Abstract: We indicate that the problem of prioritized criteria arises in situations in which there exists a relationship between the criteria so that lack of satisfaction by the higher priority criteria cannot be readily compensated for by satisfaction by lower priority criteria. Typical of this situation is the relationship between safety and cost. We consider the problem of criteria aggregation in this environment. Central to our approach is the use of importance weights to enforce this prioritization imperative. We apply our use of priority based importance weights to the case where the scope of the criteria aggregation is an OWA type aggregation.

Abstract: Fuzzy relational equations play an important role in fuzzy set theory and fuzzy logic systems, from both of the theoretical and practical viewpoints. The notion of fuzzy relational equations is associated with the concept of "composition of binary relations." In this survey paper, fuzzy relational equations are studied in a general lattice-theoretic framework and classified into two basic categories according to the duality between the involved composite operations. Necessary and sufficient conditions for the solvability of fuzzy relational equations are discussed and solution sets are characterized by means of a root or crown system under some specific assumptions.

Abstract: We deal with the linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. We present a general approach to the situation the feasible set is described by an arbitrary linear interval system. Moreover, certain dependencies between the constraint matrix coefficients can be involved. As long as we are able to characterize the primal and dual solution set (the set of all possible primal and dual feasible solutions, respectively), the bounds of the objective function result from two nonlinear programming problems. We demonstrate our approach on various cases of the interval linear programming problem (with and without dependencies).

Abstract: The optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts for these kinds of problems will follow the concept of nondominated solution adopted in the multiobjective programming problems. In order to consider the differentiation of fuzzy-valued functions, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the (strongly, weakly) Pareto optimal solutions are elicited naturally by introducing the Lagrange multipliers.

Abstract: This paper reviews the credibilistic portfolio selection approaches which deal with fuzzy portfolio selection problem based on credibility measure. The reason for choosing credibility measure is given. Several mathematical definitions of risk of an investment in the portfolio are introduced. Some credibilistic portfolio selection models are presented, including mean-risk model, mean-variance model, mean-semivariance model, credibility maximization model, ?-return maximization model, entropy optimization model and game models. A hybrid intelligent algorithm for solving the optimization models is documented. In addition, as extensions of credibilistic portfolio selection approaches, the paper also gives a brief review of some hybrid portfolio selection models.

Abstract: The main aim of this paper is to discuss Fuzzy Linear Matrix Equations (shown as FLME) of the form AXB = C for finding its fuzzy solutions. In this paper, the parametric form of the fuzzy linear system is used. Necessary and sufficient conditions for the existence of the set of fuzzy solutions are derived, and a numerical procedure for calculating the solutions is designed.

Abstract: Fuzzy Optimization models and methods has been one of the most and well studied topics inside the broad area of Soft Computing. Particularly relevant is the field of fuzzy linear programming (FLP). Its applications as well as practical realizations can be found in all the real world areas. As FLP problems constitute the basis for solving fuzzy optimization problems, in this paper a basic introduction to the main models and methods in FLP is presented and, as a whole, Linear Programming problems with fuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrix are analyzed. But fuzzy sets and systems based optimization methods do not end with FLP, and hence in order to solve more complex optimization problems, fuzzy sets based Meta-heuristics are considered, and two main operative approaches described. Provided that these techniques obtain efficient and/or effective solutions, we present a fuzzy rule based methodology for coordinating Meta-heuristics and in addition, to provide intelligence, we propose a process of extraction of the knowledge to conduct the coordination of the system.

Abstract: Two linguistic-based voting systems have been introduced in recent years, namely: Majority Judgement (Balinski and Laraki in http://ceco.polytechnique.fr/jugement-majoritaire.html , 2007a) and Range Voting (Smith in http://www.math.temple.edu/~wds/homepage/rangevote.pdf , 2000). The keys for them are aggregation procedures based on the median and the arithmetic mean of the grades assessed to the alternatives, respectively. In this paper a comprehensive framework based on centered OWA operators (Yager in Soft Comput 11:631---639, 2007a) and the 2-tuple model (Herrera and Martinez in IEEE Trans Fuzzy Syst 8:746---752, 2000) is provided to enclose such distinct approaches. In addition, we show how to avoid some drawbacks of Majority Judgement and Range Voting by means of the use of suitable aggregation functions.

