Andrzej Piegat

Fuzzy Number Addition with the Application of Horizontal Membership Functions

The paper presents addition of fuzzy numbers realised with the application of the multidimensional RDM arithmetic and horizontal membership functions (MFs). Fuzzy arithmetic (FA) is a very difficult task because operations should be performed here on multidimensional information granules. Instead, a lot of FA methods use α-cuts in connection with 1-dimensional classical interval arithmetic that operates not on multidimensional granules but on 1-dimensional intervals. Such approach causes difficulties in calculations and is a reason for arithmetical paradoxes. The multidimensional approach allows for removing drawbacks and weaknesses of FA. It is possible thanks to the application of horizontal membership functions which considerably facilitate calculations because now uncertain values can be inserted directly into equations without using the extension principle. The paper shows how the addition operation can be realised on independent fuzzy numbers and on partly or fully dependent fuzzy numbers with taking into account the order relation and how to solve equations, which can be a difficult task for 1-dimensional FAs.

Computingwithwords with the Use of Inverse Rdm Models of Membership Functions

Abstract Computing with words is a way to artificial, human-like thinking. The paper shows some new possibilities of solving difficult problems of computing with words which are offered by relative-distance-measure RDM models of fuzzy membership functions. Such models are based on RDM interval arithmetic. The way of calculation with words was shown using a specific problem of flight delay formulated by Lotfi Zadeh. The problem seems easy at first sight, but according to the authors’ knowledge it has not been solved yet. Results produced with the achieved solution were tested. The investigations also showed that computing with words sometimes offers possibilities of achieving better problem solutions than with the human mind.

Comparative Analysis of MCDM Methods for Assessing the Severity of Chronic Liver Disease

The paper presents the Characteristic Objects method as a potential multi-criteria decision-making method for use in medical issues. The proposed approach is compared with TOPSIS and AHP. For this purpose, assessment of the severity of Chronic Liver Disease (CLD) is used. The simulation experiment is presented on the basis of the Model For End-Stage Liver Disease (MELD). The United Network for Organ Sharing (UNOS) and Eurotransplant use MELD for prioritizing allocation of liver transplants. MELD is calculated from creatinine, bilirubin and international normalized ratio of the prothrombin time (INR). The correctness of the selection is examined among randomly selected one million pairs of patients. The result is expressed as a percentage of agreement between the assessed method and MELD selection. The Characteristic Objects method is completely free of the rank reversal phenomenon, obtained by using the set of characteristic objects. In this approach, the assessment of each alternative is obtained on the basis of the distance from characteristic objects and their values. As a result, correctness of the selection obtained by using the Characteristic Objects method is higher than those obtained by TOPSIS or AHP techniques.

Identification of a multicriteria decision-making model using the characteristic objects method

This paper presents a new, nonlinear, multicriteria, decision-making method: the characteristic objects (COMET). This approach, which can be characterized as a fuzzy reference model, determines a measurement standard for decision-making problems.This model is distinguished by a constant set of specially chosen characteristic objects that are independent of the alternatives. After identifying a multicriteria model, this method can be used to compare any number of decisional objects (alternatives) and select the best one. In the COMET, in contrast to other methods, the rank-reversal phenomenon is not observed. Rank-reversal is a paradoxical feature in the decision-making methods, which is caused by determining the absolute evaluations of considered alternatives on the basis of the alternatives themselves. In the Analytic Hierarchy Process (AHP) method and similar methods, when a new alternative is added to the original alternative set, the evaluation base and the resulting evaluations of all objects change. A great advantage of the COMET is its ability to identify not only linear but also nonlinear multicriteria models of decision makers. This identification is based not on a ranking of component criteria of the multicriterion but on a ranking of a larger set of characteristic objects (characteristic alternatives) that are independent of the small set of alternatives analyzed in a given problem. As a result, the COMET is free of the faults of other methods.

Comparative analysis of MCDM methods for the assessment of mortality in patients with acute coronary syndrome

Multi-criteria decision-making (MCDM) methods are commonly used in many fields of research, e.g., engineering and manufacturing systems, water resources studies , medicine, and etc. However, there is no effective approach of selecting a MCDM method to problem, which is solved. The formal requirements of each MCDM method are not sufficient because most methods would seem to be appropriate for most problems. Therefore, the main purpose of the paper is a comparison of accuracy selected MCDM methods. Proposed approach is presented on the example of mortality in patients with acute coronary syndrome. Additionally, the paper presents characteristic objects method (COMET) as a potential decision making method for use in medical problems, which accuracy is compared with TOPSIS and AHP. In the experimental study, the average and standard deviation of the root mean square error of evaluations are examined for groups of randomly selected patients, each described by age, blood pressure, and heart rate. Then, the correctness of choosing the patient in the best and worst condition is also examined among randomly selected pairs. As a result of the experimental study, rankings obtained by the COMET method are distinctly more accurate than those obtained by TOPSIS or AHP techniques. The COMET method, in the opposite of others method, is completely free of the rank reversal phenomenon, which is identified as a main source of problems with evaluations accuracy.

