Ordering Policies for Periodic-Review Inventory Systems with Quantity

OPERATIONS RESEARCH Vol. 60, No. 4, July–August 2012, pp. 785–796 ISSN 0030-364X (print) ISSN 1526-5463 (online)

http://dx.doi.org/10.1287/opre.1110.1033 © 2012 INFORMS

Ordering Policies for Periodic-Review Inventory Systems with Quantity-Dependent Fixed Costs Ozgun Caliskan-Demirag Sam and Irene Black School of Business, Penn State Erie, The Behrend College, Erie, Pennsylvania 16563, [email protected]

Youhua (Frank) Chen Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong, [email protected]

Yi Yang Department of Management Science and Engineering, Zhejiang University, Hangzhou, China, [email protected]

We consider a stochastic periodic-review inventory control system in which the fixed cost depends on the order quantity. In particular, we investigate the optimal ordering policies under three fixed cost structures. The first structure is motivated by transportation and production contracts and considers two fixed costs: if the order size is within a specified limit C, then the fixed cost is K1 ; otherwise, it is K2 , where K1 ¶ K2 . The second structure contains multiple fixed costs in which the same incremental fixed cost K is incurred for any additional order quantity up to a given identical batch capacity C. In the third structure, in addition to the K incurred as in the previous case, a common fixed cost is charged for any nonzero order size. An example of the former case arises when an order is shipped with a homogeneous fleet of trucks with per-truck fixed costs. A situation in which a fixed administrative cost plus a quantity-dependent trucking cost is incurred for each shipment exemplifies the latter case. For the first cost structure, we separate the analysis according to the conditions (1) K1 ¶ K2 ¶ 2K1 and (2) K1 ¶ K2 . Under condition (1), we introduce a new concept called C-4K1 1 K2 5-convexity, which enables us to almost completely characterize the optimal ordering policy. Under the general condition (2), we utilize a modified notion to provide a partial characterization of the optimal policy and propose a heuristic policy that performs well under a wide variety of model parameters. For the second cost structure, we show that it is optimal to order an integer multiple of the batch capacity to raise the inventory level to a specified range or band of length C, and then to order an additional full or partial batch size depending on the cost function, with no ordering required above the band. We also characterize a similar optimal policy for the third cost structure. Using different techniques, our study extends or redevelops several existing results in the literature. Subject classifications: periodic-review inventory systems; (s1 S) policies; C-4K1 1 K2 5-convexity; strong K-convexity; dynamic programming. Area of review: Manufacturing, Service, and Supply Chain Operations. History: Received August 2010; revisions received March 2011, August 2011, September 2011; accepted October 2011. Published online in Articles in Advance July 24, 2012.

1. Introduction

and so on. An example of this cost structure is described by Bigham (1986) in the handling and storing of a hazardous waste material at Safeco Corporation in Texas. During the order receiving operation, shipments are off-loaded within a specified time period, and depending on the size of the shipment, one or more crews are assigned to the off-loading task. The fixed costs arise from labor costs. The specific cost structure occurs because only integer numbers of crews are assigned, the off-loading capacity of each crew might be different, and the assignment of additional crews necessitates additional supervisory personnel at extra cost. The resulting fixed cost then takes a step-function structure

In most inventory and production systems, certain setup activities are performed each time an order is placed or a production run is initiated. Sometimes, the total cost of these activities, which is commonly referred to as the fixed or setup cost, is constant for any positive quantity. In many other practical applications, however, fixed costs might take more complex forms as a function of the order or production quantity. In this paper, we consider the periodic replenishment of a single product with such “quantity-dependent fixed costs,” where the objective in each period is to decide the order quantity that minimizes total costs. Frequently, quantity-dependent fixed costs grow as the order quantity increases. With one particular cost structure, if the order quantity is within a certain range, then a relatively small fixed cost is incurred; whereas if the order quantity exceeds this range but remains below the upper limit of another range, then a larger fixed cost is incurred,

c4z5 =

n X

Ki 16Ci−1 < z ¶ Ci 71

(1)

i=1

where z is the order quantity, Ki is the fixed cost of ordering a quantity between the values Ci−1 and Ci such that Ki ¶ Ki+1 , C0 = 0, and 1607 is an indicator function with a 785

Caliskan-Demirag, Chen, and Yang: Inventory Systems with Quantity-Dependent Fixed Costs

786

Operations Research 60(4), pp. 785–796, © 2012 INFORMS

value equaling one if the statement in brackets is true and zero otherwise. A special case of this fixed-cost structure can be observed when the quantity is ordered in batches of a given capacity C and each batch carries an individual fixed cost K. In the transportation context, items are often shipped with a homogeneous fleet of vehicles (e.g., trucks, containers, or carloads), and a fixed cost is charged for each vehicle. In this setting, a batch may contain C or fewer units—that is, it may be fully or partially filled—but K is incurred in both cases. The total fixed cost of an order then depends on the number of batches used to ship the entire order quantity z and is given by c4z5 = Kz/C1

(2)

where z/C represents the number of batches and x is the smallest integer that is greater than or equal to x (see Lippman 1969 for more applications of this cost structure). Another special case of (1) arises in certain production settings. For example, Robinson and Sahin (2001) describe several applications in the food, chemical, and pharmaceutical industries where an equipment setup cost k is incurred each time the production of a specific item is scheduled, and an additional setup cost K is incurred for each day or partial day of production because of, for example, equipment cleaning or day-lot inspection. For a given daily production capacity C, the total fixed cost of producing quantity z is then given by c4z5 = k + Kz/C0

