Some approximations in multi‐item, multi‐echelon inventory systems for recoverable items

The optimization problem as formulated in the METRIC model takes the form of minimizing the expected number of total system backorders in a two‐echelon inventory system subject to a budget constraint. The system contains recoverable items – items subject to repair when they fail. To solve this problem, one needs to find the optimal Lagrangian multiplier associated with the given budget constraint. For any large‐scale inventory system, this task is computationally not trivial. Fox and Landi proposed one method that was a significant improvement over the original METRIC algorithm. In this report we first develop a method for estimating the value of the optimal Lagrangian multiplier used in the Fox‐Landi algorithm, present alternative ways for determining stock levels, and compare these proposed approaches with the Fox‐Landi algorithm, using two hypothetical inventory systems – one having 3 bases and 75 items, the other 5 bases and 125 items. The comparison shows that the computational time can be reduced by nearly 50 percent. Another factor that contributes to the higher requirement for computational time in obtaining the solution to two‐echelon inventory systems is that it has to allocate stock optimally to the depot as well as to bases for a given total‐system stock level. This essentially requires the evaluation of every possible combination of depot and base stock levels – a time‐consuming process for many practical inventory problems with a sizable system stock level. This report also suggests a simple approximation method for estimating the optimal depot stock level. When this method was applied to the same two hypotetical inventory systems indicated above, it was found that the estimate of optimal depot stock is quite close to the optimal value in all cases. Furthermore, the increase in expected system backorders using the estimated depot stock levels rather than the optimal levels is generally small.