Pareto fronts-driven Multi-Objective Cuckoo Search for 5G Network Optimization

  • Wang, Junyan (School of Computer Science & Technology, Taiyuan University of Science and Technology)
  • Received : 2020.02.24
  • Accepted : 2020.05.30
  • Published : 2020.07.31


5G network optimization problem is a challenging optimization problem in the practical engineering applications. In this paper, to tackle this issue, Pareto fronts-driven Multi-Objective Cuckoo Search (PMOCS) is proposed based on Cuckoo Search. Firstly, the original global search manner is upgraded to a new form, which is aimed to strengthening the convergence. Then, the original local search manner is modified to highlight the diversity. To test the overall performance of PMOCS, PMOCS is test on three test suits against several classical comparison methods. Experimental results demonstrate that PMOCS exhibits outstanding performance. Further experiments on the 5G network optimization problem indicates that PMOCS is promising compared with other methods.


1. Introduction

Optimization is ubiquitous in real-life disciplines, such as identity authentication scheme optimization[1], breast cancer classification[2,3], wireless sensor network optimization[4],personalized recommendation system[5], water resource optimization[6], and so on. The purpose of optimization is to find the best solution to an optimization problem[7,8]. General optimization algorithms include three categories: evolutionary algorithms, swarm-based algorithms and trajectory-based algorithms. CS[9] is a typical swarm-based method, which has gained a lot of attentions since it was published. Up to now, many variants of CS have been proposed for various purposes, which can be summarized as follows:

Theoretical research on various forms is the first category. Binary CS algorithms have been published in many papers. To tackle binary optimization, Gherboudj et al.[10] designed a discrete binary CS, where cuckoos are represented with a family of bits. Binary CS can be applied to many practical issues, such as multiple destination routing problem[11], job shop scheduling[12], and flow-shop scheduling problems[13]. Discrete cuckoo search is next research topic. Travelling Salesman Problem(TSP)[14] is a popular benchmark problem,which requires salesman to return to finishing point with minimal trip length. To tackle this TSP, Ouyang et al.[15] designed a discrete version of CS, where all individuals are located on the surface of a sphere. Also for TSP, Jati et al.[16] proposed two-phase discrete cuckoo search algorithm, including discrete step size and a random step length. Combining CS with other effective strategies is another research aspect. Abdul et al. [17] incorporated roulette wheeland inertia weight strategies to CS for strengthening the search ability. To improve the convergence, Giridhar et al.[18] introduced crossover operation to CS.

Hybridizing CS with other strategies is the second category, which is able to make up the drawbacks of CS, as well as incorporating the advantage of CS to other methods. Genetic algorithm(GA) is a popular method, which has shown great search ability. To redundancy allocation problems, Kanagaraj et al.[19] designed a version of CS with GA by integrating genetic operator CS. Further, Kanagaraj et al.[20] empirically proved the efficiency of the combination of CS and GA. The advantage of differential evolution lies in exploring the search space. Wang et al.[21] proposed to strengthen CS by embedding DE to CS. Nancharaiah et al.[22] designed a hybrid technique combining CS and Ant Colony Optimization (ACO), where ACO is responsible to update the pheromone, while CS is used as a local search method.

In terms of application of CS, many papers have been reported. In medical domain, to determine the best parameter settings, Liu et al.[23,24] employed CS to optimize the initial parameter of the kernel function of support vector machine(SVM). To diagnose diabetes,Giveki et al.[25, 26, 27] proposed a novel approach, which mainly combined SVM and CS. To increase the cover rate of c-means defect in medical image segmentation, Manikandan etal.[28] utilized CS and PSO to achieve quick convergence. In information retrieval domain,finding relevant information is of great importance. To clearly organize and summarize the retrieval information, Zaw et al.[29] proposed a cuckoo search clustering method. To extract meaningful section of satellite image, Bhandari et al.[30] incorporated CS to wind-driven optimization algorithm by introducing Kapur’s entropy.

