Stock rationing under a profit satisficing objective

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Stock rationing under a profit satisficing objective$ Roberto Pinto n CELS – Department of Management, Information and Production Engineering, University of Bergamo, Viale Marconi 5, 24044 Dalmine (Bergamo), Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 25 July 2013 Accepted 19 December 2015

This paper addresses a rationing problem with a profit satisficing objective in a company operating many retail stores through a centralized procurement. General rationing problems arise when the available stock or capacity cannot guarantee the possibility to satisfy the demand in full, and different decisions about the allocation of the available resources may lead to different profit results. Therefore, the appropriate allocation of the stock or capacity can have a substantial impact on the company’s profit. Unlike other works in the rationing area, this paper considers a profit satisficing objective, which entails maximizing the probability of achieving a pre-specified profit target. This type of objective is sometimes preferable to maximizing the expected profit. The problem is modeled in an analytical form, for which closed-form solutions are extremely hard to compute. Thus, the conditions for achieving the satisficing objective are discussed, and two heuristic procedures are compared: one exploiting the structure of the problem and resulting in a greedy, marginal unit allocation; the other, based on the Nelder–Mead derivative-free method. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Rationing Profit satisficing Single-period problem Greedy heuristic Marginal analysis Derivative-free Monte Carlo integration

1. Introduction Managing and allocating scarce resources is an emerging problem in many supply chain contexts. In particular, allocating common resources, such as the product stock or the production capacity, to competing activities or actors with uncertain outcomes and needs is a recurring challenge in many business settings [1]. In this respect, a rationing problem arises whenever the available stock or capacity does not guarantee the possibility to satisfy the demand in full, and different decisions about the allocation of the available resources may lead to different profit results. In this case, the decision maker in the company has to discern, according to the relative importance of the served customers [2], which orders should be filled (either partially or completely) and which orders should be rejected, to achieve a prespecified objective, such as profit maximization. Thus, in shortage situations, customers are put “on allocation”, and the available resources are distributed according to the appropriate rules [3]. In this paper, we consider a rationing problem arising in the context of a company that manages the procurement of a scarce resource (i.e., a product) through a centralized purchasing department and then distributes the resource to many stores that, in turn, sell it to the final market. This may be the case for large ☆ n

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distribution firms (i.e., Wal-Mart, 7-Eleven, and Leroy Merlin) that procure the products at a centralized level and then allocate them to their retail stores. In this context, the necessity for solving a rationing problem may be linked to the characteristics of the resource (i.e., a scarce resource), to temporary situations that limit the availability of the resource (i.e., a supplier disruption), or to other constraints that affect the possibility of procuring larger quantities. Each retailer contributes in varying degrees to the company’s bottom line. Incorrect allocation may lead to situations in which some retailers are out of stock, while others have excessive, unsold stock. Thus, the company must carefully decide how to allocate the available stock among the retailers to pursue a given objective. The main contribution of this paper is to provide analysis and insight into the rationing problem when companies pursue a profit satisficing objective, namely, the objective of maximizing the probability of achieving a pre-specified profit target. Considering the analytical complexity of the problem, no closed-form solution may be available. Therefore, we introduce and discuss two solution approaches based on marginal analysis, numerical optimization and a derivative-free search method, namely, the Nelder–Mead algorithm. To address these aspects, the paper is organized as follows: In Section 2, we briefly discuss the background of the problem addressed, while in Section 3 the problem is formulated. In Section 4, we discuss the satisficing conditions, that is, the conditions that should be met to attain a pre-specified objective, while in Section 5 we illustrate the

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Please cite this article as: Pinto R. Stock rationing under a profit satisficing objective. Omega (2016), http://dx.doi.org/10.1016/j. omega.2015.12.008i

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2

heuristic procedures aiming to solve the maximization problem. We then report the results of numerical testing in Section 6, discussing the benefits and drawbacks of the two procedures. Finally, in Section 7, we report the conclusions, discuss the limitations, and propose future directions for improvement.

2. Background In a context in which a centrally procured scarce resource must be distributed to different retail stores to reach the final market, the allocation decision is particularly relevant in situations in which the central decision maker has only one opportunity to allocate the stock (i.e., single-period problems) because of the brevity of the selling season or the shortness of the product shelf life (i.e., style goods, fashion apparel, Christmas toys, and dairy or perishable products). In such contexts, there may be no future opportunities to recover from incorrect allocations. In allocating a limited stock, a company may pursue various objectives. For example, Balakrishnan et al. [4] argued that the objective in capacity rationing problems in make-to-order manufacturing firms is rather similar to the objective in perishable asset revenue management (PARM) problems, which is typical of the service operations management field. In any situation of fixed capacity and a perishable service or product, firms want to avoid spoilage of the service or product, pursuing the objective of realizing the most revenue possible while facing an uncertain demand [5]. Papier and Thonemann [6] considered the case of a rental company offering two service levels (classic and premium) entailing different prices and types of delivery guarantee. In this setting, the company has to decide under which conditions it should ration its limited fleet capacity to classic customers with the objective of increasing the service level of premium customers while meeting the guarantee. Pinto [7] discussed an expected profit maximization case, modeled as a newsvendor-like problem, presenting a multi-step algorithm for stock allocation with service level constraints, while Klein and Kolb [8] considered a firm that wants to optimally allocate fixed and limited capacity to heterogeneous customer segments with the aim of maximizing customer equity, defined as the total value of all current and future customers. Through a Markov decision process formulation, the authors studied the trade-off between short-term attainable revenues and long-term customer relationships. Further examples of objectives are given in the context of centralized decision-making by Fang [9], who illustrated a new approach for performing resource allocation based on efficiency analysis with the aim of optimizing the operations of all of the units, simultaneously reducing the total inputs on a global basis, and Karsu and Morton [10], who discussed the balanced distribution of a common resource, providing a bicriteria framework to think about trading balance off against efficiency. Finally, Turgay et al. [11] formulated a robust stochastic dynamic program to investigate the maximization of the expected total profit in a system serving different customer classes assuming demand and production rates characterized by uncertainty. The maximization of the expected profit is probably one of the most common and perhaps intuitive optimization objectives adopted, as it involves cost minimization [12] that represents a necessary but not sufficient condition for profit maximization. In general, the expected profit maximization objective is used because it is based on risk neutrality [13]. Unlike the previous cases, in this paper, we address the rationing problem using the objective of profit satisficing, namely, the objective of maximizing the probability of achieving a prespecified profit target [13,14]. The reasons motivating this assumption stem from the realization that, in many managerial situations, a budgeted profit is

established, and the disutility resulting from not achieving this targeted profit level is much larger than the rewards for overachieving. A manager may then be interested primarily in maximizing the probability of meeting the budget, regardless of whether the target level is exceeded or barely attained [14]. As argued in [15], budget attainment is sometimes a more accurate characterization of the decision-making process. Profit satisficing is not a new topic: earlier contributions date back to the 1950s, with the often-cited works by Lanzillotti [16] and Simon [17]. Although different contributions can be found addressing the satisficing objective in different contexts such as inventory placement [18], newsboy-like settings [19,20,13], and supply chain contracts [21–23], to the best of the authors’ knowledge, this objective has never been considered in stock rationing. Therefore, it represents the major contribution of this paper.

