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2ENGINEERING SCIENCES
APPLICATION OF FUZZY ELECTRE METHOD WITH TRAPEZOIDAL 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/30082021/7654
ABSTRACT
The article is devoted to the problem of multi-criteria decision-making. 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, product and their various modifications, the second group includes - AHP, ELECTRE, TOPSIS, PROMETHEE, ELECTRE. For many problems assessment of the criteria implemented by experts and presented in linguistic form. The effective approach for dealing with linguistic information is fuzzy set theory proposed by L. Zadeh. In this paper is proposed fuzzy ELECTRE method. This method is presented in details. As application problem is used the equipment selection problem 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. (2021) Application of Fuzzy Electre Method with Trapezoidal Fuzzy Numbers. World Science. 8(69). doi: 10.31435/rsglobal_ws/30082021/7654
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. 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 - Analytical 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) [5-17]. The work [3] provides information about popularity of various methods of multi-criteria decision-making. This paper discusses the ELECTRE method.
The ELECTRE method was developed by group of the French scientists led by professor B. Rua at the end of 60th years This method was very popular for solving multi-criteria problem under certain conditions. In general the ELECTRE method is based on the approach of pairwise comparison of alternatives.
ARTICLE INFO
Received: 16 June 2021 Accepted: 05 August 2021 Published: 30 August 2021
KEYWORDS
multi-criteria decision making, alternative, criterion, fuzzy ELECTRE method, pairwise, concordance, discordance.
The fuzzy ELECTRE method was developed by Chen in 2006 [6] for problem with linguistic uncertainty.
2. Description of the method.
We consider the problem where decision DM makes decisions in linguistic form. Consider all stages of fuzzy ELECTRE method:
1. First we define linguistic variables for criterion weight importance (Table 1) and the decisions with fuzzy trapezoidal numbers (Table 2).
Table 1. Linguistic variables of criterion weights
Linguistic Variables Trapezoidal Fuzzy Numbers
Very Low (VL) (0,0.1,0.2.0.3)
Low (L) (0.1,0.3,0.45,0.7)
Medium (ML) (0.4,0.5,0.7,0.8)
High (H) (0.5,0.6,0.75,0.85)
Very High (VH) (0.6,0.7,0.8,0.9)
Table 2. Linguistic variables for the decision
Linguistic Variables Trapezoidal Fuzzy Numbers
Very Poor (VP) (0,1,2,3)
Poor (P) (1,3,4.7)
Medium Poor (MP) (4,5,7,8)
Good (G) (7,8,9.9.25)
Very Good (VG) (9, 9.25, 9.5,10)
2. Present the linguistic decisions as the matrix of outcomes (alternatives - criteria) n -number of criteria m - number of alternatives (Table 3).
able 3. MCDM problem representation
Ci C2. C3 r
Ai X11 X1n
A2 %22 X23 X2n
A3 X32 X33 %3n
Am Xm2 Xm3 Xmn
Where Xij = (a^, bij, c^j, dij) is fuzzy trapezoidal representation of linguistic terms.
3. Calculate normalized matrix
R = (rij),i = 1,2, ...m; j = 1,2, ■■■n
4. The normalized fuzzy decision matrix is calculated with the formulas given below, where J and J^epresent the maximization criteria set, and minimization criteria set respectively.
(1)
fij = (^ , 71 ,TT. ),(2) d*j = maxidij,] e J (3)
a*j = minatj,] e J1 (4)
5. Calculate weighted decision matrix
V = (vij),i = 1,2,...n (5)
-O-ij b1L cJI dij.
u] ' j* dj } j* d* ' d*'
,aj * a.j * a.j * aj s
' cu ' a-ij'
Where
V(j = Vu ,i = 1,2, ...,m; j = 1,2,
.n
6. Determine concordance set ]c (set is all criteria in which alternative k is superior than alternative I) can be determined by following criteria
Ckl = {j^kj > Vlj]
(6)
7. Determine discordance set Ja (set is all criteria in which alternative k not is superior than alternative I) can be determined by following criteria
dki = ti^kj <vij} (7)
8. Determine the concordance matrix where elements is calculated by formula
Ckl=ZjeCkl™j (8)
9. Determine the discordance matrix where elements is calculated by formula
maxlvkrvijl
dkl =
jZDkl
maxlvkj—vijl ]
10. Determine average concordance index
m(m-1)
11. Determine average discordance index
D. =
1
m(m-1)
Lk=1^l=1ckl
Lk=1^l=1akl
12. Determine Boolean concordance matrix F
if cki > c* then fki = 1 oth.ewh.ise fki = 0
13. Determine Boolean discordance matrix G
if dki > d* then gki = 1 othewhise gki =
(9)
(10) (11)
(12) (13)
14. Calculate global preference matrix E by multiplication E = FG
15. Determine alternative with max preference by calculation sum of preference indexes by row of global matrix
3. Practical example.
As practice problem we consider equipment selection problem with following 4 criteria and 3 alternatives:
C1- price
C2- noise level
C3- usability
C4- dimension
As seen for C3 optimal decision is maximum for other three criteria is minimum.
