On the Convexity of Discrete (r, Q) and (s, T ) Inventory Systems

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1 On the Convexity of Discrete r, Q and s, T Inventory Systems Jing-Sheng Song Mingzheng Wang Hanqin Zhang The Fuqua School of Business, Duke University, Durham, NC 27708, USA Dalian University of Technology, Dalian, 6024, China Academy of Math. and Syst. Sci., Chinese Academy of Sciences, Beijing, 00080, China jssong@duke.edu mzhwang@dlut.edu.cn hanqin@amt.ac.cn March 3, 2008 Abstract We consider single-item r, Q and s, T inventory systems with discrete demand processes. While most of the inventory literature studies continuous approximations of these models and establishes joint convexity properties of the policy parameters in the continuous space, we show that these properties no longer hold in the discrete space, in the sense of linear interpolation extension and L -convexity. This nonconvexity can lead to failure of optimization techniques based on local optimality. It can also make certain comparative properties established previously using continuous variables invalid. Keywords: r, Q policy; s, T policy; discrete convexity; inventory/production Introduction We investigate the convexity properties of two basic single-item inventory systems with discrete i.e., integer valued demand processes, by applying the recently developed discrete convexity concepts see, e.g., Murota The first system we consider is a continuous-review r, Q inventory system where r is the reorder point and Q is the order quantity. The demand process is a Poisson process with rate λ. Unsatisfied demand is backlogged. The replenishment leadtimes are exogenous and sequential see Zipkin 2000, Chapter 7 for detail, having identical distribution as the generic random variable L. Let D denote the steady-state leadtime demand, IN the net inventory inventory on-hand - backorders, and IP the inventory position outstanding orders + net inventory, respectively. Then IN IP D, and IP is uniformly distributed on {r +,, r + Q}, which is independent of D. Consequently, the average number of

2 backorders is Br, Q E[IN Q and the average on-hand inventory equals r+q yr+ E[D y + Ir, Q E[IN + r + Q + /2 E[D + Br, Q. 2 To avoid the trivial case and without loss of generality, we assume that PrD 0 <. There are linear inventory-holding and backorder-penalty costs at unit rates h > 0 and p > 0, respectively. Let K be the fixed ordering cost for each replenishment order. Then, the expected long-run average system cost is cr, Q λk Q + h Ir, Q + p Br, Q. 3 The second system we consider is a periodic-review s, T system, where T is the reorder interval and s is the order-up-to level. Other features are similar to the r, Q system; see details in 3. For the r, Q system, to find the optimal r, Q that minimizes cr, Q, a natural approach is to establish the joint convexity of c in r, Q, so that a local optimum is guaranteed to be global optimal, and therefore a global optimum can be found by descent algorithms. However, while the notion of convexity has been well established a long time ago for functions on a continuous vector space, it was not so for functions defined on a integer vector space termed discrete convexity until more recently see, e.g., Murota This partly explains why in the inventory literature most studies on the properties of r, Q systems assume continuous demands. It was also argued that the cost function for continuous demands is an adequate approximation of the cost function for discrete demands when the order quantity Q is not too small Zheng 992. For the continuous-demand model, it was shown that, as long as the cumulative demand is a nondecreasing stochastic process with stationary increments and continuous sample paths, IP is uniformly distributed on the interval r, r + Q, and D and IP are independent of each other Serfozo and Stidham 978, Zipkin 986. Under these conditions, Zipkin 986 proves that Br, Q and Ir, Q are jointly convex in r, Q. His proof is given by explicitly expressing Br, Q in terms of integrations involving the probability distribution function of D, taking second partial derivatives and then showing the Hessian matrix is nonnegative definite. Zhang 998 provides a simpler proof for the same result. One essential step 2