3. Problem formulation In this section, we outline the problem formulation providing the necessary assumptions. To this end, consider a company that distributes a single product to the final market through a commercial channel composed of n Z 2 proprietary retailers. The company implements a centralized procurement strategy, and has a stock of A units to be distributed completely to the retailers before the selling season starts. We consider a single-period setting, in which the allocation decision cannot be revised during the selling season. We also exclude the possibility of partial allocation followed by a second allocation after the demand is revealed. Due to these assumptions, it is rational to distribute the whole quantity A among the retailers because each unit kept at the central level will not contribute to attaining the company’s satisficing objective. The retailers sell their allotted quantity to the market until depletion, and the unmet demand is lost. Each retailer sells the product to the final market at a fixed, exogenous price p known in advance (at least, just before the selling period). We assume a unique selling price p for all retailers as common for the context and the type of company under discussion. The cost of each unit for the company is c. At the end of the selling season, the unsold quantities are scrapped. For the sake of simplicity of exposition and without loss of generality, we assume no salvage value for the leftovers and no stock-out (goodwill) cost. We also assume that the transportation costs to the retailers are non-differential or negligible. For products with a short life cycle, the stock-holding costs are assumed to be negligible. Each retailer i faces a stochastic demand Di with known density f i and cumulative distribution F i defined in ½0; αi Þ (or ½0; αi ) to exclude negative demand. Further, we assume that the demands are independent because retailers enjoy local monopolies due to their geographical dispersion and are not in direct competition. These assumptions are commonly supported in the considered problem setting because the company has control over the location of its retailers; therefore, the company would not locate the retailers too close to each other. In this study, we do not explicitly consider the presence of competitor retailers from other companies. Alternatively, we can consider that the demand distributions f i already account for the presence of competitors in the region. We define the allocation of the stock A among n retailers as a real-valued vector Q ¼ ðQ 1 ; …; Q n Þ, where Q i Z 0 is the quantity allotted to the ith retailer. We assume that a simple rounding of values in Q does not introduce significant deviations from optimality. This assumption especially holds when the components of Q are not too small (i.e., few units) and the price is not

Please cite this article as: Pinto R. Stock rationing under a profit satisficing objective. Omega (2016), http://dx.doi.org/10.1016/j. omega.2015.12.008i

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substantially high (i.e., as in the case of limited edition, luxury products, which in general are sold using different channels). Under the introduced assumptions, by allotting the quantities P Q i to the retailers, such that i Q i r A, the overall profit of the system composed by the company and the retailers is expressed as

π ðQ 1 ; …; Q n Þ ¼ p U

n X

minðQ i ; Di Þ c U A

ð1Þ

i¼1

where minðQ i ; Di Þ is the actual quantity sold by the ith retailer at the end of the selling season. We are interested in determining the allocation Q ¼ ðQ 1 ; …; Q n Þ such that the probability P of attaining the given target profit B is maximized, that is max

Pðπ ðQ Þ Z BÞ

Q ¼ ðQ 1 ;…;Q n Þ

words, the maximum profit is attained when all retailers sell their allotted quantity in full. Therefore, we are interested in the probability of the event: D1 Z Q 1 \ D2 Z Q 2 \ … \ Dn Z Q n ð3Þ P provided that i Q i ¼ A. Thus, the allocation Q must maximize the probability of selling the whole quantity Q i at each retailer. Assuming independent demands, the probability of the event in Eq. (3) can be written as P ðD1 Z Q 1 \ D2 ZQ 2 \ … \ Dn Z Q n Þ ¼ P ðD1 Z Q 1 Þ∙P ðD2 Z Q 2 Þ∙…∙P ðDn Z Q n Þ n

¼ ∏ ½1  F i ðQ i Þ

ð2Þ

P subject to the general availability constraint i Q i r A. As is immediately verified, the target profit B is bounded from above by the maximum profit attainable BMAX ¼ ðp  cÞ∙A. Considering the structure of the problem expressed by Eq. (2) P and the availability constraint i Q i r A, we can state the following: Proposition 1. Considering a system of n retailers sharing a centralized procurement and facing independent demands Di in the intervals ½0; αi Þ, a rationing problem arises when the centrally available stock A cannot guarantee the a priori satisfaction of all the demands in full. In this sense, a rationing problem formally exists if P and only if i αi 4 A. P Proof. Let us assume i αi r A. Then, it is possible to find an allocation Q such that Q i Z αi 8 i. Therefore, the quantity A can be distributed among the retailers without raising rationing issues because each Q i exceeds the maximum value of the related P demand distribution f i . Conversely, if i αi 4 A, then it is possible to find an allocation Q such that for some retailers the inequality Q i o αi holds, implying the risk of not fulfilling the demand in full. Solutions with Q i 4 αi are implicitly ruled out because the quantity Q i  αi will never be sold at retailer i and can then be conveniently moved to another retailer j where Q j o αj : When a rationing problem exists in the sense defined by ProP position 1, we can consider the positive difference i αi  A as a measure of the potentially uncovered demand, or demand at risk. As a consequence, to minimize such a risk, we would allocate the P whole quantity A, leading to the availability constraint i Q i ¼ A. Thus, we will use this formulation in the remainder of the paper. Given Eqs. (1) and (2), we start discussing the cases B ¼ BMAX and B o BMAX separately in the next section, due to the different implications these cases entail.

4. Satisficing conditions Given the problem formulation in the previous section, we discuss the conditions for attaining the satisficing objective in Eq. (2). Due to the uniqueness of the selling price p, these conditions are based on the probability of selling a given quantity of product at each retailer confronted with a known demand distribution. 4.1. Case B ¼ BMAX Case B ¼ BMAX requires maximizing the probability of attaining the maximum profit. According to our assumptions, since we do not consider the cost of lost sales, such maximum profit is attained n P minðQ i ; Di Þ ¼ A, that is, when Di Z Q i 8 i. In other only when

3

ð4Þ

i¼1

The maximization of Eq. (4) is subject to the availability constraint and the non-negativity constraints: P Qi ¼ A i

Qi Z0

8 i ¼ 1…n

ð5Þ

The problem described by Eqs. (4) and (5) can be addressed via Lagrange multipliers, as illustrated in Example 1. Example 1. Let us consider the case of two retailers, each facing a uniformly distributed demand in the interval ½0; αi  with i ¼ 1; 2. Thus, the cumulative probability distribution of the demand at each retailer is F i ðxÞ ¼ αxi , and Eq. (4) can be written as max     P ðD1 Z Q 1 \ D2 ZQ 2 Þ ¼ max 1  Q 1 =α1 U 1  Q 2 =α2 . Applying the Lagrangian multipliers method with some algebraic manipulations, we obtain the optimal solution Q 1 ¼ A þ α21  α2 ; Q 2 ¼ A þ α22  α1 for A Z jα1  α2 j (while we have to check the boundary values otherwise), that for α1 ¼ α2 gives Q 1 ¼ Q 2 ¼ A2. The same procedure can be followed for problems of size n 4 2, and similar results can be obtained with other distributions, either analytically or numerically. 4.2. Case B o BMAX The case B oBMAX entails more intricacies in formulating the satisficing conditions. In fact, while in the case B ¼ BMAX the only suitable event is the one such that Di ZQ i 8 i, in case B o BMAX it is possible to have solutions where Di o Q i for some i as long as P p U minðQ i ; Di Þ  c U A Z B. In fact, to achieve the target profit B  i  B o BMAX ; selling the total quantity A is not strictly requested. For the sake of clarity, we start discussing the case n ¼ 2; we then generalize to higher problem dimensions. In the following analysis, because the term c∙A is constant and does not depend upon the decision about Q (i.e., it represents a non-differential sunk cost), to ease the notation and w.l.o.g, we remove the term c∙A from the formulation, accommodating the target B accordingly. Let us consider two retailers facing demands distributed in the intervals ½0; α1 Þ and ½0; α2 Þ. To find a solution to the general problem expressed by Eq. (2) subject to constraints (5), we have to define the selling levels that allow us to attain the profit B. Because P the total sales level is expressed as minðQ i ; Di Þ, we have to i

investigate the distribution of the sum of the minimum between the allotted quantities Q and the demand, represented by the random variables Di . We can isolate the events that represent a solution to the problem. Given n ¼ 2, such events are

