The Fuzzy Economic Order Quantity Problem with a Finite Production Rate and Backorders

The track of developing Economic Order Quantity (EOQ) models with uncertainties described as fuzzy numbers has been very lucrative. In this paper, a fuzzy Economic Production Quantity (EPQ) model is developed to address a specific problem in a theoretical setting. Not only is the production time finite, but also backorders are allowed. The uncertainties, in the industrial context, come from the fact that the production availability is uncertain as well as the demand. These uncertainties will be handled with fuzzy numbers and the analytical solution to the optimization problem will be obtained. A theoretical example from the process industry is also given to illustrate the new model. 1. Introduction The earliest models of batch-production were derived from the basic Economic Order Quantity (EOQ) model in the early 20th century. During this time, mathematical methods started emerging to optimize the size of the inventory and the orders [1], and since then, there have been an increasing number of contributions that complement the basic model in different ways. One of them is the extension of finite production rate and another one is when backordering is allowed. The EOQ models are most often used in a continuous-review setting and it is assumed that the inventory can be monitored every moment in time. Decision making under uncertainty is nothing new. In Liberatore [2], an EOQ model with backorders is derived through probabilistic means. However, the uncertainties in many supply chains today are inherent fuzzy [3]. This comes from the fact that there are seldom statistical data to support the calculations, but the uncertainty distributions have to be based on expert opinions only. This is typically the case for new products, and other products with very large seasonal variations, for example. In these cases it is often possible to use fuzzy numbers instead of probabilistic approaches [4, 5]. There are many research contributions in this line of research. For instance, Ouyang et al. [6, 7] allowed the lead times to be decision variables. Salameh and Jaber [8] introduced a model that captured also the defective rate of the goods. Chang [9] worked out some fuzzy modifications of this model. A good review of this research track is found in [10]. Another set of results in this line of research is found in Jaber et al. [11], and Khan et al. [12], where the learning aspect of the inspection of quality was taken explicitly into consideration and Khan et al. [13], where the inspection errors (as well as the imperfect items) also were modeled. There is also a track