Optimizing a supply chain problem with nonlinear penalty costs for early and late delivery under generalized lead time distribution

Abstract

This article presents a tangible vendor–buyer cooperative strategy that benefits both, the vendor and the buyer where the demand is deterministic constant, and the delivery lead time follows a general distribution. To build a realistic coordination mechanism, a delivery tolerance time range is specified beyond which two different types of nonlinear penalty costs termed as an early delivery penalty cost and late delivery penalty cost are assessed. Shortages are allowed for a short time-span. The penalty costs are taken as a product of a linear function of delivery lead time and a nonlinear function of the delivery lot size. The problem is formulated as a multi-variable mixed-integer nonlinear programming (MINLP) problem and the objective of this research is to achieve the minimum integrated expected cost where decision variables are: reorder point, delivery lot size, number of deliveries, and delivery time thresholds. Since closed-form solutions are not immediately obtained, different search procedures are employed to resolve issues relating to an integer solution. Numerical results are provided for uniform, exponential and normal distributions of delivery lead time to establish the general model.