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Effective inventory control policies with a minimum order quantity and batch ordering
Int. J. Production Economics 168 (2015) 21–30
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Effective inventory control policies with a minimum order quantity and batch ordering Han Zhu n, Xing Liu, Youhua (Frank) Chen Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong
art ic l e i nf o
a b s t r a c t
Article history: Received 21 March 2014 Accepted 5 June 2015 Available online 18 June 2015
In this paper, we consider a single-item periodic-review stochastic inventory system with both minimum order quantity (MOQ) and batch ordering requirements. In each time period, the firm can order either none or at least as much as the MOQ. At the same time, if an order is placed, the order quantity is required to be an integral multiple of a given specific batch size. We first adopt a heuristic policy which is specified by two parameters (s; t). Applying a discrete time Markov chain approach, we compute the system cost and optimize this (s; t) policy under the long-run average cost criterion. We also consider a simpler one-parameter policy, the so-called S policy, which is a special case of the (s; t) policy. In an intensive numerical study, we find that (1) both policies perform well in comparison with other policies and (2) the S policy also performs well and is compatible with the (s; t) policy; only in a few cases where demand variation is small, the latter outperforms the former significantly. We also evaluate the effects of some important parameters on system performance. & 2015 Elsevier B.V. All rights reserved.
Keywords: Minimum order quantity Batch ordering Stochastic inventory systems Markov chain Heuristic policies
1. Introduction In industries, minimum order quantity (MOQ) and batch ordering, applied independently or simultaneously, are two common requirements made by suppliers, both of which can help companies take advantage of economies of scale and hence reduce costs. The MOQ requirement means that the order quantity must equal or exceed a specified level, if an order is placed. The batch ordering requirement means that the order quantity must be an integral multiple of a specified given batch size. The application of a MOQ is common in practice. With the prevalence of e-commerce, MOQs are becoming more and more common in our lives, especially in online business-to-business sourcing portals such as alibaba.com, where suppliers often set such requirements. MOQs are also applied in manufacturing industries for products that have short lifetimes or long leadtimes. A well-known example is Sport Obermeyer, a fashion sport skiwear manufacturer, which has a minimum production level of 600 garments in Hong Kong and 1200 garments in China per order (Zhao and Katehakis, 2006). In fact, MOQ requirements are quite common in China and other low cost manufacturing countries. Low profit margins force manufacturers to pursue large production quantities to break even. On the other hand, batch ordering is
n
Corresponding author. Tel.: þ 852 54826926. E-mail addresses: [email protected] (H. Zhu), [email protected] (X. Liu), [email protected] (Y. Chen). http://dx.doi.org/10.1016/j.ijpe.2015.06.008 0925-5273/& 2015 Elsevier B.V. All rights reserved.
also a ubiquitous requirement in industries, because materials often flow in fixed batch sizes in supply chains. For example, raw materials usually arrive at factories in full truckloads, work-inprocess is often processed in convenient lot sizes between production stages, and finished goods may be transported in full containers from suppliers to warehouses or distribution centers. Therefore, it is of no surprise that suppliers who apply a MOQ may also require batch ordering. Indeed, our decision to jointly consider both MOQ and batch ordering requirements in this paper is largely motivated by our experience with a wholesale company in Hong Kong. For a variety of products, the firm first replenishes its stock from suppliers and then sells to retail customers, and for most of these products, the firm stipulates both MOQ and batch ordering requirements. The coexistence of a MOQ and batch ordering has a two-sided effect. On one hand, requiring a MOQ and batch ordering simultaneously helps suppliers reduce the risk of uncertainty and achieve economies of scale. On the other hand, the requirements may have a negative effect on buyers’ inventory control, especially when MOQs are relatively large compared with their demand, which is not unusual in practice. Managers in such situations need principles or tools to help control their inventory. However, to the best of our knowledge, no research has investigated inventory systems with both MOQ and batch ordering requirements. Thus, the primary goal of this paper is to fill this gap in the literature. In this paper, we consider a single product stochastic periodic-review inventory system with both MOQ and batch ordering requirements. The selling firm can make a decision at the beginning of
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H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
each time period after reviewing the inventory position. When the firm decides to place an order, the order quantity must satisfy both the MOQ and the batch ordering constraints, where we assume the MOQ is an integral multiple of the batch size. The leftover inventory is carried to the next period and incurs a holding cost, whereas unsatisfied demand is fully backlogged and incurs a backordering cost. The total costs consist of the linear ordering cost, the holding cost, and the backordering cost. The objective is to minimize the long-run average cost of the system. The optimal policy for the system with only the MOQ requirement, which is partially characterized by Zhao and Katehakis (2006), is rather complicated, even without batch ordering. Therefore, for inventory systems with both MOQ and batch ordering requirements, it is necessary to propose some effective heuristic policies, which is the major contribution of our work. Facing the MOQ requirement, many companies apply the (s,S) type policy to control inventories in practice (Zhou et al., 2007). Based on this, we first propose a two-parameter policy with a similar structure, i.e., the (s; t) policy, where s o t o s þ M and M represents the MOQ. The (s; t) policy operates as follows: at the beginning of each period, if the inventory position is less than or equal to s, order a quantity that is just sufficient to bring the inventory position to s þ M or above (the inventory position after ordering can be larger than s þ M, because the order quantity must also satisfy the batch ordering requirement); if the inventory position exceeds s but is no more than t, order exactly M; otherwise, order nothing. We identify the bounds for the optimal t, and propose algorithms to find the optimal values of t and s. We also examine a simpler and more easy-to-use policy, i.e., the S policy, which is a special case of the (s; t) policy. The S policy operates in the same way as the (s; t) policy with s ¼ S M and t ¼ S 1. The numerical study shows that both these polices have close-to-optimal performance in most cases and that there is an overwhelming preponderance to the best (s,S) policy over all examples. The remainder of this paper is organized as follows. The literature on MOQ and batch ordering is discussed in Section 2. In Section 3, the model description and notations are presented. In Section 4, we propose a two-parameter ðs; tÞ policy and present algorithms to optimize the policy. A simpler one-parameter policy is introduced in Section 5. Numerical examples are conducted in Section 6 to measure the effectiveness of these two policies by comparing them with other policies. Finally, Section 7 concludes the paper by summarizing the findings.
2. Literature review The existing research on stochastic inventory systems is quite extensive. Here, we mention only a few of the most relevant papers. Many papers focus on problems associated with batch ordering or MOQ separately. The literature related to our paper can be divided into two areas: (1) supply chain inventory management with batch ordering and (2) supply chain inventory management with MOQ. In the area of batch ordering, Veinott (1965) shows the optimality of the (R,Q) policy for a periodic-review inventory system with batch ordering and no fixed ordering cost. This (R, Q) policy operates as follows: at the beginning of each period, if the inventory position is less than the reorder point R, order the smallest integral multiple of the batch size Q that will bring the inventory position to at least R; otherwise order nothing. Chen (2000) generalizes Veinott's result to multi-echelon systems’ settings and demonstrates the optimality of (R,nQ) policies for multi-stage serial and assembly systems where materials flow in fixed batches and the stochastic demands are stationary over time. Chao and Zhou (2009) find the optimal inventory control policy for
a multi-echelon serial system with batch ordering and fixed replenishment intervals. They derive a distribution-function solution for its optimal control parameters and design an efficient algorithm for computing those parameters. Huh and Janakiraman (2012) extend the work of Veinott (1965) and Chen (2000) by demonstrating the optimality of echelon (R,nQ) policies for multiechelon serial systems with nested batch ordering and nonstationary demands. Although it is not optimal in some complex inventory systems with batch ordering, the reorder point, lot-size ordering policy is easy to implement. For this reason, numerous heuristic policies have been proposed, see for example Gallego (1998), Axsäter and Zhang (1999), Shang and Zhou (2010). In the area of MOQ, Zhao and Katehakis (2006) introduce the concept of M-increasing function and first partially characterize the optimal policy for multiperiod inventory systems with MOQ. For the uncharacterized part, the authors give easily computable upper bounds and asymptotic lower bounds for these intervals. However, for the characterized part, the optimal policy is complexly structured and difficult to implement in practice. Zhou et al. (2009) study a periodic-review inventory system where an additional fixed shipping cost is imposed whenever the order quantity is less than a specified free shipping quantity. The authors characterize the structure properties of the optimal inventory control policy for the single-period model and propose a heuristic policy for multi-period inventory systems. Bradford and Katehakis (2007) study a system where a single supplier has contractual obligations to provide a minimum amount and a maximum amount to all retailers. The authors show that all retailers can be partitioned into three disjoints subsets and provide the optimal allocation for each subset. For other references on MOQ, the reader is referred to Porras and Dekker (2006), Bradford and Katehakis (2006), Okhrin and Richter (2011), Mangione and Penz (2012). The most closely related papers to our work are Zhou et al. (2007) and Kiesmuller et al. (2011). Zhou et al. (2007) propose a two-parameter heuristic policy for a stochastic inventory system with MOQ requirement and demonstrate that the performance of this policy is close to the optimal policy except for a few cases when the coefficient of the demand distribution is very small. Kiesmuller et al. (2011) propose a simpler policy, which has only one parameter S. This policy works as follows: no order is placed when the inventory position is not less than the level S; otherwise an order is placed to raise the inventory to S. However, if this order is smaller than the MOQ, the order quantity is increased to the MOQ. The authors show the effectiveness of this policy and develop simple newsvendor inequalities for near-optimal policy parameters. However, both Zhou et al. (2007) and Kiesmuller et al. (2011) do not consider batch ordering. To the best of our knowledge, our paper is the first to study stochastic inventory system with both MOQ and batch ordering requirements. To combine the two requirements, we need to tackle the problem of selecting an order quantity that satisfies both the constraints simultaneously. In a system with only the MOQ constraint, the order quantity can be any integer that is larger than or equal to the MOQ. However, with the addition of batch ordering, the firm has to either round up or round down the order quantity to an integral multiple of the given batch size. Therefore, in our model, the order quantities are subject to two kinds of jumps, which makes the analysis much more difficult.
3. Model description We consider a periodic-review inventory system for a single item with stochastic demand. The demand D in each period is an independent identically distributed (i.i.d.) random variable. The retailer replenishes its stock from a supplier. In our model, when
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
an order is made, we assume that the supplier has two requirements regarding to the order quantity:
Minimum Order Quantity (MOQ)
The order quantity must be at least M, the given minimum order quantity. Batch Ordering The order quantity must be an integral multiple of Q, the given batch size.
We also assume that M is an integral multiple of the Q and M 4 Q . If M r Q , the problem is reduced to that of Veinott (1965), the optimal policy of which can easily be computed. Obviously, M is an eligible option for the order quantity. When an order is made, the order quantity must be in the form M þ iQ , where i is a nonnegative integer. The sequence of events is as follows. At the beginning of each period, after reviewing the initial inventory position, the retailer decides whether to make a placement. After the order placement, the demand is realized. The demand is satisfied according to the first-come-first-service rule and unsatisfied demand will be backlogged. At the end of each period, the inventory cost is evaluated. Excess inventory will generate an inventory holding cost h per unit per period and a penalty cost p per unit per period will be incurred for the unsatisfied demand units. In our model, we consider linear variable cost and no fixed ordering cost. We assume an infinite planning horizon and stationary setting. We do not take into consideration the fixed ordering cost, because it usually plays the same role as MOQ in helping firms achieve the economies of scale. In practice, MOQ can replace the fixed ordering cost in most cases. The average cost criterion is used to evaluate the inventory system and the objective is to minimize the long-run average cost of the system. We set the purchasing cost equal to zero because the linear ordering costs can be ignored under the average cost criterion, see e.g., Zheng and Federgruen (1991). Without loss of generality, we set the lead-time equal to zero. Our model can be easily extended to systems with positive lead times using the standard method in Heyman and Sobel (1984). In the remainder of this paper, the following notations will be used M Q Dn qn h p xn yn Zþ ½a; b
minimum order quantity batch size demand in period n order quantity in period n holding cost per unit per period penalty cost per unit per period the inventory position before ordering in period n the inventory position after ordering in period n maxð0; ZÞ the integer numbers between a and b (if a and b are integers, they are included) The expected cost function C n ðyn Þ in period n can be written as
C n ðyn Þ ¼ hE½ðyn Dn Þ þ þ pE½ðDn yn Þ þ ;
ð1Þ where yn and Dn are both integers. We can easily find that C n yn is convex and C n yn - þ 1 as j yn j -1. In the remainder of this paper, we omit the subscript when there is no ambiguity. Let yn be a minimizer of C(y). We also assume that x, M, and Q are all integers.
