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In recent decades, development of effective methods for optimizing a set of conflicted objective functions has been absorbing an increasing interest from researchers. This refers to the essence of real-life engineering systems and complex natural mechanisms which are generally multi-modal, non-convex and multi-criterion. Until now, several deterministic and stochastic methods have been proposed to cope with such complex systems. Advanced soft computational methods such as evolutionary games (cooperative and non-cooperative), Pareto-based techniques, fuzzy evolutionary methods, cooperative bio-inspired algorithms and neuro-evolutionary systems have effectively come to the aid of researchers to build up efficient paradigms with application to vector optimization. The paper aims to discuss this issue.

A novel hybrid algorithm called synchronous self-learning Pareto strategy (SSLPS) is presented for the sake of vector optimization. The method is the ensemble of evolutionary algorithms (EA), swarm intelligence (SI), adaptive version of self-organizing map (CSOM) and a data shuffling mechanism. EA are powerful numerical optimization algorithms capable of finding a global extreme point over a wide exploration domain. SI techniques (the swarm of bees in our case) can improve both intensification and robustness of exploration. CSOM network is an unsupervised learning methodology which learns the characteristics of non-dominated solutions and, thus, enhances the quality of the Pareto front.

To prove the effectiveness of the proposed method, the authors engage a set of well-known benchmark functions and some well-known rival optimization methods. Additionally, SSLPS is employed for optimal design of shape memory alloy actuator as a nonlinear multi-modal real-world engineering problem. The experiments show the acceptable potential of SSLPS for handling both numerical and engineering multi-objective problems.

To the author’s best knowledge, the proposed algorithm is among the rare multi-objective methods which fosters the use of automated unsupervised learning for increasing the intensity of Pareto front (while preserving the diversity). Also, the research evaluates the power of hybridization of SI and EA for efficient search.

Project managers work to ensure successful project completion within the shortest period and at the lowest cost. One of the main tasks of a project manager in the planning phase is to generate the project time–cost curve, and furthermore, to determine the most appropriate schedule for the construction process. Numerous existing time–cost tradeoff analysis models have focused on solving a simple project representation without regarding for typical activity and project characteristics. This study aims to present a novel approach called “multiple-objective social group optimization” (MOSGO) for optimizing time–cost decisions in generalized construction projects.

In this paper, a novel MOGSO to mimic the time–cost tradeoff problem in generalized construction projects is proposed. The MOSGO has slightly modified the mechanism operation from the original algorithm to be a free-parameter algorithm and to enhance the exploring and exploiting balance in an optimization algorithm. The evidential reasoning technique is used to rank the global optimal obtained non-dominated solutions to help decision makers reach a single compromise solution.

Two case studies of real construction projects were investigated and the performance of MOSGO was compared to those of widely considered multiple-objective evolutionary algorithms. The comparison results indicated that the MOSGO approach is a powerful, efficient and effective tool in finding the time–cost curve. In addition, the multi-criteria decision-making approaches were applied to identify the best schedule for project implementation.

Accordingly, the first major practical contribution of the present research is that it provides a tool for handling real-world construction projects by considering all types of construction project. The second important implication of this study derives from research finding on the hybridization multiple-objective and multi-criteria techniques to help project managers in facilitating the time–cost tradeoff (TCT) problems easily. The third implication stems from the wide-range application of the proposed model TCT.

The model can be used in early stages of the construction process to help project managers in selecting an appropriate plan for whole project lifecycle.

The proposal model can be applied to multi-objective contexts in diversified fields. Moreover, the model is also a useful reference for future research.

This paper makes contributions to extant literature by: introducing a method for making TCT models applicable to actual projects by considering general activity precedence relations; developing a novel MOSGO algorithm to solving TCT problems in multi-objective context by a single simulation; and facilitating the TCT problems to project managers by using multi-criteria decision-making approaches.

Today’s, supply chain production and distribution of products to improve the customer satisfaction in the shortest possible time by paying the minimum cost, has become the most important challenge in global market. On the other hand, minimizing the total cost of the transportation and distribution is one of the critical items for companies. To handle this challenge, this paper aims to present a multi-objective multi-facility model of green closed-loop supply chain (GCLSC) under uncertain environment. In this model, the proposed GCLSC considers three classes in case of the leading chain and three classes in terms of the recursive chain. The objectives are to maximize the total profit of the GCLSC, satisfaction of demand, the satisfactions of the customers and getting to the proper cost of the consumers, distribution centers and recursive centers.

Then, this model is designed by considering several products under several periods regarding the recovery possibility of products. Finally, to evaluate the proposed model, several numerical examples are randomly designed and then solved using non-dominated sorting genetic algorithm and non-dominated ranking genetic algorithm. Then, they are ranked by TOPSIS along with analytical hierarchy process so-called analytic hierarchy process-technique for order of preference by similarity to ideal solution (AHP-TOPSIS).

