FUZZY INDUCED AGGREGATION OPERATORS IN DECISION

MAKING WITH DEMPSTER-SHAFER BELIEF STRUCTURE

José M. Merigó and Montserrat Casanovas

Department of Business Administration, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain

Keywords: Aggregation operators, Decision making, Dempster-Shafer theory of evidence, Financial decision making.

Abstract: We develop a new approach for decision making with Dempster-Shafer theory of evidence when the

available information is uncertain and it can be assessed with fuzzy numbers. With this approach, we are

able to represent the problem without losing relevant information, so the decision maker knows exactly

which are the different alternatives and their consequences. For doing so, we suggest the use of different

types of fuzzy induced aggregation operators in the problem. As a result, we get new types of fuzzy induced

aggregation operators such as the belief structure – fuzzy induced ordered weighted averaging (BS-FIOWA)

operator. We also develop an application of the new approach in a financial decision making problem.

1 INTRODUCTION

The Dempster-Shafer (D-S) theory of evidence

(Dempster, 1967; Shafer, 1976) provides a unifying

framework for representing uncertainty because it

includes the situations of risk and ignorance as

special cases. For further reading on the D-S theory,

see (Yager and Liu, 2008).

Usually, when using the D-S theory it is assumed

that the available information are exact numbers

(Engemann et al., 1994; Merigó and Casanovas,

2007; Yager, 1992; 2004). However, this may not be

the real situation found in the decision making

problem because often, the available information is

vague or imprecise and it is not possible to analyze it

with exact numbers. Then, a better approach may be

the use of fuzzy numbers (FN) because it considers

the best and worst possible scenarios and a lot of

others that could occur. When using FNs, we will

follow the ideas of (Chang and Zadeh, 1972; Dubois

and Prade, 1980; Kaufmann and Gupta, 1985).

Going a step further, the aim of this paper is to

suggest the use of different types of fuzzy induced

aggregation operators for aggregating the informa-

tion in decision making with D-S theory. The reason

for using various types of aggregation operators is

that we want to show that the fuzzy decision making

problem with D-S theory can be modelled in

different ways depending on the interests of the

decision maker. We will use the fuzzy induced

ordered weighed averaging (FIOWA) operator

because it provides a parameterized family of

aggregation operators that include the fuzzy

maximum, the fuzzy minimum, the fuzzy average

(FA), the fuzzy weighted average (FWA) and the

fuzzy OWA (FOWA), among others. Then, we will

get a new aggregation operator that we will call the

belief structure - FIOWA (BS-FIOWA) operator.

We also develop an application of this new model in

a business decision making problem.

In order to do so, the remainder of the paper is

organized as follows. In Section 2, we briefly

describe some basic concepts. In Section 3, we

present the new approach about using fuzzy induced

aggregation operators in decision making with D-S

theory. Finally, in Section 4 we develop an

application of the new approach.

2 PRELIMINARIES

2.1 Fuzzy Numbers

The FN was introduced by (Chang and Zadeh,

1972). Since then, it has been studied by a lot of

authors such as (Kaufmann and Gupta, 1985).

A FN is a fuzzy subset (Zadeh, 1965) of a

universe of discourse that is both convex and normal

(Kaufmann and Gupta, 1985). Note that the FN may

be considered as a generalization of the interval

number (Moore, 1966) although it is not strictly the

548

M. Merigó J. and Casanovas M. (2008).

FUZZY INDUCED AGGREGATION OPERATORS IN DECISION MAKING WITH DEMPSTER-SHAFER BELIEF STRUCTURE.

In Proceedings of the Tenth International Conference on Enterprise Information Systems - AIDSS, pages 548-552

DOI: 10.5220/0001711105480552

Copyright

c

SciTePress

same because the interval numbers may have

different meanings.

In the literature, we find a wide range of FNs

(Kaufmann and Gupta, 1985). For example, a

trapezoidal FN (TpFN) A of a universe of discourse

R can be characterized by a trapezoidal membership

function

),( aaA = such that

).()(

),()(

344

121

aaaa

aaaa

−−=

−+=

αα

αα

(1)

where

α

∈ [0, 1] and parameterized by (a

1

, a

2

, a

3

,

a

4

) where a

1

≤ a

2

≤ a

3

≤ a

4

, are real values. Note that

if a

1

= a

2

= a

3

= a

4

, then, the FN is a crisp value and

if a

2

= a

3

, the FN is represented by a triangular FN

(TFN). Note that the TFN can be parameterized by

(a

1

, a

2

, a

4

).

