Intuitionistic Fuzzy Rule-Base Model for the Time Dependent Traveling Salesman Problem

The Traveling Salesman Problem is a well-known combinatorial optimization problem. There are many different extensions and modifications of the original problem, such as The Time Dependent Traveling Salesman Problem, this specific extension of the original Traveling Salesman Problem towards more realistic traffic conditions assessment. In the Time Dependent Traveling Salesman Problem the “distances” (costs) between nodes vary in time, they are considered longer during the rush hour period or in the traffic jam region, e.g. the city centre. In this article we introduce an even more realistic approach, the Intuitionistic Fuzzy Time Dependent Traveling Salesman Problem. It is an extension of the Time Dependent Traveling Salesman Problem with the additional notion of intuitionistic fuzzy sets (which is a generalization of the original fuzzy sets). Our goal is to give a useful extended, alternative model instead of the original abstract problem. By demonstrating that the addition of intuitionistic fuzzy elements to quantify the intangible jam factors creates an inference system that approximates the tour cost in a more practical way. Hence, we are one step closer to offering a more realistic solution for the generalized Traveling Salesman Problem. The results of two simple toy examples showed the general effectiveness of the model

THE TIME DEPENDENT TRAVELLING SALESMAN PROBLEM WITH TIME WINDOWS

none4This paper deals with an exact algorithm for the Time-Dependent Traveling Salesman Problem with Time Windows (TDTSPTW) with continuous piecewise linear cost functions. There are two main research streams that can benefit from efficient exact algorithms for TDTSPTW. The first concerns determining optimal vehicle route planning taking traffic congestion into account explicitly. The latter deals with sequence dependent set-up single machine scheduling problems minimizing total completion times or total tardiness. The contribution of this paper is twofold. First, it is proved that the Asymmetric Traveling Salesman Problem with Time Windows is optimal for the TDTSPTW, if all the arcs share a common congestion pattern. Second, an integer linear programming model is formulated for TDTSPTW, and valid inequalities are then embedded into a branch-and-cut algorithm. Preliminary results show that the proposed algorithm is able to solve instances with up to 67 vertices.A. Arigliano; G. Ghiani; E. Guerriero; Antonio Domenico GriecoA., Arigliano; Ghiani, Gianpaolo; Guerriero, Emanuela; Grieco, Antonio Domenic

On an iterative procedure for solving a routing problem with constraints

The generalized precedence constrained traveling salesman problem is considered in the case when travel costs depend explicitly on the list of tasks that have not been performed (by the time of the travel). The original routing problem with dependent variables is represented in terms of an equivalent extremal problem with independent variables. An iterative method based on this representation is proposed for solving the original problem. The algorithm based on this method is implemented as a computer program. © 2013 Pleiades Publishing, Ltd

Multi-Stop Routing Optimization: A Genetic Algorithm Approach

In this research, we investigate and propose new operators to improve Genetic Algorithm’s performance to solve the multi-stop routing problem. In a multi-stop route, a user starts at point x, visits all destinations exactly once, and then return to the same starting point. In this thesis, we are interested in two types of this problem. The first type is when the distance among destinations is fixed. In this case, it is called static traveling salesman problem. The second type is when the cost among destinations is affected by traffic congestion. Thus, the time among destinations changes during the day. In this case, it is called time-dependent traveling salesman problem. This research proposes new improvements on genetic algorithm to solve each of these two optimization problems. First, the Travelling Salesman Problem (TSP) is one of the most important and attractive combinatorial optimization problems. There are many meta-heuristic algorithms that can solve this problem. In this paper, we use a Genetic Algorithm (GA) to solve it. GA uses different operators: selection, crossover, and mutation. Sequential Constructive Crossover (SCX) and Bidirectional Circular Constructive Crossover (BCSCX) are efficient to solve TSP. Here, we propose a modification to these crossovers. The experimental results show that our proposed adjustment is superior to SCX and BCSCX as well as to other conventional crossovers (e.g. Order Crossover (OX), Cycle Crossover (CX), and Partially Mapped Crossover (PMX)) in term of solution quality and convergence speed. Furthermore, the GA solver, that is improved by applying inexpensive local search operators, can produce solutions that have much better quality within reasonable computational time. Second, the Time-Dependent Traveling Salesman Problem (TDTSP) is an interesting problem and has an impact on real-life applications such as a delivery system. In this problem, time among destinations fluctuates during the day due to traffic, weather, accidents, or other events. Thus, it is important to recommend a tour that can save driver’s time and resources. In this research, we propose a Multi-Population Genetic Algorithm (MGA) where each population has different crossovers. We compare the proposed MG against Single-Population Genetic Algorithm (SGA) in terms of tour time solution quality. Our finding is that MGA outperforms SGA. Our method is tested against real-world traffic data [1] where there are 200 different instances with different numbers of destinations. For all tested instances, MGA is superior on average by at least 10% (for instances with size less than 50) and 20% (for instances of size 50) better tour time solution compared to SGA with OX and SGA with PMX operators, and at least 4% better tour time compared toga with SCX operator

