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1 European Journal of Operational Research 214 (2011) Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: Invited Review A survey of deterministic models for the EOQ and EPQ with partial backordering David W. Pentico a,, Matthew J. Drake b a Management Science, Palumbo-Donahue School of Business, Duquesne University, Pittsburgh, PA , USA b Supply Chain Management, Palumbo-Donahue School of Business, Duquesne University, Pittsburgh, PA , USA article info abstract Article history: Received 23 January 2010 Accepted 26 January 2011 Available online 1 February 2011 Keywords: Inventory Partial backordering Lot sizing Models for the basic deterministic EOQ or EPQ problem with partial backordering or backlogging make all the assumptions of the classic EOQ or EPQ model with full backordering except that only a fraction of the demand during the stockout period is backordered. In this survey we review deterministic models that have been developed over the past 40 years that address the basic models and extensions that add other considerations, such as pricing, perishable or deteriorating inventory, time-varying or stock-dependent demand, quantity discounts, or multiple-warehouses. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction The first, and by far the best known, inventory model is the classic square-root economic order quantity (EOQ) model developed by Harris [43]. While this model has been criticized for its unrealistic assumptions, it has been widely and successfully used in practice. Possibly more important, it forms the basis for many other models that relax one or more of those assumptions. The earliest paper we could find that relaxed one of these basic assumptions was Taft [106], who used a finite production rate, leading to the basic economic production quantity (EPQ) model, also known as the economic manufacturing quantity (EMQ), economic lot size (ELS), or production lot size (PLS) model. Relaxation of the basic EOQ and EPQ models assumption that stockouts are not permitted led to the development of models for the two basic cases for stockouts backorders and lost sales. Zipkin [151] discusses these two basic model types for the EOQ, which are easily generalized to the EPQ. What took longer to develop were models that recognized that, while some customers are willing to wait for delivery, others are not. Either these customers will cancel their orders, or the supplier will have to fill them within the normal delivery time by using more expensive alternative supply methods. The first models for this variation on the basic EOQ model partial backordering or backlogging appeared in 1967 [38] and 1970 [7], although neither author showed how to solve his model. The first paper that developed a model for the basic EOQ with partial backordering (EOQ PBO) and a solution procedure for it appeared in 1973 Corresponding author. Tel.: ; fax: addresses: pentico@duq.edu (D.W. Pentico), Drake987@duq.edu (M.J. Drake). [60], with others taking somewhat different approaches appearing up to the present time. These models are summarized in Section 3.1. Comparable models for the basic EPQ with partial backordering (EPQ PBO), the first of which appeared in 1987 [58], are summarized in Section 5.1. Over the 40-plus years since the first basic EOQ PBO model appeared, many authors have developed models that have relaxed additional assumptions of the basic EOQ PBO and EPQ PBO models, including such things as time- or backlog-dependent backordering probabilities, inventory deterioration, time- or inventorydependent demand functions, and quantity discounts. Other authors have examined scenarios that included making pricing decisions or multiple items. Approximately 90% of this work has appeared since 1990 and about 75% since In this paper we provide brief summaries of most of the deterministic models in this area, focusing on the assumptions behind the models and the approaches taken to developing and solving them. Stochastic models will be covered in a second survey. We will not attempt to describe the models in detail. However, it will be necessary to use symbols for some of the models characteristics. Unfortunately, as in most areas of quantitative analysis, there is no consistency in how symbols are used in the literature in this area. For example, b has been used to mean the maximum backorder level, the maximum stockout level, the unit cost of a backorder, the probability a unit of demand will be backordered, and the probability a unit of demand will not be backordered. At the risk of making it a little more difficult to translate directly from our summary to the original paper s discussion, we will use, for the most part, a single set of notation, given in Table 1, in order to make it easier to see the similarities and differences among the papers. (Note, however, that an author may use one or more of the symbols defined in this table to mean something completely /$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi: /j.ejor

2 180 D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) Table 1 Summary of symbols used and their meanings. Symbol b b M B D F p P Q S T T 1 T 2 t d s s M U V X Meaning Fraction of demand backordered during a stockout, which may depend on the time until the next replenishment order is received (b(s)) Maximum value of b(s) over a stockout interval Maximum backorder level Demand per period, which may depend on time (D(t)), price (D(p)), both (D(p,t)), or something else, such as the inventory level Fraction of demand filled from stock, fill rate Unit selling price, which may depend on time (p(t)) Production rate, which may depend on time (P(t)) Order quantity Maximum stockout level Length of an inventory cycle Length of time during an inventory cycle for which there is inventory Length of time during an inventory cycle for which there is a stockout Length of time the start of the deterioration of inventory is delayed Time remaining until the next replenishment order is received Maximum amount of time any customer will wait Total demand during an inventory cycle Maximum inventory level Fictitious demand rate = V/T different that what it means in this survey.) Where a paper uses variables or coefficients that do not appear in other papers, we use the original paper s notation for them, defining these symbols as they are used. The structure of this paper is as follows: After a brief discussion of various models for determining the percentage of demand backordered at any given time during the stockout interval (Section 2), we will review models for the basic EOQ PBO and no additional complicating factors (Section 3). In Section 4 we will cover models that include, in addition to partial backordering, additional features, such as deteriorating inventory, non-constant demand patterns, an uncertain replenishment quantity, pricing, and multiple warehouses. In Section 5 we will review models with a finite production rate, beginning with the basic EPQ PBO and moving onto models that include additional features. 