تصمیم گیری های تعیین اندازه دسته تولید چندگانه با عملکرد هندسی قطع شده و زمان تولید متغیر

This study examines a multiple lot-sizing problem for a single-stage production system with an interrupted geometric distribution, which is distinguished in involving variable production lead-time. In a finite number of setups, this study determined the optimal lot-size for each period that minimizes total expected cost. The following cost items are considered in optimum lot-sizing decisions: setup cost, variable production cost, inventory holding cost, and shortage cost. A dynamic programming model is formulated in which the duration between current time and due date is a stage variable, and remaining demand and work-in-process status are state variables. This study then presents an algorithm for solving the dynamic programming problem. Additionally, this study examines how total expected costs of optimal lot-sizing decisions vary when parameters are changed. Numerical results show that the optimum lot-size as a function of demand is not always monotonic.

Multiple lot-sizing production-to-order (MLPO) problems have been studied for several decades (Bowman, 1955). Such problems typically arise from variations in production yield. Consider a production system with an uncertain process yield. To fulfill a particular customer demand, lots may need to be released several times to minimize total expected costs. The MLPO problem is to determine the optimal lot-size for each possible lot release. This study describes and formulates a single-stage MLPO problem with one salient feature—uncertain production lead-time. According to Yano (1987), this feature may arise due to many factors such as unreliable vendors, unreliable transportation time, job queuing, machine breakdowns, and rework. Uncertain lead-time characteristic has seldom been considered in MLPO studies; although it has been examined in production control studies (Hsu, Wee, & Teng, 2007). In this study, we assume production lead-time is a random variable; the probability for one period is p and that for two periods is 1 − p. In the MLPO problem, process yield follows an interrupted geometric (IG) distribution. The delivery agreement includes due dates; that is, customers will not accept products after delivery due dates, and salvage values of products are negligible. In contrast, finished goods produced ahead of the due date become inventory and incur holding costs. The following cost items are included: setup cost, variable production cost, inventory holding cost, and shortage cost. An example of the MLPO problem in this study is a process of drawing special steel coils. The manufacturing process has two operations: pickling and wire drawing. The pickling operation removes rust from steel coils. The processing time required for pickling a steel coil varies. In practice, a steel coil undergoes one or two pickling operations depending on the duration the coil has been in air. The drawing operation reduces the size of the input coil. Drawing speed is very fast. All coils in a lot are inspected when the whole lot is complete. The drawing operation involves a die that is worn gradually over time. When this die is excessively worn, the output does not meet specifications. This implies that the integrated drawing process follows an IG distribution, and production lead-time for a lot from release to output takes one or two periods. Special steels are customized products that in most cases cannot be sold to other customers. Thus, we assume product salvage value is negligible. This study develops a dynamic programming (DP) approach to solve the MLPO problem. Several lemmas are proposed to reduce the DP problem solution space. Numerical experiments show that the optimum lot-size, as a function of demand, is not necessarily monotonic. This study experimentally investigated how total expected costs of optimal lot-sizing decisions vary when various parameters change. The remainder of this paper is organized as follows. A literature review is given in Section 2. Section 3 presents the MLPO problem as a DP model by including a simple example to facilitate understanding the formulation. Lemmas for reducing the DP solution space are presented in Section 4. An algorithm for solving the DP is presented in Section 5. Numerical examples are given in Section 6. Conclusions are provided in Section 7.

This study addresses a new single-stage MLPO problem, which is distinguished by its inclusion of one salient feature—production lead time is uncertain with two possible outcomes. That is, production lead time is either one or two periods. Such a problem has appeared in various production processes, such as when the drawing of steel coils; however, it has scarcely been studied in literature. This study formulates the MLPO problem as a dynamic problem and examines its properties via numerical experiments. Some properties of decision variables (optimal lot-sizes) are summarized as follow. First, the optimal lot-size at any period is less than or equal to remaining demand, as proved in Lemma 1. Second, optimal lot-size as a function of demand is not necessarily monotonic. Third, optimal lot-size with variable lead-time tends to be larger than that with a fixed lead-time. Properties of decision parameters T and p are also summarized. Total cost appears to decreases with T, implying that the unit selling price can be lowered when customers accept an extended lead-time. While the value ofp is large enough, the higher is p, the lower the total expected cost tends to be. This implies that a production system would be more cost-competitive if its production lead-time could become shorter, in terms of probability. With the proposed DP model, this study determined the lowest total expected cost for any production scenario, and in turn determined the appropriate quoted price. If salvage costs for after-due products are not negligible, some lemmas in this work may not be valid. Therefore, one possible extension is to develop an MLPO model that includes substantial salvage costs. Another extension is to investigate the MLPO problem with more than two possible outcomes in production lead-time. The proposed approach appears to be applicable to such an extension; however, formulating and solving a relatively much more complex DP problem is challenging. Additionally, some other extensions include investigating different probability distributions for modeling process yield and a scenario of a multiple-stage production system.