2.2 Fuzzy Induced OWA Operator

The FIOWA (or FN-IOWA) operator was

introduced by S.J. Chen and S.M. Chen (2003). It is

an extension of the OWA operator (Yager, 1988;

Yager and Kacprzyk, 1997) that uses uncertain

information represented by FNs. It also uses a

reordering process different from the values of the

arguments. In this case, the reordering step is based

on order inducing variables. It is defined as follows.

Definition 1. Let Ψ be the set of FN. A FIOWA

operator of dimension n is a mapping FIOWA: Ψ

n

→

Ψ that has an associated weighting vector W of

dimension n such that w

j

∈ [0, 1] and

∑

=

=

n

j

j

w

1

1

,

then:

FIOWA(〈u

1

,ã

1

〉, …, 〈u

n

,ã

n

〉) =

∑

=

n

j

jj

bw

1

(2)

where b

j

is the ã

i

value of the FIOWA pair 〈u

i

, ã

i

〉

having the jth largest u

i

, u

i

is the order inducing

variable and ã

i

is the argument variable represented

in the form FN.

2.3 Dempster-Shafer Theory of

Evidence

The D-S theory provides a unifying framework for

representing uncertainty as it can include the

situations of risk and ignorance as special cases. It is

defined as follows.

Definition 2. A D-S belief structure defined on a

space X consists of a collection of n nonnull subsets

of X, B

j

for j = 1,…,n, called focal elements and a

mapping m, called the basic probability assignment,

defined as, m: 2

X

→ [0, 1] such that:

(1) m(B

j

) ∈ [0, 1].

(2)

)(

1

∑

=

n

j

j

Bm = 1. (3)

(3) m(A) = 0, ∀ A ≠ B

j.

.

3 FIOWA OPERATORS IN

DECISION MAKING WITH D-S

THEORY OF EVIDENCE

In this Section, we describe the process to follow

when using fuzzy induced aggregation operators in

decision making with D-S theory.

3.1 Decision Making Approach

A new method for decision making with D-S theory

is possible by using FN aggregation operators in the

problem. Going a step further, we see that it is

possible to use fuzzy induced aggregation operators

such as the FIOWA operator. Note it is also possible

to consider other cases such as the use of different

types of fuzzy induced generalized means and fuzzy

induced quasi-arithmetic means. The motivation for

using FNs appears because sometimes, the available

information is not clear and it is necessary to assess

it with another approach such as the use of FNs.

Although the information is uncertain and it is

difficult to take decisions with it, at least we can

represent the best and worst possible scenarios and

the possibility that the internal values of the fuzzy

interval will occur. The decision process can be

summarized as follows.

Assume we have a decision problem in which

we have a collection of alternatives {A

1

, …, A

q

} with

states of nature {S

1

, …, S

n

}. ã

ih

is the uncertain

payoff, given in the form of FNs, to the decision

maker if he selects alternative A

i

and the state of

nature is S

h

. The knowledge of the state of nature is

captured in terms of a belief structure m with focal

elements B

1

, …, B

r

and associated with each of these

focal elements is a weight m(B

k

). The objective of

the problem is to select the alternative which gives

the best result to the decision maker. In order to do

so, we should follow the following steps:

Step 1: Calculate the uncertain payoff matrix.

Step 2: Calculate the belief function m about the

states of nature.

FUZZY INDUCED AGGREGATION OPERATORS IN DECISION MAKING WITH DEMPSTER-SHAFER BELIEF

STRUCTURE

549

Step 3: Calculate the attitudinal character of the

decision maker

α

(W) (Yager, 1988).

Step 4: Calculate the collection of weights, w, to

be used in the FIOWA aggregation for each different

cardinality of focal elements. (Merigó, 2007; Yager,

1988; 1993).

Step 5: Determine the uncertain payoff

collection, M

ik

, if we select alternative A

i

and the

focal element B

k

occurs, for all the values of i and k.

Hence M

ik

= {a

ih

| S

h

∈ B

k

}.