2. Models for determining the backordering percentage One of the variations on the basic EOQ PBO and EPQ PBO models is the inclusion of time- or backlog-dependent backordering probabilities. Since this is a significant way in which the models to be discussed differ from one another, we will briefly discuss the approaches to determining the backordering percentage that have appeared in the literature. With two exceptions, all of the models prior to 1996 assumed that the percentage of demand backordered is constant over the length of the stockout interval. One exception is Montgomery et al. [60], in which the primary development was for a constant b, but which also discussed what we call the Linear 1 model for time-dependent b(s). They noted that they tried other, unidentified, forms forb(s), but they were more difficult to analyze. The other exception is Mak [57], who assumed that b is a random variable with a constant mean. Both of these exceptions are discussed in Section Backordering percentage based on the time to replenishment A number of authors, most notably Abad [1 5], have noted that it is more realistic to assume that more customers are willing to wait if the waiting time is short. That is, they assume that b is a function of s, the time remaining until replenishment. We provide brief descriptions of the models proposed for b(s), discussing them in the order in which they first appeared. Table 2 summarizes the equations for b(s) for each model type. (Note that, as we stated in Section 1 with reference to Table 1, where the notation used in this paper is defined, the notation we use in discussing the equations in Table 2 is not necessarily the same as that used by the authors of the papers using these models for b(s).) 1. Linear 1: The first model for a non-constant b, which we call Linear 1, was covered in Montgomery et al. [60], along with their constant-b model. b(s) has an initial value, b 0, at the time the stockout begins (i.e., when s has its maximum value) and increases linearly until it reaches its maximum value, b M, when the replenishment order arrives (when s = 0). Linear 1 was also analyzed by San José et al. [82]. In 1996 Abad [1] was the first to introduce both the Exponential and the Rational models for b(s). Although he provided a brief rationale there ( Since in general, consumers do not like to wait, we assume that b(s) is a decreasing function of s.), Abad [3] gives a more extensive rationale for using a b(s) function like these. After noting that the conventional approach to modeling the backlogging phenomenon requires the use of the backorder cost and the lost sale cost, he states that these costs...are difficult to estimate in practice [, so a] new approach in which customers are considered impatient...and the fraction of demand that gets backlogged at a given point in time is a decreasing function of waiting time will be used instead. While not justifying either the exponential or the rational form explicitly, Abad [1] used both types. In both, b(s) increases toward its maximum value, b M, which Abad calls the backordering intensity, at an increasing rate as s decreases toward 0, the actual rate being determined by a, the backordering resistance. He also noted that as a? 1 (i.e., customers are highly impatient), b(s)? Exponential: b(s) is b M multiplied by a negative exponential function of s. Abad [1] introduced this model originally, but Papachristos and Skouri [67] were the first to refer to it as exponential. San José et al. [80,81] analyzed this form for the basic EOQ with partial backordering, while Abad s models also included deteriorating inventory and pricing decisions and Papachristos and Skouri included deterioration. 3. Rational: b(s) isb M divided by a positive linear function of s. In addition to Abad, who used it first, San José et al. [80] and Sicilia et al. [98] used this form and were the first to use this name for it. The exponential and rational forms of b(s) are the only forms that have been used extensively by other authors. As we discuss the Table 2 Equations for functional forms of b(s). Form of b(s) Equation Range for s Linear 1 b(s)=b M (b M b 0 )(s/t 2 ) 06 s 6 T 2 Exponential b(s)=b M e as, a >0 06 s Rational b(s)=b M /(1 + as), a >0 06 s Linear 2 b(s)=b M (b M /s M )s 0 6 s 6 s M Step b(s)=1 0 6 s 6 s M Mixed exponential bðs 1 ; s 2 Þ¼b 1 e a1s1 þ b 2 e a2s2 ; b 1 ; b 2 P 0; b 1 þ b 2 6 1; a 1 ; a 2 > 0 06 s 1, s 2

3 D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) EOQ PBO and EPQ PBO models that include other features in Sections 4 and 5, we will identify which form of b(s) the authors used. The remaining forms for b(s) to be discussed have been analyzed by only one set of authors: San José, Sicilia, and García-Laguna. 