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Multi-objective is a complex problem that appears in real life while these objectives are conflicting. The swarm intelligence algorithm is often used to solve such multi-objective problems. Due to its strong search ability and convergence ability, particle swarm optimization algorithm is proposed, and the multi-objective particle swarm optimization algorithm is used to solve multi-objective optimization problems. However, the particles of particle swarm optimization algorithm are easy to fall into local optimization because of their fast convergence. Uneven distribution and poor diversity are the two key drawbacks of the Pareto front of multi-objective particle swarm optimization algorithm. Therefore, this paper aims to propose an improved multi-objective particle swarm optimization algorithm using adaptive Cauchy mutation and improved crowding distance.

In this paper, the proposed algorithm uses adaptive Cauchy mutation and improved crowding distance to perturb the particles in the population in a dynamic way in order to help the particles trapped in the local optimization jump out of it which improves the convergence performance consequently.

In order to solve the problems of uneven distribution and poor diversity in the Pareto front of multi-objective particle swarm optimization algorithm, this paper uses adaptive Cauchy mutation and improved crowding distance to help the particles trapped in the local optimization jump out of the local optimization. Experimental results show that the proposed algorithm has obvious advantages in convergence performance for nine benchmark functions compared with other multi-objective optimization algorithms.

In order to help the particles trapped in the local optimization jump out of the local optimization which improves the convergence performance consequently, this paper proposes an improved multi-objective particle swarm optimization algorithm using adaptive Cauchy mutation and improved crowding distance.

In line with computational technological advances, obtaining optimal solutions for engineering problems has become attractive research topics in various disciplines and real engineering systems having multiple objectives. Therefore, it is aimed to ensure that the multiple objectives are simultaneously optimized by considering them among the trade-offs. Furthermore, the practical means of solving those problems are principally concentrated on handling various complicated constraints. The purpose of this paper is to suggest an algorithm based on symbiotic organisms search (SOS), which mimics the symbiotic reciprocal influence scheme adopted by organisms to live on and breed within the ecosystem, for constrained multi-objective engineering design problems.

Though the general performance of SOS algorithm was previously well demonstrated for ordinary single objective optimization problems, its efficacy on multi-objective real engineering problems will be decisive about the performance. The SOS algorithm is, hence, implemented to obtain the optimal solutions of challengingly constrained multi-objective engineering design problems using the Pareto optimality concept.

Four well-known mixed constrained multi-objective engineering design problems and a real-world complex constrained multilayer dielectric filter design problem are tackled to demonstrate the precision and stability of the multi-objective SOS (MOSOS) algorithm. Also, the comparison of the obtained results with some other well-known metaheuristics illustrates the validity and robustness of the proposed algorithm.

The algorithmic performance of the MOSOS on the challengingly constrained multi-objective multidisciplinary engineering design problems with constraint-handling approach is successfully demonstrated with respect to the obtained outperforming final optimal designs.

This study aims to propose a new airport gate assignment method to effectively improve the comprehensive operation capacity and efficiency of hub airport. Gate assignment is one of the most important tasks for airport ground operations, which assigns appropriate airport gates with high efficiency reasonable arrangement.

In this paper, on the basis of analyzing the characteristics of airport gates and flights, an efficient multi-objective optimization model of airport gate assignment based on the objectives of the most balanced idle time, the shortest walking distances of passengers and the least number of flights at apron is constructed. Then an improved ant colony optimization (ICQACO) algorithm based on the ant colony collaborative strategy and pheromone update strategy is designed to solve the constructed model to fast realize the gate assignment and obtain a rational and effective gate assignment result for all flights in the different period.

In the designed ICQACO algorithm, the ant colony collaborative strategy is used to avoid the rapid convergence to the local optimal solution, and the pheromone update strategy is used to quickly increase the pheromone amount, eliminate the interference of the poor path and greatly accelerate the convergence speed.