In this paper, to tackle a 5G network optimization problem [31,32], Pareto fronts-driven Multi-Objective Cuckoo Search(PMOCS) is proposed. Then three test suits are used to verify the proposed method. Then, the 5G network optimization problem is tackled with the proposed method. Thus, according to the summaries above, this paper falls into the third categories. The highlights of this paper are presented as follows:

1) Firstly, this paper proposed a multi-objective version of CS. To upgrade CS, this paper redesigns the global search manner and local search equation. Moreover, the meanings behind each upgrade are also explained.

2) To assess the proposed method, this paper utilizes two popular test suits in later experiment, as well as several comparison methods. Detailed experimental results,including tables and figures, are presented and analyzed.

3) Further, this paper applies PMOCS to a 5G network optimization. Experimental results are analyzed to show the superiority of PMOCS.

This paper is originazed as follows. Basic definitions and related work are introduced in Section 2. Following that, PMOCS is detailed in section 3. In next section, two experiments are conducted, including an empirical verification on benchmark problem and the 5G network optimization problem.

2. Preliminaries

2.1 Basic definitions

Generally, typical multi-objective optimization problems(MOPs)[4, 33, 34] have two or three objectives, which are formulated with the following expression:

\(\begin{array}{c} \min F(x)=\left[f_{1}(x), f_{2}(x), \ldots, f_{M}(x)\right] \\ x=\left(x_{1}, x_{2}, \ldots, x_{D}\right) \in R^{D} \end{array}\)       (1)

where fM(x) indicates the M-th sub-objective function, x means the decision vector. For solution x1 and x2 , x1 dominates x2 if and only if ∀i ∈ {1, 2,3,...,M} : fi(x1) ≤ fi(x2), ∃i∈{1,2,3,...,M}: fi (x1) i (x2) .The domination relationship between x1 and xcan be expressed as x1 < x2 .

2.2 Related Work

CS[9], as a typical swarm-based method, mainly mimics the breeding behavior of cuckoo. Cuckoo is able to intelligently choose host’s nests to lay their eggs. Usually, Offspring have higher survival rates. Yang et al.[9] abstracts these behaviors with the following formulations. 

Assuming Xi is the position of cuckoo i , Xi is updated with the following global update

\(X_{i}\left(t^{\prime}\right)=X_{i}(t)+\alpha \oplus \operatorname{Levy}(\lambda)\)       (2)

where ⊕ means entry-wise multiplications and α>0 indicates the step size, which is defined with following equation:

α =α0 × (Xi(t) - Xbest)       (3)

where α0 = 0.01, Xbest indicates the global best position in current population. Levy( ) λ in Eq.(2) satisfies Eq.(4):

Levy(λ)~µ = t−λ        (4)

where 1<λ<3.

However, the egg lay by cuckoo i is generally discovered and abandoned with probability a p .In this case, new egg should be lay again with the following manner:

 Xi(t+1) =  Xi(t') + r × ( Xk(t') - Xj(t'))        (5)

where both  Xk(t') and Xj(t')  present two random cuckoo birds. r indicates a random value in [0 1]. The flowchart of CS is expressed with Fig. 1.

E1KOBZ_2020_v14n7_2800_f0001.png 이미지

Fig. 1. Flowchart of CS

Obviously, CS is originally designed for single-objective optimization problems because the global best in Eq.(2) does not exist in multi-objective optimization problems (MOPs)[35, 36, 37], instead of Pareto sets. Over the past decades, numerous researchers are dedicated to apply CS to MOPs. For example, Hanoun et al.[38] designed a Pareto archived multi-objective CS to tackle a multi-objective job shop scheduling problem. Zhang et al.[39] proposed a multi-objective CS by introducing a non-dominated sorting strategy and a dynamical local search strategy to CS. Coelho et al.[40] employed the nearest neighbor density estimation method to redesign CS for MOPs. Wang et al.[41] incorporated thenon-dominated sorting strategy of NSGA-II into CS for tackling optimal design of water distribution system. To solve multi-objective community detection problem, Zhou et al.[42] redefined a location update strategy and a abandon operator in discrete form. To optimize the heat transfer in plate-finheat exchangers, Wang et al.[43] designed a non-uniform mutation operator and a differential evolution operator to strengthen the convergence of CS. To extend CS to multi-objective problems with more than three objectives, Cui et al.[44] proposed to update each dimension of each cuckoo with different probabilities.

