A comprehensive review on NSGA-II for multi-objective combinatorial optimization problems

Received March 9, 2021, accepted March 17, 2021, date of publication April 2, 2021, date of current version April 20, 2021. Digital Object Identifier 10.1109/ACCESS.2021.3070634 A Comprehensive Review on NSGA-II for Multi-Objective Combinatorial Optimization Problems SHANU VERMA 1, (Graduate Student Member, IEEE), MILLIE PANT1, AND VACLAV SNASEL 2, (Senior Member, IEEE) 1Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India 2Department of Computer Science, VSB—Technical University of Ostrava, 70800 Ostrava, Czech Republic Corresponding author: Millie Pant ([email protected].ac.in) This work was supported in part by the CSIR, Ministry of Science and Technology, Government of India, under Grant 09/143(0966)/2019-EMR-I, and in part by the Metaheuristics framework for combinatorial optimization problems (META-MO-COPS) under Grant DST/INT/Czech/P-12/2019. ABSTRACT This paper provides an extensive review of the popular multi-objective optimization algorithm NSGA-II for selected combinatorial optimization problems viz. assignment problem, allocation problem, travelling salesman problem, vehicle routing problem, scheduling problem, and knapsack problem. It is identified that based on the manner in which NSGA-II has been implemented for solving the aforementioned group of problems, there can be three categories: Conventional NSGA-II, where the authors have implemented the basic version of NSGA-II, without making any changes in the operators; the second one is Modified NSGA-II, where the researchers have implemented NSGA-II after making some changes into it and finally, Hybrid NSGA-II variants, where the researchers have hybridized the conventional and modified NSGA-II with some other technique. The article analyses the modifications in NSGA-II and also discusses the various performance assessment techniques used by the researchers, i.e., test instances, performance metrics, statistical tests, case studies, benchmarking with other state-of-the-art algorithms. Additionally, the paper also provides a brief bibliometric analysis based on the work done in this study. INDEX TERMS NSGA-II, combinatorial optimization, multi-objective optimization, genetic algorithms. I. INTRODUCTION The Non-Dominated Sorting Genetic Algorithm (NSGA-II) is a powerful decision space exploration engine based on Genetic Algorithm (GA) for solving Multi-objective Optimization Problems (MOOPs). It was initially proposed by Deb et al. [1] in the year 2000 in ‘International Conference on Parallel Problem Solving from Nature.’ In 2002, it was published as a full-length research article in the journal, IEEE Transactions on Evolutionary Computation [2], and since then, it has been cited more than 20630 times according to IEEE Xplore. Presumably, NSGA-II is the 4th most cited journal article in the database of IEEE Xplore. Further, as per Google scholar, it has been cited more than 35240 times, out of which more than 19600 citations are from web of science. This information is sufficient to show the popularity of NSGA-II for The associate editor coordinating the review of this manuscript and approving it for publication was Hisao Ishibuchi . solving MOOPs. During its 20 years of existence, NSGA-II has been implemented on a wide range of MOOPs having continuous as well as discrete variables. However, to the best of the authors’ knowledge, there is no comprehensive review of NSGA-II, which can guide the researchers working in this area. To bridge this gap and to acquaint the readers with the versatility of NSGA-II, in this paper, the focus is on the application of NSGA-II and its variants on selected Combinatorial Optimization Problems (COPS): assignment problem, allocation problem, travelling salesman problem (TSP), vehicle routing problem (VRP), scheduling problem, and knapsack problem. We have particularly chosen COPS as in the broad world of optimization, COPs are considered to be one of the most challenging and complex problems. Since most COPs are NP-hard in nature, the computational complexity for solving these problems increases as the problem size increases. Therefore, for such problems, approximate methods such as metaheuristics approaches are preferred over classical methods [3]. VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 57757 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS A. COMBINATORIAL OPTIMIZATION PROBLEMS (COPS) In the early ’70s, when metaheuristics such as evolutionary algorithms were proposed to solve COPs, they did not have a significant influence due to the lack of efficient computational facilities [4]. However, with the passage of time and with the advent of fast and high-end computational facilities, many metaheuristics were proposed for solving complex COPs. Literature reveals that a lot of attention has been given to COPs with multiple objectives due to their challenging nature and practical utility. Multi-objective Combinatorial Optimization Problems (MOCOPs) have been surveyed from time to time by various researchers. In 1994, Ulungu & Teghem [5] surveyed and suggested the adaptations of metaheuristics to multi-objective environment as a direction of future research. Thereafter, researchers proposed effective heuristics and meta-heuristics to solve MOCOPs. Coello Coello et al. [6] gave a detailed overview of multiobjective combinatorial optimization that included relevant definitions and the basic idea of using metaheuristic for MOCOP. An overview of hybrid approximation methods for MOCOPs can be found in [7] and [8]. According to [8], an approximation algorithm for MOCOPs should necessarily be hybridized, i.e., it should be a combination of an evolutionary algorithm, a neighbourhood search algorithm (local search procedure), and problem-specific components because the universal method for a problem cannot perform better than the technique specially tailored for it. More recently, Liu et al. [9] reviewed the literature of multi-objective metaheuristics for multi-objective discrete optimization problems (MODOPs) and provided information about its application areas, test instances, and performance metrics. As a direction of future research, the author suggested reviewing various metaheuristics for different type of MODOPs. The review studies on COPs in the existing literature either discussed both single and multi-objective versions of a specific COP or only its multi-objective version along with the solution approaches. Some of the reviewed single-objective COPs by the researchers focus on specific areas like blocking flowshop scheduling problem (FSP) [10], permutation FSP [11], non-permutation FSP [12], agricultural land-use allocation [13], facility location problems [14], emergency material scheduling[15], allocation of distributed generation [16], location-routing problems [17], resource allocation for CRAN in 5G and beyond networks [18], VRP [19], cell formation problem [20] and many more. The selected papers are divided into three categories that are identified based on the implementation of NSGA-II to MOCOPs and are-conventional NSGA-II, modified NSGA-II, and hybrid NSGA-II. The papers under the first category used NSGA-II in its traditional form with the same crossover, mutation, and selection operators [2]. The papers under the second category modified the conventional NSGA-II, mainly in terms of the initialization, selection scheme, crossover and mutation operators, crowding distance operator, constraint handling technique, or some other criteria. Furthermore, the third category contains the papers in which either the conventional NSGA-II or modified NSGA-II is hybridized with a heuristic strategy, a local search operator, a machine learning technique, or another single/multiobjective optimization algorithm. On the other, those papers in which multi-criteria decision-making (MCDM) techniques are applied to the trade-off solutions obtained using conventional NSGA-II are not considered in the third category. In this study, an MCDM technique is viewed as a postoptimization technique for selecting the best-compromised solution. Such papers are studied under the first category and analyzed separately in terms of applied decision-making methods. In the selected papers, the benchmarking of NSGA-II based algorithms with other state-of-the-art algorithms, the algorithms used for hybridization with NSGA-II, the methods used for the post-Pareto optimality analysis, the number of objective functions involved in the papers are discussed. Additionally, the test instance or datasets, case studies, performance metrics, and statistical tests used in the papers are also analyzed. In summary, this paper makes the following contributions: Provides a detailed analysis of NSGA-II and its variants for solving six selected MOCOPs. Discusses the modifications in NSGA-II algorithms. Provides a brief bibliometric analysis including information about post-Pareto optimality analysis, number of objective functions, test instances, case studies, performance metrics and statistical tests. Future research directions in this field. Accordingly, the remaining of the paper is organized as follows: Section 2 contains the background of MOOP, MOCOP, concept of Pareto dominance, and NSGA-II, including its basic structure and working procedure. Section 3 provides the research