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Majority of chapters of this book show usefulness of greedy like algorithms for solving various combinatorial optimization problems. The aim of this chapter is to warn the reader that not always a greedy like approach is a good option and, in certain cases, it is a very bad option being sometimes among the worst possible options. Our message is not a discouragement from using greedy like algorithms altogether; we think that for every combinatorial optimization problem of importance, researchers and practitioners should simply investigate the appropriateness of greedy like algorithms and the existence of better alternatives to them (considering both quality of solution and running time). In many cases, especially when the running time must be very short, the conclusion may still be that the most practical of known approaches is a greedy like algorithm. The Traveling Salesman and Multidimensional Assignment Problems are optimization problems for which greedy like approaches are usually not very successful. We demonstrate this by providing both theoretical and experimental results on greedy like algorithms as well as on some other algorithms that produce (in theory and/or in experiments) much better results without spending significantly more time. There are some general theoretical results that indicate that there are, in fact, many combinatorial optimization problems for which greedy like algorithms are not the best option even among fast construction heuristics, see, e.g., [3, 5, 17]. We will not consider these general results in order to avoid most mathematical details that are not necessary for understanding the results of this chapter. For this reason we will not give proofs here apart from two simple proofs: that of Theorem 8 which shows that some instances on which the greedy algorithm fails are not exotic in a sense and that of Theorem 11 since Theorem 11 is a new result. It is not a trivial question whether a certain algorithm is greedy like or not. In the next section we define an independence system and give the classic definition of the greedy algorithm for such a system. We extend this definition to so-called greedy type algorithms that include such well-known algorithms as the Prim’s algorithm for the minimum spanning tree problem and the nearest neighbor algorithm for the traveling salesman problem. We use the term ‘greedy like’ in an informal way and we include in this class simple and fast construction heuristics that seem to us to be of greedy nature. O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg

In this paper we propose to discuss some relationships between the classical Traveling Salesman Problem (TSP), Litlewood-Paley theory, and harmonic measure. This circle of ideas is also closely related to the theory of Cauchy integrals on Lipschitz graphs, and this aspect is discussed more fully in the paper of David and Semmes [2] in this proceedings. The main differences between the subjects in [2] and this paper are that the results here are valid for one dimensional sets, whereas [2] treats d dimensional sets.

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n / log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, M\k{a}dry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem., Comment: 16 pages, 3 figures

A promising new delivery model involves the use of a delivery truck that collaborates with a drone to make deliveries. Effectively combining a truck and a drone gives rise to a new planning problem that is known as the traveling salesman problem with drone (TSP-D). This paper presents exact solution approaches for the TSP-D based on dynamic programming and provides an experimental comparison of these approaches. Our numerical experiments show that our approach can solve larger problems than the mathematical programming approaches that have been presented in the literature thus far. Moreover, we show that restrictions on the number of locations the truck can visit while the drone is away can help significantly reduce the solution times while having relatively little impact on the overall solution quality.

The Metric Travelling Salesman Problem is a subcase of the Travelling Salesman Problem (TSP), where the triangle inequality holds. It is a key problem in combinatorial optimization. Solutions of the Metric TSP are generally used for costs minimization tasks in logistics, manufacturing, genetics and other fields. Since this problem is NP-hard, heuristic algorithms providing near optimal solutions in polynomial time will be considered instead of the exact ones. The aim of this article is to experimentally find Pareto optimal heuristics for Metric TSP under criteria of error rate and run time efficiency. Two real-life kinds of inputs are intercompared - VLSI Data Sets based on very large scale integration schemes and National TSPs that use geographic coordinates of cities. This paper provides an overview and prior estimates of seventeen heuristic algorithms implemented in C++ and tested on both data sets. The details of the research methodology are provided, the computational scenario is presented. In the course of computational experiments, the comparative figures are obtained and on their basis multi-objective optimization is provided. Overall, the group of Pareto-optimal algorithms for different N consists of some of the MC, SC, NN, DENN, CI, GRD, CI + 2-Opt, GRD + 2-Opt, CHR and LKH heuristics.

