The family traveling salesman problem with incompatibility constraints

Abstract

In this article, we propose a new variant of the family traveling salesman problem (FTSP). The FTSP is an NP-hard problem that generalizes the traveling salesman problem. In the FTSP, the set of cities is partitioned into several families and one wants to find the minimum cost route that visits a given number of cities in each family. A new variant arises by introducing incompatibilities between families, that is, cities from incompatible families cannot be visited in the same route, and it is called the FTSP with incompatibility constraints (FTSP-IC). We propose compact and non-compact mixed integer linear programming models for the FTSP-IC where the incompatibility constraints are modeled by defining paths in a compatibility graph for each family. Additionally, we present a set of valid inequalities motivated by the incompatible families. The non-compact models and the valid inequalities were tested using a branch-and-cut framework. To evaluate the models we generated incompatibility matrices for benchmark instances of the FTSP, which are available in the website http://familytsp.rd.ciencias.ulisboa.pt/. With the branch-and-cut algorithm, we were able to obtain the optimal value of instances with up to 127 nodes. We also developed an ant colony optimization (ACO) algorithm and an iterated local search algorithm (ILS) to address the largest sized instances. Both metaheuristics are able to provide solutions for instances that the branch-and-cut algorithm cannot address and to improve the majority of the upper bounds obtained by the branch-and-cut algorithm in less computational time.

The multi‐depot family traveling salesman problem and clustered variants: Mathematical formulations and branch‐&‐cut based methods

“…Later on, [13] developed heuristic methods for the FTSP and obtained the best known upper bounds from the literature for the larger benchmark instances with unknown optimal value. Recently, Bernardino and Paias [12] solved a variant of the FTSP with additional incompatibility constraints.…”

“…More precisely, the generic formulation for the SC-MDFTSP is the generic formulation of the MDFTSP given above with generic constraints (11). With respect to the HC-MDFTSP, we add generic constraints (12) to the generic formulation of the MDFTSP. We recall that a feasible solution for the HC-MDFTSP is also feasible for the SC-MDFTSP since generic constraints (12) also impose that nodes of the same family have to be visited in the same route, that is, the condition modeled by generic constraints (11).…”

“…We recall that a feasible solution for the HC-MDFTSP is also feasible for the SC-MDFTSP since generic constraints (12) also impose that nodes of the same family have to be visited in the same route, that is, the condition modeled by generic constraints (11). Nevertheless, despite being more restrictive, generic constraints (12) are easier to model than generic constraints (11). For this reason, we discuss how to model generic constraints (12) in Section 3.4 and address generic constraints (11) in Sections 3.5 and 3.6.…”

“…Nevertheless, despite being more restrictive, generic constraints (12) are easier to model than generic constraints (11). For this reason, we discuss how to model generic constraints (12) in Section 3.4 and address generic constraints (11) in Sections 3.5 and 3.6.…”

“…The instance has two depots, namely 𝑑 1 and 𝑑 2 , and four families. Family 1, represented with the white color, has three members (nodes 1-3), family 2, represented with the light gray color, has four members (nodes 4-7), family 3, represented by the dark gray color, has four members (nodes [8][9][10][11] and family 4, represented by the black color, has three family members (nodes [12][13][14]. The required number of visits for the families are: v 1 = 2; v 2 = 3; v 3 = 3, and v 4 = 1.…”