3 of Zhang s proof is that IP has a continuous uniform distribution. Based on the joint convexity results, Zheng 992 develops a graphically intuitive optimality condition for the continuous r, Q system, and compares the exact optimal solution with the approximate solution obtained from using the EOQ formula. Rao 2003 extends the above convexity results and sensitivity analysis to the continuous-time, continuous-demand s, T system with compound Poisson and Brownian motion demand processes. With the theory of discrete convexity, it is interesting to see whether the above results still hold for discrete r, Q and s, T systems. This is especially important when Q and T are small so the continuous-demand approximation is not effective. Note that when these policy parameters take integer values, the methods used by the above mentioned works for continuous models are no longer valid, so direct verification of discrete convexity is needed. It is worth mentioning that recently some other authors have also applied discrete convexity concepts in inventory systems; see, for example, Lu and Song 2005 on assemble-to-order systems and Zipkin 2006 on lost-sales models. In this note we examine the convexity of discrete r, Q and s, T systems under two notions. The first is the convexity of the linear interpolation of the discrete functions; it is also known as multimodularity. The second is L -convexity, which guarantees a local optimum to be global optimal. From the definitions of these two notions see details in Section 2, neither one of them can imply the other. However, Murota 2005 shows that the two properties are equivalent subject to a unimodular coordinate transformation. That is, a multimodular function can be represented by a L -convex function through a unimodular coordinate transformation, and vice versa. In our analysis, we do not make such transformations. That is, we work with the original definitions directly. In Section 2, we show that Br, Q, Ir, Q and cr, Q are not jointly convex in the sense of linear interpolation extension or in the sense of L -convexity. As pointed out by Veinott 964, this nonconvexity can lead to a failure of the optimization technique suggested in Hadley and Whitin 963. In addition, when Br, Q and/or Ir, Q are used in a model to optimize over r, Q, a local optimal solution may not be globally optimal. On the other hand, the optimization algorithm by Federguren and Zheng 992 for the discrete r, Q system, which depends only on the unimodality of the cost function, is still valid. In Section 3, we show that some properties established by Zheng 992 for the continuous demand r, Q model fail to hold in the discrete r, Q system. For example, unlike in the continues-demand model, the loss of optimality by using the EOQ approximation for Q in the discrete-demand model can exceed 2.5%. 3

4 Finally, in Section 4, we show similar results for the periodic-review s, T system with Poisson demand. 2 Nonconvexity of Discrete r, Q Systems 2. Linear Interpolation For any given function f defined on S Z Z + {0, ±, ±2, } {0,, 2, }, a function f defined on R R + is called its extension if f agrees with f on S. Let f li be the linear interpolation extension of f on R R +. The function f is called convex in the sense of the linear interpolation extension if its linear interpolation extension f li is convex. see Altman et al or Murota 2005 define Let e, 0, e 2 0, and d 2 e e 2,. For a function f defined on S, e fx, x 2 fx, x 2 fx, x 2, e fx, x 2 fx +, x 2 fx, x 2, e2 fx, x 2 fx, x 2 + fx, x 2, d2 fx e fx e2 fx. Let F { e, d 2, e 2 }. The following is a necessary and sufficient condition for convexity; its proof is given by Altman et al Lemma. Altman et al The real-valued function f defined on S is convex in the sense of linear interpolation extension if and only if ξi ξj fx, y 0, 4 when x, y, x, y + ξ i, x, y + ξ j and x, y + ξ i + ξ j are all elements in S for ξ i F and ξ j F. Remark. In some literature, a function f defined on S is called multimodular if 4 holds see, e.g., Hajek 985, Glasseerman and Yao 994, and Altman et al Thus, the convexity in the sense of the linear interpolation extension is equivalent to multimodularity. Now we prove the following result by using the above lemma. Proposition. The functions Br, Q, Ir, Q and cr, Q are not jointly convex in the sense of linear interpolation extension, or, equivalently, not multimodular. 4