 Event E1 : D1 Z Q 1 \ D2 Z Q 2 . This event occurs when the entire quantity A is sold, attaining the profit BMAX , therefore exceeding the requested profit B. Because demands are independent, as

i¼1

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already shown for the case B ¼ BMAX ; the probability of this event is P ðE1 Þ ¼ P ðD1 ZQ 1 Þ∙P ðD2 Z Q 2 Þ ¼ ½1  F 1 ðQ 1 Þ∙½1  F 2 ðQ 2 Þ

ð6Þ

 Event E2 : D1 o Q 1 \ D2 Z Q 2 . In this case, we attain the profit B only if D1 þ Q 2 Z Bp, that is, if Bp  Q 2 rD1 o Q 1 . The probability of this event is therefore  þ B Q2 P ðE 2 Þ ¼ P r D1 o Q 1 U P ðD2 Z Q 2 Þ p    þ  B Q2 ¼ F 1 ðQ 1 Þ F 1 U ½1 F 2 ðQ 2 Þ ð7Þ p



h iþ   where Bp  Q 2 stands for max 0; Bp  Q 2 . Event E3 : D1 Z Q 1 \ D2 oQ 2 . This event is symmetrical to event E2 ; therefore  þ B Q1 r D2 o Q 2 U P ðD1 Z Q 1 Þ P ðE 3 Þ ¼ P p    þ  B Q1 ¼ F 2 ðQ 2 Þ  F 2 ð8Þ U ½1  F 1 ðQ 1 Þ p

 Event E4 : D1 o Q 1 \ D2 oQ 2 . In this case, profit B is attained only if D1 þ D2 Z Bp. The probability P ðE4 Þ involves the probability distributions of the random variables and the distribution of the sum of the random variables. We can formally write P ðE4 Þ as follows:  B P ðE4 Þ ¼ P D1 o Q 1 \ D2 o Q 2 \ D1 þ D2 Z p Z ¼ f 1 ðD1 Þ U f 2 ðD2 Þ dD1 dD2 ð9Þ Φ

where the integral is evaluated over the region Φ ¼ ðD1 ; D2 Þj  B D1 o Q 1 ; D2 oQ 2 ; D1 þD2 Z p . Depending on the distribution functions f i , the integral in P ðE4 Þ can be computed analytically or require numerical approximation. As can be verified, events Ej are mutually exclusive; therefore, P for n ¼ 2, we can write P ðπ ðQ Þ Z BÞ ¼ 4j ¼ 1 P Ej . In n ¼ 2 dimensions, the four conditions above can be visualized, as depicted in Fig. 1. The grayed areas represent the values of the demands D1 and D2 that allow the attainment of profit B.

Fig. 2. Value of the probability P as a function of Q1.

Example 2. Let us consider the same problem setting as in Example 1. Referring to Fig. 1, the probability of the event E4 can be expressed as ! Z Q2 Z Q1 Z min Q 1 ;B Z Q 2 p 1 dD2 dD1 þ dD2 dD1 P ðE 4 Þ ¼

þ B B α1 ∙α2 min Q 1 ;Bp 0 p Q2 p  D1 ð10Þ Considering Eqs. (6), (7) and (8) for the events E1 , E2 and E3 respectively, and substituting Q 2 ¼ A  Q 1 we obtain a single equation in the single variable Q 1 for the probability P ðπ ðQ Þ Z BÞ ¼ P ðπ ðQ 1 ; A  Q 1 Þ Z BÞ. For example, Fig. 2 depicts the values of P ðπ ðQ 1 ; A Q 1 Þ ZBÞ when α1 ¼ α2 ¼ 1000, B ¼ 1100, A ¼ 1400 and p ¼ 1; as it is possible to show with a bracket search in ½max ð0; A α2 Þ; α1 , there is a maximum at Q 1 ¼ Q 2 ¼ A2 ¼ 700. The left limit of the interval for the bracket search is obtained from the observation that Q 1 must be non-negative and Q 1 ¼ A  Q 2 Z A  α2 since there is no advantage in allotting a quantity Q 2 4 α2 to retailer 2. In the general case with n 4 2, however, it may be difficult to devise a similar expression even with the uniform distribution due to the difficulties in defining the limits of integration in Eq. (10). At a closer inspection of Fig. 1 and related equations, it is possible to infer the following: if the demand is “too low” at either retailer, there is no possibility to attain the desired profit. If, for example, D1 o Bp  Q 2 , then the maximum profit attainable is   p∙ðD1 þ Q 2 Þ o p∙ Bp  Q 2 þ Q 2 ¼ B. Conversely, if Bp Q 2 r 0, then for any value of D1 Z 0 there is the possibility to attain the profit B, as long as D1 þD2 Z Bp. In fact, if D2 Z Q 2 Z Bp, the target profit B is attained by definition, whereas if D2 o Q 2 , we have to assess the probability that the sum of D1 and D2 is large enough to attain the target profit (Fig. 3a). The same can be said in case Bp  Q 1 r0, thus leading to the following (Fig. 3b): the probability of Proposition 2. If Bp  Q i r 0 8 iA f1; 2g, then   attaining the required profit B is P ðπ ðQ Þ ZBÞ ¼ P D1 þ D2 Z Bp . Proof. If Bp  Q i r 0 8 i A f1; 2g, from Eqs. (6) to (9), we can write P ðπ ðQ Þ Z BÞ ¼ ½1  F 1 ðQ 1 Þ∙½1  F 2 ðQ 2 Þ þ F 1 ðQ 1 Þ U ½1  F 2 ðQ 2 Þ Z þ F 2 ðQ 2 Þ U ½1 F 1 ðQ 1 Þ þ

Fig. 1. Graphical representation of the demand values allowing the attainment of the profit B.

Φ

f 1 ðD1 Þ U f 2 ðD2 Þ dD1 dD2 ð11Þ

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As a consequence of Proposition 2, it is possible to verify that, as long as the condition Bp  Q i r 0 holds 8 i A f1; 2g, the probability P does not change upon changes in the allocation Q . Further, the following condition applies: Proposition 3. If Bp r A2, then there exists at least one allocation Q ¼

 Q 1 ; Q 2 such that Bp  Q i r 0 holds true 8 i A f1; 2g. Proof. Let us first prove that if Bp 4 A2, then the condition Bp  Q i r0 8 iA f1; 2g cannot be verified. In fact, we would have Q i Z Bp 4 A A A 2 8 i A f1; 2g, leading to Q 1 4 2 and Q 2 4 2, thus violating the constraint Q 1 þ Q 2 ¼ A.