As no-free lunch theorem[45] points out, no one method performs the best on all kinds of problems. Although various multi-objective CS[39, 46, 47] have been proposed and have exhibited outstanding performance in practical problems, they may be not suitable to the 5G network optimization problem[31]. Therefore, this paper specially designs a Pareto fronts-driven Multi-Objective Cuckoo Search algorithm (PMOCS).

3. Proposed method

In MOPs, there generally exist multiple Pareto fronts due to the conflict nature of various objectives. In principle, the former Pareto front is better than the later Pareto front regarding the convergence. Thus, to utilize the leading information inspired from the different Pareto fronts, the global manner is upgraded with Eq.(6):

Xi(t) = Xi(t) + 0.01 x α x L x rG x (Xk(t) - Xlaterj(t))       (6)

where α = 1.0 , L is generated with Levy distribution. rfollows the standard Gaussian distribution. Both Xk(t) and Xi(t)  come from the same Pareto front, while Xlaterj(t) comes from the Pareto front which is inferior to Xk(t) ’s Pareto front.

The reason for designing Eq.(6) can be illustrated with Fig. 2. Assuming there are two Pareto fronts in objective space, individual P1 and P2 belong to the first and second Pareto front, respectively. It can be said that P1 is superior to P2 in terms of convergence. Thus, P- P2 is able to provide the leading information to individuals to be updated as the arrow in Fig. 2 shows. In other words, P- P2  is used to facilitate the convergence by providing the convergent information.

E1KOBZ_2020_v14n7_2800_f0002.png 이미지

Fig. 2. Illustration of Eq.(6)

The local search manner is upgraded as follows:

Xi(t+1) = Xi(t) + r x (Xk(t) - Xsj(t))       (7)

where both Xk(t) and Xj(t) come from the first Pareto front. Note that, to further strengthen the diversity in later search stage, this paper employs a tournament selection strategy to generate Xk(t) and Xj(t). For completeness, the basic framework of the tournament selection strategy is presented in Fig. 3.

E1KOBZ_2020_v14n7_2800_f0004.png 이미지

Fig. 3. Illustration of tournament selection strategy

Eq.(7) is used to strengthen the diversity. As illustrated in Fig. 4. Let P1 and P3 be two individuals from the first Pareto front, P1 - P3 indicates the black arrow, which means individuals will be guided to search more diverse potential sections

E1KOBZ_2020_v14n7_2800_f0003.png 이미지

Fig. 4. Illustration of Eq.(7)

Updating population is a vital operation which can ensure that high-quality cuckoos pass to the next generation. PMOCS employs the non-dominated sorting strategy and crowding distance[48, 49, 50] to update the population. As Fig. 5 exhibits, parent population Pt and offspring population Qt are firstly merged into one population # of 2N cuckoos. After that, the non-dominated sorting strategy is used to divide # into multiple Pareto fronts. Then, N cuckoos are selected from the # . Note that, if only partial cuckoos in certain Pareto front are included into the next generation, the crowding distance is used as the indicator to measure each individual’s crowding degrees. Less crowded individuals are preferred and included into the next generation.

E1KOBZ_2020_v14n7_2800_f0005.png 이미지

Fig. 5. Illustration of population update

The flowchart of PMOCS is presented in Fig. 6. As can be seen, firstly, the population is initialized after defining the related paramters, such as the discover probability. Then, Cuckoo popistions are updated with Eq.(6) if the stopping criterion is not reached. After that, some cuckoo positions are updated with Eq.(7). Following that, the resultant cuckoos are updated in combintion with previous cuckoos for the next generation. If the stopping criterion is not reached, repeat the procedures above. Otherwiese, output the Pareto set.

