Fuzzy Similarity based Fuzzy TOPSIS with Multi-distances

Pasi Luukka

, Mario Fedrizzi

, Leoncie Niyigena

and Mikael Collan

School of Business, Lappeenranta University of Technology, Lappeenranta, Finland

Department of Industrial Engineering, University of Trento, Via Mesiano 77, I-38123 Trento, Italy

Laboratory of Applied Mathematics, Lappeenranta University of Technology, Lappeenranta, Finland

Keywords:

Fuzzy Similarity, Fuzzy TOPSIS, Multi-distances, OWA, O’Hagan’s Method.

Abstract:

This article introduces a new extension to fuzzy similarity based fuzzy TOPSIS that uses multi-distance in

ranking. OWA is used in the aggregation process. For the weight generation in OWA the O’Hagan’s method

is used to ﬁnd optimal weights. Several different, predeﬁned orness values are tested. The presented method

is numerically applied to a research & development project selection problem.

1 INTRODUCTION

This paper investigates and presents a new exten-

sion of the fuzzy similarity based fuzzy Technique

for Order Performance by Similarity to Ideal Solution

(fuzzy TOPSIS). Fuzzy TOPSIS was originally intro-

duced by Chen in (Chen, 2000) and later extended

to include trapezoidal fuzzy numbers in (Chen et al.,

2006). In these contributions a vertex based fuzzy

distance method was used as a measure of distance

from (”similarity to”) the ideal solutions. A similarity

measure based version of fuzzy TOPSIS was intro-

duced in (Luukka, 2011), where the similarity (dis-

tance from) to the ideal solutions is calculated by us-

ing a fuzzy similarity measure. This strain of research

was continued by (Niyigena et al., 2012), where two

different fuzzy similarity measures were considered

and by (Collan and Luukka, 2013), where four fuzzy

similarity measure based fuzzy TOPSIS variants and

a way of holistic overall ranking of projects was pre-

sented.

The fuzzy TOPSIS uses fuzzy numbers as inputs

and is thus able to incorporate inaccurate and impre-

cise information in the analysis (not having to sim-

plify reality by using crisp numbers). The main dif-

ference in using the fuzzy similarity measures and the

(crisp) distance measures in the TOPSIS environment

with fuzzy numbers is that fuzzy similarity measures

can take into consideration more of the information

that is stored in the fuzzy number, e.g. with regards

to the perimeter and the area of the fuzzy number.

The crisp distance measures defuzzify the fuzzy num-

ber in order to calculate a distance between the re-

sulting crisp number and the ideal solution. Using

a crisp distance based measure may cause a loss of

relevant information. The fuzzy similarity measure

used here is introduced in Hejazi et al (Hejazi et al.,

2011) and can take into account the perimeter and

area of fuzzy numbers. This similarity measure was

previously studied in the context of fuzzy similarity

based TOPSIS method in (Niyigena et al., 2012) and

in (Collan and Luukka, 2013).

The new contribution of this paper concentrates

on the application of multi-distances in creating ad-

ditional information for project ranking by similar-

ity coefﬁcients, after they have been analyzed with

fuzzy similarity measure based fuzzy TOPSIS. Multi-

distances are used in analyzing the ”level” of similar-

ity between analyzed criteria. High level of similarity

between criteria means a low multi-distance and can

be interpreted as homogeneity or consistency of, e.g.,

performance or expectations. Such information may

be valuable in the analysis and offers an additional

differentiator between objects. Multi-distances were

examined by Martin and Mayor (Martin and Mayor,

2010), and presented as a generalization of the notion

of distance. Martin and Mayor proposed the construc-

tion of multi-distances by means of OWA functions in

(Martin et al., 2011). Using the multi-distance in the

aggregation will add a step of pairwise distance mea-

surement of similarities between criteria (values) in

the procedure.

Use of multi-distances with fuzzy TOPSIS is, to

the best of our knowledge a new approach.

The remainder of the paper is organized as fol-

lows. In Section 2 the fuzzy similarity relation

193

Luukka P., Fedrizzi M., Niyigena L. and Collan M..

