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5SOFTWARE SELECTION ON BASE OF FUZZY AHP METHOD
Salimov Vagif Hasan Oglu, Ph.D., assoc. prof. of "Computer engineering" department, Azerbaijan state oil and industry university, Baku, Azerbaijan Republic, ORCIDID: https://orcid.org/0000-0002-0590-5437
Damirova Javida Rizvan Qizi, Ph.D., assoc. prof. of "Control and system engineering" department, Azerbaijan State Oil and Industry University, Baku, Azerbaijan Republic, ORCID ID: https://orcid.org/0000-0003-4142-8014
DOI: https://doi.org/10.31435/rsglobal_wos/30012021/7395
ARTICLE INFO
Received: 25 November 2020 Accepted: 15 January 2021 Published: 30 January 2021
KEYWORDS
multi-criteria decision making, alternative, criterion, fuzzy, AHP method, pairwise comparison, preference index, aggregation, software.
ABSTRACT
The article is devoted to the problem of multi-criteria decision making. As application problem is used the software selection problem. The analysis of existing methods for solving this problem is given. As a method for solving this problem, the most popular fuzzy AHP method (Analytic Hierarchy Process) is proposed. This method use original algorithm for pairwise comparison of criteria and alternatives. The issues of practical implementation of this method are discussed in details. The results of the solution test problem at all stages are presented.
Citation: Salimov Vagif Hasan Oglu, Damirova Javida Rizvan Qizi. (2021) Software Selection on Base of Fuzzy AHP Method. International Academy Journal Web of Scholar. 1(51). doi: 10.31435/rsglobal_wos/30012021/7395
Copyright: © 2021 Salimov Vagif Hasan Oglu, Damirova Javida Rizvan Qizi. 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. The problem of multi-criteria decision making - (MCDM) is one of the actual problem in the theory of decision making [1-2]. From a mathematical point of view, it belongs to the class of vector optimization problems. The criteria can be divided into two groups: the criteria for which the maximum value is optimal and the criteria for which the minimum value is optimal. MCDM problems can be solved with an accuracy of many non-dominated alternatives or many trade-offs. Obtaining a single solution can only be implemented on the basis of some compromise scheme that reflects the preferences of the decision maker (DM). Methods for solving this problem can be divided into two large groups: methods using the aggregation of all alternatives according to all criteria and the solution of the resulting single criterion problem, the second group is associated with the procedure of pairwise comparisons and stepwise aggregation. The first group includes methods: weighted average sum, weighted average product and their various modifications [3-4], the second group includes -Analytic Hierarchy Process (AHP), Elimination and Choice Translating Reality (ELECTRE), The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Preference Ranking Organization Method (PROMETHEE) [5-15]. This paper discusses the fuzzy AHP method.
The method of AHP (Analytic hierarchy process) was proposed in the early 80's one of the greatest authorities in the field of operations research professor at Pittsburgh University (USA) Thomas Saaty.
An important part of all decision making algorithms is the process of determining the weighting coefficients of the criteria. In many methods, these coefficients are assigned by an expert, which does not always lead to adequate values. A main feature of AHP method is original procedure for calculating weighting coefficients criteria and alternatives on the basis of a single procedure paired comparisons.
Another feature of the AHP method is the consistent use of structural approach to the problem of decision making.
The decision making problem is presented as a hierarchical structure - goal-criteria-alternatives. At present, AHP is the most popular method for solving multi-criteria decision making problems [3]. AHP's popularity is largely due to the use of intuitive technology paired comparisons and procedures weighted average. Let's consider the description of AHP.
2. Description of the method
Suppose MCDM is given in the form of a matrix of outcomes (alternatives - criteria) m -number of criteria n - number of alternatives.
Ci C2 C3 Cm
Ai
A2
A3
An
Fig. 1. MCDM problem representation
AHP is implemented in the form of a sequential multi-stage procedure. At first stage DM builds a matrix of paired comparisons of criteria, for identifying the rank criterion and accordingly the weight criteria for calculating global assessment. The ranks of each criterion are calculated on base of the preference scale and the corresponding indices. Each pair can be defined on a linguistic scale, mapped to an interval (1-9).
T.Saaty proposed an original scale for evaluating paired comparisons
• 1 - the criteria are of equal importance,
• 3 - one criterion is somewhat more important than the other,
• 5 - one criterion is significantly more important than another,
• 7 - one criterion is undeniably more important than another
• 9 - one criterion is absolutely more important than another.
The matrix of pairwise comparisons of criteria (m x m) is presented in Fig. 2.
