Appling the Roulette Wheel Selection Approach to Address the Issues of Premature Convergence and Stagnation in the Discrete Differential Evolution Algorithm

Te discrete diferential evolution (DDE) algorithm is an evolutionary algorithm (EA) that has efectively solved challenging optimization problems. However, like many other EAs


Introduction
Nowadays, most studies have focused on fnding optimization techniques that can obtain an optimum (or near optimum) solution to complex combinatorial optimization problems (COPs) within a moderate computational efort. In general, there are two groups of real-world problems: optimization problems and decision problems. Te decision problems are the problems that can be solved by answering yes or no. While the optimization problems are the problems whose solutions involve determining the optimum (maximum or minimum) solution of the problem [1]. Furthermore, the optimization problems have been classifed into two categories: discrete problems that have discrete variables and continuous problems that have continuous variables. In the same context, the COPs belong to discrete optimization problems.
One of the most important types of COPs is the quadratic assignment problem (QAP) which involves discrete variables and a set of feasible solutions [1]. It was initially defned as a mathematical model related to economic activities in [2] that aim to identify the best way to allocate locations for facilities so that every facility is mapped to only one location while every location is mapped to only one facility so as to minimize the total distance multiplied by the corresponding fows.
Te problem can be stated as given n facilities and n locations (n is the problem size); a distance matrix (D) consisting of distances between every pair of locations; a fow matrix (F) consisting of trafc fows between every pair of facilities. A solution (π) is a permutation or one-to-one mapping of facilities to locations, and the objective function is defned by the sum of distances between all assigned pairs of facilities multiplied by the corresponding fows. In other words, the QAP can be stated as follows: Exact and approximate algorithms are currently the most popular methods for solving the QAP. Finding an optimal solution of the large size of the QAP using exact algorithms is a great challenge. Generally, these methods require large computational time, and hence, they are used to solve only very small-sized problem instances [3]. Most large-sized problem instances remain nearly intractable using exact algorithms [4]. Tis reason has motivated the researchers to use approximate methods to fnd better solutions to the QAP instances within a moderate computational efort. On the other hand, there are two types of approximate algorithms: heuristic and metaheuristic algorithms. Generally, the study [5] gave the defnitions of those algorithms as follows: a heuristic algorithm is an approach to problem-solving that uses a practical method that is not guaranteed to be optimal. In this situation, the heuristics are treated as ways ofered to search and obtain better solutions, while metaheuristics are a set of intelligent strategies to enhance the efciency of heuristic procedures.
Evolutionary algorithms (EAs) are one of the best metaheuristic algorithms and have the advantage of global exploration due to the diversity of the population. Despite those advantages of the EAs possessed, however, they still sufer from the problems of premature convergence and stagnation. Amongst EAs, the diferential evolution (DE) algorithm is considered the recent algorithm in EAs, and it is very competitive approach for solving optimization problems [6]. Additionally, unlike the exact algorithms, DE algorithms are found to be very efcient in fnding solutions that are optimal (or near optimal) within a moderate efort for large-sized problem instances. As these algorithms were initially developed for solving continuous optimization problems, so as to deal with the QAP, some studies have suggested improving the DE algorithm to the discrete DE (DDE) algorithm [7,8].
Te main goal of this study is to answer the following question: how to handle the premature convergence and stagnation issues in the discrete diferential evolution algorithm (DDE)? In order to achieve that, the following specifc objectives are designed to guide the study as well: (i) to utilize a quantitative measurement of premature convergence based on the degree of nonmatching between the population solutions, then the population is divided into individual groups based on the degree of nonmatching between the population solutions and the best solution; (ii) to apply the roulette wheel selection (RWS) method to consider whether a greater nonmatching degree is of higher ftness to select a population of individual groups to be able to generate a new solution with more opportunities to avoid the occurrence of the premature convergence; and (iii) to utilize another technique when those proposed techniques in (i) and (ii) cannot succeed in preventing the stagnation for a set of solutions through regenerating those solutions after waiting for the defned number of iterations defned in the experiment setting. Tis research work has employed the DDE algorithm to solve the quadratic assignment problem (QAP) as a standard to evaluate our results and their efect on avoiding premature convergence and stagnation issues, which led to the enhancement of the accuracy of the algorithm. Our comparative study based on the statistical analysis shows that the DDE algorithm that uses the proposed techniques is more efcient than the traditional DDE algorithm and the state-of-the-art methods.
Te layout of this paper is presented as follows: Section 2 presents the research problem; related works are introduced in Section 3; the materials and methods are presented in Section 4; while the computational results and discussion are reported in Section 5, whilst Section 6 reports the statistical analysis, whereas theoretical analysis and critical explanation of the performance of enhancement proposed are introduced in Section 7. Finally, Section 8 presents the conclusion.