4. The two-parameter heuristic policy As above-mentioned, Zhao and Katehakis (2006) find the structure of the optimal policy for the system with a MOQ to be rather
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complex and conclude that such an optimal policy is not practically implementable. The presence of batch ordering makes the problem even more complicated. Therefore, it is necessary to develop some easily implementable polices that have good performance. Based on the analysis of multi-period stochastic inventory system with a MOQ (see Zhou et al., 2007) and the optimal policy structure for batch ordering (see Veinott, 1965), we propose a modified ðs; tÞ policy: given an initial inventory position xn and two integer parameters s and t, where s o t o s þ M, the order quantity qn is 8 > < M þ mQ if xn r s; if s oxn r t; qn ¼ yn xn ¼ M ð2Þ > :0 if xn 4 t: where m Z 1, and m is the unique integer such that 0 o yn ðs þ MÞ r Q . That is, when xn is not larger than s, order up to yn, such that s þM o yn r s þ M þQ ; when xn is larger than s but does not exceed t, order exactly M; and when xn is above t, do not make an order. This policy is an extension of the (s; t) policy proposed in Zhou et al. (2007), where no batch ordering exists. For this reason, we simply call our modified (s; t) policy in the remainder of this paper. To identify the optimal policy parameters s and t that minimize the long-run average cost, we use a discrete time Markov chain with transition matrix P and analyze the system under two cases: Δ ZQ and Δ o Q , where Δ is defined as the difference between t and s, i.e., Δ ¼ t s. 4.1. Case 1: Δ ZQ Under the condition Δ Z Q , Δ has a finite state space ½Q ; M 1. In our (s; t) policy, the inventory position after order placement in period n þ 1 is yn þ 1 ¼ xn þ 1 þqn þ 1 ¼ yn Dn þqn þ 1 , so 8 if yn Dn 4t; > < yn Dn D þ M if s o yn Dn r t; y yn þ 1 ¼ ð3Þ n n > : y D þ M þ mQ if y D rs: n n n n We can see that fyi g is a discrete time Markov chain (DTMC) and has the finite state space ½t þ 1; t þ M. The state space can be split into three segments: 1. ½t þ 1; s þ M: if yn þ 1 A ½t þ 1; s þ M, it means that qn þ 1 ¼ 0, and Dn ¼ yn yn þ 1 . 2. ½s þ M þ 1; s þ M þ Q : if yn þ 1 A ½s þ M þ 1; s þ M þ Q , there are three possibilities: qn þ 1 ¼ 0, qn þ 1 ¼ M, or qn þ 1 ¼ M þ mQ . The three terms correspond to Dn ¼ yn yn þ 1 , Dn ¼ yn yn þ 1 þM, and Dn ¼ yn yn þ 1 þ M þ mQ , respectively. 3. ½s þ M þ Q þ 1; t þ M: if yn þ 1 A ½s þ M þ Q þ 1; t þ M, then qn þ 1 ¼ 0 or qn þ 1 ¼ M, and Dn ¼ yn yn þ 1 , or Dn ¼ yn yn þ 1 þM. It is easy to compute the transition probabilities P i;j ¼ Probðyn þ 1 ¼ jj yn ¼ iÞ, and hence the transition matrix P: 8 pði jÞ þ for j A ½t þ 1; s þ M > > > > > 8 iA ½t þ 1; t þ M; > > > X 1 > > > < pði þ M þ mQ jÞ þ pði jÞ þ for j A ½s þ M þ1; s þ M þ Q ð4Þ P i;j ¼ m ¼ 0 > > > 8 iA ½t þ 1; t þ M; > > > > > for j A ½s þ M þQ þ 1; t þ M pði þ M jÞ þ pði jÞ þ > > > : 8 iA ½t þ 1; t þ M; where pk ¼ ProbðDn ¼ kÞ and pði jÞ þ equals to pi j if iZ j, and zero otherwise. For convenience of notation, m is allowed to take the value of 0 in the expression. Because the Markov chain is irreducible and positive recurrent, ! the unique steady state probabilities π ¼ fπ 1 ; π 2 ; …; π M g exist, where πi denotes the long-run average proportion of time in
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H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
which the inventory position y is t þi. Because the Markov chain is also aperiodic, πi is also the limiting probability that the chain is in state i. Let n-1, we can have 8 M X > > < πi ¼ 1 ð5Þ i¼1 > > :! ! πP¼ π Therefore, we can calculate the stationary probabilities by solving the linear equation (5). Before exploring the properties of the (s; t) policy, we must point out that given the MOQ M, the batch size Q, and the demand distribution, the transition matrix P and sta! tionary probabilities π depend only on Δ, and are independent of s. Note that a given Δ has and only has one corresponding P and ! π . Now we can calculate the long-run average cost for this case: LðΔ; tÞ ¼
M X
π i Cðt þiÞ:
ð6Þ
i¼1
We can derive the following proposition: Proposition 1. For a given Δ Z Q , LðΔ; tÞ is convex in t. Proof. CðÞ is convex in t for each i, so is their sum. Therefore, LðΔ; tÞ is convex in t for a given Δ. □ Let t n1 be the value at which L reaches its minimum for a given
Δ ZQ . Note that t n1 is in fact a function of Δ. For convenience of
notation, we use t n1 to denote the corresponding optimal t for a given Δ Z Q . Proposition 2. Given Δ Z Q , t n1 satisfies yn M r t n1 o yn r t n1 þ M. Proof. Based on the definitions of t n1 and yn, we prove the proposition by contradiction. First, assume t n1 4 yn . Because C(y) is non-decreasing when y4 yn , Cðt n1 Þ 4Cðyn Þ, and hence Cðt n1 þ iÞ 4Cðyn þ iÞ, 8 i A ½1; M. Therefore, we should have P PM n1 n n LðΔ; t n1 Þ ¼ M i ¼ 1 π i Cðt þ iÞ 4 i ¼ 1 π i Cðy þ iÞ ¼ LðΔ; y Þ. This contradicts the definition of t n1 , and hence t n1 r yn . Second, assume P PM n1 n t n1 ¼ yn . Then LðΔ; t n1 Þ ¼ M i ¼ 1 π i Cðt þ iÞ ¼ i ¼ 1 π i Cðy þ iÞ 4 PM PM n1 n1 n i ¼ 1 π i Cðy þ i 1Þ ¼ i ¼ 1 π i Cðt þ i 1Þ ¼ LðΔ; t 1Þ. This also contradicts the definition of t n1 . We thus now have t n1 o yn . Similarly, we can prove that yn M r t n1 again by contradiction. Assume that t n1 o yn M. Because C(y) is convex and yn is the minimizer of C(y), it can be easily seen that Cðt n1 Þ 4 Cðyn MÞ 4 Cðyn Þ. Therefore, Cðt n1 þ iÞ 4 Cðyn M þ iÞ, 8 iA ½1; M, and hence LðΔ; t n1 Þ 4 LðΔ; yn MÞ. This means that t n1 is not optimal, which contradicts the definition of t n1 . Therefore, we have yn r t n1 þ M. □ For a given Δ, the preceding propositions help us narrow the search space of t n1 . Based on the propositions, we design an n efficient algorithm to compute Δ and the corresponding t n1 that minimize the long-run average cost, see Algorithm 1 in the Appendix. All algorithms are relegated to the Appendix. When computing the transition matrix P, we find there are many repeated calculations. To avoid calculating the same probability repeatedly, we also provide a recursive method to get P for this case. For a given Δ, the corresponding transition matrix P can be divided into three parts by column, and each part is a submatrix. The first part is the first ðM ΔÞ columns of P with P i;j ¼ pði jÞ þ in this part. The second part is a submatrix with Q columns consecutive to the first part. In the second part, P P i;j ¼ pði jÞ þ þ 1 m ¼ 0 pði þ M þ mQ jÞ . The third part is the last Δ Q columns with P i;j ¼ pði þ M jÞ þpði jÞ þ . With an abuse of notation, let Δ P denote the transition matrix for a given Δ. If Δ ¼ Q , we directly Δ Δ calculate P by (4). Otherwise, we can calculate P recursively. Δ1 Now, assume we already know P . Algorithm 2 enables us to Δ calculate P recursively.
The motivation for Algorithm 2 is quite simple: P i;j depends only Δ1 on the value of i j. If P Δ are in the same part, then they i;j and P i;j Δ1 Δ1 have an identical expression and hence P Δ . If P Δ i;j ¼ P i;j i;j and P i;j Δ Δ1 are not in the same part, then P i;j and P i;j þ 1 (if exists) must be in the Δ1 same part. With i j being a constant, P Δ i;j ¼ P i þ 1;j þ 1 for i A ½1; M 1, jA ½1; M 1. For the case i¼ M or j¼M, we calculate P Δ i;j separately. 4.2. Case 2: Δ o Q At the beginning of this subsection, we must point out that as we show in the numerical study, Case 2 does arise. Under the condition Δ o Q , Δ has a finite state space ½1; Q 1. Similar to the case Δ Z Q , the inventory position after a possible ordering is still a discrete time Markov chain, and the Markov chain has the finite state space ½t þ 1; s þ M þ Q in this case. The state space can also be split into three segments: 1. yn þ 1 A ½t þ1; s þ M: if yn þ 1 A ½t þ 1; s þ M, it means that qn þ 1 ¼ 0, hence Dn ¼ yn yn þ 1 . 2. yn þ 1 A ½s þM þ 1; t þ M: if yn þ 1 A ½s þ M þ 1; t þ M, it means that qn þ 1 ¼ 0; M; or M þ mQ . The three terms correspond to three possibilities Dn ¼ yn yn þ 1 ; yn yn þ 1 þ M; and yn yn þ 1 þ M þ mQ , respectively. Recall that m is the largest integer such that 0 o yn þ 1 ðs þ MÞ rQ and m Z 1. 3. yn þ 1 A ½t þM þ 1; s þ M þ Q : if yn þ 1 A ½t þ M þ 1; s þM þ Q , it means that qn þ 1 ¼ 0 or M, then Dn ¼ yn yn þ 1 , or Dn ¼ yn yn þ 1 þ M þ mQ . The transition probabilities P i;j ¼ Probðyn þ 1 ¼ jj yn ¼ iÞ can be calculated easily and the transition matrix P is 8 for j A ½t þ 1; s þ M pði jÞ þ > > > > > 8 iA ½t þ 1; s þ M þQ ; > > > 1 > X > > > pði þ M þ mQ jÞ þ pði jÞ þ for j A ½s þ M þ1; t þ M > > < m¼0 ð7Þ P i;j ¼ 8 iA ½t þ 1; s þ M þQ ; > > > > X 1 > > > > pði þ M þ mQ jÞ þ pði jÞ þ for j A ½t þ M þ1; s þ M þ Q > > > > m¼1 > > : 8 iA ½t þ 1; s þ M þQ ; Again, let t-1, the limiting probabilities are the steady state ! probabilities π ¼ fπ 1 ; π 2 …π s þ M þ Q t g. We can calculate the stationary probabilities by solving the linear equations ( sþM þQ t Σi ¼ 1 πi ¼ 1 ð8Þ ! π P¼! π ! The transition matrix P and stationary probabilities π still only depend on Δ, and are independent of s. Now we have the long-run average cost for this case: LðΔ; tÞ ¼
M þQ ðt sÞ X i¼1
π i Cðt þ iÞ ¼
Mþ Q Δ X
π i Cðt þ iÞ:
ð9Þ
i¼1
The following propositions are useful in searching for the optimal parameters. Proposition 3. For a given Δ o Q , LðΔ; tÞ is convex in t. Proof. Given Δ o Q , M þQ Δ is fixed, and LðΔ; tÞ is convex, because it is the summation of several convex functions π i Cðt þ iÞ. □ For a given Δ o Q ; let t n2 be the value of t at which LðΔ; tÞ reaches minimum. Proposition 4. Given Δ oQ , t n2 satisfies yn M Q þ Δ r t n2 o yn r t n2 þ M þ Q Δ.
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
Proof. By
contradiction,
assume
t n2 Z yn ,
Cðt n2 þ i 1Þ; 8 i A ½1; M þ Q Δ, and LðΔ; t n2 Þ ¼
PM þ Q Δ
then
PM þ Q i¼1
Cðt n2 þ iÞ 4
Δπ i Cðt n2 þ iÞ 4
π i Cðt n2 þi 1Þ ¼ LðΔ; t n2 1Þ. This contradicts the definition of t , so t n2 o yn . Now, assume yn M Q þ Δ 4 t n2 , then Cðt n2 þ iÞ 4Cðyn M Q þ Δ þ iÞ; 8 i A ½1; M þQ Δ. Then we have LðΔ; t n2 Þ 4 LðΔ; yn M Q þ ΔÞ, which contradicts the definition of t n2 . Therefore, yn M Q þ Δ r t n2 oyn .□ i¼1
n2
Based on these propositions, we provide Algorithm 3 to n compute Δ and the corresponding t n2 for this case. We also propose a recursive method for calculating P for this case. In this case, P is a ðM Δ þ Q Þ ðM Δ þ Q Þ matrix. Obviously, the size of P will decrease as Δ increases. P can also be divided into three parts in this case. The first part is the first M Δ columns, with P i;j ¼ pði jÞ þ in this part. The second part is a submatrix with Δ columns P consecutive to the first part. In this part, P i;j ¼ pði jÞ þ þ 1 jÞ . The third part is the last Q Δ m ¼ 0 pði þ M þ mQ P columns, with P i;j ¼ pði jÞ þ þ 1 m ¼ 1 pði þ M þ mQ jÞ . Similar to the case of Δ Z Q , we also provide a recursive algorithm (Algorithm Δ 4) to calculate P. Again with an abuse of notation, we write P to denote the corresponding P for a given Δ. n1 n2 Now, we can get two pairs of solutions ðΔ ; t n1 Þ and ðΔ ; t n2 Þ for two cases separately. To get the global optimal solution, we only need to compare the two pairs and find the optimal n n pair of ðΔ ; t n Þ that minimizes LðΔ; tÞ, i.e., ðΔ ; t n Þ ¼ n1 n1 n2 n2 arg minfLðΔ ; t Þ; LðΔ ; t Þg. Algorithm 5 describes the whole (s; t) policy optimization.
5. The one-parameter heuristic policy In this section, we develop a simpler heuristic policy: when the initial inventory position is not less than S, no order is placed; when the initial inventory position is less than S but exceeds S M, order exactly M; otherwise, order M þ mQ , where m is the smallest integer such that xn þM þ mQ 4 S. The motivation for this policy is quite intuitive. Veinott (1965) has shown the optimality of the (R, Q) policy for batch ordering. The key principle of the (R,Q) policy is to order the smallest integral multiple of Q to bring the inventory position above the reorder point R. In our model, there is a MOQ constraint, so the order quantity cannot be less than M. Based on this, the order quantity in period n is qn, such that 8 > < M þ mQ if xn r S M; if S M o xn o S; qn ¼ yn xn ¼ M ð10Þ > :0 if xn Z S: We need to point out that when there is no MOQ constraint, i.e., M ¼0, the modified S policy reduces to the simple (R,Q) policy. Or when Q¼1, the modified S policy reduces to the S policy of Kiesmuller et al. (2011). For this reason, we simply call the modified S policy the S policy. In our model, because M 4 Q , the state space of the inventory level after ordering in the S policy is ½S; S þM 1. The inventory position after ordering is a discrete time Markov chain, and we can calculate the transition probability P i;j ¼ Probðyn þ 1 ¼ jj yn ¼ iÞ and transition matrix P, where
P i;j ¼
8 pði jÞ þ > > > > > > > > 1 > X > > < pði þ M þ mQ jÞ þ pði jÞ þ m¼0 > > > > > > > > pði þ M jÞ þpði jÞ þ > > > :
for j ¼ S;
We can also use the same method as that of the (s; t) policy to ! calculate the stationary probabilities π by (5) and hence the longP run average cost can be calculated by LðSÞ ¼ M i ¼ 1 π i CðS þ i 1Þ. Note that the S policy is in fact a special case of the (s; t) policy, where s ¼ S M and t ¼ S 1. Therefore, Propositions 1 and 2 still hold for the S policy and the policy optimization is simpler: because this is a special case of the (s; t) policy where Δ ¼ t s ¼ M 1, we can calculate the transition matrix and stationary probabilities, and then identify the optimal Sn that minimizes L(S) from the set ½yn M þ 1; yn as we can do for the general case Δ Z Q . The complexity of solving the linear equation (5) is OðM 3 Þ, if a Gaussian elimination is used. Because most of the calculation lies in this part, the total complexity of the S policy is OðM 3 Þ, while the complexity of the (s; t) policy is OðM 4 Þ, because Δ can take M different values.
