Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 102 (2016) 218 – 222
12th International Conference on Application of Fuzzy Systems and Soft Computing, ICAFS 2016, 29-30 August 2016, Vienna, Austria
Multi-attribute decision making based on z-valuation Rafig R.Aliyev* Durham University, UK
Abstract In this paper we investigate multi-attribute decision making problem, where the attribute values are Z-numbers, and the weight information on attributes are partially reliable. The presented method is based on overall criteria positive ideal and negative ideal solution of alternatives and distance between Z-vectors. Final decision alternative is selected on basis of degree of membership of candidates belonging to the positive ideal solution. A numerical example on multi-attribute decision making for Web Services selection is given to illustrate the solution processes of the suggested method. ©2016 2016Published The Authors. Published Elsevier B.V. © by Elsevier B.V. by This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility ofthe Organizing Committee of ICAFS 2016. Peer-review under responsibility of the Organizing Committee of ICAFS 2016 Keywords:Decision making; Z-number; Ideal solution; Distance between Z-vectors; Web services selection.
1. Introduction The process of multi-attribute criteria decision making (MADM) is to find the best all of the existing alternatives. The use of one or another multi-attribute decision theory depends mainly on decision making situations. One of the widely used theories to model human decisions is of fuzzy set theory. Some of the most popular theories that emerged for uncertainty modeling were fuzzy sets1,2,3,4 and possibility theory5,6 , the rough set theory7, Dempster8, 5 and Shafer’s10 evidence theories. It is needed to find the relevant methodology suitable for a particular problem11,12.Some of these methods are the Analytical Hierarchy Process (AHP)13 , Analytic Network Process (ANP)14, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)15, Multi-criteria Optimization and Compromise Solution (VIKOR)16, Simple Additive Weighting Method (SAW)17, Elimination Et Choice Translating EReality (ELECTRE)18, Preference Ranking Organization METHODS for Enrichment Evaluations (PROMETHEE)19, Fuzzy expert systems20 etc. Unfortunately, up to day there are scarce research on multi-attribute
*
Corresponding author, Tel: +99412 493 45 38 E-mail address:
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1877-0509 © 2016 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016 doi:10.1016/j.procs.2016.09.393
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decision making under Z-environment21 . In this paper, we suggest a new approach to study of multi-attribute decision analysis using Z-number concept. The rest of paper is structured as fallows. In Section 2 we present some prerequisites material on Z-number. In Section 3 we describe the statement of the problem and the suggested approach to MADM with Z-information. In Section 4 we illustrate an application of the suggested approach to a real-world investment problem. Finally, conclusions are given Section 5. 2. Preliminaries Definition 1.A Continuous Z-number22.A continuous Z-number is an ordered pair Z ( A, B ) where A is a continuous fuzzy number which describes a fuzzy constraint on values that a random variable X may take: X is A and B is a continuous fuzzy number with a membership function P B : > 0,1@ o > 0,1@ , which describes a fuzzy constraint on the probability measure of A : P ( A) is B Definition 2. A discrete Z-number23,24,25,26. A discrete Z-number is an ordered pair Z ( A, B ) where A is a discrete fuzzy number which describes a fuzzy constraint on values that a random variable X may take: X is A , and B is a discrete fuzzy number with a membership function P B : ^b1 ,..., bn ` o > 0,1@ , ^b1 ,..., bn ` > 0,1@ , which describes a fuzzy constraint on the probability measure of A : P ( A) is B . Definition3. A distance between Z-number-valued vectors. The distance between Z-number valued vectors is Z1
( Z11 , Z12 ,..., Z1n ) and Z 2
( Z 21 , Z 22 ,..., Z 2 n ) defined as
D( Z1 , Z 2 )
max i
d ( Z1i , Z 2i )
1 m § 1 n · L L R R L L R R ¨ n 1 ¦ a1iDk a2iD k a1iDk a2iD k m 1 ¦ b1iD k b2iD k b1iD k b2iD k ¸ , k 1 k 1 © ¹
where aiLD
1,..., n
d ( Z1i , Z 2i ) ,
(1)
^
min AiD , aiRD
max AiD , biLD
`
^
min BiD , biRD
max BiD .
