (3.18). The example considers the following parameters: r = 5, c = 1, µ =

1 2,

σ2 =

1 12 ,

ρ = 0.4, and α = 0.5. The optimal max-min solution is

SC

q1∗ = q2∗ = q ∗ = 0.7, based on the LP method, and the heuristic solution is q˜α = 0.698. The difference between q ∗ and q˜α is negligible. The distribution that produces the worst-case scenario satisfies Pr(D1 = 0.04, D2 = 1.04) = 0.05, Pr(D1 = 1.04, D2 = 0.04) = 0.05, Pr(D1 = 0.38, D2 = 0.38) = 0.7390, and

M AN U

Pr(D1 = 1.04, D2 = 1.04) = 0.1610.

1.6

1.6

1.2

D2

0.8

0.4

0 0

0.4

TE D

D2

1.2

0.8

1.2

1.6

0.8

0.4

0 0

EP

D1

0.4

0.8

1.2

1.6

D1  

(b) With ρ = −0.8

(a) With ρ = 0.4

AC C

Figure 7: Characteristics of the worst-case distributions

Figure 7a) depicts the supports of the worst-case distribution, and each possible realization is marked by a gray . The marginal distribution of each Di is a three-point distribution. When each Di is a three-point distribution, there are nine possible combinations of (D1 , D2 ). However, of these nine possible combinations, only four can have positive probabilities. Interestingly, the four combinations that have positive probabilities correspond to cases A, C1, C2, and D in Table 1. We use the same set of parameters but set the correlation coefficient to ρ = −0.8, which is less than 24

ACCEPTED MANUSCRIPT

the threshold ρ0 = −0.25, to determine what happens if ρ < ρ0 and α = 0. The demand distribution that produces the worst-case scenario is the following: Pr(D1 = 0.41, D2 = 0.43) = 0.555, Pr(D1 = 0.99, D2 = 0.20) = 0.238, and Pr(D1 = 0.17, D2 = 1.04) = 0.207. In this particular joint distribution, the

RI PT

marginal distribution of Di is also a three-point distribution. Figure 7b) shows that the three combinations of (D1 , D2 ) that have positive probabilities correspond to cases A, C1, and C2 in Table 1.

6.

CONCLUSIONS

SC

We revisit a classic inventory model with stock-out substitution by applying the max-min criterion rather than the expected profit criterion. Our approach makes use of the limited information on product demands

M AN U

and is suitable for the circumstance in which exact demand distributions can not be accurately estimated. After formulating the optimization model as a two-stage model, we find that the closed-form solution of the second stage is intractable except for two special cases. We develop a heuristic solution based on these two special cases. An extensive numerical study indicates that the performance of the heuristic solution is nearly optimal over a wide range of parameters.

A few promising paths to extend our analyses are presented here. First, we can consider that the number

TE D

of products is N ≥ 2 and the cost parameters are asymmetric. The notations, however, will become fairly complicated. In addition, the LP method will involve nN constraints, which could pose a challenge for a numerical study. Nevertheless, the idea behind our heuristic solution can be a useful guide for developing

EP

similar heuristic solutions. Second, some articles consider minimizing the maximum regret rather than maximizing the worst-case expected profit. For instance, Yue, et al. (2006) extended Scarf’s model by changing the robust decision rule from max-min to min-max regret. Perakis and Roels (2008) formulated the

AC C

newsvendor model with min-max regret as a moment problem. Investigating the effects of robust decision rule changes could offer interesting future research.

ACKNOWLEDGEMENTS The authors would like to thank the two anonymous referees for their constructive and helpful comments that have substantially improved the paper. The constructive comments and industrial insights provided by

25

ACCEPTED MANUSCRIPT

Dr. Gregory Hill (the Head of Business Analytics at Brightstar Australia) are gratefully acknowledged. The second author is supported in part by the University of Macau under MYRG2014-00057-FBA.

RI PT

APPENDIX Proof of Lemma 1:

To derive the worst-case profit function vwst (q), we need to identify a particular demand distribution that

model as follows: L = max E (D − q)+ subject to the constraints Z Z Z

∞

0 ∞

0 ∞

M AN U

F ⊂Ω

SC

maximizes the expected lost sales quantity, which equals E(D − q)+ . To this end, we solve an optimization

(A-1)

dF (x) = 1,

(A-2)

xdF (x) = µ,

(A-3)

x2 dF (x) = µ2 + σ 2 ,

(A-4)

TE D

0

The first constraint (A-2) states that the total probabilities equal 1; the second constraint (A-3) states that the mean demand equals µ; and the third constraint (A-4) implies that the variance of the demand equals σ 2 . As

EP

Scarf (1958) suggested, solving the dual problem of equation (A-1) is more convenient. We formulate the dual problem as follows. The objective is to minimize

(A-5)

y0 + y1 D + y2 D2 ≥ (D − q)+ , for any D ≥ 0.