Consider application of fuzzy ELECTRE method for this problem. All computations were performed in Ms Excel.
1. Presentation of decisions in linguistic decision matrix (Table 4)
Table . Linguistic decision matrix
C1 C2 C3 C4
A1 VG G VG MP
¿2 MP G G VG
A3 G VG MP G
The vector of criteria importance is presented as follows
w = (ML, H, VH, H) 2. Convert linguistic presentation in trapezoidal fuzzy numbers (Table 5)
1
Table 5. linguistic presentation in trapezoidal fuzzy numbers
Ci C2 C3 C4
A1 (9, 9.25, 9.5,10) (7,8,9.9.25) (9, 9.25, 9.5,10) (4,5,7,8)
A2 (0.4,0.5,0.7,0.8) (7,8,9.9.25) (4,5,7,8) (9, 9.25, 9.5,10)
A3 (7,8,9.9.25) (9, 9.25, 9.5,10) (4,5,7,8) (7,8,9.9.25)
w = (0.4,0.5,0.7,0.8) (0.5,0.6,0.75,0.85) (0.6,0.7,0.8,0.9) (0.5,0.6,0.75,0.85)
3. Calculate normalized fuzzy decision matrix by corresponding formulas (Table 6)
Ci C2 C3 C4
Ai (0.40,0.42,0.43,0.44) (0.76, 0.78,0.88,1) (0.9,0.93,0.95,1) (0.5,0.57,0.8,1)
A2 (0.5, 0.57, 0.8, 1) (0.76,0.78,0.88, 1) (0.7,0.8,0.9,0.93) (0.4,0.42,0.43,0.44)
A3 (0.43,0.44,0.5,0.57) (0.7,0.74,0.76, 0.78) (0.4,0.5,0.7,0.8) (0.43,0.44,0.5,0.57)
4. Calculate weighted normalized fuzzy decision matrix (Table 7)
Table 7. Weighted normalized fuzzy decision ^ matrix
Ci C2 C3 C4
Ai (0.16,0.21,0.3, 0.36) (0.38,0.47,0.66,0.8) (0.54,0.65,0.76,0.9) (0.25,0.34,0.60.0.85))
A2 (0.2,0.29,0.56,0.8) (0.38,0.47,0.66,0.8) (0.42,0.56,0.72,0.83) (0.2,0.25,0.32,0.38)
A3 (0.17,0.22,0.35,0.46) (0.35,0.44,0.57,0.62) (0.24.0.35,0.56,0.72) (0.22,0.27,0.38,0.49)
For ranking alternatives we have used following method / 7/
1 1 R(A)=-(a + b+-(d-c))
Let Âi and Âj two fuzzy numbers,
(i) R(Âi) > R(Âj) then Ât > Âj (ii) R(Ai) < R(Aj) then Ât < Âj (13)
(iii) R(Âi) = R(Âj) then Ât = Âj 5. Determine concordance and discordance sets
For determine concordance and discordance sets we use formulas (6) and (7) As result we have got set of concordance and discordance sets (Table 8)
Table 8. Concordance and discordance sets
Concordance set Discordance set
C(1,2)=(2,3,4) D(1,2)=(1)
C(1,3)=(2,3,4) D(1,3)=(1)
C(2,1)=(1,2) D(2,1)=(3,4)
C(2,3)=(1,2,3) D(2,3)=(4)
C(3,1)=(1) D(3,1)=(2,3,4)
C(3,2)=(4) D(3,2)=(1,2,3)
Calculate concordance indexes matrix by formula (8). (Table 9)
Table 9. Concordance indexes matrix
A1 ¿2 ¿3
A1 (0,0,0,0) (1.6,1.9,2.3,2.6) 1.6,1.9,2.3,2.6)
A2 (1.5,1.8,2.25,2.55) (0,0,0,0) (0.42,0.56,0.72,0.83)
A3 (0.4,0.5,0.7,0.8) (0.5,0.6,0.75,0.85) (0,0,0,0)
and a discordance indexes matrix by formula (9) (Table 10)
Table 10. Discordance indexes matrix
A1 A2 ¿3
A1 (0,0,0,0) (0.8,0.89,0.93,0.94) (0.033,0.033,0.25,0.56)
A2 (1,1,1,1) (0,0,0,0) (0.111,0.095,0.375,1)
A3 (1,1,1,1) (1,1,1,1) (0,0,0,0)
Next we calculate average concordance index by formula (10) and average discordance index (11) respectively. Calculate Boolean preference concordance matrix F (Table 11)
Table 11. Boolean preference concordance matrix
A1 A2 ¿3
A1 0 1 1
A2 1 0 0
A3 0 0 0
and Boolean preference discordance matrix G (Table 12).