4. Linear 2: Although the linear 1 form is relatively easy to solve and does provide for an increase in the percentage of customers who are willing to backorder as the time to replenishment decreases, it assumes that the supplier would use the same b 0 without knowing how long the stockout interval will be. While, as Montgomery et al. [60] suggested, linear 1 is probably a satisfactory approximation in many cases, the linear 2 form could be better in other cases since it allows the initial value of b to be determined by the length of the stockout period. In linear 2, b(s) = 0 until a time s M before the time at which the replenishment order will arrive and then increases linearly until it reaches its maximum value, b M,ats = 0. All demands occurring more than s M periods before the replenishment order is received will be lost sales if there is no inventory. San José et al. [79] introduced and analyzed this form. 5. Step: This form is like linear 2 in that b(s) = 0 until a time s M before the time at which the replenishment order will arrive, but it differs in that the step form then immediately increases to b(s) = 1 until the replenishment order is received rather than gradually increasing as linear 2 does. San José et al. [80] proposed and analyzed this form. The final b(s) form to be identified differs significantly from the preceding ones in that, rather thanb(s) being a non-decreasing function of s, it is U- or dish-shaped. 6. Mixed exponential: The basic assumption behind the mixed exponential form, analyzed only in Sicilia et al. [99], is that there are two types of customers. For some of them the reluctance to backorder decreases as the replenishment time approaches. However, there are others who are the opposite of this: they are more likely to backorder if the wait is longer. Sicilia et al. suggested that this might be the case if the customer believes that this will ensure that he or she will get a higher quality item. (Note: It seems to the authors of this survey that the likelihood of this type of reasoning applying for the types of items that would be managed by a basic EOQ model or one of its variants is extremely low, so that the applicability of this form would be rare.) The combination of these two types of customers results in a backordering-likelihood function that is the sum of two exponential forms. One, with a maximum value of b 1, is based on s (referred to as s 1 ), and the other, with a maximum value of b 2, is based on T 2 s (referred to as s 2 ), the amount of time from when the stockout begins until the unit of demand that will wait for s 1 periods arrives. Adding the two exponential forms gives a dish-shaped function. In Section 3.3 we will discuss each of these b(s) forms somewhat more extensively. We note that, in addition to the six forms for b(s) identified, some authors have used more general forms for which no equation is given, but general properties are stated Backordering percentage based on the size of the backlog All of the forms for b(s) discussed in Section 2.1 began with the assumption that the percentage of demand during the stockout period will be backordered is a function of s, the time remaining to the receipt of the replenishment order. In this section we discuss modeling backordering or backlogging as a function of the size of the existing backlog. Padmanabhan and Vrat [64,65], Ouyang et al. [61], Chu et al. [24], and Dye et al. [33] assumed that the probability of backordering is negatively related to the size of the existing backlog when a demand arrives. That is, when the existing backlog is small, which is the case very soon after the stockout interval begins, the probability a new demand will be backordered is high and when the existing backlog is large, which would be the case as the stockout interval nears its end, the probability a new demand will be backordered is low. Papers [64,24,33] will be reviewed in Section 3.3, [65] will be reviewed in Section 4.2.3, and [61] in Section Comment: All the forms for b(s) considered here, except the constant-b model, the step form, and the mixed exponential form, assumed that the probability of backordering will be higher (lower) if the replenishment time is closer (further away). One implication of that assumption, not only for linear 1 and 2, the exponential and the rational, but also for the step form and the constant-b models, and even for the mixed exponential form as s approaches 0, is that when the backlog is high, the probability that the next demand will be backordered does not decrease; if anything, it increases. The reason for this is clear: The backlog is increasing because it is getting closer to the time of replenishment and, since replenishment is instantaneous in the EOQ model, all the backorders will be filled soon and simultaneously. For the models in [64,65,61,24,33], the reverse is true. As noted above, these models assume that the probability of backordering will be higher when the stockout period has just begun and will become lower as the stockout continues and the replenishment time gets closer, so that the customer will wait for a shorter time. It is possible that this might be true for fashionable commodities, as suggested by Ouyang et al. [61], although they provide no justification for that statement. Even then, those are not the type of items that would generally be controlled by EOQ-type models since their lifetime tends to be short and demand is more stochastic. The relationship between the longer line and lower probability of waiting makes sense for a queuing system using a first come-first served queue discipline. A customer arriving when the queue is long may balk because she knows that her wait will be longer due to all the customers to be served before her. It does not make a lot of sense for an EOQ system, where a longer line implies that service will be sooner, not later. 3. Deterministic EOQ PBO models The deterministic EOQ PBO models satisfy all, or at least most, of the basic assumptions of the classic EOQ model with full backordering except that only a percentage of the demand when the supplier is out of stock will be backordered. In Section 3.1 we briefly summarize papers that developed models for the basic deterministic EOQ PBO model with a constant b. In Section 3.2 we consider several papers that addressed the issues of the sensitivity of those models or that identified potential problems with their formulation or solution. In Section 3.3 we examine basic models that included one of the time-based backordering percentage forms identified in Section 2. In Section 3.4 we consider models that used a backlog-based probability of backordering Basic deterministic EOQ PBO models with constant b All the models reviewed in this section are single-item models that assumed that all parameters are known and constant over an infinite time horizon, that replenishment is instantaneous with a known lead time, and that the cost to place and receive an order is a constant, independent of the size of the order. Where they differ is in their choice of variables, their assumptions about the backordering rate, and their cost structures. All but one assumed that b is known and constant. Most made the usual EOQ model assumptions about costs: (1) the holding cost is an amount per unit per year, (2) the backorder cost is an amount per unit per year plus,

