Enhanced multiobjective particle swarm optimization in combination with adaptive weighted gradient-based searching

This article proposes a new multiobjective optimization method for structural problems based on multiobjective particle swarm optimization (MOPSO). A gradient-based optimization method is combined with MOPSO to alleviate constraint-handling difficulties. In this method, a group of particles is divided into two groups—a dominated solution group and a non-dominated solution group. The gradient-based method, utilizing a weighting coefficient method, is applied to the latter to conduct local searching that yields superior non-dominated solutions. In order to enhance the efficiency of exploration in a multiple objective function space, the weighting coefficients are adaptively assigned considering the distribution of non-dominated solutions. A linear optimization problem is solved to determine the optimal weighting coefficients for each non-dominated solution at each iteration. Finally, numerical and structural optimization problems are solved by the proposed method to verify the optimization efficiency.     

A TABOO SEARCH BASED APPROACH TO FIND THE PARETO OPTIMAL SET IN MULTIPLE OBJECTIVE OPTIMIZATION

Taboo search is a heuristic optimization technique which works with a neighbourhood of solutions to optimize a given objective function. It is generally applied to single objective optimization problems. Taboo search has the potential for solving multiple objective optimization (MOO) problems, because it works with more than one solution at a time, and this gives it the opportunity to evaluate multiple objective functions simultaneously. In this paper, a taboo search based algorithm is developed to find Pareto optimal solutions in multiple objective optimization problems. The developed algorithm has been tested with a number of problems and compared with other techniques. Results obtained from this work have proved that a taboo search based algorithm can find Pareto optimal solutions in MOO effectively.     

Hierarchical multiobjective routing model in Multiprotocol Label Switching networks with two service classes – a Pareto archive strategy

The article begins by reviewing a two-level hierarchical multicriteria routing model for Multiprotocol Label Switching networks with two service classes (QoS, i.e. with Quality of Service requirements, and Best Effort services) and alternative routing, as well as the foundations of a heuristic resolution approach, previously proposed by the authors. Afterwards a new variant of this heuristic approach, which includes a Pareto archive strategy, is described. In this archive, non-dominated solutions obtained throughout the heuristic are kept. At the end of the main procedure of the heuristic, these solutions are evaluated and a final solution for the routing problem is chosen using a reference point-based approach. The application of this procedure to two test networks will show, with analytic and discrete-event simulation models, that, in certain initial conditions, this approach provides improvements in the final results concerning the top-level objective functions, especially in more ‘difficult’ situations detected through sensitivity analysis.     

A particle swarm optimizer for constrained multi-objective engineering design problems

This article presents a particle swarm optimizer (PSO) capable of handling constrained multi-objective optimization problems. The latter occur frequently in engineering design, especially when cost and performance are simultaneously optimized. The proposed algorithm combines the swarm intelligence fundamentals with elements from bio-inspired algorithms. A distinctive feature of the algorithm is the utilization of an arithmetic recombination operator, which allows interaction between non-dominated particles. Furthermore, there is no utilization of an external archive to store optimal solutions. The PSO algorithm is applied to multi-objective optimization benchmark problems and also to constrained multi-objective engineering design problems. The algorithmic effectiveness is demonstrated through comparisons of the PSO results with those obtained from other evolutionary optimization algorithms. The proposed particle swarm optimizer was able to perform in a very satisfactory manner in problems with multiple constraints and/or high dimensionality. Promising results were also obtained for a multi-objective engineering design problem with mixed variables.     

A SIMULATED ANNEALING ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION

This paper describes a novel implementation of the Simulated Annealing algorithm designed to explore the trade-off between multiple objectives in optimization problems. During search, the algorithm maintains and updates an archive of non-dominated solutions between each of the competing objectives. At the end of search, the final archive corresponds to a number of optimal solutions from which the designer may choose a particular configuration. A new acceptance probability formulation based on an annealing schedule with multiple temperatures (one for each objective) is proposed along with a novel restart strategy. The performance of the algorithm is demonstrated on three examples. It is concluded that the proposed algorithm offers an effective and easily implemented method for exploring the trade-off in multiobjective optimization problems.     

