Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.     

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.     

Two efficient algorithms for constructing almost even approximations of the Pareto front in multi-objective optimization problems

In this article, two algorithms are proposed for constructing almost even approximations of the Pareto front of multi-objective optimization problems. The first algorithm is a hybrid of the ε-constraint and Pascoletti–Serafini scalarization methods for solving bi-objective problems. The second is a modification of the successive Pareto optimization (SPO) algorithm for solving three-objective problems. In these algorithms, the MATLAB fmincon solver is used to solve single-objective optimization problems, which returns a local optimal solution. Some metrics are considered to evaluate the quality of approximations obtained by the suggested algorithms on six test problems, and their results are compared with other algorithms (normal constraint, weighted constraint, SPO, differential evolution, multi-objective evolutionary algorithm/decomposition–differential evolution, non-dominated sorting genetic algorithm-II and S-metric selection evolutionary multi-objective algorithm). Experimental results show that the proposed algorithms provide almost even approximations of the whole Pareto front, and better quality of approximation and CPU time compared with established algorithms.     

RAMS+C informed decision-making with application to multi-objective optimization of technical specifications and maintenance using genetic algorithms

The role of technical specifications and maintenance (TSM) activities at nuclear power plants (NPP) aims to increase reliability, availability and maintainability (RAM) of Safety-Related Equipment, which, in turn, must yield to an improved level of plant safety. However, more resources (e.g. costs, task force, etc.) have to be assigned in above areas to achieve better scores in reliability, availability, maintainability and safety (RAMS). Current situation at NPP shows different programs implemented at the plant that aim to the improvement of particular TSM-related parameters where the decision-making process is based on the assessment of the impact of the change proposed on a subgroup of RAMS+C attributes.This paper briefly reviews the role of TSM and two main groups of improvement programs at NPP, which suggest the convenience of considering the approach proposed in this paper for the Integrated Multi-Criteria Decision-Making on changes to TSM-related parameters based on RAMS+C criteria as a whole, as it can be seem as a decision-making process more consistent with the role and synergic effects of TSM and the objectives and goals of current improvement programs at NPP. The case of application to the Emergency Diesel Generator system demonstrates the viability and significance of the proposed approach for the Multi-objective Optimization of TSM-related parameters using a Genetic Algorithm.     

Reliability-based performance simulation for optimized pavement maintenance

Roadway pavement maintenance is essential for driver safety and highway infrastructure efficiency. However, regular preventive maintenance and rehabilitation (M&R) activities are extremely costly. Unfortunately, the funds available for the M&R of highway pavement are often given lower priority compared to other national development policies, therefore, available funds must be allocated wisely. Maintenance strategies are typically implemented by optimizing only the cost whilst the reliability of facility performance is neglected. This study proposes a novel algorithm using multi-objective particle swarm optimization (MOPSO) technique to evaluate the cost-reliability tradeoff in a flexible maintenance strategy based on non-dominant solutions. Moreover, a probabilistic model for regression parameters is employed to assess reliability-based performance. A numerical example of a highway pavement project is illustrated to demonstrate the efficacy of the proposed MOPSO algorithms. The analytical results show that the proposed approach can help decision makers to optimize roadway maintenance plans.     

Reliability optimization of series-parallel systems with a choice of redundancy strategies using a genetic algorithm

This paper proposes a genetic algorithm (GA) for a redundancy allocation problem for the series-parallel system when the redundancy strategy can be chosen for individual subsystems. Majority of the solution methods for the general redundancy allocation problems assume that the redundancy strategy for each subsystem is predetermined and fixed. In general, active redundancy has received more attention in the past. However, in practice both active and cold-standby redundancies may be used within a particular system design and the choice of the redundancy strategy becomes an additional decision variable. Thus, the problem is to select the best redundancy strategy, component, and redundancy level for each subsystem in order to maximize the system reliability under system-level constraints. This belongs to the NP-hard class of problems. Due to its complexity, it is so difficult to optimally solve such a problem by using traditional optimization tools. It is demonstrated in this paper that GA is an efficient method for solving this type of problems. Finally, computational results for a typical scenario are presented and the robustness of the proposed algorithm is discussed.     

Optimal choice of wastewater treatment train by multi-objective optimization

The typical domestic wastewater treatment train consists of some combination of unit operations for preliminary, primary, secondary, tertiary, and advanced treatment, and residual management, with many options being available for each type of unit operation. The challenge is to select treatment trains for which the extent and reliability of treatment are high, whereas the capital, operation and maintenance (O&M) costs of the treatment and land area requirement are low. This proposition has been formulated as a multi-objective optimization problem, and solved using the evolutionary/genetic optimization technique. The inputs required are the capital costs, O&M costs, land area requirements, and reliabilities of the unit operations of various types. In addition, overall environmental cost (E) corresponding to various treatment trains is input as a normalized parameter, which can take values in the range 0–100, with E being 100 corresponding to the ‘no treatment’ option. In other cases, E is a function of both treatment train efficiency and reliability. The problem was solved to determine the Pareto optimal, i.e. ‘no worse’ than each other, set of solutions under three conditions, viz. when E was not constrained, and for E<75, and E<50. Correctness of the algorithm was probed through a threefold analysis, (1) by solving a simplified two-objective problem, (2) by demonstrating the efficiency of the algorithm in picking up ‘sure-optimal’ solutions, i.e. solutions deliberately made optimal through manipulation of input data, and (3) by demonstrating that the set of optimal solutions remains approximately the same irrespective of the variations in the initial population size chosen for the genetic operations.     

