An EPQ Model with Unit Production Cost and Set-Up Cost as Functions of Production Rate

Extensive research has been devoted to economic production quantity (EPQ) problem. However, no attention has been paid to problems where unit production and set-up costs must be considered as functions of production rate. In this paper, we address the problem of determining the optimal production quantity and rate of production in which unit production and set-up costs are assumed to be continuous functions of production rate. Based on the traditional economic production quantity (EPQ) formula, the cost function associated with this model is proved to be nonconvex and a procedure is proposed to solve this problem. Finally, utility of the model is presented using some numerical examples and the results are analyzed. 1. Introduction The economic production quantity (EPQ) model has been widely used in practice because of its simplicity. However, there are some drawbacks in the assumption of the original EPQ model and many researchers have tried to improve it with different viewpoints. Recently, the classical EPQ model has been generalized in many directions. Some authors extended the EPQ model by incorporating the effect of learning in setups and process quality. Also, set-up time reduction on production run length and varying parameters have received significant attention. The relationship between set-up cost and production run length is also influenced by the learning and forgetting effects. The effect of learning and forgetting in setups and in product quality is investigated by Jaber and Bonney [1]. Porteus studied the effect of process deterioration on the optimal production cycle time [2]. Darwish generalized the EPQ model by considering a relationship between the set-up cost and the production run length [3]. Jaber investigated the lot sizing problem for reduction in setups with reworks and interruptions to restore the process to an “in-control’’ state [4]. Unlike the model presented by Khouja [5], he considered that the set-up cost and defect rate decrease as the number of restoration activities increases. Afshar-Nadjafi and Abbasi considered an EPQ model with depreciation cost and process quality cost as continuous functions of time [6]. Freimer et al. studied the effect of imperfect yield on EPQ decisions. They considered set-up cost reductions and process quality improvements as types of investments in the production processes [7]. Furthermore, the classical EPQ model has been investigated in many other ways; for example, the effect of varying production rate on the EPQ model was investigated by Khouja [8]. Huang introduced the EPQ model under