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1ENGINEERING SCIENCES
SOFTWARE SELECTION ON BASE OF SUGENO INTEGRAL
Salimov Vagif Hasan Oglu,
Ph.D., Azerbaijan Republic, Baku, Azerbaijan state oil and industry university, assoc. prof. of «Computer engineering» department, ORCID ID: https://orcid.org/0000-0002-0590-5437
DOI: https://doi.org/10.31435/rsglobal_ws/30092020/7166 ARTICLE INFO ABSTRACT
Received: 10 July 2020 The article is devoted to the problem of software selection. As a rule, this
Accepted: 17 August 2020 tasks are formalized as models of multi-criteria decision making
PubHshed: 3° September 2020 (MCDM). The peculiarity of this problem is in the fact that the
evaluation criteria, is generally defined by linguistic expert. This requires KEYWORDS the use of special methods, in particularly, the theory of fuzzy sets. To
software, fuzzy measure, solve the problem, an approach based on the a fuzzy measure was used.
fuzzy Sugeno measure, In general, a fuzzy measure allows one to take into account the effect of
Sugeno integral. the mutual influence of criteria. The main difficulty lies in identifying
this measure. The fuzzy Sugeno measure and Sugeno integral were used. An example of solving the indicated problem is given.
Citation: Salimov Vagif. (2020) Software Selection on Base of Sugeno Integral. World Science. 7(59). doi: 10.3143 5/rsglobal_ws/30092020/7166
Copyright: © 2020 Salimov Vagif. 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.
Software selection problem based on the formal methods is very actual [1-3]. As a rule, this tasks are formalized as models of multi-criteria decision making (MCDM). The peculiarity of this problem lies in the fact that the evaluation criteria, is generally defined by linguistic expert. This requires the use of special methods, in particular, the theory of fuzzy sets.
One of the problems MCDM - this is a problem taken as interrelation between criteria, i.e. the problem is non-additivity. Using these criteria such as the simple arithmetic mean, a weighted arithmetic mean, the geometric mean and a weighted geometric mean, median, mode, and others can lead to incorrect result. One of the approaches to solving this problem is the use of non-additive criteria, in particular, a fuzzy measure. The concept of fuzzy measure, based on the work of Choquet [4] in 1974 introduced by M. Sugeno [5]. In 1989, M. Sugeno [6] proposed the concept of X fuzzy measure and a new method of aggregation is the Sugeno integral.
In general, a fuzzy measure allows one to take into account the effect of the mutual influence of criteria. The main difficulty lies in identifying this measure. There are several approaches to solving this problem [7 -9]. One of the approaches to solving this problem is the formalization of the expert's preferences based on the use of Shapley coefficients and maximization of entropy. Another approach to identifying a fuzzy measure [10] is an approach based on minimizing the squared differences between the Choquet integral and global estimates of alternatives.
In this paper, we will use the Sugeno aggregation function based on the fuzzy X Sugeno measure.
2. Preliminaries.
Let's consider the basic concepts of the theory of fuzzy measure.
Fuzzy measure. Consider a variety of criteria X = [x1,x2,x3 .... }. P(X). is a collection of all potential subsets that can be built on base of X
A fuzzy measure is a mapping ^(X): ^ [0,1] that satisfies the following two conditions:
(i) ^(0) = 0 and ^(X) = 1
(ii) if AcB then ¡¿(A) <)n(B)
In general, a fuzzy measure is non-additive
A fuzzy measure ^(A) of a subset of criteria A has the meaning of a weighting coefficients and indicates the degree of importance of a given subset of criteria. If ^(A U B) = ^(A) + ^(B) then the measure is additive If ^(A U B) > ^(A) + ^(B) then the measure is super - additive If ^(A U B) < ^(A) + ^(B) then the measure is sub - additive
Fuzzy X- Sugeno measure: A set of criteria X = [x1,x2,x3, ...xn} is given and consist of n criteria and a real number X E (-1, o). A fuzzy X- Sugeno measure is a function gA(X): ^ [0,1] that satisfies the following two conditions: 9x(X) = 1
If A, B cT then gx(AuB) = gA(A) + gA(B) + XgA(A)gA(B) and AnB = 0
The parameter X can be determined from the equation
X + 1 = n7l=1(Xgi + 1)
If -1< X < 0 then ^l=1gt > g(X) - super-additive measure If X = 0 then ^l=1gt = g(X) - additive measure If X >0 then ^l=1gt < g(X) - sub- additive measure
Sugeno Integral:
Suppose a g fuzzy measure is defined on a set X, then the Sugeno integral for the function f:X^[0,^] will have the form
I fdg = max(m\n(f(xi),g(xi)))
J 1<l<n
subject to the condition
f(Xi) < f(X2) < f(x3) .......f(xn) f(x0) = 0
Consider software selection problem by 4 criteria X = (x1,x2,x3,x4), which x1-functionality, x2- cost, x3- reliability, x4 - convenience
To set the linguistic expert assessments, define following linguistic terms: Very low -0.2, Low -0.4, Medium - 0.5, High -0.7, Very high -0.9