methodology for the study. Section 4 addresses the literature survey, including NSGA-II implementation to MOCOPs, performance assessment, performed case studies, statistical analysis, and post-Pareto optimality analysis. Section 5 is about the analysis of modifications in NSGA-II. Section 6 discusses the bibliometric analysis, and lastly, Section 7 provides the conclusion and future directions drawn from this study. II. BACKGROUND In this section, we describe the concepts of MOOP, MOCOP, and Pareto dominance. The basic structure and procedure of NSGA-II are also discussed. A. MULTI-OBJECTIVE OPTIMIZATION PROBLEM A MOOP includes a set of ndecision variables, kobjective functions, and a set of (minequality and pequality) constraints. The optimization goal isMin/Max y=f(x)=(f1(x),f2(x),...,fk(x)),k≥2 (1) Subject to gi(x)≤0,i=1,2,...,m(2) hj(x)=0,j=1,2,...,p(3) 57758 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS where x=(x1,x2,...,xn) is an n-dimensional decision vector in X⊆Rn(Ris the set of real numbers), y is a k-dimensional objective vector in Rk,fdefines the mapping function, giis the ith inequality constraint, and hjis the jth equality constraint. Further, (2-3) determine the set of all feasible solutions X. B. MULTI-OBJECTIVE COMBINATORIAL OPTIMIZATION PROBLEM The MOCOPs form a particular class of MOOPs, that can be formulated as: Min/Max y=f(x)=(f1(x),f2(x), . . . fk(x)),k≥2 (4) Subject to gi(x)≤0,i=1,2,...,m(5) hj(x)=0,j=1,2,...,p(6) where x=(x1,x2,...,xn)∈Dis an n-dimensional vector in decision-space D=D1×D2×. . . ×Dn(Dnis the domain of xn), yis a k-dimensional objective vector in Rk,fis the mapping function, giis the ith inequality constraint, hjis the jth equality constraint. The other variables, k, m, and p, represent the numbers of objective functions, inequality constraints, and equality constraints, respectively. The set Sis the set of all feasible solutions that satisfy (5-6) and may describe a combinatorial structure such as spanning trees of a graph, paths, and matching. Some examples of COPs are assignment problem, allocation problem, scheduling problem, VRP, TSP, knapsack problem, sum of subset problem, network design problem, graph-colouring problem, location-routing problem, and facility location problem. C. CONCEPT OF PARETO DOMINANCE Let x1and x2be the two feasible solutions of the multiobjective minimization problem (1). The solution x1can be viewed as better than x2if the following conditions hold: 1. fj(x1)≤fj(x2) for all j= {1,2,...,k} 2. fj(x1)<fj(x2) for at least one j= {1,2,...,k} where kis the number of objective functions, fj(x) is the jth value of an objective function for decision vector x. In this case, we say that x1dominates x2(or x2is dominated by x1): x1is better than x2. The relation ‘<’ (or ‘>’ for maximization problem) can be denoted as a dominance operator G. x1Gx2represents x1dominates x2. When a solution xof (1) is not dominated by any other feasible solutions, it is called a Pareto optimal solution. The set of all Pareto optimal solutions are referred to as a Pareto set. The objective vector corresponding to the Pareto set is defined as a Pareto front, as shown in Fig. 1. D. NSGA-II NSGA-II is an improved version of the non-dominated sorting genetic algorithm (NSGA) [21], which has been criticized by researchers due to its limitations such as the absence of elitism, the need to define sharing parameter for diversity preservation, and its high computational complexity. On the other hand, the design of NSGA-II exhibits the property of elitism and does not need any sharing parameter. It uses the FIGURE 1. Pareto dominance. crowding distance operator for the mechanism of diversity preservation. Moreover, it is computationally fast and lives up to its name ‘Fast Elitist NSGA-II.’ The overall complexity of NSGA-II is at most O(MN2), where M is the number of objective functions, and N is the population size. 1) BASIC STRUCTURE OF NSGA-II The philosophy of NSGA-II is based on four main principles, which are: Non-Dominated Sorting, Elite Preserving Operator, Crowding Distance and Selection Operator. These are described in brief in the following subsections. a: NON-DOMINATED SORTING In this procedure, the population members are sorted using the concept of Pareto dominance. The process of non-dominated sorting begins with assigning the first rank to the non-dominated members of the initial population. These first ranked members are then placed in the first front and removed from the initial population. After that, the non-dominating sorting procedure is performed on the remaining population members. Further, the non-dominated members of the remaining population are assigned the second rank and placed in the second front. This process continues until the whole population members are put on different fronts according to their ranks, as shown in Fig. 2 (a). b: ELITE-PRESERVING OPERATOR Elite preserving strategy retains the elite solutions of a population by directly transferring them to the next generation. In other words, the non-dominated solutions found in each generation move on to the next generations till some solutions dominate them. c: CROWDING DISTANCE The crowding distance is calculated to estimate the density of solutions surrounding a particular solution. It is the average distance of two solutions on either side of the solution along each of the objectives. On comparing two solutions with different crowding distances, the solution with the large crowded distance is considered to be present in a less crowded region. The crowded distance of the ith solution is the average side-length of the cuboid, as shown in Fig. 2(b). If fi jis the jth value of an objective function for the ith individual and, fmax jand fmin jare the maximum and minimum values VOLUME 9, 2021 57759 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS FIGURE 2. Non-dominated sorting procedure and Crowding distance calculation. respectively of jth objective function among all the individuals. Then, the crowding distance of ith individual is defined as the average distance of two nearest solutions on either side, as given in (7). cd(i)= k X i=1 fi+1 j−fi−1 j fmax j−fmin j (7) where kis the number of objective functions. d: SELECTION OPERATOR The population for the next generation is selected using a crowded tournament selection operator, which uses the rank of the population members and their crowding distances for the selection. The rule for selecting one out of two population members for the next generation is- (i) If both the population members are of different ranks, then the one with the better rank is selected for the next generation (ii) If both the population members are of the same ranks, then the one with the higher crowding distance is selected for the next generation. 2) PROCEDURE OF NSGA-II The procedure of NSGA-II begins with generating an initial population Ptof size N. Then, a new population Qtis created after performing crossover and mutation operations on the population Pt. After that, the population Ptand Qtare combined to form a new population Rt, and the non-dominated sorting procedure is performed on Rt. Then, the population members of Rtare ranked into different fronts according to their non-domination levels. The next process is to select N members from Rtto create the next population Pt+1. If the size of the first front is greater than or equal to N, then only N members are selected from the least crowded region of the first front to form Pt+1. On the contrary, if the size of the first front is less than equal to N, then the members of the first front are directly transferred to the next generation, and the remaining members are taken from the least crowded region of the second front and added to Pt+1. If the size of Pt+1is still less than N, then the same procedure is followed for the next consecutive fronts until the size of Pt+1becomes equal to N. The populations Pt+2,Pt+3, Pt+4, ..., for the next generations are constructed using the same procedure until the stopping criteria are not satisfied. The working of NSGA-II is shown in Fig. 3. FIGURE 3. Procedure of NSGA-II. III. RESEARCH METHODOLOGY A. MATERIAL COLLECTION This study is conducted using research databases till April 2020 from Science Direct, Taylor & Francis Online, Wiley Online Library, Springer, and IEEE Xplore. The database of Science Direct was searched for article type ‘Research articles’ with search terms ‘NSGA-II, travelling salesman problem,’ ‘NSGA-II, assignment problem,’ ‘NSGA-II, allocation problem,’ ‘NSGA-II, knapsack problem,’ ‘NSGA-II, vehicle routing problem’ and ‘NSGA-II, scheduling problem.’ The database of Springer was searched for article and conference paper using advance search option with the word ‘NSGA-II’ and exact phrases ‘travelling salesman problem,’ ‘assignment problem,’ ‘allocation problem,’ ‘knapsack problem,’ ‘vehicle routing problem’ and ‘scheduling problem.’ The database of Taylor & Francis Online was searched using words’ ‘NSGA-II + travelling salesman problem’, ‘NSGA-II +assignment problem,’ ‘NSGA-II +allocation problem,’ ‘NSGA-II +knapsack problem,’ ‘NSGA-II +vehicle routing problem,’ and ‘NSGA-II +scheduling problem’ anywhere in the articles. The database of IEEE Xplore was advance searched using the search term ‘NSGA-II and travelling salesman problem,’ ‘NSGA-II and assignment problem,’ ‘NSGA-II and allocation problem,’ ‘NSGA-II and knapsack problem,’ ‘NSGA-II and vehicle routing problem’ and ‘NSGA-II and scheduling problem’ anywhere in the metadata. The database of Wiley Online Library was searched for journal papers with the same procedure as used for the IEEE Xplore database. The details of the search results are given in Table 1. Out of these, the papers focused on using conventional NSGA-II/ modified NSGA-II/hybrid NSGA-II for solving MOCOPs are considered for the study. Further, only those journals articles are selected, which are published in Science citation index expanded (SCIE) and emerging sources citation index (ESCI) indexed journals. In total, 169 papers got selected for review, of which 135 are journal papers, and the rest 34 are conference papers. The list of journals in which 57760 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 1. Database search results. selected papers are published is shown in Table 2 with their latest impact factors and quartile ranking based on JCR 2019. The number of papers based on different publishers is shown in Table 3. The maximum number of papers are from the Elsevier publisher. B. CATEGORIZATION OF SELECTED PAPERS The reviewed papers are initially classified into three categories, conventional NSGA-II, modified NSGA-II, and hybrid NSGA-II based on the implementation of NSGA-II. These three categories are further classified into different sub-categories based on the type of MOCOPs, as shown in Table 4. The miscellaneous category includes the papers based on the combinations of the above six MOCOPs. IV. REVIEW OF LITERATURE In COPs’ literature, many approximation algorithms are used for solving the MOCOPs, such as, multi-objective discrete artificial bee colony (ABC) [22]–[27]; multi-objective ant colony optimization (MOACO) [28]; improved artificial immune algorithm [29]; MOEA/D [30]; multi-objective memetic algorithm [31]–[34]; water wave optimization [35]; modified particle swarm optimization (PSO) [36], [37]; multi-objective hybrid immune algorithm [38]; GA [39]; grey wolf optimization [40]; cooperative swarm intelligence algorithm for MODOP [41]; multi-objective fruit fly optimization algorithm [42]; multi-objective discrete virus optimization algorithm [43]; NSGA-II & SPEA-II [44] and subpopulation based multi-objective evolutionary algorithm [45]. As this study is focussed on reviewing NSGA-II for MOCOPs, a detailed view of NSGA-II implementations for selected MOCOPs is given in next sub-sections. TABLE 2. List of journals. VOLUME 9, 2021 57761 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 3. Number of papers from different publishers. TABLE 4. Classification of papers based on different categories. A. NSGA-II FOR MOCOPS In the literature, NSGA-II was applied to solve the MOCOPs in conventional, modified, or hybrid forms. The modifications are often done due to the unsuitability of the conventional NSGA-II for the problems in terms of chromosome representation, crossover operator, and mutation operator. The modification and hybridization aim to improve the efficiency of NSGA-II for a particular problem or a class of problems. These improvements in NSGA-II need validation, which is done by comparing it with other state-of-the-art algorithms, including NSGA-II. In this section, the implementation of NSGA-II to six selected MOCOPs is discussed. The basic definitions of these problems are given in Table 5. From the searched literature, the problems considered under the category of assignment problems are generalized assignment problem [46], cell formulation problem [47]–[49], critical job assignment problem [50], load balancing of network traffic [51], platform-assignment problem [52], [53], frequency assignment problem [54], multi-stage weapon target assignment (MWTA) problem [55]–[57], location problem [58], [59], land-use optimization [60], optimal configuration selection [61], optimal transmission line assignment [62], multi-skilled worker assignment problem [63], spectrum assignment problem [64] and multi-stage assignment optimization for emergency rescue teams [65]. The problems categorized as allocation problems are resource allocation problem (RAP) [66]–[72], redundancy allocation problem (denoted as ReAP) [73]–[85], reliability redundancy allocation problem [86], buffer allocation [87], order allocation planning [88], land allocation [89], allocation of D-STATCOM in DSs [90], channel allocation in mobile computing [91], agile team allocation problem [92] and continuous berth allocation problem [93]. In the category of knapsack problems, the problems such as search based requirements selection [94], RAP [95], [96], optimal selection of safety measures in oil and gas facilities [97] and multi/many objective knapsack problem [98]–[104] are included. The pickup and delivery problem [105], multiple TSP [106], GEO satellite mission planning problem [107], and [108]–[110], [45] come under the category of TSPs. The other problems such as pollution routing problem [111], [112], ship weather routing problem [113], urban freight transportation planning problem [114], many-objective dynamic route planning [115], VRP with demand responsive transport [116], [117], VRP with time windows [118], and other routing problems [119]–[124] are considered under the category of VRPs. Lastly, the scheduling problems which is the most discussed category of MOCOPs consists of open shop scheduling problem [125], [126], job shop scheduling problem (JSSP) [127]–[132], FSP [133]–[138], project scheduling problem (PSP) [139], resource constrained PSP (RCPSP) [140]–[145], timetabling problem [146], cross-docking scheduling problem [147], task scheduling problem [148]–[154], machine scheduling problem [155]–[161], satellite range scheduling problem [162], multi-objective satellite data transmission scheduling problem [163], satellite scheduling of large areal tasks [164], operating room scheduling [165], [166], harvest scheduling problem [167], energy-efficiency scheduling problem [168], [169] scheduling of demand response programs [170], [171], economic dispatch problem [172], order scheduling problem [173], preventive maintenance scheduling [174], [175] resource-constrained discrete timecost-resource optimization [176], time-cost-quality tradeoff problem [177], [178], production-distribution scheduling problem, [179], job scheduling in computational grid [180], contraflow scheduling problem [181], practical scheduling release times in steel plants [182], real-time routing selection [183], resource scheduling in fog computing [184], inter-site earthmoving optimization [185], process planning and scheduling [186]–[188], cross-trained workers scheduling [189], cross-docking scheduling [190], workforce scheduling problem [191], multi-objective optimized operation of integrated energy system with hydrogen storage [192] and multi-objective integrated optimization of configuration generation and scheduling [193]. The remaining problems, which are combinations of the above six MOCOPs, are considered under the category of miscellaneous problems. The problems included in this category are resource allocation supply chain scheduling and VRP [194], resource allocation and activity scheduling for fourth-party logistics [195], sustainable hub location-scheduling problem for perishable food supply chain [196], routing and scheduling of ships [197], location-routing problem [118], [198]–[202], integrated maintenance scheduling and VRP [203], travelling thief problem [204], lock scheduling and berth allocation [205], assignment-allocation [206], nursing home location-allocation problem [207], location-allocation 57762 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 5. Definitions of combinatorial problems. TABLE 6. Summary of reviewed papers using conventional NSGA-II for MOCOPs. problem [208], high-level synthesis problem [209], industrial hazardous waste locationrouting problem [210], and multiobjective RWA network design problem [211]. 