Ranking alternatives using fuzzy numbers

We investigate the problem of employing expert opinion to rank alternatives across a set of criteria. The experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue's fuzzy ranking. This leads to a partition of the alternatives into sets H₁, H₂,… where H₁ contains the highest ranked issues, H₂ has all the second highest ranked alternatives, etc. The total ranking process is shown to possess a number of important properties. An example is presented to illustrate the method.     

Ranking residential properties by a multicriteria single price model

This study describes an application of the multicriteria single price model (Ballestero) to the ranking of alternatives. By a generalization of the original model, the equilibrium set of alternatives can be characterized from the viewpoints, respectively, of the demander and the supplier, and from that the efficiency index can be calculated. We demonstrate how, in a state of equilibrium, the two viewpoints result inevitably in inverse orders of ranking. In contrast with other proposals for full ranking of alternatives, the method used in the present study (i) assumes a moderate attitude on the part of the decision maker towards risk, with a robust axiomatic basis; (ii) assigns weights to the criteria independently of which alternative is being evaluated and the attitude (optimistic or pessimistic) of the decision maker; (iii) produces a cardinal hierarchy of the alternatives and not just an ordinal one. The model is illustrated by a sample of residential properties in the city of Valencia, Spain.     

Interactive multicriteria procedure exploiting a knowledge-based module to select electricity production alternatives: The CASTART system

For some decisional problems, it is difficult to have a set of potential alternatives a priori. This is true, for instance, when the object of the decision is dependent on an external environment, as in the case of disturbance situations in industrial processes. Computer assistance must support, first, the elaboration of a potential alternatives set and, second, the decision-making activity which will determine the decision elements leading to recovery from the disturbance. This paper deals with such a situation encountered in an electricity production context. An interactive multicriteria procedure exploiting a knowledge-based module to select electricity production alternatives is presented. Section 1 introduces the electricity production-consumption framework and identifies the Spinning Reserve Mobilization (SRM) function. Next, our contribution to the systematization and the experimentation of a decision-making process which supports the dispatcher confronted with an SRM situation is described. The decision-making process includes an intelligent module and a decisional procedure. The module and the procedure, and their interconnections, are developed in Section 2. Section 3 describes the DSS called CASTART which implements this process. An overview of DSS functionalities is given, as well as the results obtained by a dispatcher working in cooperation with the system, in the case of a simulated production incident.     

The multicriteria big cube small cube method

In this paper we propose the big cube small cube (BCSC) technique for multicriteria optimization problems. The output of our algorithm results in a set which consists of epsilon efficient solutions and which contains all efficient solutions.     

Ranking multi-criterion river basin planning alternatives using fuzzy numbers

The methodology proposed by Anand Raj and Nagesh Kumar [5] to rank the river basin planning and development alternatives under multi-criterion environment using fuzzy numbers is applied to a case study. The purpose is to find the most suitable planning of reservoirs with their associated purposes aimed at the development of one of the major peninsular river basins (Krishna river basin) in India. A set of 7 alternative systems with 8 main objectives, which are further subdivided into 18 criteria, are considered for ordering or ranking them employing the opinion (preference structure) of three experts: an acadamician, a field engineer and an official from Ministry of Water Resources, using fuzzy numbers. The fuzzy weights (w_i) of alternatives (A_i) are computed using standard fuzzy arithmetic. The concepts of maximizing set and minimizing set are introduced to decide total utility or order value of each of the alternatives.     

An external penalty-type method for multicriteria

We propose an extension of the classical real-valued external penalty method to the multicriteria optimization setting. As its single objective counterpart, it also requires an external penalty function for the constraint set, as well as an exogenous divergent sequence of nonnegative real numbers, the so-called penalty parameters, but, differently from the scalar procedure, the vector-valued method uses an auxiliary function, which can be chosen among large classes of “monotonic” real-valued mappings. We analyze the properties of the auxiliary functions in those classes and exhibit some examples. The convergence results are similar to those of the scalar-valued method, and depending on the kind of auxiliary function used in the implementation, under standard assumptions, the generated infeasible sequences converge to weak Pareto or Pareto optimal points. We also propose an implementable local version of the external penalization method and study its convergence results.     

An approximate solution of linear multicriteria control problems through the multicriteria simplex method

This paper deals with multicriteria, linear, continuous optimal control problems through the application of the multicriteria simplex method. Since the system is linear, the state variables at a given time can be expressed in terms of definite integrals. Upon using standard numerical integration formulas, the definite integrals are approximately expressed by means of weighted sums of the integrands. Then, after introducing some suitable auxiliary variables, approximate linear multicriteria programming problems are formulated. Illustrative numerical examples are provided to indicate the efficiency of the proposed method.The author is indebted to Professor Y. Sawaragi of Kyoto University for his constant encouragement. The author also wishes to thank T. Suwa for his cooperation in preparing the programming for the digital computer.     

Sensitivity analysis for the ELECTRE multicriteria method

In this paper, we develop sensitivity ranges for parameters used in the ELECTRE I multicriteria decision method. Parameters studied include criteria weights as well as significance thresholds. For these parameters, intervals are determined in which parameter changes will not affect the set of non-outranked alternatives. The approach developed is illustrated by a numerical example and possible applications are discussed.