(3)

This model can also be motivated in an inventory control context by interpreting k as the fixed administrative cost incurred for each order and K as the standard fixed cost charged per batch, similar to model (2). A simpler yet important application of quantitydependent fixed costs occurs when there are two fixed costs K1 and K2 (K1 ¶ K2 ) and the incurred fixed cost depends on a threshold value C. This structure can be seen in transportation and production contracts with certain volume restrictions. For example, Chao and Zipkin (2008) describe a case in which the fixed cost is incurred if the order quantity is above the contracted volume, which is similar to a penalty for exceeding the restrictions, and no fixed cost is incurred otherwise. A more general version of this case is that if the order quantity is below C, then the fixed cost is K1 ; otherwise it is K2 . Then, the fixed cost of ordering quantity z is c4z5 = K1 160 < z ¶ C7 + K2 16z > C70

(4)

The underlying motivation for this cost structure is that as the order quantity increases beyond a threshold value, additional administrative and operational actions must be performed that lead to a larger fixed cost, that is, K1 ¶ K2 .

In this study, we aim to contribute to the understanding of ordering decisions under quantity-dependent fixed costs. To this end, we analyze four models originating from the general structure in (1). In all the models, we adopt the framework of a finite-horizon, periodic-review inventory control system with random demand and backordering. We begin our analysis with the two fixed costs structure and consider the model given by (4) under the conditions K1 ¶ K2 ¶ 2K1 and K1 ¶ K2 separately. The first condition represents a special case that can arise in situations in which certain economies of scale apply. For example, dispatching a large-capacity truck at a high fixed cost can be more cost effective than dispatching two identical, smaller capacity trucks, each with lower fixed costs. The second condition corresponds to the general case of the two fixed costs structure. By utilizing the properties of C-4K1 1 K2 5convexity and strong K-convexity, respectively, we provide partial characterizations of the optimal policy under each condition. We then analyze the multiple fixed costs structure and derive our analytical results by focusing on the models given by (2) and (3), where the total fixed cost function exhibits constant fixed cost increments. For the various models under study, we provide the following partial characterizations of the optimal ordering policy. • In the two fixed costs settings with the function (4), we use five critical points, s1 s 00 1 s 0 1 s1 1 and S, where s ¶ s 00 ¶ s 0 ¶ s1 ¶ S. For the special case where K1 ¶ K2 ¶ 2K1 , the optimal policy suggests the following decisions. (a) If the starting inventory level x is less than s, then order more than C units to raise the inventory level to S. (b) If x is between s and s 00 , then order exactly C units. (c) If x is between s 00 and s 0 , then order no more than C units, which might include a quantity of zero, so that the resulting inventory level remains below S. (d) If x is between s 0 and s1 , then order C units or less to raise the inventory level to S. (e) If x is s1 or larger, then do not order. Note that this characterization is complete, except in the third decision region. The optimal policy for the more general model, where K1 ¶ K2 differs only in the last decision region, where an order quantity between 0 and C units is specified. • For the multiple fixed costs setting with constant fixedcost increments, we represent the optimal policy in terms of a 6Y − C1 Y 5 band structure. For the model of (2), the following optimal decisions are suggested. (a) If the starting inventory level x is below Y − C, then order at least 4Y − x5/CC units so that the resulting inventory is first raised to the range 6Y − C1 Y 5, where x is the largest integer that is smaller than or equal to x; then order an additional quantity between 0 and C. (b) If the starting inventory level is Y or larger, then do not order. In the last model with the cost structure (3), the optimal policy is slightly modified for x below Y − C such that

Caliskan-Demirag, Chen, and Yang: Inventory Systems with Quantity-Dependent Fixed Costs Operations Research 60(4), pp. 785–796, © 2012 INFORMS

after the inventory level reaches the 6Y − C1 Y 5 band, an additional quantity between 0 and mC may be ordered, where m ¶ 4K + k5/K. The remainder of this paper is organized as follows. We review the related literature in §2. Section 3 presents our analysis of the two fixed costs structure. In §3.1, we focus on the special case under the condition K1 ¶ K2 ¶ 2K1 , where we first give a formal definition of C-4K1 1 K2 5convexity and prove several preliminary results used to characterize the optimal policy. In §3.2, we show the optimal policy results for the general case and present insights from our computational studies. Section 4 presents the analysis for the multiple fixed costs structure. We first illustrate the optimal policy with a numerical example, and then derive analytical results for the two special cases of constant fixedcost increments with and without additional administrative cost. Finally, we summarize our conclusions and discuss possible extensions in §5. All the proofs are provided in the online appendix. An electronic companion to this paper is available as part of the online version that can be found at http://dx.doi.org/10.1287/opre.1110.1033.

2. Literature Review In inventory ordering models, the fixed cost component takes its simplest form when a constant K is charged for each order. Scarf (1960) and Veinott (1966) were the earliest studies to incorporate this fixed cost structure into the analysis of stochastic periodic-review inventory control, and they showed under certain assumptions that the optimal policy is of (s1 S) type. Later studies focused on general convex or concave ordering cost functions, such as those of Fox et al. (2006), Henig et al. (1997), Porteus (1971), and Yu and Benjaafar (2011). However, there have been relatively few studies that explicitly model fixed costs of more complex structures. Lippman (1969) considers a subadditive and nondecreasing ordering cost function that includes the form of constant fixed cost increments in (2) as a special case. For the general case, Lippman (1969) characterizes the point below which it is optimal to order and another point beyond which no ordering is necessary but gives no specific decision between these two points. In the special case of (2), Lippman (1969) provides a characterization of the optimal policy, but the result is derived under an additional assumption about the cost structure, and the characterization is incomplete, leaving a rather large region unspecified. Our analysis of this case is based on a different technical concept and provides results that reduce the unspecified region into a single range or band. Consequently, we are able to establish a more complete characterization of the optimal policy by identifying the critical point below which it is optimal to order and by showing that no more than one batch capacity is ordered from within the band. The fixed cost structure in (2) was later studied by Iwaniec (1979), and more recently by Alp et al. (2009).

