We address the Joint Replenishment Problem (JRP) where, in the presence of joint setup costs, dynamic lot sizing schedules need to be determined for m items over a planning horizon of N periods, with general time-varying cost and demand parameters. We develop a new, so-called, partitioning heuristic for this problem, which partitions the complete horizon of N periods into several relatively small intervals, specifies an associated joint replenishment problem for each of these, and solves them via a new, efficient branch-and-bound method. The efficiency of the branch-and-bound method is due to the use of a new, tight lower bound to evaluate the nodes of the tree, a new branching rule, and a new upper bound for the cost of the entire problem. The partitioning heuristic can be implemented with complexity 0(mN2loglogN). It can be designed to guarantee an e-optimal solution for any e > 0, provided that some of the model parameters are uniformly bounded from above or below. In particular, the heuristic is asymptotically optimal as N approaches infinity for any fixed number of items m, and it remains asymptotically optimal when both m and N are simultaneously increased to infinity. Most importantly, a numerical study shows that the partitioning heuristic performs exceptionally well. Even for small problems, the average optimality gap is only 0.38% and in none of the problem categories is it larger than 0.78%.
Federgruen, Awi, and Michal Tzur. "The joint replenishment problem with time-varying costs and demands: Efficient, asymptotic and ε-optimal solutions." Operations Research 42, no. 6 (1994): 1067-1086.
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