(A-6)

AC C


L = min y0 + y1 µ + y2 µ2 + σ 2 ,

and the constraint is

The signs of the dual decision variables yi (for i = 1, 2, 3) are unrestricted. It is important to investigate the properties of constraint (A-6). Figure 8 depicts the left-hand side (LHS)

26

ACCEPTED MANUSCRIPT

D

RI PT

RHS

SC

LHS

M AN U

Figure 8: Illustration of the LHS and RHS of constraint (A-6)

and the right-hand side (RHS) of constraint (A-6). In particular, with any given (yi , q), the LHS of constraint (A-6) is a quadratic function of D, whereas the RHS is a piece wise linear function of D. At the optimum, the quadratic curve and the convex curve are tangent at two distinct points. Let D = l and D = h be the two distinct tangent points and m, n be the corresponding probabilities, respectively. We have two cases to consider.

TE D

Case 1) l 6= 0. The tangent requirements imply that

y12 − 4y0 y2 = 0 and (y1 − 1)2 − 4y2 (y0 + q) = 0,

(A-7)

EP

where we use the properties of the roots for the quadratic equations y0 + y1 D + y2 D2 = 0 and y0 + y1 D +

AC C

y2 D2 = D − q, respectively. In addition, the Karush–Kuhn–Tucker (KKT) conditions indicate that            

m + n = 1, µ − ml − nh = 0,

µ2 + σ 2 = ml2 + nh2 ,       y1 + 2y2 l = 0,      y1 + 2y2 h = 1.

We emphasize that without equation (A-7), the KKT conditions alone are insufficient to derive the optimal

27

ACCEPTED MANUSCRIPT

solution. We find that

l = q−

q q (µ − q)2 + σ 2 and h = q + (µ − q)2 + σ 2 ,

To ensure that l > 0, q >

µ2 +σ 2 2µ

RI PT

(q − µ) 1 (q − µ) 1 +q and n = − q . 2 2 2 2 2 2 4 (µ − q) + 4σ 4 (µ − q) + 4σ

m =

must be satisfied. We conclude that if q >

distribution satisfies +√

1 2

−√

(q−µ)

and

SC

1 2

the worst-case

4(µ−q)2 +4σ 2 (q−µ)

4(µ−q)2 +4σ

(A-8)

. 2

M AN U

   q  2 2   Pr D = q − (µ − q) + σ =   q  2 2  (µ − q) + σ = Pr D = q + 

µ2 +σ 2 2µ ,

Case 2) l = 0, which implies y0 = 0. We can solve the optimal solution using similar procedures. We omit the details but show that the worst-case distribution satisfies

   Pr (D = 0) = 2σ2 2 and µ +σ   2 +σ 2 2  µ  Pr D = = µ2µ+σ2 µ µ2 +σ 2 2µ .

TE D

The second case requires q ≤

(A-9)

Using equations (A-8) and (A-9), we find that the upper bound of the expected lost sales equals

EP

   µ − 2µ2 q 2 √ µ +σ L= 2 2   (µ−q) +σ +µ−q 2

if

µ2 +σ 2 2µ

if q >

≤ q,

µ2 +σ 2 2µ

(A-10)

.

AC C

By substituting equation (A-10) into equation (3.1), we can derive the worst-case expected profit function shown in equation (3.3). By optimizing equation (3.3), we find the optimal solution shown in equation (3.4) and the optimal objective value shown in equation (3.5). It is important to derive the worst-case demand distribution because we need this result to prove Proposition 2. However, we emphasize that Case 2) is less important if the max-min inventory level is positive. In particular, if

r−c c

condition that q >

>

σ2 , µ2

µ2 +σ 2 2µ

the positive optimal max-min inventory level in equation (3.4) must satisfy the

. Therefore, in the proof of Proposition 1, only Case 1) is used.

