A Group Theoretic Tabu Search Approach to the Traveling Salesman Problem

The traveling salesman problem (TSP) is a combinatorial optimization problem that is mathematically modeled as a binary integer program. The TSP is a very important problem for the operations research academician and practitioner. This research demonstrates a Group Theoretic Tabu Search (GTTS) Java algorithm for the TSP. The tabu search metaheuristic continuously finds near-optimal solutions to the TSP under various different implementations. Algebraic group theory offers a more formal mathematical setting to study the TSP providing a theoretical foundation for describing tabu search. Specifically, this thesis uses the Symmetric Group on n letters, S(n), which is the set of all n! permutations on n letters whose binary operation is permutation multiplication, to describe the TSP solution space. Thus, the TSP is studied as a permutation problem rather than an integer program by applying the principles of group theory to define the tabu search move and neighborhood structure. The group theoretic concept of conjugation (an operation involving two group elements) simplifies the move definition as well as the intensification and diversification strategies. Conjugation in GTTS diversifies the search by allowing large rearrangement moves within a tour in a single move operation. Empirical results are presented along with the theoretical motivations for the research

A tabu search algorithm for scheduling a single robot in a job-shop environment

We consider a single-machine scheduling problem which arises as a subproblem in a job-shop environment where the jobs have to be transported between the machines by a single transport robot. The robot scheduling problem may be regarded as a generalization of the travelling-salesman problem with time windows, where additionally generalized precedence constraints have to be respected. The objective is to determine a sequence of all nodes and corresponding starting times in the given time windows in such a way that all generalized precedence relations are respected and the sum of all travelling and waiting times is minimized. We present a local search algorithm for this problem where an appropriate neighborhood structure is defined using problem-specific properties. In order to make the search process more efficient, we apply some techniques which accelerate the evaluation of the solutions in the proposed neighbourhood considerably. Computational results are presented for test data arising from job-shop instances with a single transport robot

A Lin-Kernighan Heuristic for Single Row Facility Layout

The single row facility layout problem (SRFLP) is the problem of arranging facilities with given lengths on a line, while minimizing the weighted sum of the distances between all pairs of facilities. The problem is known to be NP-hard. In this paper, we present a neighborhood search heuristic called LK-INSERT which uses a Lin-Kernighan neighborhood structure built on insertion neighborhoods. To the best of our knowledge this is the first such heuristic for the SRFLP. Our computational experiments show that LK-INSERT is competitive and improves the best known solutions for several large sized benchmark SRFLP instances.

Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

Solving Medium to Large Sized Euclidean Generalized Minimum Spanning Tree Problems

The generalized minimum spanning tree problem is a generalization of the minimum spanning tree problem. This network design problems finds several practical applications, especially when one considers the design of a large-capacity backbone network connecting several individual networks. In this paper we study the performance of six neighborhood search heuristics based on tabu search and variable neighborhood search on this problem domain. Our principal finding is that a tabu search heuristic almost always provides the best quality solution for small to medium sized instances within short execution times while variable neighborhood decomposition search provides the best quality solutions for most large instances.