`
(2)
2. Statement of the problem and its solution.
^ A1 , A2 ,..., An `
Assume that A
is a set of alternatives and C
^C1 , C2 ,..., Cm `
is a set o attributes. Every
attribute C j , j 1, m is characterised by weight W j assigned by expert or decision maker. As we deal with Z-
information valued decision inviroment, the characteristic of the alternative Ai , i 1, n on attribute C j j 1, m is
described by the form Ai
^(Z ( A , B i1
i1
), Z ( Ai 2 , Bi 2 ),..., Z ( Aij , Bij ), Z ( Aim , Bim )`
(3)
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where Z ( Aij , Bij ) is evaluation of an alternative Ai with respect to a attribute C j . Value of attributes and weights of attributes are usually derived from decision maker or experts and are vague and characterized with partial reliability. In this case, the weights W j , i 1, m are represented as
^Z ( A
Wj
w j
, B wj )` , j 1, m
(4)
Where Awj is value of weight of j -th is attribute, B wj is reliability of this value. Hence, we can represent decision matrix Dnmx as Table 1. Table 1.
Dnxm
A1
A2
An
> Z ( A21 , B21 )@ > Z ( A22 , B22 )@
C2 > Z ( A12 , B12 )@
C1 > Z ( A11 , B11 )@
> Z ( An1 , Bn1 )@ > Z ( An 2 , Bn 2 )@
Cm > Z ( A1m , B1m )@
> Z ( A2 m , B2 m )@
> Z ( Anm , Bnm )@
The common approach in the MADM is the use of the utility theories. This approach leads to transformation of a vector-valued alternative to a scalar-valued quantity. This transformation leads to loss of information. It is related to restrictive assumptions on preferences underlying utility models. In human decision it is not needed to use artificial transformation. In this case we will use the concept of positive and negative ideal point in multi-attribute decision making15. We present an ideal Z-point for attributes as Aidp
Z A
id p1
id id , B idp1 , Z Aidp 2 , B idp 2 ,..., Z Apm , B pm .
(5)
A negative attributes ideal point will be discribed as Aidp
Z A
id N1
, BNid1 , Z ANid2 , BNid2 ,..., Z ANidm , BNidm
Solution at the stated decision making problem, i.e. choice best alternative among A
(6)
^ A1 , A2 ,..., An `
consist of
the following steps: 1. Weighted distances d ip i -th alternative and positive ideal solutions (5) is defined by (1). 2. Weighted distances diN between i -th alternative and negative ideal solutions (5) is defined by (1). 3. Degree of membership ri , i 1, n ,of each alternatives belonging to the positive ideal solution is calculated. For this (7 ) is used27:
ri
1 § dip · 1 ¨ ¸ © diN ¹
(7)
Rafig R. Aliyev / Procedia Computer Science 102 (2016) 218 – 222
4. Final decision alternative is selected as max(ri ) , i 1, n 3. Practical example
We consider multi-attribute decision making for Web services selection problem21 . Today a wide variety of services are offered that can satisfy quality of services for agents. The number of options, i.e. Web services is 8 A1 , A2 , A3 … A8 . An agent has to make a decision taking into account 5 attributes C1 (cost), C2 (time), C3 (reliability), C1 (availability), C5 (repetition). In this case all 8 alternatives are evaluated under 5 attributes by Z-numbers. Components of these Z-numbers are presented by triangle fuzzy number and scaled decision matrix shown in Tables 2,3. Table 2. Decision matrix A1 A2 A3 A4 A5 A6 A7 A8