4 182 D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) possibly, a fixed cost per unit, and (3) the lost sale cost is an amount per unit. These papers will be reviewed in Section Papers that used alternative cost structures will be reviewed in Section Models using the classical cost structure The earliest papers that specifically modeled the basic deterministic EOQ PBO were Fabrycky and Banks [38] and Ali [7]. Both developed cost functions to be minimized, but neither provided a solution procedure. Montgomery et al. [60] included a fixed cost per unit short, which is the same whether that unit is backordered or not, and a cost per year per unit backordered. Their decision variables were Q (the order quantity) and S (the maximum stockout level), which they replaced by two new variables: U (the total demand during an inventory cycle = Q +(1 b)s) and V (the maximum inventory level = Q bs). Their solution procedure found V (U) and then optimized U, although they recognized that, because the cost function is not convex, their solution could not be guaranteed to be optimal. They determined a critical value for b, above which partial backordering is optimal. Leung [52] showed that an algebraic approach called the complete squares method could be used to determine the optimal values of U and S for Montgomery et al. s [60] model, even though that model s quadratic cost function is not convex. As a by-product of using this method, Leung determined the same critical value for b as in Montgomery et al. and was able to determine the conditions under which the optimal strategy is to partially backorder all demand or to not allow backordering at all. In addition, by choosing appropriate values for different parameters, he was able to develop formulas equivalent to those in Mak [57] (to be reviewed later in this section) and for variations on the basic EOQ scenario previously developed by other authors (see [52] for references to other authors who have used the complete squares method for models without partial backordering). Rosenberg [76] used the same cost structure as Montgomery et al. and also used Q and S as the initial decision variables, which he replaced by T (the cycle length) and X (a fictitious demand rate, X = V/T). He used a two-stage solution procedure, first finding X (T) and then T. Rosenberg also determined a critical value for b, above which partial backordering is optimal. Except for slight changes in notation, Park presented the same model in [70,71]. He dropped the fixed cost for a backordered unit and used U (R in [71]) and S as the decision variables. He proved that his cost function is convex as long as U P S P 0 and developed formulas for S and U (S). Park also determined a critical value for b, above which partial backordering is optimal. Wee [121], with the same cost structure as Park [70,71], used T and T 2 (the length of the stockout phase of an inventory cycle) as the decision variables. He proved his objective function is convex and determined equations for T 2 and T (T 2 ). Wee also determined a critical value for b, above which partial backordering is optimal. Although he never referred to partial backordering or any of the other terms typically used, Urban [115] developed a basic EOQ PBO model by assuming the demand rate is a constant D during the in-stock period of an inventory cycle and changes to kd during the stockout period. While his model allows for the possibility that k >1, k = 1 means full backordering, and having 0 < k < 1 corresponds to partial backordering with k equal to a constant b. The objective is profit maximization and his decision variables are the maximum (initial) inventory level, which he calls S (not the same as S in Table 1), and the most-negative shortage (backorder) level or reorder point, which he calls s (B in Table 1). A difference from the other models is that he does not include a goodwill cost for lost sales, recognizing (indirectly) only the lost profit component of a lost sale cost. Pentico and Drake [72] used T and F (the fraction of demand filled from stock or the fill rate) as the decision variables. They used the same cost structure as Park [70,71] and Wee [121], but included a fixed cost per backordered unit in an Appendix. They proved that their solution is optimal for T > 0 and 0 < F 6 1 and derived equations for T and F (T). They also determined a critical value for b, above which partial backordering is optimal and below which the optimal solution is to either use the basic EOQ model without backordering or not stock the item at all, whichever costs less. San José et al. [80] developed a general approach to analyzing deterministic EOQ PBO models with the cost structure used by Montgomery et al. [60] and Rosenberg [76] when the value of b(s) is a non-decreasing function of s, the time until the replenishment order is available to fill the backorders (see Section 2.1). This will be discussed in Section 3.3, but is mentioned here because it applies when b is a constant. Mak [57] considered the basic problem without a fixed cost per unit backordered, but with the added complication that, while the mean percentage of demand backordered during the stockout period is given by the constant b, the actual amount backordered is a random variable, with its mean given by bs and a standard deviation. He considered two cases: (1) the standard deviation of B is independent of the cumulative shortage S (r B is a constant), and (2) the standard deviation of B is proportional to the cumulative shortage (r BjS = as). As Rosenberg [76] did, Mak used Q and S as the initial decision variables, but then changed to T and X. His two-stage solution procedure was to find X (T) and then optimize over T. Mak s equations are identical to Rosenberg s except they do not include the fixed cost per unit