Firefly Algorithm (FA) is a nature-inspired optimization algorithm that can be successfully applied to continuous optimization problems. However, lot of practical problems are formulated as discrete optimization problems. In this paper a hybrid discrete firefly algorithm (HDFA) is proposed to solve the multi-objective flexible job shop scheduling problem (FJSP). FJSP is an extension of the classical job shop scheduling problem that allows an operation to be processed by any machine from a given set along different routes. Three minimization objectives -the maximum completion time, the workload of the critical machine and the total workload of all machines are considered simultaneously. This paper also proposes firefly algorithm's discretization which consists of constructing a suitable conversion of the continuous functions as attractiveness, distance and movement, into new discrete functions. In the proposed algorithm discrete firefly algorithm (DFA) is combined with local search (LS) method to enhance the searching accuracy and information sharing among fireflies. The experimental results on the well-known benchmark instances and comparison with other recently published algorithms shows that the proposed algorithm is feasible and an effective approach for the multi-objective flexible job shop scheduling problems.

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The application of metaheuristic algorithms to combinatorial optimization problems is on the rise and is growing rapidly now than ever before. In this paper the historical context and the conducive environment that accelerated this particular trend of inspiring analogies or metaphors from various natural phenomena are analysed. We have implemented the Ant System Model and the other variants of ACO including the 3-Opt, Max-Min, Elitist and the Rank Based Systems as mentioned in their original works and we converse the missing pieces of Dorigo's Ant System Model. Extensive analysis of the variants on Travelling Salesman Problem and Job Shop Scheduling Problem shows how much they really contribute towards obtaining better solutions. The stochastic nature of these algorithms has been preserved to the maximum extent to keep the implementations as generic as possible. We observe that stochastic implementations show greater resistance to changes in parameter values, still obtaining near optimal solutions. We report how Polynomial Turing Reduction helps us to solve Job Shop Scheduling Problem without making considerable changes in the implementation of Travelling Salesman Problem, which could be extended to solve other NPHard problems. We elaborate on the various parallelization options based on the constraints enforced by strong scaling (fixed size problem) and weak scaling (fixed time problem). Also Electronic supplementary material The online version of this article (doi:10.1007/s10462-015-9441-y) contains supplementary material, which is available to authorized users.The code is licensed and made available under the open source Creative Commons License to enable constructive criticism, reproducible research and to ensure maximum dissemination of the research work. The code, results and graphs are provided as supplementary materials which will also be made available as a project in Google code and Git-Hub. B Anandkumar PrakasamA. Prakasam, N. Savarimuthu we elaborate on how probabilistic behaviour helps us to strike a balance between intensification and diversification of the search space.

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