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Zusammenfassung In dieser Arbeit wird ein Ansatz zur Sensitivitätsanalyse für das Multikriteria-Entscheidungsverfahren ELECTRE I vorgestellt. Für die Gewichte der Kriterien sowie die im Verfahren verwendeten Signifikanzschwellen werden jeweils Intervalle konstruiert, innerhalb derer Parameteränderungen keine Änderungen im Ergebnis des Verfahrens bewirken. Die entwickelte Sensitivitätsanalyse wird anhand eines Beispiels illustriert und mögliche Anwendungen des Konzeptes werden aufgezeigt.

     

Solving large-scale multicriteria problems by the decomposition method

The problem of constructing a set of effective vector estimates under a large number of partial performance criteria is examined.     

Ranking players by “power” in weighted voting systems: The multi-relation method

A new quantitative definition of the “power index” is proposed for a voter in weighted voting systems (WV-schemes), where the voters a priori do not have equal rights. The proposed “power index” is generated in a self-consistent manner from information about the number of times that a voter enters any winning coalition in the WV-scheme. Explicit formulas are derived for computing the “power index”. The discussion is illustrated with prototype examples. __________ Translated from Prikladnaya Matematika i Informatika, No. 25, pp. 81–98, 2007.     

The self regulation problem as an inexact steepest descent method for multicriteria optimization

In this paper we study an inexact steepest descent method for multicriteria optimization whose step-size comes with Armijo’s rule. We show that this method is well-defined. Moreover, by assuming the quasi-convexity of the multicriteria function, we prove full convergence of any generated sequence to a Pareto critical point. As an application, we offer a model for the Psychology’s self regulation problem, using a recent variational rationality approach.     

Business failure prediction using the UTADIS multicriteria analysis method

Business failure prediction is one of the most essential problems in the field of financial management. The research on developing quantitative business failure prediction models has been focused on building discriminant models to distinguish among failed and non-failed firms. Several researchers in this field have proposed multivariate statistical discrimination techniques. This paper explores the applicability of multicriteria analysis to predict business failure. Four preference disaggregation methods, namely the UTADIS method and three of its variants, are compared to three well-known multivariate statistical and econometric techniques, namely discriminant analysis, logit and probit analyses. A basic (learning) sample and a holdout (testing) sample are used to perform the comparison. Through this comparison, the relative performance of all the aforementioned methods is investigated regarding their discriminating and predicting ability.     

The set of all nondominated solutions in linear cases and a multicriteria simplex method

In this note we are interested in the properties of, and methods for locating the set of all nondominated solutions of multiple linear criteria defined over a polyhedron. We first show that the set of all dominated solutions is convex and that the set of all nondominated solutions is a subset of the convex hull of the nondominated extreme points. When the domination cone is polyhedral, we derive a necessary and sufficient condition for a point to be nondominated. The condition is stronger than that of Ref. [1] and enables us to give a simple proof that the set of all nondominated extreme points indeed is connected. In order to locate the entire set of all nondominated extreme points, we derive a generalized version of simplex method—multicriteria simplex method. In addition to some useful results, a necessary and sufficient condition for an extreme point to be nondominated is derived. Examples and computer experience are also given. Finally, we focus on how to generate the entire set of all nondominated solutions through the set of all nondominated extreme points. A decomposition theorem and some necessary and sufficient conditions for a face to be nondominated are derived. We then describe a systematic way to identify the entire set of all nondominated solutions. Through examples, we show that in fact our procedure is quite efficient.     

Regularized extragradient method in multicriteria control problems with inaccurate data

The class of Hilbert space multicriteria optimization problems considered in the paper includes control problems for various dynamical systems with lumped as well as distributed parameters. An equilibrium point is sought under the assumption that the criteria and their derivatives are known approximately. We use a regularized extragradient method and prove its convergence. As a sample application of the general theory, we consider a control problem for a parabolic equation with two criteria.     

Regularized extragradient method for solving parametric multicriteria equilibrium programming problem

A regularized extragradient method is designed for solving unstable multicriteria equilibrium programming problems. The convergence of the method is investigated, and a regularizing operator is constructed.     

The domination property in multicriteria optimization

The domination property applies in a multicriteria problem if each feasible alternative is either efficient or else is dominated by an efficient alternative. Sufficient conditions for the property to hold are presented. We show that if certain common conditions are met then proper efficiency implies the domination property and that under convexity conditions, efficiency, proper efficiency, and the domination property are equivalent.     

Interdisciplinary graduate programs: application of the MACBETH multicriteria method for assessing their performances

Annals of Operations Research - This paper, employing the multicriteria method of decision support MACBETH, proposes a model for the evaluation of interdisciplinary graduate programs, based on...     

Continuous extragradient method for a parametric multicriteria equilibrium programming problem

We consider a multicriteria equilibrium programming problem including, as special cases, the mathematical programming problem, the problem of finding a saddle point, the multicriteria problem of finding a Pareto point, the minimization problem with an equilibrium choice of an admissible set, etc. We suggest a continuous version of the extragradient method with prediction and analyze its convergence.