A note on the estimation of the Pareto efficient set for multiobjective matrix permutation problems

There are a number of important problems in quantitative psychology that require the identification of a permutation of the n rows and columns of an n × n proximity matrix. These problems encompass applications such as unidimensional scaling, paired‐comparison ranking, and anti‐Robinson forms. The importance of simultaneously incorporating multiple objective criteria in matrix permutation applications is well recognized in the literature; however, to date, there has been a reliance on weighted‐sum approaches that transform the multiobjective problem into a single‐objective optimization problem. Although exact solutions to these single‐objective problems produce supported Pareto efficient solutions to the multiobjective problem, many interesting unsupported Pareto efficient solutions may be missed. We illustrate the limitation of the weighted‐sum approach with an example from the psychological literature and devise an effective heuristic algorithm for estimating both the supported and unsupported solutions of the Pareto efficient set.     

Trade‐off ranking method for multi‐criteria decision analysis

Multi‐criteria decision analysis presumes trade‐off between different criteria. As a result, the optimal solution is not unique and can be represented by the Pareto frontier in the objective space. Each Pareto solution is a compromise between different objectives. Despite a limited number of Pareto optimal solutions, the decision‐maker eventually has to choose only one option. Such a choice has to be made with the use of additional preferences not included in the original formulation of the optimization problem. The paper represents a new approach to an automatic ranking that can help the decision‐maker. In contrast to the other methodologies, the proposed method is based on the minimization of trade‐off between different Pareto solutions. To be realized, the approach presumes the existence of a well‐distributed Pareto set representing the entire Pareto frontier. In the paper, such a set is generated with the use of the directed search domain algorithm. The method is applied to a number of test cases and compared against two existing alternative approaches.     

Detecting local convexity on the pareto surface

Recent regain in interest in multi‐criteria optimization approaches to provide a designer with multiple solutions to select from and support decisions has led to various methodologies to generate such solutions and possibly approximate the Pareto set. This paper introduces the notions of H‐ and w‐convexities, and develops a simple method to identify local convexity of Pareto hyper‐surfaces since their shape can dictate the choice of the method to use to obtain Pareto solutions and possibly to build an approximation of that set. The method is based on comparing the results from the weighting method to those of the Tchebycheff method at any point on the Pareto hyper‐surface. If, under some conditions, the points obtained from the two methods are identical or not, a local convexity or its lack can be assumed at that location and in its immediate neighbourhood. A numerical example is included. Copyright © 2002 John Wiley & Sons, Ltd.     

A unified approach to multiple‐set canonical correlation analysis and principal components analysis

Multiple‐set canonical correlation analysis and principal components analysis are popular data reduction techniques in various fields, including psychology. Both techniques aim to extract a series of weighted composites or components of observed variables for the purpose of data reduction. However, their objectives of performing data reduction are different. Multiple‐set canonical correlation analysis focuses on describing the association among several sets of variables through data reduction, whereas principal components analysis concentrates on explaining the maximum variance of a single set of variables. In this paper, we provide a unified framework that combines these seemingly incompatible techniques. The proposed approach embraces the two techniques as special cases. More importantly, it permits a compromise between the techniques in yielding solutions. For instance, we may obtain components in such a way that they maximize the association among multiple data sets, while also accounting for the variance of each data set. We develop a single optimization function for parameter estimation, which is a weighted sum of two criteria for multiple‐set canonical correlation analysis and principal components analysis. We minimize this function analytically. We conduct simulation studies to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of functional neuroimaging data to illustrate its empirical usefulness.     

A parameter analysis method for the weight‐set to satisfy preference orders of alternatives in additive multi‐criteria value models

In the context of the additive multi‐criteria value model, this paper investigates how the set of criteria weights (weight‐set hereafter) can be determined according to the preference orders of alternatives given by the decision maker. A construction method is proposed for the weight‐set for different intervals of β, where β is a differential amount of value between the preference information on two alternatives. The results of this paper are important for sensitivity analysis in multi‐criteria decision making (MCDM) problems and multi‐criteria group decision analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimal selection of E‐learning websites using multiattribute decision‐making approaches

A computational quantitative model based on weighted Euclidean distance‐based approximation and complex proportional assessment has been developed for the evaluation, selection, and ranking of various E‐learning websites in ascending or descending order based on their Euclidean distance value from the optimal website. The E‐learning website with rank 1 is considered the optimal selection on the particular dataset under consideration. The problem of the E‐learning website Selection, Evaluation and Ranking is modeled as a multiattribute decision‐making problem in which various interrelated attributes collectively termed as ranking criteria are identified to make the evaluation of available alternatives. In this research, 5 most popular E‐learning websites related to the C programming language for the software development have been considered to show the utility of developed model. Further, the concept of methodology validation strengthens this research by comparing the obtained results with the existing multiattribute decision‐making approach as analytical hierarchy process method.     

