Pareto dominance based Multiobjective Cohort Intelligence algorithm

Highlights

• Multiple features are strategically implemented in MOCI to enhance its capability.

• Dual stopping criteria implemented to assess performance of all the algorithms in stringent manner.

• Performance is assessed over 8 algorithms, 5 benchmark test suite, 7 real problems with 3 metrics.

• Framework for exploratory analysis of performance metrics is proposed.

• MOCI performance is statistically validated and verified.

Abstract

In the recent days, several novel and specialized algorithms are coming up for solving particular class of problems. However, their performance on new benchmark or real-world problem remains unsure. This paper proposes a novel Multiobjective Cohort Intelligence (MOCI) algorithm. It is based on Pareto dominance and coevolutionary design principles to achieve efficient, effective, productive and robust performance. The capability of MOCI algorithm is enhanced through use of multiple features for balance of exploration versus exploitation, search towards promising region and avoidance of search stagnation. The performance of MOCI is assessed against the state-of-the-art algorithms, such as: ARMOEA, CMOPSO, hpaEA, LMOCSO, LSMOF, NMPSO and WOFSMPSO across multiple test suites including Classical, ZDT, DTLZ, WFG and UF. The performance assessment is conducted with truly uncorrelated performance metrics. In this regard, an exploratory approach of multiple correlation analysis is proposed. Performance of MOCI algorithm is statistically verified and validated using PROMETHEE-II and nonparametric statistical tests. MOCI is capable of achieving well converged and diversified solutions on most of the test as well as real world problems. The success of MOCI is attributed to multiple features incorporated in the algorithm. In the future, MOCI could be applied to challenging problems in engineering and management.

Introduction

An optimization problem that contains 2 to 3 conflicting objectives is referred to as Multiobjective Optimization Problem (MOOP). These MOOP differ from Single Objective Optimization Problems (SOOPs) in terms of problem-complexity and the requirement of compromise solutions by the Decision Maker (DM). In order to solve MOOP, it is useful to use Multiobjective Optimization Algorithm (MOOA). They exist in variety of types and hence, it is necessary to review them from bird’s eye-view. In this connection, all the MOOAs are classified and the representative algorithms are shown in Fig. 1. These MOOAs can be classified as follows. 1) Pareto Dominance Based Algorithms: These algorithms utilize the mechanism of Pareto dominance in selecting the Nondominated Solutions (NDS), 2) Pareto Non-Dominance Based algorithms: These algorithms utilize mechanisms other than the dominance relation in selecting the NDS, and 3) Hybrid MOOAs: These algorithms are combination of two or more algorithms of the same or different genre.

1) Pareto Dominance Based Algorithms are early developed MOOAs, which are able to explore a set of NDS throughout the entire Pareto front in a single run of the algorithm. Therefore, these algorithms are more popular among researchers. The first generation MOOAs were designed for the purpose of simplicity, whereas, the second generation MOOAs were designed for the purpose of efficiency [1]. The first and second generation MOOAs laid the foundation of MOO as a distinct research area. The third generation MOOAs were designed for effectiveness in order to achieve improved convergence and diversity. The fourth generation MOOAs will be designed for efficiency, effectiveness, productivity and robustness.

Besides Pareto dominance relation, there are many other dominance relations useful in the design of MOOAs. In view with the large amount of Pareto dominance based algorithms, only the state-of-the-art algorithms are briefly reviewed here. CMOPSO [2] is a Competitive Mechanism based Multiobjective Particle Swarm Optimizer. It is basically proposed for faster convergence towards true Pareto front. It selects leader candidate on the basis of competitive comparison between two randomly selected individuals. However, it is observed that the convergence and diversity of CMOPSO is adversely affected due to random leader selection approach. Moreover, it utilized only one performance metric for comparative performance assessment. The use of single performance metric may not reveal the true performance of any algorithm. NSGAIISDR [3] is proposed as an enhancement in dominance relation over original NSGA-II. In this work, eight different dominance relations are employed and their performances are evaluated against the proposed strong dominance relation. This strong dominance relation is basically a combination of an angle based dominance relation and niche formulation. However, when this algorithm is set to perform within limited number of iterations, it does not perform well. GPAWOA [4] is a Guided Population Archive Whale Optimization Algorithm. It is applied over multiobjective benchmark and real-world problems. Although, the archived individuals are used to guide the search, it could not achieve the diversity of the solutions over entire Pareto front. Therefore, there is a strong need to design an algorithm using Pareto dominance relation that could fetch desired number of NDS within the provided number of iterations.