E1KOBZ_2020_v14n7_2800_f0006.png 이미지

Fig. 6. Flowchart of PMOCS

4. Experimental Results and Analysis

The section is divided into two experiments. One experiment is to verify PMOCS with benchmark problems, and the second one is to apply PMOCS to the 5G network optimization problem. Before that, experimental settings are introduced.

4.1 Experimental settings

To verify PMOCS, various methods, including HMOCS[39], NSGA-II[48], NNIA[51] andMOPSO[52] are used as comparison algorithms. All parameters related to comparison methods are presented in Table 1, which follow the settings of original papers. The performance indicator is IGD[53], which is able to measure both the convergence and diversity. The test suites are ZDT[54], DTLZ[55] and MaF[56], all of which are presented in Table 2. The discover probability a p is set to 0.3. All algorithms run 20 times.

Table 1. Parameter settings

E1KOBZ_2020_v14n7_2800_t0001.png 이미지

Table 2. Illustration of test problems

E1KOBZ_2020_v14n7_2800_t0002.png 이미지

4.2 Experimental results and analysis

Table 3 exhibits the experimental results, where the best values are highlighted in boldface. The values in parentheses are variances. From Table 3, we can observe that PMOCS perform better on all test instances except ZDT2 and MaF2. On ZDT2, PMOCS is obviously inferior toNSGA-II. On MaF2, although PMOCS is slightly worse than HMOCS, there is no evident difference between them. Therefore, PMOCS is empirically better than other comparison

Table 3. Comparison of experimental results methods.

E1KOBZ_2020_v14n7_2800_t0003.png 이미지

Fig. 7 presents the comparison of Pareto fronts obtained by PMOCS and HMOCS on all test problems. It can be seen that there is not obvious differences on ZDT1, ZDT4, DTLZ1, MaF1and MaF2. Regarding ZDT2 and ZDT3, PMOCS is evidently superior to HMOCS because certain regions are missing for HMOCS. On DTLZ2, the fact can be seen that PMOCS is better than HMOCS regarding the convergence.

E1KOBZ_2020_v14n7_2800_f0007.png 이미지

Fig. 7. Comparison of Pareto fronts on test problems

4.3 5G Network Optimization

More convenient cellular networks is coming with the introduction of 5G networks. Over the past few decades, various methods have been proposed in the technological preparations for 5G network. Massive multiple-input multiple-output (MIMO) system is one of key enabler of 5G network. Massive MIMO is proposed based on the idea of deploying large arrays with a lot of active antennas at base stations (BSs) and serve limited number of users. Massive MIMO is robust to the distortions caused by hardware imperfections[57].

The aim of this section is to optimize the downlink transmission of a missive MIMO system[21], which has two conflicting objectives, high average user rates and high energy efficiency. As Fig. 8 shows, assume there are N antennas and K single-antenna users. B MHz =10 indicates the bandwidth. The emitted power per BS is P Watt. σ2 = 10-13W is the average noise power. The size of each cell is 250 250 × meters. That is to say, A = 0.252km2.

E1KOBZ_2020_v14n7_2800_f0008.png 이미지

Fig. 8. Illustration of missive MIMO system

Average user rate ( f1 ) and energy efficiency ( f2 )[58] are defined with the following equations:

\(\left\{\begin{array}{l} f_{1}=B\left(1-\frac{K}{\Upsilon}\right) \log _{2}\left(1+\frac{\frac{P}{K}(N-K)}{\sigma^{2} \Lambda_{1}+P \Lambda_{2}}\right) \\ f_{2}=\frac{K \times f_{1}}{P_{\text {total }}} \\ \text { s.t. } P_{\text {total }}=\frac{P}{\eta}+N C_{N}+K C_{K}+\frac{C_{\text {precoding }}}{L}+C_{0} \end{array}\right.\)      (8)

where Λ1 = 1.72 x 109 , Λ2 = 0.54 [59], CN = 0.5W , Ck = 0.3W ,  C0 =10W , Cprecoding = \(3 K^{2} N \frac{B}{\Upsilon}\) , L = 12.8Gflops / W. From Eq.(8), it can be seen that the average user rate should be as high as possible, while energy efficiency should be as low as possible. The variables to be optimized are K, N and P, and they follows the following constraints:

\(\begin{array}{l} x=~\left[\begin{array}{lll} K & N & P \end{array}\right] \\ \text { s.t. }\left\{\begin{array}{l} 1 \leq K \leq \frac{N}{2} \\ 2 \leq N \leq N_{\text {max }} \\ 0 \leq P \leq N P_{\max } \end{array}\right. \end{array}\)       (9)

where Nmax = 500 and Pmax = 20W .

To test the performance of PMOCS on practical problems, PMOCS, HMOCS, NNIA,NSGA-II and MOPSO are applied to the 5G network optimization. All the parameters remains the same as illustrated above. Maximal iteration is set to 2000. Fig. 9 plots the comparison

E1KOBZ_2020_v14n7_2800_f0009.png 이미지

Fig. 9. Comparison of experimental results on the 5G network optimization problem

From Fig. 9, we can observe that the Pareto front obtained by PMOCS is able to cover the most area. However, compared with PMOCS, NSGA-II performs less satisfied due to the discontinuous Pareto front. Further, HMOCS is relatively better than NNIA and MOPSO in terms of final Pareto fronts. In summary, PMOCS is empirically promising in tackling practical problems.

5. Conclusion

In this paper, to tackle the 5G network optimization problem, PMOCS is proposed. This paper firstly upgrades the global search manner to a new version, which is able to generate the evolutionary pressure to the ideal Pareto front. Then, the local search manner is improved to guide the population to various promising areas. The experimental results on three test suits show that PMOCS is better than other comparison methods. Further, the experiment results on the 5G network optimization problem demonstrate that PMOCS is more effective in comparison with other methods. Next work will concentrate on the further improvement ofPMOCS.results.


This paper is supported by National Natural Science Foundation of China (Grant No. 61806138).