1) CONVENTIONAL NSGA-II FOR MOCOPS This section presents a detailed study of the conventional NSGA-II implementation to MOCOPS. The summary of the related literature is shown in Table 6. a: ASSIGNMENT PROBLEM In [49], Azadeh et al. utilized NSGA-II to solve a large-sized cell formation problem, a traditional problem of assignment of parts, operators, and machines to the cells. In this study, the operators’ personality and decision-making styles, expertise in dealing with machines, and job security are also incorporated, which demonstrates the novelty of the proposed model. The results were validated using NSGA-II, multi-objective PSO (MOPSO), weighted sum method (WSM), and epsilon constraint method (ECM). Out of these methods, the metaheuristic approaches outperformed the classical approaches. Zhang et al. [65] used NSGA-II to optimize multi-stage assignment for emergency rescue teams in the disaster chain. The proposed NSGA-II performed better than GA when compared using scenarios designed for experiments. In [53], NSGA-II outperformed the Greedy randomized adaptive search procedure (GRASP) to solve a multi-objective oil platform location problem. Other implementations of NSGA-II to assignment problems include the load balancing of network traffic [51] and the bi-objective facility location problem [58]. b: ALLOCATION PROBLEM Attar et al. [80] suggested NSGA-II for free distributed repairable multi-state availability-ReAP. The proposed approach was compared with the strength Pareto evolutionary algorithm (SPEA-II) under cold standby and hot standby scenarios using accuracy and diversity metrics. The findings obtained from the statistical analysis showed that NSGA-II was superior to SPEA-II c: VEHICLE ROUTING PROBLEM In [122], the researchers proposed NSGA-II with a novel framework to solve multi-objective capacitated VRP. The reference value method and ranking objective method were used to prune the size of the Pareto optimal solutions according to the preferences of the decision-maker. The efficiency of the algorithm was demonstrated using Solomon datasets, a standard instance for the capacitated VRP. d: SCHEDULING PROBLEM In [154], the researchers compared NSGA-II and NSPSO for Distributed heterogeneous computing systems on benchmark instances using standard performance metrics. VOLUME 9, 2021 57763 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS The compromised optimal schedules obtained by NSGA-II have better quality than the other approach. The authors in [174] proposed a new multi-objective nonlinear model for preventive maintenance scheduling for offshore wind farms. Here, NSGA-II was implemented to obtain the trade-off between two conflicting objectives, maximum utilization, and minimum costs. In [185], NSGA-II was used to plan the inter-site earthmoving in which it deals with two conflicting objectives of minimizing earthmoving costs associated with cut and fill sites along with satisfactory construction schedules. In [171], NSGA-II configurations based on different number of function evaluations were applied to solve the scheduling problem of demand resource programs in retail electricity markets. Here, the performance of the algorithm increases as the number of function evaluations increases. e: KNAPSACK PROBLEM In [97], two mathematical models for optimal selection of safety measures in oil and gas facilities are developed, solutions of which are obtained using NSGA-II f: MISCELLANEOUS Rabbani et al. [199] applied NSGA-II and MOPSO to the newly proposed industrial waste location-routing problem, which includes three objectives, minimization of total costs, total risk, and transportation risk for all routes. The above algorithms are applied to randomly-generated test instances for comparison based on the quantity of non-dominated solutions, CPU time, spacing, and diversity metrics using a statistical t-test. Out of these, NSGA-II outperformed MOPSO for the first three performance measures. De et al. in [197] introduced a new MINLP model for the routing and scheduling of ships and solved it using NSGA-II and MOPSO. The Problem instances are developed for the verification of the proposed model and comparison of the algorithms. NSGA-II was claimed to be computationally more efficient than MOPSO for large-sized instances. 2) MODIFIED NSGA-II FOR MOCOPS In some MOOPs, NSGA-II cannot be applied directly due to their different problem structure. For solving such problems, NSGA-II can be modified according to their requirement. Also, the performance and efficiency of conventional NSGA-II can be further improved using modifications related to initial population generation, mutation and crossover operator, crowding distance operator, selection mechanism, constraint handling technique, and many other criteria. This section presents a detailed study of the implementation of modified NSGA-II to MOCOPS. The summary of the related literature is shown in Table 7. a: ASSIGNMENT PROBLEM Lian et al. [63] proposed a new multi-objective model for the assignment of multi-skilled workers to seru production systems, considering heterogeneous workers with different work skills. NSGA-II based algorithm was tested on medium size numerical examples and applied to the proposed model. The seru swap crossover strategy and a mutation strategy designed according to the given problem were used as genetic operators for the proposed NSGA-II. Li et al. [55] performed a comparative analysis of two algorithms, adaptive MOEA/D (AMOEA/D) and, adaptive NSGA-II to solve the MWTA problem. When comparing the efficiency of the proposed algorithms on MWTA instances, the adaptive NSGA-II was found better than the adaptive MOEA/D algorithm. Juan et al. [56] employed NSGA-II to solve a multi-objective dynamic weapon-target assignment problem (DWTA) and compared it with Monte Carlo random sampling method on DWTA instances. Jie et al. [57] solved a multi-objective missile-target assignment problem using three multi-objective optimization methods MOEA/D, DMOEA-εC, and NSGA-II. The computational experiments are performed, and the results claimed that DMOEA-εC and NSGA-II could find more non-dominated solutions. Lin & Yehwas [62] used NSGA-II in integration with the Technique for order of preference by similarity to ideal solution (TOPSIS) to optimally assign transmission lines to computer/communication networks with minimum cost and maximum network reliability. The uniform and simple mutations principles and SPX were used for the proposed NSGA-II. Martínez-Vargas et al. [64] proposed NSGA-II for efficient bandwidth distribution to the spectrum sharing network. Four-point binary crossover, Laplace crossover, and non-uniform mutation were used as NSGA-II genetic operators. The proposed algorithm was compared with the weighted sum approach (WSA) and parallel cell coordinate system adaptive multi-objective PSO (pccsAMOPSO) using different SA cases where NSGA-II outperformed all the comparative algorithms. Cao et al. [60] suggested NSGA-II for the spatial optimization problem of optimal land-use allocation. The single parent crossover and two mutation operators, mutation of patch cells, and mutation by constraint steering were used as genetic operators of NSGA-II. Goyal et al. [61] proposed a two-phase decision framework for optimal configuration selection for the reconfigurable manufacturing system (RMS). In the first phase, NSGA-II was used with a two-point crossover (TPX) to obtain the Pareto optimal solutions, and in the second phase, the entropy weight method and TOPSIS were used to rank the Pareto optimal solutions. The apparent trade-off obtained using the above procedure helped in enhancing the decision quality in the RMS. Azadeh et al. [47] used NSGA-II and MOPSO to solve a newly proposed model for cell formation and worker assignment problem in the field of dynamic cellular manufacturing systems. Test problems were randomly generated for the validation and verification of the proposed model and solution methods. The performance metrics such as the quantity of non-dominated points, CPU time, spacing & diversity metrics were used to compare the algorithms. Based on the comparison results, the authors claimed to use NSGA-II to provide 57764 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 7. Summary of reviewed papers using modified NSGA-II for MOCOPs. VOLUME 9, 2021 57765 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS Li et al. [212] proposed PD-NSGA-II embedded with properties of non-dominated solutions provided by scheduling experts to solve the multi-objective production-distribution scheduling problem with a single machine for production and multiple vehicles for the delivery of products. The PD-NSGA-II outperformed the SPEA algorithm on various sized test problems. The study in [200] proposed a new multi-objective location-routing model for disaster relief distribution in postearthquake, including the location of distribution centres, vehicle routing, and scheduling. NSGA-II proposed to solve the problem has TPX and reverse sequence mutation in place of standard genetic operators. Out of the two algorithms, NSGA-II and Non-dominated sorting differential algorithm (NSDE) algorithms used to solve this problem, NSGA-II outperformed NSDE in most cases. In [198], a new transportation location-routing problem was formulated, and the methods named scatter tabu search procedure for non-linear multi-objective optimization (SSPMO), NSGA-II, WSM, and ECM were implemented to achieve a trade-off between the two conflicting objectives, cost and route balance. Computational experiments on randomly generated instances showed that NSGA-II provides efficient solutions for large instances compared to other approaches. Pilato et al. [209] proposed NSGA-II for high-level synthesis (scheduling, resource allocation, and binding) of the field-programmable gate array. Here, the authors used binary crossover and unary mutation in the proposed algorithm. Blank et al. [204] proposed NSGA-II for a complex biobjective travelling thief problem, a combination of a TSP and a knapsack problem. The genetic operators used are SPX and bit-flip mutation (BFM). The proposed NSGA-II performed better than a greedy algorithm, an independent sub-problem algorithm (ISA), and ISA-local on randomly generated test instances. In the field of system protection, Khanduzi et al. [206] applied the NSGA-II and Weighted metric method to solve a new integrated assignment allocation model with two objectives that maximize the sum of the efficiency of DMUs and minimize the total distance between customers and facilities. The genetic operators used in NSGA-II are TPX, single and inversion mutations. Both the above methods were compared on 48 randomly generated test instances using two performance measures, CPU time and dominance criteria, and as a result, NSGA-II outperformed the exact method in terms of computational time. In [196], NSGA-II performed better than the improved ECM on large-sized instances for the location-scheduling problem in the perishable food supply chain. The proposed NSGA-II has SPX and shift and exchange mutation for performing genetic operations. 