Multi-objective scheduling problems: Determination of pruned Pareto sets

There are often multiple competing objectives for industrial scheduling and production planning problems. Two practical methods are presented to efficiently identify promising solutions from among a Pareto optimal set for multi-objective scheduling problems. Generally, multi-objective optimization problems can be solved by combining the objectives into a single objective using equivalent cost conversions, utility theory, etc., or by determination of a Pareto optimal set. Pareto optimal sets or representative subsets can be found by using a multi-objective genetic algorithm or by other means. Then, in practice, the decision maker ultimately has to select one solution from this set for system implementation. However, the Pareto optimal set is often large and cumbersome, making the post-Pareto analysis phase potentially difficult, especially as the number of objectives increase. Our research involves the post Pareto analysis phase, and two methods are presented to filter the Pareto optimal set to determine a subset of promising or desirable solutions. The first method is pruning using non-numerical objective function ranking preferences. The second approach involves pruning by using data clustering. The k-means algorithm is used to find clusters of similar solutions in the Pareto optimal set. The clustered data allows the decision maker to have just k general solutions from which to choose. These methods are general, and they are demonstrated using two multi-objective problems involving the scheduling of the bottleneck operation of a printed wiring board manufacturing line and a more general scheduling problem.     

RDS-NSGA-II: a memetic algorithm for reference point based multi-objective optimization

Reference point based optimization offers tools for the effective treatment of preference based multi-objective optimization problems, e.g. when the decision-maker has a rough idea about the target objective values. For the numerical solution of such problems, specialized evolutionary strategies have become popular, despite their possible slow convergence rates. Hybridizing such evolutionary algorithms with local search techniques have been shown to produce faster and more reliable algorithms. In this article, the directed search (DS) method is adapted to the context of reference point optimization problems, making this variant, called RDS, a well-suited option for integration into evolutionary algorithms. Numerical results on academic test problems with up to five objectives demonstrate the benefit of the novel hybrid (i.e. the same approximation quality can be obtained more efficiently by the new algorithm), using the state-of-the-art algorithm R-NSGA-II for this coupling. This represents an advantage when treating costly-to-evaluate real-world engineering design problems.     

Constrained multi-objective optimization using constrained non-dominated sorting combined with an improved hybrid multi-objective evolutionary algorithm

Constrained multi-objective optimization problems (cMOPs) are complex because the optimizer should balance not only between exploration and exploitation, but also between feasibility and optimality. This article suggests a parameter-free constraint handling approach called constrained non-dominated sorting (CNS). In CNS, each solution in a population is assigned a constrained non-dominated rank based on its constraint violation degree and Pareto rank. An improved hybrid multi-objective optimization algorithm called cMOEA/H for solving cMOPs is proposed. Additionally, a dynamic resource allocation mechanism is adopted by cMOEA/H to spare more computational efforts for those relatively hard sub-problems. cMOEA/H is first compared with the baseline algorithm using an existing constraint handling mechanism, verifying the advantages of the proposed constraint handling mechanism. Then cMOEA/H is compared with some classic constrained multi-objective optimizers, experimental results indicating that cMOEA/H could be a competitive alternative for solving cMOPs. Finally, the characteristics of cMOEA/H are studied.     

A joint reliability-redundancy optimization approach for multi-state series-parallel systems

In this paper, we present a practical approach for the joint reliability-redundancy optimization of multi-state series-parallel systems. In addition to determining the optimal redundancy level for each parallel subsystem, this approach also aims at finding the optimal values for the variables that affect the component state distributions in each subsystem. The key point is that technical and organizational actions can affect the state transition rates of a multi-state component, and thus affect the state distribution of the component and the availability of the system. Taking this into consideration, we present an approach for determining the optimal versions and numbers of components and the optimal set of technical and organizational actions for each subsystem of a multi-state series-parallel system, so as to minimize the system cost while satisfying the system availability constraint. The approach might be considered to be the multi-state version of the joint system reliability-redundancy optimization methods.     

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.     

An exact algorithm for preventive maintenance planning of series-parallel systems

Reliability is a meaningful parameter in assessing the performance of systems such as chemical processing facilities, power plant, aircrafts, ships, etc. In the literature, reliability optimization is widely considered during the system design phase and it is carried out by an opportune selection of both system components and redundancy. On the other hand, the problem of maintaining a required level of reliability by an opportune maintenance policy has been poorly examined. The paper tackles this problem for a system whose major components can be maintained only during a planned system downtime. An exact algorithm is proposed in order to single out the set of components that must be maintained to guarantee a required reliability level up to the next planned stop with the minimum cost. In order to verify the algorithm effectiveness, it has been applied to a complex real case regarding ship maintenance.     

