Are Base-stock Policies Optimal in Inventory Problems with Multiple Delivery Modes?

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1 Are Base-stoc Policies Optimal in Inventory Problems with Multiple Delivery Modes? Qi Feng School of Management, the University of Texas at Dallas Richardson, TX , USA Guillermo Gallego Department of Industrial Engineering and Operations Research The Columbia University in the City of New Yor, USA Suresh P. Sethi School of Management, the University of Texas at Dallas Richardson, TX , USA Houmin Yan Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong Hanqin Zhang Academy of Mathematics and Systems Sciences, Academia Sinica Beijing, , China March 2, 2005 We present a periodic review inventory model with multiple delivery modes. While base-stoc policies are optimal for one or two consecutive delivery modes, they are not so otherwise. For multiple consecutive delivery modes, we show that only the fastest two modes have optimal base stocs, and provide a simple counterexample to show that the remaining ones do not. We investigate why the base-stoc policy is or is not optimal in different situations. This paper is an abridged version of Feng et al. (2004. Subject Classifications: Inventory/production: uncertainty, multiple delivery modes, base-stoc policies.

2 1. Introduction In this note, we construct a counterexample to establish that a base-stoc policy is not optimal in general for inventory models with multiple delivery modes. In most of the studies on inventory models with replenishment leadtime options, it is assumed that there are two consecutive procurement modes available (e.g., Neuts 1964, Lawson et al. 2000, Sethi et al. 2001, Muharremoglu et. al That is, the leadtimes of the two modes vary by exactly one period. This assumption leads to the optimality of a base-stoc policy. Whittemore et al. (1997 formulate a problem for general two delivery modes and derive explicit formulas for the optimal ordering quantities when the modes are consecutive. They also comment that the optimal policy for two nonconsecutive modes may not have a simple structure. In the case of three consecutive delivery modes, Fuuda (1964 first considers a special case when orders are placed only every other period, in which case a base-stoc policy is shown to be optimal. Zhang (1996 extends Fuuda s model by allowing three consecutive modes ordered every period, and claims the optimality of a base-stoc policy under certain conditions. Her claim is untrue as evident from the counterexample presented in this note. Feng et al. (2005 show that there exist optimal base-stoc levels for the two faster modes in an inventory model with three consecutive delivery modes and demand forecast updates. We formulate a general problem in Section 2. We analyze the policy structure through an example in Section, and discuss the separability of the cost functions in Section 4. We conclude our study in Section Problem Formulation We consider a finite horizon periodic review inventory system with N consecutive delivery modes. In the case of non-consecutive delivery modes, one can insert fictitious delivery modes as suggested in Sethi et al. (2001 to transform the problem into one with consecutive modes by setting the cost for any fictitious mode to equal that for the next faster real mode. Setting costs this way implies that we only consider policies that do not issue orders using fictitious modes. An order Q i l in period l via the ith delivery mode, i = 1, 2,..., N, is an order associated with a leadtime of i periods, and is termed the type i order. We denote ql i as the realized 1

3 value of Q i l. A reference (pre-order inventory position for an order Qi l is defined as the sum of on-hand inventory in period l plus all previous and current orders placed before Q i l and to be delivered by the time Q i l is delivered, i.e., at the end of period l + i 1. A post-order inventory position for an order Q i l is defined as the sum of its reference inventory position and Q i l. The notation is given below and the inventory positions are presented in Table 1. Q i = the amount of type i order that is placed at the beginning of period and will arrive at the end of period + i 1, 1 T, 1 i N; c i = the unit procurement cost of type i order in period, 1 T, 1 i N; D = the demand in period (materialized after delivery of orders, 1 T; x = the inventory/baclog level at the beginning of period, 1 T; p i = N j=i+1 q j +i j, the amount of in-transit orders at the beginning of period that will arrive at the end of period + i 1, 1 T, 1 i N; y = x + p 1, the reference inventory position for Q 1 at the beginning of period, 1 T; z i = the post-order inventory position for Q i viewed at the beginning of period, 1 T, 1 i N; H ( = the inventory holding/baclog cost in period assessed on the beginning inventory x, 1 T; H T+1 ( = the costs for the ending inventory/baclog. We assume that that E[D ] < for each. Furthermore, the inventory cost functions H (x is nonnegative convex with H (0 = 0 for each. The objective is to choose a sequence of orders {Q 1,..., QN } 1 T so as to minimize the total expected cost given by { ( T [ N ]} J 1 x 1, p 1 1,..., p1 N 1 ; {Q 1,..., Q N } 1 T = H 1 (x 1 + E c j Qj + H +1(X +1, (1 subject to the inventory dynamics Let W (y, p 2 =1 X +1 = x + p 1 + Q1 D, 1 T. (2,.., pn 1 denote the cost-to-go function in period. Then the function satisfies 2 j=1