Research Problem
Tere are two types of metaheuristic algorithms, one of them called single-solution based metaheuristics, while the second type is called population-solution metaheuristics. Premature convergence is a situation where the convergence of the population to local optimum causes the algorithm to lose the diversity of the population [9]. Te category of the EAs is among the most important metaheuristic algorithms under the population-solution metaheuristics category, and they have a global search capacity and are very suitable for solving complex optimization problems [10]. However, they still sufer from the premature convergence issue, where all individuals have the same learning direction during the iterative process [11], which is the main cause of the stagnation issue, thus decreasing the algorithm efciency.
Te problem statement is as follows: the set of existing solutions for this issue uses the objective function of the problem as the base to design the ftness function used in selection operations such as the RWS. Terefore, as the ftness does not directly express the diversity of the population, there is a need to design a method that directly expresses like a nonmatching degree to measure the diversity of solution. Tis way the probability of the occurrence of premature convergence is reduced through increased diversity of the population.

Related Works
Regarding the DE algorithm for the COPs, nowadays, most of the researchers are interested in either developing a better algorithm or improving an existing algorithm. Furthermore, it is found that many literatures have tried to change the algorithms for continuous space to discrete space as later contains a large number of problems such as scheduling problem. In [12], a study for solving the discrete space problem is presented by adding a truncation method for rounding the rational values of a DE population such that the parameters of objective function would become discrete.
Te advantages of the DE algorithm attracted researchers to improve it to deal with discrete optimization problems, such as the QAP. In [13], an approach associated with DDE is proposed for calculating variances in the fow-shop preparation problem. As reported, the algorithm is not found to be efective because of using low mutation probability (0.20). But, the DDE algorithm operation is found to be more useful and efective if a local search method is incorporated. Initially, the DE approach was adapted to solve COPs and then applied to solve the QAP [7]. In [8], swap and insertion mutations are used along with the local search method for improving the DDE algorithm. Te modifed DDE algorithm using local search showed better results for two types of sparse and dense QAPLIB instances (https://coral.ise.lehigh.edu/data-sets/qaplib/). Premature convergence and stagnation issues have been distinguished in the literature by the study [14], and the phrase "stagnation issue" describes a circumstance in which the optimum-seeking process stalls before locating a globally optimal solution. Stasis typically happens almost for no apparent cause. Te population remains diversifed and unconverted after stagnation, unlike premature convergence, but the optimization process no longer advances.
Investigation in the DE algorithms on the premature convergence and stagnation issues has been carried out in [9] as follows: when the population is converged to the local optimum, that means, the algorithm has lost the diversity of the population, then the convergence is premature convergence. On the other hand, when the algorithm becomes incapable of generating better new solutions (new ofspring) by the evolutionary process; that is, it loses the capability to improve the solutions, then it is a stagnation problem. In this context, crossover operator plays an efcient role in the algorithm to achieve the balance between the convergence speed and population diversity [15]. Recently, an efcient crossover operator named uniform like crossover (ULX) applied by the study [16] for the DDE algorithm to solve the QAP.
Most DE algorithms developed for the scheduling problems require a transformation method for encoding the solutions as vectors, which are decoded only at the evaluation time. Tese kinds of schemes were used in several DE algorithms in various studies [17,18]. Another kind of the DE is known as the DDE, in which the evolutionary operators are developed depending on discrete permutation representation. Furthermore, it can skip the transformation operation between integer and real solution representations. However, the research on the DDE algorithm is too little, and they are generally used for solving the fow-shop scheduling problems [19,20].
