Basic Differential Evolution Research Articles

Multi-operator opposition-based learning with the neighborhood structure for numerical optimization problems and its applications

Opposition-based learning (OBL) is an effective strategy that adjusts the population to accelerate the convergence of the algorithm. However, OBL involves two phases (generation and calculation) that are seldom mentioned simultaneously. Besides, individual information in OBL is not fully used to guide evolution. First, a novel neighborhood generation strategy is proposed to fully use individual information. There are numerous variants of the neighborhood structure. Three prominent neighborhood structures are designed to demonstrate their benefits in various dimensions. Second, a multi-operator calculation method with two distributions is introduced to reach the equilibrium between diversification and intensification. Consequently, this paper proposes three multi-operator neighborhood-based OBL variants (MNOBLs). Compared with the traditional OBL variants, MNOBLs generate the opposite population by using a neighborhood of the individuals. A well-balanced calculation method can lead to good convergence behavior. Subsequently, the proposed MNOBLs are embedded into the basic differential evolution (DE) algorithm to obtain the proposed algorithms. The extensive and rigorous numerical experiments are designed at three levels: strategies, algorithms, and functions. Specifically, the effectiveness and efficiency of the proposed algorithms are verified by comparing them with 13 state-of-the-art DE variants and four metaheuristics on the CEC 2017, and CEC 2022 test suites, and two real-world optimization problems. The comprehensive results show that the proposed OBL variants provide promising results for most problems.

Parameter tuning for software fault prediction with different variants of differential evolution

The cost of software testing could be reduced if faulty entities were identified prior to the testing phase, which is possible with software fault prediction (SFP). In most SFP models, machine learning (ML) methods are used, and one aspect of improving prediction accuracy with these methods is tuning their control parameters. However, parameter tuning has not been addressed properly in the field of software analytics, and the conventional methods (such as basic Differential Evolution, Random Search, and Grid Search) are either not up-to-date, or suffer from shortcomings, such as the inability to benefit from prior experience, or are overly expensive. This study aims to examine and propose parameter tuners, called DEPTs, based on different variants of Differential Evolution for SFP with the Swift-Finalize strategy (to reduce runtime), which in addition to being up-to-date, have overcome many of the challenges associated with common methods. An experimental framework was developed to compare DEPTs with three widely used parameter tuners, applied to four common data miners, on 10 open-source projects, and to evaluate the performance of DEPTs, we used eight performance measures. According to our results, the three tuners out of five DEPTs improved prediction accuracy in more than 70% of tuned cases, and occasionally, they exceeded benchmark methods by over 10% in case of G-measure. The DEPTs took reasonable amounts of time to tune parameters for SFP as well.

Airline capacity distribution under financial budget and resource consideration

Capacity distribution is a challenging issue for an airline under financial budget and resource consideration. It is a large-scale optimization problem covering both long-term planning and short-term operating arrangements. This study investigates on the airline capacity distribution problem with financial budget and resource consideration. It contains subproblems of financial budget arrangement, fleet introduction, and fleet assignment. Among them, financial budget is arranged in multiple decision periods, fleet introduction is decided under fixed time points, while fleet assignment is decided under all available time points. To tackle this problem, an integer programming model is formulated for descriptions. Then, an integrated algorithm of modified Variable Neighborhood Search (VNS) and Branch-and-bound (B&B) strategy is developed to find solutions. In detail, a greedy heuristic approach is utilized to generate an initial solution for fleet introduction, the modified B&B strategy is utilized to generate the optimal solution for fleet assignment and the modified VNS is applied to update current solution for a new one with better quality. In addition, budget limit checks are added for financial budget arrangements. Finally, the hybrid algorithm is tested on efficiency and stability. It is also compared to other algorithms which replace the modified VNS by basic VNS, differential evolution and genetic algorithm. Computational results show that performance of our approach is powerful in terms of objective value, convergence speed and stability.