Fuzzy Similarity based Fuzzy TOPSIS with Multi-distances.

DOI: 10.5220/0004552601930200

In Proceedings of the 5th International Joint Conference on Computational Intelligence (FCTA-2013), pages 193-200

ISBN: 978-989-8565-77-8

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

between fuzzy numbers, the OWA operator, multi-

distances, and total ordering of fuzzy numbers are in-

troduced. Section 3 is devoted to the description of

the new approach to fuzzy TOPSIS based on fuzzy

similarity and multi-distances. A numerical example

is introduced in Section 4 and some conclusions in

section 5 close the paper.

2 PRELIMINARIES

In this section some preliminary mathematical con-

cepts, used in the MCDM method, are shortly intro-

duced. They include: fuzzy similarity measures, the

OWA-operator, and one often-used method to gener-

ate the weights for the OWA operator the O’Hagan’s

method. Multi-distances are deﬁned, i.e. following

the work of Martin and Mayor (Martin and Mayor,

2010) and the relationship between the OWA-operator

and multi-distances is presented. Additionally, a way

to ﬁnd a total ordering for fuzzy numbers is shortly

introduced.

2.1 Fuzzy Similarity of Fuzzy Numbers

By focusing on uncertain objects like in fuzzy sets

or fuzzy numbers, the notion of a fuzzy subset gen-

eralizes that of the classical subset, where the con-

cept of similarity can be considered as a many-

valued generalization of the classical notion of equiv-

alence (Zadeh, 1971). As an equivalence relation is a

familiar way to classify similar objects, fuzzy similar-

ity is an equivalence relation that can be used to clas-

sify multi-valued objects (Niyigena et al., 2012). The

similarity measures’ concept is of high importance in

this work, and it is deﬁned as follow:

Deﬁnition 1. For any fuzzy subset F 6=

0 of R

, and

for any elements A,B ∈ F the similarity measure func-

tion is deﬁned as (Shepard, 1987):

s(A,B) : F × F → [0, 1]

The deﬁned similarity measures s satisfying the

following properties for any x,y,z ∈ F,

• s(x, x) = s(y,y), ∀x, y ∈ F ( Reﬂexivity )

• s(x, y) ≤ s(y, y), ∀x,y ∈ F ( Minimality )

• s(x, y) = s(y, x) ( Symmetry )

• If s(x,y) = s(x, z) it implies that s(x, y) = s(x, z) =

s(y, z) (Transitivity)

Since fuzzy numbers can be considered to be a

certain type of restricted fuzzy sets, the similarity

measures for generalized fuzzy numbers come from

similarity measures for fuzzy sets.

Represented by Chen (Chen, 1985), a gener-

alized trapezoidal fuzzy number’s notation is

A =

(a,b,c,d;w), where a,b,c and d are real values and

0 < w ≤ 1. Its membership function µ

satisﬁes the

following conditions (Chen, 1985):

1. µ

is a continuous mapping from the universe of

discourse X to the closed interval in [0,1]

2. µ

= 0, where −∞ < x ≤ a

3. µ

is monotonically increasing in [a,b]

4. µ

= w, where b ≤ x ≤ c

5. µ

is monotonically decreasing in [c,d]

6. µ

= 0, where d ≤ x < ∞

Due to the ﬁt and the applicability of similarity

measures in the context of decision-making, various

similarity measures have been proposed for the calcu-

lation the degree of similarity between fuzzy numbers

of (Chen, 1985). In this work, a rather recently intro-

duced similarity measure by Hejazi et al in (Hejazi

et al., 2011) is used. The similarity measure takes in

consideration the perimeter and area of fuzzy num-

bers. The similarity measure is denoted s(M, N), and

involves fuzzy numbers M = (m

;ω

) and

N = (n

;ω

) with 0 ≤ m

≤ m

≤

≤ 1, 0 ≤ n

≤ n

≤ 1, and M(x

)

and N(x

) their corresponding membership functions

with i ∈ {1,2, 3, 4} for generalized trapezoidal fuzzy

numbers, where ω

and ω

are their corresponding

heights. The deﬁnition is as follows:

s(M, N) = (1 −

∑

i=1

− n

)

min(p(m), p(n))

max(p(m), p(n))

min(a(m),a(n)) + min(ω

,ω

)

max(a(m),a(n)) + max(ω

,ω

)