  1. Z. Cui, F. Xue, S. Zhang, X., Y. Cao, W. Zhang and J. Chen, "A Hybrid Block Chain-Based Identity Authentication Scheme for Multi-WSN," IEEE Transactions on Services Computing, vol.13, pp.241-251, 2020.
  2. M. N. Sudha, S. Selvarajan and M. Suganthi, "Feature selection using improved lion optimisation algorithm for breast cancer classification," International Journal of Bio-Inspired Computation, vol. 14, 2019.
  3. Z. Shen, Y. Niu; Y. Zuo, Q. Xie and Z. Chen, "Power control of wind energy conversion system under multiple operating regimes with deep residual recurrent neural network: theory and experiment," International Journal of Computing Science and Mathematics, vol. 10, 2019.
  4. X. Cai, P. Wang, L. Du, Z. Cui, W. Zhang and J. Chen, "Multi-objective 3-Dimensional DV-Hop Localization Algorithm with NSGA-II," IEEE Sensors Journal, vol.19, no.21, pp.10003-10015, 2019.
  5. Z. Cui, X. Xu, F. Xue, X. Cai, Y. Cao, W. Zhang and J. Chen, "Personalized Recommendation System based on Collaborative Filtering for IoT Scenarios," IEEE Transactions on Services Computing, pp. 1-1, 2020.
  6. H. Wang, W. Wang, Z. Cui, X. Zhou, J. Zhao and Y. Li, "A new dynamic firefly algorithm for demand estimation of water resources," Information Sciences, vol. 438, pp.95-106, 2018.
  7. X. Cai, H. W., Z. Cui, J. Cai, Y. Xue and L. Wang, "Bat algorithm with triangle-flipping strategy for numerical optimization," International Journal of Machine Learning and Cybernetics, vol.9, no.2, pp.199-215, 2018.
  8. B. Niu, J. Liu, and L. Tan, "Multi-swarm cooperative multi-objective bacterial foraging optimization," International Journal of Bio-Inspired Computation, vol. 13, pp. 21-31, 2019.
  9. X. Yang and S. Deb, "Cuckoo search via Levy flights," in Proc. of World Congress on Nature and Biologically Inspired Computing, India, pp.210-214, 2009.
  10. A. Gherboudj, A. Layeb and S. Chikhi, "Solving 0-1 knapsack problems by a discrete binary version of cuckoo search algorithm," International Journal of Bio-Inspired Computing, vol.4, no.4, pp.229-236, 2012.
  11. Z. Zhan and J. Zhang, "Discrete particle swarm optimization for multiple destination routing problems," in Proc. of Workshops on Applications of Evolutionary Computation, Springer, pp.117-122, 2009.
  12. P. Pongchairerks, "Particle swarm optimization algorithm applied to scheduling problems," Scienceasia, vol.35, no.1, pp.89-94, 2009.
  13. C. Liao, C. Tseng and P. Luarn, "A discrete version of particle swarm optimization for flowshop scheduling problems," Computers & Operations Research, vol. 34, no.10, pp.3099-3111, 2007.
  14. X. Ouyang, Y. Zhou, Q. Luo andH. Chen, "A novel discrete cuckoo search algorithm for spherical traveling salesman problem," Applied Mathematics & Information Sciences, vol.7, no.2, pp.777-784, 2013.
  15. A. Ouaarab, B. Ahiod and X. Yang, "Discrete cuckoo search algorithm for the travelling salesman problem," Neural Computing and Applications, vol.24, no.7-8, pp.1659-1669, 2014.
  16. G. K. Jati, H.M. Manurung, et al., "Discrete cuckoo search for traveling salesman problem," in Proc. of 7th International Conference on Computing and Convergence Technology, IEEE, pp.993-997, 2012.
  17. K. N. Abdul Rani, M. Abdul Malek and N. Siew-Chin, "Nature-Inspired Cuckoo Search Algorithm for Side Lobe Suppression in a Symmetric Linear Antenna Array," Radio engineering, vol.21, no.3, pp.865-874, 2012.
  18. M. S. Giridhar, S. Sivanagaraju, C. V. Suresh and P. Reddy, "Analyzing the multi-objective analytical aspects of distribution systems with multiple multi-type compensators using modified cuckoo search algorithm," International Journal of Parallel, vol.32, no.6, pp.549-571, 2017.
  19. X. Liu and H. Fu, "PSO-based support vector machine with cuckoo search technique for clinical disease diagnoses," Scientific World Journal, vol.2014, p.7, 2014.
  20. D. Giveki, H. Salimi, G. Bahmanyar and Y. Khademian, "Automatic Detection of Diabetes Diagnosis Using Feature Weighted Support Vector Machines Based on Mutual Information and Modified Cuckoo Search," arXiv:1201.2173, 2012.
