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

Computational Numerical Solution for Traveling Salesman Problem

This paper examined and analysed the desire of Traveling Salesman Problem (TSP) to find the cheapest way of visiting all given set of cities and returning to the starting point.     We presented a unique decomposition approach model for TSP in which the requirements and features of practical application in communication network, road transportation and supply chains are put into consideration.  We used a Mathematical Modeling solution with the application of Ant Colony Search Algorithm (ACSA) approach for result computation. In our approach, different Agents were created for difference purposes.   Information agent gathered information about best tour and detected the solution agent that arrived at a given point with information message containing details of where the solution agent has come from as well as best tour cost.  The place ant performs local pheromone decay on the relevant links.   This help to avoid random visit to irrelevant edges and allows the place ant to calculate the cost of tour of all place ants including the latest pheromone level on the links to each of the place ants. The solution agent uses available information to decide  which node to visit next and informs the place ant of  its decision to move to a given destination and update better tour  previously sampled while information about where to go next also obtained.       The place ant updates its pheromone value for that link using the equivalent of the algorithm for local pheromone update.    The cycle continues until solution agent arrives at its destination. The main advantage of our approach is that it permits the use of mixed integer programming and combinatorial optimization techniques to compute real optimal routing path, solving the problem in practice by returning actual shortest route with its numerical value and not the best effort result as provided by some previous models and analytical methods. The implementation was carried out using C# programming language.  Data used were generated and the performance evaluation of the model was carried out through simulation using Matlab 7.0.  The result shows that by considering all possible paths between a node as the source and another as the destination, all possible routes for a particular journey with shortest route in each case were generated. Keywords: Ant Colony, Combinatorial Optimization, Mixed Integer Programming, Pheromone, Search Algorithm and Traveling Salesman

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

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 Application of Ant Colony Optimization

The application of advanced analytics in science and technology is rapidly expanding, and developing optimization technics is critical to this expansion. Instead of relying on dated procedures, researchers can reap greater rewards by utilizing cutting-edge optimization techniques like population-based metaheuristic models, which can quickly generate a solution with acceptable quality. Ant Colony Optimization (ACO) is one the most critical and widely used models among heuristics and meta-heuristics. This book discusses ACO applications in Hybrid Electric Vehicles (HEVs), multi-robot systems, wireless multi-hop networks, and preventive, predictive maintenance

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

Computational Numerical Solution for Traveling Salesman Problem

This paper examined and analyzed the desire of Traveling Salesman Problem (TSP) to find the cheapest way of visiting all given set of cities and returning to the starting point.     We presented a unique decomposition approach model for TSP in which the requirements and features of practical application in communication network, road transportation and supply chains are put into consideration.  We used a Mathematical Modeling solution with the application of Ant Colony Search Algorithm (ACSA) approach for result computation.  In our approach, different Agents were created for difference purposes.   Information agent gathered information about best tour and detected the solution agent that arrived at a given point with information message containing details of where the solution agent has come from as well as best tour cost.  The place ant performs local pheromone decay on the relevant links.   This help to avoid random visit to irrelevant edges and allows the place ant to calculate the cost of tour of all place ants including the latest pheromone level on the links to each of the place ants. The solution agent uses available information to decide  which node to visit next and informs the place ant of  its decision to move to a given destination and update better tour  previously sampled while information about where to go next also obtained.  The place ant updates its pheromone value for that link using the equivalent of the algorithm for local pheromone update.  The cycle continues until solution agent arrives at its destination. The main advantage of our approach is that it permits the use of mixed integer programming and combinatorial optimization techniques to compute real optimal routing path, solving the problem in practice by returning actual shortest route with its numerical value and not the best effort result as provided by some previous models and analytical methods. The implementation was carried out using C# programming language.  Data used were generated and the performance evaluation of the model was carried out through simulation using Matlab 7.0.  The result shows that by considering all possible paths between a node as the source and another as the destination, all possible routes for a particular journey with shortest route in each case were generated. Keywords: Ant Colony, Combinatorial Optimization, Mixed Integer Programming, Pheromone, Search Algorithm and Traveling Salesman

Traveling Salesman Problem

The idea behind TSP was conceived by Austrian mathematician Karl Menger in mid 1930s who invited the research community to consider a problem from the everyday life from a mathematical point of view. A traveling salesman has to visit exactly once each one of a list of m cities and then return to the home city. He knows the cost of traveling from any city i to any other city j. Thus, which is the tour of least possible cost the salesman can take? In this book the problem of finding algorithmic technique leading to good/optimal solutions for TSP (or for some other strictly related problems) is considered. TSP is a very attractive problem for the research community because it arises as a natural subproblem in many applications concerning the every day life. Indeed, each application, in which an optimal ordering of a number of items has to be chosen in a way that the total cost of a solution is determined by adding up the costs arising from two successively items, can be modelled as a TSP instance. Thus, studying TSP can never be considered as an abstract research with no real importance