6. Numerical study In this section, we conduct numerical experiments to test the performance of the (s; t) policy and the S policy. We consider two different discrete demand distributions:
Normal distribution
To ensure nonnegative demand, we truncate the normal distribution at zero to avoid negative demand. Poisson distribution
We conduct numerical studies with respect to the following parameters: the batch size Q, the expected demand per period EðDÞ, the critical ratio p=ðp þ hÞ, and the demand coefficient of variation (we can omit this factor in the case of a Poisson distribution). We assume M¼30 unless otherwise specified. Other parameters are chosen as follows. The holding cost h ¼1 is fixed. Batch size Q varies as 3, 5, 6, 10, and 15. EðDÞ takes the values of 10, 15, 20, 30, and 40; p=ðh þ pÞ varies as 0.80, 0.85, 0.90, and 0.95, and the demand coefficient of variation (c.v.) of the normal distribution takes the values of 0.1, 0.2, 0.3, and 0.4. The complete set of parameter values is given in Table 1. All possible combinations of the parameters give us 400 instances for normally distributed demand, and 100 instances for Poisson distributed demand. We must point that we can get the same results as Zhou et al. (2007) and Kiesmuller et al. (2011), if the same parameter values are selected and Q is fixed to be 1. To better illustrate the performance of the (s; t) policy and the S policy, we compare them with two other policies. The first policy is “the optimal policy” that achieves the minimal average cost among all admissible policies. We use value iteration to compute the optimal long-run average cost and the optimal policy is computed as follows. We initially compute the minimal average cost of a certain number of periods. Then we keep increasing a fixed number of periods, computing the minimal average cost of these periods, and comparing the deviation of the two costs. The iteration does not end until the deviation is insensitive to the increments of periods. We compare the long-run average costs of the (s; t) policy and the S policy to the optimal cost. Denote the average cost of the (s; t) policy by C s;t , and CS for the S policy, and the optimal cost by COPT. For each instance, we use G1 and G3 to
8 i A ½S; S þM 1; for j A ½S þ 1; S þ Q ; Table 1 Base parameter values for the numerical experiments.
8 i A ½S; S þM 1; for j A ½S þ Q þ 1; S þ M 1; 8 i A ½S; S þM 1: ð11Þ
25
h ¼ 1; M ¼ 30; Q A f3; 5; 6; 10; 15g; EðDÞ A f10; 15; 20; 30; 40g; p=ðh þ pÞA f0; 80; 0:85; 0:90; 0:95g; c:v: A f0:1; 0:2; 0:3; 0:4g for normal distribution
26
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
denote the gaps between the costs of these policies, as follows: C s;t C OPT n100% C OPT
G1 ¼
Table 3 Performance of the (s; t) policy (Poisson distribution). Factor
Value
avg G1
max G1
Q
3 5 6 10 15
0.34 0.31 0.29 0.26 0.07
2.60 2.47 2.41 2.09 0.94
EðDÞ
10 15 20 30 40
0.00 0.05 0.01 1.21 0.00
p=ðh þ pÞ
0.8 0.85 0.9 0.95
0.08 0.19 0.31 0.44
and C S C OPT n100% C OPT
G3 ¼
Another alternative is the (s,S) policy with S s ¼ M due to the minimum order quantity constraint. This policy is often used in practice to control inventories when there is a minimum order quantity constraint, see Zhou et al. (2007), Robb and Silver (1998) and Kiesmuller et al. (2011). Due to the batch size constraint, in our (s, S) policy, when the inventory position is less than or equal to s, an order is placed to bring the inventory position above S but no more than S þ Q ; otherwise do not order. If xn denotes the inventory position in period n before ordering, the order quantity qn can be described as follows: ( qn ¼
0; M þm Q;
xn 4 s xn r s:
C s;S C s;t n100% C s;t
and G4 ¼
min G2
max G2
47.64 53.91 56.76 64.46 67.41
17.39 20.44 22.04 28.83 39.35
122.72 134.40 141.50 156.88 150.11
0.00 0.21 0.01 2.60 0.01
27.59 31.55 30.88 75.15 125.00
17.39 22.11 19.66 59.32 71.37
44.98 46.03 49.73 90.01 156.88
0.49 1.18 1.94 2.60
59.82 59.45 58.48 54.39
19.32 19.06 18.32 17.39
156.88 154.80 145.42 133.18
6.1. Performance of the (s; t) policy
where m is the smallest integer subject to xn þ qn 4 S. Note that this (s, S) policy is in fact a special case of our (s; t) policy where Δ ¼ t s ¼ 0 (assume Δ is allowed to be 0). For each instance, we also test the performance of the (s; t) policy and the S policy by comparing them to the best (s,S) policy. Denote the average cost for the best (s,S) policy by C s;S , and use G2 and G4 to denote the gaps between the costs of these policies, as follows: G2 ¼
avg G2
C s;S C S n100% CS
We calculate the average gap, the maximal gap, and the minimal gap, which are denoted by avg Gj , min Gj , and max Gj ðj ¼ 1; 2; 3; 4Þ, respectively. Table 2 Performance of the (s; t) policy (normal distribution). Factor
Value
avg G1
max G1
avg G2
min G2
max G2
Q
3 5 6 10 15
1.43 1.32 1.29 0.91 0.71
25.11 24.32 23.52 20.33 19.03
37.15 41.76 44.68 54.72 59.45
1.11 3.75 7.23 25.65 30.18
150.98 182.92 235.18 266.56 228.50
EðDÞ
10 15 20 30 40
0.10 0.52 0.71 4.29 0.04
2.12 5.63 6.05 25.11 0.20
29.56 34.56 27.82 66.52 79.28
16.50 17.83 1.11 29.27 18.40
49.49 53.79 70.01 150.98 266.56
c.v.
0.1 0.2 0.3 0.4
4.41 0.08 0.01 0.03
25.11 1.12 0.01 0.20
72.31 53.40 36.72 27.77
1.11 18.26 17.95 16.50
266.56 112.41 58.03 42.91
p=ðh þ pÞ
0.8 0.85 0.9 0.95
0.70 0.96 1.28 1.59
13.59 16.87 20.86 25.11
49.23 48.51 46.94 45.52
4.20 2.19 1.11 1.56
228.50 266.56 255.01 235.18
The numerical results of the examples of the (s; t) policy for normally distributed demand are given in Table 2. Table 2 summarizes the gaps G1 between the optimal policy and the (s; t) policy, and the gaps G2 between the (s; t) policy and the best (s,S) policy. In each row, only one parameter is fixed while all others vary. For example, for Q¼6, all the other parameters in this row vary and result in 80 instances of normal demand. It can be seen that the performance of the (s; t) policy is quite close to that of the optimal polices in most cases. For example, at c. v. ¼ 0.2, the maximum G1 is 1.12%. G1 tends to decrease as Q increases. When c.v. takes the value of 0.2, 0.3, or 0.4, the average G1 is less than 0.1%. These results indicate that the (s; t) policy performs better when the c.v. values are relatively large. Table 2 shows that the performance of the (s; t) policy is close to the optimal performance on average for normally distributed demand. However, there are still a few cases in which the (s; t) policy performs much worse than the optimal policy. When the coefficient variation of demand is small, e.g., c.v. ¼ 0.1, the maximum gap G1 between the optimal policy and the (s; t) policy is 25.11% when Q¼ 3, EðDÞ ¼ 30, and p=ðh þ pÞ ¼ 0:95. This is the worst case over all 400 instances. Table 2 also demonstrates that the (s; t) policy performs much better than the best (s,S) policy over all examples. The gap, G2, between the (s; t) policy and the best (s,S) policy can be quite substantial. For example, the maximum G2 is 266.56% at Q¼10, EðDÞ ¼ 40, c.v. ¼ 0.1, and p=ðh þ pÞ ¼ 0:85. The average G2 with respect to c.v. ¼ 0.4 is 27.77%, while the average G2 with respect to the other parameters are all larger than this value. These results indicate that the (s; t) policy always outperforms the best (s,S) policy. These observations are consistent with those of Zhou et al. (2007), who do not consider batch ordering. By conducting numerical examples, Zhou et al. (2007) show that their twoparameter policy always outperforms the best (s,S) policy and the performance of their policy improves for normally distributed demand as c.v. increases. We next present the results of the examples for Poisson distributed demand as shown in Table 3. The gap, G1, between the optimal policy and the (s; t) policy is not larger than 2.60% over all instances. As Q increases, G1 tends to decrease. The results in Table 3 indicate that the (s; t) policy achieves nearly minimal cost for Poisson distributed demand.
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
Table 4 Performance of the S policy (normal distribution).
Table 6 Performance of the two policies for different values of M.
Factor
Value
avg G3
max G3
avg G4
min G4
max G4
Factor
Q
3 5 6 10 15
4.38 3.64 3.34 1.96 1.16
40.93 35.28 33.33 23.67 19.47
32.91 38.36 41.64 53.09 58.70
0.73 3.75 7.23 24.18 29.34
102.28 182.92 235.18 266.56 228.50
Q
10 15 20 30 40
0.48 2.51 1.05 9.88 0.56
5.50 16.08 7.75 40.93 2.21
29.09 31.99 27.40 57.64 78.59
16.49 17.77 0.73 23.91 15.94
49.49 50.28 70.01 102.28 266.56
0.1 0.2 0.3 0.4
8.53 1.15 0.90 1.01
40.93 9.51 6.61 5.19
66.09 51.69 35.45 26.54
0.73 18.21 17.94 15.94
266.56 112.24 55.94 42.91
0.8 0.85 0.9 0.95
3.03 2.97 2.89 2.70
40.93 38.86 36.15 33.82
45.55 45.50 44.67 44.06
4.20 2.19 0.34 0.73
228.50 266.56 255.01 235.18
min G4
max G4
EðDÞ
c.v.
p=ðh þ pÞ
27
Table 5 Performance of the S policy (Poisson distribution). Factor
Value
avg G3
max G3
Q
3 5 6 10 15
1.83 1.48 1.32 0.81 0.27
10.22 8.36 7.45 4.65 1.86
45.19 51.96 55.04 63.53 67.06
17.39 20.44 22.04 28.83 39.35
122.72 134.40 141.50 156.88 150.11
EðDÞ
10 15 20 30 40
0.00 0.05 0.16 5.50 0.00
0.01 0.25 0.38 10.22 0.01
27.59 31.54 30.69 67.96 125.00
17.39 22.07 19.48 57.28 71.36
44.98 46.03 49.72 78.31 156.88
p=ðh þ pÞ
0.8 0.85 0.9 0.95
1.34 1.20 1.08 0.95
10.22 9.21 8.35 7.18
57.62 57.76 57.24 53.61
19.31 19.05 18.32 17.39
156.88 154.80 145.42 133.18
avg G4
Table 3 also demonstrates that the (s; t) policy outperforms the best (s,S) policy for Poisson distributed demand. The maximum gap G2 ranges from 44.98% to 156.88%. As Q increases, G2 tends to increase. Even though G2 tends to decrease as p=ðh þ pÞ increases, the minimum G2 is still as large as 17.39%. These results show that the (s; t) policy performs much better than the best (s,S) policy for Poisson distributed demand.
6.2. Performance of the S policy The numerical results of the examples of the S policy for normally distributed demand are given in Table 4. Table 4 shows that the performance of the S policy is close to that of the optimal policy except for a few cases at c.v. ¼ 0.1. For example, the maximum G3 are 9.51%, 6.61%, and 5.19%, at c.v. ¼ 0.2, 0.3, and 0.4, respectively. The corresponding average G3 are all at the 1% level. G3 tends to decrease, as Q increases. This is identical with our former analysis for the (s; t) policy, recalling that the S policy is a special case. Table 4 indicates that the S policy performs well in the cases with relatively large values of demand coefficient variation. However, in the cases with small values of demand coefficient variation, e.g. at c.v. ¼ 0.1, the S policy may not perform very well. For example, the average G3 is 8.53%, and the maximum G3 is 40.93% at c.v. ¼ 0.1.
Value
avg G1
max G1
avg G3
max G3
3 5 6
0.51 0.90 0.37
9.00 21.77 6.83
0.52 0.91 0.38
15.53 29.22 6.83
EðDÞ
5 10 15 20 25
0.01 0.78 1.13 0.91 0.15
0.33 18.38 20.73 21.77 3.24
0.01 0.78 1.13 0.92 0.16
0.33 19.66 22.73 29.22 6.95
c.v.