  21. P. Manikandan and S. Selvarajan, "A hybrid optimization algorithm based on cuckoo search and PSO for data clustering," International Review on Computers and Software, vol.8, no.9, pp.2278-2287, 2013.
  22. M. Zaw and E. Mon, "Web Document Clustering by Using PSO-Based Cuckoo Search Clustering Algorithm," Recent Advances in Swarm Intelligence and Evolutionary Computation, pp 263-281, 2014.
  23. R. Chouder and N. Benhamidouche, "New exact solutions to nonlinear diffusion equation that occurs in image processing," International Journal of Computing Science and Mathematics, vol. 10, 2019.
  24. A.K. Bhandari, V.K. Singh, A. Kumar and G.K. Singh, "Cuckoo search algorithm and wind driven optimization based study of satellite image segmentation for multilevel thresholding using Kapur's entropy," Expert System with Application, vol.41, no.7, pp.3538-3560, 2014.
  25. G. Kanagaraj, S. Ponnambalam and N. Jawahar, "A hybrid cuckoo search and genetic algorithm for reliability-redundancy allocation problems," Computer and Industrial Engineering, vol.66, no.4, pp.1115-1124, 2013.
  26. M. Behroozifar and F. Ahmadpour, "A study on spectral methods for linear and nonlinear fractional differential equations," International Journal of Computing Science and Mathematics, vol. 10, 2019.
  27. S. Asghari and N. J. Navimipour, "Cloud service composition using an inverted ant colony optimisation algorithm," International Journal of Bio-Inspired Computation, vol. 13, 2019.
  28. G. Kanagaraj, S. Ponnambalam, N. Jawahar and J.M. Nilakantan, "An effective hybrid cuckoo search and genetic algorithm for constrained engineering design optimization, Engineering Optimization," Engineering Optimization, vol.46, no.10, pp.1331-1351, 2014.
  29. G. Wang, L. Guo, H. Duan, L. Liu, H. Wang, and B. Wang, "A hybrid meta-heuristic DE/CS algorithm for UCAV path planning," Scientific World Journal, vol. 2012, pp.583973, 2012.
  30. B. Nancharaiah, and B.C. Mohan, "Hybrid optimization using ant colony optimization and cuckoo search in MANET routing," in Proc. of 2014 International Conference on Communications and Signal Processing (ICCSP), IEEE, pp.1729-1734, 2014.
  31. E. Bjornson, E. Jorswieck, and M Debbah, "Multiobjective Signal Processing Optimization: The way to balance conflicting metrics in 5G systems," IEEE Signal Processing Magazine, vol.31, no.6, pp.14-23, 2014.
  32. R. S. Parpinelli, G. F. Plichoski, R. S. Silva and P. Narloch, "A review of techniques for online control of parameters in swarm intelligence and evolutionary computation algorithms," International Journal of Bio-Inspired Computation, vol. 13, pp. 1-20, 2019.
  33. Y. Liu, "Solving nonlinear system of second-order boundary value problems using a newly constructed scaling function," International Journal of Computing Science and Mathematics, vol. 10, 2019.
  34. S. Negi and S. B. Singh, "Fuzzy reliability evaluation of linear m-consecutive weighted-k-out-of-r-from-n: F systems," International Journal of Computing Science and Mathematics, vol.10, 2019.
  35. Z. Cui, J. Zhang, D. Wu, X. Cai, H. Wang, W. Zhang and J. Chen, "Hybrid Many-Objective Particle Swarm Optimization Algorithm for Green Coal Production Problem," Information Sciences, vol.518, pp.256-271, 2020.
  36. Z. Cui, L. Du, P. Wang, X. Cai and W. Zhang, "Malicious code detection based on CNNs and multi-objective algorithm," Journal of Parallel and Distributed Computing, vol.129, pp.50-58, 2019.
  37. Z. Cui, J. Zhang, Y. Wang, Y. Cao, X. Cai, W. Zhang and J. Chen, "A pigeon-inspired optimization algorithm for many-objective optimization problems," SCIENCE CHINA Information Sciences, vol.62, no.7, pp.131-138, 2019.
  38. S. Hanoun, S. Nahavandi, D. Creighton and H. Kull, "Solving a multi-objective job shop scheduling problem using Pareto archived cuckoo search," in Proc. of 17th Emerging Technologies Factory Automation (ETFA), IEEE, pp.1-8, 2012.