backordered and they do include terms to reflect the variance of the amount backordered. Mak developed the same condition on the minimum value of b for which partial backordering is optimal as in Park [70,71], Wee [121], and Pentico and Drake [72]. Ifr B or r BjS = 0 (i.e., there is no uncertainty about the amount backordered), then Mak s equation for T is identical to the one in Pentico and Drake [72] and his equation for X (T) is identical to their D F Models using alternative cost structures In a series of papers, San José et al. considered three variations on the problem with constant b in which the cost structure differed from the classical cost structures used in the papers in Section San José et al. [84] included both fixed and timedependent unit costs for backordering (as in [60,76]) and for lost sales, a cost structure that is not considered by any other authors. Their decision variables were U and B (the maximum backorder quantity). In San José et al. [83], the holding cost per unit is a constant per unit time, lost sales have a fixed cost per unit and the cost of a backorder is an increasing quadratic function of the waiting time. Their decision variables were T 1 (the length of the in-stock phase during an inventory cycle) and T 2. In San José et al. [78] they used a constant unit holding cost per period, but they assumed that the backordering cost per unit has a fixed component plus a time-dependent component that is nondecreasing, continuous and positive. As in many of their other models, the decision variables were T 1 and T 2, but the objective was profit maximization rather than cost minimization. In [78,83,84], they used the combinations of two control parameters to determine: (1) whether the optimal policy was to not stock the item at all, to stock but not allow backorders, or to allow partial backordering, and (2) how to find the optimal values of the decision variables. Hu et al. [46] considered the constant-b problem when the unit backorder cost increases linearly with the duration of the shortage.

5 D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) Their decision variables were T and T 1. They established a critical value for the holding cost per unit, at or below which the optimal solution is to allow no backordering and above which partial backordering is optimal. The optimal solution is found by solving a complicated equation for T 1 and then using that value of T 1 in the quadratic formula to solve for T 2, and thus for T. They noted that an explicit closed form solution for the optimal policy may not be obtained, but it can be solved for by a numerical search Sensitivity of the basic model and related issues While some of the authors discussed in Section 3.1 performed some basic sensitivity analysis on their models, we review here papers that only address the sensitivity issue. But first, we consider a paper that raised issues about whether the specification of the conditions under which a model gives optimum results is complete. Whitin [128], commenting on papers by Montgomery et al. [60], Rosenberg [76], and Park [71], noted that none of those authors mentioned a simple but necessary economic constraint on the optimality of the proposed solution: [a]n item will be bought or sold only if positive profits result. Using the example in [76], he showed that ignoring this constraint meant that the example solution in [76], which met the condition for the optimality of partial backordering given in the paper, actually resulted in a negative profit, in which case the optimal solution would be to not stock the item at all. He also showed that, in the example in [71], whether the presumably optimal solution met this constraint or not depended on how the cost of a lost sale is divided between the lost profit and the rest of the cost of a stockout. We note that the formal sensitivity analysis studies found considered only the effects of changing b, not the effects of changing any of the other parameters in the models. As noted above, most of the authors who developed models with a constant b also determined a condition that can be used to determine whether partial backordering is optimal or not. These conditions, whether expressed in these terms or not, can all be translated into a condition on the minimum value of b. For Park [70,71], Wee[121], Pentico and Drake [72] and Mak [57], none of whom included a fixed cost per unit backordered in their models, this minimum value of b for which partial backordering is optimal is given by b ¼ 1 average cost per period of using the basic EOQ model : average cost per period of all lost sales For Montgomery et al. [60] and Rosenberg [76], who included a fixed cost per unit backordered, the formula for b is more complicated due to that cost. b is important in the following. Chu and Chung [23] stated that sensitivity analyses for models of inventory systems that analyze the results of solving those models for a variety of numerical cases are questionable since different conclusions may be made if different sets of numerical examples are analyzed. They illustrated this concern by considering the results of the experiment-based sensitivity analysis performed by Park [70]. Limiting their attention to cases in which b P 0, which both Park and they argued is the only situation that needs to be considered, Chu and Chung conducted a formal mathematical analysis that concluded that Park was correct in finding that Q (b) increases and the optimal cost decreases as b increases, but they found that S (b) increases with b only if another condition is met. This condition is an upper bound for the ratio of the unit holding cost per period to the unit backorder cost per period that is based on a more complicated expression involving b. Yang [137] extended Chu and Chung s [23] work to consider what happens if b < 0. While this means that the cost of all lost sales is less than the cost of using the basic EOQ with no backordering, which was Park s [70] and Chu and Chung s justification for limiting their analyses to b P 0, it does not necessarily mean that having all lost sales is better than using the EOQ PBO. Yang addressed two basic issues. First, he extended the work by Park and Chu and Chung to more completely justify the