The results indicated that non-dominated ranked genetic algorithm (NRGA) algorithm outperforms non-dominated sorting genetic algorithm (NSGA-II) algorithm in terms of computation times. However, in other metrics, any significant difference was not seen. At the end, to rank the algorithms, a multi-criterion decision technique was used. The obtained results of this method indicated that NSGA-II had better performance than ones obtained by NRGA.

This study is motivated by the need of integrating the leading supply chain and retrogressive supply chain. In short, the highlights of the differences of this research with the mentioned studies are as follows: developing multi-objective multi-facility model of fuzzy GCLSC under uncertain environment and integrating the leading supply chain and retrogressive supply chain.

The purpose of this research is to study the Multi-Skill Resource-Constrained Multi-Project Scheduling Problem (MSRCMPSP), where (1) durations of activities depend on the familiarity levels of assigned workers, (2) more efficient workers demand higher per-day salaries, (3) projects have different due dates and (4) the budget of each period varies over time. The proposed model is bi-objective, and its objectives are minimization of completion times and costs of all projects, simultaneously.

This paper proposes a two-phase approach based on the Statistical Process Control (SPC) to solve this problem. This approach aims to develop a control chart so as to monitor the performance of an optimizer during the optimization process. In the first phase, a multi-objective statistical model has been used to obtain control limits of this chart. To solve this model, a Multi-Objective Greedy Randomized Adaptive Search Procedure (MOGRASP) has been hired. In the second phase, the MSRCMPSP is solved via a New Version of the Multi-Objective Variable Neighborhood Search Algorithm (NV-MOVNS). In each iteration, the developed control chart monitors the performance of the NV-MOVNS to obtain proper solutions. When the control chart warns about an out-of control state, a new procedure based on the Conway’s Game of Life, which is a cellular automaton, is used to bring the algorithm back to the in-control state.

The proposed two-phase approach has been used in solving several standard test problems available in the literature. The results are compared with the outputs of some other methods to assess the efficiency of this approach. Comparisons imply the high efficiency of the proposed approach in solving test problems with different sizes.

The proposed model and approach have been used to schedule multiple projects of a construction company in Iran. The outputs show that both the model and the NV-MOVNS can be used in real-world multi-project scheduling problems.

Due to the numerous numbers of studies reviewed in this research, the authors discovered that there are few researches on the multi-skill resource-constrained multi-project scheduling problem (MSRCMPSP) with the aforementioned characteristics. Moreover, none of the previous researches proposed an SPC-based solution approach for meta-heuristics in order to solve the MSRCMPSP.

The aim of this paper is to compare different methods for multiobjective optimisation with respect to the same benchmark problem.

The following methods are considered: equilibrium of gradients GB, multi‐individual multi‐objective evolution‐strategy MOESTRA, and goal attainment GATT, respectively. They are applied to the shape design of the pole pitch of an electrical machine, with the aim of maximizing the air‐gap induction and minimizing the stray field in the winding.

The same initial solution of the benchmark is considered for all methods. The final solution of GB and MOESTRA dominates the initial one because both objectives are improved. GB is the most expensive method and therefore is not suited when several non‐dominated solutions should be identified. Accordingly, the trade‐off curve was approximated by means of 15 non‐dominated solutions, resorting to MOESTRA.

A multi‐objective multi‐individual evolution strategy of lowest order has proven to be cost‐effective in identifying a set of non‐dominated solutions, so helping the designer to investigate the structure of the objective space. The requirements in terms of timing and facilities appear to be compatible with the resources of an automated design centre of an industrial company.

A probabilistic non-dominated sorting genetic algorithm (P-NSGA) for multi-objective optimization under uncertainty is presented. The purpose of this algorithm is to create a tight coupling between the optimization and uncertainty procedures, use all of the possible probabilistic information to drive the optimizer, and leverage high-performance parallel computing.

This algorithm is a generalization of a classical genetic algorithm for multi-objective optimization (NSGA-II) by Deb et al. The proposed algorithm relies on the use of all possible information in the probabilistic domain summarized by the cumulative distribution functions (CDFs) of the objective functions. Several analytic test functions are used to benchmark this algorithm, but only the results of the Fonseca-Fleming test function are shown. An industrial application is presented to show that P-NSGA can be used for multi-objective shape optimization of a Formula 1 tire brake duct, taking into account the geometrical uncertainties associated with the rotating rubber tire and uncertain inflow conditions.

This algorithm is shown to have deterministic consistency (i.e. it turns back to the original NSGA-II) when the objective functions are deterministic. When the quality of the CDF is increased (either using more points or higher fidelity resolution), the convergence behavior improves. Since all the information regarding uncertainty quantification is preserved, all the different types of Pareto fronts that exist in the probabilistic framework (e.g. mean value Pareto, mean value penalty Pareto, etc.) are shown to be generated a posteriori. An adaptive sampling approach and parallel computing (in both the uncertainty and optimization algorithms) are shown to have several fold speed-up in selecting optimal solutions under uncertainty.

