New tighter polynomial length formulations for the asymmetric traveling salesman problem with and without precedence constraints

We propose a new formulation for the asymmetric traveling salesman problem, with and without precedence relationships, which employs a polynomial number of subtour elimination constraints that imply an exponential subset of certain relaxed Dantzig-Fulkerson-Johnson subtour constraints. Promising computational results are presented, particularly in the presence of precedence constraints.     

Iterative patching and the asymmetric traveling salesman problem

Although Branch-and-Bound (BnB) methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. In this paper, we investigate iterative patching, a technique in which a fixed patching procedure is applied at each node of the BnB search tree for the Asymmetric Traveling Salesman Problem. Computational experiments show that iterative patching results in general in search trees that are smaller than the classical BnB trees, and that solution times are lower for usual random and sparse instances. Furthermore, it turns out that, on average, iterative patching with the Contract-or-Patch procedure of Glover, Gutin, Yeo and Zverovich (2001) and the Karp–Steele procedure are the fastest, and that ‘iterative’ Modified Karp–Steele patching generates the smallest search trees.     

An efficient genetic algorithm for the traveling salesman problem with precedence constraints

The traveling salesman problem with precedence constraints (TSPPC) is one of the most difficult combinatorial optimization problems. In this paper, an efficient genetic algorithm (GA) to solve the TSPPC is presented. The key concept of the proposed GA is a topological sort (TS), which is defined as an ordering of vertices in a directed graph. Also, a new crossover operation is developed for the proposed GA. The results of numerical experiments show that the proposed GA produces an optimal solution and shows superior performance compared to the traditional algorithms.     

Genetic algorithm for asymmetric traveling salesman problem with imprecise travel times

This paper presents a variant of the asymmetric traveling salesman problem (ATSP) in which the traveling time between each pair of cities is represented by an interval of values (wherein the actual travel time is expected to lie) instead of a fixed (deterministic) value as in the classical ATSP. Here the ATSP (with interval objective) is formulated using the usual interval arithmetic. To solve the interval ATSP (I-ATSP), a genetic algorithm with interval valued fitness function is proposed. For this purpose, the existing revised definition of order relations between interval numbers for the case of pessimistic decision making is used. The proposed algorithm is based on a previously published work and includes some new features of the basic genetic operators. To analyze the performance and effectiveness of the proposed algorithm and different genetic operators, computational studies of the proposed algorithm on some randomly generated test problems are reported.     

Revisiting dynamic programming for precedence-constrained traveling salesman problem and its time-dependent generalization

The precedence constrained traveling salesman problem (TSP-PC), or the sequential ordering problem (SOP), consists of finding an optimal TSP tour that will also satisfy the namesake precedence constraints, typically specified as a partial order or a directed acyclic graph. Its dynamic programming (DP) solution was proposed as early as 1979, however, by late 1990s, it mostly fell out of use in plain TSP-PC. Revisiting this method, we are able to close one of the long-standing TSPLIB SOP problem instances, ry48p.3.sop, and provide improved bounds on its time complexity. Harnessing the “omnivorous” nature of DP, we prove the validity of DP optimality principle for TSP-PC with both (i) abstract cost aggregation function, which may be the arithmetic + operation as in “ordinary” TSP or $\max$ as in Bottleneck TSP, or any other left-associative nondecreasing in the first argument operation and (ii) travel cost functions depending on the set of pending tasks (“sequence dependence”). Using the latter generalization, we close several TD-SOP (time-dependent TSP-PC) instances based on TSPLIB SOP as proposed by Kinable et al., including rbg253a.sop. Through the restricted DP heuristic, which was originally formulated for time-dependent TSP by Malandraki and Dial, we improve the state-of-the-art upper bounds for all yet unsolved TSPLIB-based TD-SOP instances, including those with more than 100 cities. We also improve worst-case complexity estimates for DP in TSP-PC.     

The effect of strengthened linear formulations on improving the lower bounds for the part families with precedence constraints problem

The part families with precedence constraints problem (PFP) arises in industry, when flexible manufacturing systems are designed within a group technology approach. The aim of this problem is to arrange parts into families by imposing capacity constraints, concerning both the number of parts and processing times, besides precedence constraints in the building of families.     

A parallel branch and bound algorithm for solving large asymmetric traveling salesman problems

A parallel branch and bound algorithm that solves the asymmetric traveling salesman problem to optimality is described. The algorithm uses an assignment problem based lower bounding technique, subtour elimination branching rules, and a subtour patching algorithm as an upper bounding procedure. The algorithm is organized around a data flow framework for parallel branch and bound. The algorithm begins by converting the cost matrix to a sparser version in such a fashion as to retain the optimality of the final solution. Computational results are presented for three different classes of problem instances: (1) matrix elements drawn from a uniform distribution of integers for instances of size 250 to 10 000 cities, (2) instances of size 250 to 1000 cities that concentrate small elements in the upper left portion of the cost matrix, and (3) instances of size 300 to 3000 cities that are designed to confound neighborhood search heuristics.     