(1)

Where the values p(m) and p(n) represent the perime-

ters of the trapezoidal fuzzy numbers M and N, and

are deﬁned as:

p(m) =

− m

)

+ ω

− m

)

+ ω

+ (m

− m

) + (m

− m

)

and

p(n) =

− n

)

+ ω

− n

)

+ ω

+ (n

− n

) + (n

− n

)

The values a(m) and a(n) represent the areas of

the trapezoidal fuzzy numbers M and N, they are de-

ﬁned as:

a(m) =

− m

+ m

− m

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and

a(m) =

− n

+ n

− n

Notice that the result of the above similarity mea-

sure s(M,N) belongs to the unit interval [0, 1] and

the larger the value of the similarity measure is, the

stronger is the similarity between the fuzzy numbers

M and N.

2.2 The OWA Operator

In 1988 Yager introduced a new aggregation operator,

called ordered weighted averaging operator(OWA)

(Yager, 1988) and formalized it as follows:

An ordered weighted averaging (OWA) operator

of dimension m is a mapping R

→ R that has

associated weighting vector W = [w

,...,w

] of

dimension m with

∑

i=1

= 1 , w

∈ [0,1] and 1 ≤ i ≤ m

such that:

OWA(a

,...,a

) =

∑

i=1

(2)

where b

is the i−th largest element of the collec-

tion of objects a

,...,a

. The function value

OWA(a

,...,a

) determines the aggregated values

of arguments a

,...,a

. One of the measures

related to the OWA is the so called ”orness” measure.

For a given weighting vector W = [w

,...,w

]

the measure of orness of the OWA aggregation

operator for W is given as

orness(W ) =

m − 1

∑

i=1

(m − i)w

. (3)

It can be observed that the weighting vector

has an important role in the OWA operator; next

the O’Hagan’s method for generating the weights

is shortly presented. In 1988 O’Hagan (O’Hagan,

1988) introduced a technique for computing the

weights in OWA. The procedure for obtaining the

aggregation assumes a predeﬁned degree of orness -

the weights are obtained by maximizing the entropy

−

∑

i=1

ln(w

). The solution is based on the con-

strained optimization problem

maximize −

∑

i=1

ln(w

)

subject to α =

m − 1

∑

i=1

(m − 1)w

∑

i=1

= 1 and w

≥ 0.

The above constrained optimization problem can

be solved by using different methods. Here an ana-

lytical solution introduced by (Full

er and Majlander,

2001) is used. Below this weighting scheme is pre-

sented:

a. if m = 2, implies that w

= α and w

= 1 − α

b. if α = 0 or α = 1 implies that the correspond-

ing weighting vectors are w = (0,...0, 1) or w =

(1,0,...,0) respectively.

c. if m ≥ 3 and 0 ≤ α ≤ 1 then, we have,



m−i

· w

i−1



m−1

((m−1)·α−m).w

(m−1)·α+1−m·w

[(m − 1) · α + 1 − m · w

]

= ((m − 1) ·

α)

m−1

· [((m − 1) · α − m) · w

+ 1]

For m ≥ 3, the weights are computed by obtaining the

ﬁrst weight, followed by the last weight of the weight-

ing vector, before other weights are computed.

2.3 Multi-distances

A multi-distance is a representation of the notion of

multi-argument distances. The set X is a union of all

m-dimensional lists of elements of X , multi-distance

is deﬁned as a function D : X → [0, ∞) on a non empty

set X provided that the following properties are satis-

ﬁed for all m and x

,...,x

,y ∈ X

c1. D(x

,...,x

) = 0 if and only if x

= x

for all

i, j = 1,2,...,m

c2. D(x

,...,x

) = D(x

σ(1)

σ(2)

,...,x

σ(m)

) for any

permutation σ of i, j = 1,2,...,m

c3. D(x

,...,x

) ≤ D(x

,y) + D(x

,y) + ... +

D(x

,y).

