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.     

The traveling salesman problem: A duality approach

In this study we formulate the dual of the traveling salesman problem, which extends in a natural way the dual problem of Held and Karp to the nonsymmetric case. The dual problem is solved by a subgradient optimization technique. A branch and bound scheme with stepped fathoming is then used to find optimal and suboptimal tours. Computational experience for the algorithm is presented.This author's work was partially supported by NSF Grant #GK-38337.     

On the high multiplicity traveling salesman problem

This paper considers a version of the traveling salesman problem where the cities are to be visited multiple times. Each city has its own required number of visits. We investigate how the optimal solution and its objective value change when the numbers of visits are increased by a common multiplicator. In addition, we derive lower bounds on values of the multiplicator beyond which further increase does not improve the average tour length. Moreover, we show how and when the structure of an optimal tour length can be derived from tours with smaller multiplicities.     

Uncertain multiobjective traveling salesman problem

Traveling salesman problem is a fundamental combinatorial optimization model studied in the operations research community for nearly half a century, yet there is surprisingly little literature that addresses uncertainty and multiple objectives in it. A novel TSP variation, called uncertain multiobjective TSP (UMTSP) with uncertain variables on the arc, is proposed in this paper on the basis of uncertainty theory, and a new solution approach named uncertain approach is applied to obtain Pareto efficient route in UMTSP. Considering the uncertain and combinatorial nature of UMTSP, a new ABC algorithm inserted with reverse operator, crossover operator and mutation operator is designed to this problem, which outperforms other algorithms through the performance comparison on three benchmark TSPs. Finally, a new benchmark UMTSP case study is presented to illustrate the construction and solution of UMTSP, which shows that the optimal route in deterministic TSP can be a poor route in UMTSP.     

Genetic algorithms for the traveling salesman problem

This paper is a survey of genetic algorithms for the traveling salesman problem. Genetic algorithms are randomized search techniques that simulate some of the processes observed in natural evolution. In this paper, a simple genetic algorithm is introduced, and various extensions are presented to solve the traveling salesman problem. Computational results are also reported for both random and classical problems taken from the operations research literature.     

Continuous line drawings via the traveling salesman problem

We describe how to use the traveling salesman problem to create continuous line drawings of target pictures.     

On the nearest neighbor rule for the traveling salesman problem

Rosenkrantz et al. (SIAM J. Comput. 6 (1977) 563) and Johnson and Papadimitriou (in: E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (Eds.), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, Chichester, 1985, pp. 145-180, (Chapter 5)) constructed families of TSP instances with n cities for which the nearest neighbor rule yields a tour-length that is a factor above the length of the optimal tour.We describe two new families of TSP instances, for which the nearest neighbor rule shows the same bad behavior. The instances in the first family are graphical, and the instances in the second family are Euclidean. Our construction and our arguments are extremely simple and suitable for classroom use.     

Exploiting planarity in separation routines for the symmetric traveling salesman problem

At present, the most successful approach for solving large-scale instances of the Symmetric Traveling Salesman Problem to optimality is branch-and-cut. The success of branch-and-cut is due in large part to the availability of effective separation procedures; that is, routines for identifying violated linear constraints.

For two particular classes of constraints, known as comb and domino-parity constraints, it has been shown that separation becomes easier when the underlying graph is planar. We continue this line of research by showing how to exploit planarity in the separation of three other classes of constraints: subtour elimination, 2-matching and simple domino-parity constraints.     