result that, when b 6 b and b P 0, the optimal solution is to use the basic EOQ without backordering. Second, he showed that when b < 0, then the upper bound for the ratio of the unit holding cost per period to the unit backorder cost per period determined by Chu and Chung [23] is decisive in determining the optimal inventory policy. While Chu and Chung [23] and Yang [137] both based their sensitivity analyses on Park s [70] model, Leung [53] based his on Montgomery et al. [60]. Other than the choice of decision variables, the significant difference between the models in Park and Montgomery et al. is that the latter included a fixed cost per unit backordered, whereas Park did not. However, Leung s model differed from Montgomery et al. s in that his fixed penalty cost per unit backordered is less than the fixed penalty cost (exclusive of the lost profit) of a lost sale. Leung derived the conditions for b under which the optimal solution is to use the equations for partial backordering. He then conducted a sensitivity analysis on the decision variables and the optimal cost, determining the range of b within which each would be monotonically increasing or decreasing Basic deterministic EOQ PBO models with b a function of the time to replenishment In Section 3.1 we reviewed models in which a constant fraction b of the demand during the stockout period will be backordered, with the remainder being lost sales. In this section we review models in which b is a function of the time remaining until a replenishment order is received. We also review some papers that model the backordering rate as a function of the size of the backlog. As discussed in Section 2.1, there are a number of ways in which authors have assumed that b will change with the time remaining until a replenishment order is received. Brief descriptions of the functional forms ofb(s), where s is the time remaining until a replenishment order is received, and of the solution procedures used follow. Table 2 summarizes the equations for b(s) for each case Linear 1 Montgomery et al. [60] noted that, while many different functional forms for the manner in which the mixture of backorders and lost sales will occur are possible and the identification of the proper functional form may be difficult in practice... the constant or linear ratio (linear 1, in our categorization) models are probably satisfactory approximations. Certainly, the linear 1 form is the easiest to analyze other than the constant-b model reviewed in Section 3.1. Montgomery et al. did not show their analysis, but did state that the results are basically the same as for the constant-b model with the substitution of two new parameters in their equations, both based on using b ¼ðb 0 þ 1Þ=2, the average value of b over the time the stockout exists, instead of b. (Note: They assumed, as most authors have, that b M = 1.) San José et al. [82] expressed the concept of the linear 1 form somewhat differently than Montgomery et al. [60] did, saying that they assume that the fraction of the customers who are not willing to wait is proportional to the ratio between the waiting time (i.e., s) and the length of the shortage cycle (i.e., T 2 ). However, expressing the decrease in 1 b(s) in this way is the same as expressing the increase in b(s) as the linear 1 form does. As in most of their other work, San José et al. defined two control parameters as functions of the basic parameters in the model and then identified, for the different combinations of those control parameters, when the basic EOQ model without backordering is optimal, when

6 184 D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) all sales should be lost, and when partial backordering is optimal. They also gave, for the latter case, the equations for the optimal values of T and T 2, which were their decision variables Exponential b(s) isb M multiplied by a positive exponential function of s; i.e., b(s)=b M exp( as), with with s P 0, a > 0, and 0 6 b M 6 1. This is one of the four forms considered by San José et al. in [80]. Their decision variables were T 2 and T. While the authors described the general procedure for solving for the optimal values of T 2 and T for all four of the forms of b(s) that they identified in [80], they did not specifically address the solution for the exponential form. San José et al. [81] did address the exponential form in detail, using T 1 and T 2, rather than T 2 and T, as the decision variables. They defined four control parameters and identified, for their various combinations, what the form of an optimal solution is. Since the equation to be solved when partial backordering is optimal includes exponential terms, determining T 2 for those cases requires the use of a numerical method Rational b(s) is b M divided by a positive linear function of s; i.e., b(s)=b M /(1 + as), with s P 0, a > 0, and 0 6 b M 6 1. This is another of the four forms considered by San José et al. in [80]. Their decision variables were T 2 and T. For the rational form of b(s), finding the optimal value of T 2 for one of the combinations of the two control parameters they defined involves solving an equation that includes ln(1 + at 2 ), which requires using a search procedure rather than evaluating a closed-form expression. Sicilia et al. [98] combined the rational form of b(s) with a backorder cost function that is a quadratic and increasing function of s. As in their other papers discussed here as San José et al. [79,80,83], their decision variables were T 2 and T. Due to the more complicated backordering cost function, they used four control parameters. For most combinations of the control parameters, the optimal solution is to either use the basic EOQ without backordering or not stock the item at all. For the remaining cases, finding the optimal value of T 2 involves solving an equation that includes ln(1 + at 2 ), which, as noted, requires using a search procedure rather than substituting the problem s parameters into a closedform expression Linear 2 Unlike the linear 1 form, in which b(s) has an initial value of b 0 when the stockout interval begins, in the linear 2 form b(s) = 0 until s M periods before the end of the inventory cycle, after which it increases in a linear fashion until it