Research Article  Open Access
Dieudonné Nijimbere, Songzheng Zhao, Haichao Liu, Bo Peng, Aijun Zhang, "A Hybrid Metaheuristic of Integrating Estimation of Distribution Algorithm with Tabu Search for the MaxMean Dispersion Problem", Mathematical Problems in Engineering, vol. 2019, Article ID 7104702, 16 pages, 2019. https://doi.org/10.1155/2019/7104702
A Hybrid Metaheuristic of Integrating Estimation of Distribution Algorithm with Tabu Search for the MaxMean Dispersion Problem
Abstract
This paper presents a hybrid metaheuristic that combines estimation of distribution algorithm with tabu search (EDATS) for solving the maxmean dispersion problem. The proposed EDATS algorithm essentially alternates between an EDA procedure for search diversification and a tabu search procedure for search intensification. The designed EDA procedure maintains an elite set of high quality solutions, based on which a conditional preference probability model is built for generating new diversified solutions. The tabu search procedure uses a fast 1flip move operator for solution improvement. Experimental results on benchmark instances with variables ranging from 500 to 5000 disclose that our EDATS algorithm competes favorably with stateoftheart algorithms in the literature. Additional analysis on the parameter sensitivity and the merit of the EDA procedure as well as the search balance between intensification and diversification sheds light on the effectiveness of the algorithm.
1. Introduction
Dispersion problems are a typical class of combinatorial optimization problems, which can be categorized as efficiencybased objectives and equitybased objectives [1]. Equitybased dispersion problems studied in the literature include the maxmean dispersion problem that maximizes the mean of the aggregate dispersion, the maxmin diversity problem that maximizes the minimum dispersion [2, 3], the minimum differential dispersion problem that minimizes the difference between the maximum aggregate dispersion and minimum aggregate dispersion [4, 5], the balanced quadratic optimization problem that minimizes the difference between the maximum dispersion and the minimum dispersion [6], etc. One of the wellknown efficiencybased dispersion problems is maxsum dispersion, which requires maximizing the aggregate dispersion [7, 8].
The maxmean dispersion problem (MaxMean DP) has received extensive attention in recent years [1, 9]. It is proved to be strong NPhard and has many important applications including architectural space planning, sentiment analysis, social networks, pollution control, and web pages ranks. Let be a set of elements and be a distance matrix. The maxmean dispersion problem (MaxMean DP) is to find a subset to maximize the distance function , where denotes the size of the set . It is noteworthy that a solution is a subset of any cardinality. Let if the element is in and otherwise. The maxmean dispersion problem can be formulated as follows:
For solving the MaxMean DP, various approaches have been proposed in the literature. Marti and Sandoya (2013) [10] proposed a GRASP with path relinking algorithm, where GRASP uses a randomized greedy strategy for initial solution construction and a variable neighborhood descent method for solution improvement. Moreover, the path relinking is a postprocessing procedure to control the search diversification. Della Croce et al. (2014) [11] developed a twophase hybrid heuristic, where the first phase solves a mixed integer programming formulation to obtain an initial set of solutions and the second phase employs a local branch strategy to refine the quality of solutions. An extended version of this algorithm is presented in Della Croce et al. (2016) [12], where a path relinking phase is added to further enhance the quality of solutions. Carrasco et al. (2015) [13] proposed a dynamic tabu search algorithm, where various shortterm and longterm memory structures are integrated to enhance the performance of tabu search. Lai and Hao (2016) [9] proposed a tabu searchbased memetic algorithm that integrates a tabu search method for discovering good local optima and a random crossover operator to generate welldiversified offspring solutions. Brimberg et al. (2017) [14] developed a variable neighborhood search algorithm that examines multiple neighborhood structures based on add, drop, and swap moves and picks one of them in a probability way to perform a shaking procedure. Garraffa et al. (2017) [15] embed SDP relaxation and cutting planes into a branch and bound framework to reach optimal solutions.