We say that D is a strong multi-distance if it satis-

ﬁes c1,c2, and

D(~x

,~x

,...,~x

) ≤ D(~x

,~y) + D(~x

,~y) + ... +

D(~x

,~y). for all ~x

,~x

,...,~x

,~y ∈ X

In application contexts, the estimation of distances

between more than two elements of the set X can

be constructed using multi-distances by means of the

OWA operator as suggested by Martin and Mayor

(Martin and Mayor, 2010).

,··· ,x

) = OWA

(d(x

),d(x

),...,

d(x

m−1

)) (4)

In this case, elements x

,··· ,x

are obtained

from the similarity measure (1), and the distance ap-

plied is d(x,y) = |x − y|.

FuzzySimilaritybasedFuzzyTOPSISwithMulti-distances

195

2.4 Total Ordering of Fuzzy Numbers

Set inclusion of fuzzy sets is only a partial order,

where all fuzzy sets are not comparable. Kaufman

and Gupta (Kaufman and Gupta, 1988) propose that

when trying to ﬁnd a total order or linear order for

fuzzy numbers, where all fuzzy numbers and fuzzy

intervals are comparable, we have to ﬁrst check that

it is possible to ﬁnd a linear order by giving differ-

ent emphases to different properties of fuzzy sets. To

reach a total order or a linear order of fuzzy numbers,

an importance order must be given to three criteria. If

the ﬁrst criterion does not give a unique linear order,

then the second criterion should be used. One contin-

ues in this way as long as it is needed. The description

of the three different criteria is given below:

The removal: Let us consider an ordinary number

k ∈ R and a fuzzy number A. The left side removal

of A with respect to k, denoted by R

(A,k), is de-

ﬁned as the area bounded by k and the left side

of the fuzzy number A. Similarly, the right side

removal, R

(A,k) is deﬁned. The removal of the

fuzzy number A with respect to k is deﬁned as the

mean of R

(A,k) and R

(A,k). Thus,

R(A,k) =

(A,k) + R

(A,k)). (5)

The position of k can be located anywhere on the

x-axis including k = 0. By deﬁnition, the areas are

positive quantities, but here they are evaluated by

integration taking into account the position (nega-

tive, zero, or positive) of the variable x; therefore,

R(A,k) can be positive, negative or null.

Our ﬁrst criterion, therefore, will be this removal

with respect to k. However, two different fuzzy

numbers can have the same removal with respect

to the same k. In fact, this criterion decomposes a

set of fuzzy numbers into classes having the same

removal number.

The removal number R(A, k) deﬁned in this crite-

rion, relocated to k = 0 is equivalent to an ”ordi-

nary representative” of the fuzzy number. In the

case of a triangular fuzzy number this ordinary

representative is given by:

A =

+ 2a

+ a

, (6)

where A = (a

The mode: In each class of fuzzy numbers, one

should look for the mode, and these modes will

generate sub-classes. If the fuzzy numbers under

consideration have a non-unique mode, one takes

the mean position of the modal values. It must

be noted that this is only one way of obtaining

sub-classes, and one may need the following third

divergence criterion for further sub-classiﬁcation.

The divergence: The consideration of the diver-

gence around the mode in each sub-class leads to

the sub-sub-classes, and this criterion may be suf-

ﬁcient to obtain the ﬁnal linear ordering of fuzzy

numbers.

When one orders fuzzy numbers to size order, one

proceeds as follows. Apply the above presented cri-

teria in the exact given order, such that if the unique

linear order is not obtained then move to the next cri-

terion.

