Heuristics for multi-item two-echelon spare parts inventory control subject to aggregate and individual service measures

We consider a multi-item two-echelon spare parts inventory system in which the central warehouse operates under a (Q, R) policy and local warehouses implement (S−1,S) policy. The objective is to find the policy parameters minimizing expected system-wide inventory holding and fixed ordering subject to aggregate and individual response time constraints. Using an exact evaluation we provide a very efficient and effective heuristic, and also a tight lower bound for real-world, large-scale two-echelon spare parts inventory problems. An extensive numerical study reveals that as the number of parts increases – which is usually the case in practice – the relative gap between the cost of the heuristic solution and the lower bound approaches zero. In line with our findings, we show that the heuristic and the lower bound are asymptotically optimal and asymptotically tight, respectively, in the number of parts. In practice, this means we can solve real-life problems with large numbers of items optimally. We propose an alternative approach between system and item approaches, which are based on setting individual and aggregate service level constraints, respectively. Using our alternative approach, we show that it is possible to keep the cost benefit of using aggregate service levels while avoiding long individual response times. We also show that the well-known sequential determination of policy parameters, i.e., determining the batch sizes first, and then finding the other policy parameters using those batch sizes, which is known for its high performance in single-item models, performs relatively poor for multi-item systems.



modeling different cases in practice. By using our computationally efficient close-to-optimal heuristic, we numerically show that a hybrid approach can capture much of the cost benefit of system approach while avoiding the poor service performance at the item level.
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We use our solution method to implement – and also to test the performance of –the widely used sequential determination of policy parameters for large-scale multi-item two-echelon systems. We propose three alternative heuristics, in which the batch sizes are determined offline while the other policy parameters are found by using our method. Contrary to very promising results obtained by using our approach, we observe that the three heuristics based on sequential determination of policy parameters performs relatively poor in our setting. This implies the observation that the performance of sequentially determining the policy parameters is satisfactory for single-echelon problems (Axsäter, 1996, Gallego, 1998, Silver, Pyke, Peterson, 1998 and Zheng, 1992) does not hold for real-world, large-scale multi-item two-echelon settings.


Our solution method is based on applying Lagrangian decomposition and column generation to obtain a lower bound and then using a greedy algorithm to generate feasible policy parameters from that lower bound. Such a method is shown to perform well in different problem settings of multi-item two-echelon inventory systems such as (Topan, Bayındır, 2012, Topan, Bayındır, Tan, 2010 and Wong, Kranenburg, van Houtum, Cattrysse, 2007). Compared to these papers, we have the following main differences: (i) we consider a general system with an alternative hybrid approach, which has system and item approaches as special cases. (ii) We propose a new, two-step greedy algorithm to generate a feasible solution from the Lagrangian lower bound. We show that the solution generated by our two-step greedy algorithm is asymptotically optimal in the number of parts. This makes our greedy algorithm and also our heuristic unique in the literature. (iii) In line with this finding, we also show that the lower bound obtained by Lagrangian decomposition and column generation method is asymptotically tight. Particularly, unlike Wong et al. (2007), we consider a batch ordering policy; in contrast to Topan and Bayındır (2012), who consider a system with a batch ordering policy under a compound Poisson demand, our analysis is based on the exact evaluation of the system; contrary to Topan et al. (2010), our method can solve large-scale, practical size problems. Apart from these differences, we also provide alternative heuristics based on sequential determination of the policy parameters. Table 1 summarizes the differences between our paper and the papers that are most relevant to our paper.