مدل مقدار تولید اقتصادی قطعی با تقاضا و هزینه زمان متغیر

In today’s time-based competition, the unit cost of a high-tech product declines significantly over its short product life cycle. Consequently, in this paper, we relax the traditional economic production quantity model to allow for time-varying cost. We then prove that the optimal production schedule uniquely exists. In addition, we also show that the total cost is a convex function of the number of replenishments, which reduces the search for the optimal solution to finding a local minimum. Furthermore, we characterize the influences of both demand and cost over the length of production run time and the economic production quantity.

The classical economic production quantity (EPQ) model is widely used principally because it is so simple to use and apply. However, a major problem in using the EPQ is that it assumes not only a constant demand rate but also a fixed unit purchasing cost. As we know from a product life cycle, the demand rate remains stable only in the maturity stage. Moreover, in time-based competition today, the unit cost of a high-tech product declines significantly over its short product life cycle. For example, the cost of a personal computer drops constantly as shown in Lee et al. [1]. Therefore, using the EPQ formulation in stages other than the maturity stage or for a product with short product life cycle will cause varying magnitudes of error. In addition, the cost of purchases as a percentage of sales is often substantial (52% for all industry) as shown in Heizer and Render [2]. Consequently, adding the purchasing strategy into EPQ model is vital. For example, when the gasoline price is costly going up, we adjust it by driving less (or buying fuel-saving cars), and vice versa. Similarly, a manufacturer needs to adjust its production strategy when the cost is fluctuating with time.

In this paper, we assume that not only the demand function but also the purchase cost is positive and fluctuating with time. Consequently, our model is suitable for any given time horizon in any product life cycle including high-tech products. In Section 3, we prove that the optimal production schedule not only exists but also is unique. In addition, we also show that the total relevant cost associated with the proposed inventory system is a convex function of the number of replenishments. Hence, the search for the optimal number of replenishments is simplified to find a local minimum. Furthermore, we provide an accurate starting value for searching the optimal replenishment number, which is more computationally efficient than enumerative methods. In Section 4, we characterize the influences of both demand and unit purchase cost over the length of production run time and the economic production quantity. Finally, in Section 5, the computationally numerical results not only profoundly justify our theoretical results but also clearly show that our proposed model has the potential to operate significantly cheaper than the traditional EPQ model.