Estimation of distribution algorithms (EDA) are a class of populationbased algorithms [16–20], which have been demonstrated to be effective for solving many optimization problems. One of the most distinct features of EDA lies in the use of global statistical information extracted from a set of highfitness solutions to build a probability model. During the population evolution, EDA continually updates the probability model for producing new offspring solutions. Aickelin et al. [21] proposed an approach based on EDA with antminer for solving the nurse rostering problem, where a Bayesian network is constructed for implementing EDA. Peña et al. [22] proposed a hybrid evolutionary algorithm that combines genetic algorithm with EDA for solving multiple optimization problems, where EDA employs a probabilistic graph model to express the dependencies among different variables. Zhang et al. [23] proposed a guided local search with EDA to solve the quadratic assignment problem, in which EDA is used for the mutation operator to generate offspring solutions.
In this paper, we propose the first hybrid metaheuristic that combines EDA with tabu search (EDATS) for tackling the MaxMean DP. Basically, our proposed EDATS algorithm follows a multistart search framework, where each start includes an initial solution construction phase and a local search phase. The initial solution construction phase borrows the EDA idea of using statistical information extracted from a set of solutions to construct highquality and welldiversified solutions. The local search phase uses a simple but effective tabu search method to improve the initial solutions. Experimental results on a total of 140 instances with variables ranging from 500 to 5000 indicate that our proposed algorithm performs significantly better than most reference algorithms in the literature. Furthermore, additional evidence is provided to shed light on the effectiveness of the algorithm.
The rest of the paper is organized as follows. Section 2 describes the proposed EDATS algorithm. Section 3 presents experimental comparisons with stateoftheart algorithms in the literature and additional analysis on the effectiveness of the algorithm. Concluding remarks are given in Section 4.
2. Algorithm
This section presents the proposed EDATS algorithm, including the general scheme, the elite set initialiation, the EDA guided solution construction, and the tabu search solution improvement, as well as the elite set updating and rebuilding.
2.1. Main Scheme
The general scheme of our proposed EDATS algorithm is shown in Algorithm 1. At first, we create an elite set ES of highquality and welldiversifed solutions and record the best solution in . Then, we build a conditional preference probability matrix where each entry denotes the conditional preference probability of assigning a variable with value 1 under the condition that the variable is assigned with value 1. By referring to the information recorded in , we construct a new solution to launch a new round of search exploitation. For this purpose, we employ a tabu search solution improvement method to refine the quality of the starting solution. If this refined solution satisifies the updating criterion, an elite set updating method is triggered. Otherwise, we detect if a rebuilding method is necessary and execute it if it is. The abovementioned procedures are repeated until the elapsed time surpasses the given time limit.
(1) Input: a distance matrix  
(2) Output: The best solution  
(3) EliteSetInitialization() (see Section 2.2)  
(4) Record the best solution in  
(5) CPPM CPPMatrixCalculation(ES) (see Section 2.3.1)  
(6) while the elapsed time does not surpass the given time limit do  
(7) EDAGuidedSolutionConstruction(, ) (see Section 2.3.2)  
(8) TabuSearch() (see Section 2.4)  
(9) if then  
(10)  
(11) end if  
(12) Find the solution getting the worst objective value in  
(13) () EliteSetUpdating() (see Section 2.5)  
(14) if then  
(15)  
(16) else  
(17)  
(18) end if  
(19) if then  
(20) EliteSetRebuilding() (see Section 2.5)  
(21) end if  
(22) if undergoes updates for times then  
(23) CPPMatrixCalculation() (see Section 2.3.1)  
(24) end if  
(25) end while 
2.2. Elite Set Initialization
Algorithm 2 gives the outline of the elite set initialization procedure. The elite set consists of a set of highquality and welldiversified solutions. To obtain each solution in , we first generate a random initial solution and then submit it to a tabu search procedure for solution improvement. The random initial solution is obtained by assigning each variable with the equal probability of 0.5 to receive value 1 or value 0. The tabu search procedure employs a fast but effective 1flip move operator to refine the solution within a short computional time (see Section 2.4). If a newly obtained solution after optimization is the same as a solution in , it is forbidden to join in . In this way, we guarantee that all the solutions in are representative.
