Linguistic Quantifiers Research Articles

A fuzzy-stochastic OWA model for robust multi-criteria decision making

All realistic Multi-Criteria Decision Making (MCDM) problems face various kinds of uncertainty. Since the evaluations of alternatives with respect to the criteria are uncertain they will be assumed to have stochastic nature. To obtain the uncertain optimism degree of the decision maker fuzzy linguistic quantifiers will be used. Then a new approach for fuzzy-stochastic modeling of MCDM problems will be introduced by merging the stochastic and fuzzy approaches into the OWA operator. The results of the new approach, entitled FSOWA, give the expected value and the variance of the combined goodness measure for each alternative. Robust decision depends on the combined goodness measures of alternatives and also on the variations of these measures under uncertainty. In order to combine these two characteristics a composite goodness measure will be defined. The theoretical results will be illustrated in a watershed management problem. By using this measure will give more sensitive decisions to the stakeholders whose optimism degrees are different than that of the decision maker. FSOWA can be used for robust decision making on the competitive alternatives under uncertainty.

Linguistic summarization of time series using a fuzzy quantifier driven aggregation

We propose new types of linguistic summaries of time-series data that extend those proposed in our previous papers. The proposed summaries of time series refer to the summaries of trends identified here with straight line segments of a piecewise linear approximation of time series. We first show how to construct such an approximation. Then we employ a set of features (attributes) to characterize the trends such as the slope of the line segment, the goodness of approximation and the length of the trend. The derivation of a linguistic summary of a time series is then related to a linguistic quantifier driven aggregation of trends. For this purpose we employ the classic Zadeh's calculus of linguistically quantified propositions but, extending our previous works, with different t-norms in addition to the basic minimum. We show an application to the analysis of time-series data on daily quotations of an investment fund over an eight year period, present some interesting linguistic summaries obtained, and show results for different t-norms. The results are very promising.

Stochastic-fuzzy multi criteria decision making for robust water resources management

All realistic Multi Criteria Decision Making (MCDM) problems in water resources management face various kinds of uncertainty. In this study the evaluations of the alternatives with respect to the criteria will be assumed to be stochastic. Fuzzy linguistic quantifiers will be used to obtain the uncertain optimism degree of the Decision Maker (DM). A new approach for stochastic-fuzzy modeling of MCDM problems will be then introduced by merging the stochastic and fuzzy approaches into the Ordered Weighted Averaging (OWA) operator. The results of the new approach, entitled SFOWA, give the expected value and the variance of the combined goodness measure for each alternative, which are essential for robust decision making. In order to combine these two characteristics, a composite goodness measure will be defined. By using this measure the model will give more sensitive decisions to the stakeholders whose optimism degrees are different than that of the decision maker. The methodology will be illustrated by using a water resources management problem in the Central Tisza River in Hungary. Finally, SFOWA will be compared to other methods known from the literature to show its suitability for MCDM problems under uncertainty.

Prenex normal form in linguistic quantifiers modeled by Sugeno integrals

In this paper, Ying's prenex normal form theorem for linguistic quantifiers is strengthened so that it behaves like the one in the classical logic. This result further shows elegant logical properties of Ying's framework.

Applying 2-Tuple Multigranularity Linguistic Variables to Determine the Supply Performance in Dynamic Environment Based on Product-Oriented Strategy

Supply performance assessment is not only the core mechanism for supplier management, but also the major task for enterprises in implementing supply chain management. Consequently, supply chain system implementation should rely on the results of supply performance assessments to supervise suppliers. However, supply performance assessment is always concerned with numerous supply behaviors, and the information obtained from these supply behaviors often varies during the measurement timeframe. Because of uncertainty and inaccuracy, the numerical data are not appropriate for demonstrating dynamic information. This study employs 2-tuple multigranularity linguistic variables to deal with dynamic information appropriately and assess supply performance, and even permits decision makers to introduce different linguistic terms (linguistic scale) based on consideration of supply behavior characteristics. The 2-tuple multigranularity linguistic variables present the information via linguistic and numeric results, both where the linguistic result could express the variation of dynamic information appropriately, and where the numerical result could further reflect the difference based on the same linguistic result to improve the sensitivity. Additionally, this study also employs a fuzzy linguistic quantifier to simulate the decision criteria based on the product strategy to further obtain the modified linguistic ordered weighted averaging operator with maximum entropy that will be exercised to aggregate the 2-tuple linguistic information. This investigation provides decision makers with rapid access to complete, and integrated information on the practical supply performance of suppliers as appropriate.

