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We consider minimax single-facility location problems in multidimensional spaces with Chebyshev and rectilinear distances. Both unconstrained problems and problems with constraints imposed on the feasible location area are under examination. We start with the description of the location problems in a standard form, and then represent them in the framework of tropical (idempotent) algebra as constrained tropical optimization problems. These problems involve the minimization of non-linear objective functions defined on vectors over an idempotent semifield, subject to vector inequality and equality constraints. We apply methods and results of tropical optimization to obtain direct, explicit solutions to the problems. To solve the problem, we introduce a variable to represent the minimum value of the objective function, and then reduce the optimization problem to an inequality with the new variable in the role of a parameter. The existence conditions for the solution of the inequality serve to evaluate the parameter, whereas the solutions of the inequality are taken as a complete solution to the problem. We use the results obtained to derive solutions of the location problems of interest in a closed form, which is ready for immediate computation. Extensions of the approach to solve other problems, including minimax multi-facility location problems, are discussed. Numerical solutions of example problems are given, and graphical illustrations are presented.
|State||Published - Jul 2016|
|Event||28th European Conference on Operational Research - Poznan, Poland|
Duration: 3 Jul 2016 → 6 Jul 2016
|Conference||28th European Conference on Operational Research|
|Period||3/07/16 → 6/07/16|
Scopus subject areas
- Management Science and Operations Research
- Control and Optimization
- Algebra and Number Theory
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Николай Кимович Кривулин (Speaker)4 Jul 2016
Activity: Talk types › Oral presentation