ELSEVIER

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. © 2016Publishedby ElsevierB.V. This isanopenaccess 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

Keywords:Decision making; Z-number; Ideal solution; Distance between Z-vectors; Web services selection.

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Available online at www.sciencedirect.com

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Procedia Computer Science 102 (2016) 218 - 222

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: rafig.aliyev@hotmail.com

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

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

andB is a continuous fuzzy number with a membership function juB : [0,1] ^ [0,1], which describes a fuzzy constraint on the probability measure of A : P( A) isB

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 ,

andB is a discrete fuzzy number with a membership function juB : {b1,...,bn}—»[0,1], {b1,...,bn}c[0,1], which describes a fuzzy constraint on the probability measure of A :

P(A)isB .

Definition3. A distance between Z-number-valued vectors. The distance between Z-number valued vectors is Z1 = (Z11,Z12,...,Z1n) and Z2 = (Z21,Z22,...,Z2n) defined as

where aL = min A a, aR = max A a, bL = min B a, bR = max B a .

ia i ' ia i ' ia i ' ia i

2. Statement of the problem and its solution.

Assume that A = {A1,A2,...,An} is a set of alternatives and C = {C1,C2,...,Cm} is a set o attributes. Every attribute Cj, j = 1, m is characterised by weight Wj assigned by expert or decision maker. As we deal with Z-information valued decision inviroment, the characteristic of the alternative At, i = 1, n on attribute Cj = 1, m jis described by the form

i»^ Z2 ) = maxi=1.....n d(Z1i, Z2i) ,

i =1,...,n

d (Z,, Z 2i)

where Z(Aij, Btj) is evaluation of an alternative At with respect to a attribute Cj. 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 Wj ,i = 1,m are represented as

Wj = {Z (Aj, Bw)}, j - 1,m (4)

Where Aj is value of weight of j -th is attribute, Bj is reliability of this value. Hence, we can represent decision matrix Dnmx as Table 1.

Table 1.

C C2 - Cm

A L z (An, Bu)J [ Z (A12, B12) J ^ \Z (A1m , B1m )J

A [ z (A21, B21)] [ Z (A22, B22)] ^ [Z(A2m , Bm )]

A„ [ Z (An1, Bm)] [ Z (A, 2, B„ 2)] ••• [ 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

Ap =(z(A£,Bpi),Z(A£,Bp2),...,Z(Apm,Bpm)) • (5)

A negative attributes ideal point will be discribed as

Ap =(Z (A^, BN1), Z (A£ , BN2),..., Z (A^, Bl)) (6)

Solution at the stated decision making problem, i.e. choice best alternative among A - {A1, A^,..., An} consist of the following steps:

1. Weighted distances ip i -th alternative and positive ideal solutions (5) is defined by (1).

2. Weighted distances iN between i -th alternative and negative ideal solutions (5) is defined by (1).

3. Degree of membership r, i -1, n ,of each alternatives belonging to the

positive ideal solution is calculated. For this (7 ) is used27:

fd ^ 1 + '

4. Final decision alternative is selected as max(r(),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 ... A88. 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

c, c, C3

Ai (0.45 0.5 0.55) (0.5 0.6 0.7) (0.441 0.49 0.539)( 0.5 0.6 0.7) (0.621 0.69 0.759))( 0.5 0.6 0.7)

A2 (0.126 0.14 0.154) (0.5 0.6 0.7) (0.531 0.59 0.649)( 0.5 0.6 0.7) (0.423 0.47 0.517))( 0.5 0.6 0.7)

A3 (0.225 0.25 0.275)( 0.5 0.6 0.7) (0.711 0.79 0.869) ( 0.7 0.8 0.9) (0.27 0.3 0.33))( 0.5 0.6 0.7)

A4 (0.612 0.68 0.748)( 0.5 0.6 0.7) (0.603 0.67 0.737)( 0.5 0.6 0.7) (0.378 0.42 0.462))( 0.5 0.6 0.7)

As (0.333 0.37 0.407) (0.5 0.6 0.7) (0.225 0.25 0.275 )( 0.5 0.6 0.7) (0.522 0.58 0.638))( 0.5 0.6 0.7)

Ae (0.432 0.48 0.528)( 0.5 0.6 0.7) (0.549 0.61 0.671))( 0.5 0.6 0.7) (0.621 0.69 0.759))( 0.5 0.6 0.7)

A7 (0.738 0.82 0.902)( 0.7 0.8 0.9) (0.324 0.36 0.396))( 0.5 0.6 0.7) (0.522 0.58 0.638))( 0.5 0.6 0.7)

a8 (0.531 0.59 0.649)( 0.5 0.6 0.7) (0.378 0.42 0.462) ( 0.5 0.6 0.7) (0.648 0.72 0.792)(0.7 0.8 0.9)

Table 3. Decision matrix

Ai (0.702 0.78 0.858) (0.5 0.6 0.7) (0.126 0.14 0.154)( 0.5 0.6 0.7)

A2 (0.585 0.65 0.715) (0.5 0.6 0.7) (0.828 0.92 1.012)( 0.7 0.8 0.9)

A3 (0.747 0.83 0.913)( 0.7 0.8 0.9) (0.576 0.64 0.704) ( 0.5 0.6 0.7)

A4 (0.405 0.45 0.495)( 0.5 0.6 0.7) (0.342 0.38 0.418)( 0.5 0.6 0.7)

As (0.351 0.39 0.429) (0.5 0.6 0.7) (0.243 0.27 0.297 )( 0.5 0.6 0.7)

Ae (0.621 0.69 0.759)( 0.5 0.6 0.7) (0.702 0.78 0.858)( 0.5 0.6 0.7)

A7 (0.216 0.24 0.264)( 0.7 0.8 0.9) (0.324 0.36 0.396))( 0.5 0.6 0.7)

A8 (0.522 0.58 0.638)( 0.5 0.6 0.7) (0.252 0.28 0.308) ( 0.5 0.6 0.7)

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

Apd = ((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

A'd = ((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 , = 0.32 d 2 = 0.42 d 3 = 0.375 d 4 = 0.24

n 1 n/. ni n 4

dn5 = 0.315 dn6 = 0.261 d n7 = 0.246 d„s = 0.274

Analogously we have obtained weighted distances between Z-vectors of alternatives and negative ideal solution Z- vector:

dN1 = 0.18 dN 2 = 0.32 dN3 = 0.24 dN4 = 0.27

dN5 = 0.114 dN 2 = 0.22 dN7 = 0.429 dN8 = 0.225

The membership degree r, i = 1,8 are calculated according to (7 ) and have obtained

rx = 0.27 r2 = 0.37 r, = 0.29 r = 0.56 r5 = 0.12 r6 = 0.42 r = 0.75 r8 = 0.4

The final decision is determined as

max(r1,r2,r,,rA,r5,r6,r7,r8 ) = 0.75

The best alternative is A

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.

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