Determination of the Optimum Values of the Natural Frequency of the Nanocomposite Multilayer Organic Solar Cell by using Basic Differential Evolution Optimization Algorithms

By the analytical approach, the equation of natural frequency of nanocomposite multilayer organic solar cell (NMOSC) is analyzed in this work. The NMOSC is composed of 6 layers: Al, LiF, P3HT: PCBM, PEDOT: PSS, ITO and Glass. By using the classical plate theory, the Hooke’s Law, the geometrical compatibility equation and the nonlinear equilibrium equations are proposed. These equations combined with the Galerkin method in order to investigate the effect of elastic foundations, geometrical parameters and initial imperfection on the natural frequency. Besides, in order to determine the maximum natural frequency of NMOSC, basic differential evolution optimization algorithm (DE) was used with five variables: elastic foundations include Winkeler foudation and Pasternak foudation, temperature, width and length. In the numerical results, the influence of length to width ratio, imperfection, elastic foundations and temperature increment were evaluated in detail. The optimal results from DE algorithm are shown and compared.

A population state evaluation-based improvement framework for differential evolution

Differential evolution (DE) is one of the most efficient evolutionary algorithms for solving numerical optimization problems; however, it still suffers from premature convergence and stagnation. To address these problems, we propose a population state evaluation (PSE)-based improvement framework that can be freely embedded into various existing DE variants and population-based metaheuristic algorithms. The PSE framework comprises two population state evaluation mechanisms: one for tracking the optimization state of the population during evolution and the other for evaluating the distribution state of individuals in the population to identify the specific problem (premature convergence or stagnation) encountered by the corresponding algorithm. In addition, we design two intervention operations (dispersion and aggregation) to address premature convergence and stagnation. To verify the effectiveness of the PSE framework, we conduct comparison experiments using nine algorithms (including two basic DE algorithms, six state-of-the-art DE variants, and one non-DE algorithm) to optimize four real-world problems and 59 test functions from the IEEE CEC 2014 and IEEE CEC 2017 testbeds. The experimental results show that the PSE framework can significantly improve the optimization performance of the existing algorithms.

Optimal Trading in Local Electricity Market Based on an Enhanced Evolutionary Algorithm with Distribution Estimation

The evolution from passive to active distribution network, the ability to actively control distributed energy resources and to perform bilateral energy exchange has significantly complicated the market transactions in recent years due to activities among consumers and prosumers. Local electricity market (LEM) is challenged to optimize welfare as well as reduction of emissions by involving participants in energy transactions. In this paper, the trading strategy is modeled as a single-objective optimization problem to determine each agent's optimal energy trading. Modern heuristic optimization techniques have proven their efficiency and robustness in large optimization problems and thus, an enhanced artificial bee colony (EABC) incorporating a distribution estimation algorithm is proposed, which guides the search for optimum by building and sampling explicit probabilistic models of promising candidate solutions. With the help of a distribution estimation algorithm, the EABC eliminates many tuning parameters in the original ABC and increases its exploitation to obtain more robust and competitive performance. A comparative study is conducted with the basic ABC, differential evolution, and particle swarm optimization, and the EABC demonstrates its efficiency on the case study where nine agents are involved in trading energy in the day-ahead LEM.

Investigation of Mechanical and Tribological Characteristics of Medical Grade Ti6al4v Titanium Alloy in Addition with Corrosion Study for Wire EDM Process

Metals used in biomedical applications are frequently coated to prevent oxidation and metallic ion release, both of which can be harmful due to toxic effects. To prevent these adverse effects of metals, researchers have focused their efforts on developing various coating techniques, facilitating surface coating, or obtaining functional surfaces. (WEDM) is now considered a difficult method of obtaining functional surfaces for medical applications. The properties of the surface and subsurface layers obtained by the WEDM method are particularly interesting in this regard. The analysis utilised RSM-based computational technique to evaluate the WEDM characteristics (MRR SR) of Ti6Al4V Titanium Alloy in biomedical applications. The biggest drawback of the material in the biomedical industry, which includes orthopedic applications and dental implants, would be that it releases harmful atoms such as iron, chromium, and nickel into the bodily fluid environment. To combat the problems, a hydroxyapatite layer applied to the metal implant improves biocompatibility, osteocompatibility, and antimicrobial properties. By comparing the Modified Differential Evolution (MDE) approach to the basic differential evolution (DE) optimization strategy, the effectiveness of the MDE approach has been established. According to the cyclic polarized test, the Hap coated Titanium material had better corrosion resistance than the pure sample. The Hap coated titanium material has a higher zone of inhibition than the pure sample. The next step is to synthesis hydroxyapatite from cuttlebone, which is then electrodeposited onto titanium. FTIR, electrochemical tests, FESEM, and SEM were used to describe the coated sample, as well as an antibacterial test using E. Coli and B. Cereus bacteria. Because it is porous, the Hap coating helps bone tissue growth by preventing detrimental metal ions from escaping into the biological medium. The corrosion inhibition efficiency of the coated sample was performed in SBF (NaHCO3—0.35 g/L, MgCl2.6H2O—0.30 g/L, CaCl2·2H2O—0.37 g/L, K2HPO4. 3H2O—0.23 g/L, Na2SO4—0.071 g/L, NaCl—7.99 g/L, KCl—0.22 g/L, Tris’s buffer—6.063 g/L) solution using the potentiodynamic polarisation method and the solution pH was maintained.

