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.     

MULTIOBJECTIVE DESIGN OPTIMIZATION BY AN EVOLUTIONARY ALGORITHM

This paper presents an evolutionary algorithm for generic multiobjective design optimization problems. The algorithm is based on nondominance of solutions in the objective and constraint space and uses effective mating strategies to improve solutions that are weak in either. Since the methodology is based on nondominance, scaling and aggregation affecting conventional penalty function methods for constraint handling does not arise. The algorithm incorporates intelligent partner selection for cooperative mating. The diversification strategy is based on niching which results in a wide spread of solutions in the parametric space. Results of the algorithm for the design examples clearly illustrate the efficiency of the algorithm in solving multidisciplinary design optimization problems.     

A novel evolutionary meta-heuristic for the multi-objective optimization of real-world water distribution networks

Genetic algorithms are currently one of the state-of-the-art meta-heuristic techniques for the optimization of large engineering systems such as the design and rehabilitation of water distribution networks. They are capable of finding near-optimal cost solutions to these problems given certain cost and hydraulic parameters. Recently, multi-objective genetic algorithms have become prevalent in the water industry due to the conflicting nature of these hydraulic and cost objectives. The Pareto-front of solutions can aid decision makers in the water industry as it provides a set of design solutions which can be examined by experienced engineers. However, multi-objective genetic algorithms tend to require a large number of objective function evaluations to arrive at an acceptable Pareto-front. This article investigates a novel hybrid cellular automaton and genetic approach to multi-objective optimization (known as CAMOGA). The proposed method is applied to two large, real-world networks taken from the UK water industry. The results show that the proposed cellular automaton approach can provide a good approximation of the Pareto-front with very few network simulations, and that CAMOGA outperforms the standard multi-objective genetic algorithm in terms of efficiency in discovering similar Pareto-fronts.     

A study on evolutionary multi-objective optimization for flow geometry design

In this paper, we investigate three recently proposed multi-objective optimization algorithms with respect to their application to a design-optimization task in fluid dynamics. The usual approach to render optimization problems is to accumulate multiple objectives into one objective by a linear combination and optimize the resulting single-objective problem. This has severe drawbacks such that full information about design alternatives will not become visible. The multi-objective optimization algorithms NSGA-II, SPEA2 and Femo are successfully applied to a demanding shape optimizing problem in fluid dynamics. The algorithm performance will be compared on the basis of the results obtained.     

Multiobjective optimization by genetic algorithms: application to safety systems

When attempting to optimize the design of engineered systems, the analyst is frequently faced with the demand of achieving several targets (e.g. low costs, high revenues, high reliability, low accident risks), some of which may very well be in conflict. At the same time, several requirements (e.g. maximum allowable weight, volume etc.) should also be satisfied. This kind of problem is usually tackled by focusing the optimization on a single objective which may be a weighed combination of some of the targets of the design problem and imposing some constraints to satisfy the other targets and requirements. This approach, however, introduces a strong arbitrariness in the definition of the weights and constraints levels and a criticizable homogenization of physically different targets, usually all translated in monetary terms.The purpose of this paper is to present an approach to optimization in which every target is considered as a separate objective to be optimized. For an efficient search through the solution space we use a multiobjective genetic algorithm which allows us to identify a set of Pareto optimal solutions providing the decision maker with the complete spectrum of optimal solutions with respect to the various targets. Based on this information, the decision maker can select the best compromise among these objectives, without a priori introducing arbitrary weights.     

On the use of metamodel-assisted,multi-objective evolutionary algorithms

This article is concerned with the optimal use of metamodels in the context of multi-objective evolutionary algorithms which are based on computationally expensive function evaluations. The goal is to capture Pareto fronts of optimal solutions with the minimum computational cost. In each generation during the evolution, the metamodels act as filters that distinguish the most promising individuals, which will solely undergo exact and costly evaluations. By means of the so-called inexact pre-evaluation phase, based on continuously updated local metamodels, most of the non-promising individuals are put aside without aggravating the overall cost. The gain achieved through this technique is amazing in single-objective problems. However, with more than one objective, noticeable performance degradation occurs. This article scrutinizes the role of metamodels in multi-objective evolutionary algorithms and proposes ways to overcome expected weaknesses and improve their performance. Minimization of mathematical functions as well as aerodynamic shape optimization problems are used for demonstration purposes.     