3 FUZZY SIMILARITY BASED

FUZZY TOPSIS WITH

MULTI-DISTANCES

Fuzzy extension to the Technique for Order Perfor-

mance by Similarity to Ideal Solution (TOPSIS) was

presented by Chen (Chen, 2000) and it has been ex-

tended to solve problems involving trapezoidal fuzzy

numbers and applied, e.g., to solving the supplier se-

lection problems (Chen et al., 2006). It is a Multiple

Criteria Decision Making (MCDM) method (Chen

et al., 2006) (Socorro and Lamata, 2007) useful in

ranking objects, based on the similarity of the object

characteristics to the characteristics of an ideal object

(ideal solution). The method is based on the idea that

objects are ranked higher the shorter their distance is

from the Fuzzy Positive Ideal Solution (FPIS) and the

further away they are from the Fuzzy Negative Ideal

Solution (FNIS). One advantage of having extended

the TOPSIS method to the fuzzy environment is that a

linguistic assessment can be properly used, instead of

being constrained into using only numerical values;

linguistic variables can be mapped to corresponding

fuzzy numbers (Chen, 2000),(Chen et al., 2006).

Solution to the project selection problem, when

using the T OPSIS approach, can be presented by

considering a situation of a ﬁnite set of projects

P = {P

|i = 1, 2, ..., m} which need to be evaluated

by a committee of decision-makers D = {D

|l =

1,2,...,k}, by considering a ﬁnite set of given crite-

ria C = {C

| j = 1, 2, ..., n}.

Let us consider a decision matrix representing a set

of performance ratings of each alternative project

,i = 1, 2, ...,m with respect to each criterion C

, j =

1,2,...,n, as follows (Cui et al., 2011):

X =







... x

... ... ... ...

... x







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196

Let us also assume the weight w

of the j−th crite-

rion C

, such that the weight vector is represented as

follows:

W =



, w

, . . . , w



Where m rows represent m possible alternatives, n

columns represent n relevant criteria, and x

i j

repre-

sent the performance rating of the i−th project P

with

respect to the j−th criterion C

. The above fuzzy

ratings for each decision-maker D

,l = 1,2, ..., k are

represented by positive trapezoidal fuzzy numbers

= (a

),l = 1, 2, ..., k with the respective

membership function µ

(x). As the rating

) is for the l−th decision-maker, the ag-

gregated fuzzy number that can stand for all decision-

makers’ rating is:

R = (a, b, c, d)

with:

a = min

}, b =

∑

l=1

, c =

∑

l=1

, d =

max

}. The fuzzy rating and importance

weight of the l−th decision-maker can respec-

tively be represented by x

i jl

= (a

i jl

) and

ˆw = (w

jl1

jl2

jl3

jl4

) with i = 1, 2, ..., m; j =

1,2,...,n. Then, the aggregated fuzzy ratings x

i j

alternatives with respect to each criterion are:

i j

= (a

i j

calculated as: a

i j

= min

i jl

}, b

i j

∑

l=1

i jl

, c

i j

∑

l=1

i jl

, d

i j

= max

i jl

}. The aggregated fuzzy

weight ˆw

of each criterion can be calculated as :

ˆw

= (w

)

with w

= min

jl1

}, w

∑

l=1

jl2

, w

∑

l=1

jl3

, w

= max

jl4

}. After aggregation the

decision matrix and the weight vector are of the fol-

lowing form X = {x

i j

}

m×n

and W = {w

}

1×n

, where

i = 1, 2, ..., m and j = 1,2,...,n.

These matrices’ elements are given by positive trape-

zoidal fuzzy numbers as :

i j

= (a

i j

) and w

= (w

The linear scale transformation is used to transform

the various criteria scales into comparable scales in

order to avert overly complex mathematical opera-

tions in a decision process. By dividing the set of

criteria into beneﬁt criteria B, where the larger the

rating, the greater the preference and cost criteria C,

where the smaller the rating, the greater the prefer-

ence. A normalization method designed to preserve

the property in which the elements are normalized

trapezoidal fuzzy numbers is used. The normalized

value of x

i j

is r

i j

, and the normalized fuzzy decision

matrix is then represented as:

R = [r

i j

]

m×n

(7)

with

i j

= (

i j

), j ∈ B

i j

= (

−

i j

−

i j

−

i j

−

i j

), j ∈ C

where d

= max

i j

}, j ∈ B and a

−

= min

i j

}, j ∈

C. The weighted normalized value of r

i j

is called v

i j

and by considering the importance of each criterion,

the weighted normalized fuzzy decision matrix is rep-

resented as:

V = [v

i j

]

m×n

(8)

where v

i j

= r

i j

· w

. For all i, j, the elements v

i j

are

now normalized positive trapezoidal fuzzy numbers.

