Partial Evaluation and Efficient Discarding for the Maximal Covering Location Problem

Abstract

The maximal covering location problem attempts to locate a limited number of facilities in order to maximize the coverage over a set of demand nodes. This problem is NP-Hard and it has been often addressed by using metaheuristics, where the execution time directly depends on the number of evaluations of the objective function. In this article, the principles of efficient discarding and partial evaluation are applied to obtain more efficient versions of the objective function of this problem, i.e. not-approximate surrogate objective functions. An experimental study is presented to compare the surrogate functions in terms of number of distance comparisons and runtime. The results show that (on average) the best surrogate function is more than 5 times faster than the original function in general, and more than 8 times faster in the largest instances. This proposal allows for a more efficient metaheuristic solution based on swap operators.

A partial evaluation approach for the School Bus Routing Problem

“…Metaheuristics, despite being approximate methods that do not guarantee an optimal solution, they allow for good solutions to be reached in a reasonable time [6] . However, metaheuristics tend to have a high computational cost that is directly related to the number of times the objective function of the problem must be evaluated [9] .…”

“…Surrogate functions are a way to solve the problem of the computational cost of metaheuristics with less expensive alternatives when evaluating the objective function and/or the restrictions [16] . These functions use several techniques and can be an exact or approximate variant of the objective function and/or constraints [9] .…”

“…One of the exact approaches to surrogate functions is partial evaluation, which is applied in this work to reduce the computational cost of the metaheuristic approach. Partial evaluation is based on the principle of evaluating only the part of the solution that was modified, and it has obtained good results in NP-Complete optimization problems [9] .…”

“…Metaheuristics, in the search for solutions, must evaluate the objective function many times (hundreds, thousands of times), each one with different configurations of the solution [8] . A large number of evaluations of the objective function in an optimization problem increases the computational cost [9] . A large number of evaluated solutions are created as a result of transformations of previous solutions with the use of mutation operators [10] .…”

“…There are approaches to obtaining substitute functions for objective function that tend to lower computational costs. One of these approaches is partial evaluation (PE), which focuses on the variations introduced by the operators [8] , [9] . Thus, in each evaluation of the objective function, only the modified part of the solution is taken into account, taking advantage of previous evaluations [8] .…”