VIS: An artificial immune network for multi-objective optimization

The aim of this work is to propose and validate a novel multi-objective optimization algorithm based on the emulation of the behaviour of the immune system. The rationale of this work is that the artificial immune system has, in its elementary structure, the main features required by other multi-objective evolutionary algorithms described in the literature, such as diversity preservation, memory, adaptivity, and elitism. The proposed approach is compared with three multi-objective evolutionary algorithms that are representative of the state of the art in multi-objective optimization. Algorithms are tested on six standard problems (both unconstrained and constrained) and comparisons are carried out using three different metrics. Results show that the proposed approach has very good performances and can become a valid alternative to standard algorithms for solving multi-objective optimization problems.     

Multi-objective optimization of generalized reliability design problems using feature models—A concept for early design stages

Reliability optimization problems such as the redundancy allocation problem (RAP) have been of considerable interest in the past. However, due to the restrictions of the design space formulation, they may not be applicable in all practical design problems. A method with high modelling freedom for rapid design screening is desirable, especially in early design stages. This work presents a novel approach to reliability optimization. Feature modelling, a specification method originating from software engineering, is applied for the fast specification and enumeration of complex design spaces. It is shown how feature models can not only describe arbitrary RAPs but also much more complex design problems. The design screening is accomplished by a multi-objective evolutionary algorithm for probabilistic objectives. Comparing averages or medians may hide the true characteristics of this distributions. Therefore the algorithm uses solely the probability of a system dominating another to achieve the Pareto optimal set. We illustrate the approach by specifying a RAP and a more complex design space and screening them with the evolutionary algorithm.     

Evolutionary algorithms applied to reliable communication network design

Several evolutionary algorithms (EAs) applied to a wide class of communication network design problems modelled under the generalized Steiner problem (GSP) are evaluated. In order to provide a fault-tolerant design, a solution to this problem consists of a preset number of independent paths linking each pair of potentially communicating terminal nodes. This usually requires considering intermediate non-terminal nodes (Steiner nodes), which are used to ensure path redundancy, while trying to minimize the overall cost. The GSP is an NP-hard problem for which few algorithms have been proposed. This article presents a comparative study of pure and hybrid EAs applied to the GSP, codified over MALLBA, a general purpose library for combinatorial optimization. The algorithms were tested on several GSPs, and asset efficient numerical results are reported for both serial and distributed models of the evaluated algorithms.     

Inverse modelling of multi-objective thermodynamically optimized turbojet engines using GMDH-type neural networks and evolutionary algorithms

A novel approach is presented in this article for obtaining inverse mapping of thermodynamically Pareto-optimized ideal turbojet engines using group method of data handling (GMDH)-type neural networks and evolutionary algorithms (EAs). EAs are used in two different aspects. Firstly, multi-objective EAs (non–dominated sorting genetic algorithm-II) with a new diversity preserving mechanism are used for Pareto-based optimization of the thermodynamic cycle of ideal turbojet engines considering four important conflicting thermodynamic objectives, namely, specific thrust ({ST}), specific fuel consumption ({SFC}), propulsive efficiency (η_p), and thermal efficiency (η_t). The best obtained Pareto front, as a result, is a data table representing data pairs of non-dominated vectors of design variables, which are Mach number and pressure ratio, and the corresponding four objective functions. Secondly, EAs and singular value decomposition are deployed simultaneously for optimal design of both connectivity configuration and the values of coefficients, respectively, involved in GMDH-type neural networks which are used for the inverse modelling of the input–output data table obtained as the best Pareto front. Therefore, two different polynomial relations among the four thermo-mechanical objectives and both Mach number and pressure ratio are searched using that Pareto front. The results obtained in this paper are very promising and show that such important relationships may exist and could be discovered using both multi-objective EAs and evolutionarily designed GMDH-type neural networks.     