Table 12. Boolean preference discordance matrix
A1 A2 ¿3
A1 0 1 0
A2 1 0 0
A3 1 1 0
Finally calculate and global preferences matrix E (Table 13)
Table 13. Global pref erences matrix
A1 A2 ¿3
A1 0 1 0
A2 1 0 0
A3 0 0 0
As we see two alternatives result we have alternatives A1 and A2 have same preference, it means that problem have two solutions.
Conclusions. The article is devoted to the problem of multi-criteria decision making for equipment selection. The analysis of existing methods for solving this problem is given. The fuzzy ELECTRE is used as a method for solving this problem. The issues of practical implementation of this method are discussed in details.
As practical problem the equipment selection problem with 4 criteria and 3 alternatives is considered. The results of the solution at all stages are presented.
REFERENCES
1. Hwang, CL Yoon, K. (1981), Multiple Attribute Decision Making: Methods and Applications. New York: Springer- Verlag.
2. V. Belton and T. Stewart (2002), Multiple criteria decision analysis: an integrated approach. Springer Science &Business Media.
3. A. Mardani, A. Jusoh, Khalil MD Nor, Z. Khalifah, N. Zakwan, A. Valipour (2015), Multiple criteria decision-making techniques and their applications - a review of the literature from 2000 to 2014, ISSN: 1331-677X (Print)., https://doi.org/10.1080/1331677X.2015.1075139
4. Chakraborty, S., & Zavadskas (2014), EK Applications of WASPAS method in manufacturing decision making. Informatica, 25 (1), 1-20. https://doi.org/10.15388/Informatica.2014.01
5. A. T. Almeida (2007), "Multicriteria decision model for outsourcing contracts selection based on utility function and ELECTRE method," Computers and Operations Research, vol. 34, no. 12, pp. 3569-3574. https://doi.org/10.1016/j.cor.2006.01.003
6. Chen, C. T., Lin, C. T. and Huang, S. F. (2006), "A Fuzzy Approach for Supplier Evaluati on and Selection in Supply Chain Management" International Journal of Production Economics, 102(2):289-301. https://doi.org/10.1016/jijpe.2005.03.009
7. X. Wang and E. Triantaphyllou (2008), "Ranking irregularities when evaluating alternatives by using some ELECTRE methods," Omega, vol. 36, no. 1, pp. 45-63. https://doi.org/ 10.1016/j.omega.2005.12.003
8. L. Fei, J. Xia, Y. Feng, and L. Liu (2019), "An ELECTRE-Based Multiple Criteria Decision Making Method for Supplier Selection Using Dempster-Shafer Theory," IEEE Access, vol. 7, pp. 84701-84716. https://doi.org/10.1109/ACCESS.2019.2924945
9. M. C. Wu and T. Y. Chen (2009), "The ELECTRE multi criteria analysis approach based on intuitionistic fuzzy sets," in IEEE International Conference on Fuzzy Systems, pp. 1383-1388. https://doi.org/10.1016Zj.eswa.2011.04.010
10. F. Dammak, L. Baccour, A. Ben Ayed, and A. M. Alimi (2017), "ELECTRE method using interval-valued intuitionistic fuzzy sets and possibility theory for multi-criteria decision making problem resolution," in IEEE International Conference on Fuzzy Systems. https://doi.org/10.1109/FUZZ-IEEE.2017.8015408
11. Almeida, A. T. (2007), Multicriteria decision model for outsourcing contracts selection based on utility function and ELECTRE. Computers & Operations Research, 34 (12), 3569-3574. https://doi.org/10.1016/j.cor.2006.01.003
12. Asghari, F., Amidian, A. A., Muhammadi, J. and Rabiee, H. (2010), "A Fuzzy ELECTRE Approach for Evaluating Mobile Payment Business Models" The Fourth International Conference on Management of eCommerce and e-Government, October 23-24, China. https://doi.org/10.1109/ICMeCG.2010.78
13. Hatami-Marbini, A. and Tavana, M. (2011), "An Extension of the ELECTRE I Method for Group Decision-Making under a Fuzzy Environment" Omega, 39:373-386. https://doi.org/10.1016/j.omega.2010.09.001
14. Vahdani B. and Hadipour H. (2011) "Extension of the ELECTRE Method Based on Interval-valued Fuzzy Sets" Soft Computing, 15: 569-579. https://doi.org/10.1007/s00500-010-0563-5