 To prove that when Bp r A2, at least one allocation Q ¼ Q 1 ; Q 2 such that Bp Q 1 r 0 and Bp  Q 2 r 0 exists, let us define L ¼ A2  Bp Z 0. By construction, any   two numbers θ1 and θ2 belonging to the interval A2  L; A2 þ L and such that θ1 þ θ2 ¼ A represent a valid solution for the condition to hold. Clearly, the allocation is unique when L ¼ 0.▫ 4.3. The general case B rBMAX with n Z 2 The discussion presented in the previous section can be extended to higher dimensions, involving more than two retailers. To this end, examining events Ej , we can recognize the following pattern: E1 is the event in which no retailer has leftovers, while events E2 and E3 are the events in which exactly one retailer at a time has leftovers; finally, event E4 is the event in which both retailers have leftovers. Extending this pattern to n 42 retailers, we have

 One event representing the case of no leftovers.  One event representing the case of all retailers with leftovers.  n events representing the case in which exactly one retailer has leftovers.

 A number of events representing the case in which exactly two Fig. 3. Change in the feasible region depending on Q i values.



Simplifying, we obtain P ðπ ðQ Þ Z BÞ ¼ 1  F 1 ðQ 1 Þ UF 2 ðQ 2 Þ þ

Z Φ

f 1 ðD1 Þ Uf 2 ðD2 Þ dD1 dD2

ð12Þ

The term 1  F 1 ðQ 1 Þ∙F 2 ðQ 2 Þ is the probability that at least one retailer sells the whole quantity, that is, 1  P ðD1 r Q 1 \ D2 r Q 2 Þ (thus equivalent to the probability of events E1 , E2 and E3 in Fig. 3b), while the integral term is the probability that both retailers sell less than their allotment, still attaining the target profit

together.

We

can write the integral term as P   ðD1 r Q 1 \ D2 r Q 2 Þ  P D1 þ D2 r Bp (equivalent to the probability

of E4 ). Therefore, summing and simplifying, we obtain P ðπ ðQ Þ Z BÞ     ¼ 1  P D1 þ D2 r Bp ¼ P D1 þD2 Z Bp .▫ In general terms, Proposition 2 states that, if it is possible to allot the available quantity A to the 2 retailers in such a way that B p  Q i r0 8 i A f1; 2g, the probability P ðπ ðQ Þ Z BÞ can be calculated through the probability of the sum of the random variables representing the demand. In fact, in the case represented in Fig. 3b, we are interested in all the couples ðD1 ; D2 Þ such that D1 þD2 Z Bp, regardless of the value of D1 and D2 with respect to Q 1 and Q 2 . Vice versa, if the condition Bp  Q i r 0 8 i A f1; 2g does not hold, there are couples ðD1 ; D2 Þ such that D1 þ D2 o Bp; thus, it is required to consider the minimum between Di and Q i .

out of n retailers have leftovers. As combinatorics tells us, this  n number is . 2 A number of events representing the case in which exactly three  n out of n retailers have leftovers, that is, , and so forth. 3

Therefore, the total number of events to be considered is given by the sum of all the possible combinations of retailers with and without leftovers at the end of the season. Given n Z2 retailers,  n n P ¼ 2n , and we can express Eq. (2) in the such a number is i i¼0 general case as 2 X P Ej n

P ðπ ðQ Þ Z BÞ ¼

ð13Þ

j¼1

4.4. Generalization and resulting formula for the probability P ðπ ðQ Þ Z BÞ To evaluate the probability P ðπ ðQ Þ Z BÞ in the general case, we must consider all the possible events of interest according to Eq. (13) and evaluate their probability. In this section, we provide a general formula expressing the probability P in an analytical form. To this end, let S ¼ f1; …; ng be the set of n retailers, P ðSÞ the

powerset of S, and H ðSÞ ¼ ω A P ðSÞjjωj 4 1g, that is, the powerset of S except the empty set Ø and all the subsets of cardinality 1.

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Summarizing the discussion from the previous sections, we can write ! X minðQ i ; Di Þ Z B ¼ GðS; Q ; B; pÞ ð14Þ P pU iAS

where the function Gð∙Þ is defined as

82 02 3 þ 13 X< X 4F j Q j  F j @4B  Q i 5 A5 GðS; Q ; B; pÞ ¼ ∏ ½1  F i ðQ i Þ þ : p iAS jAS i A Snfjg

) ∏ ½1  F i ðQ i Þ

i A Snfjg

þ

8 > Z X < >

ω A HðSÞ: Φω

9 > =   ∏ f i ðDi Þ dD1 …dDn > ; iAS ð15Þ

P

B iQ i Z p

and 0 otherwise. for The region of integration Φω of the last summand in Eq. (15) is 8 < Φω ¼ ðD1 ; …; Dn Þj0 rDi r Q i 8 i A ω; Dj 4 Q j 8 j A S n ω; : 9 X X B= ð16Þ Dk þ Qh Z p; kAω

h A Snω

In Eq. (15), the first summand represents the case Di Z Q i 8 i, that is, the probability of selling the whole quantity A. The second summand, instead, represents the cases in which exactly one retailer j at a time sells less than the allotted quantity Q j (i.e., Dj rQ j ), while the others sell their whole allotment. Finally, the third summand represents the cases in which two or more retailers, represented by the subset ω, are not able to sell their allotment in full. In fact, at a closer analysis, the set H ðSÞ represents all the possible subsets of two or more retailers with leftovers.  As we n Pn can immediately infer, the cardinality of H ðSÞ is ¼ i¼2 i n 2  ðn þ 1Þ. The existence of a closed-form solution of the integrals involved in (15) depends on the distribution functions f i and on the limits of integration, as shown in Example 2. In the eventuality that a closed-form solution does not exist, or is too hard to find, or the limits of integration are hard to define analytically in higher dimensions, the integrals in Eq. (15) should be evaluated numerically, for example, adopting a Monte Carlo (MC) approach. The MC method is a well-known approach suitable for numeric integration, where a multi-dimensional integral is estimated via a sample mean computed over m independent, random samples. An appealing aspect of MC integration is that it provides a probabilistic error bound  proportional to the number of samples used, in the 1 form O m  2 , independent of the number n of dimensions considered [24]. Although the MC method is not the only numerical method for integral approximation, the analysis and discussion of the method’s benefits and limitations are beyond the scope of this paper. Therefore, we posit MC integration is a valid method for solving the formulated problem, as further discussed in Section 5. Finally, we can extend to n-dimensional problems the result provided in Proposition 2. In fact, we can generalize Proposition 2 as follows: with jSj Z 2, the probability of ! P B Di Z p . attaining the required profit B is P ðπ ðQ Þ Z BÞ ¼ P Proposition 2.* If

B p  Q i r 0 8 i A S,

iAS

In fact, if the condition Bp  Q i r 0 8 iA S holds, any n-tuple ðD1 ; …; Dn Þ such that D1 þ … þ Dn Z Bp represents a feasible solution, regardless of the value of each Di with respect to Q i . Similar to Proposition 3, we can also state the following:

Proposition 3.* If Bp r jASj, then there exists at least one allocation Q

 ¼ Q 1 ; …; Q S such that Bp  Q i r 0 holds true 8 i A S. The proofs of Propositions 2* and 3* are the natural extension to higher dimensions of the proofs of Propositions 2 and 3. Example 3. To illustrate the application of Eq. (15), let us consider a set S ¼ f1; 2; 3g of three retailers. The first term of Gð∙Þ is the probability of selling the whole quantity, that is ð17Þ ∏ ½1  F i ðQ i Þ ¼ ½1  F 1 ðQ 1 Þ U ½1  F 2 ðQ 2 Þ U ½1  F 3 ðQ 3 Þ:

iAS

For the further summands in Eq. (15), let us start with j ¼ 1 as the retailer that does not sell the whole quantity Q 1 . The profit B is attained only if

 All the other retailers in S (that is, all retailers i A S n f1g) sell their whole allotment Q i , and

 hRetailer 1 isells at least a quantity D1 between Q 1 and þ B p Q 2  Q 3

so that D1 þ Q 2 þ Q 3 Z Bp.