Tere are many studies that have focused on the issue of premature convergence and how to prevent its occurrence, especially in evolutionary algorithms such as the genetic algorithm, where population diversity helps prevent this issue. Te studies [21,22] have summarized some methods that dealt with this issue as follows ("restricted selection, dynamic application of mutation, constraints for crossover and mutation probabilities, stochastic universal sampling, variable ftness assignment, population partial reinitialization, individuals grouping methods, restricted mating, zymogenesis, species conserving techniques, ranking sort based on Pareto dominance, local search based on diversity, elitist technique, and dynamic genetic clustering algorithm").
On the other hand, most studies stated that the selection of parents was based on the value of the objective function, as they would be selected if they had the best value of the objective function after applying some restriction on the selection process. A recent study [23] reported the selection operators used in the EAs as follows: RWS, elitism selection (ES), tournament selection (TOS), stochastic universal sampling (SUS), linear rank selection (LRS), exponential rank selection (ERS), and the truncation selection (TRS).
Te researchers in the study [24] discussed a stagnation issue in the DE algorithm and how parents choose in the DE algorithm based on previous studies included in their study, where summarized the method proposed by them according to the ftness values of the solutions in the existing population, solutions with higher ftness values have a greater chance of being chosen as parents. Te choice of parents is made depending on how far of the present population's solutions are from one another. It is more likely that the solutions with short distances will be chosen as parents. Tus, the parents that been chosen from the most recently updated solutions will maximize the likelihood of producing successful solutions.
Te review [25] has proposed tracking mechanism (TM) and backtracking mechanism (BTM) when population of the DE algorithm tends to stagnate or undergo premature convergence, which prevents it from achieving the global optimum. In order to address those issues used the TM to encourage population convergence when the population entered a state of stagnation and used BTM to restore the population's diversity when it entered a state of premature convergence. Te study [26] has addressed premature convergence and stagnation issues in the DE algorithm as follows: the mutation operator signifcantly impacts diferential evolution's (DE) efectiveness. Misconfgured mutation techniques and control parameters might result in premature convergence owing to overexploitation or stagnation due to overexploration. An efcient DE algorithm must strike a balance between exploration and exploitation. Te enhanced DE (EDE) for truss design presented in this Applied Computational Intelligence and Soft Computing work makes use of two novel strategies-integrated mutation and adaptive mutation factor strategies-to achieve a fair balance between the exploration and exploitation of DE.
On the other hand, computing now includes quantum computing (QC), and the quantum-inspired DE (QDE) fully uses the QC's rapidity and the DE's optimization capabilities. Moreover, various research studies suggested an enhanced QDE with multiple methods (MSIQDE) in the literature. However, it still has poor search accuracy and premature convergence. Hence, the study [27] has resolved these issues with the MSIQDE through a new diferential mutation, and a technique of a diference vector is suggested to improve the searchability and descent ability. A new multipopulation mutation evolution method is created to assure the relative independence of each subpopulation and the population variety. Te quantum chromosome is mapped from a unit space to a solution space using the viable solution space transformation approach to arrive at the best outcome.
Recently, the study [28] has provided an efective DE version, OLELS-DE, by designing orthogonal learning and elites local search algorithms, efectively addressing DE's stagnation and premature convergence issues. A population diversity estimation technique is used to empirically differentiate between these two circumstances once the stagnation or premature convergence phenomena has been found by keeping track of the best individual's update condition during the evolution.
Another recent study [29] has covered the topics of premature convergence and stagnation in the DE algorithm, which are still thought to be unresolved problems by researchers. Te literature [29] aims to (1) provide a new insight into function landscape analysis with domain transform (DT); (2) alleviate the problems of premature convergence and stagnation, which frequently occur on complicated multimodal function landscape; and (3) construct a new searching paradigm based on DT. Te DT from signal processing and communication felds to evolutionary computation is introduced. Te domain transform-based evolutionary optimization (DTEO) technique will be presented in this part before being used to develop noiseless and noisy optimization on DE, respectively.

