DECISION MAKING BASED ON DUALITY BETWEEN POSITIVE

AND NEGATIVE EVALUATIONS

Rumiko Azuma

Graduate School of Engineering and Science, University of the Ryukyus, Nishihara, Japan

Hayao Miyagi, Yui Miyagi

Faculty of Engineering, University of the Ryukyus, Nishihara, Japan

Keywords:

Decision-making, Dual hierarchy, Reachability.

Abstract:

This paper proposes a model of dual hierarchy to study a decision-making structure with positive and negative.

It is necessary to evaluate a decision in negative stand point, such as insufﬁcient, hatred, pressure, as a cause

of fraud, as well as positive evaluation. In order to treat the positive and negative elements, we propose a

model of dual hierarchy process which can evaluate from positive and negative points. Moreover, a technique

to judge a consistency of dual evaluation is presented, using a concept of reachability matrix.

1 INTRODUCTION

On the decision-making problem under risk environ-

ment, it is important subjects how a risk is quantiﬁed

and modeled. There is a fraud as one of the risks that

exist in the company organization etc., and it appears

as the serious problem.

AHP(Analytic Hierarchy Process) proposed by

Saaty (T.L.Saaty, 1980) is one of the decision-making

method which evaluates human’s subjective feeling.

AHP for risk management (Azuma and Miyagi, 2009)

is a research treating the risk in the decision mak-

ing problem. In the research, degree of satisfaction

is regarded as positive utility, and degree of risk is

regarded as negative utility. It is based on positive

decision-making same as existing methods because

negative utility depends on statistical data. However,

in fraud prevention assessment, it is necessary to eval-

uate a decision in negative stand point, such as insuf-

ﬁcient, hatred, pressure, as a cause of fraud, as well

as positive evaluation.

In this paper, in order to solve this problem,

we discuss a development of decision-making model

which can be analyze the human’s double psychology

in fraud prevention.

2 DUALITY IN ORDER

RELATION

In decision-making process, there are some cases that

decision maker can not allow a rational and logical

consistent evaluation. As an example, there is the

deadlock in a three-cornered tie, and that is the order

relation of preferences is uncertain. Such discrepancy

can be discovered by calculating consistency index.

However, if the order relation is not the deadlock in

a three-cornered tie, it is difﬁcult to discover a dis-

crepancy. To discover the discrepancy in decision-

maker, it is the one of way to check a fraud action. In

the present study, in order to develop a new approach

to discover a discrepancy under the human’s subcon-

scious, we focus on the duality relation of order as

follows:

When the elements of the ordered set X = x

i

satis-

ﬁes the following three rules, the set X has the former

theorem p, at the same time, it has new theorem p

∗

that replaced all relations of order.

• x ≤ x (reﬂexivity)

• If x ≤ y and y ≤ x then x = yiasymmetryj

• If x ≤ y and y ≤ z then x ≤ zitransitivityj

This concept is the duality theorem (Ataka, 1966).

That is, p

∗

is the duality theorem of p. In this paper,

we propose the procedure that the discrepancy gener-

685

Azuma R., Miyagi H. and Miyagi Y..

DECISION MAKING BASED ON DUALITY BETWEEN POSITIVE AND NEGATIVE EVALUATIONS.

DOI: 10.5220/0003287006850688

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 685-688

ISBN: 978-989-8425-40-9

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

ated in the decision-making process can be shown by

utilizing the concept.

3 DUAL STRUCTURED

DECISION-MAKING MODEL

Existing decision making system is type of selecting

superiority. However, it is not easy to deal with nega-

tive elements such as ”pressure”, ”recognition of op-

portunity” and ”self-justiﬁcation” which may appear

in fraud problem. In particular, ”pressure” and ”self-

justiﬁcation” inﬂuence psychological factor of deci-

sion maker.

Then, we propose inferiority decision procedure

on opposite point of superiority in order to treat hu-

man’s negative psychology. We believe there is a

close relationship between superiority model and in-

feriority model because they are in the relation of

the ﬂip side. In the present study, we discuss both

decision-making mechanism and propose the model

with dual construction in Fig. 1.

Figure 1: Dual hierarchy process.

Flowchart for proposed model is described as fol-

lows:

Step 1) decision-maker is asked to evaluate the de-

cision criteria and alternatives and to make up a pair-

wise comparison matrix in positive or negative.

Step 2) the consistency index between positive

matrix and negative matrix is calculated.

Step 3) If the judgments are consistent, ﬁnal eval-

uation is obtained.

3.1 Positive Pairwise Comparison

Matrix

We deﬁne two matrices in order to propose new de-

cision model which has dual construction. One is the

superiority pairwise comparison matrix (positive ma-

trix) and another one is the inferiority pairwise com-

parison matrix (negative matrix).

The positive matrix is described by the ratio scale

of criterion, the same as AHP. The ordered set of

sequenced item by positive evaluation is deﬁned as

X = x

1

,...,x

n

, in an certain decision problem. If the

magnitude relation of x

i

and x

j

is x

i

≥ x

j

and its ra-

tio is ω

i

: ω

j

, then positive matrix P having ω

i

/ω

j

is

constructed:

P =

ω

1

/ω

1

... ω

1

/ω

j

... ω

1

/ω

n

ω

2

/ω

1

... ω

2

/ω

j

... ω

2

/ω

n

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ω

n

/ω

1

... ω

n

/ω

j

... ω

n

/ω

n

(1)

The relation between matrix P and its eigenvector

is given as

Pω = nω (2)

where

ω

T

= [ω

1

,ω

2

,...,ω

n

].