C1 (0.450.50.55)(0.50.60.7) (0.1260.140.154)(0.50.60.7) (0.2250.250.275)(0.50.60.7) (0.6120.680.748)(0.50.60.7) (0.3330.370.407)(0.50.60.7) (0.4320.480.528)(0.50.60.7) (0.7380.820.902)(0.70.80.9) (0.5310.590.649)(0.50.60.7)
C2 (0.4410.490.539)(0.50.60.7) (0.5310.590.649)(0.50.60.7) (0.7110.790.869)(0.70.80.9) (0.6030.670.737)(0.50.60.7) (0.2250.250.275)(0.50.60.7) (0.5490.610.671))(0.50.60.7) (0.3240.360.396))(0.50.60.7) (0.3780.420.462)(0.50.60.7)
C3 (0.6210.690.759))(0.50.60.7) (0.4230.470.517))(0.50.60.7) (0.270.30.33))(0.50.60.7) (0.3780.420.462))(0.50.60.7) (0.5220.580.638))(0.50.60.7) (0.6210.690.759))(0.50.60.7) (0.5220.580.638))(0.50.60.7) (0.6480.720.792)(0.70.80.9)
Table 3. Decision matrix
C4
C5
A1
(0.7020.780.858)(0.50.60.7)
(0.1260.140.154)(0.50.60.7)
A2
(0.5850.650.715)(0.50.60.7)
(0.8280.921.012)(0.70.80.9)
A3
(0.7470.830.913)(0.70.80.9)
(0.5760.640.704)(0.50.60.7)
A4 A5 A6 A7 A8
(0.4050.450.495)(0.50.60.7) (0.3510.390.429)(0.50.60.7) (0.6210.690.759)(0.50.60.7) (0.2160.240.264)(0.70.80.9) (0.5220.580.638)(0.50.60.7)
(0.3420.380.418)(0.50.60.7) (0.2430.270.297)(0.50.60.7) (0.7020.780.858)(0.50.60.7) (0.3240.360.396))(0.50.60.7) (0.2520.280.308)(0.50.60.7)
For the simplicity the weights vector of the 5 attributes is given as weight for C1 is W1 0.3 , for C2 is W2 0.2 , for C3 is W3 0.12 , for C4 is W4 0.18 and for C5 is W2 0.2 The positive ideal alternative is presented as Aidp ((0.738 0.82 0.902)(0.7 0.8 0.9),(0.711 0.79 0.869)(0.7 0.8 0.9),(0.648 0.72 0.792)(0.7 0.8 0.9),(0.747 0.83 0.913)(0.7 0.8 0.9),(0.828 0.92 1.012) (0.7 0.8 0.9)) The negative ideal alternative is presented as ANid ((0.126 0.14 0.154)(0.5 0.6 0.7), (0.225 0.25 0.275)(0.5 0.6 0.7), (0.27 0.3 0.33)(0.5 0.6 0.7), (0.216 0.24 0.264)(0.5 0.6 0.7), (0.126 0.14 0.154)(0.5 0.6 0.7)) According to the (1 )-(2 ) weighted distances between Z-vectors of alternatives and positive ideal solution Z-vector are obtained as d p1 0.32 d p 2 0.42 d p 3 0.375 d p 4 0.24 d p 5 0.315 d p 6 0.261 d p 7 0.246 d p 8 0.274 Analogously we have obtained weighted distances between Z-vectors of alternatives and negative ideal solution Z- vector:
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d N 1 0.18 d N 2 0.32 d N 3 0.24 dN 4 d N 5 0.114 d N 6 0.22 d N 7 0.429 dN 8 The membership degree ri , i 1,8 are calculated according to (7 ) and have obtained r1 0.27 r2 0.37 r3 0.29 r4 0.56 r5 0.12 r6 0.42 r7 0.75 r8 0.4 The final decision is determined as max(r1 , r2 , r3 , r4 , r5 , r6 , r7 , r8 ) 0.75 The best alternative is A7
0.27 0.225
4. Conclusion
Despite a lot of methods have been developed to deal with interval and fuzzy MADM unfortunately, today research on multi-attribute decision making under Z-information is scarce. The mentioned above dictated to create new approach for MADM under decision situation where decision relevant information are characterized by fuzzy uncertainty and partial reliability. For this purpose we have suggested MADM procedure based on overall criteria positive ideal and negative ideal solutions of alternatives, distance between Z-vectors and Z-information processing. Numerical example on MADM Web services selection problem demonstrates applicability and efficiency of proposed method. References 1. Zadeh LA. Fuzzy sets. Information and control. 1965;8: 338-353. 