An efficient particle swarm approach for mixed-integer programming in reliability-redundancy optimization applications

The reliability-redundancy optimization problems can involve the selection of components with multiple choices and redundancy levels that produce maximum benefits, and are subject to the cost, weight, and volume constraints. Many classical mathematical methods have failed in handling nonconvexities and nonsmoothness in reliability-redundancy optimization problems. As an alternative to the classical optimization approaches, the meta-heuristics have been given much attention by many researchers due to their ability to find an almost global optimal solutions. One of these meta-heuristics is the particle swarm optimization (PSO). PSO is a population-based heuristic optimization technique inspired by social behavior of bird flocking and fish schooling. This paper presents an efficient PSO algorithm based on Gaussian distribution and chaotic sequence (PSO-GC) to solve the reliability-redundancy optimization problems. In this context, two examples in reliability-redundancy design problems are evaluated. Simulation results demonstrate that the proposed PSO-GC is a promising optimization technique. PSO-GC performs well for the two examples of mixed-integer programming in reliability-redundancy applications considered in this paper. The solutions obtained by the PSO-GC are better than the previously best-known solutions available in the recent literature.     

Recent advances in system reliability optimization driven by importance measures

System reliability optimization problems have been widely discussed to maximize system reliability with resource constraints. Birnbaum importance is a well-known method for evaluating the effect of component reliability on system reliability. Many importance measures (IMs) are extended for binary, multistate, and continuous systems from different aspects based on the Birnbaum importance. Recently, these IMs have been applied in allocating limited resources to the component to maximize system performance. Therefore, the significance of Birnbaum importance is illustrated from the perspective of probability principle and gradient geometrical sense. Furthermore, the equations of various extended IMs are provided subsequently. The rules for simple optimization problems are summarized to enhance system reliability by using ranking or heuristic methods based on IMs. The importance-based optimization algorithms for complex or large-scale systems are generalized to obtain remarkable solutions by using IM-based local search or simplification methods. Furthermore, a general framework driven by IM is developed to solve optimization problems. Finally, some challenges in system reliability optimization that need to be solved in the future are presented.     

An orthogonal design based constrained evolutionary optimization algorithm

Solving constrained optimization problems (COPs) via evolutionary algorithms (EAs) has attracted much attention. In this article, an orthogonal design based constrained optimization evolutionary algorithm (ODCOEA) to tackle COPs is proposed. In principle, ODCOEA belongs to a class of steady state evolutionary algorithms. In the evolutionary process, several individuals are chosen from the population as parents and orthogonal design is applied to pairs of parents to produce a set of representative offspring. Then, after combining the offspring generated by different pairs of parents, non-dominated individuals are chosen. Subsequently, from the parent’s perspective, it is decided whether a non-dominated individual replaces a selected parent. Finally, ODCOEA incorporates an improved BGA mutation operator to facilitate the diversity of the population. The proposed ODCOEA is effectively applied to 12 benchmark test functions. The computational experiments show that ODCOEA not only quickly converges to optimal or near-optimal solutions, but also displays a very high performance compared with another two state-of-the-art techniques.     

A new heuristic for the flowshop scheduling problem to minimize makespan and maximum tardiness

This paper presents a new heuristic for solving the flowshop scheduling problem that aims to minimize makespan and maximize tardiness. The algorithm is able to take into account the aforementioned performance measures, finding a set of non-dominated solutions representing the Pareto front. This method is based on the integration of two different techniques: a multi-criteria decision-making method and a constructive heuristic procedure developed for makespan minimization in flowshop scheduling problems. In particular, the technique for order preference by similarity of ideal solution (TOPSIS) algorithm is integrated with the Nawaz–Enscore–Ham (NEH) heuristic to generate a set of potential scheduling solutions. To assess the proposed heuristic's performance, comparison with the best performing multi-objective genetic local search (MOGLS) algorithm proposed in literature is carried out. The test is executed on a large number of random problems characterized by different numbers of machines and jobs. The results show that the new heuristic frequently exceeds the MOGLS results in terms of both non-dominated solutions, set quality and computational time. In particular, the improvement becomes more and more significant as the number of jobs in the problem increases.     