Next, the ideal solutions must be determined and

taken from the given criteria which are linguistically

expressed, they are commonly referred to as Fuzzy

Positive Ideal Solution (FPIS) and Fuzzy Negative

Ideal Solution (FNIS). By considering a ﬁnite set of

given criteria C = {C

| j = 1, 2, ..., n}, the ways to se-

lect the FPIS(P

) and the FNIS(P

−

) come from the

weighted normalized decision matrix V = (v

i j

)

m×n

where the obtained weighted normalized values v

i j

are fuzzy numbers expressed as:

i j

= (v

i j1

i j2

i j3

i j4

)

The fuzzy positive-ideal solution P

and the fuzzy

negative-ideal solution P

−

, respectively are:

= [v

,...,v

] (9)

−

= [v

−

,...,v

−

] (10)

One way for choosing the FPIS (P

) and the FNIS

−

) have been explained in (Luukka, 2011), and is

given as follows:

Every element of P

is the maximum for all i

weighted normalized value , and every element of P

−

is the minimum for all i weighted normalized value:

= (max

i j1

,max

i j2

,max

i j3

,max

i j4

) (11)

−

= (min

i j1

,min

i j2

,min

i j3

,min

i j4

) (12)

This was also considered in our approach. The simi-

larity measure between each project and the ideal so-

lutions P

and P

−

will be needed later, when calculat-

ing the closeness coefﬁcients to determine the ranking

order of all possible alternative projects.

The similarities s

from the positive and negative

ideal solution are calculated as:

= {s

),s

),··· ,s

)} (13)

−

= {s

−

),s

−

),··· ,s

)} (14)

where for similarity we used the similarity mea-

sure from equation (1).

FuzzySimilaritybasedFuzzyTOPSISwithMulti-distances

197

Table 1: Evaluation of R & D projects.

Project C

(53,62, 68, 78) (43,50, 55, 63) (115,128, 128, 141) 0.06

(83,98, 108, 123) (85,100, 110, 125) (126,140,140, 154) 0.0594

(157,185, 204, 231) (170,200, 220, 250) (170,189, 189, 208) 18

(204,240, 268, 300) (170,200, 220, 250) (164,182, 182, 200) 0.54

(259,305, 336, 381) (510,600, 660, 750) (209,232, 232, 255) 3.10

(85,100, 110, 125) (85,100, 110, 125) (185,206,206, 227) 5

(259,305, 336, 381) (510,600, 660, 750) (209,232, 232, 255) 3.10

(94,110, 121, 138) (85,100, 110, 125) (177,197,197, 217) 1.58

(140,165, 182, 206) (153,180, 198, 225) (238,264, 264, 290) 17.15

(190,223, 245, 279) (323,380, 418, 475) (257,285, 285, 314) 1.65

(60,70, 77, 88) (68,80, 88, 100) (148,164, 164, 180) 10.03

(91,107, 118, 134) (85,100, 110, 125) (144,160,160, 176) 2.39

(247,290, 319, 363) (34,40, 44, 50) (297,330, 330, 363) 0

(370,435, 479, 544) (595,700, 770, 875) (338,375, 375, 413) 278.25

(166,195, 215, 244) (425,500, 550, 625) (279,310, 310, 341) 320.25

(221,260, 286, 325) (255,300, 330, 375) (315,350, 350, 385) 39.66

(235,277, 305, 346) (298,350, 385, 438) (311,346, 346, 381) 72.48

(281,330, 363, 413) (468,550, 605, 688) (331,368, 368, 405) 231

(344,405, 446, 506) (680,800, 880, 1000) (365,406, 406, 447) 414.75

(451,530, 583, 663) (978, 1150, 1265,1438) (394, 438,438, 482) 651

These similarity vectors are then aggregated using

OWA, as follows:

= OWA

,··· ,s

) (15)

−

= OWA

−

,··· ,s

−

) (16)

Besides this we also aggregate s

vector by using

multi-distance as

,·· · , s

) = OWA

(d(s

),d(s

...,d(s

i(n−1)

)) (17)

In the closeness coefﬁcient we now want to take

both into account, the similarities from the positive

and the negative ideal solution but also the multi-

distance. This is now done by modifying the close-

ness coefﬁcient in form given in equation (18). The

closeness coefﬁcients of the alternative project P

with

respect to the positive ideal solution by using the dis-

tance matrix (CC

) are deﬁned as:

−

+ D

+ S

−

,i = 1, 2, ..., m (18)

Next we rank the projects by closeness coefﬁ-

cients, now using ascending order.

For all i = 1, 2, ..., m and j = 1, 2, ..., n. Differ-

ent steps for the given TOPSIS algorithm can be pre-

sented as follows:

Step1: Form a decision-makers’ committee, and

identify the evaluation criteria.

Step2: Choose the appropriate linguistic variables

for the importance weight of the criteria and the

linguistic ratings for alternative projects.

Step3: Aggregate the weight of criteria to get the ag-

gregated fuzzy weight ˆw

of the criterion C

and

join the decision-makers’ ratings to get the aggre-

gated fuzzy rating x

i j

of the project P

in consid-

eration of the criterion C

Step4: Construct the fuzzy decision matrix and the

normalized fuzzy decision matrix.

Step5: Construct the weighted normalized fuzzy de-

cision matrix.

Step6: Determine the fuzzy positive (and negative)

ideal solution FPIS (and FNIS).

Step7: Construct the similarity matrix by calculating

the similarity measure of each project from the

FPIS (and FNIS).

Step8: Calculate aggregated similarity values for

each project wrt. FPIS and FNIS by using OWA.

Step9: Calculate multi-distance value for each

project wrt. FPIS.

Step10: Calculate the closeness coefﬁcient for each

project in order to determine the projects’ ranking

order.

4 NUMERICAL EXAMPLE

This numerical example uses the same data that is also

used in (Hassanzadeh et al., 2012). A pharmaceutical

company can select a certain number of projects to in-

vest in from among 20 R & D projects. Criteria that

are used in the example come from costs, revenues,

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budget constraints, and real option values (ROV) cal-

culated for each project by using the pay-off method

for real option valuation (Collan et al., 2009); these

are represented as trapezoidal fuzzy numbers. First

and third criteria are cost criterias and second and

fourth, beneﬁt criterias.

In Table 1 one can see evaluations of the differ-

ent criteria by using trapezoidal fuzzy numbers. The

fourth (ROV) criterion is carried out in computations

as a fuzzy number of form A = (a

), where

= a

For different α values, the following Table 2

shows the computed closeness coefﬁcients and the

rankings for each of the twenty projects for three dif-

ferent orness values α. In computation of fuzzy num-

bers a larger set of values of α was used. There α

values were α = 0.1, 0.2, · · · , 0.9, 1 .

Table 2: Projects’ closeness coefﬁcients values and rank-

ings, for α = 0.1, 0.5, 1.

Project CC

(α=0.1)

Rank CC

(α=0.5)

Rank CC

(α=1)

Rank

0.83 19 0.710 8 0.709 8

0.813 13 0.728 11 0.728 11

0.79 10 0.741 15 0.741 15

0.80 11 0.751 19 0.751 19

0.61 2 0.614 2 0.615 2

0.821 17 0.74 14 0.739 14

0.819 15 0.717 10 0.716 10

0.824 18 0.744 17 0.743 17

0.809 12 0.751 18 0.75 18

0.68 6 0.681 7 0.682 7

0.81 14 0.716 9 0.715 9

0.82 16 0.737 13 0.736 13

0.85 20 0.798 20 0.797 20

0.64 4 0.674 6 0.674 6

0.58 1 0.612 1 0.613 1

0.77 9 0.742 16 0.743 16

0.75 8 0.729 12 0.729 12

0.64 3 0.672 5 0.673 5

0.68 5 0.661 4 0.662 4

0.71 7 0.652 3 0.651 3

In Table 3, the minimum, mean, and the maximum

rankings from our experimental setup are summa-

rized. These are then used in the formation of trian-

gular fuzzy numbers for each project.

