A Hybrid Firefly - VNS Algorithm for the Permutation Flowshop Scheduling Problem Andromachi Taxidou 1 , Ioannis Karafyllidis 1 , Magdalene Marinaki 1 , Yannis Marinakis 1(B ) , and Athanasios Migdalas 2 1 School of Production Engineering and Management, Technical University of Crete, Chania, Greece machi taxidou@hotmail.com, Io Karafi@hotmail.com, magda@dssl.tuc.gr, marinakis@ergasya.tuc.gr 2 Industrial Logistics, Lule˚ a Technical University, 97187 Lule˚ a, Sweden athmig@ltu.se Abstract. In this paper a Permutation Flowshop Scheduling Problem is solved using a hybridization of the Firefly algorithm with Variable Neighborhood Search algorithm. The Permutation Flowshop Scheduling Problem (PFSP) is one of the most computationally complex problems. It belongs to the class of combinatorial optimization problems charac- terized as NP-hard. In order to find high quality solutions in reason- able computational time, heuristic and metaheuristic algorithms have been used for solving the problem. The proposed method, Hybrid Firefly Variable Neighborhood Search algorithm, uses in the local search phase of the algorithm a number of local search algorithms, 1-0 relocate, 1-1 exchange and 2-opt. In order to test the effectiveness and efficiency of the proposed method we used a set of benchmark instances of different sizes from the literature. Keywords: Permutation Flowshop Scheduling Problem · Firefly algorithm · Variable Neighborhood Search 1 Introduction The flowshop scheduling problem was proposed by Johnson [6] and it is a widely studied scheduling problem. In this problem, there is a set of jobs (task or items) that must be processed from a set of machines in the same order. All jobs are independent and must start their process from the same starting machine. The machines are constantly available. The purpose of the process is to find the sequence for the processing of the jobs in the machines in order to optimize a given criterion. In the literature, the most frequently used criterion is the minimization of the maximum completion time (makespan) [6]. The Firefly Algorithm (FA), proposed by Yang [24], is a nature inspired algorithm and it depends on the glowing behavior of fireflies. Each firefly has c  Springer Nature Switzerland AG 2019 A. Sifaleras et al. (Eds.): ICVNS 2018, LNCS 11328, pp. 274–286, 2019. https://doi.org/10.1007/978-3-030-15843-9_21