787 The former study investigates an ordering policy that allows only full batches, where the order size is an integer multiple of the batch capacity, and derives the conditions under which this policy is optimal. Alp et al. (2009) show that such a restriction on the optimal policy can cause a substantial loss of efficiency, which is also demonstrated in our study, and develop bounds and heuristic policies for the problem where partially filled batches are allowed. Before we review the literature related to the two fixed costs case, we note that we are not aware of any optimal policy results for the general multiple fixed cost function (1) in the context of stochastic-demand, periodic-review inventory ordering. When demand is deterministic, Bigham (1986) and Gupta (1994) integrate this fixed cost structure into the classical economic order quantity (EOQ) model. In a similar setting, Robinson and Sahin (2001) analyze the fixed cost structure given in (3). Motivated by certain transportation and production contracts, Chao and Zipkin (2008) consider a fixed cost function that is neither convex nor concave. More specifically, the fixed cost is incurred only if the order quantity exceeds a threshold value, hence the cost function can be written as c4z5 = K16z > C7 and is a special case of our fixed cost function in (4) where K1 = 0. Chao and Zipkin (2008) apply the property of K-convexity and partially characterize the optimal policy, which is presented with three critical points, by dividing the state space into five decision regions. Understandably, our model can be reduced to theirs. With the aid of new techniques, the resulting optimal policy in our model is more complex. The two fixed costs problem is essentially an uncapacitated problem, because there is no limit on the order size. However, this problem can be considered as an an approximation of the situation in which there are two facilities, one with a capacity of C and another with a relatively large capacity compared with the possible order size. Here, the literature on the capacitated periodic-review inventory problem becomes relevant. We review only the studies that are related to our work from a technical point of view, which include those of Gallego and Scheller-Wolf (2000), Shaoxiang and Lambrecht (1996), and Shaoxiang (2004). The first two papers study the problem in a finite-horizon setting and provide partial characterizations of the optimal policy. Both models can be regarded as special cases of (4). Shaoxiang and Lambrecht (1996) show that the optimal policy is not generally of an (s1 S) type but rather follows an X-Y band structure. However, they do not give a complete characterization within the band. By introducing the technical concepts of CK-convexity and strong CK-convexity, Gallego and Scheller-Wolf (2000) provide a more complete characterization of the optimal policy. Shaoxiang (2004) extends the work of Shaoxiang and Lambrecht (1996) into an infinite-horizon setting. In this case, the analysis is based on a new concept called (C1 K)-convexity and reveals that the optimal policy continues to exhibit an X-Y band structure. Furthermore, the length of the X-Y band is shown to


788 be bounded by the capacity C, and it is optimal to order at most C units when the initial inventory level lies within the band. Our result for the special case of (2) basically follows the X-Y band structure of Shaoxiang and Lambrecht (1996). Distinct from the studies in this stream of the literature, which assume a constant (single) fixed cost, we consider multiple fixed costs in a step-function structure. From the technical perspective, the studies of Gallego and Scheller-Wolf (2000) and Shaoxiang (2004) relate to our study as follows. We derive the analytical results in each of our models by applying a specific convexity structure. The newly introduced C-(K1 1 K2 )-convexity, which we use for the special case of model (4), is rooted in the notion of CKconvexity introduced by Gallego and Scheller-Wolf (2000). In the general case where K1 ¶ K2 , strong K-convexity is utilized. This property is a more restricted version of the strong CK-convexity also introduced by Gallego and Scheller-Wolf (2000). The notion of 4C1 K5-convexity that facilitates the analysis of the model in (2) is derived by Shaoxiang (2004). Finally, in the case with the cost structure (3), we apply CK-convexity, which is a more general concept than 4C1 K5-convexity. Our research contributes to the stochastic periodic-review inventory control literature by analyzing different forms of quantity-dependent fixed costs that have applications in many industrial settings. Methodologically, we contribute by introducing a new class of functions and by extending or redeveloping several results in the existing literature using different techniques. Our analysis demonstrates that the concepts of CK-convexity (Gallego and Scheller-Wolf 2000) and (C1 K)-convexity (Shaoxiang 2004) for capacitated inventory problems can be utilized in more complex settings with multiple fixed costs. Similarly, our notions of strong K-convexity and C-(K1 1 K2 )-convexity might find application in other problems.