Step 6: Calculate the fuzzy induced aggregated

payoff, V

ik

= FIOWA(M

ik

), using Eq. (2), for all the

values of i and k.

Step 7: For each alternative, calculate the

generalized expected value, C

i

, where:

∑

=

=

r

r

kiki

BmVC

1

)(

(4)

Step 8: Select the alternative with the largest C

i

as the optimal.

3.2 Using FIOWA Operators in Belief

Structures

Analyzing the aggregation in Steps 6 and 7 of the

previous subsection, it is possible to formulate in

one equation the whole aggregation process. We will

call this process the belief structure – FIOWA (BS-

FIOWA) aggregation. It can be defined as follows.

Definition 3. A BS-FIOWA operator is defined by

∑∑

=

==

r

k

q

j

jjki

k

k

kk

bwBmC

11

)(

(5)

where w

j

k

is the weighting vector of the kth focal

element such that

1

1

=

∑

=

n

j

j

k

w and w

j

k

∈ [0,1], b

j

k

is the j

k

th largest of the ã

i

k

and the ã

i

k

are FNs, and

m(B

k

) is the basic probability assignment.

Note that q

k

refers to the cardinality of each focal

element and r is the total number of focal elements.

The BS-FIOWA operator is monotonic, commu-

tative, bounded and idempotent.

Note that it is possible to distinguish between

descending (BS-DFIOWA) and ascending (BS-

AFIOWA) orders.

3.3 Families of BS-FIOWA Operators

By using a different manifestation in the weighting

vector of the FIOWA operator, we are able to deve-

lop different families of FIOWA and BS-FIOWA

operators. As we can see in Definition 3, each focal

element uses a different weighting vector in the

aggregation with the FIOWA operator. Therefore,

the analysis needs to be done individually.

For example, the maximum is obtained if w

p

= 1

and w

j

= 0, for all j ≠ p, and u

p

= Max{ã

i

}. The fuzzy

minimum is obtained if w

p

= 1 and w

j

= 0, for all j ≠

p, and u

p

= Min{ã

i

}. The FA is found when w

j

= 1/n,

for all ã

i

. The FWA is obtained if u

i

> u

i+1

, for all i,

and the FOWA operator is obtained if the ordered

position of u

i

is the same than the ordered position of

b

j

such that b

j

is the jth largest of ã

i

.

Other families of FIOWA operators could be

used in the BS-FIOWA operator such as the step-

FIOWA, and the olympic-FIOWA, among others.

For more information, see (Merigó, 2007).

For example, the step-FIOWA operator is found

when w

k

= 1 and w

j

= 0, for all j ≠ k. The olympic-

FIOWA operator is found if w

1

= w

n

= 0, and for all

others w

j

= 1/(n − 2).

Finally, if we assume that all the focal elements

use the same weighting vector, then, we can refer to

these families as the BS-fuzzy maximum, the BS-

fuzzy minimum, the BS-FA, the BS-FWA, the BS-

S-FIOWA, the BS-olympic-FIOWA, etc.

4 APPLICATION IN FINANCIAL

DECISION MAKING

In the following, we are going to develop an

application of the new approach in a decision

making problem. We will analyze the selection of

financial strategies where an enterprise is looking for

its optimal financial strategy for the next year. Note

that other applications could be developed such as

the selection of human resources, etc.

Assume an enterprise is planning its financial

strategy for the next year and considers 4 possible

financial strategies to follow: {A

1

, A

2

, A

3

, A

4

}.

In order to evaluate these financial strategies, the

group of experts considers that the key factor is the

economic situation of the company for the next year.

After careful analysis, the experts have considered

five possible situations that could happen in the

future: S

1

= Very bad, S

2

= Bad, S

3

= Regular, S

4

=

Good, S

5

= Very good.

ICEIS 2008 - International Conference on Enterprise Information Systems

550

Table 1: Fuzzy payoff matrix.