Comparative evaluation of the performance of improved genetic algorithms and differential evolution for in-core fuel management of a research reactor

This paper presents a comparative evaluation of the performance of Genetic Algorithm (GA) and Differential Evolution (DE) algorithm applied to in-core fuel management of the DNRR research reactor. Two GA variants corresponding to two selection operators, i.e., tournament (GA1) and roulette wheel (GA2) selections, respectively, with two-point crossover and scramble mutation were implemented for the ICFM problem. A comprehensive survey of the GA control parameters such as population size, crossover-type, mutation probability, and elitist archive size has been conducted to optimize the performance of the GAs. The basic DE was implemented with a standard mutation strategy DE/rand/1/bin. Numerical computations were performed based on the DNRR research reactor core loaded with 100 highly enriched uranium fuel (HEU) bundles for evaluating the performance of the GA and DE algorithms. Two main objectives were included in the fitness function to maximize the fuel cycle length and flatten the power distribution. The performance of the two GA variants and the basic DE was investigated with the same population size, fitness function, and convergence criterion. Each method was performed with 50 independent runs, and the best fitness values were collected for statistical analysis using Kruskal–Wallis and Mann–Whitney tests in comparison among the three methods. The statistical analysis shows that the performance of GA1 with tournament selection and DE are not significantly different and are better than GA2 with roulette wheel selection. DE is stable and efficient in exploring the search space to approach the global optimal solution in most runs. While, GA1 and GA2 were trapped at local optima by about 26% and 38%, respectively. However, the best solutions obtained with GA1 and GA2 after 50 independent runs are better than that obtained with DE in term of fitness values. This suggests an improvement of the basic DE is needed to maintain the potential good solutions during the evolution process.

Fractional Order Darwinian Pigeon-Inspired Optimization for Multi-UAV Swarm Controller

This paper presents a novel multiple unmanned aerial vehicle (UAV) swarm controller based on the fractional calculus theory. This controller is designed based on fractional order Darwinian pigeon-inspired optimization (FODPIO) and PID algorithm. Several comparative simulations are conducted in the paper. The simulation results reveal that FODPIObased multi-UAV formation controller is superior to the basic PIO and differential evolution (DE) method. The fractional coeffcient in FODPIO algorithm makes it effective optimization with fast convergence rate, small overshoot, and better stability. Therefore, the controller proposed in this paper is feasible and robust.

An adaptive regeneration framework based on search space adjustment for differential evolution

Differential evolution (DE) is a well-known evolutionary algorithm with simple operation and excellent performance, which has been applied to solve various optimization problems. To alleviate the problem of premature convergence or population stagnation faced by DE algorithm, this paper proposes an adaptive regeneration framework based on search space adjustment (ARSA), which can be easily embedded into various DE variants. When one individual cannot get improved for several generations, the ARSA framework will be triggered to randomly generate a substitute individual from a dynamic search space which is determined by a given adjustment mechanism controlled by two different levels of parameters. The trigger condition for each individual is adaptively controlled by its status in the current population. The space adjustment mechanism contains two strategies, one focuses on global exploration while the other on local exploitation. Moreover, the ARSA framework does not add any parameters that need to be pre-set, and all the included parameters are adaptive. To verify the availability of ARSA framework for solving complex optimization problems, thirty functions with different dimensions from IEEE CEC 2017 test platform and three real-life problems are employed for comparative experiments. The experimental results indicate that our ARSA framework notably improves the performance of two basic DE algorithms and six state-of-the-art DE variants.