0.1 0.2 0.3 0.4
2.29 0.08 0.01 0.01
21.77 1.08 0.03 0.02
2.31 0.09 0.02 0.02
29.22 6.95 5.37 4.33
p
0.8 0.85 0.9 0.95
0.45 0.54 0.67 0.72
11.91 14.24 18.79 21.77
0.47 0.56 0.68 0.73
29.22 26.93 26.34 25.01
M
10 15 20 12 18 24
0.73 0.83 1.15 0.24 0.40 0.69
18.38 20.73 21.77 7.01 8.31 9.00
0.73 0.83 1.16 0.24 0.40 0.70
19.66 22.73 29.22 7.01 15.53 9.04
Table 4 also demonstrates that the S policy outperforms the best (s,S) policy for normally distributed demand in most instances. Over all 400 instances, there is only one exception, where G4 ¼ 0.73% at Q¼3, EðDÞ ¼ 20, c.v. ¼ 0.1, and p=ðh þ pÞ ¼ 0:95. In other instances, G4 can be substantial. For example, the average G4 with respect to Q varies between 32.91% and 58.70%. As Q increases, G4 tends to increase. Recalling that G3 tends to decrease as Q increases, this may lead to the conclusion that the S policy performs better when Q is relatively large. We next test the performance of the S policy for Poisson distributed demand as shown in Table 5. We find that the average G3 is not larger than 5.50% over all instances. The maximum G3 is 10.22%, which is attained at Q¼3, EðDÞ ¼ 30, and p=ðhþ pÞ ¼ 0:8. Table 5 also shows that the S policy performs better than the best (s,S) policy for Poisson distributed demand in all instances. The gap, G4, between the S policy and the best feasible (s,S) policy is substantial. Over all 400 instances, G4 varies between 17.39% and 156.88%. These observations are consistent with Kiesmuller et al. (2011), recalling that when Q¼1, our S policy reduces to the oneparameter policy of Kiesmuller et al. (2011), who also show, by the way of numerical study, the superiority of the one-parameter policy to the (s,S) policy. The above-mentioned numerical examples are all taken when M¼30. We next present the results of the (s; t) policy and the S policy for different values of M. In the remainder of this subsection, Q varies as 3, 5, and 6 and M can take different values. Because M is an integral multiple of Q, we let M take the values of 10, 15, and 20 when Q¼ 5, otherwise M varies as 12, 18, and 24. EðDÞ varies as 5, 10, 15, 20, and 25. Other parameters take the same values as shown in Table 1, resulting in a total of 720 instances for normally distributed demand. Table 6 shows the performance of the (s; t) policy and the S policy with different values of M. From Table 6, we can draw several similar conclusions to the case with M¼30. First, both the (s; t) policy and the S policy have good performance close the optimal policy in most cases. For example, the average G1 and G3 with respect to different M are all at the 1% level. Second, the (s; t) policy and the S policy perform much worse than the optimal policy when the demand variation is
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H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
14
Table 7 Comparison of the (s; t) policy and the S policy at Q ¼5. p=ðh þ pÞ
EðDÞ ¼10
EðDÞ¼ 15
EðDÞ¼ 20
EðDÞ¼ 30
EðDÞ¼ 40
Normal c.v.¼ 0.1
0.80 0.85 0.90 0.95
2.07 2.73 2.88 1.36
10.82 10.76 10.25 7.84
0.00 0.00 0.00 0.36
20.42 15.35 10.05 4.66
0.00 0.00 0.00 0.00
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
1.25 1.53 1.60 1.64
0.13 0.15 0.09 0.02
7.75 6.34 4.87 3.40
0.09 0.07 0.05 0.03
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
0.02 0.02 0.01 0.01
0.69 0.53 0.49 0.31
5.35 4.64 4.00 3.05
1.14 1.06 0.90 0.68
Normal c.v.¼ 0.4
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
0.16 0.12 0.10 0.04
1.37 1.19 0.88 0.64
4.33 3.86 3.36 2.70
1.86 1.67 1.48 1.22
Poisson
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
0.01 0.01 0.01 0.00
0.26 0.29 0.21 0.10
7.86 6.23 5.01 3.22
0.00 0.00 0.00 0.00
Normal c.v.¼ 0.2
Normal c.v.¼ 0.3
Normalized Average Cost
G5
(s,k) c.v.=0.1 (s,k) c.v.=0.2 (s,k) c.v.=0.3 (s,k) c.v.=0.4
12 10 8
6 4 2
0 5
10
15
20
25
30
40
Fig. 1. The impact of EðDÞ in the (s; t) policy with p=ðh þ pÞ ¼ 0:9 and Q ¼ 5.
25 S S S S
Normalized Average Cost
20
small, i.e. c.v. ¼ 0.1. The average G1 and G3 are 2.29% and 2.31%, respectively, at c.v. ¼ 0.1, while they are all less than the 0.1% at other c.v. levels. The maximum G1 and G3 are 21.77% and 29.22%, which are also much larger than the values at higher c.v. levels. We should also note that for a fixed Q, G1 and G3 tend to increase as M increases. For example, when Q¼ 5, the average G1 increases from 0.73% to 1.15% as M increases from 10 to 20.
35
E(D)
c.v.=0.1 c.v.=0.2 c.v.=0.3 c.v.=0.4
15
10
5
6.3. Sensitivity analysis In this section, we first compare the performances of the (s; t) policy and the S policy. In fact, in some instances, they have the same performance, because the S policy is a special case of the (s; t) policy. However, we are still interested in the differences between the policies over all instances. We use G5 to denote the gap between the (s; t) policy and the S policy as follows: G5 ¼
C S C s;t n100% C s;t
To better illustrate the differences, we take all the instances at Q¼5 as an example, and enumerate the results in Table 7. It can be seen that the (s; t) policy does not perform significantly better than the S policy except for a few cases with small values of demand coefficient variation. For example, when c.v.¼ 0.1, the maximum G5 is 20.42% at EðDÞ ¼ 30. However, when c.v. is relatively large, the S policy performs nearly as well as the (s; t) policy. The maximum G5 is less than 8%, and in many instances these two policies have the same performance. These results indicate that the S policy is a recommendable substitute for the (s; t) policy at a high c.v. level. This is also consistent with the finding of Kiesmuller et al. (2011) that the performance of the one-parameter policy is close to the performance of the two-parameter policy when there is no batch ordering constraint. We next study the effect of the EðDÞ. To better illustrate the effect of EðDÞ, we plot in Fig. 1 the long-run average costs (normalized by the global minimum of C(y)) of the (s; t) policy as a function of EðDÞ under different levels of c.v. with a fixed M¼30. Fig. 1 illustrates that for a given M,
when EðDÞ is relatively small, e.g., less than M, a lower level of demand variation leads to a higher decreasing rate of average costs as EðDÞ increases.
0
5
10
15
20
25
30
35
40
E(D)
Fig. 2. The impact of EðDÞ in the S policy with p=ðh þ pÞ ¼ 0:8 and Q¼ 3.
when EðDÞ is relatively large, e.g., larger than M, the average cost is not affected significantly by the different demand variation levels as EðDÞ increases.
The reason for this is that as EðDÞ increases, the constraint of the minimum order quantity becomes looser and looser. Similar results can be drawn for the S policy. We plot the results for the S policy in Fig. 2. Before ending this section, we must point out that the case of Δ o Q can appear, although Δ ZQ in most of our numerical examples. For example, for normally distributed demand with EðDÞ ¼ 100 and c:v: ¼ 1, we can get the optimal Δ ¼1 and the optimal t ¼183, when M¼8, Q¼4, and p=ðh þ pÞ ¼ 0:8.
7. Conclusion In this paper, we design an algorithm for a heuristic twoparameter policy, the (s; t) policy, to control stochastic inventories with minimum order quantity and batch size constraints. Applying a Markov chain approach, we compute the system costs and provide recursive algorithms to optimize the policy under the long-run average cost criterion. We also develop the computational procedure for a simpler policy, the S policy, which is
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
motivated by the (R,Q) policy in batch ordering and is a special case of the (s; t) policy. Numerical studies are conducted to demonstrate the effectiveness of these two policies with respect to the optimal policy and the (s,S) policy for both normally and Poisson distributed demand. Overall, the S policy has a good performance close to that of the (s; t) policy; only in a few cases with small demand variation, the latter outperforms the former significantly. Acknowledgments The authors acknowledge the financial support from Hong Kong Research Grants Council Project no. 9041952. Appendix Algorithm 1. Policy optimization for Δ Z Q . 1: Set Ln1 ’Inf; 2: for Δ ¼ Q to M 1 do 3: calculate P by (4); ! 4: calculate π by (5); n 5: for t ¼ y M to yn 1 do 6: calculate LðΔ; tÞ by (6); 7:
if LðΔ; tÞ o Ln1 then
8: Ln1 ’LðΔ; tÞ, t n1 ’t, Δ ’Δ; 9: end if 10: end for 11: end for n1
Algorithm 2. Calculating P for Δ ZQ . 1: if Δ ¼ Q then 2: calculate P by (4) 3: else 4: for i¼1 to M do 5: for j¼ 1 to M Δ do Δ1 6: PΔ ; i;j ’P i;j 7: end for 8: end for 9: j’M Δ þ 1; 10: for i ¼1 to M 1 do
11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29:
Δ
Δ1
P i;j ’P i þ 1;j þ 1 ; end for i’M; Δ 1 P1 PΔ þ m ¼ 0 pði þ M þ mQ jÞ ; i;j ’P i;j for i ¼1 to M do for j ¼ M Δ þ 2 to M Δ þ Q do
Δ1 PΔ ; i;j ’P i;j end for end for if Δ ¼ Q þ 1 then for i¼1to M 1 do
PΔ i;j ’ProbðD ¼ iÞ; end for i’M; PΔ i;j ’ProbðD ¼ MÞ þ ProbðD ¼ 0Þ; else j’M Δ þ Q þ 1; for i¼1 to M 1 do Δ1 PΔ i;j ’P i þ 1;j þ 1 ;
29
30: 31:
end for i’M;
32: 33: 34:
PΔ i;j ’ProbðD ¼ Δ Q 1Þ þ ProbðD ¼ M þ Δ Q 1Þ; for i¼ 1 to M do for j ¼ M Δ þ Q þ 2 to M do
35: 36: 37: 38: 39:
Δ1 PΔ ; i;j ’P i;j end for end for end if end if
Algorithm 3. Policy optimization for Δ oQ . 1: Set Ln2 ’Inf ; 2: for Δ ¼ 1 to Q 1 do 3: calculate P by (7); ! 4: calculate π by (8); n 5: for t ¼ y M Q þ Δ to yn 1 do 6: calculate LðΔ; tÞ by (9); 7:
if LðΔ; tÞ o Ln2 then
8: Ln2 ’LðΔ; tÞ, t n2 ’t, Δ ’Δ; 9: end if 10: end for 11: end for n2
Algorithm 4. Calculating P for Δ o Q . 1: if Q¼1 then 2: calculate P by (7) 3: else 4: for i¼1 to M Δ þ Q do 5: for t j ¼1 to M Δ þ Q do 6: if j ¼ M Δ þ 1 then 7: 8:
Δ1 PΔ i;j ’P i þ 1;j þ 1 ; else
Δ1 9: PΔ ; i;j ’P i;j 10: end if 11: end for 12: end for 13: end if
Algorithm 5. Policy optimization for the (s; t) policy. 1: Set Ln1 ’ Inf, Ln2 ’Inf; 2: for Δ ¼ Q to M 1 do 3: calculate P by Algorithm 2; ! 4: calculate π by (5); n 5: for t ¼ y M to yn 1 do 6: calculate LðΔ; tÞ by (6); 7:
if LðΔ; tÞ o Ln1 then
8: Ln1 ’LðΔ; tÞ, t n1 ’t, Δ ’Δ; 9: end if 10: end for 11: end for 12: for Δ ¼ 1 to Q 1 do 13: calculate P by Algorithm 3; ! 14: calculate π by (8); n 15: for t ¼ y M Q þ Δ to yn 1 do 16: calculate LðΔ; tÞ by (9); n1
17:
if LðΔ; tÞ o Ln2 then
18: 19:
Ln2 ’LðΔ; tÞ, t n2 ’t, Δ ’Δ; end if n2
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H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
20: end for 21: end for 22: if Ln1 4 Ln2 then 23: Ln ’Ln2 , t n ’t n2 , Δ ’Δ ; 24: else n
25:
n2
Ln ’Ln1 , t n ’t n1 , Δ ’Δ ; n
n1
26: end if References Axsäter, S., Zhang, W., 1999. A joint replenishment policy for multi-echelon inventory control. Int. J. Prod. Econ. 59, 243–250. Bradford, P.G., Katehakis, M.N., 2006. Constrained inventory allocation. In: 2006 Proceedings of 10th WSEAS International Conference on Applied Mathematics, pp. 504–507. Bradford, P.G., Katehakis, M.N., 2007. Constrained inventory allocation and its applications. WSEAS Trans. Math. 6 (2), 263–270. Chao, X., Zhou, S.X., 2009. Optimal policy for a multiechelon inventory system with batch ordering and fixed replenishment intervals. Oper. Res. 57 (2), 377–390. Chen, F., 2000. Optimal policies for multi-echelon inventory problems with batch ordering. Oper. Res. 48 (3), 376–389. Gallego, G., 1998. New bounds and heuristics for (q,r) policies. Manag. Sci. 44, 219–233. Heyman, D., Sobel, M. 1984. Stochastic Models in Operations Research, vol. 2. McGraw Hill, New York.