(1) Input: a distance matrix  
(2) Output: an initial elite set  
(3) ,  
(4) repeat  
(5) RandInitialSol()  
(6) TabuSearch()  
(7) if is not a duplicate of any solutions in then  
(8)  
(9)  
(10) end if  
(11) until 
2.3. EDA Guided Solution Construction
2.3.1. The CPP Matrix Calculation
Definition 1. Let denote the subset where each solution in receives value 1 for the variable , denote the size of , denote the objective function value of the solution , and denote the minimum objective function value among all the solutions in . Then, the objective preference of the variable is defined as
Definition 2. Given the sets and , the selection probability of the variable is defined as
Definition 3. Given the selection probability and the objective preference , the preference probability of selecting the variable is defined as
Definition 4. Let denote the number of solutions in where both variables and receive value 1. Then, the interaction selection probability associated with the variables and is defined as
Definition 5. Given the selection probability and the preference probability as well as the interaction selection probability , the conditional preference probability of the variable upon having selected the variable is defined asThe procedure to calculate the CPP matrix is shown in Algorithm 3.
(1) Input: an elite set and the objective function value of each solution in  
(2) Output: a conditional preference probability matrix CPPM  
(3) for to do  
(4) Calculate the objective preference of the variable according to Eq. (4)  
(5) Calculate the selection probability of the variable according to Eq. (5)  
(6) Calculate the preference probability of each variable according to Eq. (6)  
(7) end for  
(8) for to do  
(9) for to do  
(10) Calculate the conditional preference probability according to Eq. (7) and (8)  
(11) Set  
(12) end for  
(13) end for 
In order to illustrate the CPP matrix construction, Table 1 gives a small instance to show how to calculate the information stored in the memory structures.
(a) Step 0: Given the following problem instance  
 
(b) Step l: Produce three high quality solutions to form the elite set  
 
 

2.3.2. EDA Guided Solution Construction Phase
Algorithm 4 shows the outline of the EDA guided solution construction phase. When creating a new starting solution, the first added variable is determined according to the preference probability information. Specifically, we first construct a preference candidate set in which the preference probability of each variable surpasses the threshold value , where and denote the minimum and maximum preference probability among all the variables, respectively. Then, we select a variable from randomly and assign it with value 1. Each time a variable is added in the solution, it is removed from and results in the size of decreased by 1. If is empty, we select variables from its complementary set.
(1) Input: and  
(2) Output: a generated solution  
(3) ,  
(4) repeat  
(5)  
(6) if then  
(7) Construct a preference candidate set  
(8) Randomly choose a variable from  
(9) else  
(10) Build a set of variables getting the maximum conditional preference probability  
(11) Build another set of variables, where each variable satisfies  
(12) if then  
(13) Randomly choose a variable from  
(14) else  
(15) Randomly choose a variable from  
(16) end if  
(17) end if  
(18) Assign the chosen variable to be value 1 to enlarge the current solution  
(19) Calculate the objective value obtained in this iteration  
(20) until or 
Once a variable is added in the solution, the selection of its next variable for enlarging the solution relies on the conditional preference probability. The conditional preference probability of selecting a variable upon having selected the variable is obtained from the CPP matrix, which is calculated according to Algorithm 3. For each constructive step, we build a set that includes all variables receiving the maximum conditional preference probability and a set that is composed of all the variables satisfying the condition . Then, we choose a variable from with the probability of and choose a variable from with the probability of . Finally, the chosen variable is assigned with value 1 to enlarge the current solution. The abovementioned constructive procedure repeats at least times and terminates when the objective value of the resulting solution begins to deteriorate.
It is noteworthy that we do not calculate a new CPP matrix each time the elite set ES is updated but wait until the ES updating reaches a specified number of times setting to be the size of ES because of the following reasons. First, it is time consuming to calculate a CPP matrix and performing fewer calculation will save computational efforts. Second, the CPP matrix calculation based on the old ES is demonstrated to be effective to provide statistical information for promising solutions construction according to preliminary experiments.
2.4. Tabu Search
Tabu search is an advanced local searchbased metaheuristic, which has been demonstrated to be highly effective for solving the MaxMean DP [9]. Since the EDA phase needs highquality solutions to maintain the elite set, we use a tabu search phase to improve solutions to local optimum. The general components of tabu search include move operators, an evaluation function, and a solution selection strategy, as well as tabu and aspiration rules.
