On the distribution-free continuous review (Q, r, L) inventory model with lead-time-dependent partial backlogging

ABSTRACT In this article, we study a distribution-free continuous review inventory system that assembles lost sales and backorders. The order quantity, the reorder point and lead-time are decision variables of the problem. The study under consideration assumes that purchasing cost is order dependent, partial backlogging is lead-time dependent and the distribution of lead-time demand is known partially. The objective of the paper is twofold. First, in the random framework, we provide the optimal order quantity, reorder level and lead-time simultaneously to acquire significant savings in the total costs of the model than Sana and Goyal ((q, r, l) model for stochastic demand with lead-time dependent partial backlogging. Annals of Operations Research. 233(1), 401–410, 2015). Second, we extend the crisp model to the fuzzy random environment employing the uncertain demand rate as a fuzzy random variable. We develop the mathematical methodologies to obtain the optimal solutions such that the total expected cost of the inventory system is minimized in both the cases. An algorithm is proposed with the help of the developed methodologies. We employ the proposed algorithm to solve two numerical examples to demonstrate the effectiveness of the methods.