New integer linear programming formulation for the traveling salesman problem with time windows: minimizing tour duration with waiting times

The travelling salesman problem, being one of the most attractive and well-studied combinatorial optimization problems, has many variants, one of which is called ‘travelling salesman problem with Time Windows (TSPTW)’. In this problem, each city (nodes, customers) must be visited within a time window defined by the earliest and the latest time. In TSPTW, the traveller has to wait at a city if he/she arrives early; thus waiting times directly affect the duration of a tour. It would be useful to develop a new model solvable by any optimizer directly. In this paper, we propose a new integer linear programming formulation having O(n²) binary variables and O(n²) constraints, where (n) equals the number of nodes of the underlying graph. The objective function is stated to minimize the total travel time plus the total waiting time. A computational comparison is made on a suite of test problems with 20 and 40 nodes. The performances of the proposed and existing formulations are analysed with respect to linear programming relaxations and the CPU times. The new formulation considerably outperforms the existing one with respect to both the performance criteria. Adaptation of our formulation to the multi-traveller case and some additional restrictions for special situations are illustrated.     

A new lower bound for the geometric traveling salesman problem in terms of discrepancy

We give a new lower bound for the shortest hamiltonian path through n points of [0,1]^d in terms of the discrepancy of these n points. This improves an earlier result by Steele.     

A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets

Given any family of valid inequalities for the asymmetric traveling salesman polytopeP(G) defined on the complete digraphG, we show that all members of are facet defining if the primitive members of (usually a small subclass) are. Based on this result we then introduce a general procedure for identifying new classes of facet inducing inequalities forP(G) by lifting inequalities that are facet inducing forP(G), whereG is some induced subgraph ofG. Unlike traditional lifting, where the lifted coefficients are calculated one by one and their value depends on the lifting sequence, our lifting procedure replaces nodes ofG with cliques ofG and uses closed form expressions for calculating the coefficients of the new arcs, which are sequence-independent. We also introduce a new class of facet inducing inequalities, the class of SD (source-destination) inequalities, which subsumes as special cases most known families of facet defining inequalities.Research supported by Grant DDM-8901495 of the National Science Foundation and Contract N00014-85-K-0198 of the U.S. Office of Naval Research.Research supported by M.U.R.S.T., Italy.     

A new class of cutting planes for the symmetric travelling salesman problem

A comprehensive class of cutting planes for the symmetric travelling salesman problem (TSP) is proposed which contains the known comb inequalities, the path inequalities and the 3-star constraints as special cases. Its relation to the clique tree inequalities is discussed. The cutting planes are shown to be valid for a relaxed version of the TSP, the travelling salesman problem on a road network, and—under certain conditions—to define facets of the polyhedron associated with this problem.     

A threshold accepting heuristic with intense local search for the solution of special instances of the traveling salesman problem

In real life scheduling, variations of the standard traveling salesman problem are very often encountered. The aim of this work is to present a new heuristic method for solving three such special instances with a common approach. The proposed algorithm uses a variant of the threshold accepting method, enhanced with intense local search, while the candidate solutions are produced through an insertion heuristic scheme. The main characteristic of the algorithm is that it does not require modifications and parameter tuning in order to cope with the three different problems. Computational results on a variety of real life and artificial problems are presented at the end of this work and prove the efficiency and the ascendancy of the proposed method over other algorithms found in the literature.     

Heuristic approaches for solving transit vehicle scheduling problem with route and fueling time constraints

Electric bus scheduling problem can be defined as vehicle scheduling problem with route and fueling time constraints (VSPRFTC). Every vehicle’s travel miles (route time) after charging is limited, thus the vehicle must be recharged after taking several trips and the minimal charging time (fueling time) must be satisfied. A multiple ant colony algorithm (ACA) was presented to solve VSPRFTC based on ACA used to solve traveling salesman problem (TSP), a new metaheuristic approach inspired by the foraging behavior of real colonies of ants. The VSPRFTC considered in this paper minimizes a multiple, hierarchical objective function: the first objective is to minimize the number of tours (or vehicles) and the second is to minimize the total deadhead time. New improvement of ACA as well as detailed operating steps was provided on the basis of former algorithm. Then in order to settle contradiction between accelerating convergence and avoiding prematurity or stagnation, improvement on route construction rule and Pheromone updating rule was adopted. A group feasible trip sets (blocks) had been produced after the process of applying ACA. In dealing with the fueling time constraint a bipartite graphic model and its optimization algorithm are developed for trip set connecting in a hub and spoke network system to minimize the number of vehicle required. The maximum matching of the bipartite graph is obtained by calculating the maximum inflow with the Ford–Fulkerson algorithm. At last, an example was analyzed to demonstrate the correctness of the application of this algorithm. It proved to be more efficient and robust in solving this problem.