We use a popular 1flip move operator in solving many binary quadratic programming problems, which consists in changing the value of a variable to its complement when transforming the current solution into its neighbor solutions in the search space. The evaluation function is the same as the objective function defined in (1) to measure the solution quality. For each iteration, we use the best improvement strategy that picks the one among all the neighborhood solutions with the best evaluation function value to perform the move. Given that each iteration has a total of neighborhood solutions (where denotes the number of the variables) and the identification of the best solution is time consuming if we use (1) to calculate the objective value, hence, we adopt a quick evaluation technique to reduce the computational efforts.
Specifically, we use an additional vector to record the potential of each variable , defined as
Then, the move gain of flipping the variable can be represented aswhere denotes the objective value of the solution and denotes the number of the selected elements in the current solution, i.e., .
It can also be written as
After flipping a variable to be its complement , the potential is updated as
Using the abovementioned method, the computational complexity of each iteration is reduced to .
Tabu rule requires that each time a move is performed, its reverse move is prohibited from being performed during the following specified number of iterations (called tabu tenure). A dynamic tabu tenure management strategy that divides the iterations into different intervals and uses a specified order of different tabu tenures for these intervals is employed. This strategy was first proposed in [24] for solving the graph partitioning problem and demonstrated to be effective in [9] for the MaxMean DP. Furthermore, an aspiration rule overrides the tabu rule if performing a tabu move leads to a better solution than the best solution found so far.
The proposed tabu search phase repeatedly carries out the following iterations until the best solution can not be improved for a consecutive number of iterations , called tabu search depth. To be specific, all the moves are first categorized as tabu and nontabu according to the tabu and aspiration rules. Then, the objective function value of each neighborhood solution caused by the 1flip move is quickly evaluated according to (11) and (12). Finally, either the nontabu move with the best objective value or a tabu move that satisfies the aspiration rule is chosen to be performed to move to the next solution.
2.5. Elite Set Updating and Rebuilding
We employ a popular qualitybased elite set updating strategy, where the elite set updating happens whenever the newly generated solution gets a better objective value than the worst solution in . In this case, the worst solution is replaced by this solution to execute the updating procedure. Otherwise, this new solution is disregared and updating fails. When does not undergo successful updates for continuous times, we rebuild to restart the search. The rebuilding strategy we use is to keep the best solution found so far in and generate the other solutions in the same way as initializing .
3. Computational Results and Comparisons
3.1. Benchmarks and Experimental Protocols
In order to evaluate our proposed EDATS algorithm, we use 7 different sizes of benchmarks = 500, 750, 1000, 1500, 2000, 3000, and 5000 with a total of 140 instances, where each size of benchmarks includes 20 instances from two groups (TypeI and TypeII). These instances are widely used in the literature for performance comparisons among algorithms and can be downloaded from the websites (http://www.optsicom.es/edp, http://www.mi.sanu.ac.rs/~nenad/edp, http://www.info.univangers.fr/pub/hao/maxmeandp.html).
Similar to the other reference algorithms, we use time limits as the stopping condition, which are set to be 100 seconds, 1000 seconds, and 2000 seconds for problem instances with , , and , respectively. Our EDATS algorithm is programmed in C++ and compiled using GNU g++ with −O3 flag on an Intel Xeon E5440@2.83GHz CPU. Given the random nature of our EDATS algorithm, we give 20 independent runs for each instance.
3.2. Parameter Sensitivity Analysis
Our EDATS algorithm includes seven parameters, which are the size of the elite set , the threshold coefficient in determining the preference candidate set, the threshold coefficient in determining the conditional preference candidate set, the probability to choose a variable from the maximum conditional preference set, the minimum number of added variables when constructing a solution, the tabu search depth , and the number of continuous unsuccessful elite set updates . For most algorithms that maintain an elite set of solutions, a small value between 10 and 20 is usually a preferable setting [5, 8, 9] for . The settings of these parameters are listed in Table 2, where the parameters in the tabu search phase use the recommended settings in [9] and the parameters , , and in the EDA guided solution construction phase are determined based on the following preliminary experiments.