Huh, W.T., Janakiraman, G., 2012. Technical note on optimal policies for inventory systems with batch ordering. Oper. Res. 60, 797–802. Kiesmuller, G., de Kok, A., Daba, S., 2011. Single item inventory control under periodic review and a minimum order quantity. Int. J. Prod. Econ. 133, 280–285. Hellion, B., Mangione, F., Penz, B., 2012. A polynomial time algorithm to solve the single-item capacitated lot sizing problem with minimum order quantities and concave costs. Eur. J. Oper. Res. 222 (1), 10–16. Okhrin, I., Richter, K., 2011. An O(T3) algorithm for the capacitated lot sizing problem with minimum order quantities. Eur. J. Oper. Res. 211 (3), 507–514. Porras, E., Dekker, R., 2006. An efficient optimal solution method for the joint replenishment problem with minimum order quantities. Eur. J. Oper. Res. 174 (3), 1595–1615. Robb, D.J., Silver, E.A., 1998. Inventory management with periodic ordering and minimum order quantities. J. Oper. Res. Soc. 1998, 1085–1094. Shang, K.H., Zhou, S.X., 2010. Optimal and heuristic echelon ðr; nq; tÞ policies in serial inventory systems with fixed costs. Oper. Res. 58, 414–427. Veinott, A., 1965. The optimal inventory policy for batch ordering. Oper. Res. 13, 424–432. Zhao, Y., Katehakis, M., 2006. On the structure of optimal ordering policies for stochastic inventory systems with minimum order quantity. Prob. Eng. Inf. Sci. 20, 257. Zheng, Y.-S., Federgruen, A., 1991. Finding optimal (s,S) policies is about as simple as evaluating a single policy. Oper. Res. 39, 654–665. Zhou, B., Zhao, Y., Katehakis, M., 2007. Effective control policies for stochastic inventory systems with a minimum order quantity and linear costs. Int. J. Prod. Econ. 106, 523–531. Zhou, B., Katehakis, M., Zhao, Y., 2009. Managing stochastic inventory systems with free shipping option. Eur. J. Oper. Res. 196 (1), 186–197.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Effective inventory control policies with a minimum order quantity and batch ordering Han Zhu n, Xing Liu, Youhua (Frank) Chen Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong
art ic l e i nf o
a b s t r a c t
Article history: Received 21 March 2014 Accepted 5 June 2015 Available online 18 June 2015
In this paper, we consider a single-item periodic-review stochastic inventory system with both minimum order quantity (MOQ) and batch ordering requirements. In each time period, the firm can order either none or at least as much as the MOQ. At the same time, if an order is placed, the order quantity is required to be an integral multiple of a given specific batch size. We first adopt a heuristic policy which is specified by two parameters (s; t). Applying a discrete time Markov chain approach, we compute the system cost and optimize this (s; t) policy under the long-run average cost criterion. We also consider a simpler one-parameter policy, the so-called S policy, which is a special case of the (s; t) policy. In an intensive numerical study, we find that (1) both policies perform well in comparison with other policies and (2) the S policy also performs well and is compatible with the (s; t) policy; only in a few cases where demand variation is small, the latter outperforms the former significantly. We also evaluate the effects of some important parameters on system performance. & 2015 Elsevier B.V. All rights reserved.
Keywords: Minimum order quantity Batch ordering Stochastic inventory systems Markov chain Heuristic policies
1. Introduction In industries, minimum order quantity (MOQ) and batch ordering, applied independently or simultaneously, are two common requirements made by suppliers, both of which can help companies take advantage of economies of scale and hence reduce costs. The MOQ requirement means that the order quantity must equal or exceed a specified level, if an order is placed. The batch ordering requirement means that the order quantity must be an integral multiple of a specified given batch size. The application of a MOQ is common in practice. With the prevalence of e-commerce, MOQs are becoming more and more common in our lives, especially in online business-to-business sourcing portals such as alibaba.com, where suppliers often set such requirements. MOQs are also applied in manufacturing industries for products that have short lifetimes or long leadtimes. A well-known example is Sport Obermeyer, a fashion sport skiwear manufacturer, which has a minimum production level of 600 garments in Hong Kong and 1200 garments in China per order (Zhao and Katehakis, 2006). In fact, MOQ requirements are quite common in China and other low cost manufacturing countries. Low profit margins force manufacturers to pursue large production quantities to break even. On the other hand, batch ordering is
n
Corresponding author. Tel.: þ 852 54826926. E-mail addresses: [email protected] (H. Zhu), [email protected] (X. Liu), [email protected] (Y. Chen). http://dx.doi.org/10.1016/j.ijpe.2015.06.008 0925-5273/& 2015 Elsevier B.V. All rights reserved.
also a ubiquitous requirement in industries, because materials often flow in fixed batch sizes in supply chains. For example, raw materials usually arrive at factories in full truckloads, work-inprocess is often processed in convenient lot sizes between production stages, and finished goods may be transported in full containers from suppliers to warehouses or distribution centers. Therefore, it is of no surprise that suppliers who apply a MOQ may also require batch ordering. Indeed, our decision to jointly consider both MOQ and batch ordering requirements in this paper is largely motivated by our experience with a wholesale company in Hong Kong. For a variety of products, the firm first replenishes its stock from suppliers and then sells to retail customers, and for most of these products, the firm stipulates both MOQ and batch ordering requirements. The coexistence of a MOQ and batch ordering has a two-sided effect. On one hand, requiring a MOQ and batch ordering simultaneously helps suppliers reduce the risk of uncertainty and achieve economies of scale. On the other hand, the requirements may have a negative effect on buyers’ inventory control, especially when MOQs are relatively large compared with their demand, which is not unusual in practice. Managers in such situations need principles or tools to help control their inventory. However, to the best of our knowledge, no research has investigated inventory systems with both MOQ and batch ordering requirements. Thus, the primary goal of this paper is to fill this gap in the literature. In this paper, we consider a single product stochastic periodic-review inventory system with both MOQ and batch ordering requirements. The selling firm can make a decision at the beginning of
22
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
each time period after reviewing the inventory position. When the firm decides to place an order, the order quantity must satisfy both the MOQ and the batch ordering constraints, where we assume the MOQ is an integral multiple of the batch size. The leftover inventory is carried to the next period and incurs a holding cost, whereas unsatisfied demand is fully backlogged and incurs a backordering cost. The total costs consist of the linear ordering cost, the holding cost, and the backordering cost. The objective is to minimize the long-run average cost of the system. The optimal policy for the system with only the MOQ requirement, which is partially characterized by Zhao and Katehakis (2006), is rather complicated, even without batch ordering. Therefore, for inventory systems with both MOQ and batch ordering requirements, it is necessary to propose some effective heuristic policies, which is the major contribution of our work. Facing the MOQ requirement, many companies apply the (s,S) type policy to control inventories in practice (Zhou et al., 2007). Based on this, we first propose a two-parameter policy with a similar structure, i.e., the (s; t) policy, where s o t o s þ M and M represents the MOQ. The (s; t) policy operates as follows: at the beginning of each period, if the inventory position is less than or equal to s, order a quantity that is just sufficient to bring the inventory position to s þ M or above (the inventory position after ordering can be larger than s þ M, because the order quantity must also satisfy the batch ordering requirement); if the inventory position exceeds s but is no more than t, order exactly M; otherwise, order nothing. We identify the bounds for the optimal t, and propose algorithms to find the optimal values of t and s. We also examine a simpler and more easy-to-use policy, i.e., the S policy, which is a special case of the (s; t) policy. The S policy operates in the same way as the (s; t) policy with s ¼ S M and t ¼ S 1. The numerical study shows that both these polices have close-to-optimal performance in most cases and that there is an overwhelming preponderance to the best (s,S) policy over all examples. The remainder of this paper is organized as follows. The literature on MOQ and batch ordering is discussed in Section 2. In Section 3, the model description and notations are presented. In Section 4, we propose a two-parameter ðs; tÞ policy and present algorithms to optimize the policy. A simpler one-parameter policy is introduced in Section 5. Numerical examples are conducted in Section 6 to measure the effectiveness of these two policies by comparing them with other policies. Finally, Section 7 concludes the paper by summarizing the findings.
2. Literature review The existing research on stochastic inventory systems is quite extensive. Here, we mention only a few of the most relevant papers. Many papers focus on problems associated with batch ordering or MOQ separately. The literature related to our paper can be divided into two areas: (1) supply chain inventory management with batch ordering and (2) supply chain inventory management with MOQ. In the area of batch ordering, Veinott (1965) shows the optimality of the (R,Q) policy for a periodic-review inventory system with batch ordering and no fixed ordering cost. This (R, Q) policy operates as follows: at the beginning of each period, if the inventory position is less than the reorder point R, order the smallest integral multiple of the batch size Q that will bring the inventory position to at least R; otherwise order nothing. Chen (2000) generalizes Veinott's result to multi-echelon systems’ settings and demonstrates the optimality of (R,nQ) policies for multi-stage serial and assembly systems where materials flow in fixed batches and the stochastic demands are stationary over time. Chao and Zhou (2009) find the optimal inventory control policy for
a multi-echelon serial system with batch ordering and fixed replenishment intervals. They derive a distribution-function solution for its optimal control parameters and design an efficient algorithm for computing those parameters. Huh and Janakiraman (2012) extend the work of Veinott (1965) and Chen (2000) by demonstrating the optimality of echelon (R,nQ) policies for multiechelon serial systems with nested batch ordering and nonstationary demands. Although it is not optimal in some complex inventory systems with batch ordering, the reorder point, lot-size ordering policy is easy to implement. For this reason, numerous heuristic policies have been proposed, see for example Gallego (1998), Axsäter and Zhang (1999), Shang and Zhou (2010). In the area of MOQ, Zhao and Katehakis (2006) introduce the concept of M-increasing function and first partially characterize the optimal policy for multiperiod inventory systems with MOQ. For the uncharacterized part, the authors give easily computable upper bounds and asymptotic lower bounds for these intervals. However, for the characterized part, the optimal policy is complexly structured and difficult to implement in practice. Zhou et al. (2009) study a periodic-review inventory system where an additional fixed shipping cost is imposed whenever the order quantity is less than a specified free shipping quantity. The authors characterize the structure properties of the optimal inventory control policy for the single-period model and propose a heuristic policy for multi-period inventory systems. Bradford and Katehakis (2007) study a system where a single supplier has contractual obligations to provide a minimum amount and a maximum amount to all retailers. The authors show that all retailers can be partitioned into three disjoints subsets and provide the optimal allocation for each subset. For other references on MOQ, the reader is referred to Porras and Dekker (2006), Bradford and Katehakis (2006), Okhrin and Richter (2011), Mangione and Penz (2012). The most closely related papers to our work are Zhou et al. (2007) and Kiesmuller et al. (2011). Zhou et al. (2007) propose a two-parameter heuristic policy for a stochastic inventory system with MOQ requirement and demonstrate that the performance of this policy is close to the optimal policy except for a few cases when the coefficient of the demand distribution is very small. Kiesmuller et al. (2011) propose a simpler policy, which has only one parameter S. This policy works as follows: no order is placed when the inventory position is not less than the level S; otherwise an order is placed to raise the inventory to S. However, if this order is smaller than the MOQ, the order quantity is increased to the MOQ. The authors show the effectiveness of this policy and develop simple newsvendor inequalities for near-optimal policy parameters. However, both Zhou et al. (2007) and Kiesmuller et al. (2011) do not consider batch ordering. To the best of our knowledge, our paper is the first to study stochastic inventory system with both MOQ and batch ordering requirements. To combine the two requirements, we need to tackle the problem of selecting an order quantity that satisfies both the constraints simultaneously. In a system with only the MOQ constraint, the order quantity can be any integer that is larger than or equal to the MOQ. However, with the addition of batch ordering, the firm has to either round up or round down the order quantity to an integral multiple of the given batch size. Therefore, in our model, the order quantities are subject to two kinds of jumps, which makes the analysis much more difficult.
3. Model description We consider a periodic-review inventory system for a single item with stochastic demand. The demand D in each period is an independent identically distributed (i.i.d.) random variable. The retailer replenishes its stock from a supplier. In our model, when
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
an order is made, we assume that the supplier has two requirements regarding to the order quantity:
Minimum Order Quantity (MOQ)
The order quantity must be at least M, the given minimum order quantity. Batch Ordering The order quantity must be an integral multiple of Q, the given batch size.
We also assume that M is an integral multiple of the Q and M 4 Q . If M r Q , the problem is reduced to that of Veinott (1965), the optimal policy of which can easily be computed. Obviously, M is an eligible option for the order quantity. When an order is made, the order quantity must be in the form M þ iQ , where i is a nonnegative integer. The sequence of events is as follows. At the beginning of each period, after reviewing the initial inventory position, the retailer decides whether to make a placement. After the order placement, the demand is realized. The demand is satisfied according to the first-come-first-service rule and unsatisfied demand will be backlogged. At the end of each period, the inventory cost is evaluated. Excess inventory will generate an inventory holding cost h per unit per period and a penalty cost p per unit per period will be incurred for the unsatisfied demand units. In our model, we consider linear variable cost and no fixed ordering cost. We assume an infinite planning horizon and stationary setting. We do not take into consideration the fixed ordering cost, because it usually plays the same role as MOQ in helping firms achieve the economies of scale. In practice, MOQ can replace the fixed ordering cost in most cases. The average cost criterion is used to evaluate the inventory system and the objective is to minimize the long-run average cost of the system. We set the purchasing cost equal to zero because the linear ordering costs can be ignored under the average cost criterion, see e.g., Zheng and Federgruen (1991). Without loss of generality, we set the lead-time equal to zero. Our model can be easily extended to systems with positive lead times using the standard method in Heyman and Sobel (1984). In the remainder of this paper, the following notations will be used M Q Dn qn h p xn yn Zþ ½a; b
minimum order quantity batch size demand in period n order quantity in period n holding cost per unit per period penalty cost per unit per period the inventory position before ordering in period n the inventory position after ordering in period n maxð0; ZÞ the integer numbers between a and b (if a and b are integers, they are included) The expected cost function C n ðyn Þ in period n can be written as
C n ðyn Þ ¼ hE½ðyn Dn Þ þ þ pE½ðDn yn Þ þ ;
ð1Þ where yn and Dn are both integers. We can easily find that C n yn is convex and C n yn - þ 1 as j yn j -1. In the remainder of this paper, we omit the subscript when there is no ambiguity. Let yn be a minimizer of C(y). We also assume that x, M, and Q are all integers.