The actual flight data from Guangzhou Baiyun airport of China is selected to verify the feasibility and effectiveness of the constructed multi-objective optimization model and the designed ICQACO algorithm. The experimental results show that the designed ICQACO algorithm can increase the pheromone amount, accelerate the convergence speed and avoid to fall into the local optimal solution. The constructed multi-objective optimization model can effectively improve the comprehensive operation capacity and efficiency. This study is a very meaningful work for airport gate assignment.

An efficient multi-objective optimization model for hub airport gate assignment problem is proposed in this paper. An improved ant colony optimization algorithm based on ant colony collaborative strategy and the pheromone update strategy is deeply studied to speed up the convergence and avoid to fall into the local optimal solution.

In the vast majority of published papers, the optimal reactive power dispatch (ORPD) problem is dealt as a single-objective optimization; however, optimization with a single objective is insufficient to achieve better operation performance of power systems. Multi-objective ORPD (MOORPD) aims to minimize simultaneously either the active power losses and voltage stability index, or the active power losses and the voltage deviation. The purpose of this paper is to propose multi-objective ant lion optimization (MOALO) algorithm to solve multi-objective ORPD problem considering large-scale power system in an effort to achieve a good performance with stable and secure operation of electric power systems.

A MOALO algorithm is presented and applied to solve the MOORPD problem. Fuzzy set theory was implemented to identify the best compromise solution from the set of the non-dominated solutions. A comparison with enhanced version of multi-objective particle swarm optimization (MOEPSO) algorithm and original (MOPSO) algorithm confirms the solutions. An in-depth analysis on the findings was conducted and the feasibility of solutions were fully verified and discussed.

Three test systems – the IEEE 30-bus, IEEE 57-bus and large-scale IEEE 300-bus – were used to examine the efficiency of the proposed algorithm. The findings obtained amply confirmed the superiority of the proposed approach over the multi-objective enhanced PSO and basic version of MOPSO. In addition to that, the algorithm is benefitted from good distributions of the non-dominated solutions and also guarantees the feasibility of solutions.

The proposed algorithm is applied to solve three versions of ORPD problem, active power losses, voltage deviation and voltage stability index, considering large -scale power system IEEE 300 bus.

The purpose of this paper is to realize a multi‐objective location‐routing network optimization in reverse logistics using particle swarm optimization based on grey relational analysis with entropy weight.

Real world network design problems are often characterized by multi‐objective in reverse logistics. This has recently been considered as an additional objective for facility location problem or vehicle routing problem in reverse logistics network design. Both of them are shown to be NP‐hard. Hence, location‐routing problem (LRP) with multi‐objective is more complicated integrated problem, and it is NP‐hard too. Due to the fact that NP‐hard model cannot be solved directly, grey relational analysis and entropy weight were added to particle swarm optimization to decision among the objectives. Then, a mathematics model about multi‐objective LRP of reverse logistics has been constructed, and a proposed hybrid particle swarm optimization with grey relational analysis and entropy weight has been developed to resolve it. An example is also computed in the last part of the paper.

The results are convincing: not only that particle swarm optimization and grey relational analysis can be used to resolve multi‐objective location‐routing model, but also that entropy and grey relational analysis can be combined to decide weights of objectives.

The method exposed in the paper can be used to deal with multi‐objective LRP in reverse logistics, and multi‐objective network optimization result could be helpful for logistics efficiency and practicability.

The paper succeeds in realising both a constructed multi‐objective model about location‐routing of reverse logistics and a multi‐objective solution algorithm about particle swarm optimization and future stage by using one of the newest developed theories: grey relational analysis.

The purpose of this paper is to propose a multi-objective differential evolution algorithm named as MOMR-DE to resolve multicast routing problem. In mobile ad hoc network (MANET), multicast routing is a non-deterministic polynomial -complete problem that deals with the various objectives and constraints. Quality of service (QoS) in the multicast routing problem mainly depends on cost, delay, jitter and bandwidth. So the cost, delay, jitter and bandwidth are always considered as multi-objective for designing multicast routing protocols. However, mobile node battery energy is finite and the network lifetime depends on node battery energy. If the battery power consumption is high in any one of the nodes, the chances of network’s life reduction due to path breaks are also more. On the other hand, node’s battery energy had to be consumed to guarantee high-level QoS in multicast routing to transmit correct data anywhere and at any time. Hence, the network lifetime should be considered as one objective of the multi-objective in the multicast routing problem.