Specifically, we choose 20 most challenging problem instances and vary with a step size of 0.05, with a step size of 0.05, and with a step size of 0.1. During each run, we only change the setting of one parameter while keeping the settings of the other parameters unchanged. For each chosen instance, we give 20 independent runs and record the average objective values.
In order to avoid many tabulated results, we report the results of the Friedman statistical testings and examine if different parameter settings present significant differences. Our experimental findings disclose that different settings of the parameter present significant difference with the value of 0.01639 and setting yields better results. In addition, the parameters and are not sensitive with the values of 0.3369 and 0.0686, respectively.
3.3. Experimental Results
In this section, we report experimental comparisons between our EDATS algorithm and the best algorithms in the literature on solving different sizes of benchmarks.
Table 3 shows results of different algorithms on solving the instances with 500 variables, including a path relinking algorithm (PR) [10], a hybrid heuristic (HH) [11], a tabu search algorithm (TPTS) [13], a hybrid threephase approach (HTP) [12], a tabu searchbased memetic algorithm (TSMA) [9], a variable neighborhood search approach (GVNS) [14], and our EDATS algorithm. The results of other algorithms are directly extracted from [9, 14]. The first column represents the name of each instance. Columns 2 to 5 report the best solution values found by the algorithms PR, HH, HTP, and TPTS, respectively. The average solution values and computational time of these algorithms are not available in the literature. Columns 6 to 14 report computational results of the algorithms TSMA, GVNS, and EDATS, where the three columns under each algorithm report the best solution values , the average solution values , and the computational time , respectively.

To make the computational times of TSMA, GVNS, and EDATS comparable, we use the PassMark performance test results to evaluate the performance differences between different computing platforms. We find that the mark of the computer where GVNS runs on (Intel(R) Core(TM) i72600@3.4GHz) is 8203 and the mark of the computer (Intel Xeon E5440@2.83GHz) where TSMA and EDATS run is 3976. Hence, the time column of GVNS is multiplied by 2.06 when comparing with TSMA and EDATS.
As can be seen from Table 3, EDATS and TSMA perform the same best in terms of the best solution values, performing better than the other algorithms PR, HH, HTP, TPTS, and GVNS. In particular, the average solution values found by both EDATS and TSMA are the same as the best solution value for each instance, performing better than GVNS. The wilcoxon statistical testings performed for EDATS and each other algorithm indicate that EDATS is significantly better than PR, HH, TS, and HTP with the values less than 0.05 in terms of the best solution values. Although EDATS is not significantly better than GNVS with respect to the best solution values, it performs significantly better than GVNS with respect to the average solution values by obtaining the value of 0.001656. Finally, the normalized average CPU time 0.71 seconds of EDATS is shorter against 52.08 seconds of GVNS and 1.17 seconds of TSMA.
Since PR, HH, and HTP do not report results for instances with more than 500 variables, we are not able to include comparisons with these algorithms in Tables 4–9. Table 4 compares EDATS with TPTS, GVNS, and TSMA for solving instances with 750 variables. From Table 4, we observe that EDATS and TSMA perform much better than GVNS and TPTS in terms of the best solution values. In addition, both EDATS and TSMA are able to reach the same results for all the 20 independent runs, dominating GVNS and TPTS in terms of the average solution values. Wilcoxon statistical testings in terms of the best solution values indicate that EDATS is significantly better than GVNS and TPTS by obtaining the values of 0.0001428 and 0.009152, respectively. In terms of the average solutions, EDATS is also significantly better than GVNS by obtaining the values of 0.000213. The normalized average CPU time 2.09 seconds of EDATS is shorter compared to 250.15 seconds of GVNS and 3.38 seconds of TSMA.






Table 5 compares EDATS with TPTS, GVNS, and TSMA for solving instances with 1000 variables. We find that EDATS and TSMA perform much better than GVNS and TPTS, as observed in Table 4. Statistical testing indicates that EDATS is significantly better than TPTS by obtaining the values of 0.00009556. Moreover, EDATS is significantly better than GVNS in terms of both best and average results, by obtaining the values of 0.03603 and 0.00009502, respectively. The normalized average CPU time 2.46 seconds of EDATS is shorter compared to 435.30 seconds of GVNS and 4.07 seconds of TSMA.