Swarm algorithms for single- and multi-objective optimization problems incorporating sensitivity analysis

Swarm algorithms such as particle swarm optimization (PSO) are non-gradient probabilistic optimization algorithms that have been successfully applied for global searches in complex problems such as multi-peak problems. However, application of these algorithms to structural and mechanical optimization problems still remains a complex matter since local optimization capability is still inferior to general numerical optimization methods. This article discusses new swarm metaphors that incorporate design sensitivities concerning objective and constraint functions and are applicable to structural and mechanical design optimization problems. Single- and multi-objective optimization techniques using swarm algorithms are combined with a gradient-based method. In the proposed techniques, swarm optimization algorithms and a sequential linear programming (SLP) method are conducted simultaneously. Finally, truss structure design optimization problems are solved by the proposed hybrid method to verify the optimization efficiency.     

Particle swarm optimization approach for multi-objective composite box-beam design

This paper presents a multi-agent search technique to design an optimal composite box-beam helicopter rotor blade. The search technique is called particle swarm optimization (‘inspired by the choreography of a bird flock’). The continuous geometry parameters (cross-sectional dimensions) and discrete ply angles of the box-beams are considered as design variables. The objective of the design problem is to achieve (a) specified stiffness value and (b) maximum elastic coupling. The presence of maximum elastic coupling in the composite box-beam increases the aero-elastic stability of the helicopter rotor blade. The multi-objective design problem is formulated as a combinatorial optimization problem and solved collectively using particle swarm optimization technique. The optimal geometry and ply angles are obtained for a composite box-beam design with ply angle discretizations of 10°, 15° and 45°. The performance and computational efficiency of the proposed particle swarm optimization approach is compared with various genetic algorithm based design approaches. The simulation results clearly show that the particle swarm optimization algorithm provides better solutions in terms of performance and computational time than the genetic algorithm based approaches.     

矿物质粉体对砂浆及混凝土Cl-渗透性的影响

研究了不同水胶比、不同矿物质粉体掺量的砂浆和混凝土，经标准养护至56天、90天时的导电量。在相同水胶比和相同矿物质粉体掺量下，混凝土的导电量远低于砂浆的导电量。含矿物质粉体的砂浆及混凝土的导电量均低于基准砂浆及混凝土的导电量。导电量随水胶比的降低而降低，也随龄期的增长而降低。     

Function-transformation methods for multi-objective optimization

It is useful with multi-objective optimization (MOO) to transform the objective functions such that they all have similar units and orders of magnitude. This article evaluates various transformation methods using simple example problems. Viewing these methods as different means to restrict function values sheds light on how the methods perform. The weighted sum approach for MOO is used to study how well different methods aid in depicting the Pareto optimal set. Whereas using unrestricted weights is well suited for providing a single solution that reflects preferences, it is found that using a convex combination of functions is desirable when generating the Pareto set. In addition, it is shown that some transformation methods are detrimental to the process of generating a diverse spread of points, and criteria are proposed for determining when the methods fail to generate an accurate representation of the Pareto set. Advantages of using a simple normalization–modification are demonstrated.     

Multi-objective optimization based on meta-modeling by using support vector regression

Practical engineering design problems have a black-box objective function whose forms are not explicitly known in terms of design variables. In those problems, it is very important to make the number of function evaluations as few as possible in finding an optimal solution. So, in this paper, we propose a multi-objective optimization method based on meta-modeling predicting a form of each objective function by using support vector regression. In addition, we discuss a way how to select additional experimental data for sequentially revising a form of objective function. Finally, we illustrate the effectiveness of the proposed method through some numerical examples.     