4. The two-parameter heuristic policy As above-mentioned, Zhao and Katehakis (2006) find the structure of the optimal policy for the system with a MOQ to be rather
23
complex and conclude that such an optimal policy is not practically implementable. The presence of batch ordering makes the problem even more complicated. Therefore, it is necessary to develop some easily implementable polices that have good performance. Based on the analysis of multi-period stochastic inventory system with a MOQ (see Zhou et al., 2007) and the optimal policy structure for batch ordering (see Veinott, 1965), we propose a modified ðs; tÞ policy: given an initial inventory position xn and two integer parameters s and t, where s o t o s þ M, the order quantity qn is 8 > < M þ mQ if xn r s; if s oxn r t; qn ¼ yn xn ¼ M ð2Þ > :0 if xn 4 t: where m Z 1, and m is the unique integer such that 0 o yn ðs þ MÞ r Q . That is, when xn is not larger than s, order up to yn, such that s þM o yn r s þ M þQ ; when xn is larger than s but does not exceed t, order exactly M; and when xn is above t, do not make an order. This policy is an extension of the (s; t) policy proposed in Zhou et al. (2007), where no batch ordering exists. For this reason, we simply call our modified (s; t) policy in the remainder of this paper. To identify the optimal policy parameters s and t that minimize the long-run average cost, we use a discrete time Markov chain with transition matrix P and analyze the system under two cases: Δ ZQ and Δ o Q , where Δ is defined as the difference between t and s, i.e., Δ ¼ t s. 4.1. Case 1: Δ ZQ Under the condition Δ Z Q , Δ has a finite state space ½Q ; M 1. In our (s; t) policy, the inventory position after order placement in period n þ 1 is yn þ 1 ¼ xn þ 1 þqn þ 1 ¼ yn Dn þqn þ 1 , so 8 if yn Dn 4t; > < yn Dn D þ M if s o yn Dn r t; y yn þ 1 ¼ ð3Þ n n > : y D þ M þ mQ if y D rs: n n n n We can see that fyi g is a discrete time Markov chain (DTMC) and has the finite state space ½t þ 1; t þ M. The state space can be split into three segments: 1. ½t þ 1; s þ M: if yn þ 1 A ½t þ 1; s þ M, it means that qn þ 1 ¼ 0, and Dn ¼ yn yn þ 1 . 2. ½s þ M þ 1; s þ M þ Q : if yn þ 1 A ½s þ M þ 1; s þ M þ Q , there are three possibilities: qn þ 1 ¼ 0, qn þ 1 ¼ M, or qn þ 1 ¼ M þ mQ . The three terms correspond to Dn ¼ yn yn þ 1 , Dn ¼ yn yn þ 1 þM, and Dn ¼ yn yn þ 1 þ M þ mQ , respectively. 3. ½s þ M þ Q þ 1; t þ M: if yn þ 1 A ½s þ M þ Q þ 1; t þ M, then qn þ 1 ¼ 0 or qn þ 1 ¼ M, and Dn ¼ yn yn þ 1 , or Dn ¼ yn yn þ 1 þM. It is easy to compute the transition probabilities P i;j ¼ Probðyn þ 1 ¼ jj yn ¼ iÞ, and hence the transition matrix P: 8 pði jÞ þ for j A ½t þ 1; s þ M > > > > > 8 iA ½t þ 1; t þ M; > > > X 1 > > > < pði þ M þ mQ jÞ þ pði jÞ þ for j A ½s þ M þ1; s þ M þ Q ð4Þ P i;j ¼ m ¼ 0 > > > 8 iA ½t þ 1; t þ M; > > > > > for j A ½s þ M þQ þ 1; t þ M pði þ M jÞ þ pði jÞ þ > > > : 8 iA ½t þ 1; t þ M; where pk ¼ ProbðDn ¼ kÞ and pði jÞ þ equals to pi j if iZ j, and zero otherwise. For convenience of notation, m is allowed to take the value of 0 in the expression. Because the Markov chain is irreducible and positive recurrent, ! the unique steady state probabilities π ¼ fπ 1 ; π 2 ; …; π M g exist, where πi denotes the long-run average proportion of time in
24
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
which the inventory position y is t þi. Because the Markov chain is also aperiodic, πi is also the limiting probability that the chain is in state i. Let n-1, we can have 8 M X > > < πi ¼ 1 ð5Þ i¼1 > > :! ! πP¼ π Therefore, we can calculate the stationary probabilities by solving the linear equation (5). Before exploring the properties of the (s; t) policy, we must point out that given the MOQ M, the batch size Q, and the demand distribution, the transition matrix P and sta! tionary probabilities π depend only on Δ, and are independent of s. Note that a given Δ has and only has one corresponding P and ! π . Now we can calculate the long-run average cost for this case: LðΔ; tÞ ¼
M X
π i Cðt þiÞ:
ð6Þ
i¼1
We can derive the following proposition: Proposition 1. For a given Δ Z Q , LðΔ; tÞ is convex in t. Proof. CðÞ is convex in t for each i, so is their sum. Therefore, LðΔ; tÞ is convex in t for a given Δ. □ Let t n1 be the value at which L reaches its minimum for a given
Δ ZQ . Note that t n1 is in fact a function of Δ. For convenience of
notation, we use t n1 to denote the corresponding optimal t for a given Δ Z Q . Proposition 2. Given Δ Z Q , t n1 satisfies yn M r t n1 o yn r t n1 þ M. Proof. Based on the definitions of t n1 and yn, we prove the proposition by contradiction. First, assume t n1 4 yn . Because C(y) is non-decreasing when y4 yn , Cðt n1 Þ 4Cðyn Þ, and hence Cðt n1 þ iÞ 4Cðyn þ iÞ, 8 i A ½1; M. Therefore, we should have P PM n1 n n LðΔ; t n1 Þ ¼ M i ¼ 1 π i Cðt þ iÞ 4 i ¼ 1 π i Cðy þ iÞ ¼ LðΔ; y Þ. This contradicts the definition of t n1 , and hence t n1 r yn . Second, assume P PM n1 n t n1 ¼ yn . Then LðΔ; t n1 Þ ¼ M i ¼ 1 π i Cðt þ iÞ ¼ i ¼ 1 π i Cðy þ iÞ 4 PM PM n1 n1 n i ¼ 1 π i Cðy þ i 1Þ ¼ i ¼ 1 π i Cðt þ i 1Þ ¼ LðΔ; t 1Þ. This also contradicts the definition of t n1 . We thus now have t n1 o yn . Similarly, we can prove that yn M r t n1 again by contradiction. Assume that t n1 o yn M. Because C(y) is convex and yn is the minimizer of C(y), it can be easily seen that Cðt n1 Þ 4 Cðyn MÞ 4 Cðyn Þ. Therefore, Cðt n1 þ iÞ 4 Cðyn M þ iÞ, 8 iA ½1; M, and hence LðΔ; t n1 Þ 4 LðΔ; yn MÞ. This means that t n1 is not optimal, which contradicts the definition of t n1 . Therefore, we have yn r t n1 þ M. □ For a given Δ, the preceding propositions help us narrow the search space of t n1 . Based on the propositions, we design an n efficient algorithm to compute Δ and the corresponding t n1 that minimize the long-run average cost, see Algorithm 1 in the Appendix. All algorithms are relegated to the Appendix. When computing the transition matrix P, we find there are many repeated calculations. To avoid calculating the same probability repeatedly, we also provide a recursive method to get P for this case. For a given Δ, the corresponding transition matrix P can be divided into three parts by column, and each part is a submatrix. The first part is the first ðM ΔÞ columns of P with P i;j ¼ pði jÞ þ in this part. The second part is a submatrix with Q columns consecutive to the first part. In the second part, P P i;j ¼ pði jÞ þ þ 1 m ¼ 0 pði þ M þ mQ jÞ . The third part is the last Δ Q columns with P i;j ¼ pði þ M jÞ þpði jÞ þ . With an abuse of notation, let Δ P denote the transition matrix for a given Δ. If Δ ¼ Q , we directly Δ Δ calculate P by (4). Otherwise, we can calculate P recursively. Δ1 Now, assume we already know P . Algorithm 2 enables us to Δ calculate P recursively.
The motivation for Algorithm 2 is quite simple: P i;j depends only Δ1 on the value of i j. If P Δ are in the same part, then they i;j and P i;j Δ1 Δ1 have an identical expression and hence P Δ . If P Δ i;j ¼ P i;j i;j and P i;j Δ Δ1 are not in the same part, then P i;j and P i;j þ 1 (if exists) must be in the Δ1 same part. With i j being a constant, P Δ i;j ¼ P i þ 1;j þ 1 for i A ½1; M 1, jA ½1; M 1. For the case i¼ M or j¼M, we calculate P Δ i;j separately. 4.2. Case 2: Δ o Q At the beginning of this subsection, we must point out that as we show in the numerical study, Case 2 does arise. Under the condition Δ o Q , Δ has a finite state space ½1; Q 1. Similar to the case Δ Z Q , the inventory position after a possible ordering is still a discrete time Markov chain, and the Markov chain has the finite state space ½t þ 1; s þ M þ Q in this case. The state space can also be split into three segments: 1. yn þ 1 A ½t þ1; s þ M: if yn þ 1 A ½t þ 1; s þ M, it means that qn þ 1 ¼ 0, hence Dn ¼ yn yn þ 1 . 2. yn þ 1 A ½s þM þ 1; t þ M: if yn þ 1 A ½s þ M þ 1; t þ M, it means that qn þ 1 ¼ 0; M; or M þ mQ . The three terms correspond to three possibilities Dn ¼ yn yn þ 1 ; yn yn þ 1 þ M; and yn yn þ 1 þ M þ mQ , respectively. Recall that m is the largest integer such that 0 o yn þ 1 ðs þ MÞ rQ and m Z 1. 3. yn þ 1 A ½t þM þ 1; s þ M þ Q : if yn þ 1 A ½t þ M þ 1; s þM þ Q , it means that qn þ 1 ¼ 0 or M, then Dn ¼ yn yn þ 1 , or Dn ¼ yn yn þ 1 þ M þ mQ . The transition probabilities P i;j ¼ Probðyn þ 1 ¼ jj yn ¼ iÞ can be calculated easily and the transition matrix P is 8 for j A ½t þ 1; s þ M pði jÞ þ > > > > > 8 iA ½t þ 1; s þ M þQ ; > > > 1 > X > > > pði þ M þ mQ jÞ þ pði jÞ þ for j A ½s þ M þ1; t þ M > > < m¼0 ð7Þ P i;j ¼ 8 iA ½t þ 1; s þ M þQ ; > > > > X 1 > > > > pði þ M þ mQ jÞ þ pði jÞ þ for j A ½t þ M þ1; s þ M þ Q > > > > m¼1 > > : 8 iA ½t þ 1; s þ M þQ ; Again, let t-1, the limiting probabilities are the steady state ! probabilities π ¼ fπ 1 ; π 2 …π s þ M þ Q t g. We can calculate the stationary probabilities by solving the linear equations ( sþM þQ t Σi ¼ 1 πi ¼ 1 ð8Þ ! π P¼! π ! The transition matrix P and stationary probabilities π still only depend on Δ, and are independent of s. Now we have the long-run average cost for this case: LðΔ; tÞ ¼
M þQ ðt sÞ X i¼1
π i Cðt þ iÞ ¼
Mþ Q Δ X
π i Cðt þ iÞ:
ð9Þ
i¼1
The following propositions are useful in searching for the optimal parameters. Proposition 3. For a given Δ o Q , LðΔ; tÞ is convex in t. Proof. Given Δ o Q , M þQ Δ is fixed, and LðΔ; tÞ is convex, because it is the summation of several convex functions π i Cðt þ iÞ. □ For a given Δ o Q ; let t n2 be the value of t at which LðΔ; tÞ reaches minimum. Proposition 4. Given Δ oQ , t n2 satisfies yn M Q þ Δ r t n2 o yn r t n2 þ M þ Q Δ.
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
Proof. By
contradiction,
assume
t n2 Z yn ,
Cðt n2 þ i 1Þ; 8 i A ½1; M þ Q Δ, and LðΔ; t n2 Þ ¼
PM þ Q Δ
then
PM þ Q i¼1
Cðt n2 þ iÞ 4
Δπ i Cðt n2 þ iÞ 4
π i Cðt n2 þi 1Þ ¼ LðΔ; t n2 1Þ. This contradicts the definition of t , so t n2 o yn . Now, assume yn M Q þ Δ 4 t n2 , then Cðt n2 þ iÞ 4Cðyn M Q þ Δ þ iÞ; 8 i A ½1; M þQ Δ. Then we have LðΔ; t n2 Þ 4 LðΔ; yn M Q þ ΔÞ, which contradicts the definition of t n2 . Therefore, yn M Q þ Δ r t n2 oyn .□ i¼1
n2
Based on these propositions, we provide Algorithm 3 to n compute Δ and the corresponding t n2 for this case. We also propose a recursive method for calculating P for this case. In this case, P is a ðM Δ þ Q Þ ðM Δ þ Q Þ matrix. Obviously, the size of P will decrease as Δ increases. P can also be divided into three parts in this case. The first part is the first M Δ columns, with P i;j ¼ pði jÞ þ in this part. The second part is a submatrix with Δ columns P consecutive to the first part. In this part, P i;j ¼ pði jÞ þ þ 1 jÞ . The third part is the last Q Δ m ¼ 0 pði þ M þ mQ P columns, with P i;j ¼ pði jÞ þ þ 1 m ¼ 1 pði þ M þ mQ jÞ . Similar to the case of Δ Z Q , we also provide a recursive algorithm (Algorithm Δ 4) to calculate P. Again with an abuse of notation, we write P to denote the corresponding P for a given Δ. n1 n2 Now, we can get two pairs of solutions ðΔ ; t n1 Þ and ðΔ ; t n2 Þ for two cases separately. To get the global optimal solution, we only need to compare the two pairs and find the optimal n n pair of ðΔ ; t n Þ that minimizes LðΔ; tÞ, i.e., ðΔ ; t n Þ ¼ n1 n1 n2 n2 arg minfLðΔ ; t Þ; LðΔ ; t Þg. Algorithm 5 describes the whole (s; t) policy optimization.