4 Table 1: Inventory positions viewed at the beginning of period. Period l Post-order inventory position of Q i l : Order placed in period y to be delivered at the beginning of period l : N 2 + N 1 z 1 z 2 z z N 1 Q 1 q 2 1 q 2 Q 2 q 1 Q z N History of orders to be delivered at the beginning q 4 q 5 4 of period l : q N 1 N+2 q 4 2 q q N 1 N+ q 4 1 q q N 1 N+4 Q N 1 In-transit orders to be delivered at the beginning of period l q N N+1 q N N+2 p 1 È = p 2 Nj=2 q j È = Nj= q j +1 j +2 j q N N+ p È = Nj=4 q j + j q 1 N p N 1 = q 1 N Q N the dynamic programming equation W (y, p 2,.., p N 1 = min z 1 y, z j zj 1 +p j, j=2,...,n { c 1 (z 1 y + N j=2 c j (zj zj 1 p j + EH +1(z 1 D ( } +EW +1 z 2 D, z z2,..., zn zn 1, 1 T 1. ( One can show, along the lines of the proof of Theorem 4. in Sethi et al. (2001, that there exist functions z i (y, p 2,..., pn, 1 i N, which minimize the right-hand side in (.. An Example In this section, we examine a three period example with three consecutive modes. There are three types of orders, namely, fast Q 1, medium Q 2, and slow Q, with stationary ordering costs c 1 =, c 2 = 2, and c = 1, respectively. The demand D 1 for the first period follows uniform distribution over [0, 20], and the demands for the last two periods are deterministic D 2 = D = 20. Holding and baclog costs at the end of each period are given by H 4 (x = { 0 if x 0, 10x if x < 0; H (x = { x if x 0, 4x if x < 0; H 2 (x = { 2x if x 0, 4x if x < 0..1 The Optimal Ordering Policies The optimal cost function for the third period is given by W (y = { 0 if y 20, 60 y if y < 20.

5 The optimal base-stoc level is z 1 = 20. The optimal cost function for period 2 is given by 100 2q1 y 2 if q1 20, y 2 20, W 2 (y 2, q1 = 20 2q1 + y 2 if 20 < y 2 40 q1, 60 y 2 if q1 > 20, y 2 20, 60 + y 2 if y 2 > max{20, 40 q1}. Here, p 2 2 = q 1. The optimal base-stoc levels are ( z1 2, z2 2 = (20, 40. The optimal cost function for period 1 is given by W 1 (y 1,q0 = (4 40 2q 0 y 1 if y 1 10,q 40 0, (a 85 2q 0 6y (y 1 2 if 10 < y 1 min{20, q 0 }, (b 205 2q 0 if 20 < y 1 q 0, (c 150 9q (q 0 2 7y q 0 y (y 1 2 if max{20, q 0 } < y 1 0 q0, (d 105 6q (q 0 2 4y q 0 y (y 1 2 if max{20,0 q0 } < y 1 40 q0, (e q0 + 4y 1 if max{20,40 q0 } < y 1 50 q0, (f q (q 0 2 y q 0 y (y 1 2 if max{20,50 q0 } < y 1 60 q0, (g q0 + 5y 1 if y 1 > max{20,60 q0 }, (h 80 4q (q 0 2 y 1 if y q 0, 40 < q 0 80, (i 210 9q (q 0 2 1y q 0 y (y 1 2 if max{ q 0, q 0 } < y 1 min{20,0 q0 }, (j 165 6q (q y q 0 y (y 1 2 if max{ q 0,0 q 0 } < y 1 min{20,40 q0 }, ( 5 + 2q0 2y (y 1 2 if max{0,40 q0 } < y 1 min{20,50 q0 }, (l 10 q (q 0 2 7y q 0 y (y 1 2 if max{0,50 q0 } < y 1 min{20,60 q0 }, (m 50 + q0 y (y 1 2 if y 1 > max{0,60 q0 }, (n q 0 y 1 if y 1 5 q0,q 0 > 5, (o q (q 0 2 y 1 if y q 0, 80 < q 0 5, (p 165 6q (q y q 0 y (y 1 2 if 5 q0 < y 1 min{0,0 q0 }, (q 5 + 2q0 2y 1 if 0 q0 < y 1 min{0,40 q0 }, (r 10 q (q 0 2 7y q 0 y (y 1 2 if 50 q0 < y 1 min{0,60 q0 }, (s 50 + q0 y 1 if 60 q0 < y 1 0. (t In Fig. 1., we depict the various regions involved in defining W 1 (y 1, q 0 in (4. Here, p2 1 = q 0. There are optimal base-stoc levels for the first two modes, which are given by.2 Discussion (5 q0, 5 if q 0 5, ( z 1, 1 z 1 2 (14 2 = 5 q 0, q 0 if 80 q 0 < 5, ( q 0, 50 + q 0 if 40 q 0 < 80, (10, if 0 q 0 < 40. We focus on analyzing the optimal decision for Q 1. Fig. 2 shows how Q 1 changes for given the reference inventory position z1 2, which does not follow a base-stoc structure (This also 4