28

ACCEPTED MANUSCRIPT

Reformulation Based on Active Constraints Suppose there are four tangent points, the worst-case distribution is a four-point joint distribution with

Table 2: Four-Point Distribution D1 x1 w1 u1 v1

D2 x2 w2 u2 v2

Joint Probability pa pb pc pd

Pre-Condition (x1 , x2 ) ∈ D (w1 , w2 ) ∈ C1 (u1 , u2 ) ∈ C2 (v1 , v2 ) ∈ A, B1, B2

SC

Case I II III IV

RI PT

parameters shown in Table 2.

M AN U

A, B1, B2, C1, C2, D in the above table are the six cases in Table 1. The moment constraints include:    pa + pb + pc + pd = 1,        pa x1 + pb w1 + pc u1 + pd v1 = µ1 ,       pa x2 + pb w2 + pc u2 + pd v2 = µ2 ,

(A-11)

TE D

  pa x21 + pb w12 + pc u21 + pd v12 = µ21 + σ12 ,        pa x22 + pb w22 + pc u22 + pd v22 = µ22 + σ22 ,       pa x1 x2 + pb w1 w2 + pc u1 u2 + pd v1 v2 = µ1 µ2 + ρσ1 σ2 . To ensure that each point of the four-point distribution represents a tangent point, there are three condi-

   y1 + 2y3 x1 + y5 x2 = 0,    y2 + 2y4 x2 + y5 x1 = 0,      y0 + y1 x1 + y2 x2 + y3 x21 + y4 x22 + y5 x1 x2 = 0.

(A-12)

AC C

as follows:

EP

tions, two for the tangent condition and one for the equality condition. For case I, the three constraints are

For case II, the constraints are as follows:    y1 + 2y3 w1 + y5 w2 = 1 − α,    y2 + 2y4 w2 + y5 w1 = 0,      y0 + y1 w1 + y2 w2 + y3 w2 + y4 w2 + y5 w1 w2 = (1 − α)(w1 − q1 ). 1 2

29

(A-13)

ACCEPTED MANUSCRIPT

For case III, the constraints are as follows:

(A-14)

RI PT

   y1 + 2y3 u1 + y5 u2 = 0,    y2 + 2y4 u2 + y5 u1 = 1 − α,      y0 + y1 u1 + y2 u2 + y3 u2 + y4 u2 + y5 u1 u2 = (1 − α)(u2 − q2 ). 1 2 For case IV, the constraints are as follows:

SC

   y1 + 2y3 v1 + y5 v2 = 1,    y2 + 2y4 v2 + y5 v1 = 1,      y0 + y1 v1 + y2 v2 + y3 v 2 + y4 v 2 + y5 v1 v2 = v1 + v2 − q1 − q2 . 1 2

(A-15)

M AN U

When the correlation coefficient ρ is less than the threshold ρl defined in Section 4.1, the worst-case distribution may become a three-point joint distribution. In this case, we need to remove case IV from the analysis. The modified system involves 15 variables and 15 equations.

Proof of Proposition 1:

TE D

Consider a standard LP model with an objective Z = min cx and constraints Ax ≥ b and x ≥ 0. It is well-known that the optimal objective value Z ∗ is convex decreasing when b decreases to zero. After applying the discrete demand scenario approximation, we find that the inventory level q appears in the RHS of constraint (3.19) with a negative sign. When q increases, the RHS of constraint (3.19) decreases.

EP

The aforementioned result of the LP model indicates that the optimal objective value of equation (3.16) is convex decreasing with respect to q. By letting n → ∞ and δ → 0, the convex and decreasing properties

AC C

are preserved.

We now examine the objective function in equation (3.18). With symmetric inventory levels, the first term of (3.18) represents the procurement cost and equals −2cq, which is concave with respect to q. The second term of (3.18) is a constant. The third term equals the objective value of equation (3.16) multiplied by −r. Furthermore, the third term is concave and increasing with respect to q. In summary, the objective function of (3.18) is concave with respect to q. Finally, when q → ∞, the worst-case expected lost sales approach zero and the objective function of

30

ACCEPTED MANUSCRIPT

(3.18) approaches negative infinity. Therefore, the optimal solution q ∗ is finite.