A New Class of Cycle Inequality for the Time-Dependent Traveling Salesman Problem

The Time-Dependent Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem, where the cost for travel between two nodes is dependent on the nodes and their position in the tour. Inequalities for the Asymmetric TSP can be easily extended to the TDTSP, but the added time information can be used to strengthen these inequalities. We look at extending the Lifted Cycle Inequalities, a large family of inequalities for the ATSP. We define a new inequality, the Extended Cycle (X-cycle) Inequality, based on cycles in the graph. We extend the results of Balas and Fischetti for Lifted Cycle Inequalities to define Lifted X-cycle Inequalities. We show that the Lifted X-cycle Inequalities include some inequalities which define facets of the submissive of the TDTS Polytope

Algorithms for Variants of Routing Problems

In this thesis, we propose mathematical optimization models and algorithms for variants of routing problems. The first contribution consists of models and algorithms for the Traveling Salesman Problem with Time-dependent Service times (TSP-TS). We propose a new Mixed Integer Programming model and develop a multi-operator genetic algorithm and two Branch-and-Cut methods, based on the proposed model. The algorithms are tested on benchmark symmetric and asymmetric instances from the literature, and compared with an existing approach, showing the effectiveness of the proposed algorithms. The second work concerns the Pollution Traveling Salesman Problem (PTSP). We present a Mixed Integer Programming model for the PTSP and two mataheuristic algorithms: an Iterated Local Search algorithm and a Multi-operator Genetic algorithm. We performed extensive computational experiments on benchmark instances. The last contribution considers a rich version of the Waste Collection Problem (WCP) with multiple depots and stochastic demands using Horizontal Cooperation strategies. We developed a hybrid algorithm combining metaheuristics with simulation. We tested the proposed algorithm on a set of large-sized WCP instances in non-cooperative scenarios and cooperative scenarios

Restricted Dynamic Programming Heuristic for Precedence Constrained Bottleneck Generalized TSP

We develop a restricted dynamical programming heuristic for a complicated traveling salesman problem: a) cities are grouped into clusters, resp. Generalized TSP; b) precedence constraints are imposed on the order of visiting the clusters, resp. Precedence Constrained TSP; c) the costs of moving to the next cluster and doing the required job inside one are aggregated in a minimax manner, resp. Bottleneck TSP; d) all the costs may depend on the sequence of previously visited clusters, resp. Sequence-Dependent TSP or Time Dependent TSP. Such multiplicity of constraints complicates the use of mixed integer-linear programming, while dynamic programming (DP) benefits from them; the latter may be supplemented with a branch-and-bound strategy, which necessitates a “DP-compliant” heuristic. The proposed heuristic always yields a feasible solution, which is not always the case with heuristics, and its precision may be tuned until it becomes the exact DP

Defining Asymptotic Parallel Time Complexity of Data-dependent Algorithms

The scientific research community has reached a stage of maturity where its strong need for high-performance computing has diffused into also everyday life of engineering and industry algorithms. In efforts to satisfy this need, parallel computers provide an efficient and economical way to solve large-scale and/or time-constrained problems. As a consequence, the end-users of these systems have a vested interest in defining the asymptotic time complexity of parallel algorithms to predict their performance on a particular parallel computer. The asymptotic parallel time complexity of data-dependent algorithms depends on the number of processors, data size, and other parameters. Discovering the main other parameters is a challenging problem and the clue in obtaining a good estimate of performance order. Great examples of these types of applications are sorting algorithms, searching algorithms and solvers of the traveling salesman problem (TSP). This article encompasses all the knowledge discovery aspects to the problem of defining the asymptotic parallel time complexity of datadependent algorithms. The knowledge discovery methodology begins by designing a considerable number of experiments and measuring their execution times. Then, an interactive and iterative process explores data in search of patterns and/or relationships detecting some parameters that affect performance. Knowing the key parameters which characterise time complexity, it becomes possible to hypothesise to restart the process and to produce a subsequent improved time complexity model. Finally, the methodology predicts the performance order for new data sets on a particular parallel computer by replacing a numerical identification. As a case of study, a global pruning traveling salesman problem implementation (GP-TSP) has been chosen to analyze the influence of indeterminism in performance prediction of data-dependent parallel algorithms, and also to show the usefulness of the defined knowledge discovery methodology. The subsequent hypotheses generated to define the asymptotic parallel time complexity of the TSP were corroborated one by one. The experimental results confirm the expected capability of the proposed methodology; the predictions of performance time order were rather good comparing with real execution time (in the order of 85%)