Abstract: Since Song and Chissom (Fuzzy Set Syst 54:1---9, 1993a) first proposed the structure of fuzzy time series forecast, researchers have devoted themselves to related studies. Among these studies, Hwang et al. (Fuzzy Set Syst 100:217---228, 1998) revised Song and Chissom's method, and generated better forecasted results. In their method, however, several factors that affect the accuracy of forecast are not taken into consideration, such as levels of window base, length of interval, degrees of membership values, and the existence of outliers. Focusing on these factors, this study proposes an improved fuzzy time series forecasting method. The improved method can provide decision-makers with more precise forecasted values. Two numerical examples are employed to illustrate the proposed method, as well as to compare the forecasting accuracy of the proposed method with that of two fuzzy forecasting methods. The results of the comparison indicate that the proposed method produces more accurate forecasting results.

Abstract: In this paper, credibilistic logic is introduced as a new branch of uncertain logic system by explaining the truth value of fuzzy formula as credibility value. First, credibilistic truth value is introduced on the basis of fuzzy proposition and fuzzy formula, and the consistency between credibilistic logic and classical logic is proved on the basis of some important properties about truth values. Furthermore, a credibilistic modus ponens and a credibilistic modus tollens are presented. Finally, a comparison between credibilistic logic and possibilistic logic is given.

Abstract: This special issue encompasses five papers devoted to the recent developments in the field of Computing with Words and Decision Making. The issue originated from presentations at the “8th International FLINS Conference on Computational Intelligence in Decision and Control” that was held in Madrid, Spain, September 21–24th, 2008. Every paper for the special issue was reviewed by at least two referees and finally five papers were accepted according to the referees’ evaluations. This special issue is focused in Computing with Words and Decision Making. The submissions include two review and position papers on Computing with Words and Perceptual Computing. The first paper, “Historical Reflections and New Positions on Perceptual Computing”, by Mendel, traces the historical origins of perceptual computing. It also takes the position that interval type-2 fuzzy sets, and not type-1 fuzzy sets, should be used in perceptual computing. Finally, it proposes some testable guidelines for when a solution (e.g., perceptual computing) may be branded computing with words. The second paper, “Computing with Words in Decision Making: Foundations, Trends and Prospects”, reviews how the Computing with Words methodology has been used in the Decision Making field to create and enrich decision models in which the information that is provided and manipulated has a qualitative nature. Herrera et al. present an overview of the Computing with Words methodology and they

Abstract: This note discusses three types of solutions for a system of interval-valued fuzzy relational equations with max-T composition and illustrates their relations to the solutions of a system of fuzzy relational inequalities with max-T composition. It validates the major claims appeared in Fuzzy Optimization and Decision Making, 2 (2003) 41---60; 4 (2005) 331---349.

Abstract: In this paper, we extend Guo and Xia's necessary condition which has been presented by Guo and Xia (Fuzzy optimizat Decis Mak 5: 33---47, 2006) in order to study the finitely many constraints of fuzzy relation inequalities and optimize a linear objective function on this region which is defined by the fuzzy max---min operator. The new condition provides a means for removing the unnecessary paths resulting from Guo and Xia's paths. Also, an algorithm and two numerical examples are offered to abbreviate and illustrate the steps of the resolution process of the problem.