6 Figure 1: The cost function for period 1. y 1 = 60 q 0 y 1 = 50 q 0 t s q 0 n y 1 = 40 q0 y 1 = 5 q0 y 1 = q 0 y 1 = q 0 y 1 = q 0 r q o p i a m l j b y 1 = 0 q0 f c d e y 1 g h Figure 2: The function Q 1 (z2 1. Q 1 e f g 20 0 a (,20 c Q 1 (z2 1 b d z 2 1 gives a counterexample to Lemma 1 in Zhang To further explore the reason, we consider the optimal post-order inventory positions in period 1: (z1 1,z1 2,z 1 = (5 ( 10,, 10 if y 1 10, ( y 1,, 10 if 10 < y 1 q 0, ( y 1, y 1 + q0, y 1 + q if q 0 < y 1 0 q0, if 0 q ( y 1, y 1 + q0, 50 if 0 q 0 < y 1 50 q0, 0 < 40 ; ( y 1, y 1 + q0, y 1 + q0 if y 1 50 q0, ( q 0, 50 + q 0, q 0 if y q 0, ( y 1, y 1 + q0, y 1 + q if 1 2 q 0 < y 1 0 q0, ( y 1, y 1 + q0, 50 if 0 q 0 < y 1 50 q0, if 40 ( y 1, y 1 + q0, y 1 + q0 if y 1 50 q0, q 0 < 80 ; ( q 0, 50 + q 0, 50 if y q 0, ( y 1, y 1 + q 0, 50 if q 0 < y 1 50 q 0, ( y 1, y 1 + q 0, y 1 + q 0 if y 1 50 q 0, (5 q 0, 5, 50 if y 1 5 q 0, ( y 1, y 1 + q 0, 50 if 5 q 0 < y 1 50 q 0, ( y 1, y 1 + q 0, y 1 + q 0 if y 1 50 q 0, if 80 q 0 < 5; if q 0 5; 5

7 It seems that the ideal order-up-to level for Q 1 would be 50, if the ordering policy for the slow mode were to follow a base-stoc policy. However, this cannot always be achieved when q0 is low (q 0 80/. Table 2 illustrates the optimal ordering decisions when 0 q 0 < 40/. If y 1 < / q 0, we have to order a positive Q 1 2 in period 2 and 20 units of Q 1. However, the inventory position z1 does not account for the value of Q 1 2. Thus, the optimal decision is not to bring z1 up to 50. When y 1 / q0, there is no fast order placed in period 2. As a result, the optimal inventory position z 1 should be as close to the target level 50 as possible. One may thin that if we modify the reference inventory position for Q 1 to include the anticipated order Q 1 2, then we could restore the base-stoc policy. However, this would not wor in the case of general stochastic demand. Table 2: When 0 q 0 < 40 Period y 1 (, 10] (10, q 0 [ q 0,0 q 0 ] (0 q 0,40 q 0 ] (40 q 0, 50 q 0 ] (50 q 0, 1 z Q y z y 1 y 1 y 1 y 1 y 1 z 1 + q q0 y 1 + q0 y 1 + q0 y 1 + q0 y 1 + q0 y 1 + q0 z 1 2 Q q 0 y 1 q z1 2 y 1 + q0 y 1 + q0 y 1 + q0 y 1 + q0 Q y 1 q0 50 y 1 q0 0 z y 1 + q y 1 + q0 d 1 d 1 y 1 + q0 d 1 y 1 + q0 d 1 y 1 + q0 d 1 y 1 + q0 d 1 2 z EQ (40 y 1 q0 2 (40 y 1 q0, z EQ 2 [max{60 y q0,0}]2 40 Period y 2 When q 0 is lower than 80/, there is a positive probability that Q1 2 > 0. Since the reference inventory position for the type order in period 1 does not tae this order into account, the optimal ordering policy in period 1 is not a base-stoc policy. When q 0 we always have Q 1 2 is greater than 80/, = 0 and the optimal policy in period 1 follows a base-stoc policy. We also observe from our example that when the type orders do not follow base-stoc policy, the order quantity Q 1 is constant 20. However, this observation cannot be generalized. If we modify the holding/baclog cost in period to be H (x = 0.5x 2, then Q 1 does not have this property any longer. The corresponding Q 1 (z2 1 is given in Fig.. 6