Determination of Pareto frontier in multi-objective maintenance optimization

The objective of a maintenance policy generally is the global maintenance cost minimization that involves not only the direct costs for both the maintenance actions and the spare parts, but also those ones due to the system stop for preventive maintenance and the downtime for failure. For some operating systems, the failure event can be dangerous so that they are asked to operate assuring a very high reliability level between two consecutive fixed stops. The present paper attempts to individuate the set of elements on which performing maintenance actions so that the system can assure the required reliability level until the next fixed stop for maintenance, minimizing both the global maintenance cost and the total maintenance time. In order to solve the previous constrained multi-objective optimization problem, an effective approach is proposed to obtain the best solutions (that is the Pareto optimal frontier) among which the decision maker will choose the more suitable one. As well known, describing the whole Pareto optimal frontier generally is a troublesome task. The paper proposes an algorithm able to rapidly overcome this problem and its effectiveness is shown by an application to a case study regarding a complex series-parallel system.     

A lexicographic weighted Tchebycheff approach for multi-constrained multi-objective optimization of the surface grinding process

In this article a multi-objective mathematical model is developed to minimize total time and cost while maximizing the production rate and surface finish quality in the grinding process. The model aims to determine optimal values of the decision variables considering process constraints. A lexicographic weighted Tchebycheff approach is developed to obtain efficient Pareto-optimal solutions of the problem in both rough and finished conditions. Utilizing a polyhedral branch-and-cut algorithm, the lexicographic weighted Tchebycheff model of the proposed multi-objective model is solved using GAMS software. The Pareto-optimal solutions provide a proper trade-off between conflicting objective functions which helps the decision maker to select the best values for the decision variables. Sensitivity analyses are performed to determine the effect of change in the grain size, grinding ratio, feed rate, labour cost per hour, length of workpiece, wheel diameter and downfeed of grinding parameters on each value of the objective function.     

Self-adaptive multi-objective harmony search for optimal design of water distribution networks

In multi-objective optimization computing, it is important to assign suitable parameters to each optimization problem to obtain better solutions. In this study, a self-adaptive multi-objective harmony search (SaMOHS) algorithm is developed to apply the parameter-setting-free technique, which is an example of a self-adaptive methodology. The SaMOHS algorithm attempts to remove some of the inconvenience from parameter setting and selects the most adaptive parameters during the iterative solution search process. To verify the proposed algorithm, an optimal least cost water distribution network design problem is applied to three different target networks. The results are compared with other well-known algorithms such as multi-objective harmony search and the non-dominated sorting genetic algorithm-II. The efficiency of the proposed algorithm is quantified by suitable performance indices. The results indicate that SaMOHS can be efficiently applied to the search for Pareto-optimal solutions in a multi-objective solution space.     

Particle-swarm harmony search for water network design

The optimal design of water distribution networks is a non-linear, multi-modal, and constrained problem classified as an NP-hard combinatorial problem. Because of the drawbacks of calculus-based algorithms, the problem has been tackled by assorted stochastic algorithms, such as the genetic algorithm, simulated annealing, tabu search, shuffled frog-leaping algorithm, ant colony optimization algorithm, harmony search, cross entropy, and scatter search. This study proposes a modified harmony search algorithm incorporating particle swarm concept. This algorithm was applied to the design of four bench-mark networks (two-loop, Hanoi, Balerma, and New York City networks), with good results.     

Multilayered composite structure design optimisation using distributed/parallel multi-objective evolutionary algorithms

This paper presents a research work on stacking sequence design optimisation for multilayered composite plate using a parallel/distributed evolutionary algorithm. The stacking sequence of fibres has a dramatic influence on the strength of multilayered composite plates. Multiple layers of fibre-reinforced material systems offer versatility in engineering material design due to the fact that the stacking sequence of each orthotropic layer can offer full advantage of superior mechanical properties. Numerical results show that the optimal composite structures have lower weight, higher stiffness and also affordable cost when compared to the extreme and intermediate composite structures. In addition, the benefits of using a parallel optimisation system are also presented.