Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.     

A conic scalarization method in multi-objective optimization

This paper presents the conic scalarization method for scalarization of nonlinear multi-objective optimization problems. We introduce a special class of monotonically increasing sublinear scalarizing functions and show that the zero sublevel set of every function from this class is a convex closed and pointed cone which contains the negative ordering cone. We introduce the notion of a separable cone and show that two closed cones (one of them is separable) having only the vertex in common can be separated by a zero sublevel set of some function from this class. It is shown that the scalar optimization problem constructed by using these functions, enables to characterize the complete set of efficient and properly efficient solutions of multi-objective problems without convexity and boundedness conditions. By choosing a suitable scalarizing parameter set consisting of a weighting vector, an augmentation parameter, and a reference point, decision maker may guarantee a most preferred efficient or properly efficient solution.     

NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point

Most interactive methods developed for solving multiobjective optimization problems sequentially generate Pareto optimal or nondominated vectors and the decision maker must always allow impairment in at least one objective function to get a new solution. The NAUTILUS method proposed is based on the assumptions that past experiences affect decision makers’ hopes and that people do not react symmetrically to gains and losses. Therefore, some decision makers may prefer to start from the worst possible objective values and to improve every objective step by step according to their preferences. In NAUTILUS, starting from the nadir point, a solution is obtained at each iteration which dominates the previous one. Although only the last solution will be Pareto optimal, the decision maker never looses sight of the Pareto optimal set, and the search is oriented so that (s)he progressively focusses on the preferred part of the Pareto optimal set. Each new solution is obtained by minimizing an achievement scalarizing function including preferences about desired improvements in objective function values. NAUTILUS is specially suitable for avoiding undesired anchoring effects, for example in negotiation support problems, or just as a means of finding an initial Pareto optimal solution for any interactive procedure. An illustrative example demonstrates how this new method iterates.     

ParEGO extensions for multi-objective optimization of expensive evaluation functions

This paper deals with multi-objective optimization in the case of expensive objective functions. Such a problem arises frequently in engineering applications where the main purpose is to find a set of optimal solutions in a limited global processing time. Several algorithms use linearly combined criteria to use directly mono-objective algorithms. Nevertheless, other algorithms, such as multi-objective evolutionary algorithm (MOEA) and model-based algorithms, propose a strategy based on Pareto dominance to optimize efficiently all criteria. A widely used model-based algorithm for multi-objective optimization is Pareto efficient global optimization (ParEGO). It combines linearly the objective functions with several random weights and maximizes the expected improvement (EI) criterion. However, this algorithm tends to favor parameter values suitable for the reduction of the surrogate model error, rather than finding non-dominated solutions. The contribution of this article is to propose an extension of the ParEGO algorithm for finding the Pareto Front by introducing a double Kriging strategy. Such an innovation allows to calculate a modified EI criterion that jointly accounts for the objective function approximation error and the probability to find Pareto Set solutions. The main feature of the resulting algorithm is to enhance the convergence speed and thus to reduce the total number of function evaluations. This new algorithm is compared against ParEGO and several MOEA algorithms on a standard benchmark problems. Finally, an automotive engineering problem allowing to illustrate the applicability of the proposed approach is given as an example of a real application: the parameter setting of an indirect tire pressure monitoring system.     

g-dominance: Reference point based dominance for multiobjective metaheuristics

One of the main tools for including decision maker (DM) preferences in the multiobjective optimization (MO) literature is the use of reference points and achievement scalarizing functions [A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in: G. Fandel, T. Gal (Eds.), Multiple-Criteria Decision Making Theory and Application, Springer-Verlag, New York, 1980, pp. 469–486.]. The core idea in these approaches is converting the original MO problem into a single-objective optimization problem through the use of a scalarizing function based on a reference point. As a result, a single efficient point adapted to the DM’s preferences is obtained. However, a single solution can be less interesting than an approximation of the efficient set around this area, as stated for example by Deb in [K. Deb, J. Sundar, N. Udaya Bhaskara Rao, S. Chaudhuri, Reference point based multiobjective optimization using evolutionary algorithms, International Journal of Computational Intelligence Research, 2(3) (2006) 273–286]. In this paper, we propose a variation of the concept of Pareto dominance, called g-dominance, which is based on the information included in a reference point and designed to be used with any MO evolutionary method or any MO metaheuristic. This concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function. On the other hand, we will show how it can be easily used with any MO evolutionary method or any MO metaheuristic (just changing the dominance concept) and, to exemplify its use, we will show some results with some state-of-the-art-methods and some test problems.     