In this paper we present a novel universal metaheuristic, Discrete Bacterial Memetic Evolutionary Algorithm (DBMEA), which is based on the combination of the Bacterial Evolutionary Algorithm and local search techniques, used for solving NP-hard optimization problems. The algorithm was tested on a series of symmetric Traveling Salesman Problems (TSP) and Traveling Salesman Problem with time windows (TSPTW) benchmarks. The size of the symmetric TSP benchmarks went up to 5 000 cities. In all cases the DBMEA algorithm produced optimal or near-optimal solutions and the difference from the known best values was within 0.16%. While for large size problems it was much faster than the Concorde solver, it was found to be slower compared to the Helsgaun-Lin-Kernighan heuristic, which is the most efficient TSP solver method. With some slight modifications the same algorithm was also tested on TSP with time windows (TSPTW) benchmark instances. In most cases the DBMEA procedure found the known best solutions, and it was again the second fastest method compared with the state-of-the-art techniques for the TSPTW. DBMEA is called efficacious because it is a universal method. It can be efficiently applied to various NP-hard optimization problems and, as in all cases, it results in the optimal or a very near-optimal solutions, while its runtime is very predictable in terms of the size of the problem, and the topology of the instance does not affect its runtime significantly. Even though heuristics developed for a particular type of problem might perform better for that restricted class, our novel method proposed here is universally applicable and may be deployed successfully for optimizing other discreet NP-hard graph search and optimization problems as well.

A colored traveling salesman problem (CTSP) is a generalization of the well-known multiple traveling salesman problem. In our prior CTSP, each salesman is allocated a particular color and each city, carrying 1, 2, or all salesmen’s colors depending on the problem types, allows any salesmen with the same color to visit exactly once. This paper presents a more common CTSP, in which city colors are diverse, i.e., each city has one to all salesmen’s colors while other elements of the problem keeps unchanged. It is a generalization of the existing CTSPs, i.e., the radial and serial ones, and can be used to model the scheduling problems with different accessibility of jobs toward executors. A city color matrix is introduced to describe the accessibility difference of cities to all salesmen. Since CTSP is NP-hard, this paper presents a variable neighborhood search (VNS) approach, instead of computationally intractable exact solutions. First, the repetitive solution space due to the dual-chromosome encoding for the prior genetic algorithms can be entirely avoided by using direct-route encoding. Then, a two-stage greedy initialization algorithm is utilized by VNS to generate the initial solution. A city removal mechanism and a reinsertion operation are introduced to change the neighborhood space of the current solution and 2-opt method is adopted for the local search. Extensive simulation is conducted and the results show that the proposed VNS is an efficient heuristics to solve CTSP.

Carrier–vehicle systems generally consist of a slow carrier (e.g., a ship) with a long operational range and a faster vehicle (e.g., an aircraft) with a limited operational range. The carrier has the role of transporting the faster vehicle and of deploying, recovering, and servicing it. The goal of the carrier–vehicle traveling salesman problem (CVTSP) is to permit the faster vehicle to visit a given collection of targets in the shortest time while using the carrier as a base for possible multiple trips. As a consequence, the carrier and vehicle should be synchronized. The visiting sequence of the targets is not given a priori. We present a mixed-integer, second-order conic programming (MISOCP) formulation for the CVTSP. Computational results are shown for the resolution of the model with commercial solvers. The MISOCP structure and the relationship to the traveling salesman problem are exploited for developing a ranking-based solution algorithm that outperforms the commercial solvers.