「あいまいと感性」伝統工芸素材の感性評価とデータ解析

The objective of this paper is to visualize relationships between evaluated objects and evaluated items by means of a fuzzy correspondence analysis, wherein evaluation fluctuations are properly treated. Every relationship is expressed by a circle corresponding to each of objects and items which illustrates its relative fuzziness. A similarity is also proposed to express numerically these relationships. In order to deal with customers' fastidious demands with some evaluation items, OWA operator and the similarity are used to show the orders of objects depending on Linguistic Quantifiers. In this paper, traditional crafts in Ishikwa Prefecture are focused to show the effectiveness of these methods.

Linguistic quantifiers based on Choquet integrals

The introduction of linguistic quantifiers has provided an important tool to model a large number of issues in intelligent systems. Ying [M.S. Ying, Linguistic quantifiers modeled by Sugeno integrals, Artificial Intelligence 170 (2006) 581–606] recently introduced a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno’s integral. Representing linguistic quantifiers by fuzzy measures, this paper evaluates linguistic quantified propositions by the Choquet integral. Some elegant logical properties of linguistic quantifiers are derived within this approach, including a prenex normal form theorem stronger than that in Ying’s model. In addition, our Choquet integral approach to the evaluation of quantified statements is compared with others, in particular with Ying’s Sugeno integral approach.

Some induced ordered weighted averaging operators and their use for solving group decision-making problems based on fuzzy preference relations

In [R.R. Yager, D.P. Filev, Operations for granular computing: Mixing words and numbers, in: Proceedings of the FUZZ-IEEE World Congress on Computational Intelligence, Anchorage, 1998, pp. 123–128] Yager and Filev introduced the Induced Ordered Weighted Averaging (IOWA) operator. In this paper, we provide some IOWA operators to aggregate fuzzy preference relations in group decision-making (GDM) problems. These IOWA operators when guided by fuzzy linguistic quantifiers allow the introduction of some semantics or meaning in the aggregation, and therefore allow for a better control over the aggregation stage developed in the resolution process of the GDM problems. In particular, we present the Importance IOWA (I-IOWA) operator, which applies the ordering of the argument values based upon the importance of the information sources; the Consistency IOWA (C-IOWA) operator, which applies the ordering of the argument values based upon the consistency of the information sources; and the Preference IOWA (P-IOWA) operator, which applies the ordering of the argument values based upon the relative preference values associated to each one of them. We provide a procedure to deal with ‘ties’ in respect to the ordering induced by the application of one of these IOWA operators; it consists of a sequential application of the above IOWA operators. We also present a selection process for GDM problems based on the concept of fuzzy majority and the above three IOWA operators. Finally, we analyse the reciprocity and consistency properties of the collective fuzzy preference relations obtained using IOWA operators.

Implementing an extension of the analytical hierarchy process using ordered weighted averaging operators with fuzzy quantifiers in ArcGIS

This paper focuses on the integration of GIS and an extension of the analytical hierarchy process (AHP) using quantifier-guided ordered weighted averaging (OWA) procedure. AHP_OWA is a multicriteria combination operator. The nature of the AHP_OWA depends on some parameters, which are expressed by means of fuzzy linguistic quantifiers. By changing the linguistic terms, AHP_OWA can generate a wide range of decision strategies. We propose a GIS-multicriteria evaluation (MCE) system through implementation of AHP_OWA within ArcGIS, capable of integrating linguistic labels within conventional AHP for spatial decision making. We suggest that the proposed GIS-MCE would simplify the definition of decision strategies and facilitate an exploratory analysis of multiple criteria by incorporating qualitative information within the analysis.

An Approach to Computing With Words Based on Canonical Characteristic Values of Linguistic Labels

Herrera and Martinez initiated a 2-tuple fuzzy linguistic representation model for computing with words (CW), which offers a computationally feasible method for aggregating linguistic information (that are represented by linguistic variables with equidistant labels) through counting indexes of the corresponding linguistic labels. Lawry introduced an alternative approach to CW based on mass assignment theory that takes into account the underlying definitions of the words. Recently, we provided a new (proportional) 2-tuple fuzzy linguistic representation model for CW that is an extension of Herrera and Martinez's model and takes into account the underlying definitions of linguistic labels of linguistic variables in the process of aggregating linguistic information by assigning canonical characteristic values (CCVs) of the corresponding linguistic labels. In this paper, we study further into CW based on CCVs of linguistic labels to provide a unifying link between Lawry's framework and Herrera and Martinez's 2-tuple framework as well as allowing for computationally feasible CW. Our approach is based on a formal definition of CCV functions of a linguistic variable (which is introduced under the context of its proportional 2-tuple linguistic representation model as continuation of our earlier works) and on a group voting model that is for the probabilistic interpretation of the (whole) semantics of the ordered linguistic terms set of an arbitrary linguistic variable. After the general framework developed in the former part of this paper, we focus on a particular linguistic variable - probability. We show that, for a linguistic probability description, the expectation of its posterior conditional probability is a canonical characteristic value of the linguistic probability description. Then, a calculus for reasoning with linguistic syllogisms and inference from linguistic information is introduced. It is investigated under the context that linguistic quantifiers in such linguistic syllogisms can be arbitrary linguistic probability description and that the related linguistic information can be linguistic facts, overfacts, or underfacts. Intrinsically, our approach to this calculus is with computing with words based on canonical characteristic values of linguistic labels.