Thus, the probability of such an event is    þ  B Q2 Q3 F 1 ðQ 1 Þ  F 1 ∙½1  F 2 ðQ 2 Þ∙½1  F 3 ðQ 3 Þ: p

ð18Þ

The same reasoning must be applied to each retailer jA S. Finally, we have to consider the cases in which more than one retailer (the case of all retailers included) sell less than the allotted quantity. As stated in the previous section, the elements in H ðSÞ ¼ ff1; 2g; f1; 3g; f2; 3g; f1; 2; 3gg represent the sets of retailers not able to sell their entire allotment. Let us start with ω ¼ f1; 2g. In this case, it is assumed that retailer 3 sells the entire quantity, and profit B is attained only if the sum of the demand at retailers 1 and 2 exceeds the quantity Bp  Q 3 so that D1 þ D2 þ Q 3 Z Bp. As discussed before, since demands are independent, the probability of this event is obtained by multiplying f i and integrating over the region   B Φf1;2g ¼ ðD1 ; D2 ; D3 ÞjD1 r Q 1 ; D2 r Q 2 ; D3 Z Q 3 ; D1 þ D2 þ Q 3 Z p ð19Þ The same reasoning must be applied to each subset ω in H ðSÞ, obtaining the final expression.

5. Heuristic procedures for the profit satisficing objective Considering the model illustrated in the previous section, our final goal is to find an allocation Q that maximizes Gð∙Þ. The main issue relies on the fact that Gð∙Þ is generally hard to optimize using an analytical approach. This is due mainly to the distribution functions f i , for which a closed-form solution of the integral may not be available, and to the definition of the integration limits in higher dimensions (i.e., for n 4 2). However, the structure of the problem suggests that a greedy or marginal allocation heuristic (i.e., a procedure which assigns available units sequentially to the retailer that has the highest probability of selling each additional unit without creating infeasibilities) may be effective [25]. Therefore, in the next sections we address this aspect defining a greedy procedure for which we provide some computational results in Sections 6.1, 6.2 and 6.3. In particular, we compared the effectiveness of the greedy procedure against the results provided by an exhaustive search. However, due to the time it requires to complete on fairly large problem instances, an exhaustive search method is not always a suitable option: therefore, we implemented an exhaustive search only for the case with three retailers.

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7

Input:

1.

Input:

2.

Output: allocation , probability

2.

Output: allocation , probability

3.

Set

3.

For in

4.

If

1.

5.

: Evaluate

6.

If

+1

is not unique, break tie randomly

7.

+1

:

4.

Generate initial solution

5.

Evaluate

6.

If stopping criteria are met:

7.

allocating

to

Go to step 12.

8.

Else:

9.

Generate next candidate solution

8.

10.

Scale candidate solution

9.

11.

Go to step 5.

10. 11. 12.

Go to step 4

12.

Else: Evaluate

according to rule (20)

Update best solution

13.

Next

14.

Show best solution and exit

Show best solution and exit Fig. 5. Nelder–Mead-based allocation procedure. Fig. 4. Greedy allocation procedure.

In order to provide a comparison term also for larger problem instances, we implemented another heuristic procedure aiming to find good results with a reasonable computational effort. This second heuristic is based on the combination of two well-known methods: (i) a Monte Carlo integration method for evaluating the numerical value of Gð∙Þ and (ii) a derivative-free search method, namely, the Nelder–Mead simplex method. We briefly introduce the greedy and the Nelder–Mead procedures in the next two sections. 5.1. The greedy (marginal allocation) procedure In this section, we introduce a simple procedure to allocate the quantity A to the retailers according to a marginal analysis. The basic idea is to allocate progressively each unit to the retailer that has the highest probability to sell it. Such a procedure can be proved to attain the optimal allocation for the maximization of the expected sales [25]. Although the objective discussed in this paper is substantially different, a marginal allocation procedure is an appealing solution approach, thanks to its simplicity, intuitive functioning and speed of execution. Therefore, we implemented and tested the greedy allocation procedure illustrated in Fig. 4 to assess its suitability for the problem discussed in this paper. In doing this, we also aimed at understanding whether the expected sales maximization and the profit satisficing were related objectives. The effectiveness of this approach with regard to the maximization of Gð∙Þ is discussed in Section 6; briefly anticipating the results presented there, we can state that, although the maximization of the expected sales represents neither a sufficient nor a necessary condition for the maximization of Gð∙Þ, it usually produces very good results. 5.2. An alternative, derivative-free heuristic In this section, we describe an approach for maximizing Gð∙Þ P subject to the availability constraint i Q i ¼ A and the nonnegativity constraints Q i Z 0 with reasonable computational effort and time, especially if compared with an exhaustive search procedure. The aim of this approach is to provide a reliable comparison term for the greedy procedure when dealing with larger problem instances than those solvable with an exhaustive approach.

To limit the computational effort and time, we opted for a derivative-free (or direct) method, a numeric optimization algorithm that neither requires nor explicitly approximates the calculation of the gradient of the objective function [26]. Compared with other, gradient-based methods, direct methods can usually better escape local optima, and usually require fewer function evaluations. Further, they are well suited when the derivatives of the function are unavailable or unreliable [27]. In our case, specifically, the computation of the gradient of the function Gð∙Þ should be performed numerically, requiring substantial computational effort and time. We selected the Nelder–Mead (NM) [28] simplex algorithm because it is one of the most used, most studied, and most robust direct search methods for unconstrained optimization applied in many research fields [29,30]. Although the NM algorithm may fail to converge to a local optimum, the algorithm may escape regions that would be basins of attraction for a pointwise descent search [31]. The NM algorithm can be used in a multi-start search, possibly reducing or overcoming the risk of non-convergence. The NM simplex algorithm provides a well-defined set of rules to generate new candidate solutions. This is achieved through a set of operations to rescale, reflect and reshape the simplex, based on the local behavior of the function [29]. These operations are associated with scalar parameters used in manipulating the vertices of the simplex: α (reflection), β (expansion), γ (contraction), and δ (shrink). The values of these parameters are customarily chosen to be α ¼1, β ¼ 2, γ ¼0.5, and δ ¼0.5. For further details on the implementation, see [29,32,33]. The procedure we devised, summarized in Fig. 5, starts allocating the available quantity A to the retailers. This initial allocation can be performed randomly or according to other, specified rules. This first allocation Q is plugged into Gð∙Þ and evaluated using the analytical calculations and the Monte Carlo integration. Then, the NM algorithm generates a new candidate solution Q~ starting from the current solution Q . This step may generate an issue: In fact, the original NM algorithm addresses unconstrained problems. This means that the variables can assume any value after the rescale and reshape operations are executed. In our case, we have to guarantee positive or zero values of Q and the availP ability constraint i Q i ¼ A. Although specific adaptations of the NM algorithm exist for box-constrained problems ensuring the positiveness of Q , they cannot address the availability constraint addressed here. Therefore, we need to modify the standard procedure to take this  constraint into account. To this end, the can didate solution Q~ ¼ Q~ 1 ; …; Q~ n emerging from an iteration of the

Please cite this article as: Pinto R. Stock rationing under a profit satisficing objective. Omega (2016), http://dx.doi.org/10.1016/j. omega.2015.12.008i

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8

NM algorithm is modified 8 ~ > < P Q i ~ UA if maxðQ j ;0Þ j Qi ¼ > : 0 if

using a proportional scaling as follows: Q~ i 4 0 Q~ i r 0

8 i ¼ 1; …; n

ð20Þ

In this way, the following outcomes are achieved:

 Negative values are converted into zeros, thus removing the problem of negative values in the solution.