The consistent traveling salesman problem aims to identify minimum-cost routes to be followed by a single vehicle so as to provide a set of customers with service that adheres to arrival-time consistency across the multiple time periods of a planning horizon. In this paper, we address this problem via a new, exact algorithm that decomposes the problem into a sequence of single-period traveling salesman problems with time windows within a branch-and-bound search procedure. The new algorithm is highly competitive, as it is able to solve to guaranteed optimality instances with up to 100 customers requiring service over a five-period horizon, effectively doubling the size of instances that were solvable by the previous state of the art. Furthermore, and in contrast to all previous approaches, the new algorithm accounts for route duration limits, whenever applicable, as well as incorporates the flexibility for the vehicle to idle before providing service to a customer. We in fact show that, for the benchmark instances we considered, the cost of implementing consistent routes can be reduced significantly if the vehicle is allowed to idle en route, further motivating the need for algorithmic schemes to incorporate this realistic option.

This paper presents the stability analysis of an optimal tour for the symmetric traveling salesman problem (TSP) by obtaining stability regions. The stability region of an optimal tour is the set of all cost changes for which that solution remains optimal and can be understood as the margin of optimality for a solution with respect to perturbations in the problem data. It is known that it is not possible to test in polynomial time whether an optimal tour remains optimal after the cost of an arbitrary set of edges changes. Therefore, this paper develops tractable methods to obtain under and over approximations of stability regions based on neighborhoods and relaxations. The application of the results to the two-neighborhood and the minimum 1 tree (M1T) relaxation are discussed in detail. For Euclidean TSPs, stability regions with respect to vertex location perturbations and the notion of safe radii and location criticalities are introduced. Benefits of this paper include insight into robustness properties of tours, minimum spanning trees, M1Ts, and fast methods to evaluate optimality after perturbations occur. Numerical examples are given to demonstrate the methods and achievable approximation quality.

A colored traveling salesman problem (CTSP) is a generalization of the well-known multiple traveling salesman problem. This paper investigate a class of CTSP, called serial CTSP (S-CTSP). Each of its salesmen has his exclusive cities and shares some cities with its neighbor(s) in a serial manner. It can be used to model the scheduling problem of multimachine engineering systems with linearly arranged machines. S-CTSP is NP-hard. Developing effective and efficient approaches to S-CTSP is important to enable its industrial applications. This paper presents a population-based incremental learning (PBIL) approach to it. After analyzing its solution space, we set up some probability matrix models to guide the individual search of the algorithm. Then, a distance penalty is introduced into the state transfer function that can select the cities with small penalty values by the Roulette method to form a good route. By adding a powerful local search operation, 2-opt, to the algorithm, we can further enhance its search ability. Extensive simulation is conducted and its results show that the augmented PBIL is effective and well outperforms the genetic algorithms and CPLEX.

This paper addresses the traveling salesman problem with arbitrary neighbourhoods, which is an NP-hard problem in combinatorial optimization and is important in operations research and theoretical computer science. Existing methods based on neighbourhood predigestion and discretization are impractical in cases in which the neighbourhoods are arbitrary. In this paper, the neighbourhoods of the traveling salesman problem are generalized to arbitrarily connected regions in planar Euclidean space. A novel approach to solving this problem is proposed, including a boundary-based encoding scheme and a double-loop hybrid algorithm based on particle swarm optimization and genetic algorithm. In the hybrid algorithm, linear descending inertia weight particle swarm optimization is adopted to search continuous visiting positions in the outer loop, and the genetic algorithm is used to optimize the discrete visiting sequence in the inner loop. The boundary-based encoding scheme can reduce the search space significantly without degrading the solution quality, and the hybrid algorithm can find a high-quality solution in a reasonable time. The computational results on both small and large instances demonstrate that the proposed approach can guarantee a high-quality solution in a reasonable time, compared with three other state-of-the-art algorithms: iterative algorithm, branch-and-bound algorithm, and upper and lower bound algorithm. Moreover, the proposed approach works efficiently in a real-world application that cannot be solved by existing algorithms.