5. The one-parameter heuristic policy In this section, we develop a simpler heuristic policy: when the initial inventory position is not less than S, no order is placed; when the initial inventory position is less than S but exceeds S M, order exactly M; otherwise, order M þ mQ , where m is the smallest integer such that xn þM þ mQ 4 S. The motivation for this policy is quite intuitive. Veinott (1965) has shown the optimality of the (R, Q) policy for batch ordering. The key principle of the (R,Q) policy is to order the smallest integral multiple of Q to bring the inventory position above the reorder point R. In our model, there is a MOQ constraint, so the order quantity cannot be less than M. Based on this, the order quantity in period n is qn, such that 8 > < M þ mQ if xn r S M; if S M o xn o S; qn ¼ yn xn ¼ M ð10Þ > :0 if xn Z S: We need to point out that when there is no MOQ constraint, i.e., M ¼0, the modified S policy reduces to the simple (R,Q) policy. Or when Q¼1, the modified S policy reduces to the S policy of Kiesmuller et al. (2011). For this reason, we simply call the modified S policy the S policy. In our model, because M 4 Q , the state space of the inventory level after ordering in the S policy is ½S; S þM 1. The inventory position after ordering is a discrete time Markov chain, and we can calculate the transition probability P i;j ¼ Probðyn þ 1 ¼ jj yn ¼ iÞ and transition matrix P, where
P i;j ¼
8 pði jÞ þ > > > > > > > > 1 > X > > < pði þ M þ mQ jÞ þ pði jÞ þ m¼0 > > > > > > > > pði þ M jÞ þpði jÞ þ > > > :
for j ¼ S;
We can also use the same method as that of the (s; t) policy to ! calculate the stationary probabilities π by (5) and hence the longP run average cost can be calculated by LðSÞ ¼ M i ¼ 1 π i CðS þ i 1Þ. Note that the S policy is in fact a special case of the (s; t) policy, where s ¼ S M and t ¼ S 1. Therefore, Propositions 1 and 2 still hold for the S policy and the policy optimization is simpler: because this is a special case of the (s; t) policy where Δ ¼ t s ¼ M 1, we can calculate the transition matrix and stationary probabilities, and then identify the optimal Sn that minimizes L(S) from the set ½yn M þ 1; yn as we can do for the general case Δ Z Q . The complexity of solving the linear equation (5) is OðM 3 Þ, if a Gaussian elimination is used. Because most of the calculation lies in this part, the total complexity of the S policy is OðM 3 Þ, while the complexity of the (s; t) policy is OðM 4 Þ, because Δ can take M different values.
6. Numerical study In this section, we conduct numerical experiments to test the performance of the (s; t) policy and the S policy. We consider two different discrete demand distributions:
Normal distribution
To ensure nonnegative demand, we truncate the normal distribution at zero to avoid negative demand. Poisson distribution
We conduct numerical studies with respect to the following parameters: the batch size Q, the expected demand per period EðDÞ, the critical ratio p=ðp þ hÞ, and the demand coefficient of variation (we can omit this factor in the case of a Poisson distribution). We assume M¼30 unless otherwise specified. Other parameters are chosen as follows. The holding cost h ¼1 is fixed. Batch size Q varies as 3, 5, 6, 10, and 15. EðDÞ takes the values of 10, 15, 20, 30, and 40; p=ðh þ pÞ varies as 0.80, 0.85, 0.90, and 0.95, and the demand coefficient of variation (c.v.) of the normal distribution takes the values of 0.1, 0.2, 0.3, and 0.4. The complete set of parameter values is given in Table 1. All possible combinations of the parameters give us 400 instances for normally distributed demand, and 100 instances for Poisson distributed demand. We must point that we can get the same results as Zhou et al. (2007) and Kiesmuller et al. (2011), if the same parameter values are selected and Q is fixed to be 1. To better illustrate the performance of the (s; t) policy and the S policy, we compare them with two other policies. The first policy is “the optimal policy” that achieves the minimal average cost among all admissible policies. We use value iteration to compute the optimal long-run average cost and the optimal policy is computed as follows. We initially compute the minimal average cost of a certain number of periods. Then we keep increasing a fixed number of periods, computing the minimal average cost of these periods, and comparing the deviation of the two costs. The iteration does not end until the deviation is insensitive to the increments of periods. We compare the long-run average costs of the (s; t) policy and the S policy to the optimal cost. Denote the average cost of the (s; t) policy by C s;t , and CS for the S policy, and the optimal cost by COPT. For each instance, we use G1 and G3 to
8 i A ½S; S þM 1; for j A ½S þ 1; S þ Q ; Table 1 Base parameter values for the numerical experiments.
8 i A ½S; S þM 1; for j A ½S þ Q þ 1; S þ M 1; 8 i A ½S; S þM 1: ð11Þ
25
h ¼ 1; M ¼ 30; Q A f3; 5; 6; 10; 15g; EðDÞ A f10; 15; 20; 30; 40g; p=ðh þ pÞA f0; 80; 0:85; 0:90; 0:95g; c:v: A f0:1; 0:2; 0:3; 0:4g for normal distribution
26
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
denote the gaps between the costs of these policies, as follows: C s;t C OPT n100% C OPT
G1 ¼
Table 3 Performance of the (s; t) policy (Poisson distribution). Factor
Value
avg G1
max G1
Q
3 5 6 10 15
0.34 0.31 0.29 0.26 0.07
2.60 2.47 2.41 2.09 0.94
EðDÞ
10 15 20 30 40
0.00 0.05 0.01 1.21 0.00
p=ðh þ pÞ
0.8 0.85 0.9 0.95
0.08 0.19 0.31 0.44
and C S C OPT n100% C OPT
G3 ¼
Another alternative is the (s,S) policy with S s ¼ M due to the minimum order quantity constraint. This policy is often used in practice to control inventories when there is a minimum order quantity constraint, see Zhou et al. (2007), Robb and Silver (1998) and Kiesmuller et al. (2011). Due to the batch size constraint, in our (s, S) policy, when the inventory position is less than or equal to s, an order is placed to bring the inventory position above S but no more than S þ Q ; otherwise do not order. If xn denotes the inventory position in period n before ordering, the order quantity qn can be described as follows: ( qn ¼
0; M þm Q;
xn 4 s xn r s:
C s;S C s;t n100% C s;t
and G4 ¼
min G2
max G2
47.64 53.91 56.76 64.46 67.41
17.39 20.44 22.04 28.83 39.35
122.72 134.40 141.50 156.88 150.11
0.00 0.21 0.01 2.60 0.01
27.59 31.55 30.88 75.15 125.00
17.39 22.11 19.66 59.32 71.37
44.98 46.03 49.73 90.01 156.88
0.49 1.18 1.94 2.60
59.82 59.45 58.48 54.39
19.32 19.06 18.32 17.39
156.88 154.80 145.42 133.18
6.1. Performance of the (s; t) policy
where m is the smallest integer subject to xn þ qn 4 S. Note that this (s, S) policy is in fact a special case of our (s; t) policy where Δ ¼ t s ¼ 0 (assume Δ is allowed to be 0). For each instance, we also test the performance of the (s; t) policy and the S policy by comparing them to the best (s,S) policy. Denote the average cost for the best (s,S) policy by C s;S , and use G2 and G4 to denote the gaps between the costs of these policies, as follows: G2 ¼
avg G2
C s;S C S n100% CS
We calculate the average gap, the maximal gap, and the minimal gap, which are denoted by avg Gj , min Gj , and max Gj ðj ¼ 1; 2; 3; 4Þ, respectively. Table 2 Performance of the (s; t) policy (normal distribution). Factor
Value
avg G1
max G1
avg G2
min G2
max G2
Q
3 5 6 10 15
1.43 1.32 1.29 0.91 0.71
25.11 24.32 23.52 20.33 19.03
37.15 41.76 44.68 54.72 59.45
1.11 3.75 7.23 25.65 30.18
150.98 182.92 235.18 266.56 228.50
EðDÞ
10 15 20 30 40
0.10 0.52 0.71 4.29 0.04
2.12 5.63 6.05 25.11 0.20
29.56 34.56 27.82 66.52 79.28
16.50 17.83 1.11 29.27 18.40
49.49 53.79 70.01 150.98 266.56
c.v.
0.1 0.2 0.3 0.4
4.41 0.08 0.01 0.03
25.11 1.12 0.01 0.20
72.31 53.40 36.72 27.77
1.11 18.26 17.95 16.50
266.56 112.41 58.03 42.91
p=ðh þ pÞ
0.8 0.85 0.9 0.95
0.70 0.96 1.28 1.59
13.59 16.87 20.86 25.11
49.23 48.51 46.94 45.52
4.20 2.19 1.11 1.56
228.50 266.56 255.01 235.18
The numerical results of the examples of the (s; t) policy for normally distributed demand are given in Table 2. Table 2 summarizes the gaps G1 between the optimal policy and the (s; t) policy, and the gaps G2 between the (s; t) policy and the best (s,S) policy. In each row, only one parameter is fixed while all others vary. For example, for Q¼6, all the other parameters in this row vary and result in 80 instances of normal demand. It can be seen that the performance of the (s; t) policy is quite close to that of the optimal polices in most cases. For example, at c. v. ¼ 0.2, the maximum G1 is 1.12%. G1 tends to decrease as Q increases. When c.v. takes the value of 0.2, 0.3, or 0.4, the average G1 is less than 0.1%. These results indicate that the (s; t) policy performs better when the c.v. values are relatively large. Table 2 shows that the performance of the (s; t) policy is close to the optimal performance on average for normally distributed demand. However, there are still a few cases in which the (s; t) policy performs much worse than the optimal policy. When the coefficient variation of demand is small, e.g., c.v. ¼ 0.1, the maximum gap G1 between the optimal policy and the (s; t) policy is 25.11% when Q¼ 3, EðDÞ ¼ 30, and p=ðh þ pÞ ¼ 0:95. This is the worst case over all 400 instances. Table 2 also demonstrates that the (s; t) policy performs much better than the best (s,S) policy over all examples. The gap, G2, between the (s; t) policy and the best (s,S) policy can be quite substantial. For example, the maximum G2 is 266.56% at Q¼10, EðDÞ ¼ 40, c.v. ¼ 0.1, and p=ðh þ pÞ ¼ 0:85. The average G2 with respect to c.v. ¼ 0.4 is 27.77%, while the average G2 with respect to the other parameters are all larger than this value. These results indicate that the (s; t) policy always outperforms the best (s,S) policy. These observations are consistent with those of Zhou et al. (2007), who do not consider batch ordering. By conducting numerical examples, Zhou et al. (2007) show that their twoparameter policy always outperforms the best (s,S) policy and the performance of their policy improves for normally distributed demand as c.v. increases. We next present the results of the examples for Poisson distributed demand as shown in Table 3. The gap, G1, between the optimal policy and the (s; t) policy is not larger than 2.60% over all instances. As Q increases, G1 tends to decrease. The results in Table 3 indicate that the (s; t) policy achieves nearly minimal cost for Poisson distributed demand.
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
Table 4 Performance of the S policy (normal distribution).
Table 6 Performance of the two policies for different values of M.
Factor
Value
avg G3
max G3
avg G4
min G4
max G4
Factor
Q
3 5 6 10 15
4.38 3.64 3.34 1.96 1.16
40.93 35.28 33.33 23.67 19.47
32.91 38.36 41.64 53.09 58.70
0.73 3.75 7.23 24.18 29.34
102.28 182.92 235.18 266.56 228.50
Q
10 15 20 30 40
0.48 2.51 1.05 9.88 0.56
5.50 16.08 7.75 40.93 2.21
29.09 31.99 27.40 57.64 78.59
16.49 17.77 0.73 23.91 15.94
49.49 50.28 70.01 102.28 266.56
0.1 0.2 0.3 0.4
8.53 1.15 0.90 1.01
40.93 9.51 6.61 5.19
66.09 51.69 35.45 26.54
0.73 18.21 17.94 15.94
266.56 112.24 55.94 42.91
0.8 0.85 0.9 0.95
3.03 2.97 2.89 2.70
40.93 38.86 36.15 33.82
45.55 45.50 44.67 44.06
4.20 2.19 0.34 0.73
228.50 266.56 255.01 235.18
min G4
max G4
EðDÞ
c.v.
p=ðh þ pÞ
27
Table 5 Performance of the S policy (Poisson distribution). Factor
Value
avg G3
max G3
Q
3 5 6 10 15
1.83 1.48 1.32 0.81 0.27
10.22 8.36 7.45 4.65 1.86
45.19 51.96 55.04 63.53 67.06
17.39 20.44 22.04 28.83 39.35
122.72 134.40 141.50 156.88 150.11
EðDÞ
10 15 20 30 40
0.00 0.05 0.16 5.50 0.00
0.01 0.25 0.38 10.22 0.01
27.59 31.54 30.69 67.96 125.00
17.39 22.07 19.48 57.28 71.36
44.98 46.03 49.72 78.31 156.88
p=ðh þ pÞ
0.8 0.85 0.9 0.95
1.34 1.20 1.08 0.95
10.22 9.21 8.35 7.18
57.62 57.76 57.24 53.61
19.31 19.05 18.32 17.39
156.88 154.80 145.42 133.18
avg G4
Table 3 also demonstrates that the (s; t) policy outperforms the best (s,S) policy for Poisson distributed demand. The maximum gap G2 ranges from 44.98% to 156.88%. As Q increases, G2 tends to increase. Even though G2 tends to decrease as p=ðh þ pÞ increases, the minimum G2 is still as large as 17.39%. These results show that the (s; t) policy performs much better than the best (s,S) policy for Poisson distributed demand.
6.2. Performance of the S policy The numerical results of the examples of the S policy for normally distributed demand are given in Table 4. Table 4 shows that the performance of the S policy is close to that of the optimal policy except for a few cases at c.v. ¼ 0.1. For example, the maximum G3 are 9.51%, 6.61%, and 5.19%, at c.v. ¼ 0.2, 0.3, and 0.4, respectively. The corresponding average G3 are all at the 1% level. G3 tends to decrease, as Q increases. This is identical with our former analysis for the (s; t) policy, recalling that the S policy is a special case. Table 4 indicates that the S policy performs well in the cases with relatively large values of demand coefficient variation. However, in the cases with small values of demand coefficient variation, e.g. at c.v. ¼ 0.1, the S policy may not perform very well. For example, the average G3 is 8.53%, and the maximum G3 is 40.93% at c.v. ¼ 0.1.
