Source: https://slidex.tips/download/customers-with-positive-demand-lead-times-place-orders-in-advance-of-their-needs
Timestamp: 2019-04-21 00:24:27+00:00

Document:
A customer who places his order l periods in advance is said to have a demand lead time l.
Notice that under centralized control, information sharing and coordination issues are not our concern. Reviews of these topics are provided by Cachon (2001) and Chen (2001).
A term coined by Eppen and Schrage (1981).
We use the term increasing in the weak sense; that is, increasing means nondecreasing.
The authors discuss this in the context of a multi-item system.
detailed review on the use of demand information, we refer the reader to Gallego and Özer (2002). The rest of this paper is organized as follows. In §2, we introduce the demand model and the dynamic programming formulation. In §3, we apply the relaxation approach to obtain a lower bound model. We establish the optimality of a state-dependent basestock policy ﬁrst for the L = 0 case followed by L > 0. We do this to separate the effect of advance demand information from that of risk pooling. In §4, we propose a heuristic to control ordering and shipping decisions for the system. In §5, we conduct a numerical study of several problem instances. First, we report on the performance of the heuristic. Next, we provide insights into a set of design problems and the beneﬁts of advance demand information. In §6, we apply the restriction approach and state our assumptions to provide a closed-form solution for the system-wide basestock level. In §7, we conclude and suggest directions for future research. All the proofs are in the appendix.
ot s ≡ 0 for s ≥ t + N because we do not observe demand information beyond the information horizon.
+ EJt+1 xt+1 vt+1 ot+1  (3) j j j j j j j where Gt yt = ct − ct+1 yt + G˜ t yt and JT +1 · · · ≡ 0 and the expectation is with respect to the demand j vectors dt . The formal construction of this dynamic program is similar to Gallego and Özer (2001).
We remark that if the inventory manager wishes to incorporate advance demand information only up to the next l + 1 periods for all retailers, then the state space would be similar to that of classical distribution systems (as in Federgruen and Zipkin 1984a). All proposed heuristics to solve the classical distribution system and the reported performance measures would apply to this special case. The inventory manager would only need to update his inventory position to account for the observed part and use modiﬁed inventory positions instead. From this point, on we consider only the problem where N > l + 1.
For two vectors Ot and Ot , Ot ≥ Ot if and only if each element of Ot is greater than or equal to the corresponding element of Ot .
Lemma 1. V˜t Xt4  Ot = Vt Xt  vt  Ot +rt Xt  vt  Ot  This result states that the algorithm in (6) is similar to the one in (4), except the constant term rt , which is deﬁned in the appendix. This term is independent of order and shipment decisions, hence it can be dropped for optimization purposes. We remark that Theorem 2 also applies to the above dynamic programming algorithm.
This allocation is referred to as myopic for two reasons. First, it minimizes the total expected cost of managing the retailer inventories at the end of period t + l (the period when the allocations are available for the retailer) and ignores the impact of this allocation on other periods. Second, it does not take into consideration the observed demand information beyond the retailers’ lead time (unlike the lower bound). Notice that this is a feasible solution to the original problem, hence it is an upper bound. In our numerical study, we use a greedy algorithm both to solve the myopic allocation and to evaluate Rt Yt deﬁned in Equation (5). To solve the allocation problem, we start with yti = xti and allocate one unit at a time to the jth retailer with the smallest current j value of the ﬁrst difference (that is, minj Gt y + 1 − j Gt y ) until all wt−L is allocated. To evaluate Rt Yt , J j j we start with Rt  j=1 ∗ yt , where ∗ yt is the minimum j of Gt · and allocate the difference Yt − Yt∗ , if positive, one unit at a time to the jth retailer with the smallest current value of the ﬁrst difference. Otherwise, we reduce the amount allocated one unit at a time from the jth retailer with the largest current value of the j j ﬁrst difference (that is, maxj Gt y − Gt y − 1 ). Similar algorithms are used by Fox (1966), Federgruen and Zipkin (1984a, b), Zipkin (2000), and Aviv and Federgruen (2001a). We remark that a capacitated distribution system, in which a warehouse is allowed to process up-to a limited quantity, can be solved by using the replenishment policy in Özer and Wei (2001) together with a myopic allocation.