Pricing and location decisions in multi-objective facility location problem with M/M/m/k queuing systems

This article presents a new multi-objective model for a facility location problem with congestion and pricing policies. This model considers situations in which immobile service facilities are congested by a stochastic demand following M/M/m/k queues. The presented model belongs to the class of mixed-integer nonlinear programming models and NP-hard problems. To solve such a hard model, a new multi-objective optimization algorithm based on a vibration theory, namely multi-objective vibration damping optimization (MOVDO), is developed. In order to tune the algorithms parameters, the Taguchi approach using a response metric is implemented. The computational results are compared with those of the non-dominated ranking genetic algorithm and non-dominated sorting genetic algorithm. The outputs demonstrate the robustness of the proposed MOVDO in large-sized problems.     

A weight-based multiobjective immune algorithm: WBMOIA

A novel immune algorithm is suggested for finding Pareto-optimal solutions to multiobjective optimization problems based on opt-aiNET, the artificial immune system algorithm for multi-modal optimization. In the proposed algorithm, a randomly weighted sum of multiple objectives is used as a fitness function, and a local search algorithm is incorporated to facilitate the exploitation of the search space. Specifically, a new truncation algorithm with similar individuals (TASI) is proposed to preserve the diversity of the population. Also, a new selection operator is presented to create the new population based on TASI. Simulation results on seven standard problems (ZDT2, ZDT6, DEB, VNT, BNH, OSY and KIT) show that the proposed algorithm is able to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the vector immune algorithm and the elitist non-dominated sorting genetic system.     

Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.     

Multi-objective feasibility enhanced particle swarm optimization

This article introduces a new method entitled multi-objective feasibility enhanced partical swarm optimization (MOFEPSO), to handle highly-constrained multi-objective optimization problems. MOFEPSO, which is based on the particle swarm optimization technique, employs repositories of non-dominated and feasible positions (or solutions) to guide feasible particle flight. Unlike its counterparts, MOFEPSO does not require any feasible solutions in the initialized swarm. Additionally, objective functions are not assessed for infeasible particles. Such particles can only fly along sensitive directions, and particles are not allowed to move to a position where any previously satisfied constraints become violated. These unique features help MOFEPSO gradually increase the overall feasibility of the swarm and to finally attain the optimal solution. In this study, multi-objective versions of a classical gear-train optimization problem are also described. For the given problems, the article comparatively evaluates the performance of MOFEPSO against several popular optimization algorithms found in the literature.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

Applying multiple objective tabu search to continuous optimization problems with a simple neighbourhood strategy

One of the first multiple objective versions of the tabu search (TS) algorithm is proposed by the author. The idea of applying TS to multiple objective optimization is inspired from its solution structure. TS works with more than one solution (neighbourhood solutions) at a time and this situation gives the opportunity to evaluate multiple objectives simultaneously in one run. The selection and updating stages are modified to enable the original TS algorithm to work with more than one objective. In this paper, the multiple objective tabu search (MOTS) algorithm is applied to multiple objective non‐linear optimization problems with continuous variables using a simple neighbourhood strategy. The algorithm is applied to four mechanical components design problems. The results are compared with several other solution techniques including multiple objective genetic algorithms. It is observed that MOTS is able to find better and much wider spread of solutions than the reported ones. Copyright © 2005 John Wiley & Sons, Ltd.     

Multi-objective planning of a multiple geostationary spacecraft refuelling mission

The multi-objective optimization of multiple geostationary spacecraft refuelling is investigated in this article. A servicing spacecraft (SSc) and a propellant depot (PD), both parked initially in geostationary Earth orbit (GEO), are utilized to refuel multiple GEO targets of known propellant demand. The capacitated SSc is expected to rendezvous with fuel-deficient GEO targets or the PD for the purpose of refuelling or getting refuelled. The multiple geostationary spacecraft refuelling problem is treated as a multi-variable combinatorial optimization problem with the principal objective of minimizing the propellant consumption and the mission duration. A two-level optimization model is built, and the design variables are the refuelling order X, the refuelling time T and the binary decision variable S. The non-dominated sorting genetic algorithm is employed to solve the up-level optimization problem. For the low-level optimization, an exact algorithm is proposed. Finally, numerical simulations are presented to illustrate the effectiveness and validity of the proposed approach.     