Tables 6 and 7 compare EDATS with GVNS for solving instances with 1500 and 2000 variables since GVNS is the only algorithm that reports results on these instances. From these two tables, we find that EDATS is able to find better solution values for 38 out of 40 instances. Moreover, the average solution values obtaind by EDATS are much better than GVNS. Further statistical testings indicate that EDATS is significantly better than GVNS in terms of best and average results. For solving instances with 1500 variables, the normalized average CPU time of EDATS and GVNS is 66.53 seconds against 525.44 seconds. For solving instances with 2000 variables, the normalized average CPU time of EDATS and GVNS is 80.66 seconds against 673.89 seconds. These results indicate that EDATS performs better than GVNS in terms of both solution quality and computational time.
Tables 8 and 9 present results of EDATS, TSMA, and TPTS for the most challenging instances. From Table 8, we find that EDATS performs as well as TSMA in terms of the best and average solution values, which are better than TPTS. Further statistical testings indicate that EDATS is significantly better than TPTS by obtaining the values of 0.000001907 but is not statistically comparable to TSMA. The normalized average CPU time of EDATS and TSMA is 154.21 seconds against 145.75 seconds.
From Table 9, we first observe that EDATS finds better results than TSMA for the instances MDPI6_5000, MDPI7_5000, and MDPII5_5000 (marked in bold) and matches results for the other instances. Statistical testings indicate that EDATS is significantly better than TPTS with the value of 0.000001907 but not better than TSMA in a significant level with the value of 0.3447. However, EDATS is significantly outperformed by TSMA in terms of the average solution values with the value of 0.001592. The normalized average CPU time of EDATS and TSMA is 1031.20 seconds against 817.16 seconds.
To conclude, these experimental comparisons demonstrate the effectiveness and efficacy of our proposed EDATS algorithm.
3.4. The Balance between Intensification and Diversification
Our proposed EDATS algorithm achieves a good balance between intensification and diversification during the search. For characterizing search intensification, we record how the best solution value changes as the population (elite set) evolves. For characterizing search diversification, we record the minimum Hamming distances (i.e., we calculate the Hamming distance for each pair of solutions in this elite set and use the minimum value of the Hamming distances to evaluate the diversification.) We perform experiments on two large instances MDPI10_3000 and MDPI3_5000. The experimental results are shown in Figures 1 and 2.
(a)
(b)
(a)
(b)
Figures 1(a) and 2(a) disclose that the best solution values are constantly improving as the population updates, which indicates good intensification of the EDATS algorithm during the search. Figures 1(b) and 2(b) show that the minimum Hamming distance fluctuates as the population updates. In some search periods, the observation that the distances between solutions in the population are decreased indicates that the EDATS algorithm focuses on intensification since highquality solutions are usually closer to each other. When the algorithm has already explored for certain periods, the search is switched to perform diversification and the distances between solutions are increased. Hence, this experiment reveals that our EDATS algorithm achieves a good balance between intensification and diversification.
3.5. Effectiveness of the EDA Component for Search Diversification
In order to verify the role of the EDA component for achieving good search diversification, we produce two EDATS variants by using popular diversification strategies in the literature to replace EDA. The first variant called RGTS uses a randomized greedy strategy and the second variant called RPTS uses a randomized perturbation strategy. In RGTS, the solution is iteratively constructed, where each iteration first builds a restricted candidate list of elements that produces a large objective gain and then randomly chooses an element from this list to join in the partial solution. Repeat this procedure until no objective gain is positive. In RPTS, the best solution in the population is perturbed, i.e., randomly changing a certain number of elements, to produce a new solution.
We perform additional experiments for RGTS and RPTS on a set of 20 most challenging instances with 3000 and 5000 variables (whose best objective values are difficult to be obtained comparing to the other instances), in which the experimental settings are the same as the previous EDATS algorithm. For each instance, we record the percent deviations of the best and average solution values obtained by RGTS and RPTS from those found by EDATS and show the experimental results in Figure 1.
