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

Modification of Ant Colony Optimization Algorithm to Solve the Traveling Salesman Problem

Traveling salesman problem (TSP) is an optimization problem in determining the optimal route of a number of nodes that will only be passed once with the initial node as the final destination. One method for solving TSP is the Ant Colony Optimization (ACO) Algorithm. ACO is inspired by ant behaviour in searching for food, where ants produce pheromones to find food sources and make a route from the colony to food that will be followed by other ants. However ACO has not been considered as the optimal method for resolving TSP. This is because ACO has several shortcomings in the computational process. Comparisons between pheromones are not yet clear, and slow computing time causes the results of ACO to be not optimal. To correct these deficiencies, modifications will be made to the ACO. Modifications are made by changing some values in the ACO, such as adjusting the number of ants by the node automatically, changing the value in the pheromone renewal, and adding value to the construction of the solution. The outcome of this research is the modification of ACO did not provide shorter computing time with a more accurate final value, thus did not provide an optimal solution. The test results in this study found that the average computation time for the last iteration of each test was 0.54 second, and for the 10 iteration computation time obtained an average of 5.54 second for four tests. The amount of memory used in four tests in this study was 440.11 mb for 10 iterations

Exploration in stochastic algorithms: An application on MAX-MIN Ant System

In this paper a definition of the exploration performed by stochastic algorithms is proposed. It is based on the observation through cluster analysis of the solutions generated during a run. The probabilities associated by an algorithm to solution components are considered. Moreover, a consequent method for quantifying the exploration is provided. Such a measurement is applied to MAX-MIN Ant System. The results of the experimental analysis allow to observe the impact of the parameters of the algorithm on the exploration.exploration, cluster analysis, MAX-MIN Ant System

A Perturbed Self-organizing Multiobjective Evolutionary Algorithm to solve Multiobjective TSP

Travelling Salesman Problem (TSP) is a very important NP-Hard problem getting focused more on these days. Having improvement on TSP, right now consider the multi-objective TSP (MOTSP), broadened occurrence of travelling salesman problem. Since TSP is NP-hard issue MOTSP is additionally a NP-hard issue. There are a lot of algorithms and methods to solve the MOTSP among which Multiobjective evolutionary algorithm based on decomposition is appropriate to solve it nowadays. This work presents a new algorithm which combines the Data Perturbation, Self-Organizing Map (SOM) and MOEA/D to solve the problem of MOTSP, named Perturbed Self-Organizing multiobjective Evolutionary Algorithm (P-SMEA). In P-SMEA Self-Organizing Map (SOM) is used extract neighborhood relationship information and with MOEA/D subproblems are generated and solved simultaneously to obtain the optimal solution. Data Perturbation is applied to avoid the local optima. So by using the P-SMEA, MOTSP can be handled efficiently. The experimental results show that P-SMEA outperforms MOEA/D and SMEA on a set of test instances

Adaptive multimodal continuous ant colony optimization

Seeking multiple optima simultaneously, which multimodal optimization aims at, has attracted increasing attention but remains challenging. Taking advantage of ant colony optimization algorithms in preserving high diversity, this paper intends to extend ant colony optimization algorithms to deal with multimodal optimization. First, combined with current niching methods, an adaptive multimodal continuous ant colony optimization algorithm is introduced. In this algorithm, an adaptive parameter adjustment is developed, which takes the difference among niches into consideration. Second, to accelerate convergence, a differential evolution mutation operator is alternatively utilized to build base vectors for ants to construct new solutions. Then, to enhance the exploitation, a local search scheme based on Gaussian distribution is self-adaptively performed around the seeds of niches. Together, the proposed algorithm affords a good balance between exploration and exploitation. Extensive experiments on 20 widely used benchmark multimodal functions are conducted to investigate the influence of each algorithmic component and results are compared with several state-of-the-art multimodal algorithms and winners of competitions on multimodal optimization. These comparisons demonstrate the competitive efficiency and effectiveness of the proposed algorithm, especially in dealing with complex problems with high numbers of local optima

The Traveling Salesman Problem with Stochastic and Correlated Customers

It is well-known that the cost of parcel delivery can be reduced by designingroutes that take into account the uncertainty surrounding customers’ presences. Thus far, routing problems with stochastic customer presences have relied on the assumption that all customer presences are independent from each other. However, the notion that demographic factors retain predictive power for parcel-delivery efficiency suggests that shared characteristics can be exploited to map dependencies between customer presences. This paper introduces the correlated probabilistic traveling salesman problem (CPTSP). The CPTSP generalizes the traveling salesman problem with stochastic customer presences, also known as the probabilistic traveling salesman problem (PTSP), to account for potentialcorrelations between customer presences. I propose a generic and flexible model formulation for the CPTSP using copulas that maintains computational and mathematical tractability in high-dimensional settings. I also present several adaptations of existing exact and heuristic frameworks to solve the CPTSP effectively. Computational experiments on real-world parcel-delivery data reveal that correlations between stochastic customer presences do not always affect route decisions, but could have a considerable impact on route costestimates