3. Two Fixed Costs: Model and Analysis We consider a firm that manages the periodic-review inventory system of a single product under stochastic demand. The time horizon is assumed to be finite and consists of T periods. At the beginning of each period t, the firm may place an order, in which case a fixed cost as shown in (4) is incurred. An order of size C or lower can be obtained at the fixed cost K1 , but ordering larger than C carries the higher fixed cost K2 . The total cost of ordering can also include a linear purchase cost at a per-unit rate c. Any unsatisfied demand is fully backordered at a backlog/penalty cost assessed at the end of a given period. Similarly, excess inventory is carried to the next period by incurring a holding cost. We present the case of zero leadtimes but note that our analysis can easily be generalized to incorporate nonzero leadtimes due to the full backordering of shortages (Scarf 1960). Consistent with the literature, we analyze our problem with a dynamic programming approach. We assume that all costs in future periods are discounted


at a rate of ¶ 1, and that demand in consecutive periods is independently distributed. For periods t = 11 21 0 0 0 1 T , xt = the inventory level at the start of period t before an order is placed, yt = the inventory level after any order is placed in period t but before the demand is realized, Dt = the nonnegative demand in period t (random variable), and L4yt 5 = the one-period expected holding and backorder cost with the order-up-to inventory level yt . We assume that the one-period cost function L4yt 5 is convex and that limyt → L4yt 5 = . For the numerical examples, it is convenient to allow the holding and backorder costs to be linear, that is L4yt 5 = hE64yt − Dt 5+ 7 + pE64Dt − yt 5+ 7, where h and p are the unit inventory holding and penalty costs per period. Let ft 4xt 5 be the total expected cost when the initial inventory in period t is xt and the optimal ordering policy is employed in the remaining t periods. We can then write the dynamic programming recursion for ft 4xt 5 as ft 4xt 5 = −cxt + inf K1 16xt < yt ¶ xt + C7 yt ¾xt

+K2 16yt > xt + C7 + Gt 4yt 5 1

(5)

where Gt 4yt 5 = cyt + L4yt 5 + E6ft−1 4yt − Dt 57. The firm’s objective is to determine the policy that maximizes fT 4x5 for all x. We assume that the terminal function f0 4x5 is convex and set the boundary conditions to f0 4x5 = 0. We also assume that c = 0, which is without loss of generality as the effect of the linear ordering cost on the optimal policy structure can be ignored after some simple transformations (see Veinott and Wagner 1965). We analyze the problem for two cases. In the first case, we assume that K1 ¶ K2 ¶ 2K1 , which allows us to exploit a new type of convexity property to derive some structural results. Our analysis in the second case considers the more general condition K1 ¶ K2 and uses a different convexity structure. Many studies have analyzed inventory models with fixed costs, and various kinds of convexity or concavity properties have been employed to characterize the optimal policy according to the model structure. In Definition 1, we give some general definitions to provide a unifying framework that connects our convexity concepts with similar concepts in the literature. Definition 1. (i) A real-valued function G is called 4K1 z5-convex for K ¾ 0 if, for all y, 0 < b < , and z ∈ 601 5, G4y + z5 + 4K1 z5 ¾ G4y5 + 4z/b58G4y5 − G4y − b59. (ii) A real-valued function G is called strong 4K1 z5convex for K ¾ 0 if, for all y, 0 < b < , 0 ¶ a < and z ∈ 601 51 G4y + z5 + 4K1 z5 ¾ G4y5 + 4z/b58G4y − a5 − G4y − a − b590


789


For convenience, we write the variables in onedimensional form, but the definitions can easily be extended to the multidimensional case. Note that 4K1 z5-convexity corresponds to the standard convexity when 4K1 z5 = 0, to the K-convexity of Scarf (1960) when 4K1 z5 = K, to the symmetric-K-convexity of Chen and Simchi-Levi (2004) when 4K1 z5 = max8z/b1 1 − z/b9K, to the 4K1 1 K2 5convexity of Ye and Duenyas (2007) when 4K1 z5 = 41 − z/b5K1 + 4z/b5K2 − min8z/b1 1 − z/b9 min8K1 1 K2 9, and to the weak 4K1 1 K2 5-convexity of Semple (2007) when 4K1 z5 = 41 − z/b5K1 + 4z/b5K2 . Similarly, strong 4K1 z5-convexity implies ordinary convexity when 4K1 z5 = 0, the strong CK-convexity of Gallego and Scheller-Wolf (2000) when 4K1 z5 = K for z ∈ 601 C7, and strong K-convexity when 4K1 z5 = K, which is a special case of strong CK-convexity with C → . 3.1. Special Case 4K1 ¶ K2 ¶ 2K1 5 In this section, we consider the special case of K1 ¶ K2 ¶ 2K1 . We first formally define C-(K1 1 K2 )-convexity and derive some preliminary results for the analysis of the optimal policy. 3.1.1. Preliminary Results: C-4K1 1 K2 5-Convexity. A common procedure in the analysis of inventory control problems is to find an appropriate convexity property, or a 4K1 z5 function with appealing properties such as laid out in Definition 1. To tackle our model, we define a particular 4K1 z5-convexity that originates from the notion of strong CK-convexity of Gallego and Scheller-Wolf (2000). Definition 2. A real-valued function G is called C-4K1 1 K2 5-convex for K1 , K2 ¾ 0 if, for all y, 0 ¶ a < , 0 < b < , and z ∈ 601 5, G4y + z5 + C 4K1 1 K2 1 z5 z ¾ G4y5 + 8G4y − a5 − G4y − a − b591 b ( K1 z ∈ 601 C71 C 4K1 1 K2 1 z5 = K2 z > C0

where (6)

Figure 1 gives an illustration of C-(K1 1 K2 )-convexity. In the graph, the points A = 4y − a − b1 G4y − a − b55, B = 4y − a1 G4y − a55, and R = 4y1 G4y55. Furthermore, for C ¾ z1 ¾ 0, E = 4y + z1 1 K1 + G4y + z1 55, and for z2 > C, F = 4y + z2 1 K2 + G4y + z2 55. Note that the points are selected such that A and B lie to the left of y, E lies between y and y + C, and F lies to the right of y + C. Geometrically, C-4K1 1 K2 5-convexity means that two lines drawn from any point R to connect points R and E and to connect points R and F both have larger slopes than a line connecting any two points A and B behind y. Note that when K2 → +, C-4K1 1 K2 5-convexity reduces to CK-convexity (Gallego and Scheller-Wolf 2000) and when K1 = K2 = K it reduces to strong K-convexity, which is used to analyze our general two fixed costs problem. We elaborate the properties of C-4K1 1 K2 5-convex functions in Lemma 1.

