Abstract
In this study, the ant colony optimization (ACO) algorithm is modified with the K-opt operation to solve the covering salesman problem(CSP) under one restriction in crisp and imprecise (fuzzy, rough) environments. A CSP involves two phases- the division of cities into groups with the selection of the visiting cities and searching of the Hamiltonian circuit through the visiting cities. But, none of the studies in the literature is made following the direct approach. Also, none of the studies in the literature gives attention to reduce the total travel distance of the unvisited cities from the visited city of a group. Moreover, there is no algorithm in the literature which provides the solution of a CSP with the specified coverage range $r$. Also, none has introduced any algorithm to solve CSPs in imprecise environments. Though algorithms are available to solve the Traveling Salesman Problems in the imprecise environments, the approach cannot deal with the problems involving fuzzy data with non-linear membership functions or the problems involving rough data where the rough estimation can not be done using Lebesgue measure. The well establish algorithm for any routing problem is the ACO, but not much attention has been paid to solve the CSP using ACOs. To overcome these limitations on the studies of the ACO on the CSPs, here, an algorithm is proposed for the division of groups of the set of cities depending upon the maximum number of cities in a group and the total number of groups. Then ACO is used to find the shortest/minimum-cost path of the problem by selecting only one visiting the city from each group without violating the restriction of the specified coverage range $r$ of the location of the unvisited cities. K-opt operation is applied periodically at the end of ACO operation to improve the quality of the best found solution so far by the ACO algorithm and to arrest any premature convergence. For the restricted problems paths are searched in such a manner that the total distance/travel cost of different unvisited cities of a group from the visited city of the group should not exceed a predefined upper limit. To solve the problem in an imprecise environment some approach is followed so that the tour is searched without transferring the imprecise optimisation problem into an equivalent crisp optimisation problem. Also, the simulation approaches in fuzzy and rough environments are proposed to deal with the CSPs with any type of estimation of the imprecise data set. Algorithm is tested with the standard benchmark crisp problems available in the literature. To test the algorithm in the imprecise environments, the imprecise instances are derived randomly from the standard crisp instances using a specified rule. Test results imply that the proposed algorithm is efficient enough in solving the CSPs in the crisp as well as in the imprecise environments.
Simulation of the algorithms and their visualization forthe solutions to the restricted problems of the cosmic dynamics of the fourteen bodies with three rings