information imbalance that this distribution system can face. We also observe that the error term decreases as the number of balanced retailers in the system increases. One ﬁnal observation is that as we increase the mean which is also the variance, the optimality gap tends to increase. Together with the numerical results reported in Aviv and Federgruen (2001a) and Federgruen and Zipkin (1984a, b), we expect an increase in the error term if we use demand distributions with larger coefﬁcient of variation. These observations suggest that the heuristic should perform fairly well for problems where 1 the retailer lead time is short relative to the supplier lead times, 2 the number of balanced retailers in the system is large, 3 the planning horizon is long, and 4 the demand variation is modest. 5.2.
t + 3) is 886%. Notice also that the system maintains lower inventory levels (zero inventory) as customers place their orders (3 > L + l periods) in advance. In this sense, advance demand information is a substitute for inventory. Based on such an analysis, the inventory manager can decide whether to invest in strategies to obtain advance demand information. Notice that advance demand information allows a fundamental bridge between build-to-stock and build-to-order systems.
narios. For a ﬁxed system lead time, the difference between any two curves in Figure 3(a) and (b) gives us the reduction in inventory-related costs and basestock levels due to advance demand information. These savings can be seen as the value of advance demand information. We also observe the reduction in base-stock levels and the costs with the system lead time. Both ﬁgures depict the joint effect of total lead time and advance demand information on the system performance.
Note. N = 3 h = 1 c = 10 p = 19.
penalty cost as reducing the retailers’ lead time by one period, and 2 has more gains than reducing supplier lead time by one period. The gain due to risk pooling and advance demand information, together, outweighs the gain from reducing the supplier lead time. Another example for the later phenomenon can be observed in Figure 3(a). While B has one period longer supply lead time than A, the inventory manager in B obtains demand information one period earlier than A’s manager. The difference between the costs in A and B can be attributed to the gains due to risk pooling. For a continuous review single location inventory control problem, Hariharan and Zipkin (1995, p. 1600) show that: “Demand lead times are, in a precise sense, the opposite of supply lead times.” Our numerical results suggest that this result carries over to the periodic review distribution system as an UB on the cost; that is, one would gain even more compared to the single location problems due to risk pooling. 5.5.
Next, we provide an explicit solution for the systemwide order-up-to level. Our aim is to gain further insights for the joint role of risk pooling and advance demand information. To obtain this closed-form solution, we assume that (1) unit holding and penalty j cost is the same for all retailers, (2) dt s follows a nor2 for mal distribution with mean 8s−t and variance 9s−t s ∈ t     t + N , (3) allocation assumption (introduced by Eppen and Schrage 1981) holds,10 and (4) we restrict the policy space to the class of stationary order-up-to S policies and allocate based on the solution of the myopic problem in Equation (7).
Allocation Assumption. In each allocation period t, the warehouse receives a sufﬁcient amount so that it can bring all retailers to an equal fractile level on their respective demand distributions; that is, stock-out probability is the same at all retailers.
by pooling demand and gaining through statistical economies of scale (but never eliminating it). Given all four assumptions, Equation (8) provides a simple closed-form solution for the system-wide base-stock level. This is a nonadaptive control mechanism unlike the solution provided by the dynamic program.
The author thanks an anonymous associate editor and the referees. The discussions during my presentations at the University of California–Berkeley, Columbia University, Stanford University were beneﬁcial. The suggestions and comments have considerably improved the exposition of this article.
that the relaxation yields a dynamic programming (DP) recursion with a state space that aggregates the state variables across the retailers. We prove this ﬁrst for the terminal period T . To distinguish the DP under relaxation from the original one given in Equation (3), we refer to it as Jt and JT +1 ≡ 0. Note that for t = T , ot is a vector of zeros because we do not accept orders beyond T + l by assumption.