2. Aliev RA, Mamedova GA, AlievRR. Fuzzy Sets Theory and its application .Tabriz: Tabriz University; 1993. 3. Aliev RA, Alizadeh AV, Guirimov BG. Unprecisiated information-based approach to decision making with imperfect information. Proc. 9th Int. Conf. on Application of Fuzzy Systems and Soft Computing 2010; 387-397. 4. Aliev RA. Fundamentals of the fuzzy logic-based generalized theory of decisions.New York, Berlin: Springer;2013. 5. Zadeh LA. A computational approach to fuzzy quantifiers in natural languages. Comput Math Appl 1983; 9:149-184. 6. Aliev RA , Pedrycz W., Huseynov OH. Decision theory with imprecise probabilities.Int J Inf Tech Decision 2012; 271-306. 7. Pawlak Z. Rough sets. Int J Comput Inf Sci 1982; 11:341-356. 8. Dempster AP. Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 1967; 325-339. 9. Dempster AP. Upper and lower probability inferences based on a sample from a finite univariate population. Biometrika 1967; 54: 515- 528. 10. Shafer G. A mathematical theory of evidence.Princeton: Princeton university press.;1976. 11. Szmidt E. Distances and similarities in intuitionistic fuzzy sets. Springer; 2014. 12Aliev.RA, Tserkovny A. Systemic approach to fuzzy logic formalization for approximate reasoning. Inform Sciences 2011; 1045-1059. 13. Saaty TL. How to make a decision: the analytic hierarchy process. Eur J Oper Res 1990; 48: 9-26. 14. Saaty TL Analytic network process. In Encyclopedia of Operations Research and Management Science Springer US 2001;117: 28-35. 15. Hwang C, Yoon K. Multiple attribute decision making methods and application,New York: Springer;1981. 16. Opricovic S.,Tzeng GH. Extended VIKOR method in comparison with outranking methods.Eur J Oper Res 2007; 178:514-529. 17. Chou SY, Chang YH, ShenCY.A fuzzy simple additive weighting system under group decision-making for facility location selection with objective/subjective attributes.Eur J Oper Res 2008; 189: 132-145. 18. Roy B. ELECTRE III: Unalgorithme de classements fon désurune représentationfloue des préférences en présence de criteres multiples. Cahiers du CERO1978; 20:3-24. 19. Brans JP, Vincke P, Mareschal B. How to select and how to rank projects: The PROMETHEE method.Eur J Oper Res 1986; 24: 228-238. 20. Aliev RA. Fuzzy expert systems.Soft computing, Prentice-Hall, Inc. 1994.p. 99-108. 21. Aliyev RR. Similarity based multi-attribute decision making under Z-information. Eighth Int. Conf. on Soft Computing with Words and Perceptions in System Analysis, Decision and Control, Antalya, Turkey. 2015. p.33-39. 22. Zadeh. LA. A note on a Z-number, Inform Sciences 2011; 181:2923-2932. 23.Aliev RA, Alizadeh AV, Huseynov OH, The arithmetic of discrete Z-numbers. Inform. Sciences 2015; 290: 134-155. 24. Aliev RA, Alizadeh AV, Huseynov OH, Jabbarova KI. Z-number based Linear Programming.Int J IntellSyst 2015; 30:563-589. 25. Aliev RA, Huseynov OH.Decision theory with imperfect information. Singapour: World Scientific; 2014. 26. Aliev RA, Huseynov OH, Aliyev RR, Alizadeh AV. The Arithmetic of Z-numbers. Theory and Applications. Singapore: World Scientific; 2015. 27. Tong H., Zhang S. A Fuzzy Multi-attribute Decision Making Algorithm for Web Services Selection Based on QoS.IEEE Asia-Pacific Conference on Services Computing 2006; p. 51 – 57.
