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.     

MONSS: A multi-objective nonlinear simplex search approach

This article presents a novel methodology for dealing with continuous box-constrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple elements of the Pareto optimal set. The search is directed by a well-distributed set of weight vectors, each of which defines a scalarization problem that is solved by deforming a simplex according to the movements described by Nelder and Mead's method. Considering an MOP with n decision variables, the simplex is constructed using n+1 solutions which minimize different scalarization problems defined by n+1 neighbor weight vectors. All solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a proper representation of the Pareto optimal front. In this article, it is shown that a well-designed strategy using just mathematical programming techniques can be competitive with respect to the state-of-the-art multi-objective evolutionary algorithms against which it was compared.     

IMMUNE NETWORK SIMULATION WITH MULTIOBJECTIVE GENETIC ALGORITHMS FOR MULTIDISCIPLINARY DESIGN OPTIMIZATION

The paper presents a method called MOGA-INS for Multidisciplinary Design Optimization (MDO) of systems that involve multiple competing objectives with a mix of continuous and discrete variables. The method is based on the Immune Network Simulation ( INS) approach that has been extended by combining it with a Multi-Objective Genetic Algorithm ( MOGA). MOGA obtains Pareto solutions for multiple objective optimization problems in an all-at-once manner. INS provides a coordination strategy for subsystems in MDO to interact and is naturally suited for genetic algorithm-based optimization methods. The MOGA-INS method is demonstrated with a speed-reducer example, formulated as a two-level two-objective design optimization problem.     

A bi-objective model for design and analysis of sustainable intermodal transportation systems: a case study of Turkey

This paper presents a mixed-integer linear optimisation model to analyse the intermodal transportation systems in the Turkish transportation industry. The solution approach includes mathematical modelling, data analysis from real-life cases and solving the resulting mathematical programming problem to minimise total transportation cost and carbon dioxide emissions by using two different exact solution methods in order to find the optimal solutions. The novel approach of this paper generates Pareto solutions quickly and allows the decision makers to identify sustainable solutions by using a newly developed solution methodology for bi-objective mixed-integer linear problems in real-life cases.     

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.     

Coordinating strategic outsourcing decisions for semiconductor assembly using a bi-objective genetic algorithm

Increasing global competition has forced high-tech companies to focus on their core competences and outsource other activities to maintain their competitive advantages in the supply chains. While most companies rely on domain experts to coordinate strategic outsourcing decisions among a number of qualified vendors with different capabilities, the present problem can be formulated into a complex nonlinear, multi-dimensional, multi-objective combinatorial optimisation problem. Focused on real settings, this study aims to fill the gap via developing a bi-objective genetic algorithm (boGA) for determining the outsourcing order allocation with nonlinear cost structure, while minimising both the total alignment gap and the total allocation cost. The proposed boGA incorporates specific random key representation to facilitate encoding and decoding. This study also develops a bi-objective Pareto solution generation algorithm to enable efficient searching of Pareto solutions in multiple ranks and designs a composite Pareto ranking selection with uniform sum rank weighting for effective selection. To estimate its validity, the proposed boGA was validated with realistic cases from a leading semiconductor company in Hsinchu Science Park in Taiwan. The optimal boGA parameters were tested using a set of experiments. Scenario analyses were conducted to evaluate the performance of the proposed algorithm under different demand conditions using the metrics in the literature. The results have shown the practical viability of the proposed algorithm to solve the present problem of monthly outsourcing decisions for the case company in practicable computation time. This algorithm can determine the near-optimal Pareto front for decision makers to further incorporate with their preferences. This study concludes with discussion of future research directions.     

Optimization of constrained multiple-objective reliability problems using evolutionary algorithms

This paper illustrates the use of multi-objective optimization to solve three types of reliability optimization problems: to find the optimal number of redundant components, find the reliability of components, and determine both their redundancy and reliability. In general, these problems have been formulated as single objective mixed-integer non-linear programming problems with one or several constraints and solved by using mathematical programming techniques or special heuristics. In this work, these problems are reformulated as multiple-objective problems (MOP) and then solved by using a second-generation Multiple-Objective Evolutionary Algorithm (MOEA) that allows handling constraints. The MOEA used in this paper (NSGA-II) demonstrates the ability to identify a set of optimal solutions (Pareto front), which provides the Decision Maker with a complete picture of the optimal solution space. Finally, the advantages of both MOP and MOEA approaches are illustrated by solving four redundancy problems taken from 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.     

A two-stage approach for multi-objective decision making with applications to system reliability optimization

This paper proposes a two-stage approach for solving multi-objective system reliability optimization problems. In this approach, a Pareto optimal solution set is initially identified at the first stage by applying a multiple objective evolutionary algorithm (MOEA). Quite often there are a large number of Pareto optimal solutions, and it is difficult, if not impossible, to effectively choose the representative solutions for the overall problem. To overcome this challenge, an integrated multiple objective selection optimization (MOSO) method is utilized at the second stage. Specifically, a self-organizing map (SOM), with the capability of preserving the topology of the data, is applied first to classify those Pareto optimal solutions into several clusters with similar properties. Then, within each cluster, the data envelopment analysis (DEA) is performed, by comparing the relative efficiency of those solutions, to determine the final representative solutions for the overall problem. Through this sequential solution identification and pruning process, the final recommended solutions to the multi-objective system reliability optimization problem can be easily determined in a more systematic and meaningful way.     