  39. M. Zhang, H. Wang, Z. Cui and J. Chen, "Hybrid multi-objective cuckoo search with dynamical local search," Memetic Computing, vol.10, no.2, pp.199-208, 2018.
  40. L. Coelho, F. Guerra and N. Batistela, "Multi-objective cuckoo search algorithm based on duffing's oscillator applied to jiles-atherton vector hysteresis parameters estimation," IEEE Transactions on Magnetics, vol.49, no.5, pp.1745-1748, 2013.
  41. Q. Wang, S. Liu and H. Wang, "Multi-objective cuckoo search for the optimal design of water distribution systems," in Proc. of International conference on civil engineering and urban planning, 2012.
  42. X. Zhou, Y. Liu and B. Li, "A multi-objective discrete cuckoo search algorithm with local search for community detection in complex networks," Modern Physics Letter B, vol.30, no.07, pp.1650080, 2016.
  43. Z. Wang and Y. Li, "Irreversibility analysis for optimization design of plate fin heat exchangers using a multi-objective cuckoo search algorithm," Energy Conversion and Management, vol.101, pp.126-135, 2015.
  44. Z Cui, M. Zhang, and H Wang. "A hybrid many-objective cuckoo search algorithm," Soft Computing, vol.10, no.2, pp.199-208, 2019.
  45. D. H. Wolpert, and W. G. Macready, "No free lunch theorems for optimization," IEEE Transactions on Evolutionary Computation, vol. 1, no.1, pp.67-82, 1997.
  46. X. Cai, X. Gao and Y. Xue, "Improved bat algorithm with optimal forage strategy and random disturbance strategy," International Journal of Bio-inspired Computation, vol.8, no.4, pp.205-214, 2016.
  47. Z. Cui, Y. Chang, J. Zhang, X. Cai and W. Zhang, "Improved NSGA-III with selection-and-elimination operator," Swarm and Evolutionary Computataion, vol.49, pp.23-33,2019.
  48. K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, vol.6, no.2, pp.182-197, 2002.
  49. P. Wang, F. Xue, H. Li, Z. Cui, L. Xie and J. Chen, "A Multi-Objective DV-Hop Localization Algorithm Based on NSGA-II in Internet of Things," Mathematics, vol.7, no.2, 184, 2019.
  50. Y. Wang, P. Wang, J. Zhang, Z. Cui, X. Cai, J. Chen, and W. Zhang, "A Novel Bat Algorithm with Multiple Strategies Coupling for Numerical Optimization," Mathematics, vol.7, no.2, 135, 2019.
  51. M. Gong, L. Jiao, H. Du, and L. Bo, "Multiobjective immune algorithm with non-dominated neighbor-based selection," Evolutionary Computation, vol.16, no.2, pp.225-255, 2008.
  52. C. A. Coello, and M. S. Lechuga, "MOPSO: A proposal for multiple objective particle swarm optimization," in Proc. of the IEEE Congress on Evolutionary Computation, pp.1051-1056, 2002.
  53. C. A. Coello, and N. C. Cortes, "Solving multiobjective optimization problems using an artificial immune system," Genetic Programming and Evolvable Machines, vol. 6, no.2, pp. 163-190, 2005.
  54. E. Zitzler, K. Deb, and L. Thiele, "Comparison of multiobjective evolutionary algorithms: Empirical results," Evolutionary computation, vol.8, no. 2, pp.173-195, 2000.
  55. K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, "Scalable test problems for evolutionary multiobjective optimization," Evolutionary Multiobjective Optimization. Theoretical Advances and Applications, pp.105-145, 2005.
  56. R. Cheng, M. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin and X. Yao, "A benchmark test suite for evolutionary many-objective optimization," Complex & Intelligent Systems, vol.3, no.1, pp.67-81, 2017.
  57. H. Yang and T. Marzetta, "Total energy efficiency of cellular large scale antenna system multiple access mobile networks," in Proc. of 2013 IEEE online conference on green communications, pp.27-32, 2013.
  58. G. Auer, V. Giannini, C. Desset, "How much energy is needed to run a wireless network?," IEEE wireless communications, vol. 18, no.5, pp.40-49, 2011.
  59. E. Bjornson, L. Sanguinetti and J. Hoydis, "Optimal Design of Energy-Efficient Multi-User MIMO Systems: Is Massive MIMO the Answer?," IEEE Transactions on Wireless Communications, vol.14, no.6, pp.3059-3075, 2015.