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

A survey on metaheuristics for stochastic combinatorial optimization

Metaheuristics are general algorithmic frameworks, often nature-inspired, designed to solve complex optimization problems, and they are a growing research area since a few decades. In recent years, metaheuristics are emerging as successful alternatives to more classical approaches also for solving optimization problems that include in their mathematical formulation uncertain, stochastic, and dynamic information. In this paper metaheuristics such as Ant Colony Optimization, Evolutionary Computation, Simulated Annealing, Tabu Search and others are introduced, and their applications to the class of Stochastic Combinatorial Optimization Problems (SCOPs) is thoroughly reviewed. Issues common to all metaheuristics, open problems, and possible directions of research are proposed and discussed. In this survey, the reader familiar to metaheuristics finds also pointers to classical algorithmic approaches to optimization under uncertainty, and useful informations to start working on this problem domain, while the reader new to metaheuristics should find a good tutorial in those metaheuristics that are currently being applied to optimization under uncertainty, and motivations for interest in this fiel

Controlling the Balance of Exploration and Exploitation in ACO Algorithm

خوارزمية النمل هي واحده من خوارزميات البحث عن الحلول المثلى ضمن فضاء واسع من الاحتمالات على نحو شبيه بطريقة النمل في البحث والتقفي لإيجاد الحلول لبعض المشاكل المعقدة التي يصعب حلها باستخدام خوارزميات الذكاء الاصطناعي التقليدية. تستخدم هذه الخوارزمية عمليه البحث في فضاء الحالات للاستنتاج حلول مختلفة اثناء عمليه البحث معتمدة على التوازن بين استكشاف حلول جديدة لتوسيع رقعة البحث وبين استغلال الحلول الجيدة لتحسين الحلول المستخرجة مسبقا. ان عمليه خلق توازن بين هاتين العمليتان يؤدي لتحسين النتائج والخروج بحلول أكثر امثليه.  هدف هذا البحث هو ايجاد قانون احتمالي أكثر ملاءمة وقادر على خلق توازن أفضل بين عمليتي الاستكشاف والاستغلال. بعد اجراء ستة تجارب مختلفة من حيث أشكال البينات تم اثبات ان التحسين في هذه الخوارزمية يؤدي الى انتاج حلول عالية الجودة من ناحية قصر طول المسار المكتشفAnt colony optimization is a meta-heuristic algorithm inspired by the foraging behavior of real ant colony. The algorithm is a population-based solution employed in different optimization problems such as classification, image processing, clustering, and so on. This paper sheds the light on the side of improving the results of traveling salesman problem produced by the algorithm. The key success that produces the valuable results is due to the two important components of exploration and exploitation. Balancing both components is the foundation of controlling search within the ACO. This paper proposes to modify the main probabilistic method to overcome the drawbacks of the exploration problem and produces global optimal results in high dimensional space. Experiments on six variant of ant colony optimization indicate that the proposed work produces high-quality results in terms of shortest route

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

The Traveling Salesman Problem: Deceptivley Easy to State; Notoriously Hard to Solve

The purpose of this thesis is to give an overview of the history of the Traveling Salesman Problem and to show how it has been an integral part of the development of the fields of Integer Programming, and Combinatorial Optimization. The thesis starts in the 1800s and progresses through current attempts on solutions of the problem. The thesis is not meant to describe in detail every attempt made, nor to describe an original solution, but to provide a high level overview of every solution attempt, and to guide the reader on what has been done, and what still can be done

Hybrid biogeography-based evolutionary algorithms

Hybrid evolutionary algorithms (EAs) are effective optimization methods that combine multiple EAs. We propose several hybrid EAs by combining some recently-developed EAs with a biogeography-based hybridization strategy. We test our hybrid EAs on the continuous optimization benchmarks from the 2013 Congress on Evolutionary Computation (CEC) and on some real-world traveling salesman problems. The new hybrid EAs include two approaches to hybridization: (1) iteration-level hybridization, in which various EAs and BBO are executed in sequence; and (2) algorithm-level hybridization, which runs various EAs independently and then exchanges information between them using ideas from biogeography. Our empirical study shows that the new hybrid EAs significantly outperforms their constituent algorithms with the selected tuning parameters and generation limits, and algorithm-level hybridization is generally better than iteration-level hybridization. Results also show that the best new hybrid algorithm in this paper is competitive with the algorithms from the 2013 CEC competition. In addition, we show that the new hybrid EAs are generally robust to tuning parameters. In summary, the contribution of this paper is the introduction of biogeography-based hybridization strategies to the EA community

Incorporating Memory and Learning Mechanisms Into Meta-RaPS

Due to the rapid increase of dimensions and complexity of real life problems, it has become more difficult to find optimal solutions using only exact mathematical methods. The need to find near-optimal solutions in an acceptable amount of time is a challenge when developing more sophisticated approaches. A proper answer to this challenge can be through the implementation of metaheuristic approaches. However, a more powerful answer might be reached by incorporating intelligence into metaheuristics. Meta-RaPS (Metaheuristic for Randomized Priority Search) is a metaheuristic that creates high quality solutions for discrete optimization problems. It is proposed that incorporating memory and learning mechanisms into Meta-RaPS, which is currently classified as a memoryless metaheuristic, can help the algorithm produce higher quality results. The proposed Meta-RaPS versions were created by taking different perspectives of learning. The first approach taken is Estimation of Distribution Algorithms (EDA), a stochastic learning technique that creates a probability distribution for each decision variable to generate new solutions. The second Meta-RaPS version was developed by utilizing a machine learning algorithm, Q Learning, which has been successfully applied to optimization problems whose output is a sequence of actions. In the third Meta-RaPS version, Path Relinking (PR) was implemented as a post-optimization method in which the new algorithm learns the good attributes by memorizing best solutions, and follows them to reach better solutions. The fourth proposed version of Meta-RaPS presented another form of learning with its ability to adaptively tune parameters. The efficiency of these approaches motivated us to redesign Meta-RaPS by removing the improvement phase and adding a more sophisticated Path Relinking method. The new Meta-RaPS could solve even the largest problems in much less time while keeping up the quality of its solutions. To evaluate their performance, all introduced versions were tested using the 0-1 Multidimensional Knapsack Problem (MKP). After comparing the proposed algorithms, Meta-RaPS PR and Meta-RaPS Q Learning appeared to be the algorithms with the best and worst performance, respectively. On the other hand, they could all show superior performance than other approaches to the 0-1 MKP in the literature