Ci C2 Cm
Ci 1 Ci /C 2 Ci /Cm
C2 C2 /Ci i C2 /Cm
C3 C3 /Ci i
i
Cm Cm /Ci Cm /C2 i
Fig. 2. Pairwise comparison of criterion matrix Here C[/ Cj , preference index of C^ criterion over Cj
In fuzzy AHP (FAHP) are used fuzzy numbers. In this article trapezoidal fuzzy numbers (TFN) are used.
Definition: Trapezoidal fuzzy number
Trapeziodal fuzzy number, A = (a, b, c, d), has following membership function:
( 0
x
a
liA{x) = <
b — a 1
x — c d — c
x < a a<x<b b < x < c
c < x < d x > d
0
V
Fig. 3. Trapezoidal fuzzy number
Let's consider basic mathematical operations with two TFT numbers: A1 and A2:
A- = (a-, b-, c-, d-) A2 = (a2, b2, C2, d-2)
Addition
A-® A2 = (a- + a.2, b- + b2, c- + c2, d- + d-2) Subtraction
A--OA2 = (a- - a.2, b- - b2, c- - c2, d- - d-2)
(Aa, Ab, Ac, Ad) A> 0 Aa) A< 0
„ , .... . -r ((Aa, Ab, Ac, Scalar multiplication AA = fa fa
Division
A--0A2 = (a1/d2,b1/b2,c1/c2,d1/a,2) inverse JL-1 (1/dl1/c,1/b.1/a)
Ranking function
For ranking alternatives is used the function proposed by R.Yager: R(A)=i(a + d-4i + 2i) Let's Ai and Aj two TFN numbers ,
(i) R(Ai) > R(Aj) then At > Aj
(ii) r(Ai) < R(Aj) then At < Aj
(iii) r(Ai) = R(Aj) then At = Aj
Paiwise prefernces are defined as follows:
1=( 1,1,1,1) - two criteria are of absolutely equal importante
1 = (0,1,1.5,2) -one criterion is little more important than the other,
3 = (2,2.5,3.5,4) - one criterion is somewhat more important than the other,
5 = (4,4.5,5.5,6) - one criterion is significantly more important than another,
7 = (6,6.5,7.5,8) - one criterion is undeniably more important than another,
9 = (8,8.5,9,9) - one criterion is absolutely more important than another.
It is important to note that if the preference Q over Cj is 5, then the preference Cj over Q is 1/5, this relationship is called inverse symmetry. Logical transitivity must also be performed. If both conditions are met then the matrix is called consistency, otherwise the inconsistency. Fuzzy inverse symmetric prefernce will be 1-1= (1/2, 1/1,1,1) 3-1= (1/4,1/3.5, 1/2.5, 1/2) 5-1 = (1/6, 1/5.5, 1/4.5, 1/4) 7-1 = (1/8, 1/7.5 , 1/6.5, 1/6) 9-1 = (1/9, 1/9, 1/ 8.5, 1/8)
If is needed intermedia values can be used: 2 = (1,1.5,2.5,3) 4 = (3,3.5,4.5,5) 6 = (5,5.5,6.5,7) 8 = (7,7.5,8.5,9)
The corresponding inversy symmetric numbers 2-1= (1/3,1/2.5, 1/1.5, 1/1) 4-1= (1/5,1/4.5, 1/3.5, 1/3) 6-1= (1/7,1/6.5, 1/5.5, 1/5) 8-1= (1/9,1/8.5, 1/7.5, 1/7)
T. Saaty proposed to use as an aggregator average geometric mean
Gi = , (1)
where: Xi = CJCj fuzzy preference index of Q over Cj
Next for any row of pairwise preferences matrix we calculate aggregated index and weights of criterias by formulas:
S = Z?=1Gi, , Wi = Si , (2)
As result we have vector of fuzzy criterias weights :
w = (w1, w2, w3 .... vim) At second stage for any criterion Ci are determined pairwise comparison matrix (Fig 4.)
Ai A2 ... . ... . An
Ai 1 h/K A11An
A2 Ä2/Ä1 1 A2/An
A3 Ä3IÄ1 1
... 1
An AJAi 1
Fig. 4. Matrix ofpairwise comparison of alternatives For any matrix are calculated normalised preference indexes (2)
On base of these indexes the consolidated preference indexes matrix is builded, (Fig. 5.)
Ci C2 C3 Cm
A i Xu X12 X13 Xlm
A 2 X21 X22 X2m
A 3
An Xm X-n2 X-nT, ^nm
Fig. 5. The consolidated matrix ofpreferences indexes of all alternatives
At third stage on base of criterion weight for any alternative are calculated global preference indexes (3):
m
^ WiXn = Bi i=i
IT=lWiX2i=B2 (3)
m
^ WiX3i = Bn i=i
At last stage on base of ranking function is detrmined the alternative with maximum of global preference index.