Reliability optimization using multiobjective ant colony system approaches

The multiobjective ant colony system (ACS) meta-heuristic has been developed to provide solutions for the reliability optimization problem of series-parallel systems. This type of problems involves selection of components with multiple choices and redundancy levels that produce maximum benefits, and is subject to the cost and weight constraints at the system level. These are very common and realistic problems encountered in conceptual design of many engineering systems. It is becoming increasingly important to develop efficient solutions to these problems because many mechanical and electrical systems are becoming more complex, even as development schedules get shorter and reliability requirements become very stringent. The multiobjective ACS algorithm offers distinct advantages to these problems compared with alternative optimization methods, and can be applied to a more diverse problem domain with respect to the type or size of the problems. Through the combination of probabilistic search, multiobjective formulation of local moves and the dynamic penalty method, the multiobjective ACSRAP, allows us to obtain an optimal design solution very frequently and more quickly than with some other heuristic approaches. The proposed algorithm was successfully applied to an engineering design problem of gearbox with multiple stages.     

A multiple surrogate assisted evolutionary algorithm for optimization involving iterative solvers

In many real-world optimization problems, the underlying objective and constraint function(s) are evaluated using computationally expensive iterative simulations such as the solvers for computational electro-magnetics, computational fluid dynamics, the finite element method, etc. The default practice is to run such simulations until convergence using termination criteria, such as maximum number of iterations, residual error thresholds or limits on computational time, to estimate the performance of a given design. This information is used to build computationally cheap approximations/surrogates which are subsequently used during the course of optimization in lieu of the actual simulations. However, it is possible to exploit information on pre-converged solutions if one has the control to abort simulations at various stages of convergence. This would mean access to various performance estimates in lower fidelities. Surrogate assisted optimization methods have rarely been used to deal with such classes of problem, where estimates at various levels of fidelity are available. In this article, a multiple surrogate assisted optimization approach is presented, where solutions are evaluated at various levels of fidelity during the course of the search. For any solution under consideration, the choice to evaluate it at an appropriate fidelity level is derived from neighbourhood information, i.e. rank correlations between performance at different fidelity levels and the highest fidelity level of the neighbouring solutions. Moreover, multiple types of surrogates are used to gain a competitive edge. The performance of the approach is illustrated using a simple 1D unconstrained analytical test function. Thereafter, the performance is further assessed using three 10D and three 20D test problems, and finally a practical design problem involving drag minimization of an unmanned underwater vehicle. The numerical experiments clearly demonstrate the benefits of the proposed approach for such classes of problem.     

Green permutation flowshop scheduling problem with sequence-dependent setup times: a case study

Increasing global energy consumption, large variations in its cost and the environmental degradation effects are good reasons for the manufacturing industries to become greener. Green shop floor scheduling is increasingly becoming a vital factor in the sustainable manufacturing. In this paper, a green permutation flowshop scheduling problem with sequence-dependent setup times is studied. Two objectives are considered including minimisation of makespan as a measure of service level and minimisation of total energy consumption as a measure of environmental sustainability. We extend a bi-objective mixed-integer linear programming model to formulate the stated problem. We develop a constructive heuristic algorithm to solve the model. The constructive heuristic algorithm includes iterated greedy (CHIG) and local search (CHLS) algorithms. We develop an efficient energy-saving method which decreases energy consumption, on average, by about 15%. To evaluate the effectiveness of the constructive heuristic algorithm, we compare it with the famous augmented ?-constraint method using various small-sized and large-sized problems. The results confirm that the heuristic algorithm obtains high-quality non-dominated solutions in comparison with the augmented ?-constraint method. Also, they show that the CHIG outperforms the CHLS. Finally, this paper follows a case-study, with in-depth analysis of the model and the constructive heuristic algorithm.     

Coupling ant colony and the degraded ceiling algorithm for the redundancy allocation problem of series–parallel systems

The redundancy allocation problem (RAP) is a well known NP-hard problem which involves the selection of elements and redundancy levels to maximize system reliability given various system-level constraints. As telecommunications and internet protocol networks, manufacturing and power systems are becoming more and more complex, while requiring short developments schedules and very high reliability, it is becoming increasingly important to develop efficient solutions to the RAP. This paper presents an efficient algorithm to solve this reliability optimization problem. The idea of a heuristic approach design is inspired from the ant colony meta-heuristic optimization method and the degraded ceiling local search technique. Our hybridization of the ant colony meta-heuristic with the degraded ceiling performs well and is competitive with the best-known heuristics for redundancy allocation. Numerical results for the 33 test problems from previous research are reported and compared. The solutions found by our approach are all better than or are in par with the well-known best solutions.