A multi-objective optimization evolutionary algorithm incorporating preference information based on fuzzy logic

A multi-objective optimization evolutionary algorithm incorporating preference information interactively is proposed. A new nine grade evaluation method is used to quantify the linguistic preferences expressed by the decision maker (DM) so as to reduce his/her cognitive overload. When comparing individuals, the classical Pareto dominance relation is commonly used, but it has difficulty in dealing with problems involving large numbers of objectives in which it gives an unmanageable and large set of Pareto optimal solutions. In order to overcome this limitation, a new outranking relation called “strength superior” which is based on the preference information is constructed via a fuzzy inference system to help the algorithm find a few solutions located in the preferred regions, and the graphical user interface is used to realize the interaction between the DM and the algorithm. The computational complexity of the proposed algorithm is analyzed theoretically, and its ability to handle preference information is validated through simulation. The influence of parameters on the performance of the algorithm is discussed and comparisons to another preference guided multi-objective evolutionary algorithm indicate that the proposed algorithm is effective in solving high dimensional optimization problems.     

基于混沌集群智能优化算法的多目标粗糙集属性约减

针对管理实践及大数据处理过程中具有多决策属性的粗糙集属性约减问题,将条件属性依赖度与知识分辨度进行结合构建属性权重,分别建立针对不同决策属性的约减目标函数,引入帕累托最优思想,将基于多决策属性的粗糙集属性约减问题转化为离散多目标优化问题。针对该问题的结构设计了具有集群智能优化思想的元胞自动机求解算法,在算法中引入基于个体的非支配解集平衡局部最优与全局最优的关系,引入混沌遗传算子增加种群多样性。以某铁路局设备安全风险处理数据为案例构建多决策属性粗糙集决策表进行优化计算并进行管理决策分析。研究发现:(1)相对于传统的NSGA-II与MO-cell算法,本文提出的算法具有更强的多目标属性挖掘性能;(2)帕累托最优思想可以较好地解释多决策属性粗糙集在管理实践中的意义。     

Synchronous and asynchronous Pareto-based multi-objective Artificial Bee Colony algorithms

Pareto-based multi-objective optimization algorithms prefer non-dominated solutions over dominated solutions and maintain as much as possible diversity in the Pareto optimal set to represent the whole Pareto-front. This paper proposes three multi-objective Artificial Bee Colony (ABC) algorithms based on synchronous and asynchronous models using Pareto-dominance and non-dominated sorting: asynchronous multi-objective ABC using only Pareto-dominance rule (A-MOABC/PD), asynchronous multi-objective ABC using non-dominated sorting procedure (A-MOABC/NS) and synchronous multi-objective ABC using non-dominated sorting procedure (S-MOABC/NS). These algorithms were investigated in terms of the inverted generational distance, hypervolume and spread performance metrics, running time, approximation to whole Pareto-front and Pareto-solutions spaces. It was shown that S-MOABC/NS is more scalable and efficient compared to its asynchronous counterpart and more efficient and robust than A-MOABC/PD. An investigation on parameter sensitivity of S-MOABC/NS was presented to relate the behavior of the algorithm to the values of the control parameters. The results of S-MOABC/NS were compared to some state-of-the art algorithms. Results show that S-MOABC/NS can provide good approximations to well distributed and high quality non-dominated fronts and can be used as a promising alternative tool to solve multi-objective problems with the advantage of being simple and employing a few control parameters.     

Saddle points and scalarizing sets in multiple objective linear programming

The main purpose of this paper is to study saddle points of the vector Lagrangian function associated with a multiple objective linear programming problem. We introduce three concepts of saddle points and establish their characterizations by solving suitable systems of equalities and inequalities. We deduce dual programs and prove a relationship between saddle points and dual solutions, which enables us to obtain an explicit expression of the scalarizing set of a given saddle point in terms of normal vectors to the value set of the problem. Finally, we present an algorithm to compute saddle points associated with non-degenerate vertices and the corresponding scalarizing sets.     