8 Figure : Q 1 (z2 1 for nonlinear holding costs. Q i h e (2.47, g f a Q 1 = z Õ z (z2 1 2 c Q 1 (z2 1 b d z Separability and the Base-Stoc Policies The example in the last section shows that the optimal ordering policies for inventory models with more than two modes are no longer base-stoc policies in general. Next, we examine the relationship between an optimal base-stoc policy and the separability of the cost functions. The next result establishes the existence of optimal base-stoc levels for the first two modes in general inventory systems with multiple delivery modes. Proposition 1 The minimand in the right-hand side of ( can be expressed as G 1 (z 1 + G 2 (z 2,..., zn, where G 1(z 1 is convex in z1 and G 2(z 2,..., zn is jointly convex in (z2,..., zn. Let (z 1 such that,..., zn be the minimizing vector in (. Then there exist two base stocs z1 and z2 z 1 = z 1 y, z 2 = z 2 (z 1 + p 2. In general, the cost-to-go function defined in ( is not separable in reference inventory positions (z 2,..., zn. If it were, it can be easily shown that there would exist optimal basestoc levels in period for type and higher orders. This may happen in some very special cases. One such example is Fuuda s (1964 model of three consecutive modes where orders are placed in every other period. There the resulting cost-to-go function is separable in the reference inventory positions, and the optimal ordering policy is a base-stoc policy. Furthermore, the follwoing proposition represents a generalization of Fuuda s (1964 result. Proposition 2 If the first mode in period + 1 is not used, then the first three modes in period have optimal base-stoc levels. 7

9 Remar 1 Note that the proof cannot be generalized for more than three modes. That is, if the first two modes in period + 1 are not used, then the fourth mode in period may not have an optimal base stoc. 5. Concluding Remars We have shown that the optimality of a base-stoc policy is closely related to the structure of the cost-to-go function. Since the cost-to-go function for each period is separable in the post-order inventory positions for the first two modes, there exist optimal base-stoc levels for the first two modes. In general, the cost-to-go function for a given period is not separable in post-order inventory positions of higher modes. Thus, the optimal ordering decision for type or higher order may not have a base-stoc structure. Our discussion shows that the base-stoc policy fails to be optimal even under very restrictive conditions, e.g., uniform demand, linear holding costs. The intuitive reason is that the optimal post-order inventory position for a mode of type or higher cannot, in general, anticipate the future order quantities. Moreover, the dependence of the optimal order quantities on the reference inventory position and in-transit orders may be quite complex. References Feng, Q., G. Gallego, Sethi, S. P., H. Yan, and H. Zhang A Periodic Review Inventory Model with Three Consecutive Delivery Modes and Forecast Updates. J. Optimiz. Theory Appl ,,,, Optimality and Nonoptimality of Base-stoc Policy in Inventory Model with Multiple Delivery Modes. (the unabridged version of this paper. University of Texas at Dallas, Richardson, TX. Fuuda, Y Optimal Policies for the Inventory Problem with Negotiable Leadtime. Management Sci Lawson, D. G. and E. L. Porteus Multistage Inventory Management with Expediting. Opns. Res Muharremoglu, A. and J. N. Tsitsili Dynamic Leadtime Management in Supply Chains, woring paper. Graduate School of Business, Columbia University, New 8