5 Proof. In order to prove that the function Br, Q is not jointly convex in r, Q, it is sufficient to show that there is a pair r 0, Q 0 S such that that is, the following result holds: d2 e2 Br 0, Q 0 > 0, Br 0 +, Q 0 + Br 0 +, Q 0 + Br 0, Q 0 + Br 0, Q > 0. 5 Note that from, Br, Q can be expressed as follows: Br, Q Q Q 2Q + + yr+ ty + tr+ yr+ + 2Q tr+ + t yprd t Q t t yprd t Q [t r t rprd t tr+q+ Using this expression, we obtain + + yr+q+ ty + tr+q+ yr+q+ t yprd t t t yprd t [t r Q t r QPrD t. 6 Br +, Q + Br +, Q t r 2t r PrD t 2Q + tr+2 t r Q 3t r Q 2PrD t 2Q + tr+q+3 [ t r 2t r PrD t 2Q 2Q tr+2 tr+q+2 2QQ + + 2QQ + t r Q 2t r Q PrD t t r 2t r PrD t tr+ tr+q+2 t r Q 2t r + Q PrD t, 5

6 and Br, Q + Br, Q + 2 t r t rprd t 2Q + tr+ t r Q 2t r Q PrD t 2Q + tr+q+2 [ t r t rprd t 2Q + 2 tr+ t r Q 3t r Q 2PrD t 2Q + 2 2Q + Q + 2 tr+q+3 2Q + Q + 2 t r t rprd t tr+ tr+q+2 t r Q 2t r + Q + PrD t. Therefore, Br +, Q + Br +, Q + Br, Q + Br, Q + 2 { t r [ Q + 2t r 2 + Qt rprd t 2QQ + Q + 2 tr+ } + [2t r Q 2t r PrD t { QQ + Q tr+q+2 t r t r Q 2PrD t tr+ tr+q+2 } t r t r Q 2PrD t r+q+ t r r + Q + 2 tprd t. 7 QQ + Q + 2 tr+ In order to prove 5, it is sufficient to find a pair r 0, Q 0 S such that t r 0 r 0 + Q tprd t > 0. 8 r 0 +Q 0 + tr 0 + 6

7 Because PrD 0 <, we have 0 < PrD j. Hence, there must exist a positive j integer number k 0 such that PrD k 0 > 0. Choosing any r 0 satisfying r 0 k 0 2 and Q 0 > 0 satisfying Q 0 > k 0 r 0, we obtain t r 0 r 0 + Q tprd t r 0 +Q 0 + tr 0 + t r 0 r 0 + Q tprd t r 0 +Q 0 + tr 0 +2 r 0 +Q 0 + tr 0 +2 PrD t PrD k 0 > 0. In view of 7 and 8, this completes the proof of the nonconvexity of Br, Q. Now we consider Ir, Q. It follows from 2 and 8 that for r 0, Q 0 with r 0 k 0 2, Q 0 > k 0 r 0 and PrD k 0 > 0, Ir 0 +, Q 0 + Ir 0 +, Q 0 + Ir 0, Q 0 + Ir 0, Q r Q Br 0 +, Q r Q 0 + Br 0 +, Q r 0 + Q Br 0, Q 0 + 2r 0 + Q Br 0, Q Br 0 +, Q 0 + Br 0 +, Q 0 + Br 0, Q 0 + Br 0, Q > 0. 9 Hence, in view of Lemma, Ir, Q is not convex. Finally, we consider cr, Q. Similar to the above, we prove that there exists a pair r 0, Q 0 such that cr 0 +, Q 0 + cr 0 +, Q 0 + cr 0, Q 0 + cr 0, Q >

8 From 3, It follows from 9 that Using 7, cr +, Q + cr +, Q + cr, Q + cr, Q + 2 λk Q + λk Q + λk Q + λk Q + 2 +h Ir +, Q + Ir +, Q + Ir, Q + Ir, Q + 2 +p Br +, Q + Br +, Q + Br, Q + Br, Q + 2. cr +, Q + cr +, Q + cr, Q + cr, Q + 2 2λK QQ + Q + 2 +p + h Br +, Q + Br +, Q + Br, Q + Br, Q + 2. cr +, Q + cr +, Q + cr, Q + cr, Q + 2 r+q+ 2λK + p + h t r r + Q + 2 tprd t. QQ + Q + 2 tr+ If we take K, λ, PrL, p 0, h, r 0 and Q 0, then cr +, Q + cr +, Q + cr, Q + cr, Q t 2 t t! e > 0. t This implies that cr, Q is not jointly convex. The nonconvexity in the sense of the linear interpolation implies that Hadley and Whitin s optimization approach does not work. 8