Distance‐Based Measures of Specialization and Balance in Multi‐Criteria: A DEA‐Integrated Method

Decision makers pursuing several distinct goals may differ in their behaviour. Some try to achieve all goals simultaneously, whereas others concentrate on only certain ones and are therefore specialized. We present a new method of measuring the specialization degree with respect to a given standard balanced relation of the goals. The measurement is integrated into data envelopment analysis (DEA) methodology, uses a non‐compensative distance measure, permits optional convex polyhedral cones as predefined balance sets and guarantees feasible solutions. Geometrically, it is based on the angle between the balance cone and the ray through the point of investigation. The proposed models can be used to support and complement the usual effectiveness and efficiency analysis in multi‐criteria decision analysis and DEA. They are motivated and illustrated by a simple numerical example, compared with an alternative method recently published and applied to real data from the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

Multi‐criteria minisum facility location problems

This paper deals with the multi‐criteria Weber (minisum) problem. Several new solution concepts are introduced, related to ideas of equilibrium between the different aspects covered by the objectives. Structural results are presented characterizing the proposed solution sets and showing the relationships existing with the efficient sets in the point objective location problem. Also, algorithms to solve this problem for the polyhedral norm case are developed. The proposed approach is more general than previously studied on multi‐criteria location problems and is particularly applicable in the real world when different scenarios must be compared. Copyright © 1999 John Wiley & Sons, Ltd.     

Constructing robust crew schedules with bicriteria optimization

Optimization‐based computer systems are used by many airlines to solve crew planning problems by constructing minimal cost tours of duty. However, today airlines do not only require cost effective solutions, but are also very interested in robust solutions. A more robust solution is understood to be one where disruptions in the schedule (due to delays) are less likely to be propagated into the future, causing delays of subsequent flights. Current scheduling systems based solely on cost do not automatically provide robust solutions. These considerations lead to a multiobjective framework, as the maximization of robustness will be in conflict with the minimization of cost. For example crew changing aircraft within a duty period is discouraged if inadequate ground time is provided. We develop a bicriteria optimization framework to generate Pareto optimal schedules for the domestic airline. A Pareto optimal schedule is one which does not allow an improvement in cost and robustness at the same time. We developed a method to solve the bicriteria problem, implemented it and tested it with actual airline data. Our results show that considerable gain in robustness can be achieved with a small increase in cost. The additional cost is mainly due to an increase in overnights, which allows for a reduction of the number of aircraft changes. Copyright © 2003 John Wiley & Sons, Ltd.     

Young children's understanding of the context‐relativity of normative rules in conventional games

We investigated young children's awareness of the context‐relative rule structure of simple games. Two contexts were established in the form of spatial locations. Familiar objects were used in their conventional way at location 1, but acquired specific functions in a rule game at location 2. A third party then performed the conventional act at either of the two locations, constituting a mistake at location 2 (experimental condition), but appropriate at location 1 (control condition). Three‐year‐olds (but not 2‐year‐olds) systematically distinguished the two conditions, spontaneously intervening with normative protest against the third party act in the experimental, but not in the control condition. Young children thus understand context‐specific rules even when the context marking is non‐linguistic. These results are discussed in the broader context of the development of social cognition and cultural learning.     

A Topsis‐Based Approach for Prioritized Aggregation in Multi‐Criteria Decision‐Making Problems

Aggregation in multi‐criteria decision‐making environments is a process of combining the values of a set of attributes into one representative value for the entire set of attributes. Many aggregation methods—ranging from the simple averaging approach to more sophisticated methods, such as ordered weighted averaging—have been applied in previous research. One challenge in aggregation arises in special cases of prioritized aggregation, wherein the prioritized relationships between attributes must be considered during aggregation. This paper presents a new approach to aggregating attributes with prioritized relationships. First, an overview of past research is conducted to identify different aggregation methods, classes and properties. Next, the concept of prioritized aggregation is explained in detail. A prioritized aggregation method utilizing the technique of order preference by similarity to ideal solution is then presented. Subsequently, the presented prioritized aggregation method is applied on an actual case study. According to the results, the aggregation method presented in this paper is, through the application of technique of order preference by similarity to ideal solution, capable of quantifying and considering the prioritized relationship between a set of attributes undergoing aggregation. Finally, conclusions are stated, and a discussion describing future work pertinent to this paper is presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Pareto‐Optimal Predictor Composite Formation: A complementary approach to alleviating the selection quality/adverse impact dilemma