Auto-Tuning Parameters of Fractional PID Controller Design for Air-Conditioning Fan Coil Unit

The traditional integer order PID controller manipulates the air-conditioning fan coil unit (FCU) that offers cooling and heating loads to each air-conditioning room in summer and winter, respectively. In order to maintain a steady indoor temperature in summer and winter, the control quality cannot meet the related requirements of air-conditioning automation, such as large overshoot, large steady state error, long regulating time, etc. In view of these factors, this paper develops a fractional order PID controller to deal with such problem associated with FCU. Then, by varying mutation factor and crossover rate of basic differential evolution algorithm adaptively, a modified differential evolution algorithm (MDEA) is designed to tune the satisfactory values of five parameters of indoor temperature fractional order PID controller. This fractional order PID control system is configured and the corresponding numerical simulation is conducted by means of MATLAB software. The results indicate that the proposed fractional order PID control system and MDEA are reliable and the related control performance indexes meet with the related requirements of comfortable air-conditioning design and control criteria.

MOEA/UE: A novel multi-objective evolutionary algorithm using a uniformly evolving scheme

In multi-objective optimization, the Pareto-optimal solutions must be widely and uniformly distributed on the Pareto-optimal front. When this condition is satisfied, decision makers can choose satisfactory solutions. In this paper, gaps in the evolving nondominated front are filled by a multi-objective evolutionary algorithm using a uniformly evolving scheme (MOEA/UE), thus generating a uniform and wide Pareto-optimal front. The MOEA/UE first calculates the distance between the points on the nondominated front and identifies those with relatively wide gaps. At the widely spaced points, the objective and constraint functions are approximated by linear functions expressed as Taylor series in auxiliary linear programs. Finally, the MOEA/UE solves these auxiliary linear programs and retains the optimal or intermediate solutions as the potentially superior individuals that will be selected in the next generation of the population. Convergence toward potentially superior solutions is accelerated by a mutation operator (a basic differential evolution operator). In constrained multi-objective optimization problems, the MOEA/UE implements a new constraint-handling strategy that pulls infeasible solutions into or nearby the feasible domain. Experimental results illustrate that MOEA/UE obtains wider and more uniform Pareto-optimal fronts and better Pareto-optimal solutions.

Nonlinear vibration and geometric optimization of nanocomposite multilayer organic solar cell under wind loading

This paper presents an analytical approach on the nonlinear vibration of nanocomposite multilayer organic solar cell (NMOSC) subjected to the combination of wind load and uniform temperature change. The NMOSC comprises of five layers, which are made of Al, P3HT:PCBM, PEDOT:PSS, Graphene and Glass. Compared to common models of the NMOSC, ITO is replaced by graphene to control the output colors of NMOSC. The nonlinear motion equations and the compatibility equation are deriving by taking into account the effect of viscous damping, von Karman nonlinearity terms, elastic foundations and initial imperfection. The formula of the wind load is improved for consideration in the general case that the direction of the wind load is not just perpendicular to the structure. Four edges of the NMOSC are assumed to be simply supported and immovable in the transverse plane. The Galerkin and Runge – Kutta methods are applied to obtain the dynamic response and the natural frequency of the NMOSC. Four optimization algorithms (Bees Algorithm, Basic differential evolution algorithm, enhanced colliding bodies optimization algorithm, social group optimization algorithm) are used to determine the maximum value of natural frequency of the NMOCS, which depends on nine variables including the geometrical parameters, elastic foundations stiffness and temperature increment. In the numerical results, the effects of elastic foundations, initial imperfection, viscous damping ratio, temperature increment, wind load and the length to width ratio on the dynamic response and natural frequency are considered in details. The maximum values of the natural frequency of the NMOSC with four algorithms are obtained and compared. The results show that the optimal value obtained by four algorithms is close to each other. However, the calculation time with enhanced colliding bodies optimization algorithm is the fastest and with Bees Algorithm is the longest of all.