Materials and Methods
In this section, all the algorithms and techniques that were used to achieve the objectives of this research work have been presented as follows:

Discrete Diferential Evaluation (DDE) Algorithm.
Te steps of DDE algorithm are stated as follows: Initialization: initialize random population matrix π � π 1 , π 2 , π 3 , . . ., π P s of size P s * N d where P s is the population size, and N d is the dimension of problem space. Each population individual must be unique. Evaluate ftness: obtain the best solution π t−1 best in the population P s by using equation (1).

Mutation: the following Equation can be used to fnd
here, π t−1 b represents the best solution from the previous generation in the target population; the P m is the mutation probability; and insert and swap are merely the single insertion and swap moves, respectively, r∈ [0, 1] is a uniform random number. Crossover: the crossover operation can be performed under the condition as shown in the following equation: where the CR and P c are crossover operation and crossover probability, respectively. Tat means, the crossover operation is used if a randomly generated number, r < P c , and then produce the individual u t i . Otherwise, the individual is selected as u t i � v t i . Selection: selection operation that depends on ftness function can be calculated by equation (4). Te selection is based on the existence of the correct amongst the test and target individuals.

Fitness Proportionate Selection
Mode. Fitness proportionate selection (FPS) is a selection method in the genetic algorithm (GA) and RWS method is one example of FPS methods. According to a summary provided by [30], the frst step of the FPS method is to calculate each individual's ftness value. Next, the individual's proportion of ftness within the entire group is calculated, which alludes to the probability that an individual is selected amid the process of selection. Te probability that the individual i is chosen, is calculated in the following.
FPS is a very efective method for a parent to be selected. Tis gives everyone the privilege of becoming a parent with a proportional probability to their ftness value. Consequently, only the higher ftness value selections are made, which are eventually propagated to the generation that follows [31]. Te RWS is one of the frst methods for selection operator that has been used successfully in many applications of EAs [32]. Figure 1 shows the steps of the RWS.
In RWS, there is a circular wheel, as outlined below, along with a fxed point for choosing chromosomes arranged along the wheel's circumference. To choose the frst parent, the area of the wheel that comes ahead of the fxed point is chosen, and this process is also applied to choose the second parent. Notably, ftter individuals with greater wheel areas will have a higher probability of being selected whenever the wheel spins, which means that the chance of selecting an individual is directly informed by its ftness value. In the example given in Table 1, the ftness value of chromosome 1 is the highest, and so it has the greatest probability of being selected in comparison to the other chromosomes. Likewise, chromosome 5 has the lowest probability of being selected. Figure 2 shows the implementation of the FPS by using RWS to select chromosomes.
Similarly, the concept of the FPS has been used in the transition rule on the ant system (AS) algorithm, but in a way that fts into the algorithmic components (such as the pheromone denoted by τ) in the transition rule. Tis is called the random proportional rule (RPR), which is an implementation of the transition rule that gives the probability that the ant k in the city r chooses to move the city s [33]. Equation (6) shows the RPR.
where τ represents the pheromone; η � 1/δ represents the inverse of the distance δ(r, s), J k (r) refers to the set of cities that have yet to be visited by ant k positioned in city r (to make the solution feasible); and the parameter (β > 0), determines the relative importance of pheromone versus distance. It is also worth noting that the probability value of the node visited previously is 0, which avoids repeated visits. On the other hand, the ant colony optimization (ACO) algorithm sufers from the premature convergence issue. For this reason, a new implementation of transition probability is presented in [34] for ACO, called global random proportional rule (GRP), to prevent this issue by enhancing a random proportional rule. Te primary purpose of GRP is to enhance exploration through an increased probability of choosing solution components with a low pheromone trail in order to use the algorithm to encourage ants to choose a new shorter route to prevent premature convergence. Equation (7) shows the GRP. It shows the meaning of x (r, s) as it refects the efect in passing on edges from r to s during the trial paths.
In our study, we use the same idea of GPR to use the FPS and implement its ftness as a nonmatching degree in order to give more opportunity for a group of solutions which can avoid failure of premature convergence. Te ftness value will formulate on the no-matching between the best solution and the existing solutions in the groups after dividing the population's solutions into those groups based on the degree of diference. Te probability of those groups is calculated in.

Proposed Enhance Discrete Diferential Evolution (EDDE)
Algorithm. Tis section includes the enhancement of the DDE algorithm to deal with the premature convergence and stagnation issues. Figure 3 shows the phases of enhancing and evaluating of proposed EDDE algorithm as follows: Te steps of the proposed enhanced DDE (EDDE) algorithm are presented as follows:

Population Initialization Stage.
Te frst step is to initialize the population (solutions) randomly. As the goal of the QAP, it assigns only one location to one facility; therefore, the same facility should not be repeated in one solution. Since beginning with a better initial population gives a better solution, several researchers use heuristic algorithms to generate better population, and we apply the sequential sampling algorithm [35] for initiating the population. It is a basic form of the sequential constructive sampling algorithm which is summarized as follows: Given a distance matrix, organize the locations in nondescending order of their distances in every row of the matrix. Beginning from allocating facility 1 to a location from frst row of the matrix, a complete allocation is constructed by allocating the residual facilities to other locations probabilistically from the residual locations (the locations that are not presently allocated to any other facility in the current allocation) in each row. Te process is continued until a complete allocation is constructed. Te probability of allocating a residual location (in a row of the matrix) is  Figure 1: Steps of roulette wheel selection [32].  Applied Computational Intelligence and Soft Computing allocated in way so that the frst residual location is assigned more probability than second one, second is more than third one, and so on. Next, for each residual location, cumulative probability is computed. Ten, a random number r∈ [0, 1] is produced, and the location that symbolizes the number in the cumulative probability range is accepted. Repeat the process until the population is full.

Finding
Best Solution π t−1 best . Te next step is to fnd the best solution, π t−1 best from the population based on the objective function by using equation (1).

