ON THEORY OF INTUITIONISTIC FUZZY SETS (OR VAGUE SETS)

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1 International Journal of Fuzzy Systems On Theory and Rough of Intuitionistic Systems Fuzzy Sets (Or Vague Sets) 113 4(2), December 2011, pp , Serials Publications, ISSN: X ON THEORY OF INTUITIONISTIC FUZZY SETS (OR VAGUE SETS) Musheer Ahmad Department of Information Technolgy, Nizwa College of Technology, Sultanate of Oman Abstract: In this paper we study about the higher order fuzzy sets specially about intuitionistic fuzzy sets (or vague sets). Keywords: Intuitionistic Fuzzy Sets (IFS), Indeterministic Part, Vague Set. 1. INTRODUCTION Different authors ([1-10], [11], [12], [13], [14], [15], [16]) from time to time have made a number of generalizations of fuzzy set theory of Zadeh [17]. While fuzzy sets are applicable to many application domains, higher order fuzzy sets may not, because of their specialization in character by birth. Application of higher order fuzzy sets makes the solution-procedure more complex, but if the complexity on computation-time, computation-volume or memory-space are not the matter of concern then a better results could be sometimes achieved. The most popular generalizations of fuzzy sets are iv-fuzzy sets, L-fuzzy sets, intuitionistic fuzzy sets (IFS), vague sets, iv-ifs, type-ii fuzzy sets, etc. Of these, the notion of intuitionistic fuzzy set theory (IFS theory) introduced by Atanassov [2] is of interest to us for our work reported in this paper. Vague sets and intuitionistic fuzzy sets are same models developed with identical philosophy as justified in [11]. 2. THEORY OF INTUITIONISTIC FUZZY SETS (IFS) An important and very potential generalization of fuzzy set theory came in the form of intuitionistic fuzzy set theory of Atanassov. All fuzzy sets can be viewed as intuitionistic fuzzy sets, but the converse is not true. Many authors have asserted that there are a large number of real life problems for which IFS theory is a more suitable tool than fuzzy set theory. In most cases of judgments, evaluation is done by human beings (or by an intelligent agent) where there is certainly a limitation of knowledge or intellectual functionaries. Naturally, every decision-maker hesitates more or less, on every evaluation activity. It is a common feature of any human being. To decide whether = 12 or not 12, the hesitation is nil because of the reason that it is a hard type of computing (not soft-computing). But to judge whether a patient has cancer or not, a doctor, (the decision-maker), will hesitate because of the fact that a fraction of evaluation may remain indeterministic to him. This is the breaking philosophy in the notion of IFS theory introduced by Atanassov. The non-membership part may have more significant importance compared to the complement of fuzzy sets. If there is no hesitation, the intuitionistic-fuzziness reduces to fuzziness and in such a case there is absolutely no need to exploit intuitionistic fuzzy tools for analyzing the problem. Let E be the set of all countries with elective governments. Assume that we know for every country x C E the percentage of the electorate that have voted for the corresponding government. Let it be denoted by M(x). Suppose that µ(x) = M(x)/100 and v(x) = 1-µ (x). This number corresponds to the part of electorate who have not voted for the government. By means of the fuzzy set theory we cannot analyse this situation in more details. However, if we define v(x) as the number of votes given to parties or persons outside the government, then we can show the part of electorate who have not voted at all and the corresponding number will be 1-µ (x)-v(x). This philosophy adds an extra edge to the Zadeh s fuzzy sets to construct the new higher order fuzzy set of type { (x, µ (x), v(x)) C E } alongwith the constraint 0 µ (x) + v(x) 1.

