Value differences using second-order distributions

Most decision models for handling vague and imprecise information are unnecessarily restrictive since they do not admit for discrimination between different beliefs in different values. This is true for classical utility theory as well as for the various interval methods that have prevailed. To allow for more refined estimates, we suggest a framework designed for evaluating decision situations considering beliefs in sets of epistemically possible utility and probability functions, as well as relations between them. The various beliefs are expressed using different kinds of belief distributions. We show that the use of such distributions allows for representation principles not requiring too hard data aggregation, but still admitting efficient evaluation of decision situations.     

A lexicographical compromise method for multiple criteria group decision problems with imprecise information

This paper addresses multiple criteria group decision making problems where each group member offers imprecise information on his/her preferences about the criteria. In particular we study the inclusion of this partial information in the decision problem when the individuals’ preferences do not provide a vector of common criteria weights and a compromise preference vector of weights has to be determined as part of the decision process in order to evaluate a finite set of alternatives. We present a method where the compromise is defined by the lexicographical minimization of the maximum disagreement between the value assigned to the alternatives by the group members and the evaluation induced by the compromise weights.     

Sequential incorporation of imprecise information in multiple criteria decision processes

In this paper we deal with multicriteria decision processes and develop tools that permit to ease the task of analysing such models. We provide a methodology to sequentially incorporate imprecise preference information which is given by means of general linear relations in the weighting coefficients. The results presented allow us to evaluate the quality of the information supplied and can be used to reduce the number of irrelevant alternatives to be presented to the decision maker (DM). Several examples based on multiple criteria linear programming illustrate the results of the paper.     

Fuzzy stochastic linear programming: Survey and future research directions

In many real-life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain. Although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist, there are no reliable transformations between them. This calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models. Fuzzy stochastic linear programming is an attempt to fulfill this need. It deals with modelling and problem solving issues related to situations where randomness and fuzziness co-occur in a linear programming framework. In this paper we provide a survey of the essential elements, methods and algorithms for this class of linear programming problems along with promising research directions. Being a survey, the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest.     

Finite approximations of data-based decision problems under imprecise probabilities

In decision theory under imprecise probabilities, discretizations are a crucial topic because many applications involve infinite sets whereas most procedures in the theory of imprecise probabilities can only be calculated for finite sets so far. The present paper develops a method for discretizing sample spaces in data-based decision theory under imprecise probabilities. The proposed method turns an original decision problem into a discretized decision problem. It is shown that any solution of the discretized decision problem approximately solves the original problem.In doing so, it is pointed out that the commonly used method of natural extension can be most instable. A way to avoid this instability is presented which is sufficient for the purpose of the paper.     

The fuzzy decision problem: An approach to the problem of testing statistical hypotheses with fuzzy information

This paper is devoted to the problems of testing statistical hypotheses about an experiment, when the available information from its sampling is `vague'. When the information supplied by the experimental sampling is exact, the problems of testing statistical hypotheses about the experiment can be regarded as a particular statistical decision problem. In addition, decision procedures may be used in problems of testing hypotheses.In a similar manner, the problem of testing statistical hypotheses about an experiment when the available sample information is vague, is approached in this paper as a particular fuzzy decision problem (as defined by Tanaka, Okuda and Asai). This approach assumes that the previous information about the experiment can be expressed by means of certain conditional probabilistic information, whereas the present information about it can be expressed by means of fuzzy information. The preceding framework allows us to extend the notion of risk function and some nonfuzzy decision procedures to the fuzzy case, and particularize them to the problem of testing.Finally, several illustrative examples are presented.     

Additive aggregation with variable interdependent parameters: the VIP analysis software

We consider the aggregation of multicriteria performances by means of an additive value function under imprecise information. The problem addressed here is the way an analysis may be conducted when the decision makers are not able to (or do not wish to) fix precise values for the importance parameters. These parameters can be seen as interdependent variables that may take several values subject to constraints. Firstly, we briefly classify some existing approaches to deal with this problem. We argue that they complement each other, each one having its merits and shortcomings. Then, we present a new decision support software—VIP analysis—which incorporates approaches belonging to different classes. It proposes a methodology of analysis based on the progressive reduction of the number of alternatives, introducing a concept of tolerance that lets the decision makers use some of the approaches in a more flexible manner.     

A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts

Group decision making is a type of decision problem in which multiple experts acting collectively, analyze problems, evaluate alternatives, and select a solution from a collection of alternatives. As the natural language is the standard representation of those concepts that humans use for communication, it seems natural that they use words (linguistic terms) instead of numerical values to provide their opinions. However, while linguistic information is readily available, it is not operational and thus it has to be made usable though expressing it in terms of information granules. To do so, Granular Computing, which has emerged as a unified and coherent framework of designing, processing, and interpretation of information granules, can be used. The aim of this paper is to present an information granulation of the linguistic information used in group decision making problems defined in heterogeneous contexts, i.e., where the experts have associated importance degrees reflecting their ability to handle the problem. The granulation of the linguistic terms is formulated as an optimization problem, solved by using the particle swarm optimization, in which a performance index is maximized by a suitable mapping of the linguistic terms on information granules formalized as sets. This performance index is expressed as a weighted aggregation of the individual consistency achieved by each expert.     