基于粒子群与聚类的多目标优化算法

针对粒子群优化算法容易陷入局部最优的问题,提出了一种基于粒子群优化与分解聚类方法相结合的多目标优化算法。算法基于参考向量分解的方法,通过聚类优选粒子策略来更新全局最优解。首先,通过每条均匀分布的参考向量对粒子进行聚类操作,来促进粒子的多样性。从每个聚类中选择一个具有最小聚合函数适应度值的粒子,以平衡收敛性和多样性。动态更新全局最优解和个体最优解,引导种群均匀分布在帕累托前沿附近。通过仿真实验,与4种粒子群多目标优化算法进行对比。实验结果表明,提出的算法在27个选定的基准测试问题中获得了20个反世代距离(IGD)最优值。     

基于差异演化算法和多属性决策的机电系统可靠性多目标优化设计

针对机电系统可靠性设计问题,以可靠性和费用(或体积等)最优为目标建立可靠性设计的多目标优化模型．提出了自适应多目标差异演化算法,该算法提出了自适应缩放因子和混沌交叉率,采用改进的快速排序方法构造Pareto最优解,采用NSGA-II的拥挤操作对档案文件进行消减．采用自适应多目标差异演化算法获得多目标问题的Pareto最优解,利用TOPSIS方法对Pareto最优解进行多属性决策．实际工程结果表明：自适应多目标差异演化算法调节参数更少,且求得的Pareto最优解分布均匀;采用基于TOPSIS的多属性决策方法得到的结果合理可行．     

Multiobjective optimization with an adaptive weight determination scheme using the concept of hyperplane

Finding an optimum design that satisfies all performances in a design problem is very challenging. To overcome this problem, multiobjective optimization methods have been researched to obtain Pareto optimum solutions. Among the different methods, the weighted sum method is widely used for its convenience. However, since the different weights do not always guarantee evenly distributed solutions on the Pareto front, the weights need to be determined systematically. Therefore, this paper presents a multiobjective optimization using a new adaptive weight determination scheme. Solutions on the Pareto front are gradually found with different weights, and the values of these weights are adaptively determined by using information from the previously obtained solutions' positions. For an n-objective problem, a hyperplane is constructed in n -dimensional space, and new weights are calculated to find the next solutions. To confirm the effectiveness of the proposed method, benchmarking problems that have different types of Pareto front are tested, and a topology optimization problem is performed as an engineering problem. A hypervolume indicator is used to quantitatively evaluate the proposed method, and it is confirmed that optimized solutions that are evenly distributed on the Pareto front can be obtained by using the proposed method.     

Multi-objective design optimization of a new space radiator

In this work different multi-objective techniques are used to the conceptual design of a new kind of space radiator. Called VESPAR (Variable Emittance Space Radiator), the radiator has an effective variable emittance which makes it able to reduce or avoid the demand for heater power to warm up equipment during cold case operations in orbit. The multi-objective approach was aimed on obtaining a radiator that minimize its mass while at the same time minimize the need for heater power during cold case. Four multi-objective algorithms were used: Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II), Multi-Objective Genetic Algorithm (MOGA), Multi-Objective Simulating Annealing (MOSA) and Multi-Objective Generalized Extremal Optimization (M-GEO). The first three algorithms were used under the modeFrontier^® optimization software package, while M-GEO is a recently proposed multi-objective implementation of the Generalized Extremal Optimization (GEO) algorithm. The Pareto frontier showing the trade-off solutions between radiator mass and heater power consumption is obtained by the four algorithms and the results compared. An assessment of the performance of M-GEO on this problem, compared to the other well-known multi-objective algorithms is also made.     

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.     

Bi-objective scheduling on uniform parallel machines considering electricity cost

This article investigates a bi-objective scheduling problem on uniform parallel machines considering electricity cost under time-dependent or time-of-use electricity tariffs, where electricity price changes with the hours within a day. The aim is to minimize simultaneously the total electricity cost and the number of machines actually used. A bi-objective mixed-integer linear programming model is first formulated for the problem. An insertion algorithm is then proposed for the single-objective scheduling problem of minimizing the total electricity cost for a given number of machines. To obtain the whole Pareto front of the problem, an iterative search framework is developed based on the proposed insertion algorithm. Computational results on real-life and randomly generated instances demonstrate that the proposed approach is quite efficient and can find high-quality Pareto fronts for large-size problems with up to 5000 jobs.     

Migration NSGA: method to improve a non-elitist searching of Pareto front,with application in magnetics

The paper describes a migration strategy to improve classical non-dominated sorting genetic algorithm (NSGA) to find optimal solution of a multi-objective problem. Migration NSGA has been tested to assess its performance using analytical functions for which the Pareto front is known in analytical form, as well as two case studies in electromagnetics, for which the Pareto front is not known a priori. This strategy improves the approximation of the Pareto-optimal solutions of a multi-objective problem by introducing new individuals in the population miming the effect of migrations.     

A new multi-objective optimization model for preventive maintenance and replacement scheduling of multi-component systems

In this article, a new multi-objective optimization model is developed to determine the optimal preventive maintenance and replacement schedules in a repairable and maintainable multi-component system. In this model, the planning horizon is divided into discrete and equally-sized periods in which three possible actions must be planned for each component, namely maintenance, replacement, or do nothing. The objective is to determine a plan of actions for each component in the system while minimizing the total cost and maximizing overall system reliability simultaneously over the planning horizon. Because of the complexity, combinatorial and highly nonlinear structure of the mathematical model, two metaheuristic solution methods, generational genetic algorithm, and a simulated annealing are applied to tackle the problem. The Pareto optimal solutions that provide good tradeoffs between the total cost and the overall reliability of the system can be obtained by the solution approach. Such a modeling approach should be useful for maintenance planners and engineers tasked with the problem of developing recommended maintenance plans for complex systems of components.     

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.