Geometric illustration of a C-(K1 1 K2 )-convex function.

Figure 1. G(x)

F

K2

E B

K1

A R y–a–b

y–a

x y

y + z1

y+C

y + z2

Lemma 1. (a) A convex function is also a C-(01 0)-convex function. (b) If f is C-(K1 1 K2 )-convex and is a positive scalar, then f is C-(K1 , K2 )-convex. (c) If f is C-(K1 1 K2 )-convex, then it is also C-(K10 , K20 )convex for any K10 ¾ K1 and K20 ¾ K2 . (d) The sum of a C-(K1 1 K2 )-convex function and a C(K10 , K20 )-convex function is C-(K1 + K10 , K2 + K20 )-convex. (e) If v is C-4K1 1 K2 5-convex, is the probability R +density of a positive random variable, and G4y5 = v4y − 545 d, then G is also C-4K1 1 K2 5-convex. 0 The stated results imply preservation properties for C-(K1 1 K2 5-convexity under certain common operators that are relatively straightforward to demonstrate. We next prove that C-4K1 1 K2 5-convexity can be preserved under a minimization operator. This nontrivial result plays a central role in deriving the structural properties of the optimal cost functions. Lemma 2. Suppose that G4x5 is a C-(K1 1 K2 )-convex function and K1 , K2 are nonnegative constants such that K1 ¶ K2 ¶ 2K1 . Then, f 4x5 = miny¾x 8K1 16x < y ¶ x + C7 + K2 16y > x + C7 + G4y59 is also C-4K1 1 K2 5-convex. We briefly note that the stated condition K2 ¶ 2K1 is necessary for the result to hold and refer readers to the proof for details. 3.1.2. Analysis of the Optimal Policy. To characterize the optimal policy, we define some critical points using the cost functions. We can easily verify that ft 4xt 5 and Gt 4xt 5 are continuous for any t = 01 11 0 0 0 1 T , and thus the critical points are well defined. Definition 3. Given the nonnegative constants C, K1 , and K2 , we define S = arg inf Gt 4x53 s1 = inf8x Gt 4x5 ¶ K1 + Gt 4S593 s2 = inf8x Gt 4x5 ¶ K2 + Gt 4S590


790


Note that we drop the subscript t from the notation of the critical points and the starting inventory level for simplicity. To interpret s1 , consider the following two actions: (1) order a quantity lower than C so that the resulting inventory level reaches S and the fixed cost is K1 , and (2) order nothing. If the first action is feasible, then s1 is the threshold level such that when x < s1 , ordering dominates not ordering. The interpretation of s2 is parallel, meaning the point below which ordering up to S with an order quantity larger than C (when feasible) dominates ordering nothing. The continuity of the functions allows the critical points to take the corresponding values, such as Gt 4s1 5 = K1 + Gt 4S5. We use s1 , s2 , and S to prove the following lemma. Lemma 3. If Gt 4x5 is a C-(K1 1 K2 )-convex function, then (i) Gt 4y5 + K1 ¾ Gt 4x5 for any x + C > y > x > s1 ; (ii) Gt 4y5 + K2 ¾ Gt 4x5 for any y > x > s2 ; (iii) s2 ¶ s1 ¶ S; and (iv) Gt 4x5 is nonincreasing in 4−1 s2 5. Figure 2 illustrates the critical points and some of the results from Lemma 3. Note that when x > s1 , it is more costly to place an order of any size than not to order, hence “order nothing” is the optimal decision in this region. However, the given set of critical points is not sufficient to characterize the optimal policy in all regions. For example, when x ¶ s2 and s2 < x ¶ s1 , it might be thought optimal to order up to S if S − x > C and S − x ¶ C, respectively, but this might not be the case. For a given initial inventory level x ¶ s2 , notice that ordering up to y rather than S might result in a lower cost, that is, Gt 4y5 + K1 as opposed to Gt 4S5 + K2 , provided that y − x ¶ C and S − x > C. Consequently, additional critical points are needed, which we define through an alternative representation of the objective function ft 4x5. Let Jt 4x5 = min 8K1 14z > 05 + Gt 4x + z591 0¶z¶C

and

(7)

Vt 4x5 = inf 8K2 14y > x5 + Jt 4y590

(8)

y¾x

We show the result and subsequently define the critical points s, s 0 , and s 00 . Figure 2.