Environmental impact assessment using the evidential reasoning approach

Environmental impact assessment (EIA) problems are often characterised by a large number of identified environmental factors that are qualitative in nature and can only be assessed on the basis of human judgments, which inevitably involve various types of uncertainties such as ignorance and fuzziness. So, EIA problems need to be modelled and analysed using methods that can handle uncertainties. The evidential reasoning (ER) approach provides such a modelling framework and analysis method. In this paper the ER approach will be applied to conduct EIA analysis for the first time. The environmental impact consequences are characterized by a set of assessment grades that are assumed to be collectively exhaustive and mutually exclusive. All assessment information, quantitative or qualitative, complete or incomplete, and precise or imprecise, is modelled using a unified framework of a belief structure. The original ER approach with a recursive ER algorithm will be introduced and a new analytical ER algorithm will be investigated which provides a means for using the ER approach in decision situations where an explicit ER aggregation function is needed such as in optimisation problems. The ER approach will be used to aggregate multiple environmental factors, resulting in an aggregated distributed assessment for each alternative policy. A numerical example and its modified version are studied to illustrate the detailed implementation process of the ER approach and demonstrate its potential applications in EIA.     

PROMETHEE-MD-2T method for project selection

Project selection is a real problem of multicriteria group decision making (MCGDM) where each decision maker expresses his/her preferences depending on the nature of the alternatives and on his/her own knowledge over them. Thus, information, as much quantitative as qualitative, coexists. The traditional methods of MCGDM developed for project selection usually discriminates in favour of quantitative information at the expense of qualitative information, and this is due to the capability to integrate this first type of information inside their procedure. In this article, two new multicriteria 2-tuple group decision methods called “Preference Ranking Organisation Method for Enrichment Evaluation Multi Decision maker 2-Tuple-I and II” (PROMETHEE-MD-2T-I and II) are presented. They are able to integrate inside their procedure both quantitative and qualitative information in an uncertain context. This has been performed by integrating a 2-tuple linguistic representation model dealing with non-homogeneous and imprecise information data made up by valued intervals, numerical and linguistic values into the aggregation operators of Promethee methods. Although they have been developed for project selection problems, these proposed methods can be applied to all kinds of decision-making problems with heterogeneous and multigranular information.     

Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider multicriteria decision problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be uncertain and/or imprecise and is provided by one or several experts. We model this evaluation as a basic belief assignment (BBA). In order to compare the different pairs of alternatives according to each criterion, the concept of first belief dominance is proposed. Additionally, criteria weights are also expressed by means of a BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical example.     

A QFD-based fuzzy MCDM approach for supplier selection

Supplier selection is a highly important multi-criteria group decision making problem, which requires a trade-off between multiple criteria exhibiting vagueness and imprecision with the involvement of a group of experts. In this paper, a fuzzy multi-criteria group decision making approach that makes use of the quality function deployment (QFD) concept is developed for supplier selection process. The proposed methodology initially identifies the features that the purchased product should possess in order to satisfy the company’s needs, and then it seeks to establish the relevant supplier assessment criteria. Moreover, the proposed algorithm enables to consider the impacts of inner dependence among supplier assessment criteria. The upper and the lower bounds of the weights of supplier assessment criteria and ratings of suppliers are computed by using the fuzzy weighted average (FWA) method. The FWA method allows for the fusion of imprecise and subjective information expressed as linguistic variables or fuzzy numbers. The method produces less imprecise and more realistic overall desirability levels, and thus it rectifies the problem of loss of information. A fuzzy number ranking method that is based on area measurement is used to obtain the final ranking of suppliers. The computational procedure of the proposed framework is illustrated through a supplier selection problem reported in an earlier study.     

Cost,Revenue and Profit Efficiency Models in Generalized Fuzzy Data Envelopment Analysis

One of the most important information given by data envelopment analysis models is the cost, revenue and profit efficiency of decision making units (DMUs). Cost efficiency is defined as the ratio of minimum costs to current costs, while revenue efficiency is defined as the ratio of maximum revenue to current revenue of the DMU. This paper presents a framework where data envelopment analysis (DEA) is used to measure cost, revenue and profit efficiency with fuzzy data. In such cases, the classical models cannot be used, because input and output data appear in the form of ranges. When the data are fuzzy, the cost, revenue and profit efficiency measures calculated from the data should be uncertain as well. Fuzzy DEA models emerge as another class of DEA models to account for imprecise inputs and outputs for DMUs. Although several approaches for solving fuzzy DEA models have been developed, numerous deficiencies including the $\alpha$-cut approaches and types of fuzzy numbers must still be improved. This scheme embraces evaluation method based on vector for proposed fuzzy model. This paper proposes generalized cost, revenue and profit efficiency models in fuzzy data envelopment analysis. The practical application of these models is illustrated by a numerical example.     