There are no existing algorithms that use the full probabilistic distribution to guide the optimizer. The method presented herein bases its sorting on real function evaluations, not merely measures (i.e. mean of the probabilistic distribution) that potentially do not exist.

This paper aims to present a novel multi-objective version of the Gorilla Troops optimizer (GTO), based on crowding distance, to achieve the optimal design of a brushless direct current motor.

In the proposed algorithm, the crowding distance technique was integrated into the GTO to perform the leader selection and also for the external archive refinement from extra non-dominated solutions. Furthermore, with a view to improving the diversity of non-dominated solutions in the external archive, mutation operator was used. For constrained problems, an efficient strategy was adopted. The proposed algorithm is referred to as CD-MOGTO.

To validate the effectiveness of the proposed approach, it was initially tested on three constrained multi-objective problems; thereafter, it was applied to optimize the design variables of brushless direct current motor to concurrently fulfill six inequality constraints, maximize efficiency and minimize total mass.

The results revealed the high potential of the proposed algorithm over different recognized algorithms in solving constrained multi-objective issues and the brushless direct current motors.

This paper aims to propose a dynamically balanced mechanism for cleaning unit used in agricultural thresher machine using a dynamically equivalent system of point masses.

The cleaning unit works on crank-rocker Grashof mechanism. This mechanism can be balanced by optimizing the inertial properties of each link. These properties are defined by the dynamic equivalent system of point masses. Parameters of these point masses define the shaking forces and moments. Hence, the multi-objective optimization problem with minimization of shaking forces and shaking moments is formulated by considering the point mass parameters as the design variables. The formulated optimization problem is solved using a posteriori approach-based algorithm i.e. the non-dominated sorting Jaya algorithm (NSJAYA) and a priori approach-based algorithms i.e. Jaya algorithm and genetic algorithm (GA) under suitable design constraints.

The mass, center of mass and inertias of each link are calculated using optimum design variables. These optimum parameters improve the dynamic performance of the cleaning unit. The optimal Pareto set for the balancing problem is measured and outlined in this paper. The designer can choose any solution from the set and balance any real planar mechanism.

The efficiency of the proposed approach is tested through the existing cleaning mechanism of the thresher machine. It is found that the NSJAYA is computationally more efficient than the GA and Jaya algorithm. ADAMS software is used for the simulation of the mechanism.

One of the main components of multi-objective, and therefore, many-objective evolutionary algorithms, is the selection mechanism. It is responsible for performing two main tasks simultaneously. First, it has to promote convergence by selecting solutions which are as close as possible to the Pareto optimal set. And second, it has to promote diversity in the solution set provided. In the current work, an exhaustive study that involves the comparison of several selection mechanisms with different features is performed. Particularly, Pareto-based and indicator-based selection schemes, which belong to well-known multi-objective optimisers, are considered. The paper aims to discuss these issues.

Each of those mechanisms is incorporated into a common multi-objective evolutionary algorithm framework. The main goal of the study is to measure the diversity preserved by each of those selection methods when addressing many-objective optimisation problems. The Walking Fish Group test suite, a set of optimisation problems with a scalable number of objective functions, is taken into account to perform the experimental evaluation.

The computational results highlight that the the reference-point-based selection scheme of the Non-dominated Sorting Genetic Algorithm III and a modified version of the Non-dominated Sorting Genetic Algorithm II, where the crowding distance is replaced by the Euclidean distance, are able to provide the best performance, not only in terms of diversity preservation, but also in terms of convergence.

The performance provided by the use of the Euclidean distance as part of the selection scheme indicates this is a promising line of research and, to the best of the knowledge, it has not been investigated yet.

This paper aims to optimally plan distributed generation (DG) and capacitor in distribution network by optimizing multiple conflicting operational objectives simultaneously so as to achieve enhanced operation of distribution system. The multi-objective optimization problem comprises three important objective functions such as minimization of total active power loss (P^loss_total), reduction of voltage deviation and balancing of current through feeder sections.

In this study, a hybrid configuration of weight improved particle swarm optimization (WIPSO) and gravitational search algorithm (GSA) called hybrid WIPSO-GSA algorithm is proposed in multi-objective problem domain. To solve multi-objective optimization problem, the proposed hybrid WIPSO-GSA algorithm is integrated with two components. The first component is fixed-sized archive that is responsible for storing a set of non-dominated pareto optimal solutions and the second component is a leader selection strategy that helps to update and identify the best compromised solution from the archive.

The proposed methodology is tested on standard 33-bus and Indian 85-bus distribution systems. The results attained using proposed multi-objective hybrid WIPSO-GSA algorithm provides potential technical and economic benefits and its best compromised solution outperforms other commonly used multi-objective techniques, thereby making it highly suitable for solving multi-objective problems.

A novel multi-objective hybrid WIPSO-GSA algorithm is proposed for optimal DG and capacitor planning in radial distribution network. The results demonstrate the usefulness of the proposed technique in improved distribution system planning and operation and also in achieving better optimized results than other existing multi-objective optimization techniques.