S

1

S

2

S

3

S

4

S

5

A

1

(50,60,70) (30,40,50) (30,40,50) (60,70,80) (40,50,60)

A

2

(10,20,30) (20,30,40) (50,60,70) (50,60,70) (80,90,100)

A

3

(30,40,50) (50,60,70) (40,50,60) (40,50,60) (40,50,60)

A

4

(60,70,80) (40,50,60) (30,40,50) (30,40,50) (30,40,50)

Depending on the uncertain situations that could

happen in the future, the experts establish the payoff

matrix. As the future states of nature are very

imprecise, the experts cannot determine exact

numbers in the payoff matrix. Instead, they use FNs

to calculate the future benefits of the enterprise

depending on the state of nature that happens in the

future and the financial strategy selected. Note that

in this example the experts use TFN. Then, they can

calculate the best and worst possible scenarios and

represent all the internal results with a membership

level. The results are shown in Table 1.

After careful analysis of the information, the

experts have obtained some probabilistic informa-

tion about which state of nature will happen in the

future. This probabilistic information is represented

by the following belief structure about the states of

nature.

Focal element

B

1

= {S

1

, S

2

, S

3

} = 0.3

B

2

= {S

3

, S

4

, S

5

} = 0.3

B

3

= {S

2

, S

3

, S

4

, S

5

} = 0.4

The attitudinal character of the enterprise is very

complex because it involves the opinion of different

members of the board of directors. Therefore, the

experts use order inducing variables for analysing

the attitudinal character of the enterprise. The results

are shown in Table 2.

Table 2: Inducing variables.

S

1

S

2

S

3

S

4

S

5

A

1

7 6 4 9 2

A

2

1 5 7 9 3

A

3

4 3 8 6 5

A

4

2 5 6 7 8

Table 3: Fuzzy aggregated results.

FA FWA FOWA FIOWA AFIOWA

V

11

(36.6,46.6,56.6) (36,46,56) (36,46,56) (36,46,56) (38,48,58)

V

12

(43.3,53.3,63.3) (43,53,63) (42,52,62) (43,53,63) (45,55,65)

V

13

(40,50,60) (42,52,62) (38,48,58) (39,49,59) (41,51,61)

V

21

(26.6,36.6,46.6) (29,39,49) (25,35,45) (25,35,45) (29,39,49)

V

22

(60,70,80) (62,72,82) (59,69,79) (62,72,82) (59,69,79)

V

23

(50,60,70) (53,63,73) (47,57,67) (50,60,70) (50,60,70)

V

31

(40,50,60) (40,50,60) (39,49,59) (41,51,61) (40,50,60)

V

32

(40,50,60) (40,50,60) (40,50,60) (40,50,60) (40,50,60)

V

33

(42.5,52.5,62.5) (42,52,62) (42,52,62) (43,53,63) (43,53,63)

V

41

(43.3,53.3,63.3) (42,52,62) (42,52,62) (45,55,65) (42,52,62)

V

42

(30,40,50) (30,40,50) (30,40,50) (30,40,50) (30,40,50)

V

43

(32.5,42.5,52.5) (32,42,52) (32,42,52) (33,43,53) (32,42,52)

Table 4: Fuzzy generalized expected value.

FA FWA FOWA FIOWA AFIOWA

A

1

(40,50,60) (40.5,50.5,60.5) (38.6,48.6,58.6) (39.3,49.3,59.3) (41.3,51.3,61.3)

A

2

(46,56,66) (48.5,58.5,68.5) (44,54,64) (46.1,56.1,66.1) (46.4, 56.4, 66.4)

A

3

(41,51,61) (40.8,50.8,60.8) (40.5,50.5,60.5) (41.5,51.5,61.5) (41.2,51.2,61.2)

A

4

(35,45,55) (34.4,44.4,54.4) (34.4,44.4,54.4) (35.7,45.7,55.7) (34.4,44.4,54.4)

FUZZY INDUCED AGGREGATION OPERATORS IN DECISION MAKING WITH DEMPSTER-SHAFER BELIEF

STRUCTURE

551

The experts establish the following weighting

vectors for the FIOWA:

Weighting vector

W

3

= (0.3, 0.3, 0.4)

W

4

= (0.2, 0.2, 0.3, 0.3)

W

5

= (0.1, 0.2, 0.2, 0.2, 0.3)

With this information, we can obtain the aggre-

gated payoffs. The results are shown in Table 3.

Once we have the aggregated results, we have to

calculate the fuzzy generalized expected value. The

results are shown in Table 4.