Dealing with imprecise information in group multicriteria decisions: a methodology and a GDSS architecture

This paper outlines a distributed GDSS suitable to be used over the Internet, based on the VIP Analysis methodology and software. VIP Analysis incorporates complementary approaches to deal with the aggregation of multicriteria performances by means of an additive value function under imprecise information. This proposed GDSS intends to support a decision panel forming a democratic decision unit, whose members wish to reach a final decision in a choice problem, based on consensus or on some majority rule. Its purpose is not to impose an aggregated model from the individual ones. Rather, the GDSS is designed to reflect to each member the consequences of his/her inputs, confronting them with analogous reflections of the group members' inputs. We propose aggregation procedures to provide a reflection of the group's inputs to each of its members, and an architecture for a GDSS implementing these procedures.     

Solution of a possibilistic multiobjective linear programming problem

The estimate of the parameters which define a conventional multiobjective decision making model is a difficult task. Normally they are either given by the Decision Maker who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from the past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets and several fuzzy approaches to multiobjective programming have been developed 1, 9, 10, 11. The fuzziness of the parameters gives rise to a problem whose solution will also be fuzzy, see 2, 3, and which is defined by its possibility distribution. Once the possibility distribution of the solution has been obtained, if the decision maker wants more precise information with respect to the decision vector, then we can pose and solve a new problem. In this case we try to find a decision vector, which approximates as much as possible the fuzzy objectives to the fuzzy solution previously obtained. In order to solve this problem we shall develop two different models from the initial solution and based on Goal Programming: an Interval Goal Programming Problem if we define the relation “as accurate as possible” based on the expected intervals of fuzzy numbers, as we showed in [4], and an ordinary Goal Programming based on the expected values of the fuzzy numbers that defined the goals. Finally, we construct algorithms that implement the above mentioned solution method. Our approach will be illustrated by means of a numerical example.     

Fuzzy random regression based multi-attribute evaluation and its application to oil palm fruit grading

Multi-attribute decision-making is usually concerned with weighting alternatives, thereby requiring weight information for decision attributes from a decision maker. However, the assignment of an attribute’s weight is sometimes difficult, and may vary from one decision maker to another. Additionally, imprecision and vagueness may affect each judgment in the decision-making process. That is, in a real application, various statistical data may be imprecise or linguistically as well as numerically vague. Given this coexistence of random and fuzzy information, the data cannot be adequately treated by simply using the formalism of random variables. To address this problem, fuzzy random variables are introduced as an integral component of regression models. Thus, in this paper, we proposed a fuzzy random multi-attribute evaluation model with confidence intervals using expectations and variances of fuzzy random variables. The proposed model is applied to oil palm fruit grading, as the quality inspection process for fruits requires a method to ensure product quality. We include simulation results and highlight the advantage of the proposed method in handling the existence of fuzzy random information.     

Computing upper and lower bounds in interval decision trees

This article presents algorithms for computing optima in decision trees with imprecise probabilities and utilities. In tree models involving uncertainty expressed as intervals and/or relations, it is necessary for the evaluation to compute the upper and lower bounds of the expected values. Already in its simplest form, computing a maximum of expectancies leads to quadratic programming (QP) problems. Unfortunately, standard optimization methods based on QP (and BLP – bilinear programming) are too slow for the evaluation of decision trees in computer tools with interactive response times. Needless to say, the problems with computational complexity are even more emphasized in multi-linear programming (MLP) problems arising from multi-level decision trees. Since standard techniques are not particularly useful for these purposes, other, non-standard algorithms must be used. The algorithms presented here enable user interaction in decision tools and are equally applicable to all multi-linear programming problems sharing the same structure as a decision tree.     

A logical characterization of coherence for imprecise probabilities

Whilst supported by compelling arguments, the representation of uncertainty by means of (subjective) probability does not enjoy a unanimous consensus. A substantial part of the relevant criticisms point to its alleged inadequacy for representing ignorance as opposed to uncertainty. The purpose of this paper is to show how a strong justification for taking belief as probability, namely the Dutch Book argument, can be extended naturally so as to provide a logical characterization of coherence for imprecise probability, a framework which is widely believed to accommodate some fundamental features of reasoning under ignorance. The appropriate logic for our purposes is an algebraizable logic whose equivalent algebraic semantics is a variety of MV-algebras with an additional internal unary operation representing upper probability (these algebras will be called UMV-algebras).