Illustration of the critical points in Definition 3 and some of the results from Lemma 3. Gt (z)

K1

x

s2

s1

y

S

K2

z

Lemma 4. For any K2 ¾ K1 , ft 4x5 = Vt 4x5, where Vt 4x5 is given in Equation (8). Definition 4. Given nonnegative constants C, K1 , and K2 , and C-4K1 1 K2 5-convex functions Gt 4x5 and Jt 4x5, we define s = inf8x Jt 4x5 ¶ K2 + Jt 4S593 s 0 = min8S − C1 s1 93 s 00 = inf8x s ¶ x ¶ s 0 1 Jt 4x5 < K1 + Gt 4x + C590 The time indices are again suppressed for convenience. Parallel to the interpretations of s1 and s2 , here s is the point below which ordering a quantity larger than C to reach S dominates ordering no more than C units, and s 00 is the point below which ordering C units dominates ordering less than C units. It is easy to verify that Jt 4S5 = Gt 4S5. The following lemma, together with Lemma 3, is vital to the characterization of the optimal policy. Lemma 5. If Gt 4x5 is a C-(K1 1 K2 )-convex function and Jt 4x5 and Gt 4x5 satisfy (7), then (i) Jt 4y5 + K2 ¾ Jt 4x5 for any y > x > s; (ii) s + C ¶ S; (iii) s ¶ s 00 ¶ s 0 ¶ s1 ; and (iv) Jt 4x5 ¾ K1 + Gt 4x + C5 for any s ¶ x < s 00 . We now use the set of critical points (s1 s 00 1 s 0 1 s1 1 S) and present the optimal policy in the five decision regions 4−1 s5, 6s1 s 00 5, 6s 00 1 s 0 5, 6s 0 1 s1 5, 6s1 1 +5. Our main result for this case is established in Theorem 1. We illustrate the critical points and optimal decisions in Figure 3. Theorem 1. (a) Gt 4x5 and ft 4x5 are C-(K1 1 K2 )-convex for all t. (b) For each t = 11 21 0 0 0 1 T , there exists an optimal policy that can be characterized by the points s ¶ s 00 ¶ s 0 ¶ s1 ¶ S in the following way. (i) Order up to S when x < s; (ii) Order exactly C when s ¶ x < s 00 ; (iii) Order no more than C when s 00 ¶ x < s 0 ; (iv) Order up to S when s 0 ¶ x < s1 ; and (v) Order nothing when x ¾ s1 . Before we proceed to the general case of two fixed costs, we make some remarks about our analysis and the results for the special case. So far, we have assumed that the terminal function f0 4x5 and the one-period holding and backorder cost function L4y5 are convex. By parts (a) and (d) of Lemma 1, we can relax the convexity assumption to C-(K1 1 K2 )-convexity. Under this assumption, our model reduces to the classical inventory model without a fixed cost when K1 = K2 = 0. It is then easy to verify from our analysis that s1 = S, and thus the base-stock policy is optimal. When K1 = K2 , our model reduces to the classical inventory model with one fixed cost, for which the optimal policy is well known to be of (s1 S) type. We can show that our optimal policy takes an 4s1 S5 form if we can verify


791


Figure 3.

Optimal policy for the case in which K1 ¶ K2 ¶ 2K1 . Order > C up to S

Order ≤ C up to y < S

Order = C up to y < S

s

s

that s1 = s when K1 = K2 . From Lemma 5, we know that s1 ¾ s, and thus it suffices to prove that s1 ¶ s. By the definition of s, this is equivalent to proving that for any x < s1 , Jt 4x5 ¾ Jt 4S5 + K2 . When x < s1 , we have n o Jt 4x5 = min Gt 4x51 K2 + min Gt 4x + z5 1 0 Y and 0 < z ¶ C. Lemma 6 is used by Shaoxiang (2004) to partially characterize the optimal policy for a variation of the classical inventory model with a single fixed cost and finite ordering capacity C. In particular, parts (i) and (ii) of the lemma imply that it is optimal to order full capacity when the inventory level is below Y − C, whereas part (iii) indicates that when the inventory level is above Y , it is optimal not to order. However, when the inventory level is between Y − C and Y , no explicit optimal decision is characterized and the ordering pattern varies depending on the particular instance (Shaoxiang and Lambrecht 1996, and Shaoxiang

Optimal policy for the instance where K1 = 20, K2 = 40, K3 = 60, C1 = 0, C2 = 10, C3 = 40.

xt ∈ Optimal decision xt ∈ Optimal decision

4−1 −217 Order up to 44 4−31 47 Order exactly 40

4−211 −167 Order exactly 40 441 97 Order up to 44

4−161 −117 Order up to 24 491 147 Order exactly 10

4−111 −67 Order exactly 40 4141 177 Order up to 24

4−61 −37 Order up to 34 4171 +5 Order nothing


794


2004). In our study, Lemma 6 provides some preliminary results for the characterization of the optimal policy under multiple fixed costs. To simplify the analysis, we define min

0¶z¶C

Optimal policy for an instance with K = 50 and C = 40.

100

8K14z > 05 + Jti−1 4x + z591

90

i = 11 21 0 0 0

(10)

and Jt0 4x5 = Gt 4x5. Here, Jti 4x5 can be interpreted as the minimum cost in period t when at most i batches are ordered. We establish in Lemma 7 that ft 4x5 = limi→+ Jti 4x5. Letting y ∗ 4xt 5 be the optimal post-order inventory level, or the order-up-to level when the initial inventory level is xt , we prove our main result in Theorem 3.

80

Order-up-to level

Jti 4x5 =

Figure 4.