Figures 3(a) and 3(b) show the best percent deviations of RGTS and RPTS from EDATS for each of the 20 chosen instances, respectively. From Figure 3(a), we find that the best percent deviations of RGTS from EDATS are nonpositive for all the instances except for 1 instance. From Figure 3(b), we find that the best percent deviations of RPTS from EDATS are nonpositive for all the instances. These observations disclose that EDATS finds better or equal best objective values than RGTS and RPTS.
(a)
(b)
(c)
(d)
Figures 3(c) and 3(d) show the average percent deviations of RGTS and RPTS from EDATS for each of the 20 chosen instances, respectively. From Figure 3(c), we find that the average percent deviations of RGTS from EDATS are negative for all the instances. From Figure 3(d), we find that the average percent deviations of RPTS from EDATS are negative for all the instances except for 1 instance. These observations disclose that EDATS finds better average objective values than RGTS and RPTS.
In conclusion, this experiment demonstrates that the incorporated EDA component plays an essential role to the effectiveness of the algorithm.
4. Conclusion
In this paper, we have presented for the first time a hybrid metaheuristic that combines estimation of distribution algorithm with tabu search (EDATS) for solving the maxmean dispersion problem. The proposed algorithm mainly relies on a dedicated EDA guided solution construction method for diversification and a tabu search optimization method for intensification. Extensive experiments disclose that our EDATS algorithm outperforms or competes favorably with stateoftheart algorithms in the literature. Additional analysis on the parameter sensitivity and the merit of the EDA procedure as well as the search balance between intensification and diversification sheds light on the effectiveness of the algorithm. Our findings motivate further investigation of hybrid metaheuristics of combining local search with other learning strategies.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research has been supported by the National Natural Science Foundation of China (NSFC) under Grant 71501157.
References
 O. A. Prokopyev, N. Kong, and D. L. MartinezTorres, “The equitable dispersion problem,” European Journal of Operational Research, vol. 197, no. 1, pp. 59–67, 2009. View at: Publisher Site  Google Scholar
 F. Della Croce, A. Grosso, and M. Locatelli, “A heuristic approach for the maxmin diversity problem based on maxclique,” Computers & Operations Research, vol. 36, no. 8, pp. 2429–2433, 2009. View at: Publisher Site  Google Scholar
 D. C. Porumbel, J. Hao, and F. Glover, “A simple and effective algorithm for the MaxMin diversity problem,” Annals of Operations Research, vol. 186, no. 1, pp. 275–293, 2011. View at: Publisher Site  Google Scholar
 R. Aringhieri, R. Cordone, and A. Grosso, “Construction and improvement algorithms for dispersion problems,” European Journal of Operational Research, vol. 242, no. 1, pp. 1–13, 2015. View at: Publisher Site  Google Scholar
 Y. Wang, Q. Wu, and F. Glover, “Effective metaheuristic algorithms for the minimum differential dispersion problem,” European Journal of Operational Research, vol. 258, no. 3, pp. 829–843, 2017. View at: Publisher Site  Google Scholar
 A. P. Punnen, S. Taghipour, D. Karapetyan, and B. Bhattacharyya, “The quadratic balanced optimization problem,” Discrete Optimization, vol. 12, pp. 47–60, 2014. View at: Publisher Site  Google Scholar
 R. Aringhieri and R. Cordone, “Comparing local search metaheuristics for the maximum diversity problem,” Journal of the Operational Research Society, vol. 62, no. 2, pp. 266–280, 2011. View at: Publisher Site  Google Scholar
 Y. Wang, J.K. Hao, F. Glover, and Z. Lü, “A tabu search based memetic search for the maximum diversity problem,” Engineering Applications of Artificial Intelligence, vol. 27, pp. 103–114, 2014. View at: Publisher Site  Google Scholar