Table 3: The minimum, mean, and the maximum rankings.

Project Minimum Mean Maximum

8 10.3 19

11 11.9 14

10 13.7 15

11 17.1 19

2 2 2

14 15.2 18

10 11 15

17 17.5 19

12 17.1 18

6 6.8 7

9 10 14

13 14.1 17

20 20 20

4 5.4 6

1 1 1

9 13.6 16

8 10.5 12

3 4.4 5

4 4.3 5

3 4.1 7

Total ordering is found for fuzzy numbers pre-

sented in Table 3 by using the method introduced by

Kaufman and Gupta (Kaufman and Gupta, 1988). For

this purpose removal number, dispersion, and modal

value are calculated in a way presented above - Table

4 presents the resulting overall ranking. According to

the result the top ﬁve projects are 15,5, 18, 19, and 20.

Table 4: Overall rankings of the R & D projects using re-

moval number, dispersion, and modal value.

Project Rank Removal no div mode

1 1 0 1

2 2 0 2

3 4.2 2 4.4

4 4.4 1 4.3

5 4.55 4 4.1

6 5.2 2 5.4

7 6.65 1 6.8

8 10.25 4 10.5

9 10.75 5 10

10 11.75 5 11

11 11.9 11 10.3

12 12.2 3 11.9

13 13.05 7 13.6

14 13.1 5 13.7

15 14.55 4 14.1

16 15.6 4 15.2

17 16.05 6 17.1

18 16.05 8 17.1

19 17.75 2 17.5

20 20 0 20

5 CONCLUSIONS

A new multiple-criteria decision making approach

was presented; it is an extension for the fuzzy simi-

larity based fuzzy TOPSIS. OWA was used for aggre-

gating similarity to fuzzy negative and positive ideal

solutions for each criterion and multi-distance was

used in collecting information about the ”similarity

of these similarities” that can be understood as a mea-

sure of homogeneity or consistency of a given project.

This has allowed the inclusion of more relevant infor-

mation than the use of a simple defuzziﬁcation. The

method was applied to a R & D project selection prob-

lem. The results are dependent on the proper selection

of α, when the weights are generated for the OWA

operator. This weight generation was done by using

O’Hagan’s method that ﬁnds the weights as an op-

timal solution for a predeﬁned (given) orness value

(α). We examined the effect of the pre-selection by

testing with a number of orness values. We presented

a way to take in to consideration the ”created” infor-

mation with different orness values by forming fuzzy

numbers from the different rankings of the projects

FuzzySimilaritybasedFuzzyTOPSISwithMulti-distances

199

created in this way. By using multidistances a mea-

sure of homogeneity of similarity of the different cri-

teria of each project to the fuzzy positive ideal solu-

tion was calculated. This was done to include infor-

mation about the consistency of the level of goodness

of projects (by the selected criteria). This informa-

tion was included in the closeness coefﬁcient that was

used in the ranking of the projects. The ﬁnal ranking

thus includes information about the goodness of each

project (as ranked by TOPSIS) and about the ”stabil-

ity” of the level of goodness of each of the criteria of

each project. The top ﬁve projects from the numerical

example were found to be 15, 5, 18, 19, and 20. No-

table from the results is that projects 15 and 5 were

always top 2 choices, but project 20 varied between

rankings 3 to 7 so that with lower values of orness

ranking was lower and after orness value 0.6 it was

always the third best choice. Forming a fuzzy number

from different rankings allows one to include differ-

ent points of view and creating an intelligent overall

ranking. Furthermore, more relevant information is

carried along in the analysis, until the ranking stage,

enabling the ranking to take more things into consid-

eration and thus being based on a more holistic view

of the problem.

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