Fuzzy random integrated vendor-buyer inventory model with imperfect-quality items and controllable lead time

In this research, an integrated vendor-buyer inventory model with a random number of defective items under fuzzy random lead time demand is developed. The length of lead time affects the competitive abilities of a business directly and it can be controlled by adding a crashing cost. It is assumed that the lead time demand is normally distributed and the shortages are permitted, which are partially backlogged. In the model, lead time demand, annual demand rate, adjustable production rate, and backorder fraction are imprecise in nature and are represented by the triangular fuzzy numbers. Signed distance method is used to defuzzify the fuzzy total inventory cost. The objective is to determine simultaneously the optimal order quantity, the lead time, and the number of deliveries, which minimise the expected total average cost. An algorithm procedure of finding the optimal solution is developed. Furthermore, numerical example and sensitivity analysis are carried out to illustrate the proposed integrated model.