Divide Population into Groups.
In this step, the population (solutions) is divided into groups based on nonmatch degree (Degrees of Diference) with the best solution π t−1 best by using Algorithm 1 as follows: Suppose the best solution π best is (1 2 3 4 5 6 8 7) and suppose the solutions found in the population are given in Table 2.
Based on the example given in Table 2, the degree of diference starts at 2, and their number is equal the size of the problem. Te degrees of the diferences can be obtained by comparing the solutions in Table 2 with the best solution as follows: solution π 1 has a degree of diference is 2 from the best solution, while solution π 2 has a degree of diference is 3 from the best solution, and solution π 3 has a degree of diference is 2 from the best solution, . . ., etc. Te last column in Table 2 shows the degree of diferences between the best solution and the solutions in population. Te population is divided into groups based on the degree of diferences which is shown in Table 3.
Each group contains several solutions that are not necessarily equal to the degree of diferences. For example, if the degree of diference is 2, then this group can contain one solution or more than one solution which difers from two degrees from the best solution.

Apply Roulette Wheel Selection (RWS) to Select the Solutions Group π t−1
gh . In this section, the solutions group π t−1 gh has been selected using the FPS that is given in equation (8) and RWS. Let the set of groups be {g 1 , g 2 , . . ., g m } and 1 ≤ h ≤ m be the number of groups that divide the population based on the degree of the nonmatching between the best solution and the solutions found in the population, S is assigned to 1 to start with the frst element of the selected group. Considering Table 3, the ftness value of group 1 is the highest that has more chance of selection than the other groups, and group 4 has very rare or no chance of selection, the implementation of the FPS is done by RWS to select the group. Table 4 presents the ftness value and probability of each group.

Crossover Operator Stage.
Tis step has been included in fnding the new solutions u t i by using crossover operator between the solutions found in the group π t−1 gh and the solution generated from the mutation stage v t i . Equation (9) shows this step.
where the P c ∈ [0,1] is the crossover probability. If the r < P c, the ULX operator [16] is used to generate the u t i otherwise the v t i is selected. Te process of the ULX is as follows: frst, Input: Best solution π t−1 best and current solutions π i Output: Nonmatch degree Nonmatch � 0 For each facility in the current solutions π i If the location of the facility not-match in π t−1 best and π i Nonmatch � nonmatch + 1 End if End For ALGORITHM 1: Nonmatching degree.

Solution
Distributions of facilities to locations Degree of diferences

Selection Operator Stage.
Te selection operator is the fnal step of the DDE algorithm which depends on the objective function to choose the best solution after the crossover step. Te new solutions π t gh are the solutions either in the u t i or in the group π t−1 gh . Equation (10) shows the selection step.

Check Solution Stagnation Stage.
Tis section included the number of trials the EDDE algorithm waits for before considering the solution is stagnated and should be regenerated randomly after 10 trials. Equation (11) shows this step.
Te steps of the proposed EDDE algorithm are illustrated through the pseudo code shown in Algorithm 2.
Te fowchart of EDDE algorithm given in Figure 4 as follows:

Computational Results and Discussion
Tis section elucidates the efciency of the improved algorithm (EDDE). In order to encode the improved algorithm, MATLAB (R2018b (9.5.0.944444), 64 bit (win64), August 28-2018, and License Number: 968398) is employed on a PC with Intel (R) Core (TM) i7-3770 CPU @3.40 GHz under MS Windows 10 and 8 GB RAM. Tis section comprises two parts: the frst part highlights the parameters used for the proposed algorithm, whereas the second part discusses the results of the study.

Parameters Tuning.
Te same parameter settings that were suggested in [8] are used for the DDE algorithm for solving the QAP. Tese parameters are related to the population size P s , mutation probability P m , probability of crossover P c , and Number of runs as given in Table 5.
Our EDDE algorithm is implemented and tested on some of the QAP instances of various sizes from the QAPLIB website and then compared with the traditional DDE algorithm. Te criteria used for comparisons are based on the gap (Relative Percent Deviation) of the solutions by the algorithms and nonmatching between the solutions.