reaches its maximum value, b M, when the replenishment order arrives. The only paper that analyzed the basic EOQ model with partial backordering andb(s) modeled by the linear 2 form is San José et al. [79], in which T 1 and T 2 were the decision variables. Given the need to concern themselves with the issue of whether T 2 is greater than s M, in which case there is an interval at the start of the stockout period during which all sales will be lost, the cost function to be minimized is more complicated than the one they used for the linear 1 form. As a result, their characterization of the nature of an optimal solution is also more complicated, using combinations of three control parameters, rather than two, as they did in [82]. For most combinations the optimal solution is to either use the basic EOQ model without backordering or to not stock the item. If partial backordering is optimal, determining the optimal value of T 2 may involve solving for the roots of a fourth-power polynomial Step Like the linear 2 form, the step form includes specifying a maximum value for s, s M, beyond which no customers will be willing to wait. The major difference between the linear 2 and step forms is that in the step form all customers will wait (i.e., b(s) = 1) as long as s is less than s M, whereas in the linear 2 form b(s) increases linearly after s < s M. San José et al. [80] addressed the step form as one of the four they considered using a general solution approach for problems in which b(s) is a non-decreasing function of s. Their decision variables were T 2 and T. The solution procedure begins by ignoring s M and solving the problem as if there were full backordering. If T 2 P s M, then the problem is solved. If not, a more complicated solution procedure is used Mixed exponential All of the forms for b(s) discussed so far, except the constant-b models in Section 3.1 and the step form in Section 3.3.5, assumed that a customer who would have to wait longer would be less likely to backorder. As discussed in Section 2.1, the basic assumption behind the mixed exponential form analyzed in Sicilia et al. [99] is that there are two types of customers, those whose reluctance to backorder decreases as the replenishment time approaches and others who are more likely to backorder if the wait is longer. The combination of these two types of customers results in a backordering-likelihood or customer-impatience function that is the sum of two exponential forms, resulting in a curve that is dish-shaped. How high the two lips of the dish are depends on the relative sizes of b 1 and b 2, but since their sum must not exceed 1, the maximum probability that a customer will wait for any value of s is no more than 1. Sicilia et al. [99] used T 1 and T 2 as the decision variables. As they did in their other work [98,78 84], they defined control parameters as functions of the basic model parameters, and identified, for each combination of them, the nature of the optimal solution. Since the equation to be solved when partial backordering is optimal includes exponential terms, determining T 2 for those cases requires the use of a numerical method Backordering likelihood based on the size of the backlog All of the forms for b(s) discussed so far began with the assumption that the probability that a demand during the stockout period will be backordered is a function of the time remaining to the receipt of the replenishment order. In Section 2.2 we discussed the concept behind basing the backordering likelihood on the size of the existing backlog. Here we discuss three papers that used that approach. Padmanabhan and Vrat [64] assumed that the probability of backordering is a negatively related to the size of the existing backlog when a demand arrives. The cost function is convex, so the optimum values for T and T 1, the decision variables, can be found by simultaneously solving a pair of equations, both of which include exponential terms, which they do numerically. Chu et al. [24] revisited the work by Padmanabhan and Vrat [64] in order to analyze and improve on their solution method. They determined that if the setup cost is less than a critical value, then T 1 and T ðt 1Þ can be obtained from two straightforward equations rather than having to solve a pair of non-linear equations simultaneously; otherwise, the best policy is to prolong the shortage period as long as possible. Dye et al. [33] noted that neither Padmanabhan and Vrat [64] nor Chu et al. [24] recognized either the cost of lost sales due to shortages or the purchase cost of backordered units in their cost function. However, in addition to including these two costs, Dye et al. made other changes to the model. They used a finite time horizon, assumed a positive log-concave demand function rather than a constant demand rate, and assumed inventory deterioration at a constant percentage. As result, their model, other than making the assumption that the backordering rate decreases as the number of backorders increases, is like the models considered in

7 D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) Section and the solution procedure is like the one described there. 4. Deterministic EOQ PBO models with additional considerations All the models reviewed in Section 3 except Dye et al. [33] have the form of a basic deterministic EOQ PBO model. Although there are differences with respect to the cost structure or whether b is a constant or a function of either the time remaining until the replenishment order will be received or the size of the existing backlog, they all have a constant demand rate and instantaneous replenishment and a constant cost structure over an infinite time horizon. In this section we review papers that introduce additional considerations, such as deteriorating or perishable inventory, demand that varies with time or the stock level, quantity discounts, inflation, uncertain receipt quantity, pricing decisions, or multiple stocking locations. We begin with models that introduce one additional consideration and then review models that have two or more added considerations. (Note: Unless defined where used, symbol definitions are given in Table 1. Descriptions of the timebased backordering functions b(s) are in Section 2.1 and Table 2.) 