In our rejoinder to the comments of Kehoe (this issue) and Potosky, Bobko and Roth (this issue) we emphasize that our proposal on Pareto‐optimal predictor composite formation is a complementary and not a competitive alternative for reducing the tension between selection quality and adverse impact. Our work addresses the decisions to be made once one has decided to use a predictor composite. We also further clarify the basic features of Pareto‐optimal tradeoffs and Pareto‐optimal composites within the context of personnel selection. In particular, we indicate that Pareto‐optimal tradeoffs between validity and adverse impact emerge because these goals are different and not because of any dualism between them.     

Surrogate‐assisted multicriteria optimization: Complexities,prospective solutions,and business case

Complexity in solving real‐world multicriteria optimization problems often stems from the fact that complex, expensive, and/or time‐consuming simulation tools or physical experiments are used to evaluate solutions to a problem. In such settings, it is common to use efficient computational models, often known as surrogates or metamodels, to approximate the outcome (objective or constraint function value) of a simulation or physical experiment. The presence of multiple objective functions poses an additional layer of complexity for surrogate‐assisted optimization. For example, complexities may relate to the appropriate selection of metamodels for the individual objective functions, extensive training time of surrogate models, or the optimal use of many‐core computers to approximate efficiently multiple objectives simultaneously. Thinking out of the box, complexity can also be shifted from approximating the individual objective functions to approximating the entire Pareto front. This leads to further complexities, namely, how to validate statistically and apply the techniques developed to real‐world problems. In this paper, we discuss emerging complexity‐related topics in surrogate‐assisted multicriteria optimization that may not be prevalent in nonsurrogate‐assisted single‐objective optimization. These complexities are motivated using several real‐world problems in which the authors were involved. We then discuss several promising future research directions and prospective solutions to tackle emerging complexities in surrogate‐assisted multicriteria optimization. Finally, we provide insights from an industrial point of view into how surrogate‐assisted multicriteria optimization techniques can be developed and applied within a collaborative business environment to tackle real‐world problems.     

Evaluating Fund Performance by Compromise Programming with Linear‐Quadratic Composite Metric: An Actual Case on The CaixaBank in Spain

This paper proposes an additive measure on the basis of compromise programming to evaluate fund performance from multiple criteria, of which the most usual are profitability and risk. This proposal is motivated by the fact that compromise programming is a sound decision support model to obtain scores of alternatives by minimizing weighted distances to an ideal point, the weights reflecting the investor's preferences for the criteria. To define the distance objective function, the linear‐quadratic composite metric is used, which combines advantages of linear and non‐linear objective functions. A critical advantage of compromise programming for fund performance evaluation is that the model can be extended to more than two financial criteria while other measures currently used (either ratio‐based or leverage‐based measures) only consider two criteria, say, profitability and risk. In the application, three investor's profiles are defined, which involve different weighting systems and lead to different fund rankings. These rankings are compared with domination relationships, the latter formulating if a fund is dominated or non‐dominated by convex combinations of other funds. Numerical tables are provided with data, computational process and results, which are analysed. Copyright © 2012 John Wiley & Sons, Ltd.     

Fourier‐Motzkin elimination method in MOLP problems

This paper introduces a method for finding the non‐dominated set in criterion space of an MOLP problem based on Fourier–Motzkin Elimination. The method incorporates a procedure that eliminates decision variables from constraints and transforms a feasible region from decision space to criterion space. While avoiding characterization of the efficient set in decision space, it finds the non‐dominated set in criterion space. After the decision maker's most preferred criterion vector has been identified, the corresponding efficient solutions can be found by backward substitution. The method was implemented in MATLAB using a tabular form and computational experiments were conducted. The results indicate that although the method performs well for relatively small problems, it can be computationally intensive for larger problems. Nonetheless, the method is intuitively appealing and it provides useful insights into the geometry and theory of MOLP. As such it makes it a valuable educational tool for students of multi‐criteria decision analysis. Copyright © 2003 John Wiley & Sons, Ltd.     