2 114 Musheer Ahmad To deal with such type of situation, Atanassov introduced the theory of intuitionistic fuzzy set theory by adding an extra dimension. One should not confuse intuitionistic fuzzy set theory of Atanassov [1-10] with the interval-valued fuzzy set theory of Zadeh [18], because they are two different notions. In interval-valued fuzzy set theory, the membership values are proposed on the philosophy of membership-property only, without considering the fact of non-membership. But in intuitionistic fuzzy set theory, both membership function and non-membershipfunction are proposed independently. In the next section we give the definition of intuitionistic fuzzy sets and then recollect some basic operations on them. 3. INTUITIONISTIC FUZZY SET (IFS) An intuitionistic fuzzy set (IFS) A in a universe of discourse E is defined as an object of the following form where the functions: µ A : E [0,1], and v A : E [0,1] A = {(x, µ A (x), v A (x)) x c E} define the degree of membership and the degree of non-membership respectively of the element x c E to be in A, and for every x c E we have the constraint 0 µ A (x) 1. Let us call this constraint 0 µ A (x) 1 by Atanassov constraint. Obviously, each ordinary fuzzy set may be written as {(x, µ A (x), 1-µ A (x)) x c E} and thus every fuzzy set is an intuitionistic fuzzy set but not conversely. The non-membership functions may have more significant importance compared to the complement of fuzzy sets while applying in real life problems. 4. INDETERMINISTIC PART Let A be an IFS in the universe E with the membership function µ A and the non-membership function v A. The amount π A (x) = 1 - (µ A (x)) is called the hesitation part or the indeterministic part,and this amount may cater to either membership value or to non-membership value or to both. Clearly, in case of ordinary fuzzy sets (Zadeh s fuzzy sets) it is presumed that π A (x) = 0, for every x c E. Every set in the sense of the set theory may be viewed as a fuzzy set and hence, as an IFS too. The figure below presents the most widely accepted geometric interpretation of an IFS. (An intuitionistic fuzzy set).

3 On Theory of Intuitionistic Fuzzy Sets (Or Vague Sets) 115 In the fuzzy set theory there are three basic ways to construct membership functions: - Employing expert knowledge, - Explicitly, on the basis of observations collected in advance and processed appropriately (e.g., by probabilistic distribution) - Analytically, by suitably chosen functions (e.g., probabilistic distribution). The two latter cases are treated in much the same way as for ordinary fuzzy sets; however, these methods are now used for the estimation of both the degree of membership and the degree of non-membership independently of a given element of a fixed universe to a subset of the same universe. It is to be noted that a correct method must honor the Atanassov Constraint. 0 µ A (x) SOME OPERATIONS ON IFSS If A and B are two intuitionistic fuzzy subsets of the set E, then A B iff x E, [µ A (x) µ B (x) and υ A (x) υ B (x)]. A B iff B A. A = B iff x E, [µ A (x) = µ B (x) and υ A (x) = υ B (x)]. A = { x, υ A (x), µ A (x) x E}. A B = { x, min (µ A (x), µ B (x)), max(υ A (x), υ B (x)) x E}. A B = { x, max(µ A (x), µ B (x)), min(υ A (x), υ B (x)) x E}. A + B = { x, µ A (x) + µ B (x)- µ B (x) µ A (x) υ B A.B = { x, µ A (x) µ B (x) + υ B (x)- υ A (x) υ B A = { x, 1-υ A A = { x, 1-υ A C(A) = { x, K, L x E} and I(A) = { x, k, l x E}, where K = k = sup µ x E A (x), L = inf υ x E A (x), inf µ x E A (x), and l = sup x E υ A (x). 6. VAGUE SETS ARE INTUITIONISTIC FUZZY SETS Fuzzy set A is defined as the set of ordered pairs A = {(u, µ A (u)): u U}, where µ A (u) is the grade of membership of element u in set A. The greater µ A (u), the greater is the truth of the statement that the element u belongs to the set A. But Gau and Buehrer [13] pointed out that this single value combines the evidence for u and the evidence against u. It does not indicate the evidence for u and the evidence against u, and it does not also indicate how much there is of each. The same is the philosophy with which Atanassov [2] originally defined intuitionistic fuzzy sets earlier Definition: Vague Set A vague set (or in short VS) A in the universe of discourse U is characterized by two membership functions given by: (i) a truth membership function t A : U [0, 1], and (ii) a false membership function f A : U [0, 1]