 X. Lai and J.K. Hao, “A tabu search based memetic algorithm for the maxmean dispersion problem,” Computers & Operations Research, vol. 72, pp. 118–127, 2016. View at: Publisher Site  Google Scholar
 R. Martí and F. Sandoya, “GRASP and path relinking for the equitable dispersion problem,” Computers & Operations Research, vol. 40, no. 12, pp. 3091–3099, 2013. View at: Publisher Site  Google Scholar
 F. Della Croce, M. Garraffa, and F. Salassa, “A hybrid heuristic approach based on a quadratic knapsack formulation for the maxmean dispersion problem,” in Combinatorial Optimization, Lecture Notes in Computer Science, pp. 186–197, 2014. View at: Publisher Site  Google Scholar
 F. Della Croce, M. Garraffa, and F. Salassa, “A hybrid threephase approach for the MaxMean Dispersion Problem,” Computers & Operations Research, vol. 71, pp. 16–22, 2016. View at: Publisher Site  Google Scholar
 R. Carrasco, A. Pham, M. Gallego, F. Gortázar, R. Martí, and A. Duarte, “Tabu search for the max–mean dispersion problem,” KnowledgeBased Systems, vol. 85, pp. 256–264, 2015. View at: Publisher Site  Google Scholar
 J. Brimberg, N. Mladenović, R. Todosijević, and D. Urošević, “Less is more: Solving the MaxMean diversity problem with variable neighborhood search,” Information Sciences, vol. 382383, pp. 179–200, 2017. View at: Publisher Site  Google Scholar
 M. Garraffa, F. Della Croce, and F. Salassa, “An exact semidefinite programming approach for the maxmean dispersion problem,” Journal of Combinatorial Optimization, vol. 34, no. 1, pp. 71–93, 2017. View at: Publisher Site  Google Scholar
 A. Arin and G. Rabadi, “Integrating estimation of distribution algorithms versus Qlearning into MetaRaPS for solving the 01 multidimensional knapsack problem,” Computers & Industrial Engineering, vol. 112, pp. 706–720, 2017. View at: Publisher Site  Google Scholar
 P. A. N. Bosman and D. Thierens, “Expanding from discrete to continuous estimation of distribution algorithms: The IDEA,” in Parallel Problem Solving from Nature PPSN VI, M. Schoenauer, K. Deb, and G. Rudolph, Eds., vol. 1917 of Lecture Notes in Computer Science, pp. 767–776, Springer, 2000. View at: Publisher Site  Google Scholar
 P. Larrañaga, “A review on estimation of distribution algorithms,” in Estimation of Distribution Algorithms, P. Larranaga and J. A. Lozano, Eds., vol. 2, pp. 57–100, Kluwer Academic Publishers, 2002. View at: Publisher Site  Google Scholar
 P. Larranaga and J. A. Lozano, Estimation of Distribution Algorithms, Kluwer Academic Publishers, 2002.
 M. Pelikan, D. E. Goldberg, and F. G. Lobo, “A survey of optimization by building and using probabilistic models,” Computational Optimization and Applications, vol. 21, no. 1, pp. 5–20, 2002. View at: Publisher Site  Google Scholar  MathSciNet
 U. Aickelin, E. K. Burke, and J. Li, “An estimation of distribution algorithm with intelligent local search for rulebased nurse rostering,” Journal of the Operational Research Society, vol. 58, no. 12, pp. 1574–1585, 2017. View at: Publisher Site  Google Scholar
 J. M. Peña, V. Robles, P. Larrañaga, V. Herves, F. Rosales, and M. S. Pérez, “GAEDA: hybrid evolutionary algorithm using genetic and estimation of distribution algorithms,” in Proceedings of the International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems (IEA/AIE 2004), R. Orchard, Ed., Lecture Notes in Artificial Intelligence, pp. 361–371, 2004. View at: Google Scholar
 Q. Zhang, J. Sun, E. Tsang, and J. Ford, “Estimation of distribution algorithm with 2opt local search for the quadratic assignment problem,” Towards a New Evolutionary Computation, pp. 281–292, 2006. View at: Google Scholar
 P. Galinier, Z. Boujbel, and M. Coutinho Fernandes, “An efficient memetic algorithm for the graph partitioning problem,” Annals of Operations Research, vol. 191, no. 1, pp. 1–22, 2011. View at: Publisher Site  Google Scholar  MathSciNet
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Copyright © 2019 Dieudonné Nijimbere et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.