The two best studied facility location problems are the p-median problem and the uncapacitated facility location problem (Daskin, Network and discrete location: models, algorithms, and applications. Wiley, New York, 1995; Mirchandani and Francis, Discrete location theory. Wiley, New York, 1990). Both seek the location of the facilities minimizing the total cost, assuming no uncertainty in costs exists, and thus all parameters are known. In most real-world location problems the demand is not certain, because it is a long-term planning decision, and thus, together with the minimization of costs, optimizing some robustness measure is sound. In this paper we address bi-objective versions of such location problems, in which the total cost, as well as the robustness associated with the demand, are optimized. A dominating set is constructed for these bi-objective nonlinear integer problems via the ε-constraint method. Computational results on test instances are presented, showing the feasibility of our approach to approximate the Pareto-optimal set.
A discrete location problem with nonlinear objective is addressed. A set of p plants is to be open to serve a given set of clients. Together with the locations, the number p of facilities is also a decision variable. The objective is to minimize the total cost, represented as the transportation cost between clients and plants, plus an increasing nonlinear function of p.Two Lagrangean relaxations are considered to derive lower bounds. Dual information is also used to design a core heuristic.Computational results are given, showing that nearly optimal solutions are obtained in short running times.
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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