MULTI‐DIMENSIONAL CONSEQUENTIALISM

This article introduces and explores a distinction between multi‐dimensional and one‐dimensional consequentialist moral theories. One‐dimensional consequentialists believe that an act's deontic status depends on just one aspect of the act, such as the sum total of wellbeing it produces, or the sum total of priority‐ or equality‐adjusted wellbeing. Multi‐dimensional consequentialists believe that an act's deontic status depends on more than one aspect. They may, for instance, believe that the sum total of wellbeing produced by an act and the degree to which the wellbeing is equally distributed in the population affect the act's deontic status independently of each other. These two aspects cannot be reduced into any single (composite) aspect. Wellbeing and equality are two separate considerations that cannot be merged into some novel entity that accurately reflects both intuitions. On the multi‐dimensional view I defend, such clashes between separate aspects are irresolvable and are best accounted for by claiming that moral rightness and wrongness are non‐binary concepts. Some acts are, literally speaking, a little bit right (because they maximise wellbeing) and a little bit wrong (because they do not maximise equality). 1     

Validity and Adverse Impact Potential of Predictor Composite Formation

Previous research on the validity and adverse impact (AI) of predictor composite formation focused on the merits of regression‐based or ad hoc composites. We argue for a broader focus. Ad hoc chosen composites are usually not Pareto‐optimal, whereas the regression‐based composite represents only one element from the total set of Pareto‐optimal composites and can, therefore, provide only limited information on the potential for validity and AI reduction of forming predictor composites when both validity and AI are of concern. In that case, other Pareto‐optimal composites may provide a better benchmark to decide on the merits of the predictor composite formation. We summarize a method to determine the set of Pareto‐optimal composites and apply the method to a representative collection of selection predictors. The application shows that the assessment of the AI and validity of predictor composite formation can differ substantially from the one arrived at when considering only regression‐based composites.     

Triple Bottomline Many‐Objective‐Based Decision Making for a Land Use Management Problem

A land use many‐objective optimization problem for a 1500‐ha farm with 315 paddocks was formulated with 14 objectives (maximizing sawlog production, pulpwood production, milksolids, beef, sheep meat, wool, carbon sequestration, water production, income and Earnings Before Interest and Tax; and minimizing costs, nitrate leaching, phosphorus loss and sedimentation). This was solved using a modified Reference‐point‐based Non‐dominated Sorting Genetic Algorithm II augmented by simulated epigenetic operations. The search space had complex variable interactions and was based on economic data and several interoperating simulation models. The solution was an approximation of a Hyperspace Pareto Frontier (HPF), where each non‐dominated trade‐off point represented a set of land‐use management actions taken within a 10‐year period and their related management options, spanning a planning period of 50 years. A trade‐off analysis was achieved using Hyper‐Radial Visualization (HRV) by collapsing the HPF into a 2‐D visualization capability through an interactive virtual reality (VR)‐based method, thereby facilitating intuitive selection of a sound compromise solution dictated by the decision makers' preferences under uncertainty conditions. Four scenarios of the HRV were considered emphasizing economic, sedimentation and nitrate leaching aspects—giving rise to a triple bottomline (i.e. the economic, environmental and social complex, where the social aspect is represented by the preferences of the various stakeholders). Highlights of the proposed approach are the development of an innovative epigenetics‐based multi‐objective optimizer, uncertainty incorporation in the search space data and decision making on a multi‐dimensional space through a VR‐simulation‐based visual steering process controlled at its core by a multi‐criterion decision making‐based process. This approach has widespread applicability to many other ‘wicked’ societal problem‐solving tasks. Copyright © 2014 John Wiley & Sons, Ltd.     

Comparing graphic and symbolic classification in interactive multiobjective optimization

In interactive multiobjective optimization systems, the classification of objective functions is a convenient way to direct the solution process in order to search for new, more satisfactory, solutions in the set of Pareto optimal solutions. Classification means that the decision maker assigns the objective functions into classes depending on what kind of changes in their values (in relation to the current values) are desirable. Here we study the role of user interfaces in implementing classification in multiobjective optimization software and how classification should be realized. In this way, we want to pay attention to the usability of multiobjective optimization software. Typically, this topic has not been of interest in the multiobjective optimization literature. However, usability aspect is important because in interactive classification‐based multiobjective optimization methods, the classification is the core of the solution process. We can say that the more convenient the classification is, the more efficient the system or the method is and the better it supports the work of the decision maker. We report experiments with two classification options, graphic and symbolic ones, which are used in connection with an interactive multiobjective optimization system WWW‐NIMBUS. The ideas and conclusions given are applicable for other interactive classification‐based method, as well. Copyright © 2005 John Wiley & Sons, Ltd.