3) HYBRID NSGA-II FOR MOCOPS NSGA-II is good at solving problems with large search spaces but traditionally requires substantial computational effort to find the true Pareto front. Therefore, other search methods are combined with conventional NSGA-II/ modified NSGA-II to push the non-dominated solutions towards the real Pareto optimal solutions with an acceptable computational cost. This section presents a detailed study of hybrid NSGA-II implementations to solve MOCOPs. The summary of the related literature is shown in Table 8. a: ASSIGNMENT PROBLEM In [48], Niakan et al. proposed a novel multi-objective dynamic cell formation problem considering worker’s assignment, social, economic, and environmental aspects. The authors developed a hybrid algorithm combining NSGA-II with MOSA involving the standard crossover and single, multi, and inversion mutations and used randomly generated test instances and performance metrics to compare it with NSGA-II and MOSA. The obtained results showed the superiority of the proposed approach over the other compared algorithms. In [59], Medaglia et al. developed a new facility location problem model and proposed two multi-objective evolutionary algorithms to obtain the set of Pareto optimal solutions. The first algorithm GA-GAH combines NSGA-II with a fast greedy fitness assignment heuristic, and the second algorithm GA-MIP combines it with a mixed-integer program (MIP) heuristic. The authors tested these two approaches on data from Boyacá’s hospital waste management network. The GA-MIP was found better than GA-GAH in terms of the SSC metric. The proposed GA-MIP again compares with the noninferior set estimation (NISE) method on publicly available instances. The GA-MIP scaled better than the NISE method on large instances and can also find non-supported solutions along with supported solutions. Cococcioni et al. [50] proposed a multi-objective model for worker’s risk perception and caution to improve workers’ occupational safety at the workplace. They presented a modified version of NSGA-II based on mutation operator only and used semi-supervised learning to generate initial populations. After that, the best Pareto optimal solution was obtained using the TOPSIS method. Finally, the validation of the proposed methodology was carried out using data collected from small manufacturing enterprises. Segredo et al. [54] suggested ways to deal with the frequency assignment problem (FAP) in designing a global system for mobile communications networks, known as automatic frequency planning and channel assignment problem. Several multi-objectivization methods integrated with NSGA-II and a novel non-destructive crossover operator (to avoid premature convergence) were proposed for the mono-objective FAP and compared with the best up-to-date sequential method on two US cities instances: Seattle and Denver. The results indicate that the proposed method has better quality and speed than other methods. b: ALLOCATION PROBLEM Guo et al. [88] merged NSGA-II with a tabu search based local improvement procedure and a self-adaptive population 57772 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS size adjustment process to solve a multi-objective order allocation problem. The modified mutation based on uniform mutation and fitness-based scanning crossover were used as genetic operators. The suggested approach performed better than the conventional NSGA-II and the industrial method on the performed experiment. Su et al. [87] also integrated NSGA-II with tabu search (TS-NSGA-II) for the buffer allocation problem of remanufacturing systems. This TS-NSGA-II performed better than the traditional NSGA-II. Britto et al. [92] proposed a hybrid approach based on NSGA-II and Mamdani fuzzy inference systems to address a team allocation problem in an agile software development project. The Mamdani fuzzy inference system was used to estimate the developer productivity. In [83], NSGA-II was hybridized with an adaptive population-based simulated annealing (APBSA) to solve a bi-objective ReAP for systems reliability. The proposed approach was compared with three commonly known methods, MOGA, NRGA, and NSGA-II, on randomly generated test instances using four performance measures, i.e., mean ideal distance, spread, coverage metric, and data envelopment analysis. The developed hybrid algorithm outperformed the other algorithms for the last three performance measures, but NSGA-II has the best mean ideal distance. Zhang et al. [72] developed an NSGA-II-TRA algorithm in which a novel heuristic was used for constraint handling in NSGA-II to solve the optimal testing RAP. Also, the Z-scorebased Euclidean distance is adopted to estimate the difference between solutions. The proposed algorithm was compared with two existing MOEAs, i.e., multi-objective differential evolution based on weighted normalized sum (WNSMODE) and harmonic distance-based non-dominated sorting genetic algorithm-II (HaD-MOEA), and was found better based on capacity, coverage, and pure diversity values. c: VEHICLE ROUTING PROBLEM Rauniyar et al. [111] incorporated a new paradigm of multi-factorial optimization into NSGA-II to handle the multi-objective pollution routing problem. Experiments were performed based on benchmark task sets. The proposed algorithm was compared with SPEA-II and conventional NSGA-II using performance measures to investigate the proposed approach’ efficiency. Simulation results showed that the proposed method was more efficient than the other methods with faster convergence. In [118], a mathematical formulation is devised for multiobjective VRP with the time windows model. Xu et al. suggested a hybrid NSGA-II in which the Or-opt heuristic was hybrid with NSGA-II to improve the quality of the solutions. The above heuristic was used in the initialization phase and also to generate mutants. This new hybrid algorithm outperformed NSGA-II on both the 30-customer and 498-customer cases. Wang et al. [119] designed a collaborative multi-depot VRP with time window assignment to reduce the time uncertainty, which causes operation challenges and extra cost to the logistics service provider. The authors presented a hybrid heuristic combining K-means clustering, Clarke–Wright (CW) saving algorithm, and an extended non-dominated sorting genetic algorithm-II (E-NSGA-II) for solutions to the model. E-NSGA-II combined NSGA-II with PMX, relocation, 2-opt∗exchange, and swap mutation and compared with NSGA-II and MOPSO for its validation in which found superior to both approaches. Wang et al. [120] introduced a new multi-objective model for collaborative multiple centres VRP with simultaneous delivery and pickup to minimize the operating costs and the number of vehicles in the network. They proposed a hybrid algorithm HNSGA-II combining NSGA-II with the K-means algorithm and used PMX and swap mutation operators. According to experimented results, HNSGA-II outperformed NSGA-II by 5.5% and MOPSO by 11.9% in terms of cost objective on modified Solomon benchmarks. HNSGA-II also has a high performance on a real case study in Chongqing city. Mandal et al. [121] developed a memetic algorithm integrating NSGA-II with a dominance-based local search procedure (DBLSP) and a clone management principle (CMP) for a bi-objective mixed capacitated general routing problem (MCGRP). Also, three well-known crossover operators (X-set), i.e., PMX, OX, and edge recombination crossover (ERX), were used to explore the different parts of the search space. The proposed approach outperformed XNSGA-II (NSGA-II with only X-set and CMP) on the experimental tests performed on standard MCGRP instances. d: TRAVELLING SALESMAN PROBLEM Chen et al. [110] suggested pNSGA-II, a hybrid NSGA-II, to overcome the limitations of multi-objective optimization algorithms based on GA, such as premature convergence and non-uniformly distributed solutions for bi-objective TSP (BTSP). In their work, NSGA-II was embedded with the Physarum-inspired computational model (PCM) in the initialization phase and the hill-climbing method. The proposed approach was compared with eight algorithms, including a typical GA based method HYGA, the multiple ant colony systems (MACS), Pareto ACO (PACO), three enhanced algorithms of PACO, i.e., pPACO, pMACS, and pBIANT, the bicriterion ant algorithm (BIANT) and NSGA-II. The benchmark instances were constructed using two single objective TSP instances available in the literature. The proposed algorithm pNSGA-II was found superior to the other algorithms using benchmark instances, performance measures, and statistical analysis. In [45], Moraes et al. proposed a subpopulation-based multi-objective approach MOEA/NSM for BTSP. This method integrated NSGA-II, SPEA-II, MOEA/D, and 2-opt local search technique was compared with NSGA-II, SPEA-II, and MOEA/D on BTSP datasets. The results showed that MOEA/NSM outperformed the other algorithms. Li et al. [108] performed a comparison among NSGA-II, MOEA/D, and their variants NSGA-II-ACO and VOLUME 9, 2021 57773 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 8. Summary of reviewed papers using hybrid NSGA-II for MOCOPs. 