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1DECISION MAKING UNDER LINGUISTIC UNCERTAINTY CONDITIONS ON BASE OF GENERALIZED FUZZY NUMBERS
Ph.D., Salimov Vagif Hasan Oglu, assoc. prof. of "Computer engineering department" Azerbaijan state oil and industry university, Baku, Azerbaijan Republic, ORCID ID https://orcid.org/0000-0002-0590-5437
DOI: https://doi.org/10.31435/rsglobal_ws/30082021/7655
ARTICLE INFO
Received: 16 June 2021 Accepted: 09 August 2021 Published: 30 August 2021
KEYWORDS
linguistic uncertanty, decision making, membership function, aggregation, multi attribute decision making, generalized fuzzy numbers.
ABSTRACT
This article is devoted to the problem of decision making under linguistic uncertainty. The effective method for modelling linguistic uncertainty is the fuzzy set theory. There are several types of fuzzy number types proposed by L. Zadeh: fuzzy type-1, fuzzy type-2, Z-numbers. Chen proposed concept of generalized fuzzy numbers. Generalized trapezoidal fuzzy numbers (GTFN) one of effective approach which can be used for modeling linguistic uncertainty. GFTN very convenient model which allow take in account second order uncertainty. GFTN are formalized and major operations are described as practical problem is considered group decision making for supplier selection. In this case the criteria assessments are expressed by experts in linguistic form. Group decision making model is presented as 2 step aggregation procedure, in first step is aggregated value of alternative by expert, in second step by criteria. Numerical example with four criteria and three alternatives are presented and solved.
Citation: Salimov Vagif Hasan Oglu. (2021) Decision Making Under Linguistic Uncertainty Conditions on Base of Generalized Fuzzy Numbers. World Science. 8(69). doi: 10.31435/rsglobal_ws/30082021/7655
Copyright: © 2021 Salimov Vagif Hasan Oglu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
1. Introduction. Decision making problem with imperfect information is very actual problem. As known in many practical cases we need to be satisfied of expert information and the linguistic assessments. One is effective method of modelling linguistic information is fuzzy set approach. The are many scientific works dedicated to applications of classical fuzzy approach which is named fuzzy type-1 proposed by L. Zadeh (1965) [1]. In 1975 L. Zadeh [2] proposed more general approach fuzzy type-2, which expands the features of classical fuzzy type-1 model. Chen in 1985 proposed generalized fuzzy set concept [3]. L. Zadeh in 2011 proposed fuzzy Z-numbers approach [4]. All these approaches allow not only modelling our imprecise knowledge about factors and also take in account our imprecision about membership function. All these models have more powerful features for modelling uncertainty [6-16].
2. Preliminaries. In this article we discuss about application of generalized trapezoidal fuzzy numbers (GTFN) for modelling MADM problem [6].
Definition: General fuzzy number. A fuzzy set A, defined on the universal set of the real numbers R, is said to be generalized fuzzy number if it is membership function has the following characteristics:
(i) ^a-R^ [0,1] is continous
(ii) ^a(x) = 0 for all x (-o, a] U[d, o)
(iii) ^a(x) is stricly increasing on [a, b]and stricly decreasing on [c, d]
(iv) ^a(x) = w,for all xe[b,c], where 0 < w < 1.
Generalized trapezoidal fuzzy number A = (a, b, c, d, w) is said to be generalized fuzzy number if its membership function is given
( 0 x — a
b — a Hx(x) = { w
x — c
d — c
x < a \ a<x<b b < x < c c < x < d x > d
given:
Fig 1. Comparison between membership function of TFN and GTFN Here W plays role of confidence level.