CONSENSUS MODELLING IN GROUP DECISION MAKING: DYNAMICAL APPROACH BASED ON FUZZY PREFERENCES

The. notion of consensus plays an important role in group decision making, particularly when the collective preference structure is generated by a dynamical aggregation process of the single individual preference structures. In this dynamical process of aggregation each single decision maker gradually transforms his/her preference structure by combining it, through iterative weighted averaging, with the preference structures of the remaining decision makers. In this way, the collective decision emerges dynamically as a result of the consensual interaction among the various decision makers in the group. From the point of view of applied mathematics, the models of consensual dynamics stand in the context of multi-agent complex systems, with interactive and nonlinear dynamics. The consensual interaction among the various agents (decision makers) acts on their state variables (the preferences) in order to optimize an appropriate measure of consensus, which can be of type 'hard' (unanimous agreement within the group of decision makers) or 'soft' (partial agreement within the group of decision makers). In this paper, we study the modelling of consensus reaching when the individual testimonies are assumed to be expressed as fuzzy preference relations. Here consensus is meant as the degree to which most of the experts agree on the preferences associated to the most relevant alternatives. First of all we derive a degree of dissensus based on linguistic quantifiers and then we introduce a form of network dynamics in which the quantifiers are represented by scaling functions. Finally, assuming that the decision makers can express their preferences in a more flexible way, i.e. by using triangular fuzzy numbers, we describe the iterative process of opinion transformation towards consensus via the gradient dynamics of a cost function expressed as a linear combination of a dissensus cost function and an inertial cost function.

Alternative Forms of Aggregation in the Analytic Hierarchy Process: Ordered Weighted Averaging Operators

The analytic hierarchy process (AHP) methodology was introduced by Thomas Saaty and has had numerous applications in a wide range of contexts. In a common three-level hierarchy, AHP involves the aggregation of criterion importance weights (or priorities) and the performance scores (or priorities) of alternatives as the sum of the products of weights and scores for each alternative. The multiplicative AHP (MAHP) involves the multiplicative aggregation of performance scores raised to the power of the criterion weights and has been seen by many individuals (notably, Freerk Lootsma and John Barzilai) as an alternative more desirable structure. It is suggested that alternative forms of aggregation of performance scores and criterion weights might be more useful, in particular the ordered weighted averaging (OWA) operator introduced by Ronald Yager. The choice of weights in an OWA operator may be guided by a linguistic quantifier involving the importance weights associated with each criterion. The geometric ordered weighted averaging (GOWA) operator is also considered as a possibility for aggregation in the MAHP. An example is given relating to the location of a Green Bridge Link (for public transport and non-motorized modes of transport) in Brisbane, Queensland, Australia. The four approaches (alternatives) to the identified site of the Green Bridge (Dutton Park) are a Full Busway, Boggo Road, Cornwall Street, and Kent Street.

On reciprocity indexes in the aggregation of fuzzy preference relations using the OWA operator

Chiclana, Herrera and Herrera-Viedma studied conditions under which the reciprocity property is maintained in the aggregation of reciprocal fuzzy preference relations using the OWA operator guided by a relative linguistic quantifier. In this note, we focus on the reciprocity in the aggregation of fuzzy preference relations (that is, the additive reciprocity is not assumed) using the OWA operator. By defining a reciprocity index for measuring the non-reciprocity degree of fuzzy preference relations, we show that a sufficient (but not necessary) condition under which a “collective” fuzzy preference relation, obtained by aggregating a set of “individual” fuzzy preference relations using an OWA operator guided by a linguistic quantifier with parameter a and b , has a reciprocity index no greater than the largest of the corresponding “individual” reciprocity indexes. Our result is helpful to complete Chiclana et al.'s decision model [F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems 97 (1998) 33–48].

Orness and parameterized RIM quantifier aggregation with OWA operators: A summary

A necessary and sufficient condition for the ordered weighted average (OWA) aggregation value of an arbitrary aggregated set to consistently increase with the orness level is proposed. The OWA operator properties associated with the orness level are extended. Then, with the generating function representation of Regular Increasing Monotone (RIM) quantifier, all these conditions and properties are extended to the case of RIM quantifiers, which can be seen as the continuous OWA operator with free dimension. Some families of consistency RIM quantifiers and their corresponding OWA operators are summarized. Some existing linguistic term RIM quantifiers are collected and two parameterized generalization forms of them are proposed, which can be useful for the selection and comparison of the linguistic quantifier in theory and applications.