70 60 50 40 30

Lemma 7. For any t = 11 21 0 0 0 1 T 1 ft 4x5 = limi→+ Jti 4x5. Theorem 3. For any t = 11 21 0 0 0 1 T 1 (i) ft 4x5, Gt 4x5, and Jti 4x5 are 4C1 K5-convex, where i = 11 21 0 0 0; (ii) there exists a Yt such that y ∗ 4xt 5 = xt when xt ¾ Yt ; ∗ y 4xt 5 = y ∗ 4xt + C5 when xt < Yt − C; and xt ¶ y ∗ 4xt 5 ¶ xt + C when Yt − C ¶ xt < Yt . Theorem 3 indicates that it is optimal to order nothing when the inventory level is above Yt . It also implies that for any two starting inventory levels below Yt that differ by a multiple of C, the optimal order quantities also differ by the same multiple of C and reach an identical postorder inventory level. To characterize the ordering policy, it suffices to find the optimal order-up-to level y ∗ 4xt 5 for Yt − C ¶ xt < Yt , which, as the theorem reveals, satisfies xt ¶ y ∗ 4xt 5 ¶ xt + C. The practical interpretation of Theorem 3 is as follows. In Case 1, where the starting inventory level is between Yt − C and Yt , order at most one partial or full batch. In Case 2, where the starting inventory level satisfies xt < Yt − C, order 4Yt − xt 5/C full batches to raise the inventory level up to the range 6Yt − C1 Yt 5. Let this new level be zt and treat zt as in Case 1 for an extra partial or full batch. Similar to the finding of the X-Y band structure by Shaoxiang and Lambrecht (1996) and Shaoxiang (2004), our results partially characterize the optimal policy for the region 6Yt − C1 Yt 5 within which ordering up to a certain local minimum might be optimal and no additional structural pattern appears to exist. We now illustrate the optimal policy with a numerical example. Corresponding to the formulation in (9), we define St = arg inf Gt 4x5. We assume that the demand in each period has a Poisson distribution with a mean = 30. We set the rest of the parameter values at T = 50, h = 1, p = 8, K = 50, C = 40, = 009, and consider t = 6. Figure 4 shows the optimal policy for this example. Here, Y6 = 26 and S6 = 66. We can see that when x6 < Y6 = 26, the order-up-to level for x6 − mC is the same as that for x6 , where m is any positive integer. Table 6 presents the optimal decision for x6 ∈ 6Y6 − C1 Y6 5 = 6−141 265 in

20 –200

–150

–100

–50

0

50

100

Starting inventory level

detail. When the starting inventory level lies in 6−141 −27 or 4111 267, it is optimal to order a full batch. However, when the starting inventory level lies in 4−21 117, it is optimal to order a partial batch to raise the inventory level up to the local minimum y = 38. Our cost model allows partial batch ordering where the fixed cost is fully incurred for each batch that is used, independent of its utilization rate. Although the partial batch ordering could potentially lead to higher fixed costs, it can be advantageous from a total cost perspective due to the added flexibility in inventory control. An alternative policy is to allow only full batches by restricting the order quantities to integer multiples of the batch capacity (e.g., Iwaniec 1979). In practice, this policy might sometimes be preferred due to the immediate savings in fixed costs and its simple structure, but its performance might be far from optimal in some cases. Consider an example that is almost identical to the previous example, except that K = 10. For t = 1, it can be verified that G1 4255 = 44043, G1 4375 = 9066, G1 4655 = 35000, and that G1 4x5 is a convex function with a minimizer at x = 37. When the starting inventory level is 25, it is clearly optimal to order up to 37, which results in the minimum cost f1 4255 = 10 + G1 4375 = 19066. In contrast, the optimal decision under the full batch policy is to order nothing, as the resulting cost 44043 4= G1 42555 is lower than that of ordering a full batch, that is, 45000 4= 10 + G1 46555. In this instance, the relative error reaches almost 126%. If we consider t = 6, then the maximum relative error is around 51.68%. These observations are consistent with the findings of Alp et al. (2009), who compare the partial and full batch ordering policies through extensive numerical studies. Other insights from their work indicate that the suboptimality of the full batch policy increases when the fixed costs (K) are small and the full batch sizes (C) are large. We note that our analysis so far assumes no limit on the number of batches that could be assigned to each order.


795


Table 6.

Optimal policy in the region 6Yt − C1 Yt 5 when K = 50 and C = 40.

xt ∈ Optimal decision

6−141 −27 Order exactly 40

In practice, there might be cases where only a finite number of batches (n) is available; for example, where there is limited fleet size in a trucking environment. To analyze our problem in such cases, we can modify the cost function in (1) by setting Kn+1 = , and we can show that the objective functions Jti 4x5, i = 11 21 0 0 0 1 n remain 4C1 K5-convex. Then by conducting a similar analysis as in Theorem 3, we can verify that the original optimal policy must be slightly modified to account for the limited availability of batches. When the starting inventory level is below Yt , it is optimal to first raise the inventory level as close to the 6Yt − C1 Yt 5 region as possible and then, if not all batches have been assigned to the order, to use another partial or full batch to fulfill the remaining order quantity. When n = 1, our problem reduces to the classical capacitated inventory model (e.g., Gallego and Scheller-Wolf 2000 and Shaoxiang 2004). As we adopt the technical approach of Shaoxiang (2004), our results for this case are naturally identical. 4.2. Special Case: Additional Fixed Cost We next consider the case with the cost structure (3), which is essentially an extension of the problem analyzed in §4.1. With this case, in addition to the standard fixed cost K associated with each batch, a common fixed cost k is incurred whenever an order is placed, or equivalently, Ci+1 −Ci = C, Ki+1 − Ki = K, and K1 = K + k. Incorporating this additional fixed cost requires the application of a different technical property called CK-convexity, which was introduced by Gallego and Scheller-Wolf (2000). CK-convexity can be formally defined by lifting the second condition of 4C1 K5convexity in Definition 5, thus making it less restricted. As a result, the length of the order-up-to band is no longer guaranteed to be less than or equal to C, meaning that more than one batch capacity could be ordered from a point in the region. Nevertheless, we can build an upper bound on the number of batches to be ordered. We state our main result in Theorem 4. Theorem 4. For any t = 11 21 0 0 0 1 T 1 (i) there exists a Xt and Yt such that it is optimal to order when xt ¶ Xt and to order nothing when xt > Yt . Moreover, y ∗ 4xt − C5 = y ∗ 4xt 5 when xt < Xt ; (ii) y ∗ 4xt 5 ¶ Yt + 4K + k5/KC for any xt ¶ Yt . Similar to the result in §4.1, we identify an Xt − Yt band structure in the optimal policy. For starting inventory levels exceeding Yt , it is optimal to order nothing; and for those below Xt , it is optimal to order such that the inventory level is first raised to the Xt − Yt band, and then an additional quantity of multiple batches mC may be ordered, where m is bounded by 4K + k5/K. For starting inventory levels within the band, at most 4K + k5/K batches may be ordered.