57774 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 8. (Continued.) Summary of reviewed papers using hybrid NSGA-II for MOCOPs. MOEA/D-ACO for solving the multi-objective TSP using a pheromone trail based probabilistic representation. The benchmark instances were constructed using two single-objective instances given in the literature. Results showed that the proposed variants performed better than corresponding traditional algorithms. However, NSGA-II performed better than MOEA/D-ACO. The researchers in [109] proposed a new model TSP model that aims to extend and combine different TSP variants to generate vendor routes in a sales territory. The authors introduced a new permutation representation and evaluated three different algorithms, i.e., NSGA-II, SPEA-II, and IVF/NSGA-II, on real scenarios. The results claimed that IVF/NSGA-II outperformed the other algorithms in most cases. Also, VOLUME 9, 2021 57775 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS the obtained solution routes reduce the route distance by 35% and increase the overall performance by 60%. Li et al. [107] proposed a hybrid algorithm HNSGA-II to solve a target mission planning model of GEO satellites. The suggested algorithm based on NSGA-II used a heuristic during initialization and mutation, embedded with TSP optimization, and performed better than NSGA-II on randomly generated instances using the hypervolume measure. The authors in [105] formulated a bi-objective selective pickup and delivery problem. They proposed a memetic algorithm based on NSGA-II along with a repair strategy for constraint handling. e: SCHEDULING PROBLEM Abdi & Zarandi [148] proposed a task scheduling method for the design optimization of heterogeneous multiprocessor embedded systems. In the optimization procedure, the authors used NSGA-II as a powerful design exploration engine. Half of the initial population was generated randomly for diversity preservation. The remaining half of the population was generated using a list scheduling-based heuristic to find solutions relatively close to the optimal solutions. Chitra et al. [152] also considered the multi-objective task scheduling problem on heterogeneous systems. Here, NSGA-II was hybridized with a local search method called the simple neighbourhood search method. This proposed algorithm provided better results than conventionalNSGA-II, conventional SPEA-II, and hybrid SPEA-II on random task graphs. In [153], the authors presented a multi-objective model for task scheduling in cloud computing with two conflicting objectives makespan and energy consumption. NSGA-II was implemented with ANN support and without ANN support to obtain the Pareto optimal solutions. According to the authors, NSGA-II with ANN support provided better solutions as compared to NSGA-II. Vilcot & Billaut [128] considered a general JSSP with objectives to minimize the makespan and the maximum lateness. The authors proposed two methods based on the NSGA-II framework with different initialization phases. One of the methods used randomly generated initial population, and the other method used initial population partially generated using tabu search. According to the performance comparison results, NSGA-II with tabu search performed better in terms of solution quality and computational time. In [131], NSGA-II was integrated with genetic programming hyper-heuristic (GPHH) to handle multi-objective dynamic flexible JSSP. The proposed algorithm performed better than two methods, i.e., SPEA-II based GPHH and WSM. Gong et al. [130] presented a multi-objective flexible JSSP with worker flexibility. Here, NSGA-II was integrated with a local search operator, a designed coding/encoding method, and adaptive genetic operators. The proposed algorithm was compared with NSGA-II, non-dominated neighbour immune algorithm (NNIA), and NNIA +LOCAL on instances constructed from traditional instances using convergence and diversity measures. The simulation results showed that the proposed memetic algorithm performed better than the other compared algorithms. Autuori et al. [129] claimed that for solving the flexible JSSP, the hybridization of the mapping method with NSGA-II was more efficient than the hybridization of simple hill-climbing local search method with NSGA-II in terms of hypervolume and set coverage metric. Wang et al. [132] proposed hybrid NSGA-II for multiobjective fuzzy flexible JSSP. In the proposed NSGA-II, the initial population was generated using machine assignment and operation sequencing rules, and a well-designed greedy chromosome algorithm was used along with two effective genetic operators. Further, the non-dominated sorting procedure was improved using a modified crowding distance measure. Also, VNS was used as a local search operator to enhance the exploitation ability of NSGA-II. The performance of the proposed algorithm was validated using benchmark instances available in the literature. In [179], NSGA-II was hybrid with a heuristic for population initialization and a heuristic crossover operator to solve a new bi-objective production-distribution supply chain scheduling model in a flowshop environment. Cai et al. [137] proposed a novel mixed-integer linear programming model for a distributed permutation FSP with transportation conditions. NSGA-II was modified in terms of population initialization, i.e., a heuristic was used to initialize the population for each objective function. The improved NSGA-II outperformed the conventional NSGA-II and SPEA-II algorithms. Zeng et al. [135] proposed a hybrid algorithm integrating NSGA-II with tabu search and a job merging strategy for a multi-objective flexible batch processing FSP in manufacturing industries. For multi-objective permutation FSP, Chiang et al. [134] developed an efficient algorithm combining NSGA-II with a problem-specific local search procedure NEH and several adaptations, including acceptance criterion and ordering strategy. Wang et al. [133] proposed a memetic algorithm based on NSGA-II to solve the multi-objective parallel FSP that includes an order encoding scheme, a heuristic for population initialization, and an embedded local search operator. Han et al. [138] improved NSGA-II to solve multiobjective lot-streaming FSP. The traditional genetic operators of NSGA-II were replaced by the estimation of distributed algorithm (EDA) (crossover) and swap and insertion mutations. Also, a restarting strategy was performed on the population when the population diversity was less than the given threshold. The proposed modified NSGA-II outperformed NSGA-II, discrete harmony search (DHS) & threshold accepting (TA) on a series of random experiments Vidal et al. [156] suggested a neuro-evolutionary algorithm integrating NSGA-II with a multi-layer perceptron neural network (for processing time estimations) to solve a complex machine scheduling problem in the custom furniture industry. The proposed approach was compared with five stateof-the-art algorithms on JSSP benchmark instances using a set coverage metric. The authors conclude that the proposed method was better than other methods in most of the test instances. Also, the estimated time obtained through 57776 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS multi-layer perceptron has better accuracy than the other regression techniques used for time estimations. Liu et al. [157] studied machine scheduling under disruption to minimize weighted discounted total completion time and deviation from the initial schedule. A quantuminspired hybrid algorithm was employed to solve the problem in which NSGA-II was hybrid with quantum computing considering qubit representation. Here, the near-optimal schedules obtained by the proposed algorithm were more effective than the conventional NSGA-II. Ramacher & Mönch [158] hybridized NSGA-II with a problem-specific heuristic to solve a machine scheduling problem with interfering job sets. Song et al. [162] proposed learning guided NSGA-II for a multi-objective satellite range scheduling problem. The algorithm contained NSGA-II and a learning mechanism to speed up the convergence. Zhang et al. [163] presented a multi-objective model for a large-scale satellite-data transmission scheduling problem. NSGA-II integrated with support vector machine (SVM) classification was proposed and compared with dominance-based NSGA-II, indicator-based evolutionary algorithm (IBEA), decomposition-based evolutionary algorithm (MOEA/D), and self-adaptive MOEA/D (SaMOEA/D) for large scale problems. The results showed that NSGA-II with SVM found more efficient solutions for large-scale problems in a reasonable amount of time. Zeng et al. [168] used NSGA-II with tabu search for energy-efficiency scheduling in paper mills to save energy. The proposed algorithm performed better than NSGA-II. Huang et al. [172] merged NSGA-II with MOPSO for joint voyage scheduling and economic dispatch for energyefficient scheduling in all-electric ships with virtual energy storage. Xiao et al. [142] proposed three hybrid algorithms for the RCPSP, each combining the electromagnetism heuristic with NSGA-II, MOEA/D, and SPEA-II, respectively. According to this study, the integration of electromagnetism was best suitable