Power-form demand and cost functions in inventory lot size models

Abstract

This paper presents a continuous-time inventory model with known time-varying demand and carrying cost functions. Backlogs are prohibited, replenishments are assumed to be instantaneous and the planning horizon is finite. The problem is to find the optimal number and schedule of replenishments, i.e. the time intervals between consecutive orders, which minimizes total carrying and replenishment costs throughout the planning horizon. First, a necessary and sufficient condition on the optimal replenishment times is derived for general demand and carrying cost functions when the number of replenishments is fixed. Then, a complete solution (optimal number and schedule of replenishments) is given for the case of power-form demand and carrying cost functions. The asymptotic properties of the solution as the planning horizon tends to infinity are also investigated. The model lends itself to a tractable parametric analysis, and generalizes several special cases already known in the literature.