9 2.2 L -Convexity We adopt the following midpoint convexity characterization of L -convexity: A function f defined on S is L -convex if and only if the following midpoint property holds for all r, Q, r 2, Q 2 S: fr, Q + fr 2, Q 2 f r + r 2, Q + Q f r + r 2, Q + Q with max{ r r 2, Q Q 2 } 2, where x x denotes the integer obtained by rounding down or up x to the nearest integer. It has been shown see Murota 2003 that L -convexity guarantees that any local optimum coincides with the global optimum. We have: Proposition 2. The functions Br, Q, Ir, Q and cr, Q are not L -convex. Proof. To prove the result for Br, Q, it suffices to show that there exists a vector r, Q S such that [ [ Br, Q Br, Q Br, Q + Br, Q + < 0. This corresponds to r, Q r, Q and r 2, Q 2 r, Q + in the above definition. Similar to 5, we have [ [ Br, Q Br, Q Br, Q + Br, Q + QQ + t rprd t + tr+ +QPrD r + Q + ED r + QQ + r+q tr+q+, t r QPrD t r t rprd t + ED r Q t0 r+q t r QPrD t + Q Q t0 r r+q t rprd t t rprd t QQ + QQ + t0 t0 r+q t rprd t < 0. tr+ 9 t0 PrD t

10 So holds. Therefore, we know Br, Q is not L -convex. Along the same lines, we can show that Ir, Q is not L -convex. Finally we consider cr, Q. Similarly, we prove that there is a pair r, Q such that cr, Q cr, Q cr, Q + cr, Q + < 0. 2 From 3, cr, Q cr, Q cr, Q + cr, Q + [ h Ir, Q Ir, Q Ir, Q + Ir, Q + [ +p Br, Q Br, Q Br, Q + Br, Q +. It follows from 2 that Ir, Q Ir, Q Ir, Q + Ir, Q + Br, Q Br, Q Br, Q + Br, Q +. Therefore, cr, Q cr, Q cr, Q + cr, Q + [ p + h Br, Q Br, Q Br, Q + Br, Q +. Consequently, the non-l -convexity of cr, Q directly follows from the non-l -convexity of Br, Q. Let S r +Q and make a variable change Q S r. We next would like to examine the convexity property of the performance measures as functions of r, S. In particular, define ˆBr, S Br, S r 3 S E[D y + S r yr+ S t yprd t S r yr+ ty [ t yprd t S r yr+ ty 0 ys+ ty t yprd t.

11 Similarly, define Îr, S Ir, S r, ĉr, S cr, S r. 4 We have: Proposition 3. ˆBr, S, Îr, S, and ĉr, S are not L -convex. Proof. Here we only consider ˆBr, S, Îr, S, and ĉr, S. According to the definition of L -convex, it suffices to prove that there exists a pari r, S such that ˆBr, S ˆBr, S < ˆBr, S + + ˆBr, S +. 5 It follows from 8 that ˆBr, S ˆBr, S ˆBr, S + ˆBr, S + S+2 t yprd t + S t yprd t S r + 3 S r yr ty yr+ ty S r + 2 S+ yr t yprd t ty tr S r + S+ yr+ ty t yprd t S+ t rprd t + PrD S + S r + 2S r + 3 S r S 2r S rs r + S r + 2S r + 3 S yr+ ty S t yprd t S+ t rprd t + S r + 2PrD S + S r + 2S r + 3 tr 22S 2r + 3 S S + t yprd t. 6 S rs r + If we take λ 5, PrL, r 0 and S 00, then yr+ ty ˆBr, S ˆBr, S ˆBr, S + ˆBr, S < 0, 7