Comparison Based on the Nonmatching between the Solutions with the Best Solution.
Te value of the nonmatching was calculated through a comparison between the arrangement of facilities occupied by the locations in the best solution and between those facilities occupied by the locations in those solutions in the population by using the following algorithm. Ten, the crossover is performed between the best solution and the selected solution in the group to produce better ofspring. So, if there is a big match between these parents then there is a chance that the ofspring will be identical to one of the two parents, and thus the premature convergence will occur. Certainly, the solutions having high degrees of nonmatching will increase the diversity of the population by creating new solutions (ofspring). Table 6 shows this comparison as follows: Te nonmatching values are computed by Algorithm 1, if the value of the average nonmatching is small this means that the solutions in the population are identical with the best solution due to a lack of diversifcation. On the contrary, if the value of the average nonmatching is high, then this means that the solutions in the population have a diference from the best solution, and in this way, the diversifcation is preserved.
Te above results were obtained by fnding the degrees of nonmatching between the solutions and the best solution in all iterations after 10 runs, then sum them and dividing by the number of iterations multiplied number of runs and multiplied size of the population. Te crossover stage in the DDE algorithm produced solutions with high degrees of matching and in some cases are completely identical to the best solution. For example, the mean of nonmatching of the solutions of the instances (Tai20a, Tai20b, Tai25a, Tai25b, Tai30a, Tai30b, Tai35a, Tai35b, and Tai40a) are 0, 0, 0, 0, 0, 0, 0, 0, and 0, respectively, whilst the mean of nonmatching of the solutions these instances are (4.136, 4.363, 4.412, 7.415, 4.419, 10.643, 4.919, 6.595, and 16.890, respectively) by using the EDDE algorithm.
Te RWS technique that was applied to select a population of individual groups for the crossover operation to give a higher probability to selected the nonmatching solution to generate solutions through this stage with a high percentage of diference with the best solution i.e., in other words, the locations of those facilities that these solutions contain are diferent enough to allow the generation of new solutions so that those locations are highly diferent degree compared to existing locations in the best solution. In this way, it is possible to preserve the diversity of the population and thus reduce the issue of premature convergence. A graphic representation of Table 6 is shown in Figure 5. For t � 1 to t max Find the best solution π t−1 best by using equation (1) Divide population to groups based on nonmatching with π t−1 best by using Algorithm 1 Determining solutions group π t−1 gh by using RWS For S � 1 to |π t−1 gh | Mutation stage by using equation (2) Crossover stage by using equation (9) Selection stage by using equation (10) Check solution stagnation stage W by using equation (11) If keep same the solution

Comparison Based on the Gap (Relative Percent Deviation). Tis section includes the comparison between DDE and EDDE algorithms based on the gap (relative percent deviation) that is given by
Regenerate the solution at random after 10 trials Mutation Operation Stage by using Equation (2) Crossover Operator Stage by using Equation (9) Selection Operator Stage by using Equation (10) Check Solution Stagnation Stage W by using Equation (11) Is there an improvement in the solution? No

Yes
Regenerate the solution at random after 10 trials Yes t > t max

No End Yes
Find the best solution in the population by using Equation (1) π t-1 best Divide Population into groups based on non-match degree with the best solution by using algorithm 1 π t-1 best Apply roulette wheel selection (RWS) to select the solutions group π t-1 gh S > | | π t-1 gh Figure 4: Flowchart of the proposed EDDE algorithm.
Applied Computational Intelligence and Soft Computing 9 where BS denotes the best solution found by any algorithm after 10 runs, while the BKS refers to the optimal solution or the best-known solution reported in QAPLIB. In this study, the EDDE algorithm addresses the issue of premature convergence which has responded positively by improving the gaps obtained by the DDE algorithm. Table 7 compares the results between the DDE and the EDDE algorithms based on the gap of the obtained solutions. Te results of this comparison show the average of the gaps obtained by the DDE algorithm before the enhancement is 3.369, which is refned to 1.297 after enhancement. Tis is conducted by applying the concept of the FPS to generate new solutions with a high degree of nonmatching in order to reduce the probability of creating solutions that are matched with the best solution by the crossover operator stage. Figure 6 shows the graphic representation of Table 7.
Te values of the gap have been calculated using equation (12), and the less value of the gap is better than the high value of the gap.
Te graphical representation of Table 7 is shown in the following fgure:

Statistical Analysis
Tis section has discussed the results of this study using the SPSS software as follows: 6.1. Normal Distribution Test. We are testing whether the data follow a normal distribution by using the following hypotheses: Null Hypothesis: the data are normally distributed. Alternative Hypothesis: the data do not follow a normal distribution.
In order to check normality, we used the one-sample Kolmogorov-Smirnov Test. Table 8 presents the results of this test as follows: In the SPSS software, the p value is labeled "Sig.," from above Table both p values are below 0.05 so we rejected the Null hypothesis in our test and accept the alternative hypothesis, Hence, this does not follow a normal distribution. Tables 9-11,  Table 9 includes descriptive statistics, Table 10 relates to  Wilcoxon Signed Ranks Test, and Table 11 includes test statistics.

