Optimal control policies for an inventory system with commitment lead time

Abstract

We consider a firm which faces a Poisson customer demand and uses a base-stock policy to replenish its inventories from an outside supplier with a fixed lead time. The firm can use a preorder strategy which allows the customers to place their orders before their actual need. The time from a customer's order until the date a product is actually needed is called commitment lead time. The firm pays a commitment cost which is strictly increasing and convex in the length of the commitment lead time. For such a system, we prove the optimality of bang-bang and all-or-nothing policies for the commitment lead time and the base-stock policy, respectively. We study the case where the commitment cost is linear in the length of the commitment lead time in detail. We show that there exists a unit commitment cost threshold which dictates the optimality of either a buy-to-order (BTO) or a buy-to-stock strategy. The unit commitment cost threshold is increasing in the unit holding and backordering costs and decreasing in the mean lead time demand. We determine the conditions on the unit commitment cost for profitability of the BTO strategy and study the case with a compound Poisson customer demand. KEYWORDS advance demand information, commitment cost, inventory management, preorder strategy 1 This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Optimal control policies for assemble-to-order systems with commitment lead time

“…From Lemma 2, we know that for all y 2 N 0 ; w 2 W; P 1 ðy; s w 2 Þ is non-decreasing in y and w. As it is shown in Ahmadi et al (2019), having P 1 ðy; s w 2 Þ being non-decreasing in both y and w implies that s w 1 is non-increasing in w. Combining the facts that (i) the inequalities in Theorem 1 hold for the optimal base-stock levels and (ii) F 2 ðyÞ and P 1 ðy; s w 2 Þ are nondecreasing in y and w for all y 2 N 0 ; w 2 W implies that when we increase the commitment lead time, each component's optimal basestock level either stays the same or it decreases by one unit, i.e., the jump size is 1.…”

“…We would like to note that F 2 ðyÞ is the cumulative distribution function of a Poisson random variable with mean l 2 ¼ kL 2 ; where L 2 depends on w. Hence, although F 2 ðyÞ seems to be a function of y only, it is also a function of w by definition. For a single location system Ahmadi et al (2019) prove that F 2 ðyÞ is non-decreasing in y and w. This result is enough to conclude that s w 2 is non-increasing in w. We refer to Ahmadi et al (2019) for the details.…”

“…We consider continuous-review base-stock policies, deterministic replenishment lead times and Poisson customer demand. The study that is most closely related to ours is Ahmadi et al (2019). Different from us, the authors study a single-location inventory system.…”

“…Assuming that the commitment lead time is the same for all customers, the authors prove the optimality of a base-stock policy. A similar setting has been studied in Ahmadi et al (2019). Assigning a cost to commitment lead time, the authors characterize the optimal preorder strategy and the corresponding optimal replenishment strategy.…”

“…As stated above, the commitment lead time is the time difference between the customer order and the corresponding demand, i.e., the customer places her order w time units before her arrival, i.e., demand. According to Hariharan and Zipkin (1995) and Ahmadi et al (2019) a single-location inventory system with commitment lead time is equivalent to a singlelocation inventory system where the commitment lead time is subtracted from the component replenishment lead time. The same is valid for the ATO system under consideration.…”