A comparison of lower bounds for the symmetric circulant traveling salesman problem

When the matrix of distances between cities is symmetric and circulant, the traveling salesman problem (TSP) reduces to the so-called symmetric circulant traveling salesman problem (SCTSP), that has applications in the design of reconfigurable networks, and in minimizing wallpaper waste. The complexity of the SCTSP is open, but conjectured to be NP-hard, and we compare different lower bounds on the optimal value that may be computed in polynomial time. We derive a new linear programming (LP) relaxation of the SCTSP from the semidefinite programming (SDP) relaxation in [E. de Klerk, D.V. Pasechnik, R. Sotirov, On semidefinite programming relaxation of the traveling salesman problem, SIAM Journal of Optimization 19 (4) (2008) 1559-1573]. Further, we discuss theoretical and empirical comparisons between this new bound and three well-known bounds from the literature, namely the Held-Karp bound [M. Held, R.M. Karp, The traveling salesman problem and minimum spanning trees, Operations Research 18 (1970) 1138-1162], the 1-tree bound, and the closed-form bound for SCTSP proposed in [J.A.A. van der Veen, Solvable cases of TSP with various objective functions, Ph.D. Thesis, Groningen University, The Netherlands, 1992].     

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.     

The multi-commodity one-to-one pickup-and-delivery traveling salesman problem

This paper concerns a generalization of the traveling salesman problem (TSP) called multi-commodity one-to-one pickup-and-delivery traveling salesman problem (m-PDTSP) in which cities correspond to customers providing or requiring known amounts of m different commodities, and the vehicle has a given upper-limit capacity. Each commodity has exactly one origin and one destination, and the vehicle must visit each customer exactly once. The problem can also be defined as the capacitated version of the classical TSP with precedence constraints. This paper presents two mixed integer linear programming models, and describes a decomposition technique for each model to find the optimal solution. Computational experiments on instances from the literature and randomly generated compare the techniques and show the effectiveness of our implementation.     

Risky traveling salesman problem

In this paper we introduce a methodology for optimizing the expected cost of routing a single vehicle which has a probability of breaking down or failing to complete some of its tasks. More specifically, a calculus is devised for finding the optimal order in which each site should be visited.     

A stochastic dynamic traveling salesman problem with hard time windows

Just-in-time (JIT) trucking service, i.e., arriving at customers within specified time windows, has become the norm for freight carriers in all stages of supply chains. In this paper, a JIT pickup/delivery problem is formulated as a stochastic dynamic traveling salesman problem with time windows (SDTSPTW). At a customer location, the vehicle either picks up goods for or delivers goods from the depot, but does not provide moving service to transfer goods from one location to another. Such routing problems are NP-hard in deterministic settings, and in our context, complicated further by the stochastic, dynamic nature of the problem. This paper develops an efficient heuristic for the SDTSPTW with hard time windows. The heuristic is shown to be useful both in controlled numerical experiments and in applying to a real-life trucking 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.     

Solving the asymmetric traveling purchaser problem

The Asymmetric Traveling Purchaser Problem (ATPP) is a generalization of the Asymmetric Traveling Salesman Problem with several applications in the routing and the scheduling contexts. This problem is defined as follows. Let us consider a set of products and a set of markets. Each market is provided with a limited amount of each product at a known price. The ATPP consists in selecting a subset of markets such that a given demand of each product can be purchased, minimizing the routing cost and the purchasing cost. The aim of this article is to evaluate the effectiveness of a branch-and-cut algorithm based on new valid inequalities. It also proposes a transformation of the ATPP into its symmetric version, so a second exact method is also presented. An extensive computational analysis on several classes of instances from literature evaluates the proposed approaches. A previous work () solves instances with up to 25 markets and 100 products, while the here-presented approaches prove optimality on instances with up to 200 markets and 200 products. Partially supported by “Ministerio de Ciencia y Tecnología” (TIC2003-05982-C05-02), and by Vicerrectorado de Investigación y Desarrollo Tecnológico de la Universidad de La Laguna.     

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.     

The traveling salesman problem with few inner points

We propose two algorithms for the planar Euclidean traveling salesman problem. The first runs in O(k!kn) time and O(k) space, and the second runs in O(2^kk²n) time and O(2^kkn) space, where n denotes the number of input points and k denotes the number of points interior to the convex hull.     

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.