Value
avg G1
max G1
avg G3
max G3
3 5 6
0.51 0.90 0.37
9.00 21.77 6.83
0.52 0.91 0.38
15.53 29.22 6.83
EðDÞ
5 10 15 20 25
0.01 0.78 1.13 0.91 0.15
0.33 18.38 20.73 21.77 3.24
0.01 0.78 1.13 0.92 0.16
0.33 19.66 22.73 29.22 6.95
c.v.
0.1 0.2 0.3 0.4
2.29 0.08 0.01 0.01
21.77 1.08 0.03 0.02
2.31 0.09 0.02 0.02
29.22 6.95 5.37 4.33
p
0.8 0.85 0.9 0.95
0.45 0.54 0.67 0.72
11.91 14.24 18.79 21.77
0.47 0.56 0.68 0.73
29.22 26.93 26.34 25.01
M
10 15 20 12 18 24
0.73 0.83 1.15 0.24 0.40 0.69
18.38 20.73 21.77 7.01 8.31 9.00
0.73 0.83 1.16 0.24 0.40 0.70
19.66 22.73 29.22 7.01 15.53 9.04
Table 4 also demonstrates that the S policy outperforms the best (s,S) policy for normally distributed demand in most instances. Over all 400 instances, there is only one exception, where G4 ¼ 0.73% at Q¼3, EðDÞ ¼ 20, c.v. ¼ 0.1, and p=ðh þ pÞ ¼ 0:95. In other instances, G4 can be substantial. For example, the average G4 with respect to Q varies between 32.91% and 58.70%. As Q increases, G4 tends to increase. Recalling that G3 tends to decrease as Q increases, this may lead to the conclusion that the S policy performs better when Q is relatively large. We next test the performance of the S policy for Poisson distributed demand as shown in Table 5. We find that the average G3 is not larger than 5.50% over all instances. The maximum G3 is 10.22%, which is attained at Q¼3, EðDÞ ¼ 30, and p=ðhþ pÞ ¼ 0:8. Table 5 also shows that the S policy performs better than the best (s,S) policy for Poisson distributed demand in all instances. The gap, G4, between the S policy and the best feasible (s,S) policy is substantial. Over all 400 instances, G4 varies between 17.39% and 156.88%. These observations are consistent with Kiesmuller et al. (2011), recalling that when Q¼1, our S policy reduces to the oneparameter policy of Kiesmuller et al. (2011), who also show, by the way of numerical study, the superiority of the one-parameter policy to the (s,S) policy. The above-mentioned numerical examples are all taken when M¼30. We next present the results of the (s; t) policy and the S policy for different values of M. In the remainder of this subsection, Q varies as 3, 5, and 6 and M can take different values. Because M is an integral multiple of Q, we let M take the values of 10, 15, and 20 when Q¼ 5, otherwise M varies as 12, 18, and 24. EðDÞ varies as 5, 10, 15, 20, and 25. Other parameters take the same values as shown in Table 1, resulting in a total of 720 instances for normally distributed demand. Table 6 shows the performance of the (s; t) policy and the S policy with different values of M. From Table 6, we can draw several similar conclusions to the case with M¼30. First, both the (s; t) policy and the S policy have good performance close the optimal policy in most cases. For example, the average G1 and G3 with respect to different M are all at the 1% level. Second, the (s; t) policy and the S policy perform much worse than the optimal policy when the demand variation is
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H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
14
Table 7 Comparison of the (s; t) policy and the S policy at Q ¼5. p=ðh þ pÞ
EðDÞ ¼10
EðDÞ¼ 15
EðDÞ¼ 20
EðDÞ¼ 30
EðDÞ¼ 40
Normal c.v.¼ 0.1
0.80 0.85 0.90 0.95
2.07 2.73 2.88 1.36
10.82 10.76 10.25 7.84
0.00 0.00 0.00 0.36
20.42 15.35 10.05 4.66
0.00 0.00 0.00 0.00
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
1.25 1.53 1.60 1.64
0.13 0.15 0.09 0.02
7.75 6.34 4.87 3.40
0.09 0.07 0.05 0.03
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
0.02 0.02 0.01 0.01
0.69 0.53 0.49 0.31
5.35 4.64 4.00 3.05
1.14 1.06 0.90 0.68
Normal c.v.¼ 0.4
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
0.16 0.12 0.10 0.04
1.37 1.19 0.88 0.64
4.33 3.86 3.36 2.70
1.86 1.67 1.48 1.22
Poisson
0.80 0.85 0.90 0.95
0.00 0.00 0.00 0.00
0.01 0.01 0.01 0.00
0.26 0.29 0.21 0.10
7.86 6.23 5.01 3.22
0.00 0.00 0.00 0.00
Normal c.v.¼ 0.2
Normal c.v.¼ 0.3
Normalized Average Cost
G5
(s,k) c.v.=0.1 (s,k) c.v.=0.2 (s,k) c.v.=0.3 (s,k) c.v.=0.4
12 10 8
6 4 2
0 5
10
15
20
25
30
40
Fig. 1. The impact of EðDÞ in the (s; t) policy with p=ðh þ pÞ ¼ 0:9 and Q ¼ 5.
25 S S S S
Normalized Average Cost
20
small, i.e. c.v. ¼ 0.1. The average G1 and G3 are 2.29% and 2.31%, respectively, at c.v. ¼ 0.1, while they are all less than the 0.1% at other c.v. levels. The maximum G1 and G3 are 21.77% and 29.22%, which are also much larger than the values at higher c.v. levels. We should also note that for a fixed Q, G1 and G3 tend to increase as M increases. For example, when Q¼ 5, the average G1 increases from 0.73% to 1.15% as M increases from 10 to 20.
35
E(D)
c.v.=0.1 c.v.=0.2 c.v.=0.3 c.v.=0.4
15
10
5
6.3. Sensitivity analysis In this section, we first compare the performances of the (s; t) policy and the S policy. In fact, in some instances, they have the same performance, because the S policy is a special case of the (s; t) policy. However, we are still interested in the differences between the policies over all instances. We use G5 to denote the gap between the (s; t) policy and the S policy as follows: G5 ¼
C S C s;t n100% C s;t
To better illustrate the differences, we take all the instances at Q¼5 as an example, and enumerate the results in Table 7. It can be seen that the (s; t) policy does not perform significantly better than the S policy except for a few cases with small values of demand coefficient variation. For example, when c.v.¼ 0.1, the maximum G5 is 20.42% at EðDÞ ¼ 30. However, when c.v. is relatively large, the S policy performs nearly as well as the (s; t) policy. The maximum G5 is less than 8%, and in many instances these two policies have the same performance. These results indicate that the S policy is a recommendable substitute for the (s; t) policy at a high c.v. level. This is also consistent with the finding of Kiesmuller et al. (2011) that the performance of the one-parameter policy is close to the performance of the two-parameter policy when there is no batch ordering constraint. We next study the effect of the EðDÞ. To better illustrate the effect of EðDÞ, we plot in Fig. 1 the long-run average costs (normalized by the global minimum of C(y)) of the (s; t) policy as a function of EðDÞ under different levels of c.v. with a fixed M¼30. Fig. 1 illustrates that for a given M,
when EðDÞ is relatively small, e.g., less than M, a lower level of demand variation leads to a higher decreasing rate of average costs as EðDÞ increases.
0
5
10
15
20
25
30
35
40
E(D)
Fig. 2. The impact of EðDÞ in the S policy with p=ðh þ pÞ ¼ 0:8 and Q¼ 3.
when EðDÞ is relatively large, e.g., larger than M, the average cost is not affected significantly by the different demand variation levels as EðDÞ increases.
The reason for this is that as EðDÞ increases, the constraint of the minimum order quantity becomes looser and looser. Similar results can be drawn for the S policy. We plot the results for the S policy in Fig. 2. Before ending this section, we must point out that the case of Δ o Q can appear, although Δ ZQ in most of our numerical examples. For example, for normally distributed demand with EðDÞ ¼ 100 and c:v: ¼ 1, we can get the optimal Δ ¼1 and the optimal t ¼183, when M¼8, Q¼4, and p=ðh þ pÞ ¼ 0:8.
7. Conclusion In this paper, we design an algorithm for a heuristic twoparameter policy, the (s; t) policy, to control stochastic inventories with minimum order quantity and batch size constraints. Applying a Markov chain approach, we compute the system costs and provide recursive algorithms to optimize the policy under the long-run average cost criterion. We also develop the computational procedure for a simpler policy, the S policy, which is
H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
motivated by the (R,Q) policy in batch ordering and is a special case of the (s; t) policy. Numerical studies are conducted to demonstrate the effectiveness of these two policies with respect to the optimal policy and the (s,S) policy for both normally and Poisson distributed demand. Overall, the S policy has a good performance close to that of the (s; t) policy; only in a few cases with small demand variation, the latter outperforms the former significantly. Acknowledgments The authors acknowledge the financial support from Hong Kong Research Grants Council Project no. 9041952. Appendix Algorithm 1. Policy optimization for Δ Z Q . 1: Set Ln1 ’Inf; 2: for Δ ¼ Q to M 1 do 3: calculate P by (4); ! 4: calculate π by (5); n 5: for t ¼ y M to yn 1 do 6: calculate LðΔ; tÞ by (6); 7:
if LðΔ; tÞ o Ln1 then
8: Ln1 ’LðΔ; tÞ, t n1 ’t, Δ ’Δ; 9: end if 10: end for 11: end for n1
Algorithm 2. Calculating P for Δ ZQ . 1: if Δ ¼ Q then 2: calculate P by (4) 3: else 4: for i¼1 to M do 5: for j¼ 1 to M Δ do Δ1 6: PΔ ; i;j ’P i;j 7: end for 8: end for 9: j’M Δ þ 1; 10: for i ¼1 to M 1 do
11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29:
Δ
Δ1
P i;j ’P i þ 1;j þ 1 ; end for i’M; Δ 1 P1 PΔ þ m ¼ 0 pði þ M þ mQ jÞ ; i;j ’P i;j for i ¼1 to M do for j ¼ M Δ þ 2 to M Δ þ Q do
Δ1 PΔ ; i;j ’P i;j end for end for if Δ ¼ Q þ 1 then for i¼1to M 1 do
PΔ i;j ’ProbðD ¼ iÞ; end for i’M; PΔ i;j ’ProbðD ¼ MÞ þ ProbðD ¼ 0Þ; else j’M Δ þ Q þ 1; for i¼1 to M 1 do Δ1 PΔ i;j ’P i þ 1;j þ 1 ;
29
30: 31:
end for i’M;
32: 33: 34:
PΔ i;j ’ProbðD ¼ Δ Q 1Þ þ ProbðD ¼ M þ Δ Q 1Þ; for i¼ 1 to M do for j ¼ M Δ þ Q þ 2 to M do
35: 36: 37: 38: 39:
Δ1 PΔ ; i;j ’P i;j end for end for end if end if
Algorithm 3. Policy optimization for Δ oQ . 1: Set Ln2 ’Inf ; 2: for Δ ¼ 1 to Q 1 do 3: calculate P by (7); ! 4: calculate π by (8); n 5: for t ¼ y M Q þ Δ to yn 1 do 6: calculate LðΔ; tÞ by (9); 7:
if LðΔ; tÞ o Ln2 then
8: Ln2 ’LðΔ; tÞ, t n2 ’t, Δ ’Δ; 9: end if 10: end for 11: end for n2
Algorithm 4. Calculating P for Δ o Q . 1: if Q¼1 then 2: calculate P by (7) 3: else 4: for i¼1 to M Δ þ Q do 5: for t j ¼1 to M Δ þ Q do 6: if j ¼ M Δ þ 1 then 7: 8:
Δ1 PΔ i;j ’P i þ 1;j þ 1 ; else
Δ1 9: PΔ ; i;j ’P i;j 10: end if 11: end for 12: end for 13: end if
Algorithm 5. Policy optimization for the (s; t) policy. 1: Set Ln1 ’ Inf, Ln2 ’Inf; 2: for Δ ¼ Q to M 1 do 3: calculate P by Algorithm 2; ! 4: calculate π by (5); n 5: for t ¼ y M to yn 1 do 6: calculate LðΔ; tÞ by (6); 7:
if LðΔ; tÞ o Ln1 then
8: Ln1 ’LðΔ; tÞ, t n1 ’t, Δ ’Δ; 9: end if 10: end for 11: end for 12: for Δ ¼ 1 to Q 1 do 13: calculate P by Algorithm 3; ! 14: calculate π by (8); n 15: for t ¼ y M Q þ Δ to yn 1 do 16: calculate LðΔ; tÞ by (9); n1
17:
if LðΔ; tÞ o Ln2 then
18: 19:
Ln2 ’LðΔ; tÞ, t n2 ’t, Δ ’Δ; end if n2
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H. Zhu et al. / Int. J. Production Economics 168 (2015) 21–30
20: end for 21: end for 22: if Ln1 4 Ln2 then 23: Ln ’Ln2 , t n ’t n2 , Δ ’Δ ; 24: else n
25:
n2
Ln ’Ln1 , t n ’t n1 , Δ ’Δ ; n
n1
26: end if References Axsäter, S., Zhang, W., 1999. A joint replenishment policy for multi-echelon inventory control. Int. J. Prod. Econ. 59, 243–250. Bradford, P.G., Katehakis, M.N., 2006. Constrained inventory allocation. In: 2006 Proceedings of 10th WSEAS International Conference on Applied Mathematics, pp. 504–507. Bradford, P.G., Katehakis, M.N., 2007. Constrained inventory allocation and its applications. WSEAS Trans. Math. 6 (2), 263–270. Chao, X., Zhou, S.X., 2009. Optimal policy for a multiechelon inventory system with batch ordering and fixed replenishment intervals. Oper. Res. 57 (2), 377–390. Chen, F., 2000. Optimal policies for multi-echelon inventory problems with batch ordering. Oper. Res. 48 (3), 376–389. Gallego, G., 1998. New bounds and heuristics for (q,r) policies. Manag. Sci. 44, 219–233. Heyman, D., Sobel, M. 1984. Stochastic Models in Operations Research, vol. 2. McGraw Hill, New York.
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