Abstract: An algorithm for the selection among n alternatives based on the evaluation of n (distinct) groups of persons according to the same m criteria is described. The evaluation of each person for each criterion is represented by a proportional ordinal 2-tuple and the overall opinion is aggregated by a pair of quantifier-guided ordered weighted averaging (OWA) aggregation and (floating) anchoring value-based ordered weighted averaging (AV-OWA) aggregation operators. An example is provided to illustrate the algorithm. The decision function of the algorithm is shown to be a Choquet integral of the associated function of two variables (corresponding to the two aggregation processes in the algorithm) which can be accomplished alternatively by a Choquet integral of Fubini type.

Abstract: Some properties of convergence for archimedean t-conorms and t-norms are investigated and a definition of independence for events, evaluated by a decomposable measure, is introduced. This definition generalizes the concept of independence provided by Kruse and Qiang for ?-additive fuzzy measures. Finally, we derive the two Borel---Cantelli lemmas in the context of the general framework considered.

Abstract: We first discuss the significant role that duality plays in many aggregation operations involving intuitionistic fuzzy subsets We then consider the extension to intuitionistic fuzzy subsets of a number of ideas from standard fuzzy subsets In particular we look at the measure of specificity We also look at the problem of alternative selection when decision criteria satisfaction is expressed using intuitionistic fuzzy subsets We introduce a decision paradigm called the method of least commitment We briefly look at the problem of defuzzification of intuitionistic fuzzy subsets

Abstract: The KKT optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts are proposed by defining an ordering relation on the class of all fuzzy numbers. Owing to this ordering relation being a partial ordering, the solution concepts proposed in this paper will follow from the similar solution concept, called Pareto optimal solution, in the conventional multiobjective programming problems. In order to consider the differentiation of fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers.

Abstract: This paper traces the historical origins of perceptual computing and credits Tong and Bonissone (IEEE Trans Syst Man Cybern 10:716---723, 1980) as being the first to originate it but under a different name. It also takes the position that interval type-2 fuzzy sets, and not type-1 fuzzy sets, should be used in perceptual computing. Finally, it proposes some testable guidelines for when a solution (e.g., perceptual computing) may be branded computing with words.

Abstract: Computing with Words (CW) methodology has been used in several different environments to narrow the differences between human reasoning and computing. As Decision Making is a typical human mental process, it seems natural to apply the CW methodology in order to create and enrich decision models in which the information that is provided and manipulated has a qualitative nature. In this paper we make a review of the developments of CW in decision making. We begin with an overview of the CW methodology and we explore different linguistic computational models that have been applied to the decision making field. Then we present an historical perspective of CW in decision making by examining the pioneer papers in the field along with its most recent applications. Finally, some current trends, open questions and prospects in the topic are pointed out.

Abstract: We introduce a novel linear order on every family of fuzzy numbers which satisfies the assumption that their modal values must be all different and must form a compact subset of $${\mathbb{R}}$$ . A distinct new feature is that our linear determined procedure employs the corresponding order of a class interval associated with a confidence measure which seems intuitively anticipated. It is worthy noting that although we start from an entirely different rationale, we introduce a fuzzy ordering which initially coincides with the one established earlier by Ramik and Rimanek. However, this fuzzy ordering does not apply when the supports of the fuzzy numbers overlap. In order to cover such cases we extent this initial fuzzy ordering to the "extended fuzzy order" (XFO). This new XFO method includes a possibility and a necessity measure which are compared with the widely accepted PD and NSD indices of D. Dubois and H. Prade. The comparison shows that our possibility and necessity measures comply better with our intuition.

Abstract: Fabric hand evaluation (FHE) is the main measure in textile material selection for fashion design and development. Fabric hand evaluation requires considering multiple evaluation aspects/criteria by a group of evaluators. Some fabric features can also be measured using instruments. The evaluation often uses linguistic terms in the weights of criteria, and the weights and judgments of evaluators. To support a FHE-based material selection, this study first develops a fabric hand-based textile material evaluation model. It then proposes a human-machine measure integrated fuzzy multi-criteria group decision-making method. A software tool is also developed, which implements the proposed method and is applied in fabric hand-based textile material evaluation.