محاسبه سیاست های موجودی چرخه تکمیل دوباره غیر ثابت تحت عرضه کننده تصادفی کالا

In this paper we address the general multi-period production/inventory problem with non-stationary stochastic demand and supplier lead-time under service level constraints. A replenishment cycle policy (Rn,Sn) is modeled, where Rn is the nth replenishment cycle length and Sn is the respective order-up-to-level. We propose a stochastic constraint programming approach for computing the optimal policy parameters. In order to do so, a dedicated global chance-constraint and the respective filtering algorithm that enforce the required service level are presented. Our numerical examples show that a stochastic supplier lead-time significantly affects policy parameters with respect to the case in which the lead-time is assumed to be deterministic or absent.

An interesting class of production/inventory control problems is the one that considers the single location, single product case under stochastic demand. One of the well-known policies that can be adopted to control such a system is the “replenishment cycle policy’ (R,S). Under the non-stationary demand assumption this policy takes the form (Rn,Sn), where Rn denotes the length of the nth replenishment cycle, and Sn the order-up-to-level value for the nth replenishment. This easy to implement inventory control policy yields at most 2N policy parameters fixed at the beginning of an N-period planning horizon. For a discussion on inventory control policies see Silver et al. (1998). The replenishment cycle policy provides an effective means of damping the planning instability. Furthermore, it is particularly appealing when items are ordered from the same supplier or require resource sharing. In such a case all items in a coordinated group can be given the same replenishment period. Periodic review also allows a reasonable prediction of the level of the workload on the staff involved and is particularly suitable for advanced planning environments. For these reasons, as stated by Silver et al. (1998), (R,S) is a popular inventory policy. Due to its combinatorial nature, the computation of (Rn,Sn) policy parameters is known to be a difficult problem to solve to optimality. An early approach proposed by Bookbinder and Tan (1988) is based on a two-step heuristic method. Tarim and Kingsman, 2004 and Tarim and Kingsman, 2006 and Tempelmeier (2007) propose a mathematical programming approach to compute policy parameters. Tarim and Smith (2008) give a computationally efficient constraint programming formulation. An exact formulation and a solution method are presented in Rossi et al. (2008). All the above mentioned works assume either zero or a fixed (deterministic) supplier lead-time (i.e., replenishment lead-time). However, the lead-time uncertainty, which in various industries is an inherent part of the business environment, has a detrimental effect on inventory systems. For this reason, there is a vast inventory control literature analysing the impact of supplier lead-time uncertainty on the ordering policy (Whybark and Williams, 1976, Speh and Wagenheim, 1978 and Nevison and Burstein, 1984). A comprehensive discussion on stochastic supplier lead-time in continuous-time inventory systems is presented in Zipkin (1986). Kaplan (1970) characterises the optimal policy for a dynamic inventory problem where the lead-time is a discrete random variable with known distribution and the demands in successive periods are assumed to form a stationary stochastic process. Since tracking all the outstanding orders through the use of dynamic programming requires a large multi-dimensional state vector, Kaplan assumes that orders do not cross in time and supplier lead time probabilities are independent of the size/number of outstanding orders (for details on order-crossover see Hayya et al., 1995). The assumption that orders do not cross in time is valid for systems where the supplier production system has a single-server queue structure operating under a FIFO policy. Nevertheless, there are settings in which this assumption is not valid and orders cross in time. This has been recently investigated in Hayya et al. (2008), Bashyam and Fu (1998) and Riezebos (2006). As Riezebos underscores, the types of industries that have a higher probability of facing order crossovers are either located upstream in the supply chain, or use natural resources, or order strategic materials from multiple suppliers or from abroad. In a case study, he showed that the potential cost savings realized by taking order crossovers into account were approximately 30%. Unfortunately, he remarks, modern ERP systems are not able to handle order crossovers effectively. In a recent work, Babaï et al. (2009) analyze a dynamic re-order point control policy for a single-stage, single-item inventory system with non-stationary demand and lead-time uncertainty. To the best of our knowledge, there is no complete or heuristic approach in the literature that addresses the computation of (Rn,Sn) policy parameters under stochastic supplier lead time and service level constraints. Computing optimal policy parameters under these assumptions is a hard problem from a computational point of view. We argue that incorporating both a non-stationary stochastic demand and a stochastic supplier lead time—without assuming that orders do not cross in time—in an optimization model is a relevant and novel contribution. In this work, we propose a stochastic constraint programming (Walsh, 2002) model for computing optimal (Rn,Sn) policy parameters under service level constraints and stochastic supplier lead times. In stochastic constraint programming, complex non-linear relations among decision and stochastic variables—such as the chance-constraints that enforce the required service level—can be effectively modeled by means of global chance-constraints ( Hnich et al., 2009). Examples of global chance-constraints applied to inventory control problems can be found in Rossi et al. (2008) and Tarim et al. (2009). Our model incorporates a dedicated global chance-constraint that enforces, for each replenishment cycle scheduled, the required non-stockout probability. The model is tested on a set of instances that are solved to optimality under a discrete stochastic supplier lead time with known distribution. The paper is organized as follows. In Section 2 we provide the formal definition of the problem and we discuss the working assumptions. In Section 3 we provide a deterministic reformulation for the chance-constraints that enforce the required service level. In Section 4 we introduce stochastic constraint programming and we discuss how it is possible to embed the deterministic reformulation of the chance-constraints within a global chance-constraint. This global chance-constraint is then enforced in the stochastic constraint programming model for computing the optimal policy parameters. In Section 5 we present our computational experience on a set of instances. Finally, in Section 6, we draw conclusions.

A novel approach for computing replenishment cycle policy parameters under non-stationary stochastic demand, stochastic lead time and service level constraints has been presented. The approach is based on SCP and it employs a dedicated global chance-constraint in order to enforce the required service level in each period. The assumptions under which we developed our approach for the stochastic lead time case proved to be less restrictive than those commonly adopted in the literature for complete methods. In particular we faced the problem of order-crossover, which is a very active research topic. Our approach merged well-known concepts such as deterministic equivalent modeling of chance-constraints and scenario-based modeling. Our computational experience showed that a stochastic supplier lead time may significantly impact the structure and the cost of the optimal replenishment cycle policy with respect to the case in which the lead time is deterministic or absent. In our future research, we aim to develop dedicated cost-based filtering algorithms able to significantly speed up the search for the optimal policy parameters.