An adaptive dimension level adjustment framework for differential evolution

Differential evolution (DE) has been recognized as one of the most popular evolutionary algorithms. There are numerous DE variants adopting multi-operators based cooperation strategy to improve their performance, but almost all of the adopted cooperation strategies are essentially implemented at the individual level or population level, and the implementation at the dimension level are scarce. In this paper, an adaptive dimension level adjustment (ADLA) framework is designed to relieve the premature convergence or stagnation problem faced by DE algorithm, which can be easily combined with diverse DE variants. When the current optimal individual cannot get improved for a given uninterrupted iterations, ADLA framework will be triggered to select some individuals at random according to specific rule and reinitialize portion of their dimensions from a dynamic search space that adjusted by a population level macroparameter and one individual level microparameter. Moreover, ADLA framework contains two reinitialization operators with different search characteristics, and the coordination between them is executed at the dimension level, which has potential advantages in balancing the global exploration ability and local exploitation ability. Extensive comparison experiments are carried out based on IEEE CEC 2014 test platform, two basic DE algorithms and six outstanding DE variants. The experimental results demonstrate that ADLA framework can memorably enhance the performance of every DE algorithm used for comparison.

ε-Constrained Differential Evolution Using an Adaptive ε-Level Control Method

Evolutionary algorithms and swarm intelligence algorithms have been widely used for constrained optimization problems for decades and numerous techniques for constraint handling have been proposed. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\varepsilon }$ </tex-math></inline-formula> -constrained method is a very effective one. In the literature, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\varepsilon }$ </tex-math></inline-formula> value was usually controlled via an exponential function, which is not competent for solving certain types of constrained optimization problems, e.g., whose global optima are located near the boundary of the feasible and infeasible regions. To solve this problem, this article proposes a new adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\varepsilon }$ </tex-math></inline-formula> control method and incorporate it into a basic differential evolution (DE) algorithm: (DE/rand/1/exp). Based on the information of constraint violation in the current population, the adaptive method controls the value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\varepsilon }$ </tex-math></inline-formula> through a simple heuristic rule. Compared with the traditional exponential function-based control methods, the proposed adaptive method can prevent the algorithm from being trapped into local optima while retaining the obtained near-optimal candidate solutions in the infeasible region for generating promising searching paths. Besides, we set the crossover rate (CR) as a more reasonable value for DE/rand/1/exp, which can enhance the efficiency significantly. The well-known 2006 IEEE Congress on Evolutionary Computation (CEC 2006) competition on real-parameter single-objective constrained optimization benchmark is adopted to evaluate the effectiveness of the proposed adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\varepsilon }$ </tex-math></inline-formula> -constrained DE. Fifteen constrained engineering optimization problems are collected from the literature to test the proposed algorithm. Moreover, the adaptive <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\varepsilon }$ </tex-math></inline-formula> control method is extended to an adaptive algorithm to solve the benchmark problems from CEC 2017. The comparison results confirm the superiority of the proposed method.

An Adaptive Differential Evolution Algorithm with Restart for Solving Continuous Optimization Problems

A new adaptive differential evolution algorithm with restart (ADE-R) is proposed as a general-purpose method for solving continuous optimization problems. Its design aims at simplicity of use, efficiency and robustness. ADE-R simulates a population evolution of real vectors using vector mixing operations with an adaptive parameter control based on the switching of two selected intervals of values for each scaling factor and crossover rate of the basic differential evolution algorithm. It also incorporates a restart technique to supply new contents to the population to prevent premature convergence and stagnation. The method is tested on several benchmark functions covering various types of functions and compared with some well-known and state-of-art methods. The experimental results show that ADE-R is effective and outperforms the compared methods.

