A comparative study of solution representations for the unrelated machines environment Ivan Vlašic ´ a,b , Marko Ðurasevic ´ a,⇑ , Domagoj Jakobovic ´ a a University of Zagreb, Faculty of Electrical Engineering and Computing, Croatia b minus5, Croatia article info Article history: Received 4 May 2019 Revised 27 March 2020 Accepted 18 May 2020 Available online 4 June 2020 Keywords: Unrelated machines environment Genetic algorithms Solution representations Scheduling abstract Scheduling problems are quite difficult to solve since in many cases no exact algorithms exist which can obtain the optimal solution in a reasonable amount of time. Therefore, these problems are often solved by using various metaheuristic methods, like genetic algorithms. To use these methods, the first step which needs to be performed is to define an encoding scheme that will be used to represent the solutions. Until now, several encoding schemes were proposed for the unrelated machines environment, each of which comes with its own benefits and drawbacks. However, the performance of metaheuristic methods depends on the applied encoding scheme. Unfortunately, no extensive research was performed in the lit- erature to compare different solution representations for the unrelated machines scheduling problem. Therefore, the choice of the solution representation used is mostly provisional and is usually not based on any existing knowledge of how it would perform on the considered problem. This can cause the algo- rithms to obtain suboptimal results, which can lead to wrong conclusions about the performance. Thus, the goal of this paper is to test seven solution representations that were used in previous studies to rep- resent solutions for the unrelated machines scheduling problem. The selected solution representations were tested for optimising four scheduling criteria, while additionally measuring the execution time of the genetic algorithm when using each of the encodings. The obtained results demonstrate that the encoding which is based on the permutation of jobs obtains the best results, making it the superior encoding scheme for this type of scheduling problem. Ó 2020 Elsevier Ltd. All rights reserved. 1. Introduction Scheduling is a process by which a certain number of jobs are allocated to a set of machines, in a way that one or more user- defined criteria are optimised (Pinedo, 2012). In the unrelated machines environment, each job needs to be allocated to only one of the available machines. However, each job has a different execution time for each of the available machines, meaning that the choice of the machine on which a job will be executed can have a high impact on the generated schedule. Scheduling in the unrelated machines environment can be found in many practical real-world examples, such as scheduling jobs on multiprocessor computers, airplanes on landing lanes in airports, operations to operating rooms in hospitals, jobs in circuit board and semiconduc- tor manufacturing, and many other (Fanjul-Peyro and Ruiz, 2012; Lee et al., 2013; Wang et al., 2013). Although scheduling problems can be solved by using various optimisation methods, heuristic and metaheuristic methods are most often used to obtain solutions. The reason is that most scheduling problems are NP-hard, there- fore an algorithm that could solve such problems optimally in a reasonable amount of time is unknown. However, metaheuristic algorithms can obtain solutions of good quality in a relatively short amount of time, which makes them suitable for solving various scheduling problems. Although a variety of different metaheuristic methods were proposed in the literature, the genetic algorithm (GA) (Goldberg, 1989; Mitchell, 1998; Eiben and Smith, 2015) represents one of the most popular and widely used metaheuristic algorithms. To be able to apply GAs for solving a certain scheduling problem, it is mandatory to define how the solutions to this problem are rep- resented in the algorithm. The choice of the solution representa- tion has a large impact not only on the effectiveness, execution time, and memory consumption of the algorithm but also on the genetic operators which can be used for adapting the solutions. Therefore, it is important to select the appropriate solution repre- sentation for solving the problem at hand. Although many different https://doi.org/10.1016/j.cor.2020.105005 0305-0548/Ó 2020 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: ivan.vlasic@outlook.com (I. Vlašic ´), marko.durasevic@fer.hr (M. Ðurasevic ´), domagoj.jakobovic@fer.hr (D. Jakobovic ´). Computers and Operations Research 123 (2020) 105005 Contents lists available at ScienceDirect Computers and Operations Research journal homepage: www.elsevier.com/locate/caor