 If the sum of the Q~ i s exceeds A, the Q~ i s are scaled down 

assigning A proportionally to their original value as determined by the NM algorithm. Similarly, if the sum of the Q~ i s is less than A, the Q~ i s are scaled up, again with a proportional allocation.

Thus, through Eq. (20) all the constraints consistently hold, regardless of the values in Q~ generated by the NM algorithm. The modified solution Q ¼ ðQ 1 ; …; Q n Þ is then plugged into Gð∙Þ and evaluated again. The process is repeated until a termination criterion is met (i.e., number of iterations, running time, size of the simplex, maximum distance between the values of the vertices). In our tests, to enforce the convergence of the procedure, we set as the main termination criterion the maximum distance ε between the vertices. We set ε, as is customary in the literature, to 0.001 [27]. Given the heuristic nature of the NM algorithm, it does not guarantee the optimality of the solution. To overcome this limitation, the search procedure is repeated T times, each starting from a different initial point (i.e., a different initial random allocation). The best solution of all runs is then retained.

6. Computational results We tested the performance of the proposed procedures using a set of randomly generated problems. To this end, we considered three main cases: (i) in the first case, we assumed a set of retailers facing the same demand distribution with the same mean and standard deviation. With this problem setting, an optimal allocation is given by allotting the same quantity to all the retailers (i.e., equal allocation); thus, we know the value of an optimal allocation and the related probability Gð∙Þ to compare with the solution provided by the proposed procedures; (ii) in the second case, we considered a set of retailers facing demands with the same distribution but with different parameters. Within this case, we performed an optimality gap analysis in the setting with three

retailers, comparing the heuristics solutions against the optimal solution found through an exhaustive search; (iii) finally, the third case extends the second case to settings with six and nine retailers, and larger values of A, for which an exhaustive search was unsuitable. In the third case, with no optimal solutions available, we compared the solutions of the two procedures against each other. In all cases, we assumed a truncated normal distribution, as an example of a general distribution with no closed-form integral expression. The heuristic procedures have been implemented in C, and the tests have been performed on a Core i5 computer. We report the results of the three cases in Sections 6.1, 6.2 and 6.3, respectively. Since the assessment of the computational effort required by the heuristics is also important from both the researcher and practitioner perspectives, we report some results in Section 6.4. 6.1. Case I: retailers facing the same demand distribution The first case allowed us to assess the convergence of the proposed procedures to a known optimal solution. The problem instances have been generated combining the following parameters:

 Number of retailers n A f3; 6; 9g each facing a truncated normal distribution in the interval ½0; 1000.

 Average demand μ A f250; 500; 750g.  Standard deviation σ of demand: The standard deviation has been calculated in proportion to the average demand μ considering 

different coefficients of variations cvA f0:1; 0:25; 0:50; 0:75; 1:0; 1:25g, that is, σ ¼ μ∙cv. Coefficient of availability a A f1:0; 1:1; 1:2; 1:3g, used to define Pn the quantity available as A ¼ a∙ i ¼ 1 μi . B p

 Ratio ρ ¼ A A f0:7; 0:8; 0:9; 0:95g, to define the target B as a fraction ρ of the availability A.

Each of the resulting 3  3  6  4  4 ¼864 problem instances has been solved 20 times with the Nelder–Mead-based procedure (NM procedure henceforth) starting from different initial points. The greedy procedure, instead, has been run only once for each instance, since its results depend upon the demand distribution and do not change from run to run (i.e., given the number of retailers, the demand distribution and the available quantity, the allocation is deterministic regardless other problem parameters). For each solution, we computed the deviation from the optimal  value as vopt  vheu j, where vopt represents the known optimal solution and vheu the respective heuristic solution, both expressed

Fig. 6. Percentage of the instance results within given deviation thresholds (deviation between the NM procedure and the optimal value, expressed in the range [0, 100]).

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in the interval ½0; 100. We set m ¼100,000 iterations for the MC integration. Overall, the proposed NM procedure converged to a solution with a maximum deviation of 1.0 in about 81% of the 20  864 tests; the overall median value is 0.1. These results vary according to the size of the problems, as shown in Fig. 6. As is possible to observe, for a given deviation, the percentage of solutions decreases as the number of retailers increases, providing preliminary evidence for the fact that the Nelder–Mead algorithm may become less efficient in higher dimensions [30,34]. To appreciate the effectiveness of the NM procedure, however, for each problem instance we analyzed the best solution out of the 20 attempts; as a result, in the 98.6% of the 864 problem instances, the heuristic was able to find at least one solution out of 20 attempts with a deviation smaller than 0.01, and in 100% of the instances, the heuristics found at least one solution with a maximum deviation of 0.2 from the real optimum, regardless of the number of involved retailers. These results confirm the effectiveness of the proposed NM procedure in finding optimal solutions in the case at hand. Considering the greedy procedure, the results were even better: on the same set of problems, the greedy procedure attained the optimal solution in 100% of the cases, allotting the same quantity to all the retailers. Although this is a rather intuitive outcome considering the structure of the procedure, it is a noteworthy result, considering that the greedy procedure requires substantially less computational time (see Section 6.4) and does not require multiple starting points. In fact, the greedy procedure has the appealing characteristic of requiring only one evaluation of the function Gð∙Þ at the end of the allocation procedure. However, the greedy procedure requires the repeated evaluations of the argmaxi ðP ðDi Z Q i þ1ÞÞ (see Fig. 4) to define the best allocation of each unit. Although such a computation individually requires a relatively small amount of time, the computational effort increases directly with the value of A and with the number of retailers. 6.2. Case II: optimality gap analysis for the 3-retailer instance with different demand distributions The case with different demand distributions can be extremely difficult to solve at optimality, as there are neither robust optimality conditions for the solutions found by the heuristics nor a convergence guarantee for the NM method. In this situation, possibly the only approach that guarantees providing the optimal solution is an exhaustive search. Exhaustive search methods are not always suitable due to the time they require to complete on large problem instances. Nonetheless, in the attempt to assess the optimality gap for at least small problem instances, we implemented an exhaustive search for the case with three retailers and A ¼ 150. To this end, given a 3-retailer problem instance with different demand distributions,

we considered all the integer values ðQ 1 ; Q 2 ; Q 3 Þ A N3 jQ 1 þ Q 2 þ Q 3 ¼ Ag (as discussed in Section 3, we assume that a simple rounding of values in the solution does not introduce significant deviations from optimality), and for each triple ðQ 1 ; Q 2 ; Q 3 Þ, we calculated the probability Gð∙Þ, thus identifying the best solution to be compared with the results of the heuristic procedures. We generated 150 random instances combining different values of the coefficient of variation and the ratio ρ. We then solved each instance through the exhaustive search and the proposed procedures, restarting the NM procedure 20 times from different initial points. The comparison of the results confirms the rather good performance displayed by the NM procedure in the case with same demand distributions: indeed, 87.1% of the solutions have a deviation smaller than 0.5 from the optimum, whereas the median value is 0.1. Further, in 98.5% of the 150