ERG-DE: An elites regeneration framework for differential evolution

Differential evolution (DE) is one of the most popular paradigms of evolutionary algorithms. Numerous variants of basic DE have been developed in the past two decades after it was first proposed. However, very few works focused on re-exploring the neighborhood area of the elite solutions, which is definitely a promising area according to the proximate optimality principle. Here, a simple yet efficient elites regeneration (ERG) framework was designed to fill this gap. The elite population in this framework is defined as a group of individuals with better fitness values and they are regenerated after the selection procedure in DE. Specifically, a new individual is produced from the search space around each elite individual (i.e. the parent individual) by sampling Gaussian or Cauchy probability models and replaces the parent if it has better fitness value. The implementation of this procedure only introduces two parameters that need to be tuned, i.e. the standard deviation for Gaussian distribution and the scale parameter for the Cauchy distribution. In the proposed framework, the elite individuals serve as the mean or location parameters of the probability models and the standard deviation and scale parameters are tuned by experiments as a small constant value. Thus, offspring individuals are generated in areas close to their corresponding elite parents. The framework allows thorough exploitation of search neighborhoods around elite individuals and ultimately helps the elite individuals escaping from local optima. Experiments results on CEC2014 benchmark revealed that ERG framework significantly increased the optimization capacity for four original DE algorithms, four classical DE variants, and two state-of-the-art DE variants. In addition, it also demonstrated competitive performance when compared with another DE framework.

An enhanced differential evolution algorithm with adaptation of switching crossover strategy for continuous optimization

Abstract Designing an efficient optimization method which also has a simple structure is generally required by users for its applications to a wide range of practical problems. In this research, an enhanced differential evolution algorithm with adaptation of switching crossover strategy (DEASC) is proposed as a general-purpose population-based optimization method for continuous optimization problems. DEASC extends the solving ability of a basic differential evolution algorithm (DE) whose performance significantly depends on user selection of the control parameters: scaling factor, crossover rate and population size. Like the original DE, the proposed method is aimed at e ciency, simplicity and robustness. The appropriate population size is selected to work in accordance with good choices of the scaling factors. Then, the switching crossover strategy of using low or high crossover rates are incorporated and adapted to suit the problem being solved. In this manner, the adaptation strategy is just a convenient add-on mechanism. To verify the performance of DEASC, it is tested on several benchmark problems of various types and di culties, and compared with some well-known methods in the literature. It is also applied to solve some practical systems of nonlinear equations. Despite its much simpler algorithmic structure, the experimental results show that DEASC greatly enhances the basic DE. It is able to solve all the test problems with fast convergence speed and overall outperforms the compared methods which have more complicated structures. In addition, DEASC also shows promising results on high dimensional test functions.

Peregrine preying pattern based differential evolution for robot path planning

Differential evolution (DE) is a simple and robust evolutionary algorithm. DE has three simple steps and it is easy to implement. It is a popular meta-heuristics that efficiently solved various complex optimization problems. This paper addressed the robot path planning problem. The robot path planning problem is considered as one of the complex optimization problem. Here a new variant of DE with new scaling factor inspired by peregrine preying pattern. The performance of anticipated algorithm is equated with basic DE and other competitive algorithms. Results proves that new variant perform well for considered problem.

Improved Differential Evolution Algorithm for Flexible Job Shop Scheduling Problems

This research project aims to study and develop the differential evolution (DE) for use in solving the flexible job shop scheduling problem (FJSP). The development of algorithms were evaluated to find the solution and the best answer, and this was subsequently compared to the meta-heuristics from the literature review. For FJSP, by comparing the problem group with the makespan and the mean relative errors (MREs), it was found that for small-sized Kacem problems, value adjusting with “DE/rand/1” and exponential crossover at position 2. Moreover, value adjusting with “DE/best/2” and exponential crossover at position 2 gave an MRE of 3.25. For medium-sized Brandimarte problems, value adjusting with “DE/best/2” and exponential crossover at position 2 gave a mean relative error of 7.11. For large-sized Dauzere-Peres and Paulli problems, value adjusting with “DE/best/2” and exponential crossover at position 2 gave an MRE of 4.20. From the comparison of the DE results with other methods, it was found that the MRE was lower than that found by Girish and Jawahar with the particle swarm optimization (PSO) method (7.75), which the improved DE was 7.11. For large-sized problems, it was found that the MRE was lower than that found by Warisa (1ST-DE) method (5.08), for which the improved DE was 4.20. The results further showed that basic DE and improved DE with jump search are effective methods compared to the other meta-heuristic methods. Hence, they can be used to solve the FJSP.