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.

In this study, Bat algorithm (BA) is modified along with K-opt operation and one newly proposed perturbation approach to solve the well known covering salesman problem (CSP). Here, along with the restriction of the radial distances of the unvisited cities from the visited cities another restriction is imposed where a priority is given to some cities for the inclusion in the tour, i.e., some clusters to be created where the prioritised cities must be the visiting cities and the corresponding CSP is named as Prioritised CSP (PCSP). In the algorithm, 3-opt and 4-opt operations are used for two different purposes. The 4-opt operation is applied for generating an initial solution set of CSP for the BA and the 3-opt operation generates some perturbed solutions of a solution. A new perturbation approach is proposed for generating neighbour solutions of a potential solution where the exchange of some cities in the tour is made and is named as K-bit exchange operation. The proposed solution approach for the CSP and PCSP is named as the modified BA embedded with K-bit exchange and K-opt operation (MBAKEKO). It is a two-stage algorithm where in the first stage of the algorithm the clustering of the cities is done with respect to a fixed visiting city of each cluster in such a manner that the distances of the other cities of the cluster must lie with in the fixed covering distance of the problem and in the second stage the BA is applied to find the minimum cost Hamiltonian circuit by passing through the visiting cities of the clusters. MBAKEKO is tested with a set of benchmark test problems with significantly large sizes from the TSPLIB. To measure the performance of MBAKEKO, its results are compared with the results of different well-known approaches for CSPs available in the literature. It is observed from the comparison studies that MBAKEKO searches the minimum cost tour for any of the considered instances compared to all other well-known algorithms in the literature. It can be concluded from the numerical studies that the performance of MBAKEKO is better with respect to the state-of-the-art algorithms available in the literature.