The Traveling Salesman Problem with Hotel Selection (TSPHS) is a variant of the classic Traveling Salesman Problem. It arises from a number of real-life applications where the maximum travel time for each “day trip” is limited. In this paper, we present a highly effective hybrid between dynamic programming and memetic algorithm for TSPHS. The main features of the proposed method include a dynamic programming approach to find an optimal hotel sequence for a given tour, three dedicated crossover operators for solution recombination, an adaptive rule for crossover selection, and a two-phase local refinement procedure that alternates between feasible and infeasible searches. Experiments on four sets of 131 benchmark instances from the literature show a remarkable performance of the proposed approach. In particular, it finds improved best solutions for 22 instances and matches the best known results for 103 instances. Additional analyses highlight the contribution of the dynamic programming approach, the joint use of crossovers and the two local search phases to the performance of the proposed algorithm.

Multi-bridge machining systems U+0028 MBMS U+0029 have gained wide applications in industry due to their high production capacity and efficiency. They contain multiple bridge machines working in parallel within their partially overlapping workspaces. Their scheduling problems can be abstracted into a serial-colored travelling salesman problem in which each salesman has some exclusive cities and some cities shared with its neighbor U+0028 s U+0029. To solve it, we develop a greedy algorithm that selects a neighboring city satisfying proximity. The algorithm allows a salesman to select randomly its shared cities and runs accordingly many times. It can thus be used to solve job scheduling problems for MBMS. Subsequently, a collision-free scheduling method is proposed to address both job scheduling and collision resolution issues of MBMS. It is an extension of the greedy algorithm by introducing time window constraints and a collision resolution mechanism. Thus, the augmented greedy algorithm can try its best to select stepwise a job for an individual machine such that no time overlaps exist between it and the job sequence of the neighboring machine dealt in the corresponding overlapping workspace; and remove such a time overlap only when it is inevitable. Finally, we conduct a case study of a large triplebridge waterjet cutting system by applying the proposed method.

International audience; Gottschalk and Vygen proved that every solution of the well-known subtour elimination linear program for traveling salesman paths is a convex combination of a set of more and more restrictive "generalized Gao trees" of the underlying graph. In this paper we give a short proof of this, as a {\em layered} convex combination of bases of a sequence of more and more restrictive matroids. Our proof implies (via the matroid partition theorem) a strongly-polynomial combinatorial algorithm for finding this convex combination. This is a new connection of the TSP to matroids, offering also a new polyhedral insight.

The present article describes a contribution to solve transportation problems with green constraints. The aim is to solve an urban traveling salesman problem where the objective function is the total emitted CO2. We start by adapting ASIF approach for calculating CO2 emissions to the urban logistics problem. Then, we solve it using ant colony optimization metaheuristic. The problem formulation and solving will both work under a web-based mapping platform. The selected problem is a real-world NP-hard transportation problem in the city of Casablanca.

We introduce a new generalization of the traveling salesman problem with pickup and delivery, that stems from applications in maritime logistics, in which each node represents a port and has a known draft limit. Each customer has a demand, characterized by a weight, and pickups and deliveries are performed by a single ship of given weight capacity. The ship is able to visit a port only if the amount of cargo it carries is compatible with the draft limit of the port. We present an integer linear programming formulation and we show how classical valid inequalities from the literature can be adapted to the considered problem. We introduce heuristic procedures and a branch-and-cut exact algorithm. We examine, through extensive computational experiments, the impact of the various cuts and the performance of the proposed algorithms.

In this paper, we present a single-commodity flow-based formulation for the high-multiplicity asymmetric traveling salesman problem (HMATSP), which is an extension of the asymmetric traveling sales...