Multicriteria group decision making under incomplete preference judgments: Using fuzzy logic with a linguistic quantifier

In the face of increasing global competition and complexity of the socioeconomic environment, many organizations employ groups in decision making. Inexact or vague preferences have been discussed in the decision-making literature with a view to relaxing the burden of preference specifications imposed on the decision makers and thus taking into account the vagueness of human judgment. In this article, we present a multiperson decision-making method using fuzzy logic with a linguistic quantifier when each group member specifies incomplete judgment possibly both in terms of the evaluation of the performance of different alternatives with respect to multiple criteria and on the criteria themselves. Allowing for incomplete judgment in the model, however, makes a clear selection of the best alternative by the group more difficult. So, further interactions with the decision makers may proceed to the extent to compensate for the initial comfort of preference specifications. These interactions, however, may not guarantee the selection of the best alternative to implement. To circumvent this deadlock situation, we present a procedure for obtaining a satisfactory solution by the use of a linguistic-quantifier-guided aggregation that implies the fuzzy majority. This is an approach that combines a prescriptive decision method via mathematical programming and a well-established approximate solution method to aggregate multiple objects. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 641–660, 2007.

Assessment of supplier performance based on product-development strategy by applying multi-granularity linguistic term sets

Measuring supplier performance possesses the properties of multi-dimension and multi-scale, it increases the complexity of the aggregation and assessment process. A focal company must develop an effective procedure for overcoming the complexity of performance measurement. Furthermore, the assessment of supplier performance must conform to the product-developed strategy of the focal company. This study is a further research deeper into the concept already introduced in Chang et al. [1] [Applying fuzzy linguistic quantifier to select supply chain partners at different phases of product life cycle. International Journal of Production Economics 2006; 100(2):348–59]; using multi-granularity linguistic variable and numerical ration scale to represent the overall supply performance. By unifying the derived information, the measurement complexity could be resolved. The fuzzy preference was constructed to adjust the consistent direction and transform information into fuzzy relationship. Finally, fuzzy linguistic quantifier guided ordered weighted aggregation (FLQG-OWA) operator with maximal entropy was computed and aggregated with all indicators to meet the current policy of the focal company.

Analysis of OWA operators in decision making for modelling the majority concept

The majority concept plays a main role in decision making processes where one of the main problems is to define a decision strategy which takes into account the individual opinions of the decision makers to produce an overall opinion which synthesizes the opinions of the majority of the decision makers. The reduction of the individual values into a representative value of majority is usually performed trough an aggregation process. The most common operator used in these processes is the OWA operator, in which the majority concept can be modelled using fuzzy logic and linguistic quantifiers. In this work the fusion processes and the semantic used for modelling the majority concept in the OWA operators are analyzed and compared in order to present different approach to obtain a feasible majority aggregation value for the decision making problem.

Decision support tool for multi-objective job shop scheduling problems with linguistically quantified decision functions

This paper presents a new tool for multi-objective job shop scheduling problems. The tool encompasses an interactive fuzzy multi-objective genetic algorithm (GA) which considers aspiration levels set by the decision maker (DM) for all the objectives. The GA's decision (fitness) function is defined as a measure of truth of a linguistically quantified statement, imprecisely specified by the DM using linguistic quantifiers such as most, few, etc., that refer to acceptable distances between the achieved objective values and the aspiration levels. The linguistic quantifiers are modelled using fuzzy sets. The developed tool is used to analyse and solve a real-world problem defined in collaboration with a pottery company. The tool provides a valuable support in performing various what-if analyses, for example, how changes of batch sizes, aspiration levels, linguistic quantifiers and the measure of acceptable distances affect the final schedule.

Linguistic quantifiers modeled by Sugeno integrals

Since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them, linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning. This paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures, and the truth value of a quantified proposition is evaluated by using Sugeno's integral. This framework allows us to have some elegant logical properties of linguistic quantifiers. We compare carefully our new model of quantification and other approaches to linguistic quantifiers. A set of criteria for linguistic quantification was proposed in the previous literature. The relationship between these criteria and the results obtained in the present paper is clarified. Some simple applications of the Sugeno's integral semantics of quantifiers are presented.

DECISION COMPROMISE MODELLING BASED ON OWA OPERATORS

This paper is concerned by the industrial application of Ordered Weighted Averaging OWA operators, introduced by Yager. These operators allow to express the type of compromises, by the notion of linguistic quantifiers, such as “most” of criteria. The interest of this method of aggregation is, beyond its simplicity of use, its evaluation of products according a unique scale. Furthermore, the weights are not fixed by criteria but by levels of performance. In this paper we present a methodology of classification of products by two approaches. The first one is based on a learning sample and the second one on linguistic quantifiers. An industrial application, from a food production, illustrates these approaches. We then discuss the classifications obtained by these two approaches and we present a comparison.