An efficient optimal solution method for the joint replenishment problem

Abstract

During the last two decades, many heuristic procedures for the joint replenishment problem have appeared in the literature. The only available optimal solution procedure was based on an enumerative approach and was computationally prohibitive. In this paper we present an alternative optimal approach based on global optimisation theory. By applying Lipschitz optimisation one can find a solution with an arbitrarily small deviation from an optimal value. An efficient procedure is presented which uses a dynamic Lipschitz constant and generates a solution in little time. The running time of this procedure grows only linearly in the number of items.

Approximation Algorithms and Hardness Results for the Joint Replenishment Problem with Constant Demands

“…An α-approximation algorithm for a minimization problem is an algorithm that produces a feasible solution with cost at most α times the optimal cost. We show that a simple modification of an algorithm by Wildeman et al [WFD97] gives polynomial-time approximation algorithms with ratios very close to or better than the current best approximations for the JRP with constant demands. We illustrate this in detail for dynamic policies in the finite horizon case and for GI.…”

An efficient optimal solution method for the joint replenishment problem with minimum order quantities

“…We derived proper lower and upper bounds for the basic cycle time T using both formulations of the JRP with constraint, with and without correction factor, and we proposed a new solution method based on the formulation of the problem given by Wildeman et al [7]. We demonstrated that an iterative procedure approach to find simultaneously the optimal values for the k j 's and T, as the one used by Viswanathan [6] to improve the bounds, does not perform well in this case.…”

“…Following the procedure of Wildeman et al [7], we can rewrite the original function TC(T,k) as follows:…”

“…Viswanathan [6] and Fung and Ma [3] used tighter bounds on T to improve the procedures by Goyal and Van Eijs. Wildeman, Frenk and Dekker [7] presented an efficient optimal solution method for large problems based on Lipschitz optimisation. We will adapt their formulation but develop a completely new method to make it suitable for minimum order quantities.…”

The Q(s,S) control policy for the joint replenishment problem extended to the case of correlation among item-demands

“…The joint replenishment problem has received considerable attention in the literature. One stream of research assumes that the demands are deterministic, cf Visvanathan (1996) and Wildeman et al (1997). Another stream assumes that demands follow a stochastic process, where the focus has been on analyzing good control policies.…”