Recently, many metaheuristic algorithms formulate the multicast routing problem as a single-objective problem, although it obviously is a multi-objective optimization problem. In the MOMR-DE, the network lifetime, cost, delay, jitter and bandwidth are considered as five objectives. Furthermore, three QoS constraints which are maximum allowed delay, maximum allowed jitter and minimum requested bandwidth are included. In addition, we modify the crossover and mutation operators to build the shortest-path multicast tree to maximize network lifetime and bandwidth, minimize cost, delay and jitter.

Two sets of experiments are conducted and compared with other algorithms for these problems. The simulation results show that our proposed method is capable of achieving faster convergence and is more preferable for multicast routing in MANET.

In MANET, most metaheuristic algorithms formulate the multicast routing problem as a single-objective problem. However, this paper proposes a multi-objective differential evolution algorithm to resolve multicast routing problem, and the proposed algorithm is capable of achieving faster convergence and more preferable for multicast routing.

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.

The interval multi-objective optimization problems (IMOPs) are universal and vital uncertain optimization problems. In this study, an interval multi-objective grey wolf optimization algorithm (GWO) based on fuzzy system is proposed to solve IMOPs effectively.

First, the classical genetic operators are embedded into the interval multi-objective GWO as local search strategies, which effectively balanced the global search ability and local development ability. Second, by constructing a fuzzy system, an effective local search activation mechanism is proposed to save computing resources as much as possible while ensuring the performance of the algorithm. The fuzzy system takes hypervolume, imprecision and number of iterations as inputs and outputs the activation index, local population size and maximum number of iterations. Then, the fuzzy inference rules are defined. It uses the activation index to determine whether to activate the local search process and sets the population size and the maximum number of iterations in the process.

The experimental results show that the proposed algorithm achieves optimal hypervolume results on 9 of the 10 benchmark test problems. The imprecision achieved on 8 test problems is significantly better than other algorithms. This means that the proposed algorithm has better performance than the commonly used interval multi-objective evolutionary algorithms. Moreover, through experiments show that the local search activation mechanism based on fuzzy system proposed in this study can effectively ensure that the local search is activated reasonably in the whole algorithm process, and reasonably allocate computing resources by adaptively setting the population size and maximum number of iterations in the local search process.

This study proposes an Interval multi-objective GWO, which could effectively balance the global search ability and local development ability. Then an effective local search activation mechanism is developed by using fuzzy inference system. It closely combines global optimization with local search, which improves the performance of the algorithm and saves computing resources.

In this work, the multi-objective optimization shuffled complex evolution is proposed. The algorithm is based on the extension of shuffled complex evolution, by incorporating two classical operators into the original algorithm: the rank ordering and crowding distance. In order to accelerate the convergence process, a Local Search strategy based on the generation of potential candidates by using Latin Hypercube method is also proposed.

The multi-objective optimization shuffled complex evolution is used to accelerate the convergence process and to reduce the number of objective function evaluations.

In general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics. From a statistical point of view, we demonstrated that differences may exist between the proposed methodology and other evolutionary strategies concerning two different metrics (convergence and diversity), for a class of benchmark functions (ZDT functions).

The development of a new numerical method to solve multi-objective optimization problems is the major contribution.

This chapter presents the survey of selected linear and mixed integer programming multi-objective portfolio optimization. The definitions of selected percentile risk measures are presented. Some contrasts and similarities of the different types of portfolio formulations are drawn out. The survey of multi-criteria methods devoted to portfolio optimization such as weighting approach, lexicographic approach, and reference point method is also presented. This survey presents the nature of the multi-objective portfolio problems focuses on a compromise between the construction of objectives, constraints, and decision variables in a portfolio and the problem complexity of the implemented mathematical models. There is always a trade-off between computational time and the size of an input data, as well as the type of mathematical programming formulation with linear and/or mixed integer variables.