Proof of Proposition 2:

RI PT

The optimal values of li and hi are associated with the two tangent points in Figure 8, if there exists a joint distribution satisfying all the moment constraints. Because the worst-case scenario requires us to maximize L in equation (3.11), by strategically omitting constraint (3.15) on covariances, we can obtain an upper bound for the optimal objective value. We call the model without constraint (3.15) the relaxed model. Let

SC

Li denote the worst-case lost sales revenue for each product by applying equation (A-10) from the single ¯ equals P2 Li . Because we omit the product model. The optimal objective value of the relaxed model, L, i=1 ¯ ≥ L∗ , where L∗ denotes the optimal objective value of equation covariance constraint, it must hold that L

M AN U

(3.11) with constraint (3.15).

The solution to the four simultaneous equations (4.1-4.3) is uniquely given by the following:

(A-16)

TE D

 p   θll = βi βj + ρ βi βj (1 − βi )(1 − βj );     p   θlh = βi (1 − βj ) − ρ βi βj (1 − βi )(1 − βj ); p   θhl = (1 − βi )βj − ρ βi βj (1 − βi )(1 − βj );       θ = (1 − β )(1 − β ) + ρpβ β (1 − β )(1 − β ). i j i j i j hh

We know that the value of the correlation ρ is in the interval [−1, 1]. Since θij in equation (A-16) is non-

EP

negative, we have the following two cases:

AC C

• If 0 ≤ ρ ≤ 1, then θll and θhh are non-negative, but to guarantee θlh ≥ and θhl ≥ 0, we require: (s

ρ ≤ min

βi (1 − βj ) , (1 − βi )βj

s

) (1 − βi )βj . βi (1 − βj )

(A-17)

• If −1 ≤ ρ ≤ 0, then θlh and θhl are non-negative, but to guarantee θll ≥ and θhh ≥ 0, we require: ( s

ρ ≥ max −

s ) βi βj (1 − βi )(1 − βj ) ,− . (1 − βi )(1 − βj ) βi βj

(A-18)

Therefore, only when conditions (A-17) and (A-18) are satisfied, does there exist a feasible solution to 31

ACCEPTED MANUSCRIPT

Equations (4.1) to (4.3). Otherwise, some of the probabilities {θll , θlh , θhl , θhh } could be negative which makes the relaxed solution infeasible for the omitted covariance constraint. Recall that Scarf’s optimal inventory level is given by equation (3.4). Because we presume that the ri −ci ci

>

σi2 , µ2i

we can substitute equation (3.4) into equation (A-8) to

RI PT

robust order quantity is positive, i.e.,

find the value of βi when the max-min inventory level is used. With some algebra, we find that

=

q

ri −ci ci

−

q

ci ri −ci



1 1 − µi ) = + r  q +q 2 q 2 2 ri −ci ci 4 (µi − qi∗ )2 + 4σi2 − + 4σi2 4 σ2i ci ri −ci q q r−ci ci − ci r−ci 1 1 . + q q ci 2 2 r−ci + r−ci

(A-19)

M AN U

ci

SC

βi =

σi 2

(qi∗

Using the expression of βi in equation (A-19) and the inequality (A-17 and A-18), we find that whenever ρl ≤ ρ ≤ ρh , the relaxed solution is feasible for the fully constrained model.

REFERENCES

TE D

Alfares, H., H. Elmorra. 2005. The distribution-free newsboy problem: extensions to the shortage penalty case. International Journal of Production Economics, 93(94),465–477. Anderson, E. J., P. Nash. 1987. Linear Programming in Infinite-Dimensional Spaces: Theory and Applica-

EP

tions. John Wiley and Sons, Chichester, UK.

Anupindi, R., M. Dada, S. Gupta. 1998. Estimation of consumer demand with stock-out based substitution:

AC C

An application to vending machine products. Marketing Science, 17(4), 406–423. Aydın, G., W. H. Hausman. 2009. The role of slotting fees in the coordination of assortment decisions. Production and Operations Management, 18(6), 635–652. Bernstein, F., A. G. K¨ok, L. Xie. 2011. The role of component commonality in product assortment decisions. Manufacturing and Service Operations Management, 13(2), 261–270. Cachon, G. P., C. Terwiesch, Y. Xu. 2005. Retail assortment planning in the presence of consumer search. Manufacturing and Services Operations Management, 7(4), 330–346.

32

ACCEPTED MANUSCRIPT

Caro, F., V. Mart´ınez-de-Alb´eniz. 2010. The impact of quick response in inventory-based competition. Manufacturing and Service Operations Management, 12(3), 409–429. Ernst, R., P. Kouvelis. 1999. The effects of selling packaged goods on inventory decisions. Management

RI PT

Science, 45(8), 1142–1155. Gallego, G., I. Moon. 1993. The distribution free newsboy problem: Review and extensions. Journal of the Operational Research Society, 44(8), 825–834.