for NSGA-II. Tao & Dong [140] suggested integrating NSGA-II with tabu search for a newly proposed multi-mode RCPSP with alternative project structures. Guo et al. [173] combined NSGA-II based multi-objective framework with an effective production process simulator to handle the order scheduling problems in production planning. Here, NSGA-II was modified in terms of chromosome representation, genetic operators and embedded with heuristic pruning and decision-making method to find a single solution from a set of Pareto optimal solutions. In [182], the authors integrated NSGA-II with a decoding heuristic and a non-dominated solution construction method (NSCM) to obtain efficient Pareto optimal solutions for multi-objective production scheduling with release time in steel plants. Wang et al. [175] solved an integrated problem of preventive maintenance and rescheduling problem for the arrival of a new job in a single machine layout using an improved The traditional gene NSGA-II. The exploration and exploitation of NSGA-II were balanced using the mutation operator of differential evolution, high-quality initial solutions, and approximation heuristic strategy (AHS). Naik et al. [186] introduced the adaptive multi-objective resource selection model for selecting the resources inside the hybrid cloud environment. In this model, NSGA-II was hybrid with gravitational search algorithm (GSA) to find the near-optimal schedule for the cloud users. Hu & Yan [213] proposed NSGA-II coupled with an EPANET simulator to solve a new multi-objective model for scheduling valves and hydrants to improve the response of drinking water contamination event. The proposed model and methodology were validated using two water distribution networks and studied the impact of different parameters on the proposed algorithm. To solve the spatial combinatorial problem in forest planning, Fotakis et al. [167] introduced a spatial operator, a local search operator, or the second kind of mutation in NSGA-II. In [166], the authors proposed a five-objective mixedinteger linear programming model for complex decisionmaking regarding the planning and scheduling of operation theatres in hospitals. The authors utilized NSGA-II with a semi-random procedure for initialization to reduce computational time. Guo et al. [190] proposed NSGA-II with a novel greedy local search to accelerate its convergence speed to solve a multi-dock cross-docking scheduling problem. The proposed algorithm outperformed the greedy strategy and approximated the enumeration algorithm for a small-size problem. For a large-size problem, the proposed algorithm was found more efficient than the enumeration algorithm. f: KNAPSACK PROBLEM Sato et al. [98] proposed a distributed parallelized NSGA-II with a migration method (e-DNSGA-II) for constrained multi-objective knapsack problems. The proposed approach was compared with conventional single CPU NSGA-II, parallel NSGA-II without migration method, and DNSGA-II using two constrained test problems. Results showed that e-DNSGA-II achieved higher hypervolume, enabled high-speed operation, improved diversity, and has an increased number of non-dominated solutions for both The traditional gene problems. Ishibuchi & Narukawa [101] proposed S-MOGLS, the hybrid algorithm of NSGA-II with the local search method, for many-objective knapsack problems. The authors explained the framework of S-MOGLS and examined its four variants based on genetic search and local search method (Weighted scalar or Pareto ranking). These four versions of S-MOGLS performed better than SPEA, NSGA-II, memetic Pareto archived evolution strategy (MPAES), and the MOGLS algorithm on many-objective test problems, especially in terms of D1R measure. g: MISCELLANEOUS Leesutthipornchai et al. [211] solved a routing and wavelength assignment problem in wavelength division multiplexing optical networks using a multi-objective evolutionary VOLUME 9, 2021 57777 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS approach. The authors hybridized NSGA-II with an algorithm named GA for routing allocation with minimum degree first for wavelength assignment (GA-MDF) to obtain the non-dominated solutions for the problem and compared it with WSM. According to the authors, although NSGA-II was computationally expensive, its solutions were more diverse than WSM. Rabbani et al. [210] studied waste management in the industrial sector and extended a deterministic industrial waste location-routing model capable of covering inventory decisions in waste treatment, considered a multi-period planning horizon, and was a stochastic model. Along with this novelty in the model, the authors integrated NSGA-II and Monte-Carlo simulation, which provide high-quality solutions in less computational time than the hybrid simulation-analytical modeling approach on randomly generated problem instances. Amini et al. [201] studied a location-arc routing problem, a combination of location problem and arc routing problem. A mathematical model was developed to minimize the two objectives, makespan and total costs. Two metaheuristics: NSGA-II and multi-objective late acceptance hill-climbing (MOLAHC) algorithm, were considered. The hybridization of NSGA-II and MOLAHC with local search (hybrid +LS) and hybridization of NSGA-II with local search (NSGAII +LS) were found better when compared with the same algorithms without using the local search. Out of these two algorithms, NSGA-II +LS was more efficient, and hybrid + LS took less computational time. The researchers in [194] proposed a new mathematical model integrating two problems: supply chain scheduling and VRP to minimize the resources and energy consumption and penalty for total tardiness. Here, NSGA-II integrated with VNS outperformed NSGA-II in obtaining the Pareto optimal solutions for the given problem Doolun et al. [208] integrated NSGA-II with five different variants of the differential evolution algorithm to solve a multi-objective location-allocation problem in a multiechelon green supply chain network. The problem was to find the location of manufacturing plants and warehouses and then allocating resources to the various stages of the supply chain. The proposed algorithms (NSDEA) was compared with the existing multi-objective hybrid PSO (MOHPSO) algorithm and outperformed it in evaluating Pareto optimal solutions B. PERFORMANCE ASSESSMENT This section discusses the test instances, case studies, performance measures, and statistical tests used by the researchers to validate the effectiveness of the NSGA-II algorithms to solve MOCOPs. 1) TEST INSTANCES The researchers used test instances to validate their proposed model and algorithms. Typically, the test instances are of three sizes, small-sized, medium-sized, and largesized instances. The performances of NSGA-II based algorithms for small-sized instances are mostly close to the exact algorithms. However, for medium and large-sized instances, exact algorithms generally fail to provide solutions in a reasonable time because of the NP-hard nature of most of the MOCOPs. Therefore, an intelligent algorithm like NSGA-II is proposed to obtain near-optimal solutions to such problems in less computational time. The test instances are generally taken from standard benchmarks/datasets available in the literature. However, for many newly proposed models of MOCOP, benchmarks are not available in the literature. The validation of such models is done through randomly generated test instances. The test instances used in the studied literature are given in Table 6–8. Link sources for some of the test instances are provided in the footnotes below the respective tables. 2) CASE STUDIES The researchers performed case studies to demonstrate the effectiveness of the NSGA-II based algorithms and their application in practice. The conducted case studies in the literature are given in Table 9. 3) PERFORMANCE MEASURES In the studied literature, a comparison with other state-of-theart algorithms quantifies the proposed NSGA-II algorithms’ effectiveness. Single-objective optimization algorithms can be easily compared using objective-values or computational times. In multi-objective optimization algorithms also, if all the objective-values of one algorithm are better than the corresponding objective-values of the other, then they can be compared using their objectives-values. However, it is difficult to compare the non-dominated sets of near-optimal solutions obtained using the multi-objective optimization algorithms. Thus, many metrics have been developed to evaluate the performance of such algorithms. They are used to compare the sets of solutions obtained by the algorithms in terms of convergence and diversity [214]. Researchers have used different performance metrics to compare their proposed NSGA-II with exact methods, random methods, single-objective and multi-objective optimization algorithms, and many other methods, as illustrated in Table 6-8. The researchers also considered computational time to compare the efficiency of two or more algorithms. Therefore, in this study, the objective function values and computational time are also treated as performance metrics. The most used performance metrics in the studied literature are given in Table 10. The other rarely used performance metrics are maximum sum of the objective-values (MaxSum) [102], [99], maximum distance Dmax [175], [157], epsilon indicator [45], [134], average quality [157], [175], average hypervolume [105], [121], width measure (M2 metric) [147], system performance [148], solution quality [54], size of reference set [62], relative percentage difference between objective-values [210], relative and absolute quality [59], overall non-dominated vector generation (ONVG) [175], number of unscheduled tasks [163], number of generations [68], number of channel reuses, user level fairness, and 57778 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 9. Case studies. stability of the communication mode of the D2D users [96], norm based pure diversity metric [72], modified mean ideal distance measure [127], maximum Pareto front error [201], M1,M2,M3 [110], k-distance [198], hole-relative size metric, µdistance metric [73], error, ratio, distance based measure, fairness [158], empirical attainment function [45], data envelopment analysis [83], data dependency threshold [68], convex hull of the approximated efficient frontier [59], convergent metric [140], capacity measure [72], average rank index, average crowding distance and the mapping pattern of solutions [89], convergent rate metric [85], infeasibility metric [85], ratio of non-dominated solutions [138], rate of achievement to two objectives simultaneously [190], and area under linear regression curves [190]. 