Nature has the ability of sustainability and improvisation for better survival. This unique characteristic reflects a pattern of optimization that inspires the computational intelligence toward different scopes of optimization: a nondeterministic optimization approach or a nature-inspired metaheuristic algorithm. To date, there are many metaheuristic algorithms introduced with good promising results and also becoming a powerful method for solving numerous optimization problems. In this paper, a new metaheuristic algorithm inspired from a plant growth system is proposed, which is defined as tree physiology optimization (TPO). A plant growth consists of two main counterparts: plant shoots and roots. Shoots extend to find better sunlight for the photosynthesis process that converts light and water supplied from the roots into energy for plant growth; at the same time, roots elongate in the opposite way in search for water and nutrients for shoot survival. The collaboration from both systems ensures plant sustainability. This idea is transformed into an optimization algorithm: shoots with defined branches find the potential solution with the help of roots variable. The shoots-branches extension enhances the search diversity and the root system amplifying the search via evaluated fitness. To demonstrate its effectiveness, two different classes of problem are evaluated: (1) a continuous benchmark test function compared to particle swarm optimization (PSO) and genetic algorithm (GA) and (2) an NP-hard problem with the traveling salesman problem (TSP) compared to GA and nearest-neighbor (NN) algorithm. The simulation results show that TPO outperforms PSO and GA in all problem characteristics (flat surface and steep-drop with a combination of many local minima and plateau). In the TSP, TPO has a comparable result to GA.

In an earlier paper (Bao and Liu [1] ), we considered a version of the clustered traveling salesman problem (CTSP), in which both the starting and ending vertex of each cluster are free to be selected, and proposed a 2.167-approximation algorithm. In this note, we first improve this approximation ratio to 1.9 by introducing a new method to define the inter-node lengths for all the nodes in Step 2 of Algorithm A of Bao and Liu [1] . Based on the above method, we then provide a 2.5-approximation algorithm for another version of CTSP where the starting vertex of each cluster is given while the ending vertex is free to be selected, which improves the previous approximation ratio of 2.643 of Guttmann-Beck et al. [5] .

This paper deals with the spherical traveling salesman problem. In this problem, all cities are located on the surface of a sphere and the cities must be visited exactly once in a tour. We propose a new and effective meta-heuristic algorithm with greedy behavior for solving this problem. The proposed algorithm is based on the discrete flower pollination algorithm, which is a bio-inspired meta-heuristic algorithm enhanced by order-based crossover, pollen discarding behavior and partial behaviors. To evaluate the proposed algorithm, it is compared with four effective existing algorithms (the genetic algorithm, two variants of the genetic algorithm and tabu search) on a set of available spherical traveling salesman instances. The results show the superiority of our algorithm in both solution quality and robustness of the solutions.

Most of the meta-heuristic algorithms are based on the natural processes. They were inspired by physical, biological, social, chemical, social–biological, biological–geography, music, and hybrid processes. In this paper, artificial atom algorithm which was inspired by one of natural processes was applied to traveling salesman problem. The obtained results have shown that for small-scale TSP, artificial atom algorithm is closer to optimum than the other compared heuristic algorithms such as tabu search, genetic algorithm, particle swarm optimization, ant colony optimization, and their different combinations.

We present a randomized O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem (ATSP). This provides the first asymptotic improvement over the long-standing Θ(log n)-approximation bound stemming from the work of Frieze et al. (1982) [Frieze AM, Galbiati G, Maffioki F (1982) On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12(1):23–39]. The key ingredient of our approach is a new connection between the approximability of the ATSP and the notion of so-called thin trees. To exploit this connection, we employ maximum entropy rounding—a novel method of randomized rounding of LP relaxations of optimization problems. We believe that this method might be of independent interest.

The methods for solving the traveling salesman problem: Monte Carlo, reduction of rows and columns, averaged coefficients for planning the flight paths of unmanned aerial vehicles, and the results of the work are analyzed. The solution of the traveling salesman problem allows to reduce the time for decision making and UAV costs when planning flights. It is established that the first two solve the problem with some errors, and, when using the Monte Carlo method, these errors tend to increase, and the method of reduction of rows and columns minimizes. When using the method of averaged coefficients, the traveling salesman problem is solved more optimally in comparison with the methods considered by the distance and time criterion for solving the problem. This method gives a significant gain (5–10 %) for these criteria. The relevance and importance of the application of this method is in civil or military operations using UAVs in the face of limited decision-making time in the planning and energy resources of the aircraft.

