A numerical study on the effect of the balance assumption in one-warehouse multi-retailer inventory systems

One-warehouse multi-retailer systems under periodic review have been studied extensively in the literature. The optimal policy has not been characterized yet. It would require solving a multi-dimensional dynamic program, which is hard due to the curse of dimensionality. In order to let the dynamic program decompose, researchers often make the so-called balance assumption. All available heuristics for periodic review distribution systems are based on some form of this assumption. For these heuristics, often further approximate steps are applied. We investigate the pure effect of the balance assumption in this paper. The balance assumption is the relaxation of a set of constraints in the original dynamic program and yields a lower bound model, which we solve exactly. This gives us a lower bound for the optimal cost of the original model. An upper bound for the true optimal cost is obtained by simulating the optimal policy for the relaxed problem with a slightly modified allocation rule. This modified policy is referred to as the LB heuristic policy. We use the relative gap between the upper and lower bound as a measure to assess the impact of the balance assumption. Based on extensive testing, we identify when the gap is small, and when not. For those instances with small gaps, both the lower bound is tight and the performance of the LB heuristic policy is close to the optimal. We also identify many practically relevant settings under which the balance assumption yields large gaps. For these instances, either the lower bound is poor or the LB heuristic policy is far from optimal, or both. In any case, it implies that more research is needed to develop better lower bounds and/or better heuristics for these instances

A numerical study on the effect of the balance assumption in one-warehouse multi-retailer inventory systems

The periodic review two-echelon divergent inventory system, consisting of a single warehouse and multiple retailers that face stochastic demand of customers, is a well-studied model in the inventory literature. Unlike in the serial and assembly structures, the optimal policy is not known due to the allocation (rationing) problem. In order to let the overall model decompose into smaller problems, researchers often make the so-called balance assumption (the allocation policy may allocate negative quantities to the retailers). This paper is a numerical study that quantifies the effect of the balance assumption on the average expected cost. The balance assumption leads to a relaxed version of the original problem, so the corresponding cost is a lower bound for the optimal cost. An upper bound can be obtained by simulating the solution of the relaxed problem under a modified allocation rule. The gap between the bounds is used as a measure to assess the impact of the balance assumption. The effect of leadtimes, holding and penalty costs, mean and variance of the demand processes, and number of retailers on the gap is identified. We determine the parameter settings that result in small gaps. Further, the relation between imbalance probability and percentage gap is analyzed numerically. Our results point out that for several relevant cases, the balance assumption may lead to large gaps unlike the generally established belief in the literature that this assumption is not a serious limitation. In the light of the insights obtained, we conclude that researchers should put more effort in developing good heuristics for these relevant cases