A stochastic joint replenishment problem considering transportation and warehouse constraints with gainsharing by Shapley Value allocation

The purpose of this paper is to introduce a heuristic approach that uses a capacitated inventory model as means for identifying a collaborative agreement between different buyers jointly replenishing multiple items from multiple vendors, thus attaining economies of scale while reducing by sharing fixed procurement and operational costs. The proposed approach is denominated Stochastic Collaborative Joint Replenishment Problem (S-CJRP) and consists of two stages. The first stage determines a cost-efficient replenishment frequency for each collaborating company in all possible coalition arrangements. To accomplish the former, an extension of the known Joint Replenishment Problem (JRP) considering real-life capacity constraints, such as stochastic demand assuming normal distribution, finite storage and transport, is solved via genetic algorithms delivering a suitable coalition. In a second stage, the Shapley Value function is established to assess and allocate the potential gains achieved by colluding in the first stage, determining a fair share distribution among players that increases the viability of such coalition. Several scenarios from a simulated numerical study illustrate average cost savings of 32.3%. 28.2% and 32.7% for 3, 4 and 5 players, respectively, considering up to 30 items for the proposed collaboration, in all cases consistently exhibiting cost reduction and increasing the proposal feasibility.