As we can see, depending on the fuzzy aggre-

gation operator used, the results and decisions may

be different. Note that in this case, our optimal

choice is the same for all the aggregation operators

but in other situations we may find different

decisions between each aggregation operator.

A further interesting issue is to establish an ordering

of the financial strategies. Note that this is very

useful when the decision maker wants to consider

more than one alternative. As we can see, depending

on the aggregation operator used, the ordering of the

financial strategies may be different. Note that in

this example the results are clear being A

2

the

optimal choice and the ordering: A

2

⎬A

3

⎬A

1

⎬A

4

excepting for the AFIOWA operator, where the

ordering is:

A

2

⎬A

1

⎬A

3

⎬A

4

.

5 CONCLUSIONS

We have studied the D-S theory of evidence in

decision making with uncertain information

represented in the form of FNs. With this approach,

we have been able to assess the information in a

more complete way because in this model we

consider the different scenarios that could happen in

the problem. For doing so, we have used different

types of fuzzy induced aggregation operators in the

decision process such as the FIOWA operator. Then,

we have obtained the BS-FIOWA operator.

We have also developed an application of the

new approach in a business decision making

problem about selection of financial strategies. We

have seen the usefulness of this approach about

using probabilities and FIOWAs in the same

problem. We have also seen that depending on the

fuzzy induced aggregation operator used the results

may lead to different decisions.

In future research, we expect to develop further

extensions to this approach by adding new

characteristics in the problem and applying it to

other decision making problems.

REFERENCES

Chang, S.S.L., Zadeh, L.A., 1972. On fuzzy mapping and

control, IEEE Transactions on Systems, Man and

Cybernetics, 2, pp. 30-34.

Chen, S.J., Chen, S.M., 2003. A new method for handling

multi-criteria fuzzy decision making problems using

FN-IOWA operators, Cybernetics and Systems, 34, pp.

109-137.

Dempster, A.P., 1967. Upper and lower probabilities

induced by a multi-valued mapping, Annals of

Mathematical Statistics, 38, pp. 325-339.

Dubois, D., Prade, H., 1980. Fuzzy Sets and Systems:

Theory and Applications, Academic Press, New York.

Engemann, K.J., Miller, H.E., Yager, R.R., 1996. Decision

making with belief structures: an application in risk

management, International Journal of Uncertainty,

Fuzziness and Knowledge-Based Systems, 4, pp. 1-26.

Kaufmann, A., Gupta, M.M., 1985. Introduction to fuzzy

arithmetic, Publications Van Nostrand, Rheinhold.

Merigó, J.M., 2007. New extensions to the OWA operator

and its application in business decision making.

Unpublished thesis (in Spanish), Department of

Business Administration, University of Barcelona.

Merigó, J.M., Casanovas, M. 2007. Decision making with

Dempster-Shafer theory using induced aggregation

operators. In Proceedings of the 4th

International

Summer School on AGOP, Ghent, Belgium, pp. 95-

100.

Moore, R.E,, 1966. Interval Analysis, Prentice Hall,

Englewood Cliffs, NJ.

Shafer, G.A., 1976. Mathematical Theory of Evidence,

Princeton University Press, Princeton, NJ.

Yager, R.R., 1988. On Ordered Weighted Averaging

Aggregation Operators in Multi-Criteria Decision

Making. IEEE Transactions on Systems, Man and

Cybernetics, B 18, pp. 183-190.

Yager, R.R., 1992. Decision Making Under Dempster-

Shafer Uncertainties, International Journal of General

Systems, 20, pp. 233-245.

Yager, R.R., 1993. Families of OWA operators. Fuzzy Sets

and Systems, 59, pp. 125-148.

Yager, R.R., 2004. Uncertainty modeling and decision

support, Reliability Engineering and System Safety,

85, pp. 341-354.

Yager, R.R., Kacprzyk, J., 1997. The Ordered Weighted

Averaging Operators: Theory and Applications.

Kluwer Academic Publishers, Norwell, MA.

Yager, R.R., Liu, L. 2008. Classic Works in the Dempster-

Shafer Theory of Belief Functions. Springer-Verlag,

Berlin.

Zadeh, L.A., 1965. Fuzzy Sets, Information and Control,

8, pp. 338-353.

ICEIS 2008 - International Conference on Enterprise Information Systems

552