Jammernegg, W., P. Kischka. 2009. Risk preferences and robust inventory decisions. International Journal

SC

of Production Economics, 118(1), 269–274.

Jiang, H., S. Netessine, S. Savin. 2011. Robust Newsvendor Competition under Asymmetric Information.

M AN U

Operations Research, 59(1), 254–261.

K¨ok, A. G., M. L. Fisher. 2007. Demand estimation and assortment optimization under substitution: Methodology and application. Operations Research, 55(6), 1001–1021. Li, Q., A. Y. Ha. 2008. Reactive capacity and inventory competition under demand substitution. IIE Transactions, 40(8), 707–717.

TE D

Li, Z. 2007, A single-period assortment optimization model. Production and Operations Management, 16(3), 369–380.

Liao, Y., A. Banerjee, C. Yan. 2011. A distribution-free newsvendor model with balking and lostsales

EP

penalty. International Journal of Production Economics, 133(1),224–227. McGillivray, A. R., E. A. Silver. 1978. Some concepts for inventory control under substitutable demands.

AC C

INFOR, 16, 47–63.

Moon, I., S. Choi. 1995. The distribution free newsboy problem with balking. The Journal of the Operational Research Society, 46(4), 537–542. Moon, I., E. A. Silver. 2000. The multi-item newsvendor problem with a budget constraint and fixed ordering costs. The Journal of the Operational Research Society, 51(6), 602–608. Netessine, S., N. Rudi. 2003. Centralized and competitive inventory models with demand substitution. Operations Research, 51(2), 329–335.

33

ACCEPTED MANUSCRIPT

Netessine, S., N. Rudi, Y. Wang. 2006. Inventory competition and incentives to back-order. IIE Transactions, 38, 883–902. Parlar, M. 1988. Game theoretic analysis of the substitutable product inventory problem with random de-

RI PT

mands. Naval Research Logistics. 35(3), 397–409. Pasternack, B. A., Z. Drezner. 1991. Optimal inventory policies for substitutable commodities with stochastic demand. Naval Research Logistics, 38, 221–240.

Perakis, G., G. Roels. 2008. Regret in the newsvendor model with partial information. Operations Research,

SC

56(1), 188–203.

Qiu, R. Z., J. Shang, X. Y. Huang. 2014. Robust inventory decision under distribution uncertainty: A

M AN U

CVaR-based optimization approach. International Journal of Production Economics. 153, 13–23. Raza, S. A. 2014. A distribution free approach to newsvendor problem with pricing. 4OR - A Quarterly Journal of Operations Research, 12(4), 335–358.

Scarf, H. 1958. A min-max solution of an inventory problem. K. Arrow, S. Karlin, H. Scarf, eds. Studies in The Mathematical Theory of Inventory and Production. Stanford University Press, California, USA,

TE D

201–209.

Smith, S. A., N. Agrawal. 2000. Management of multi-item retail inventory systems with demand substitution. Operations Research, 48(1), 50–64.

EP

Vairaktarakis, G. L. 2000. Robust multi-item newsboy models with a budget constraint. International Journal of Production Economics, 66, 213–226. van Ryzin, G., S. Mahajan. 1999, On the relationship between inventory costs and variety benefits in retail

AC C

assortment. Management Science, 45(11), 1496–1515. Wagner, M. G. 2015. Robust purchasing and information asymmetry in supply chains with a price-only contract. IIE Transactions, 47, 819–840. Wang, Q., M. Parlar. 1994. A three-person game theory model arising in stochastic inventory control theory. European Journal of Operational Research, 76(1), 83–87. Ye, T. F. 2014. Inventory management with simultaneously horizontal and vertical substitution. Internation-

34

ACCEPTED MANUSCRIPT

al Journal of Production Economics, 156, 316–324. Yue, J., B. Chen, M.-C. Wang. 2006. Expected value of distribution information for the newsvendor problem. Operations Research, 54(6), 1128–1136.

AC C

EP

TE D

M AN U

SC

Manufacturing and Service Operations Management, 10(3), 539–546.

RI PT

Zhao, X. and D. R. Atkins. 2008. Newsvendors under simultaneous price and inventory competition.

35