The traveling salesman problem (TSP) is one of the most studied problems in combinatorial optimization. Given a set of nodes and the distances between them, it consists in finding the shortest route that visits each node exactly once and returns to the first. Nevertheless, more flexible and applicable formulations of this problem exist and can be considered. The Steiner TSP (STSP) is a variant of the TSP that assumes that only a given subset of nodes must be visited by the shortest route, eventually visiting some nodes and edges more than once. In this paper, we adapt some classical TSP constructive heuristics and neighborhood structures to the STSP variant. In particular, we propose a reduced 2-opt neighborhood and we show that it leads to better results in smaller computation times. Computational results with an implementation of a GRASP heuristic using path-relinking and restarts are reported. In addition, ten large test instances are generated. All instances and their best-known solutions are made available for download and benchmarking purposes.

We consider the m-Peripatetic Salesman Problem (m-PSP) on random inputs with discrete distribution function. In this paper we present a polynomial approximation algorithm which, under certain conditions, with high probability (w.h.p.) gives optimal solution for both the m-PSP on random inputs with identical weight functions and the m-PSP with different weight functions.

This paper carries out a performance analysis of major Nature-inspired Algorithms in solving the benchmark symmetric and asymmetric Traveling Salesman’s Problems (TSP). Knowledge of the workings of the TSP is very useful in strategic management as it provides useful guidance to planners. After critical assessments of the performances of eleven algorithms consisting of two heuristics (Randomized Insertion Algorithm and the Honey Bee Mating Optimization for the Travelling Salesman’s Problem), two trajectory algorithms (Simulated Annealing and Evolutionary Simulated Annealing) and seven population-based optimization algorithms (Genetic Algorithm, Artificial Bee Colony, African Buffalo Optimization, Bat Algorithm, Particle Swarm Optimization, Ant Colony Optimization and Firefly Algorithm) in solving the 60 popular and complex benchmark symmetric Travelling Salesman’s optimization problems out of the total 118 as well as all the 18 asymmetric Travelling Salesman’s Problems test cases available in TSPLIB91. The study reveals that the African Buffalo Optimization and the Ant Colony Optimization are the best in solving the symmetric TSP, which is similar to intelligence gathering channel in the strategic management of big organizations, while the Randomized Insertion Algorithm holds the best promise in asymmetric TSP instances akin to strategic information exchange channels in strategic management.

We solve a defender-attacker-defender problem over a traveling salesman problem (TSP), in which the defender first acts to defend a subset of arcs, the attacker then interdicts a subset of undefended arcs (thus increasing their costs), and the defender solves a TSP over the remaining network. Our approach employs an exact approach augmented with a TSP restriction phase to accelerate the convergence of the algorithm. Our computational results show success for the first time in optimally solving defender-attacker-defender TSP problems.

We study a generalization of the classical traveling salesman problem, where multiple salesmen are positioned at different depots, of which only a limited number (k) can be selected to service customers. For this problem, only two 2-approximation algorithms are available in the literature. Here, we improve on these algorithms by showing that a non-trivial extension of the well-known Christofides heuristic has a tight approximation ratio of 2 − 1 / ( 2 k ) . In doing so, we develop a body of analysis which can be used to build new approximation algorithms for other vehicle routing problems.

