LCMFO: An Improved Moth-Flame Algorithm for Combinatorial Optimization Problems Heba AbdElhamid #1 , Ahmed Helmi *2 , Ibrahim Ziedan #3 # Computer and Systems Department, Faculty of Engineering Zagazig University Computer and Systems Department - Faculty of Engineering - Zagazig University - Egypt 1 softintel8@gmail.com 3 ieziedan@gmail.com * Computer and Systems Department. Faculty of Engineering Zagazig University Computer and Systems Department- Faculty of Engineering - Zagazig University - Egypt 2 amhm162@gmail.com Abstract— Combinatorial optimization problems (COPs) are challenging class of problems in the field of optimization. Permutations are preferred as solution representation scheme in most cases. Metaheuristic techniques can be used to look for good solutions for COPs with low cost. Moth-flame algorithm (MFO) is one recent population-based metaheuristic technique for continuous optimization problems. In this work improvement of MFO when used to solve COPs is studied. An improved version of MFO (called LCMFO) where Lévy-flight function is used to prepare initial solutions is proposed. Also crossover functions of genetic algorithms are used together with the basic technique of MFO to generate new solutions. Both MFO and LCMFO are tested with travelling salesman problem (TSP) as one popular COP. Experimental results show that there is a notable improvement of about 20-40% in the quality of solutions found by LCMFO over MFO only. Keyword - Combinatorial Optimization Problems (COP), Moth-Flame Optimization (MFO), Crossover Function, Lévy-Flight Distribution, Travelling Salesman Problem (TSP) I. INTRODUCTION Combinatorial optimization problems (COPs) are NP-hard problems [1],[2]. It takes time of an exponential order for an exact algorithm to find the optimal solution [3]. Many real-world instances belong to such discrete optimization class such as travelling salesman problem (TSP) [1],[4],[5], assignment problem [6],[7], constraint satisfaction problem [8], knapsack problem [9], minimum spanning trees [10], scheduling problems [11], vehicle routing problem[12], and others. For COPs it requires enumerating the whole combinatorial search space in a brute force manner if only optimal solution is required and nothing else. The most famous COP in literature is probably TSP. In TSP, one has to make a tour of  cities, starting from a root city, passing by each city just once, then returning back to the root one. Solutions of TSP are best encoded as permutations of cities [13]. Here the optimal tour has the minimal total covered distance. Assuming that travels between any pair of cities are the same (i.e. symmetric TSP) and there is no constraint that may reject some tour then it easily noted that the size of the solution space is ሺ௡ଵሻ! ଶ . This is considering the worst-case of TSP. However for large , asymmetric TSP and TSP with constraints have a search space of size Ω ቀቀ ௡ ௘ ቁ ௡ ቁ. Moreover, going beyond the brute force method and using tree- based algorithms [13] (commonly called branch and x methods including branch and bound, branch and cut, and branch and price) an exponential cost may result. Besides exact methods of COPs, approximation algorithms such as metaheuristic techniques [13] found a great interest of the research community to solve different classes of optimization problems in a reasonable time. However, there is no guarantee of reporting one optimal solution by a metaheuristic algorithm but finding a near-optimal solution in a reasonable time may be accepted. Almost all metaheuristic techniques depend on real values in the range (0,1) to encode solutions of the studied problem. Also finding a new neighbourhood of a current solution follows a systematic way during the search. Thus there is a difficulty facing metaheuristic methods when solving COPs exemplified in the different encoding schemes for solutions generated by an applied technique as well as the final solutions of the problem (i.e. solutions required by the cost function to calculate the fitness). Mapping from a real-valued solution to a permutation one degrades the quality of generated solutions regardless of how much the mapping methods are good. This decreases the convergence rate of applied algorithm. Also the ability to overcome local minima areas in the search space becomes limited. Therefore it is worthy to investigate how a metaheuristic technique is improved when solving a COP. ISSN (Print) : 2319-8613 ISSN (Online) : 0975-4024 Heba AbdElhamid et al. / International Journal of Engineering and Technology (IJET) DOI: 10.21817/ijet/2018/v10i6/181006091 Vol 10 No 6 Dec 2018-Jan 2019 1793