4 116 Musheer Ahmad where t A (u) is a lower bound of the grade of membership of u derived from the evidence for u, and f A (u) is a lower bound on the negation of u derived from the evidence against u, and t A (u) + f A (u) 1. Thus the grade of membership of u in the vague set A is bounded by a sub-interval [t A (u), 1- f A (u)] of [0,1]. This indicates that if the actual grade of membership is µ(u), then t A (u) µ(u) 1- f A (u). The vague set A is written as A = {< u, [t A (u), f A (u)] > : u U}, where the interval [t A (u), 1-f A (u)] is called the vague value of u in A and is denoted by V A (u). For example, consider an universe U = {DOG, CAT, RAT}. A vague set A of U could be A = {<DOG, [.7,.2]>, <CAT, [.3,.5]., <RAT, [.4,.6]>}. Intuitionistic Fuzzy Sets (or Vague Sets) have an Extra Edge Over Fuzzy Sets There are a number of generalizations of fuzzy sets of Zadeh done by different authors. For each generalization, one (or more) extra dimension is annexed with a more specialized type of aim and objective. Thus, a number of higher order fuzzy sets are now in literatures and are being applied into the corresponding more specialized application domains. While fuzzy sets are applicable to each of such application domains, higher order fuzzy sets can not, because of its specialization character by birth. Application of higher order fuzzy sets makes the solutionprocedure more complex, but if the complexity on computation-time, computation-volume or memory-space are not the matter of concern then a better results could be achieved. Intuitionistic fuzzy sets of Atanassov have also an extra edge over fuzzy sets. Let U be a universe, the set of all students of Calcutta School. Let A be an intuitionistic fuzzy set of all good-in-maths students of the universe U, and B be a fuzzy set of all good-in-maths students of U. Suppose that an intellectual Manager M 1 proposes the membership value µ B (x) for the element x in the fuzzy set B by his best intellectual capability. On the contrary, another intellectual Manager M 2 proposes independently two membership values t A (x) and f A (x) for the same element in the vague set A by his best intellectual capability. The amount t A (x) is the true-membership value of x and f A (x) is the false-membership value of x in the vague set A. Both M 1 and M 2 being human agents have their limitation of perception, judgment, processing-capability with real life complex situations. In the case of fuzzy set B, the manager M 1 proposes the membership value µ B (x) and proceed to his next computation. There is no higher order check for this membership value in general. In the later case, the manager M 2 proposes independently the membership values t A (x) and f A (x), and makes a check at this base-point itself by exploiting the constraint t A (x) + f A (x) 1. If it is not honored, the manager has a scope to rethink, to reshuffle his judgment procedure either on evidence against or on evidence for or on both. The two membership values are proposed independently, but they are mathematically not independent. This is the breaking philosophy of intuitionistic fuzzy sets (Gau and Buehrer s vague sets [4,11]). Since vague sets and intuitionistic fuzzy sets are same concepts as clearly justified by Bustince and Burillo in [12], in the subsequent part of this thesis, in all definitions as well as characterizations, the two terminologies vague set and intuitionistic fuzzy set have been used with same meaning and objectives. There is no confusion in this issue Definition An intuitionistic fuzzy set (or a vague set) A of a set U with t A (u)=0 and f A (u) = 1 u U is called the zero intuitionistic fuzzy set (or zero vague set) of U Definition An intuitionistic fuzzy set (or a vague set) A of a set U with t A (u)=1 and f A (u)=0 u U is called the unit intuitionistic fuzzy set (or unit vague set) of U. References [1] Angelov, P. P., Optimization in an Intuitionistic Fuzzy Environment, Fuzzy Sets and Systems, 86, (1997), [2] Atanassov, K., Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, Vol. 20, (1986), [3] Atanassov, K., New Operations Defined Over Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 61, (1994), [4] Atanassov, K., More on Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 33, (1989), [5] Atanassov, K., Remarks on the Intuitionistic Fuzzy Sets-III, Fuzzy Sets and Systems, 75, (1995), [6] Atanassov, K., Remarks on a Temporal Intuitionistic Uzzy Logic, 2nd Scientific Session of Mathematical Foundation Artificial Intelligence, Sofia IM-MFAIS, (1990), 1-5.

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