The inequality is due to the convexity of Rt x . Note that the smallest minimizer of Ht x − 3 Ot is nothing but yt∗ Ot + 3. Also yt∗ Ot + 3e1 is the smallest minimizer of Ht · Ot + 3e1 , which is convex from Part 1. From inequality (9) and Remark 1, we conclude that yt∗ Ot + 3e1 ≤ yt∗ Ot + 3. Note that   Ht yt∗ Ot  Ot  xt < yt∗ Ot  (10) Vt x Ot = −ct0 x +  H x O  x ≥ yt∗ Ot  t t Next, we prove =Vt x − 3 Ot ≤ =Vt x Ot + 3e1 for all t. To do this, we divide the domain of these two functions into four mutually exclusive, collectively exhaustive cases. Case 1. If x − 3 < yt∗ Ot and x < yt∗ Ot + 3e1 , then =Vt x − 3 Ot = 0 −ct = =Vt x Ot + 3e1 . Case 2. If x − 3 < yt∗ Ot and x ≥ yt∗ Ot + 3e1 , then =Vt x − 3 Ot = −ct0 ≤ −ct0 +=Ht x Ot +3e1 = =Vt x Ot +3e1 . The inequality is due to the fact that Ht · Ot +3e1 is increasing for x ≥ yt∗ Ot +3e1 (which is due to Parts 1 and 2). Case 3. The case x − 3 ≥ yt∗ Ot and x < yt∗ Ot + 3e1 is not possible because we already showed that yt∗ Ot + 3e1 ≤ yt∗ Ot + 3. Case 4. If x − 3 ≥ yt∗ Ot and x ≥ yt∗ Ot + 3e1 , then =Vt x − 3 Ot = 0 −ct + =Ht x − 3 Ot ≤ −ct0 + =Ht x Ot + 3e1 = =Vt x Ot + 3e1  The inequality is due to Equation (9).
Aviv, Y. 2002. A time series framework for supply chain inventory management. Oper. Res. Forthcoming. , A. Federgruen. 2001a. Capacitated multi-item inventory systems with random and seasonally ﬂuctuating demands: Implication for postponement strategies. Management Sci. 47 512–531.
, . 2001b. Design for postponement: Apprehensive characterization of its beneﬁts under unknown demand distributions. Oper. Res. 49 578–598. Billington. C., J. Amaral. 1999. Investing in product design to maximize proﬁtability through postponement. http://www.ascet. com. Cachon, G. 2001. Supply chain coordination with contracts. Working paper, University of Pennsylvania, Philadelphia, PA. Chen, F. 2000. Market segmentation, advance demand information and supply chain performance. Manufacturing Service Oper. Management 3(1) 53–67. . 2001. Information sharing and supply chain coordination. Working paper, Columbia University, New York. Dell, M. 2000. Direct from Dell: Strategies that Revolutionized an Industry. Harper Collins, New York. Eppen, G., L. Schrage. 1981. Centralized ordering policies in a multi-warehouse system with lead times and random demands. L. Schwarz, ed. Multi-Level Production/Inventory Control Systems: Theory and Practice. North-Holland, Amsterdam, The Netherlands, 51–69. Erkip, N., W. H. Hausman, S. Nahmias. 1990. Optimal centralized ordering policies in multi-echelon inventory systems with correlated demands. Management Sci. 36 381–392. Federgruen, A. 1992. Centralized planning models for multiechelon inventory systems under uncertainty, Chapter 4. S. Graves, A. Rinnooy Kan, P. Zipkin, eds. Handbook Oper. Res. Management Sci., North-Holland, Amsterdam, The Netherlands, 133–173. , P. Zipkin. 1984a. Approximations of dynamic, multi location production and inventory problems. Management Sci. 30 69–84. , . 1984b. Allocation policies and cost approximations for multi-location inventory systems. Naval Res. Logist. Quart. 31 97–129. , . 1984c. Computational issues in an inﬁnite horizon multi-echelon inventory model. Oper. Res. 32 818–832. Fox, B. 1966. Discrete optimization via marginal analysis. Management Sci. 4 36–153. Gallego, G., Ö. Özer. 2001. Integrating replenishment decisions with advance demand information. Management Sci. 47 1344–1360. , . 2002. Optimal use of demand information in supply chain management, Chapter 5. J. Song, D. Yao, eds. Supply Chain Structures: Coordination, Information and Optimization, Kluwer Academic Publishers, 119–160.
206–222. . 1980. Teaching notes on inventory theory. Stanford University, Stanford, CA. Zipkin, P. 1982. Exact and approximate cost functions for product aggregates. Management Sci. 28 1002–1012. . 2000. Foundations of inventory management. The Irwin McGraw-Hill Series, Boston, MA.
Accepted by Fangruo Chen; received February 23, 2001. This paper was with the author 8 months for 2 revisions.
Report "Customers with positive demand lead times place orders in advance of their needs. A"
Download "Customers with positive demand lead times place orders in advance of their needs. A"
Share & Embed "Customers with positive demand lead times place orders in advance of their needs. A"

References: §2
 §3
 §4
 §5
 §6
 §7