4) STATISTICAL ANALYSIS In some papers, to investigate the effectiveness of the proposed optimization model, the researchers conducted a series of experiments using available or randomly generated test instances or datasets. They performed statistical tests on the performance metrics values obtained from the compared algorithms to show the statistical significance of the VOLUME 9, 2021 57779 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS TABLE 10. Performance metrics. experimental results. Therefore, similar to performance metrics, statistical analysis is also used to measure the algorithms’ performance. The various statistical tests used in the studied literature are given in Table 11. C. POST-PARETO OPTIMALITY ANALYSIS A population-based algorithm like NSGA-II provides a set of compromised optimal solutions after solving a MOOP. The choice of the best-compromised solution among the set of all compromised solutions is made by decision-makers using the preference information. Many techniques based on MCDM, data clustering, fuzzy logic, and some other criteria that are applied to find the best-compromised solution in the studied literature is given in Table 12. V. ANALYSIS OF MODIFICATIONS IN NSGA-II Researchers modified NSGA-II mainly in terms of crossover operator (113 papers), mutation operator (119 papers), initialization procedure (27 papers), parent selection (17 papers), and constraint handling technique (7 papers) to improve its convergence and diversity. The other modifications (24 papers) are related to crowding distance calculation, external archive, non-dominated sorting algorithm, adaptive parameter, correlation-based weighted-sum fitness, parallel processing, refinement operation, elite preservation strategy, controlled elitism, and dominance relation, as illustrated in Table 13. The authors mostly modified crossover and mutation operators because these operators depend on the type of COP. In other words, the selection or design of these operators TABLE 11. Statistical tests. depends on the chromosome representation of the problem. The chromosome representation is used to represent the potential solution to a problem. The common chromosome representations for GA are binary coding (e.g., 1101101), integer coding (e.g., 1,2,5,7,8), real coding (e.g., 0.235, 0.43, 0.53, 8.5) and permutation representation (e.g.., (1 2 3), (1 3 2)). The chromosome representation is selected according to the nature of the problem. The conventional NSGA-II crossover and mutation cannot be applied to all types of COPs. Also, the inappropriate representation may result in poor performance of the algorithm. Therefore, other crossover operators, such as TPX, UX (binary coding), arithmetic crossover (real coding), PMX, OX (integer coding), 57780 VOLUME 9, 2021 S. Verma et al.: Comprehensive Review on NSGA-II for Multi-Objective COPS FIGURE 4. Crossover and mutation operators. TABLE 12. Post-Pareto optimality techniques. and mutation operators such as bit-wise mutation, bit-flip mutation (binary coding), Gaussian mutation (real coding) and inversion mutation (integer coding) are used according to the chromosome representation. The most used crossover and mutation operators are shown in Fig. 4. The modifications in the parent selection mechanism are done to improve the convergence rate of the algorithm. The modifications are related to the increase in tournament size or using the other techniques. Roulette wheel selection is the most used (5 out of 17 papers) technique among all the parent selection modifications. In this selection, a wheel is rotated, and the fixed point on the wheel’s circumference chooses a parent among all individuals, and the individual with a greater pie has a greater probability of becoming a parent, as shown in Fig. 5. Further, in conventional NSGA-II, the initial population is randomly generated. The researchers modified this procedure by generating the initial population using heuristic or problem information for rapid convergence and high-quality solutions. The researchers used CDP (used in conventional NSGA-II), penalty function strategy, and repair method for handling constraints. In the penalty function method, infeasible solutions’ fitness is reduced in proportion to the number of violated constraints. On the other hand, the repair mechanism modifies the infeasible solutions so that the violated constraints get satisfied. FIGURE 5. Roulette wheel selection. Further, researchers also hybrid other local search methods with NSGA-II to explore the solutions in a solution’ neighbourhood. These local search methods include methods such as machine learning techniques, heuristics, and other metaheuristics. The hybrid methods used in the studied literature are given in Table 8. These methods are embedded mainly in the initialization procedure and before parent selection to improve the local search or exploitation ability of NSGA-II. VI. BIBLIOMETRIC ANALYSIS The scheduling problem (42%) is the most exploited MOCOP using NSGA-II, followed by allocation problem (17%), miscellaneous (11%), assignment problem (11%), VRP (8%), knapsack problem (7%), and TSP (4%) as shown in Fig. 6. The modified NSGA-II algorithms (58%) are the most applied algorithms, followed by hybrid NSGA-II (34%) and conventional NSGA-II (8%), as shown in Fig. 7. Fig. 8 shows the distribution of papers based on the number of objective functions. Bi-objective problems, the most considered problems in the studied literature, comprise 72% of the total papers reviewed, followed by tri-objective problems (21%) and many-objective problems (6%). Further, the authors in [98] consider both bi-objective and tri-objective problems, and the authors in [117] reduced a 5-objective problem to a 3-objective problem. The benchmarking of an algorithm (using different test instances) studies the best practices of its implementation to VOLUME 9, 2021 57781 S. 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Smith, ‘‘Constraint satisfaction problems: Algorithms and applications,’’ Eur. J. Oper. Res., vol. 119, no. 3, pp. 557–581, Dec. 1999, doi: 10.1016/S0377-2217(98)00364-6. SHANU VERMA (Graduate Student Member, IEEE) received the M.Sc. degree in mathematics from the Central University of Rajasthan, India. She is currently pursuing the Ph.D. degree with the Department of Applied Science and Engineering, IIT Roorkee, India. Her research interests include multi-objective optimization, combinatorial optimization, evolutionary algorithms, swarm intelligence-based algorithms, and their applications to complex real-life problems. MILLIE PANT received the M.Sc. degree in mathematics from CCS University, Meerut, and the Ph.D. degree from the Department of Mathematics, IIT Roorkee, India. She is currently a Professor with the Department of Applied Science and Engineering, IIT Roorkee. She has authored or coauthored more than 200 articles in various journals and conferences of national and international repute. Her research interests include numerical optimization and operations research, evolutionary algorithms and supply chain management, and swarm intelligence techniques. VACLAV SNASEL (Senior Member, IEEE) is currently a Professor with the Department of Computer Science, VSB—Technical University of Ostrava, Czech Republic. He works as a Researcher and a University Teacher. He is also the Dean of the Faculty of Electrical Engineering and the Computer Science Department. He is the Head of the Research Programme IT4 Knowledge Management, European Center of Excellence IT4 Innovations. His research and development experience includes over 30 years in the industry and academia. He works in a multi-disciplinary environment involving artificial intelligence, social networks, conceptual lattice, information retrieval, semantic web, knowledge management, data compression, machine intelligence, neural networks, web intelligence, nature and bio-inspired computing, data mining, and applied to various real-world problems. VOLUME 9, 2021 57791