Bank Balance-Sheet Management: An Alternative Multi-Objective Model

Bank balance-sheet management entails considering competing and conflicting objectives such as maximization of returns and minimization of risks associated with alternative portfolio combinations. Traditional multi-objective models simply provide the decision-maker with the entire set of non-dominated solutions; the decision-maker must then choose, unaided, the best solution based on his subjective trade-offs, experience and judgement. This paper develops an alternative multi-objective balance-sheet management model which allows the explicit incorporation of the decision-maker's trade-offs between conflicting objectives, and attempts to reduce his cognitive burden while ensuring that the solution obtained belongs to the set of non-dominated points.     

Cone contraction and reference point methods for multi-criteria mixed integer optimization

Interactive approaches employing cone contraction for multi-criteria mixed integer optimization are introduced. In each iteration, the decision maker (DM) is asked to give a reference point (new aspiration levels). The subsequent Pareto optimal point is the reference point projected on the set of admissible objective vectors using a suitable scalarizing function. Thereby, the procedures solve a sequence of optimization problems with integer variables. In such a process, the DM provides additional preference information via pair-wise comparisons of Pareto optimal points identified. Using such preference information and assuming a quasiconcave and non-decreasing value function of the DM we restrict the set of admissible objective vectors by excluding subsets, which cannot improve over the solutions already found. The procedures terminate if all Pareto optimal solutions have been either generated or excluded. In this case, the best Pareto point found is an optimal solution. Such convergence is expected in the special case of pure integer optimization; indeed, numerical simulation tests with multi-criteria facility location models and knapsack problems indicate reasonably fast convergence, in particular, under a linear value function. We also propose a procedure to test whether or not a solution is a supported Pareto point (optimal under some linear value function).     

Local models—an approach to distributed multi-objective optimization

When solving real-world optimization problems, evolutionary algorithms often require a large number of fitness evaluations in order to converge to the global optima. Attempts have been made to find techniques to reduce the number of fitness function evaluations. We propose a novel framework in the context of multi-objective optimization where fitness evaluations are distributed by creating a limited number of adaptive spheres spanning the search space. These spheres move towards the global Pareto front as components of a swarm optimization system. We call this process localization. The contribution of the paper is a general framework for distributed evolutionary multi-objective optimization, in which the individuals in each sphere can be controlled by any existing evolutionary multi-objective optimization algorithm in the literature.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

A parallel multiple reference point approach for multi-objective optimization

This paper presents a multiple reference point approach for multi-objective optimization problems of discrete and combinatorial nature. When approximating the Pareto Frontier, multiple reference points can be used instead of traditional techniques. These multiple reference points can easily be implemented in a parallel algorithmic framework. The reference points can be uniformly distributed within a region that covers the Pareto Frontier. An evolutionary algorithm is based on an achievement scalarizing function that does not impose any restrictions with respect to the location of the reference points in the objective space. Computational experiments are performed on a bi-objective flow-shop scheduling problem. Results, quality measures as well as a statistical analysis are reported in the paper.     

兼顾成本与时间保证率的泊位及岸桥协同调度优化

针对集装箱班轮根据船期表按计划到离港的运行规律以及港口企业追求低运营成本的需求,本文以集装箱班轮按计划离港保证率最大和码头作业成本最低为目标,构建了泊位及岸桥协同调度多目标优化模型;设计了叠加式局部搜索算法,将其嵌入到带精英策略的非支配排序遗传算法中,经过相互交叉反馈运算,得到Pareto非劣解;采用“性价比”的概念和量化方法,选择出对港口和船公司的利益偏向最小的实施方案,解决了在Pareto解集中寻优的问题。最后,以大连港集装箱码头的生产实际为例,验证了上述优化模型及算法的合理性和有效性。     

A mathematical basis for satisficing decision making

This paper presents a conceptual and mathematical model of the process of satisficing decision making under multiple objectives in which the information about decision maker's preferences is expressed in the form of aspiration levels. The mathematical concept of a value (utility) function is modified to describe satisficing behavior; the modified value function (achievement scalarizing function) should possess the properties of order preservation and order approximation. It is shown that the mathematical basis formed using aspiration levels and achievement scalarizing functions can be used not only for satisficing decision making but also for Pareto optimization, and thus provides an alternative to approaches based on weighting coefficients or typical value functions. This mathematical basis, which can also be regarded as a generalization of the goal programming approach in multiobjective optimization, suggests pragmatic approaches to many problems in multiobjective analysis.     

