An EOQ model for fuzzy defective rate with allowable proportionate discount

In this paper, we developed both crisp and fuzzy EOQ models with proportionate discount (discount increases when number of defects decrease in each lot) for items with imperfect quality. First, we construct an optimal order quantity of crisp case. Next, proposed three different fuzzy inventory models where in the first case the defective rate is fuzzified, in the next case, both defective rate and annual demand rate are fuzzified and finally in the case of the third model all costs, defective rate and annual demand are taken to be fuzzy. Lastly, we developed the model for items with imperfect quality with inspection errors, as the inspector may commit errors while screening the lot. The probability of misclassification errors is assumed to be known. The inspection process may consist of three costs: (a) cost of inspection (b) cost of Type I errors and (c) cost of Type II errors. The defective items, classified by the inspector and the buyer, would be salvaged as a single batch that is sold at a discounted price. The objective is to find the optimal lot size for models to maximize the total profit (both for crisp and fuzzy models) and used fuzzy numbers for defective items, demand rate and/or all types of costs (exclusively for fuzzy models). We considered the triangular fuzzy numbers to represent the fuzziness of all types of costs, defective items and annual demand. Finally, the optimum order quantity is obtained using the signed distance method. A numerical example is provided to illustrate the results of the proposed models and the sensitivity analysis is conducted to know the effect of changes made for the values of different parameters on the actual lot size and the profit respectively.