A three‐stage p‐median based exact method for the optimal diversity management problem

Abstract

The optimal diversity management problem (ODMP) arises in many application fields when a company, producing a good and/or a service customizable with options, has to satisfy many different client demands with various subset of options, but only a limited number of option combinations can be produced. ODMP can be represented by a disconnected network and formulated as a large-scale p-median problem (PMP). In this article we improve a known decomposition approach where smaller PMPs, related to the network components, can be solved instead of the initial large problem. The proposed method is structured in three stages and it combines Lagrangian relaxation-based techniques, variable fixing and reduction tests, and a dynamic programming algorithm. It drastically reduces the number and the dimensions of the p-median subproblems to be solved to optimality by a MIP solver and to be combined to determine the optimal solution of the original PMP by a multiple choice knapsack problem. A sequential and a parallel implementation of the method are provided and tested. Obtained results on known and new test instances show that our approach considerably outperforms state-of-the-art algorithms for large-scale ODMPs.decomposition, Lagrangian relaxation, multiple choice knapsack, optimal diversity management, p-median, parallel computing I N T R O D U C T I O NThe optimal diversity management problem (ODMP) is a well-known optimization problem arising in many application fields, every time a company produces a good and/or a service which can be provided with options. In this case the product can be personalized by the customer who can choose different option combinations (configurations) depending on her/his needs or preferences. In this context, satisfying all the possible demands with exactly the required options would impose a company to produce all the possible configurations in advance and manage all of them at the assembly lines. Moreover, the production operations could start only after a demand is received, so providing huge delays in satisfying a request. To overcome these drawbacks, a company usually produces a limited number of opportunely chosen configurations to cover all the possible ones. In this way a demand, if no available configuration contains exactly the required options, can be satisfied by a compatible configuration, that is, by a configuration containing all the required options plus some others undemanded by the customer. This implies that a client could receive some unrequired options, so generating an over-cost for the company.

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