12 so 5 holds. Therefore, we know Br, S is not L -convex. Noting that Îr, S Îr, S Îr, S + Îr, S + ˆBr, S ˆBr, S ˆBr, S + ˆBr, S +, and 7, we have that Ĩr, S is not L -convex. Furthermore, by we have that ĉr, S λk S r + h Îr, S + p ˆBr, S, ĉr, S ĉr, S ĉr, S + ĉr, S + 22S 2r + 3λK S r + 2S r + 3 S rs r + [ +p + h ˆBr, S ˆBr, S ˆBr, S + ˆBr, S +. If we take K p + h, λ 5, PrL, r 0 and S 00, then ĉr, S ĉr, S ĉr, S + ĉr, S + p + h p + h < 0. Therefore, we know Br, S is not L -convex. Finally, it is interesting to investigate whether another variable change, i.e., letting r S Q and expressing the performance measures as functions of S, Q, could lead to convexity. More specifically, define BS, Q BS Q, Q 8 S E[D y + Q ys Q+ S Q [ Q ys Q+ ty ys Q+ ty t yprd t t yprd t 2 t yprd t. ys+ ty

13 Also, define ĨS, Q IS Q, Q, cs, Q cs Q, Q. 9 Similar to the proof of Proposition 3, we can show: Proposition 4. BS, Q, ĨS, Q, and cs, Q are not L -convex. 3 Comparison with the EOQ Model Due to its closed-form solution and robustness, the Economic Order Quantity EOQ is often used as a heuristic order quantity in r, Q systems. So, it is of interest to know how good such an approximation is. Because the EOQ model assumes deterministic demand, it is then also interest to learn how, in general, a solution from the simpler deterministic model compares to the solution for the stochastic model. Zheng 992 studies these issues for the continues-demand r, Q system. In this section we investigate whether his findings still hold for the discrete-demand system. Before stating Zheng s results, we first introduce some notation. For the continuous r, Q system, the expected long-run average cost of the system can be expressed as cr, Q λk + h r+q r Let r, Q arg min r,q cr, Q. Ey D + dy + p r+q r Q ED y + dy. 20 When the demand is deterministic, the above model reduces to the EOQ model with planned backorders, its long-run average cost equals c d r, Q λk + h r+q r y ED + dy + p r+q ED y + dy r. 2 Q Let r d, Q d arg min r,q c d r, Q. Then we have the following simple closed-form solution EOQ: Q d 2λKp + h, r d ED h hp h + p Q d. Let Q d be a heuristic order quantity and optimize over r, the resulting system cost is then c r Q d, Q d, where r Q d arg min r cr, Q d. Zheng 992 shows the following 3

14 Lemma 2. Zheng 992, Theorem 2 and Theorem 5 Q d Q and err c r Q d, Q d c r, Q c r, Q 8. In other words, for the continuous demand case, the optimal order quantity for the stochastic system is always larger than that of the corresponding constant-demand model. In addition, the optimality loss by using the EOQ heuristic order quantity will not exceed /8. Although Zheng 992 assumes a positive fixed leadtime EL, it is clear that his analysis only depends on the leadtime demand, so these results hold for the exogenous, sequential supply system assumed here. In the following, we show that there exist counterexamples to these properties in discrete r, Q systems. For the discrete deterministic r, Q system, let I d r, Q Q r+q tr+ Then its long-run average cost equals t ED +, B d r, Q Q r+q tr+ ED t +. c d r, Q λk Q + h I dr, Q + p B d r, Q. Denote r d, Q d arg max r,q c d r, Q. Example. Letting p, h 0, λ 2, K 28, and PrL, we have see 20 and 2 r Q r d Q d r Q r d Q d Thus, the order quantity from the deterministic leadtime-demand model can be greater than that from the stochastic leadtime-demand model. Example 2. Letting p 0, h 0, λ 0, K, and PrL 0.9, PrL 2 0., we have see 20 and 2 r Q rd Q d crq d, Q d cr, Q err