2) Non-dominance based algorithms are classified as follows. 2.1) Trajectory or Single Point Based Algorithms: These algorithms are based on single individual or trajectory approach to search the NDS. These algorithms are designed to explore the search space than to exploit and hence, they are mostly suitable for multiobjective integer optimization problems. MOTS [5] is a Multiobjective Tabu Search algorithm applied to solve NP-hard scheduling and resource allocation problems. MOSA/R-HH [6] is a Multiobjective Simulated Annealing based on Reseed for reinforced learning of Hyper Heuristic. MOSA/R is implemented to obtain better NDS at high level heuristic. These NDS are provided as input to low level heuristic to find the best neighbourhood NDS. Two repick schemes are proposed on the basis of the maximum and the minimum level of domination. Although, the approach is efficient, convergence of the NDS towards true Pareto front is a major concern.

2.2) Population Based Algorithms: These algorithms are based on the population consisting of many individuals to search the NDS. They are subdivided as follows. 2.2.1) Scalarization Algorithms: These algorithms convert MOOP into SOOP. According to the kind of transformation used, these algorithms are further sub-classified as follows. 2.2.1.1) Aggregation Based Algorithms: These algorithms convert the given MOOP into SOOP as a vector sum of objectives with or without the application of weights. These methods are also recognized as vector optimization methods. Enhanced Goal Attainment Method (EGAM) [7] is a modification over the classical goal attainment method. This approach requires high computational time and could not explore the well-converged and well-diversified solutions. Weighted Goal Programming (WGP) [8] is applied to solve mixed integer linear programming problem. Since, Goal programming is a satisficing method, WGP may provide dominated solutions to multiobjective integer problems.

2.2.1.2) Decomposition Based Algorithms: A given MOOP is divided into several SOOPs with the help of scalarizing functions such as Tchebycheff function, penalty based boundary intersection function, etc. Every subproblem corresponds to an individual vector produced in a set. This approach is usually combined with metaheuristics for random search, so that multiple NDS can be obtained in a single run. MFEA/D-DRA [9] is a Multiobjective multi Factorial Evolutionary Algorithm based on Decomposition and Dynamic Resource Allocation strategy. It is applied to solve benchmark problems. However, this algorithm showed a poor convergence towards true Pareto front for some benchmark problems. MLIP-MOEA/D [10] is a Multi-Layer Interaction Preference based Multi-Objective Evolutionary Algorithm through Decomposition. It is proposed with a maximin operator to obtain diverse NDS. It is based on the interaction of the DM at two layers. In spite of two layers of interaction of DM and use of maximin operator, it couldn’t search diverse NDS. In summary: large number of function evaluations and better convergence and diversity of the NDS are challenges to this class of algorithms.

2.2.2) Speciation Based Algorithms: In these algorithms, every objective is optimized by assigning specific individuals to that objective. Vector Evaluated Genetic Algorithm (VEGA) is a pioneering and viable MOOA of this class. MMACO_R [11] is a Multiobjective Multi population Ant Colony Optimization algorithm based on Reconstructing pheromone. It is proposed to solve multiobjective economic emission dispatch problem. In this work, multiple populations are implemented in order to overcome the premature convergence of ant colony optimization algorithm. However, its performance is not assessed over challenging benchmark test suites. SSMOPSO [12] is a Self-organized Speciation based Multi-Objective Particle Swarm Optimizer. It is proposed to solve specifically the multimodal MOOPs. In SSMOPSO, a special crowding distance (SCD) operator is implemented in both the decision and objective space. However, SCD increases the computational complexity of the algorithm as compared to simple crowding distance operator. In summary, search ability and computational complexity are the major issues reported with this class of MOOAs.

2.2.3) Surrogate Assisted Algorithms: In such algorithms, an approximation (generally, Kriging) model is built-up to find NDS and thereby reduce the computational burden of fitness evaluation. MOFEPSO [13] is a Multi-Objective Feasibility Enhanced Particle Swarm Optimizer guided by online surrogates based on adaptive sampling and infill criterion. It is proposed for multiobjective optimization of contaminated groundwater remediation designs. In order to solve highly constrained problems, feasible individuals are favored for search guidance. Similar to the classical Multiobjective Particle Swarm Optimization (MOPSO), convergence and diversity are also the challenge for MOFEPSO algorithm. SK-MOCBA [14] is a Stochastic Kriging metamodeling based Multiobjective Optimal Computing Budget Allocation algorithm. This algorithm is applied and validated on a limited set of benchmark functions; as a result, it couldn’t perform well on the other problems. Also, it exhibited poor convergence of NDS towards true Pareto front. In summary, kriging based metamodeling approach is common in these types of MOOAs. However, convergence and diversity are still the challenge to this class of MOOAs. The major concern of this class of MOOAs is the necessity of very high computational time to obtain a set of NDS as compared to dominance based algorithms.