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 sequential quadratically constrained quadratic programming technique for a multi-objective optimization problem

In this article a line search algorithm is proposed for solving constrained multi-objective optimization problems. At every iteration of the proposed method, a subproblem is formulated using quadratic approximation of all functions. A feasible descent direction is obtained as a solution of this subproblem. This scheme takes care some ideas of the sequential quadratically constrained quadratic programming technique for single objective optimization problems. A non-differentiable penalty function is used to restrict constraint violations at every iterating point. Convergence of the scheme is justified under the Slater constraint qualification along with some reasonable assumptions. The proposed algorithm is verified and compared with existing methods with a set of test problems. It is observed that this algorithm provides better results in most of the test problems.     

Multi-objective hierarchical genetic algorithms for multilevel redundancy allocation optimization

Multilevel redundancy allocation optimization problems (MRAOPs) occur frequently when attempting to maximize the system reliability of a hierarchical system, and almost all complex engineering systems are hierarchical. Despite their practical significance, limited research has been done concerning the solving of simple MRAOPs. These problems are not only NP hard but also involve hierarchical design variables. Genetic algorithms (GAs) have been applied in solving MRAOPs, since they are computationally efficient in solving such problems, unlike exact methods, but their applications has been confined to single-objective formulation of MRAOPs. This paper proposes a multi-objective formulation of MRAOPs and a methodology for solving such problems. In this methodology, a hierarchical GA framework for multi-objective optimization is proposed by introducing hierarchical genotype encoding for design variables. In addition, we implement the proposed approach by integrating the hierarchical genotype encoding scheme with two popular multi-objective genetic algorithms (MOGAs)—the strength Pareto evolutionary genetic algorithm (SPEA2) and the non-dominated sorting genetic algorithm (NSGA-II). In the provided numerical examples, the proposed multi-objective hierarchical approach is applied to solve two hierarchical MRAOPs, a 4- and a 3-level problems. The proposed method is compared with a single-objective optimization method that uses a hierarchical genetic algorithm (HGA), also applied to solve the 3- and 4-level problems. The results show that a multi-objective hierarchical GA (MOHGA) that includes elitism and mechanism for diversity preserving performed better than a single-objective GA that only uses elitism, when solving large-scale MRAOPs. Additionally, the experimental results show that the proposed method with NSGA-II outperformed the proposed method with SPEA2 in finding useful Pareto optimal solution sets.     

A multi-objective discrete reliability optimization problem for dissimilar-unit standby systems

A new methodology for the reliability optimization of a k dissimilar-unit nonrepairable cold-standby redundant system is introduced in this paper. Each unit is composed of a number of independent components with generalized Erlang distributions of lifetimes arranged in a series–parallel configuration. We also propose an approximate technique to extend the model to the general types of nonconstant hazard functions. To evaluate the system reliability, we apply the shortest path technique in stochastic networks. The purchase cost of each component is assumed to be an increasing function of its expected lifetime. There are multiple component choices with different distribution parameters available for replacement with each component of the system. The objective of the reliability optimization problem is to select the best components, from the set of available components, to be placed in the standby system to minimize the initial purchase cost of the system, maximize the system mean time to failure, minimize the system variance of time to failure, and also maximize the system reliability at the mission time. The goal attainment method is used to solve a discrete time approximation of the original problem.      

Pareto optimization using the struggle genetic crowding algorithm

Many real-world engineering design problems involve the simultaneous optimization of several conflicting objectives. In this paper, a method combining the struggle genetic crowding algorithm with Pareto-based population ranking is proposed to elicit trade-off frontiers. The new method has been tested on a variety of published problems, reliably locating both discontinuous Pareto frontiers as well as multiple Pareto frontiers in multi-modal search spaces. Other published multi-objective genetic algorithms are less robust in locating both global and local Pareto frontiers in a single optimization. For example, in a multi-modal test problem a previously published non-dominated sorting GA (NSGA) located the global Pareto frontier in 41% of the optimizations, while the proposed method located both global and local frontiers in all test runs. Additionally, the algorithm requires little problem specific tuning of parameters.     

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.     

Joint redundancy and maintenance optimization for multistate series–parallel systems

This paper formulates the joint redundancy and replacement schedule optimization problem generalized to multistate system, where the system and its components have a range of performance levels. Multistate system reliability is defined as the ability to maintain a specified performance level. The system elements are chosen from a list of available products on the market and the number of such elements is determined for each system component. Each element is characterized by its capacity, reliability and cost. The reliability of a system element is characterized by its lifetime distribution with the hazard rate, which increases with time. It is specified as the expected number of failures during different time intervals. The optimal system structure and the number of element replacements during the study period are defined as those which provide the desired level of system reliability with minimal sum of costs of capital investments, maintenance and unsupplied demand caused by failures. A universal generating function technique is applied to evaluate the multistate system reliability. A genetic algorithm is used as an optimization technique. Examples of determination of the optimal system structure and replacement schedule are provided.