Nonparametric Test. Tis test includes three
Te following Table presents the Wilcoxon Signed Ranks Test as follows: In order to use Wilcoxon Signed Ranks test, the following hypotheses must be used: Null hypothesis: there is no diference in the mean of the two samples. Alternative hypothesis: there is a real diference between the mean of the two samples.
From the above table, we fnd that the Wilcoxon Signed Ranks test value is −4.831, and p value is 0.000 which is less than 0.05; therefore, we rejected the null hypothesis and accepted the alternative hypothesis.

Theoretical Analysis and Critical Explanation of the Performance of Enhancement Proposed
Research indicates that metaheuristic algorithms, including EAs, are among the current state-of-the-art algorithms for addressing NP-hard problems. While EAs beneft from global exploration search, issues such as stagnation, premature convergence occur in the iterative operation and slow in the exploitation mechanisms. Although the exploitation issue in the DDE algorithm has been addressed by the recent study [36]; however, the critical reading of the literature indicates that stagnation, premature convergence issues have not been investigated in the DDE algorithm associated with EAs to solve the QAP. Terefore, further Table 5: Parameter setting of the DDE algorithm [8].
Parameter Value P s size of population 100 P m probability of mutation 0.9 P c probability of crossover 0.9 Number of Run 10 research is required to enhance the algorithm that accounts for current challenges, which is the area of contribution of our study, specifcally the avoidance of stagnation and premature convergence in the DDE algorithm. Te following sections have discussed the theoretical analysis and critical explanation of the performance of enhancement proposed.

Analysis of the Stagnation Situation for DDE and EDDE
Algorithms. Tis section included the comparison between DDE and EDDE algorithms illustrating the stagnation situation for DDE and how EDDE avoids this issue. Tis comparison applied an instance of the QAP data called "Tai12a" as a case study. Te size of this instance is 12 facilities/locations, and the optimal solution of this instance in the QAP database is 224416. Suppose the number of solutions in a population in the DDE and EDDE algorithms is 7 solutions, the results of this comparison are discussed in Tables 12 and 13 as follows:

Critical Explanation of the Performance of Enhancement Proposed.
A key point revealed by the critical analysis of the improved algorithm's performance (i.e., EDDE) is that premature convergence arises from limited population diversity. Due to this, the algorithm generates solutions that are comparable in the evolutionary process, losing its ability to enhance solutions and leading to stagnation. Tese issues were considered in this study by introducing quantitative measurement to premature convergence in the improved algorithm (EDDE) based on the degree of nonmatching between the population solutions. Ten, the population was Tai20b Tai25a  Tai25b  Tai30a  Tai30b  Tai35a  Tai35b  Tai40a  Tai40b  Tai50a  Tai50b  Tai60a  Tai60b  Tai80a  Tai80b  Tai100a  Tai100b Sko42 QAP Instances DDE algorithm EDDE algorithm Figure 5: Graphic representation of Table 6.   Tai20a  Tai20b  Tai25a  Tai25b  Tai30a  Tai30b  Tai35a  Tai35b  Tai40a  Tai40b  Tai50a  Tai50b  Tai60a  Tai60b  Tai80a  Tai80b  Tai100a  Tai100b Sko42 DDE algorithm EDDE algorithm Figure 6: Graphic representation of Table 7.      Regenerated π Best : the π Best (bold) is the best solution in the population of the DDE and EDDE algorithms. Cost: the best value of the objective function obtained by DDE and EDDE algorithms by using equation (1).
Nonmatching is the number of diferent facility locations between two solutions and calculated by using Algorithm 1. Group: a set of individuals with the same nonmatching degree as the best individual in EDDE algorithm. Gap: the best value obtained by equation (12). Solution stagnation start: the number of iterations that the solution has not changed (not improved) in the DDE algorithm, where max iteration is 1000. Waiting for solution improvement: the number of trials the EDDE algorithm waits for before considering the solution is stagnated and should be regenerated randomly (italic) the default value is 10 trials.
divided into individual groups based on the degree of nonmatching between the population solutions and the best solution. Moreover, the new utilization of the RWS has helped to a greater nonmatching degree to select a population of individual groups to be able to generate a new solution with more opportunities to avoid the occurrence of premature convergence in the improved algorithm EDDE. Figure 7 illustrates a practical example of the impact of the positive contributions of this research on the performance of the DDE algorithm (before enhancement) and EDDE algorithm (after enhancement). Te example included the solutions of two QAP instances, Tai25a and Tai25b; the results of solutions in those instances show the DDE algorithm was unable to achieve the optimal solution in both instances (Tai 25a and Tai25b), as it obtained the best gap of 3.433 and 2.672, respectively. While the EDDE algorithm has achieved the optimal solutions for those instances, hence it has achieved the best gap is 0 and 0, respectively. Te blue color in Table 7 indicates the solution by the DDE algorithm, while the red in Table 7 indicates the solution by the EDDE algorithm.

