The use of fuzzy logic in various combinatorial optimization problems

Conventional models of many combinatorial optimization problems rarely encompass real-life problems, because real-life problems usually contain a high degree of uncertainty. These uncertainties can be modeled using various methods, including fuzzy sets. Apart from the precise description from the problems’ nature, fuzzy variables can describe the problem better, improve the solution and reduce costs for decision makers. In this chapter we show how fuzzy logic can be used for modeling uncertainties in combinatorial problems and improve their quality. The focus will be on the Location Set Covering Problem (LSCP), the Maximal Covering Location Problem (MCLP) and the Minimal Covering Location Problem (MinCLP) as a special modification of the MCLP, but the same method could be applied to other problems. These problems are applicable in searching for optimal places for desired and undesired facilities under the given conditions. Each problem will be formally described with its own mathematical model and some of their instances will be solved. Firstly, small-size instances of the problems will be solved with an exact algorithm using the CPLEX optimizer tool, and when a dimension becomes too big for exact solving, the instances will be then solved with a Particle Swarm Optimization (PSO) meta-heuristic.