3. Problem solving.
Software alternatives and criterion values obtained from experts are presented in a matrix
Table 1. Criteria values for various software alternatives
Xi X4
Ai 0.7 0.5 0.4 0.9
A2 0.2 0.4 0.7 0.5
A3 0.7 0.5 0.9 0.4
A4 0.5 0.4 0.7 0.9
a5 0.2 0.5 0.9 0.7
Given expert evaluation of fuzzy measures for the individual criteria gx(xi) = 0.7, gA(x2) = 0.5, gA(X3) = 0.5, gA(X4) = 0.7
Sugeno equation for 4 criteria is
4
x+i = H(Xgi + i)
i=i We have:
X+1 = (0.7 X + 1)(0.5X + 1)(0.5X + 1)(0.7X + 1) After simplification, we obtain an algebraic equation 0.122X4 + 0.84X3 + 2.14X2 + 1.4X=0
After solving the equation in Matlab, we get the following roots: Xi = 0
X2 = -2.49 + 1.75i X3 = -2.49 - 1.75i X4 = -0.97
We choose the root Xe(-1, ao), we have X = -0.97
The result shows the presence of a generally negative relationship between the criteria, i.e. sub-additivity takes place.
Let's calculate fuzzy measures for all subsets of criteria 9x№ = 0.
gx(xi,x2) = gX(xi) + gx(x2) + Xgx(xí)gx(x2)=°.86°5 9x(xi,x3) = 9x(xi) + 9x(x3) + Xgx(xí)gx(xз)=°.86°5
9x(X±,X4) = gX(x1) + gX(X4) + XgX(x1)gX(X4)=09247 9x(X2,X3) = 9x(X2) + 3X(X3) + XgX(X2)gX(X3)=0 7575 gx(x2,x4) = gX(X2) + 9x(x4) + Xgx(X2)gx(X4)=°.86°5 9x(X3,X4) = gX(X3) + 9x(X4) + XgX(X3)gX(X4)=0.8605 gX(Xi,X2,X3) = gX(Xi,X2) + gX(X3) + XgX(Xi,X2)gX(X3)=0 9432 gX(Xi,X2,X4) = gX(Xi,X2) + gX(X4) + Xgx(Xi,X2)gx(x4)=09762 9X(xi>x3>x4) = 9X(Xi,X3) + gX(X4) + XgX(Xi,X3)gX(X4)=0.9762 9x(X2,X3,X4) = gX(X2,X3) + gX(X4) + XgX(X2,X3)gX(X4)=0.9432
gX(Xi,X2,X3,X4) = 1
The results of calculation showed fuzzy measure of paired relationships among the greatest strength has relationship between xi and x4 and the weakest among x2 and x3 the sets consisting of 3 criteria are the most interference criteria xi ,x3, x4 and the smallest x2 ,x3, x4 Calculate values of Sugeno integral for all alternatives:
Ci = f fdg = max (min(^, gx(xi, %2, X3, X4)), min (X2, gx(xi, x2> X4)
min (xi,gX(xi,X4)),min (x^gxM)) = 0,7
¿2 = f fdg = max (min(xi, gx(xv %2, X3, X4)), min (X2, gX(x2, X3, X4) min (X4, gX(x3, X4)), min (X3, gx(x3))) = 0,6
£3 = f fdg = max (min(x4, gx(xv %2, X3, X4)), min (X2, gx(xi, x2> x3) min (xi, gx(Xi, X3)), min (X3, gx(X3))) = 0,4
C4 = f fdg = max (min(%, gx(xv %2, X3, X4)), min (X2, gx(x2, X3, X4) min (x4,gx(x3,x4)),min (x3,gx(x3))) = 0,7
£5 = f fdg = max (min(xi, gx(xv %2, X3, X4)), min (X2, gx(x2, X3, X4))
min (x4,gx(x3,x4)),min (x3,gx(x3))) = 0,6 We have:
Ci=0,7 C2=0,6 C3=0,4 Q=0,7 C5=0,6 Thus are best alternatives Ai and A4 4. Conclusions.
The article is devoted to the problem of modeling the software selection process. The main attention is paid to the problem of mutual influence of selection criteria. To solve this problem, was proposed an approach based on the use of Sugeno fuzzy measure and Sugeno integral. The problem of software selection with the 4 criteria was solved.
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