Given two sets, R and B, consisting of n cities each, in the bipartite traveling salesman problem one looks for the shortest way of visiting alternately the cities of R and B, returning to the city of origin. This problem is known to be NP-hard for arbitrary sets R and B. In this paper we provide an O(n6) algorithm to solve the bipartite traveling salesman problem if the quadrangle property holds. In particular, this algorithm can be applied to solve in O(n6) time the bipartite traveling salesman problem in the following cases: S = R ∪ B is a convex point set in the plane, S = R ∪ B is the set of vertices of a simple polygon and V = R ∪ B is the set of vertices of a circular graph. For this last case, we also describe another algorithm which runs in O(n2) time.

We introduce the traveling salesman problem with forbidden neighborhoods (TSPFN). This is an extension of the Euclidean TSP in the plane where direct connections between points that are too close are forbidden. The TSPFN is motivated by an application in laser beam melting. In the production of a workpiece in several layers using this method one hopes to reduce the internal stresses of the workpiece by excluding the heating of positions that are too close. The points in this application are often arranged in some regular (grid) structure. In this paper we study optimal solutions of TSPFN instances where the points in the Euclidean plane are the points of a regular grid. Indeed, we explicitly determine the optimal values for the TSPFN and its associated path version on rectangular regular grids for different minimal distances of the points visited consecutively. For establishing lower bounds on the optimal values we use combinatorial counting arguments depending on the parities of the grid dimensions. Furthermore we provide construction schemes for optimal TSPFN tours for the considered cases.

The article justifies the necessity of reducing the measurement time of the part surfaces in the coordinate measuring machine by minimizing the trajectory of the sensor. It is established that the search of the minimal path by the sensor corresponds to the travelling salesman problem that can be solved by exact or approximate method. The branch and bound method and ant colony optimization are selected for comparison. We perform computer simulation for finding the optimal path of the sensor by controlling different numbers of points on plane, cylindrical and spherical surfaces. It is established that ant colony algorithm gives close to optimal solution and requires far less time for its search in contrast to the branch and bound algorithm. Application of the ant colony optimization will significantly improve the performance of the measurement of the part surfaces in the coordinate measuring machine.

The Probabilistic Traveling Salesman Problem (PTSP) is an extension of the classical Traveling Salesman Problem (TSP). The main difference is the stochastic presence of the customers, that is, the number of them to be visited each time is a random variable, where each customer associates with a given probability of occurrence. The resolution of problem consists in finding an a priori tour visiting all customers which minimizes the expected length over all possibilities. In this paper, we propose an exact algorithm for the solution of the PTSP. In this context, we derive new expressions for the lower bound and transitional, permanent evaluations as well as the mechanism of separation. The numerical results demonstrate the advantage of the proposed evaluations.

Genetic algorithms are evolutionary techniques used for optimization purposes according to survival of the fittest idea. These methods do not ensure optimal solutions; however, they give good approximation usually in time. The genetic algorithms are useful for NP-hard problems, especially the traveling salesman problem. The genetic algorithm depends on selection criteria, crossover, and mutation operators. To tackle the traveling salesman problem using genetic algorithms, there are various representations such as binary, path, adjacency, ordinal, and matrix representations. In this article, we propose a new crossover operator for traveling salesman problem to minimize the total distance. This approach has been linked with path representation, which is the most natural way to represent a legal tour. Computational results are also reported with some traditional path representation methods like partially mapped and order crossovers along with new cycle crossover operator for some benchmark TSPLIB instances and found improvements.

We investigate in this paper a Green Travelling Salesman Problem with Time Windows (GTSPTW), a bi-criteria variant of the classical TSPTW. The proposed GTSPTW consists of determining the vehicle’s speed in order to minimize the fuel consumption and resulting emission costs. The problem can model numerous industries as service companies and post office. We propose a linear mixed integer mathematical model for the GTSPTW. Due to the NP -hardness of the problem, we develop a hybrid bi-criteria ant colony optimization starts from a local search solution, then performs numerous neighborhoods. We compare the ability of the proposed algorithm with the exact solution using CPLEX on benchmark instances in the literature. In order to demonstrate the performance of our proposed approach, we apply it on a real case within the city of Gafsa in the south of Tunisia.