9

instances, the heuristic found at least one solution out of 20 attempts with a deviation smaller than 0.01, whereas 100% of the instances have been solved at least once with a maximum gap of 0.3 from the optimum. Considering the greedy procedure, the most interesting result emerging from this analysis is related to the fact that the greedy allocation is not dependent upon the target. Despite this, the values of the function Gð∙Þ (that instead depends upon the allocation and the target) were always very close to the optimum, confirming the goodness of the procedure with a minimum, median and maximum deviation equal to 0.11, 0.72, and 1.84, respectively. Therefore, the unique allocation provided by the greedy procedure represents a good “average” allocation leading to fairly accurate results, regardless the target pursued. 6.3. Case III: retailers facing different, random demand distributions In this case, the problem instances were generated by randomly assigning to each retailer the value of the average demand and the coefficient of variation. The definition of the availability A and the target B followed the same process discussed in Section 6.1 through the parameters a and ρ. For each problem size in the set f3; 6; 9g, we generated and solved 100 problem instances. However, in this second case, we did not have at our disposal the optimal solution to compare with. To prove the effectiveness of the NM procedure, we first investigated whether the solutions found were at least local optimums. To this end, we performed a further analysis perturbing each solution by randomly shifting small quantities between retailers and re-evaluating the resulting probability. The results of this analysis support the conclusion that the NM procedure converged to (at least) local optimums: in all cases, the perturbation of the solution provided by the heuristic lead to a solution within 0.0032  of 1the  optimal one, that is, within the theoretical error bound O m  2 for the Monte Carlo integration (see Section 4.4). With regard to the greedy procedure, despite the allocations are not dependent upon the target, the results were quite positive: the absolute deviations between the results of the NM and the greedy procedures were lower than 1.0 in about 84% of the instances, whereas there were few instances (about 6% of the sample) with deviations larger than 2.1 (Fig. 7). 6.4. Computational effort As reported in the previous sections, both the two procedures attained very good performance in terms of closeness to the optimum, with a slight superiority of the NM procedure with respect to the greedy procedure. However, the computational effort required to run the two procedures – measured in terms of CPU time on the machine used for testing – was substantially different. Regarding the computational effort, Fig. 8 and Table 1 summarize the CPU time required by the two procedures on different problem sizes (i.e. number of retailers and availability A). The results confirm the superiority of the greedy procedure in this respect; the greedy procedure is substantially faster than the NM procedure, and the difference increases significantly as the size of the problem increases. Moreover, it is also possible to appreciate that, while the time required by the greedy procedure increases linearly with the number of retailers and with the availability, the time required by the NM procedure increases steeply as the size of the problem increases. The computational effort required by the NM procedure on larger instances is also affected by a large variability, depending upon the selection of the starting point (Fig. 9).

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10

Fig. 7. Percentage of the instance results within given deviation thresholds (deviation between the greedy and the NM procedures, expressed in the range [0, 100]).

Fig. 8. CPU times comparison.

Table 1 Computational effort (CPU seconds) for different problem sizes. Availability A

Number of retailers

3 6 9

NM procedure average time per run (standard deviation)

Greedy procedure average time

1000

4000

7000

10,000

1000

4000

7000

10,000

2.98 (0.21) 14.46 (1.89) 32.41 (5.33)

3.37 (0.19) 17.16 (2.15) 39.34 (8.87)

3.67 (0.33) 16.52 (2.27) 41.44 (8.47)

3.74 (0.32) 17.63 (2.32) 44.17 (6.17)

0.28

0.96

1.66

2.35

0.55

1.92

3.31

4.69

0.82

2.88

4.94

6.98

However, the NM procedure resulted to be very effective when compared with the exhaustive search. In fact, on problem settings with three retailers and an availability A ¼ 150, it was possible to evaluate the function Gð∙Þ about 24 times per second. To provide a comparison term, each run of the NM procedure required on average 51 evaluations of Gð∙Þ, whereas with A ¼ 150 there were more than five thousands integer combinations of Q 1 , Q 2 and Q 3 requiring the evaluations of Gð∙Þ. For the sake of completeness,

Table 2 provides further details about the number of function evaluations for different sizes of the problem. 6.5. Discussion The comparison operated between the NM and the greedy procedures allowed to highlight their practical equivalence, with a slight superiority in terms of accuracy of the NM procedure at the

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cost of a substantially higher computational effort. Although the greedy procedure results in the same allocation regardless the target Bp, such an allocation represents an effective “average” solution, whose deviation, in terms of value of the function Gð∙Þ, from the results of the NM procedure (or from the optimum, when available) is usually below few percentage points. Furthermore, the negative impacts of larger errors resulted from the greedy procedure, compared to the higher accuracy resulted from the NM procedure, are counterbalanced by the benefits due to the substantial reduction in the computational time required. The numerical results suggest that the maximization of the expected sales represents neither a sufficient nor a necessary condition for the maximization of Gð∙Þ (i.e., the maximization of

the expected sales does not guarantee an allocation that maximizes Gð∙Þ; conversely, an allocation that maximizes Gð∙Þ does not necessarily coincide with the allocation that maximizes the expected sales). However, it practically produces very good results in a time-effective way. In summary, the main advantages of the greedy procedure over the NM procedure can be summarized as follows:

 Clear, step-by-step definition of the allocations: the allocations





Fig. 9. Distribution of the average running times for the NM procedure.

11

are built iteratively, comparing the marginal contribution of each unit to the overall goal. This makes the whole procedure more comprehensible if compared with the NM procedure, where the definition of candidate solutions follows rules that, although quite comprehensible as well, are basically random and, as such, possibly less trustworthy and understandable for the decision maker. Easiness of implementation: comparing the two procedures represented in Figs. 4 and 5, it is immediately evident that the NM procedure entails more activities with a higher degree of complexity. For example, the multi-start approach and the generation of the next candidate solutions require a workflow that is more complex than the workflow entailed by the greedy procedure. Therefore, the greedy procedure can be implemented more easily, even in standard spreadsheets. Low computational effort: as mentioned previously in the paper, the greedy procedure requires only one evaluation of the function Gð∙Þ for each target, at the end of the allocation procedure. Conversely, the NM procedure must evaluate the function Gð∙Þ for each candidate solution. Coupled with the necessity to run the NM procedure from different initial points to be confident that it has not been trapped in a local maximum, the computational effort required by the NM procedure is usually substantially higher. Although the number of computations required by the greedy procedure increases directly with the

Table 2 NM performance as number of evaluations of function G. Availability A Average number of function evaluations (standard deviation)

Number of retailers 3 6 9

Average number of function evaluations per second

1000

4000

7000

10,000

1000

4000

7000

10,000

71 (5.22) 178 (24.38) 267 (46.12)

80 (4.61) 212 (27.77) 323 (76.58)

87 (7.93) 204 (29.51) 341 (73.26)

88 (7.99) 218 (30.07) 364 (53.52)

23.71

23.72

23.69

23.65

12.33

12.36

12.38

12.38

8.23

8.22

8.22

8.23

Fig. 10. Maximum probabilities of achieving the target B/p.