Horizontal Membership Function and Examples of its Applications

The paper introduces horizontal membership functions (HMFs) which define a fuzzy set not in form of commonly used vertical membership functions of type μxa0=xa0f1(x) but in the horizontal form xxa0=xa0f2(μ). Until now, constructing HMFs had seemed impossible because of horizontal ambiguity of this function. Now, however, it became possible thanks to the multidimensional, RDM-interval arithmetic based on relative-distance-measure variables. HMFs enable direct introducing uncertain, interval or fuzzy variable-values in usual mathematical formulas of type yxa0=xa0f(x1,…,x2) together with crisp values, without using Zadeh’s extension principle. Thus, a relatively easy aggregation of crisp and uncertain knowledge became possible. The paper shows application of HMFs, first on example of a classical mathematical function yxa0=xa0f(x1,x2) and next, on example of a computing with words challenge problem.

Application of the Horizontal Membership Function to the Uncertain Displacement Calculation of a Composite Massless Rod Under a Tensile Load

Fuzzy arithmetic, based on Zadeh’s extension principle, is a common method applied to solve problems with uncertain parameters. The paper presents the fuzzy arithmetic operations on fuzzy numbers in a new way, using the horizontal membership functions (HMFs). The horizontal membership functions enable to introduce uncertain, interval, or fuzzy variable values together with crisp values in arithmetic operations without using Zadeh’s extension principle. Thus, a relatively easy aggregation of crisp and uncertain knowledge is possible. The numerical example of one-dimensional static problem consisting of a two-component massless rod under tensile load is considered.

Fuzzy Arithmetic Type 1 with Horizontal Membership Functions

The chapter shortly (because of the volume limitation) presents multidimensional fuzzy arithmetic based on relative-distance-measure (RDM) and horizontal membership functions which considerably facilitate calculations. This arithmetic will be denoted as MD-RDM-F one. It delivers full, multidimensional problem solutions that further enable determining, in an accurate and unique way, various representations of the solutions such as span (maximal uncertainty of the solution), cardinality distribution of possible solution values, center of gravity of the solution granule, etc. It also allows for taking into account relations and dependencies existing between variables, what is absolutely necessary e.g. in calculations with fuzzy probabilities that always should sum up to 1 or in equation system solving.

Aggregation of Inconsistent Expert Opinions with Use of Horizontal Intuitionistic Membership Functions

Single expert opinion expressed in form of an intuitionistic membership function (IMF) has uncertainty of the second order because it consists of the membership—(mu (x)) and of the non-membership function (nu (x)). Two different, considerably inconsistent expert opinions have an increased uncertainty order. Often we do not know, which of the opinion is more or less credible. Hence, IMF representing both aggregated opinions cannot be a standard IMF. It should have an increased order of uncertainty. Possibility of appropriate modeling aggregated opinions offers theory of fuzzy sets type-2 developed mainly by J. Mendel. In this paper authors show how application of this theory in connection with horizontal version of IMFs allows for constructing of an aggregated IMF of two inconsistent intuitionistic expert opinions.

Optimal estimator of hypothesis probability for data mining problems with small samples

Abstract The paper presents a new (to the best of the authors’ knowledge) estimator of probability called the “Eph √ 2 completeness estimator” along with a theoretical derivation of its optimality. The estimator is especially suitable for a small number of sample items, which is the feature of many real problems characterized by data insufficiency. The control parameter of the estimator is not assumed in an a priori, subjective way, but was determined on the basis of an optimization criterion (the least absolute errors).The estimator was compared with the universally used frequency estimator of probability and with Cestnik’s m-estimator with respect to accuracy. The comparison was realized both theoretically and experimentally. The results show the superiority of the Eph √ 2 completeness estimator over the frequency estimator for the probability interval ph ∈ (0.1, 0.9). The frequency estimator is better for ph ∈ [0, 0.1] and ph ∈ [0.9, 1].