The Optimal Tolerance Solution of the Basic Interval Linear Equation and the Explanation of the Lodwick’s Anomaly

Determining the tolerance solution (TS) of interval linear systems (ILSs) has been a task under consideration for many years. It seems, however, that this task has not been fully and unequivocally solved. This is evidenced by the multiplicity of proposed methods (which sometimes provide different results), the existence of many questions, and the emergence of strange solutions provided by, for example, Lodwick’s interval equation anomaly (LIEA). The problem of solving ILEs is probably more difficult than we think. The article presents a new method of ILSs solving, but it is limited to the simplest, basic equation [a̲,a¯]X=[b̲,b¯], which is an element of all more complex forms of ILSs. The method finds the optimal TS for this equation by using multidimensional interval arithmetic (MIA). According to the authors’ knowledge, this is a new method and it will allow researchers to solve more complex forms of ILSs and various types of nonlinear interval equations. It can also be used to solve fuzzy linear systems (FLSs). The paper presents several examples of the method applications (including one real-life case)

A decomposition approach to type 2 interval arithmetic

The classic interval has precise borders A = [a, ā] . Therefore, it can be called a type 1 interval. Because of great practical importance of such interval data, several versions of type 1 interval arithmetic have been created. However, sometimes precise borders a and ā of intervals cannot be determined in practice. If the borders are uncertain, then we have to do with type 2 intervals. A type 2 interval can be denoted as AT2 = [aL, aR], [āL, āR]. The paper presents multidimensional decomposition RDM type 2 interval arithmetic (D-RDM-T2-I arithmetic), where RDM means relative-distance measure. The decomposition approach considerably simplifies calculations and is transparent for users. Apart from this arithmetic, examples of its applications are also presented. To the authors’ best knowledge, no papers on this arithmetic exist. D-RDM-T2-I arithmetic is necessary to create type 2 fuzzy arithmetic based on horizontal µ-cuts, which the authors aim to do

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

Computing with words with the use of inverse RDM models of membership functions

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

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

Realistic Optimal Tolerant Solution of the Quadratic Interval Equation and Determining the Optimal Control Decision on the Example of Plant Fertilization

In scientific journals, it is increasingly common to find articles presenting methods for solving problems not based on idealistic mathematical models containing perfectly accurate coefficient values that cannot be obtained in practice, but on models in which coefficient values are affected by uncertainty and are expressed in the form of intervals, fuzzy numbers, etc. However, solving tasks with interval coefficients is not fully mastered, and a number of such problems cannot be solved by currently known methods. There is undeniably a research gap here. The article presents a method for solving problems governed by the quadratic interval equation and shows how to find the tolerant optimal control value of such a system. This makes it possible to solve problems that could not be solved before. The paper introduces a new concept of the degree of robustness of the control to the set of all possible multidimensional states of the system resulting from its uncertainties. The method presented in the article was applied to an example of determining the optimal value of nitrogen fertilization of a sugar beet plantation, the vegetation of which is under uncertainty. It would be unrealistic to assume precise knowledge of crop characteristics here. The proposed method allows to determine the value of fertilization, which gives a chance to obtain the desired yield for the maximum number of field conditions that can occur during the growing season

A Decomposition Approach to Type 2 Interval Arithmetic

The classic interval has precise borders A=[a_,a¯]A = \left[ {\underline{a},\bar a} \right]. Therefore, it can be called a type 1 interval. Because of great practical importance of such interval data, several versions of type 1 interval arithmetic have been created. However, sometimes precise borders a_\underline{a} and ā of intervals cannot be determined in practice. If the borders are uncertain, then we have to do with type 2 intervals. A type 2 interval can be denoted as AT2=[[a_L,a_R],[a¯L,a¯R]]{A_{T2}} = \left[ {\left[ {{\underline{a}_L},{\underline{a}_R}} \right],\left[ {{{\bar a}_L},{{\bar a}_R}} \right]} \right]. The paper presents multidimensional decomposition RDM type 2 interval arithmetic (D-RDM-T2-I arithmetic), where RDM means relative-distance measure. The decomposition approach considerably simplifies calculations and is transparent for users. Apart from this arithmetic, examples of its applications are also presented. To the authors’ best knowledge, no papers on this arithmetic exist. D-RDM-T2-I arithmetic is necessary to create type 2 fuzzy arithmetic based on horizontal μ-cuts, which the authors aim to do

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

The paper presents a new (to the best of the authors ’ knowledge) estimator of probability called the “E