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Fig. 11. Allocations as a function of the target B/p, given an availability of A ¼ 2000 units.

Fig. 12. Allocations and probabilities for varying average demand of retailer 3 (same cv for all demands).

Fig. 13. Allocations and probabilities for varying average demand of retailer 3 (same σ for all demands).

value of A and with the number of retailers, it hardly exceeds the computational time required by the NM procedure. From the practitioner perspective, the numerical procedures presented in this paper can be used to gain further insights into the problem and support decision makers in their tasks. For example, it is possible to analyze the variation in the probability function Gð∙Þ for

varying levels of the target Bp. For the sake of illustration, let us consider a case with three retailers 1, 2 and 3 facing truncated

normally distributed demands TN μ; σ in the range ½0; 1000, D1  TN ð250; 125Þ, D2  TN ð500; 375Þ, and D3  TN ð750; 187:5Þ, respectively, representing demands with a coefficient of variation (cv) equal to 0.5, 0.7 and 0.25. Repeating the NM procedure for varying levels of Bp between 0 and A and for different values of the stock

Please cite this article as: Pinto R. Stock rationing under a profit satisficing objective. Omega (2016), http://dx.doi.org/10.1016/j. omega.2015.12.008i

R. Pinto / Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎

availability A, we can describe the variation of the maximum probability Gð∙Þ and, with respect to the NM procedure, the allocations as a function of the target, as shown in Figs. 10 and 11, respectively. Clearly, the probability of attaining a given target Bp depends upon the availability A (Fig. 10). These types of charts may be helpful for managers in setting a realistic target for the pursued performance and to understand the impact of a loss/gain in the quantity A. As it is possible to infer observing the grayed area on the left of Fig. 11, for values of the target strictly smaller than A3, the allocations provided by the NM procedure seem erratic, while for values of the target greater than A3, the allocations follow a more regular pattern. This pattern can be justified by the light of Proposition 3*, which implies the existence of several alternative allocations that provide the same result in terms of the probability Gð∙Þ for “small” values of the target. Indeed, it is possible to verify that the allocations in the grayed area depend upon the starting point of the search procedure. Regardless of the starting point, the allocations follow a regular pattern for larger values of the target. Fig. 11 also reports the allocations obtained with the greedy procedure. It is worth noting that the allocations resulting from the NM procedure should be monotone in the parameter Bp; the fact that they are not strictly so in Fig. 11 has to be ascribed to several factors, such as the presence of different solutions with rather similar values of Gð∙Þ, the stochastic nature of the solution procedure, the numerical approximation of the integrals, and the stopping criterion of the heuristics. Nonetheless, considering the overall picture instead of the single number, the impact on the decision maker should be minimal. A different analysis can be performed to analyze the variations in the allocations and in the resulting probabilities as a consequence of the variation in the characteristics of the demands. To this end, let us consider two retailers facing demands D1  TN ð250; 125Þ, D2  TN ð750; 375Þ, and a third retailer facing a

generic demand D3  TN μ3 ; σ 3 . Given an availability of, say, A ¼ 2000 units and a target Bp ¼ 1600 (that is, 80% of the availability), we solved the sequence of problems obtained letting μ3 change in the interval ½150; 900 and keeping the coefficient of variation cv ¼ 0:5 the same for all the three demand distributions (i.e., setting σ 3 ¼ cv∙μ3 ). The results obtained using the greedy procedure are reported in Fig. 12 in terms of probability Gð∙Þ and allocations. A similar procedure has been performed keeping the same σ ¼ 200 for all the demand distributions and letting μ3 change as described previously. The results are reported in Fig. 13. Figs. 12 and 13 show how the allocations and the probabilities change as the mean value of the demand distribution at one retailer changes; similar analysis can be performed changing other factors or even using different demand distributions, which would impact both the allocations and the shape of the function Gð∙Þ. Clearly, it is not possible to be exhaustive in analyzing the possible results of these tests, whose interpretation may also depend on the goal of the decision maker. However, the results of these types of analysis, applied to tailored cases according to the interest of the decision makers, can provide support in investigating different scenarios and performing what-if analysis. Moreover, the greedy procedure can effectively support multiple analysis, thanks to the low computational time it usually entails.

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The objectives pursued with the allocation decision can be of different typologies. In this paper, we presented a formulation of the general problem of maximizing the probability of attaining a target profit under an availability constraint. The resulting model can be analytically intractable, and no closed-form solutions may be available. Therefore, we proposed and analyzed two heuristic procedures that exploit different approaches: one that leverages on two wellproven methods (i.e., Monte Carlo integration and the Nelder–Mead derivative-free search algorithm) and the other that exploits the structure of the problem to provide an “average” solution with good performance. For our purposes, the Nelder–Mead algorithm has been adapted to address the non-negativity constraints and the availability constraint, keeping the approach simple and of practical use. The proposed approaches are legitimated under several demand distribution functions subject to a few commonly adopted restrictions. Many of the usually adopted continuous distributions automatically fall within such restrictions. As emerges from the analysis, the greedy procedure represents a viable option for solving the profit satisficing problem with good accuracy and in a time-efficient way. The NM procedure, on the other hand, can be useful when a higher accuracy is required. The model and the solution procedures proposed can serve practitioners as a supporting tool for the allocation decision and for performing what-if analysis in setting the target profit to pursue. The problem discussed in this paper is subject to numerous assumptions aiming at reducing its complexity. Nonetheless, we foresee the possibility of further investigation, starting from the base established in this paper and reconsidering some of the assumptions. Regarding the assumptions, in the formulation of the problem, we purposely neglected the salvage value for unsold items, as well as the goodwill cost of lost sales. This allowed us to address the problem in terms of the probability of selling a certain amount of the available stock. In fact, considering the salvage value and the goodwill cost, the final profit depends not only on the amount sold but also on the amount not sold. However, including these costs requires different considerations from the mathematical point of view, which are beyond the scope of this paper and are, therefore, the object of further investigation. An alternative direction of analysis is represented by the investigation of the satisficing objective when a minimal service level (i.e., measured as the probability of being able to serve a given percentage of the demand) is imposed at the retailers. The service level constraint imposes a minimal allocation to each retailer; the interest in this aspect is derived from the fact that the profit satisficing objective and the minimal service level objective may conflict. The problem discussion is limited to the case with the same selling price p for all the retailers. Although this can be quite a common setting in the type of supply chain considered in this study, analyzing the rationing problem in a non-homogeneous prices pi context (or, more generally, in a context with different margins mi for the retailers) represents an interesting direction for further research. Similarly, including the transportation cost, which typically depends upon the retailer’s position and the quantity shipped, in the analysis could be relevant.

7. Conclusions In contexts characterized by scarcity of resources, the quest for an optimal inventory and/or capacity allocation arises often. The allocation decision may have a substantial influence on the performance of a company, especially in contexts characterized by the impossibility of recovering from previous decisions, such as in single-period problems.

Acknowledgments The author wishes to thank the anonymous referees and the Associate Editor for their helpful comments and suggestions. In particular, the idea underpinning Section 5.1 and related contents is due to Reviewer #4, who is gratefully acknowledged.

Please cite this article as: Pinto R. Stock rationing under a profit satisficing objective. Omega (2016), http://dx.doi.org/10.1016/j. omega.2015.12.008i

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Please cite this article as: Pinto R. Stock rationing under a profit satisficing objective. Omega (2016), http://dx.doi.org/10.1016/j. omega.2015.12.008i