3. Practice problem solving
As practice problem is considered software selection problem [13-14]. Main criteria are: C1- functionality, C2 - price, C3 - usability. C4 - relialability
and four alternatives are proposed.
All calculation were implemented in Ms Excel (Fig. 6).
Fig. 6. FAHP computation model in Ms Excel According to FAHP method for 4 criteria were determined pairwise comparison matrix
Cl C2 C3 C4
Cl 1 2 3 2
C2 V33 1 V22 I2
C3 V3 2 1 2
C4 V2 2 % 1
At next step on base of formulas (1) and (2) are calculated weight coefficients of all criteria: W1 =(0.23, 0.33, 0.62, 0.85) W2 = ( 0.07,0.0.9,0.17, 0.27) W3 = ( 0.11,0.17, 0.33, 0.47) W4 = ( 0.09, 0.13, 0.27,0.42)
For any criterion we determine fuzzy pairwise comparison matrix and preference indexes Ci criterion
Ai A2 A3 A4 Preference index
Ai 1 V2 V3 2 (0.09, 0.13, 0.26, 0.40)
A2 2 1 2 V2 (0.13, 0.20, 0.42, 0.62)
A3 3 V3 1 2 (0.15, 0.23, 0.45, 0.67)
A4 1/3 2 1/2 1 (0.10, 0.14, 0.30, 0.47)
C2 criterion
A1 A2 A3 A4 Preference index
A1 1 V2 3 V2 (0.10, 0.14, 0.24, 0.33)
A2 5 1 V3 2 (0.18, 0.25, 0.43, 0.58)
A3 V3 3 1 2 (0.15, 0.21, 0.39, 0.52)
A4 2 V2 V2 1 (0.10, 0.15, 0.29, 0.44)
C3 criterion
A1 A2 A3 A4 Preference index
A1 1 V3 2 5 (0.11, 0.15, 0.27, 0.38)
A2 3 1 2 2 (0.17, 0.26, 0.50, 0.70)
A3 V2 V2 1 V2 (0.11, 0.15, 0.29, 0.48)
A4 V5 V2 2 1 (0.12, 0.17, 0.31, 0.45)
C4 criterion
A1 A2 A3 A4 Preference index
A1 1 5 3 V3 (0.20, 0.26, 0.42, 0.53)
A2 V5 1 5 2 (0.16, 0.21,0.33, 0.42)
A3 V3 V5 1 V3 (0.05, 0.07, 0.11, 0.14)
A4 3 V2 3 1 (0.18, 0.25,0.42, 0.57)
We consolidate all alternative preference indexes in one matrix.
C1 C2 C3 C4
A1 (0.09,0.13, 0.26, 0.40) (0.10, 0.14, 0.24, 0.33) (0.11, 0.15, 0.27, 0.38) (0.20, 0.26, 0.42, 0.53)
A2 (0.13,0.20, 0.42, 0.62) (0.18, 0.25, 0.43, 0.58) (0.17, 0.26, 0.50, 0.70) (0.16, 0.21,0.33, 0.42)
A3 (0.15,0.23, 0.45, 0.67) (0.15, 0.21, 0.39, 0.52) (0.11, 0.15, 0.29, 0.48) (0.05, 0.07, 0.11, 0.14)
A4 (0.10, 0.14, 0.30, 0.47) (0.10, 0.15, 0.29, 0.44) (0.12, 0.17, 0.31, 0.45) (0.18, 0.25,0.42, 0.57)
For any criterion on base of formula (3) is calculated global preference indexes of all alternatives:
Bi= (0.06, 0.12, 0.41, 0.83)
B2= (0.07, 0.16, 0.59,1.19)
B3= (0.06, 0.13, 0.47, 0.99)
B4= (0.06, 0.12, 0.45, 0.97)
For any alternative is calculated ranking function:
R(A1)= 0.533554102,
R(A2)= 0.76396875,
R(A3)= 0.634708217,
R(A4)= 0.618137562
Alternative A2 have maximum value of global preference index 0.763968, so A2 alternative is optimal.
Conclusions. The article is devoted to the problem of multi-criteria decision making for software selection. The analysis of existing methods for solving this problem is given. The fuzzy AHP method is used for solving this problem. The issues of practical implementation of this method are discussed in details.
As test, the problem of software selection problem with 4 criteria and 4 alternatives is considered. The results of the solution at all stages are presented.
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