This paper presents both crisp and fuzzy EOQ models for defective items present in each lot when shortages are allowed and backorder takes place. The aim of the work is to first construct an optimal order quantity for the crisp case and then to develop the corresponding fuzzy model. In contrast to the previous inventory models, an allowable proportionate discount is incorporated for the defective items present in each lot to provide a general framework to the model. The aim of the present paper is to find the optimal order size and the expected shortage level so as to obtain the optimum total profit for both the models. The necessary and sufficient conditions for the existence and uniqueness of the optimal solutions are derived and it is also shown that under certain conditions the crisp model boils down the traditional EOQ backorder formula. For the fuzzy case, triangular fuzzy numbers are used for the defective rates and for defuzzification signed distance method is used. Finally, numerical example is provided to illustrate the solution procedure and sensitivity analysis is performed on the results to analyze the effect of the variations taken place for the parameters involved in the model.

The goal of this work is to provide an Economic Order Quantity (EOQ) model with allowable proportionate discount of imperfect quality items and cross selling effects. We first developed the ordering policy in this study by introducing cross-selling impact. Under the cross-selling effects, the frequent item sets are considered as individual entities and their corresponding EOQ can be computed further with. Moreover, when items are defective in nature the cross-selling effects remain more prominent. To establish the desired relationship among the item sets, different data mining techniques are explored initially. The work progresses further with the application of the cross-selling effects to estimate the EOQ. The work conducts a 100% screening of lot, as each lot assumed to have a fraction of imperfection. Then after variable discounts allowed for the imperfect items to find total profit. Finally, numerical analysis with a few examples is given to experience the outcomes of presented model.