An extension of the integrated production–delivery lot sizing model under stochastic delivery time with transportation cost is investigated. The delivery time is assumed to be exponentially distributed. The presumption, the optimum delivery quantity determining the minimum of the transportation cost, has been widely adopted in transportation management models. Hence, we model the transportation cost as a function of delivery quantity. We consider comprehensive costs in the optimization problem. The expected total cost per unit of time with respect to the delivery quantity is proved to be a convex function under certain feasible conditions. As a result of computational studies, our proposed production–delivery decision model has shown notably adaptable when the delivery time is random. In particular, when the producer’s and retailer’s carrying costs are low, and/or mean time of delivery and transportation costs are high, our suggested policy saves more than 9% as opposed to the scheme recommended by prior researches, which manifests highly advanced gains.