The question considered in the travelling salesman problem is the following. For a given list of cities and the distances between each pair of cities, to construct the shortest possible path that visits each city exactly once and returns to the origin city. The problem is an NP-hard problem in combinatorial optimization and is important in operations research. The traveling salesman problem is one of the most famous and heavily researched problems in theoretical computer science. We consider the version, which is the Symmetric Maximum Traveling Salesman Problem. The article describes an approximation algorithm for the maximum traveling salesman problem, based on two known polynomial time approximation algorithms. The accuracy of this algorithm is 25/33.

In this paper we study Integer Linear Programming models and develop branch-and-cut algorithms to solve the Black-and-White Traveling Salesman Problem (BWTSP) (Bourgeois, Laporte, & Semet, 2003) which is a variant of the well known Traveling Salesman Problem (TSP). Two strategies to model the BWTSP have been used in the literature. The problem is either modeled on the original graph as TSP using a single set of binary edge variables and with additional non-trivial hop and distance constraints between every pair of black nodes (see Ghiani, Laporte, & Semat, 2006) or as a sequence of constrained paths composed of white nodes connecting pairs of black nodes (see Muter, 2015). In this paper, we study and develop an intermediate approach based on the observation that it is sufficient to guarantee the required distance (and hop) limit of the path from a given black node to the next black node without explicitly stating which one it is. Thus, instead of stating the two constraints (after adding appropriately defined variables) for each pair of black nodes, they are stated for each black node only (that represents the source of each path). Based on this idea we develop several variants of position- and distance-dependent reformulations together with corresponding layered graph representations. Branch-and-cut algorithms are developed for all proposed formulations and empirically compared by an extensive computational study. The obtained results allow us to provide insights into individual advantages and disadvantages of the different models.

The traveling salesman problem has long been regarded as a challenging application for existing optimization methods as well as a benchmark application for the development of new optimization methods. As with many existing algorithms, a traditional genetic algorithm will have limited success with this problem class, particularly as the problem size increases. A rule based genetic algorithm is proposed and demonstrated on sets of traveling salesman problems of increasing size. The solution character as well as the solution efficiency is compared against a simulated annealing technique as well as a standard genetic algorithm. The rule based genetic algorithm is shown to provide superior performance for all problem sizes considered. Furthermore, a post optimal analysis provides insight into which rules were successfully applied during the solution process which allows for rule modification to further enhance performance.

An instance of Hamiltonian cycle problem can be solved by converting it to an instance of Travelling salesman problem, assigning any choice of weights to edges of the underlying graph. In this note we demonstrate that, for difficult instances, choosing the edge weights to be the resistance distance between its two incident vertices is often a good choice. We also demonstrate that arguably stronger performance arises from using the inverse of the resistance distance. Examples are provided demonstrating benefits gained from these choices.

International audience; We study a problem of minimizing the maximum number of identical workers over all cycles of a paced assembly line comprised of m stations and executing n parts of k types. There are lower and upper bounds on the workforce requirements and the cycle time constraints. We show that this problem is equivalent to the same problem without the cycle time constraints and with fixed workforce requirements. We prove that the problem is NP-hard in the strong sense if and the workforce requirements are station independent, and present an Integer Linear Programming model, an enumeration algorithm and a dynamic programming algorithm. Polynomial in k and polynomial in n algorithms for special cases with two part types or two stations are also given. Relations to the Bottleneck Traveling Salesman Problem and its generalizations are discussed.

Electrical vehicle technology has now developed. Various public transportation can start using electric power sources. However, there are battery capacity constraints. When the distance is too far, the battery capacity is not sufficient to provide the required power. In this paper, we will explain how the bottleneck travel salesman problem is applied to minimize battery capacity requirements. So that vehicles can choose the right route by minimizing the maximum distance between destinations. The optimization process is carried out with the Brute Force algorithm and python programming language.