10 Yor, NY Neuts, M An Inventory Model with an Optional Time Lag. J. SIAM Sethi, S. P., H. Yan and H. Zhang Peeling Layers of an Onion: A periodic review inventory model with multiple delivery models and forecast updates. J. Optimiz. Theory Appl Whittemore, A. S. and S. C. Saunders Optimal Inventory under Stochastic Demand with Two Supply Options. SIAM J. APPL. MATH Zhang, V. L Ordering Policies for an Inventory System with Three Supply Modes, Naval Res. Logist Appendix The proof of Proposition 1 uses the following lemma. Lemma 1 Suppose G(x 1,..., x n is jointly convex in (x 1,..., x n. Define G o (x 1,..., x = min {G(x 1,..., x n x j x j 1 +a j for j = +1,..., n} for = 1,..., N. Then G o (x 1,..., x is jointly convex in (x 1,..., x. Proof. We show that G o (x 1,..., x is jointly convex by induction on. Clearly, when = N, G o (x 1,..., x N = G(x 1,..., x N is jointly convex. Suppose that G o (x 1,..., x +1 is jointly convex in (x 1,..., x +1. Then, G o (x 1,..., x = min{g(x 1,..., x N x j x j 1 + a j for j = + 1,..., N} = min{g o (x 1,..., x +1 x +1 x + a +1 }. Now define x +1 (x 1,..., x = arg min x +1 G o (x 1,..., x +1, then G o (x 1,..., x = G (x o 1,..., x +1 (x 1,..., x (x + a +1. Clearly, G (x o 1,..., x +1 (x 1,..., x is convex since the lower envelope of a convex function is convex. Thus, G o (x 1,..., x is convex. Proof of Proposition 1. The minimand in the right-hand side of ( is easily seen to be separable in z 1 and (z2,...zn. By Lemma 1, we have W (y, p 2,..., pn 1 9

11 = min{g 1 (z 1 + G 2(z 2,..., zn z1 y, z j zj 1 + p j, j = 2,..., N} = min{g 1 (z 1 + Go 2 (z2 z1 y, z 2 z1 + p2 }. Denote z 1, z2 and ẑ1 as the unconstrained minimizer to G 1(z 1, Go 2(z 2, and G 1(z 1 + G o 2(z 1 + p2, respectively. Let z1 = z1 ẑ1 and z2 = z2. Also note that } { } min {G 1 (z 1 + Go2 (z2 = min G 1 (z z 1 y z 1 z 2 y 1 + Go 2 ( z 2 (z1 + p2. z1 +p1 The function inside min on the right-hand side is convex in z 1. We consider two cases. Case 1: z 1 + p2 z2. First note that in this case z1 ẑ1, z1 = z 1. We have z1 + p1 z2. Thus, for any z 1 y, G 1 ( z 1 y + G o 2 ( ( z 2 (y z 1 + p 2 G 1 (z 1 y + G o 2 z 2 (y z 1 + p 2. To see the last inequality, we note that the right-hand side is convex in z 1. If y z 1, the right-hand side is minimized at z 1 = z1 = z1. If y > z 1, the right-hand side is minimized at z 1 = y. Hence, the minimizer is z 1 = z1 y = z 1 y and z 2 = z2 (z1 + p2. Case 2: z 1 + p2 > z2. In this case the minimizer must be such that z1 + p2 = z2. Note that ẑ 1 < z1 and z2 < ẑ1 + p2. So z1 = ẑ1 and z2 = z2 < ẑ1 + p2. Thus, for any z1 y, G 1 (ẑ 1 y + G o 2 (ẑ1 y + p 2 = G 1(ẑ 1 y + G o 2 G 1 (z 1 y + G o 2 ( z 2 (z1 + p2. ( z 2 (ẑ1 y + p 2 Similar to Case 1, if y ẑ 1, then the right-hand side is minimized at ẑ1. Otherwise, it is minimized at y. Hence, the minimizer is z 1 z 2 (z1 + p2. = ẑ 1 y = z 1 y and z 2 Proof of Proposition 2. The cost function for period can be written as J (z 1,...zN = c1 (z1 y + N j=2 c j (zj zj 1 p j + EH +1(z 1 D +EW +1 (z 2 D, z z2,..., zn zn 1 N = c 1 (z1 y + c j (zj zj 1 p j + EH +1(z 1 D + j=2 = z 1 + p2 = 10

12 +E min z +1 z4 D, z j +1 zj zj+1 z j, +EH +2 (z 2 D D +1 { c +1 (z +1 z4 + D + N j=4 c j +1 (zj +1 zj 1 +1 zj+1 + z j +EW +2 (z D D +1, z +1 z + D + D +1,..., z N +1 zn 1 +1 }. Clearly, the last expression is separable in (z 1, z2, z. Hence, the result follows from Proposition 5 of Feng et al. (