2.2.4) Indicator Based Algorithms: These algorithms implement performance indicators to guide the search and select the NDS. Any performance indicator that predicts both the diversity and convergence simultaneously can be used as an indicator to drive an algorithm’s performance. MaOEAIGD [15] is an IGD based Evolutionary Algorithm for Many Objective optimization problems. In this algorithm, the rank and proximity distance criteria are used to select the NDS. MaOEAIGD requires about one million function evaluations to explore a Pareto front. Since, this algorithm explores NDS on the basis of IGD metric value, it shows poor performance. This is because, IGD reports the best values, when MaOEAIGD explores the end solutions of Pareto front. This results into poor diversity of MaOEAIGD algorithm. IDBEA [16] is an Indicator and crowding distance Based Evolutionary Algorithm. It is applied to solve combined heat and power emission dispatch problem. IDBEA is an enhancement over IBEA algorithm, because of use of crowding distance operator for diversity management. However, it is known that IBEA requires large number of function evaluations with high computational cost. In summary: the requirement of large number of function evaluations and high computational complexity are the major issues with this class of MOOAs.

2.2.5) Reference Point Based Algorithms: In these type of algorithms, reference vectors are produced on a hyperplane irrespective of the shape of the Pareto front. The major objectives of such algorithms are to obtain convergence and diversity of NDS. These reference points are used for guiding the search operation to obtain the NDS. LMOCSO [17] is a Large scale Multi-Objective Competitive Swarm Optimizer. It is basically designed to solve large scale optimization problems. Leader particle is selected through the competition between two randomly selected individuals to guide the search. In this work, reference vectors and environmental selection strategy are utilized to select diverse Pareto solutions. hpaEA [18] is a hyper plane assisted Evolutionary Algorithm. It is also designed to solve large scale MOOPs. In this algorithm, novel environmental selection strategy is implemented to maintain the convergence and diversity of NDS. As these algorithms are specialized for large scale optimization, their performance on small scale problems is unsure.

3) Hybrid MOOAs: In this type of algorithms two or more complementary algorithms are combined. WOFSMPSO [19] is a Weighted Optimization Framework based Speed constrained Multiobjective Particle Swarm Optimization. It is designed for solving large scale MOOPs using variable grouping and elimination techniques. This algorithm is applied over selected problems of different test suites. Although, it showed satisfactory performance with hypervolume indicator, the quality of Pareto front is poor. MOEA/D-AMG [20] is a hybridization of Decomposition based Multiobjective Evolutionary Algorithm and an Adaptive Multiple Gaussian process model. The Gaussian process model is varied by changing the standard deviation. It is used as a recombination operator in MOEA/D to obtain diverse Pareto solutions. In these type of MOOAs, appropriate cross of algorithms is a challenge. In addition, tuning of multiple algorithmic parameters, especially of different genres of algorithms is a cumbersome task.

Based on the literature review and an ongoing development of the third and fourth generation Pareto dominance based algorithms, we propose a novel Multiobjective Cohort Intelligence (MOCI) algorithm. It is based on Pareto dominance and coevolutionary principles. The major contributions of this work are as follows: 1. MOCI algorithm is enriched with the multiple features to enhance its efficiency, effectiveness, productivity and the robustness. 2. Dual stopping criteria such as maximum number of NDS solution and the maximum number of generations is proposed for stringent performance of algorithms. 3. Leader and follower cohorts are formulated with the division of population into superior and inferior candidates. Multiple leader and follower selection strategy is proposed with a due attention for their appropriate sizing to guide the search towards promising regions. 4. Pareto nondominance strategies are implemented in such a way as to increase the selection pressure in the second developmental stage. 5. Multiple performance metrics are employed and the performance results are presented with the help of the truly uncorrelated performance metrics. A framework to conduct multiple correlation analysis is proposed on the basis of criteria and attribute types. 6. Algorithms are ranked with the nonparametric statistical test. They are validated using post hoc test and verified using PROMOTHEE-II.

Section 2 presents brief review of cohort intelligence algorithm. Section 3 describes the proposed MOCI algorithm. Section 4 describes the research design employed, the performance of MOCI and competitive algorithms on five different benchmark test suites and statistical validation of their performances. Section 5 describes performance of the proposed MOCI algorithm on the real-world problems. Section 6 describes the characteristics of the research design. 7 Conclusions, 8 Future scope describes the conclusions and future scope of research work respectively.