Non-dominated sorting procedure for Pareto dominance ranking on multicore CPU and/or GPU

Evolutionary multi-objective optimization algorithms aim at finding an approximation of the Pareto set. For hard to solve problems with many conflicting objectives, the number of functions evaluations to represent the Pareto front can be large and time consuming. Parallel computing can reduce the wall-clock time of such algorithms. Previous studies tackled the parallelization of a particular evolutionary algorithm. In this research, we focus on improving one of the most time consuming procedures—the non-dominated sorting—, which is used in the state-of-the-art multi-objective genetic algorithms. Here, three parallel versions of the non-dominated sorting procedure are developed: (1) a multicore (based on Pthreads); (2) a Graphic Processing Unit (GPU) (based on CUDA interface); and (3) a hybrid (based on Pthreads and CUDA). The user can select the most suitable option to efficiently compute the non-dominated sorting procedure depending on the available hardware. Results show that the use of GPU computing provides a substantial improvement in terms of performance. The hybrid approach has the best performance when a good load balance is established among cores and GPU.     

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.     

Annealing-tabu PAES: a multi-objective hybrid meta-heuristic

Most real-life optimization problems require taking into account not one, but multiple objectives simultaneously. In most cases these objectives are in conflict, i.e. the improvement of some objectives implies the deterioration of others. In single-objective optimization there exists a global optimum, while in the multi-objective case no optimal solution is clearly defined, but rather a set of solutions. In the last decade most papers dealing with multi-objective optimization use the concept of Pareto-optimality. The goal of Pareto-based multi-objective strategies is to generate a front (set) of non-dominated solutions as an approximation to the true Pareto-optimal front. However, this front is unknown for problems with large and highly complex search spaces, which is why meta-heuristic methods have become important tools for solving this kind of problem. Hybridization in the multi-objective context is nowadays an open research area. This article presents a novel extension of the well-known Pareto archived evolution strategy (PAES) which combines simulated annealing and tabu search. Experiments on several mathematical problems show that this hybridization allows an improvement in the quality of the non-dominated solutions in comparison with PAES, and also with its extension M-PAES.     

New simulation-based frameworks for multi-objective reliability-based design optimization of structures

In the present study, two new simulation-based frameworks are proposed for multi-objective reliability-based design optimization (MORBDO). The first is based on hybrid non-dominated sorting weighted simulation method (NSWSM) in conjunction with iterative local searches that is efficient for continuous MORBDO problems. According to NSWSM, uniform samples are generated within the design space and, then, the set of feasible samples are separated. Thereafter, the non-dominated sorting operator is employed to extract the approximated Pareto front. The iterative local sample generation is then performed in order to enhance the accuracy, diversity, and increase the extent of non-dominated solutions. In the second framework, a pseudo-double loop algorithm is presented based on hybrid weighted simulation method (WSM) and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) that is efficient for problems including both discrete and continuous variables. According to hybrid WSM-NSGA-II, proper non-dominated solutions are produced in each generation of NSGA-II and, subsequently, WSM evaluates the reliability level of each candidate solution until the algorithm converges to the true Pareto solutions. The valuable characteristic of presented approaches is that only one simulation run is required for WSM during entire optimization process, even if solutions for different levels of reliability be desired. Illustrative examples indicate that NSWSM with the proposed local search strategy is more efficient for small dimension continuous problems. However, WSM-NSGA-II outperforms NSWSM in terms of solutions quality and computational efficiency, specifically for discrete MORBDOs. Employing global optimizer in WSM-NSGA-II provided more accurate results with lower samples than NSWSM.