4.1. Models with a single additional consideration Delayed backordering Abboud and Sfairy [6] developed a model for the basic EOQ in which partial backordering with a constant b is delayed for m periods after the stockout phase begins. Prior to that time all customers will backorder. The objective is to minimize the average cost per period; the decision variables are Q and t, the length of the partial backordering phase exceeding m. By examining the roots of a quadratic function for t, they determine whether (1) partial backordering is optimal and Q is determined from a simple linear function of t, (2) t = 0 and Q is determined from a square root formula that adjusts the basic EOQ formula for m, or (3) the item should not be stocked Deteriorating or perishable inventory Although deteriorating or perishable inventory is by far the most popular additional consideration in the development of EOQ PBO models, almost all the papers that include this also include other additional features. Only two papers were found that add only deteriorating inventory to the basic partial backordering model. Wee and Mercan [126] used a constant b and inventory deteriorating at a constant percentage rate. Their decision variables were T 1 and T. They proved that their cost function is convex if b exceeds a specified lower bound and the optimal solution is found by simultaneously solving a pair of non-linear equations. Shah and Shukla s [86] scenario is the same as that in Wee and Mercan except that they assumed that b(s) has a rational form. They also used T 1 and T 2 and found a solution by solving a pair of non-linear equations, although they did not prove that the solution is optimal Demand pattern Urban [115] extended his discussion of a basic EOQ model with a constant demand rate that changes to a different constant demand rate when the inventory level drops to 0 (Section 3.1), to consider the two cases in which the demand rate during the instock period is either (a) a constant that is a function of the initial stock level or (b) given by a power function of the instantaneous stock level until that reaches a specified level, at which point D becomes constant. In both cases he recognizes partial backordering by having the demand rate change to kd (which corresponds to partial backordering at a constant rate b if 0 < k < 1) when the inventory level reaches 0. As was the case with his basic EOQ PBO model in Section 3.1, the objective was profit maximization and he did not include a goodwill cost for lost sales. Hsieh et al. [45] examined Urban s [115] case (b) with two important changes: (1) b(s) is not constant but has a rational form, and (2) there is a goodwill cost per unit of lost sales. They proved that there is a unique optimal replenishment policy and gave a relatively simple procedure for finding it. Zhou et al. [150] used a finite time horizon. They assumed that the demand rate is an increasing function of time, although they also proved some results for a decreasing function. They used both the rational and exponential forms for b(s). The decision variables were (1) n, the number of complete cycles during the horizon, (2) the starting time for each cycle, which is the start of its stockout phase, and (3) each cycle s replenishment time, which is the start of its in-stock phase. They recognized that, since the demand rate changes over time, the optimal lengths of the n cycles and the relative times of the replenishments within each cycle are not necessarily the same. Their solution procedure finds solutions for n = 1, 2, etc. and identifies the value of n with the lowest cost. They could not prove their cost function is convex, but they proved results that narrow the search ranges for n and the optimal times for each cycle Uncertain replenishment quantity Most inventory control models assume that the quantity received is the same as the quantity ordered, Q. However, it is possible that the quantity received, Y, is a random variable, with E (YjQ)=aQ, where the bias factor, a, may exceed 1.0. Kalro and Gohil [50] extended Silver [100] on adjusting the basic EOQ model without backordering for an uncertain replenishment quantity to consider both full and constant-b partial backordering. They considered Silver s two cases with respect to the variability of YjQ: (1) r YjQ = r, independent of Q; (2) r YjQ = r 1 Q, where r 1 = r YjQ=1. Kalro and Gohil used the same cost structure and solution approach as Rosenberg [76] did for the basic EOQ PBO (Section 3.1), adapted for the randomness of Y. They replaced their original decision variables Q and S with: (1) the cycle length, T =(aq +(1 b)s)/d, and (2) a fictitious demand rate, X =(aq bs)/t. Their solution procedure was to find an equation for X (T) and use that to find an equation for T. They established conditions similar to Rosenberg s under which partial backordering is optimal. As in Mak [57], who included uncertainty of the quantity backordered (Section 3.1), Kalro and Gohil s results were similar to Rosenberg s, adjusted for including r YjQ Pricing The vast majority of papers about partial backordering assume that the vendor s price is given. If it is considered at all, it is because the objective is profit maximization or, if the objective is cost minimization, the price becomes part of the penalty for a lost sale. In addition to papers that also include deteriorating or perishable inventory, which will be covered later, there are a few papers that include price as a decision variable, recognizing that by giving a customer a discount when out of stock, the vendor may increase backordering and reduce lost sales, thus increasing the overall profit. Assuming that the price charged during the in-stock phase of a cycle is given, Drake and Pentico [29] modified their basic constant-b EOQ model with both fixed and time-dependent costs per unit backordered (Appendix B of [72]), which used T and F as the decision variables, to include d, the discount offered during the stockout phase, as a decision variable. They assumed that b is a linear function of d. They determined the range for d within which offering a discount to increase b is feasible by determining when b(d) is greater than b (d), the minimum value of b(d) for which