Differential Evolution Algorithm for Multiple Inter-dependent Components Traveling Thief Problem Ismail M Ali 1 , Daryl Essam 2 and Kathryn Kasmarik 3 School of Engineering and IT, University of New South Wales Canberra, Australia Email: 1 Ismail.Ali@student.adfa.edu.au, 2 d.essam@adfa.edu.au, 3 Kathryn.Kasmarik@adfa.edu.au Abstract—Differential evolution was mainly proposed for solv- ing optimization problems with continuous decision variables because of its Euclidean distance-based learning concept. This made it unsuitable for many binary and discrete problems. However, several studies approved the applicability of differential evolution algorithm for effectively solving such problems. In this paper, a new design of differential evolution, which incorporates mapping and repairing methods, modified mutation operator and local searches, is proposed to solve the complex multi- components traveling thief problems that are characterized by both binary and discrete parameters. Also, a novel initialization and repairing method, which enables differential evolution’s operators to only evolve solutions of one component and optimally distribute/update the solutions of the other one with considering the inter-dependency between both components, is introduced. To judge the performance of the proposed algorithm, 13 strongly correlated instances of traveling thief problems have been solved and the results have been compared with those from 24 self- designed and state-of-the-art algorithms. Results demonstrated the competitive performance of the proposed algorithm in terms of the quality of obtained solutions and computational time. Index Terms—Differential evolution, Traveling thief problem, Combinatorial optimization problem, Evolutionary algorithms I. I NTRODUCTION Differential Evolution (DE) was first introduced in 1997 as a method that optimizes a problem with continuous decision variables by iteratively trying to improve a candidate solution [1]. DE is a stochastic population-based search technique, which is inspired by the biological model of evolution and mimicked the natural selection model. As it has a long history of successfully solving continuous optimization problems, DE is considered a powerful tool for solving such problems [2]. In its process, DE adapted three main evolutionary operators, namely: mutation, crossover and selection to direct the search towards (near-) optimal solutions. The quality of the solutions in DE is measured by a pre-defined fitness function, called the objective function, and based on the obtained fitness values, the solutions are ranked. The Traveling Thief Problem (TTP) is an NP-hard Combi- natorial Optimization Problem (COP) with both discrete and binary decision variables [3]. TTP is a recent benchmark for problems with multiple inter-dependent components. It is a combination of two well-known optimization problems: Traveling Salesman Problem (TSP) and Knapsack Problem (KP). These two components are integrated in such a way that the optimal solution for every single component (problem) does not essentially lead to obtaining the optimal solution for TTP. This means that these two components are interdependent in the sense that a solution for one component affects the quality of the solutions for other components, and hence the quality of the solutions for the whole problem. Practically, TTP reflects the complexity of real-world applications, which contain more than single NP-hard problems. This can be widely observed in many fields, such as planning, supply chain, scheduling, routing, and transportation of water tanks [4]. Several techniques have recently been developed for solving the newly introduced TTP benchmark, which is providing many test instances of the interdependent multi-components problems of TTP [3]. The first attempt to tackle TTPs was in 2014, by Polyakovskiy et al. who applied several heuristic and meta-heuristic techniques, such as simple constructive heuristic (SH) and an iterative heuristic. In their work, they incorporated the Random Local Search (RLS) with a simple Evolutionary Algorithm (EA) to repeatedly create a single new solution and record it as the best. In the next iteration, the solution is improved by comparing the newly created one with the best of the previous iterations [5]. In the same year, another optimization problem solver, called CoSolver, was introduced to solve the TTP by decomposing it into two sub-problems and solve them by separate models while main- taining the communication between them. Then it combines the obtained solutions to create the overall TTP solution [6]. Moreover, a memetic algorithm with two-stage local search and multi-complexity reduction approaches has been proposed for solving large-scale instances of TTPs in a maximum of 10 minutes of computational time [7]. Recently, an Ant Colony Optimization (ACO) algorithm with two hybridization levels is also introduced for TTPs [8]. Further investigation of TTPs has been conducted by using a meta-heuristic technique (i.e. Genetic Algorithm (GA)) with multiple local searches, such as 2-OPT, insertion and bit-flip [24]. Because of its importance in various applications and its ability to present the real- world problems with their complexity, and although many state-of-the-art techniques have been proposed, there is still a recent and high demand to find an algorithm that can solve 978-1-7281-6929-3/20/$31.00 ©2020 IEEE