4−21 117 Order up to 38

4111 267 Order exactly 40

5. Conclusions In most of the earlier studies on periodic-review inventory problems, the fixed cost is assumed to be invariant to the order size. However, in real systems there can be situations in which the fixed cost is dependent on the order size. For example, some transportation contracts indicate that buyers bear higher fixed ordering costs when they exceed a specified contract volume C. Other examples are seen in the truck freight shipping industry, where the fixed costs increase as additional trucks are dispatched to ship an order; and in production settings, where the total number of setups increases with the lot size. Motivated by such applications, we analyze inventory problems with quantity-dependent total fixed costs. Our base model setting is a finite-horizon, periodicreview inventory control system in which demand is nonstationary and random and unsatisfied demand is backordered. We model fixed costs of a step-function type, where jumps in the fixed cost occur whenever the order quantity exceeds certain capacity limits or thresholds. In the first part of our analysis, we focus on the case with two fixed costs: a fixed cost of K1 that is incurred when the order quantity is less than or equal to C and a fixed cost of K2 ¾ K1 that is incurred otherwise. In the special case of K1 ¶ K2 ¶ 2K1 , we introduce a new technical concept called C-4K1 1 K2 5-convexity, which facilitates a partial characterization of the optimal policy. The policy is presented with five critical points over five exhaustive decision regions and has some resemblance to, but is significantly more complex than, the classical 4s1 S5 type policies. In the more general case of K1 ¶ K2 , we use another concept called strong K-convexity to partially characterize the optimal policy. We also develop a heuristic method to simplify the policy, and conduct numerical experiments to evaluate its performance, which appears highly satisfactory in a large set of test instances. In the second part, we extend our analysis to consider multiple fixed costs, for which the optimal policy can be quite complex. Although an exploration of structural results in this most general setting eludes us, we are able to derive a characterization of the optimal policy by restricting the fixed cost structure to one with constant cost increments K that occur at equally spaced points that differ by C. In other words, the order quantity is shipped in batches (e.g., truck loads or containers), each with an identical capacity C and fixed cost K. Our results for this special case reveal that the optimal policy follows an X-Y band structure, where no order is placed above Y and an order of size of at most C (a single batch) could be placed within the band. For starting inventory levels below X, it is optimal to first raise the inventory level to reach the band and then to order an additional quantity of at most a single batch if ordering


796 is necessary. In a variant of this problem, we add another fixed ordering cost k that is incurred every time an order is placed along with the standard fixed costs of the assigned batches. We show that the X-Y band type of the optimal policy structure remains similar, except that the number of batches ordered from points within the band may be higher than one but no more than 4K + k5/K. As a future study, our research into the two fixed costs case could be extended by considering the case where K1 ¾ K2 . Our preliminary analysis shows that some of the current results no longer hold under this alternative setting, and that new technical properties might be necessary to analyze the optimal policy. Another direction for future research is to incorporate a decreasing linear ordering cost function, which is likely to call for a new set of techniques. Electronic Companion An electronic companion to this paper is available as part of the online version at http://dx.doi.org/10.1287/opre.1110.1033.

Acknowledgments The authors thank the associate editor and two anonymous referees for their constructive suggestions, which helped improve the content and presentation of this article. They also thank Jian Yang for introducing Definition 1 and Jeannette Song for her insights during the initial stages of this research. All correspondence will be directed to the corresponding author, Yi Yang.

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Ozgun Caliskan-Demirag is an assistant professor of supply chain management at Sam and Irene Black School of Business, Penn State Erie, the Behrend College. She holds a Ph.D. in industrial and systems engineering from Georgia Institute of Technology. Her main research interests are in the areas of supply chain management, operations/marketing interface, inventory management, and decentralized resource allocation. Youhua (Frank) Chen is a professor of management sciences at the City University of Hong Kong. He has a long-term interest in the area of stochastic inventory models. His current interests also include dynamic pricing and revenue management, and supply-chain models with risk considerations. This paper was accepted while he was on the faculty of the Chinese University of Hong Kong. Yi Yang is an assistant professor in the Department of Management Science and Engineering Management at Zhejiang University. He received his Ph.D. degree in systems engineering and engineering management from the Chinese University of Hong Kong in 2011. His main research interests include inventory management and revenue management. This paper was accepted while he was a postdoctoral researcher at the Chinese University of Hong Kong.