15 From this, we can see the upper bound /8 does not hold any more for the discrete demand case. However, for the continuous case, we have r Q r d Q d c r Q d, Q d c r, Q err Nonconvexity of Discrete s, T Systems We now consider a periodic review s, T inventory system in which the inventory position IP t is reviewed every T periods. If, upon review, IP t is below s, then order enough to bring IP t back to s. The demand process is a Poisson process with per-period demand rate λ. That is, let D denote the demand for one period, then D has a Poisson distribution with parameter λ. The leadtime for each replenishment order is assumed to be a positive integer l. There is a fixed order cost K, a per-period unit inventory holding cost h, and a per-period unit backorder cost p. Note that the cumulative demand in t-periods, denoted by Dt, has a Poisson distribution with parameter λt. Following a similar argument in Rao 2003, the long-run average inventory and backorders under the s, T policy are Is, T T Bs, T T l+t tl+ l+t tl+ E[s Dt + T E[Dt s + T l+t tl+ n0 l+t tl+ s λt λtn s ne, λt s + Is, T. Therefore, the long-run average cost under the s, T policy is Cs, T K PrDT > 0 + his, T + pbs, T T K T e λt + his, T + pbs, T. 22 Rao 2003 proves that if s, T are continuous so l is also a real number and the above summations are integrals, then Cs, T is jointly convex in s and T. In the following, we show that this is not so for discrete systems. Proposition 5. The function Cs, T for the discrete system is not convex in the sense of linear interpolation extension nor in the sense of L -convexity. 5

16 Proof. We first analyze the linear interpolation extension. Observe that Bs, T + Bs, T + + Bs, T Bs, T l+t + [ l+t + λt s + Is, T + λt s + Is, T + T + T + tl+ tl+ [ l+t [ l+t + λt s + Is, T λt s + Is, T T T We have Furthermore, tl+ tl+ Is, T + Is, T + + Is, T Is, T. 23 Cs, T + Cs, T + + Cs, T Cs, T h + p[is, T + Is, T + + Is, T Is, T. 24 Is, T + Is, T + T + l+t + tl+ n0 l+t + [ s e λt T + tl+ s λt λtn s ne λt n. n0 T + l+t + tl+ n0 s λt λtn s ne Similarly, Hence, Is, T Is, T T l+t tl+ [ s e λt n0 λt n. Is, T + Is, T + + Is, T Is, T [ + e λl+t s [ λ n l + T + n l+t + T + T T + n0 Take s and T, then tl+ s e λt n0 λt n Cs, T + Cs, T + + Cs, T Cs, T 2 e λl+ e λ > 0, which implies that Cs, T is not convex in the sense of linear interpolation extension

17 We next examine L -convexity. Taking T in 24 and 25 gives that Cs, T + Cs, T + + Cs, T Cs, T p + h [ s e λl+2 λl + 2 n + 2 n0 p + h s e λl+ e λ λl + 2 n s + 2 n0 s [e λl+ n0 n0 λl + n λl + n. Taking λ 0., l 56, T and s 36, we get that Cs, T + Cs, T + + Cs, T Cs, T < 0. This implies that Cs, T is not L -convex. Acknowledgement: We thank Paul Zipkin for helpful comments on an earlier version of this paper. We also thank Mr. Lijian Lu and Mr. Junfei Huang for their help in the numerical examples. This research was supported in part by National Natural Science Foundation of China under award No and award No References [ Altman E., Gaujal B. and A. Hordijk Multimodularity, convexity, and optimization properties. Mathematics of Operations Research 25, [2 Brooks R.S. and J.Y. Lu On the convexity of the backorder function for a E.O.Q. policy, Management Science 5, [3 Federgruen A. and Y.S. Zheng An effecient algorithem for computing an optimal r, q policy in continuous review stochastic inventory systems, Operations Research 40, [4 Glasserman P. and D. Yao Monotone optimal control of permutable GSMPs, Mathematics of Operations Research, 9, [5 Hadley G. and T.M. Whitin Analysis of Inventory Systems, Prentice-Hall Inc., New York. [6 Hajek B Extremal splitting of point processes, Mathematics of Operations Research, 0,

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