Consider arithmetical operations on two trapezoidal GTFN numbers: A1 and A2 numbers are
At = (a-i, bt, ct, d±, wt) Al2 = (a2, b2, C2, d2, W2) Addition
A-^Q A2 = (a1 + a2, b1 + b2, c1 + c2, d1 + d2; mm^^ w2)) Subtraction
A1Q A2 = (a1 — a2, b1 — b2, c1 — c2, d1 — d2; min(w1, w2)) Scalar Multiplication _ ( (Aa, Ab, Ac, Ad; w) A> 0 \(Ad, Ac, Ab, Aa ;w) A< 0 Ranking function
For ranking alternatives we have used following centroid method /6/
/ dc — ab w c — b
(x0,90) = \a + b + c + d — ———--rr,T-(1 + ■
Ranking function
(dc) — (a + b)' 3 v (d + c) — (a + b)
))
(1)
R(A) = jx2 +y2 Let Âi and Âj two fuzzy numbers,
(iv) R(Âi) > R(Âj) thenÂi > Âj
(v) R (Âi) < r (Âj ) then Ât < Âj
(vi) r(â-) = R(Âj) thenÂi = Âj
With GTFN we can represent the crisp interval and also imprecise interval. If a=b and c=d and W ^ 1 we have imprecise interval with confidence level W. If a=b, c=d and w=1 then we have crisp interval.
3. Problem statement and solving method
Let's consider supplier selection problem with GTFN. This problem is formalized as MADM problem. Exist 3 potential suppliers At (i = 1,2,3) and their activity are described by 4 attributes: Ci- raw quality, C2 - risk factor, C3 -service level, C4 - company profile. Let's say that for decision making group of 3 experts established Ek (k = 1,2,3) and corresponding weight coefficients are determined A = (0.3 0.45 0.25)
0
V.
For 4 attributes Q (i = 1,2,3,4) are determined weight coefficients w = (0.3 0.15 0.2 0.35)
In table 1 are presented linguistic terms which will be used for alternative evaluation "Very Low" (VL), "Low" (L), "Medium" (M), "High" (H), "Very High" (VH) (Fig.2)
_Table 1. Linguistic terms for ^ alternative evaluation_
Linguistic term GTFN values
Very Low (VL) (0,0.1,0.2.0.3;0.6)
Low (L) (0.1,0.3,0.45,0.7;0.7)
Medium (0.4,0.5,0.7,0.8;0.8)
High (H) (0.5,0.6,0.75,0.85;0.85)
Very High (VH) (0.6,0.7,0.8,0.9;1)
Fig. 2. Linguistic terms for alternatives evaluation
Experts using these terms have evaluated any potential suppliers and results are presented in following tables 3-5
Table 3 A ternatives evaluation by 1st expert
Ci C2 C3 Q
Ai M H VH VH
A2 H M H H
A3 VH VH M H
Table 4 A ternatives evaluation by 2nd expert
Ci C2 C3 Q
Ai H VH H H
A2 M H VH VH
A3 H VH M VH
Table 5. Alternatives evaluation by 3rd expert
Ci C2 C3 Q
Ai M H H H
A2 H VH VH H
A3 M H M VH
First we carry out aggregation by experts using formula
and we have achieved following results: A11=(0.46,0.57,0.75,0,86;0,8) A12 =(0.55,0.65,0.77, 0.87;0.85) A13= (0.53,0.63,0.77,0.87;0.8) A14=(0.53,0.63,0.77,0.87;0.8) A21= (0.46,0.56,0.73,0.83;0.8) A22= (0.50,0.60,0.75,0.85;0.8) A23= (0.57,0.67,0.79,0.89;0.85) A24= (0.55,0.65,0.77,0.87;0.8) A31= (0.73,0.79,0.87,0.92;0.8) A32 = (0.58,0.68,0.79,0.89;0.85) A33 =(0.40,0.50,0.70,0.80;0.8) A34 = (0.57,0.67,0.79,0.89;0.85)
These results can be presented as collective decision matrix
(A11A12A13A14 A21A22A23A24 A31A32A33A34S
On next step we carry out aggregation by attributes using formula
Ai = ®4=1(<»iAij)
As result we have global evaluation of all alternatives (Table 6) Table 6. Global evaluation of all alternatives
Alternatives GTFN values
Ai (0.51,0.61,0.76,0.87;0,8)
Ä2 (0.52,0.62,0.76,0.86;0.8)
A3 (0.58,0.67,0.79,0.88;0,8)
For comparison alternative decisions we will use Rank function (1) Rank(41)=3.52> Rank(43)=3.49> Rank(42)=3.45 It means that best is supplier A1
Conclusions. In this article have been considered problem of MADM under linguistic
uncertainty. As model of decision making used group decision making approach and as model for
modeling uncertainty have been used GTFN model. As test problem for proposed model have been
used the supplier selection problem.
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