Comparison Performance of the EDDE Algorithm with the
State-of-the-Art Methods. Tis section has included the comparison between the EDDE algorithm and the state-ofthe-art methods that solved QAP. A recent study [37] discussed a performance study of metaheuristic approaches for the QAP that includes the ACO, GA, PSO, bat algorithm (BA), tabu search (TS) algorithm, and a modifed variant of the discrete PSO algorithm.

Process
Solution Nonmatching degree EDDE algorithm Select best solution (π Best ) 4 5 11 3 7 10 12 9 8 6 1 2 -Mutation stage to select the parent 1 4 3 11 5 7 10 12 9 8 6 1 2 2 Select individual as parent 2 from the group that selection by RWS 4 2 11 6 12 1 7 5 9 3 10 8 10 Crossover stage between parent 1 and parent 2 2 11 6 7 1 12 9 8 3 10 5 6 DDE algorithm Select best solution (π Best ) 4 5 11 7 1 8 12 10 9 3 6 2 -Mutation stage to select the parent 1 4 5 1 7 11 8 12 10 9 3 6 2 2 Select individual from a population as parent 2 4 5 11 7 1 8 12 10 9 3 6 2 0 Crossover stage between parent 1 and parent 2 4 5 11 7 1 8 12 10 9 3 6 2 0 It can be noted that the new solution (ofspring) generated by the EDDE algorithm (bold) by the crossover stage has a high degree of nonmatching with the best solution which means the diversity of the population will be increased. While the opposite is in the DDE algorithm that generated ofspring (italic) by the stage of crossover that degree of nonmatching was zero compared with the best solution, which means that the new solution is matched with the best solution which leads to loss of diversity.    Table 14 shows the results of the comparison between the proposed algorithm EDDE and among those algorithms that have been proposed in the recent literature that dealt with solutions of QAP instances. Te results of that comparison showed that the proposed algorithm EDDE had a more efcient performance than others to converge to the optimal solutions. Moreover, the EDDE algorithm obtained 0.004 of the best average value of the gap whilst the other algorithms (ACO, BA, GA, PSO, Modifed PSO, and TS) have obtained 0.498, 7.114, 10.990, 7.865, 6.518, and 4.266, respectively. Figure 8 shows the graphic representation of these results as follows.

Limitations of the Study.
Recently, the authors in the study [16] utilized ULX for the DDE algorithm to solve the QAP. However, there are many crossover operators suggested for the QAP in other algorithms such as the genetic algorithm [38]. Furthermore, this study used only the DDE algorithm which belongs to the EAs category although there are many known algorithms that belong to the category of EAs.

. Conclusion
Tis study aims to address the limitations of evolutionary algorithms (EAs), which can sufer from premature convergence and stagnation. Despite the advantages of EAs, these issues can hinder their ability to solve complex optimization problems (COPs). Terefore, to avoid premature convergence in the EAs by ensuring the diversity of the population considered to enhance the selection operations by proposing diversity measures between solutions, hence the enhancement of convergence to optimal solutions. We implemented our proposed enhancement DDE as an algorithm of the evolutionary algorithm category. To evaluate the performance of the DDE algorithm before and after the enhancement, it was tested on some benchmark instances from the QAPLIB website. Subsequently, we compared the results obtained by DDE and enhanced DDE algorithms based on the gap and nonmatching solutions. Our comparative study reveals that the EDDE algorithm is more efcient than the traditional DDE algorithm. Moreover, the proposed algorithm is better than the other algorithms, including ACO, GA, PSO, bat algorithm (BA), tabu search (TS), and a modifed variant of the discrete PSO algorithm in convergence to optimal solutions of some QAP instances. We suggest applying the proposed algorithm EDDE to solve other problems of the COPs in future work, such as capacitated vehicle routing problem (CVRP) and nurse scheduling problem (NSP).

Data Availability
Te data that used to support the fndings of this study are available from the corresponding author upon request. BA [34] GA [34] PSO [34] Modified PSO [34] Our proposed EDDE algorithm TS [34]  Table 14.
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