3.2 Negative Pairwise Comparison

Matrix

We consider the opposite problem which has the order

relation x

j

≥ x

i

. When the ratio scale of each item is

deﬁned as ω

j

: ω

i

, the negative matrix N constructed

by x

j

≥ x

i

is deﬁned as

N =

ω

1

/ω

1

... ω

j

/ω

2

... ω

n

/ω

1

ω

1

/ω

2

... ω

j

/ω

2

... ω

n

/ω

2

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ω

1

/ω

n

... ω

j

/ω

n

... ω

n

/ω

n

. (3)

where N = P

T

and P and N is the duality relation for

≥. Then, the relation between N and its eigenvector

is given as

Nω

′

= nω

′

(4)

where

ω

′T

= [1/ω

1

,1/ω

2

,...,1/ω

n

].

By the above deﬁnition, Eq. (4) could be derivedfrom

Eq. (2) and we acquire that both matrices is the dual-

ity relation by duality theorem: They are in the rela-

tion that the order is reversed.

In Step 1, positive matrix and negative matrix are

obtained by each evaluation of decision-maker. Se-

lect n − 1 combinations (nFnumber of elements) of

element he want in order to reduce the burden on

decision-maker’s task to compare. Then, it is not nec-

essary to check each consistency of matrix because 2

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

686

combinations is minimum number of combinations to

be able to calculate all evaluations.

All order, whether in human thinking, involves

proportionality among the parts. Thus, to create pos-

itive matrix or negative matrix, we must use ratio

scales to capture and synthesize the relations inherent

in that order. If, for example, to create positivematrix,

decision-maker is comparing each element according

to wight we ask: ”How much important is the ele-

ment i than the element j in proportion ?” or ”How

much inferior is the element i than the element j in

proportion ?” If the answer, ” i is more important than

j in proportion of 8 parts to 2” is gotten, the score of

pairwise comparison is ω

i

/ω

j

= 8/2. In positive and

negative evaluations, it is better to compare pair of el-

ements as differently as possible. Thus, the contradic-

tion between positive question and negative question

can be made easy to discover.

3.3 Consistency

In step 2, we must judge the consistency for the con-

structed matrix. There are two kinds of consistency.

One is the consistency of each matrix. Another is the

consistency of duality matrices. In this paper, Saaty’s

method (T.L.Saaty, 1980) is adopted for the former

and new method is proposed for the latter.

3.3.1 Consistency for Each Matrix

Next, consistency of each matrix is checked. The pro-

posed positive matrix and negative matrix are consis-

tent because their values are consisted by n−1 of ratio

and other values are calculated at the them.

However, if values are obtained by comparing

the pair of elements as AHP, we adopt the Saaty’s

method. Saaty proved that for consistent reciprocal

matrix, the largest eigen value is equal to the size of

comparison matrix, or λ

max

= n. Then a measure of

consistency, called Consistency Index is given as de-

viation or degree of consistency using the following

formula

CI =

λmax− n

n− 1

. (5)

3.3.2 Consistency in Duality

We propose how to judge the consistency in duality

between P and N. The consistency in duality means

that positive and negative have same order of priority

in criteria and alternatives. It is necessary to examine

the consistency of duality to discover implicit contra-

diction of the decision-maker.

We apply the concept of reachability matrix to

our procedure. If proposed consistency is satisﬁed

completely, the order relation between P and N is

duality. We deﬁned, the relation of priorities in

positive and negative matrices is not reversed and it

should be considered within the acceptable limits.

Step1) creation of adjacency matrix

Both matrix P and N can be represented by a ma-

trix M, called the adjacency matrix, as shown below.

There is a row and column for each node; M[i, j] = 1

if (i, j) element of matrix is more than 1, if (i, j) ele-

ment is otherwise M[i, j] = 0.

M

P

= [m

P

ij

]

M

N

= [m

N

ij

] (6)

m

ij

=

1 :ω

ij

≥ 1

0 :ω

ij

< 1orunknown

where M

P

is the adjacency matrix of P and M

N

is the

adjacency matrix of N.

Step2) creation of reachability matrix

The next step derives the reachability matrices R

P

and R

N

. It is calculated on gotten the adjacency ma-

trices with Boolean OR operation as follows:

R

P

=

m

∑

k=1

M

k

P

(7)

R

N

=

m

∑

k=1

M

k

N

(8)

When the number of k(≤ m) is added up, the ele-

ments of adjacency matrix M converge in speciﬁc k.

Then, the convergent matrix is reachability matrix. If

R(i, j) = 1 then it means i element is more important

than j element. if R(i, j) = 0 then i is less important

than j, respectively. Thus, R represents the relation

of important degree in n-tuple pairwise comparison.

Step3) judgement of the consistency of duality

In the last step, we judge the consistency of dual-

ity on P and N. When obtaining R

P

and R

N

in step2

have same elements, it is deﬁned by ω = 1/ω

′

that

both matrices are consistent in duality. To judge the

consistency, matrix C is deﬁned as

R

P

⊕ R

T

N

= C (9)

where operator ⊕ represents exclusive OR. As C =

O(where O is zero matrix)Cwe deﬁne the relation be-

tween P and N as being consistent. On the other hand,

as C 6= O, it suggests that there is an inconsistency of

order about i and j where 1 in C.

DECISION MAKING BASED ON DUALITY BETWEEN POSITIVE AND NEGATIVE EVALUATIONS

687

4 APPLICATION

The following example is taken from the paper pre-

sented by H.S.Rian and T.Sekiguchi (RIAN and

Takashi, 1995). Suppose that a company choices a

excellent person in three persons A

1

, A

2

and A

3

, on

decision-making for person perception. Decision cri-

teria are as follows:

• sense of responsibility (C

1

)

• inventive idea (C

2

)

• knowledge (C

3

)

Decision-maker gives two values (n = 3) through

a pairwise comparison of above criteria in positive

and negative. In positive criteria, for example, to ask

a decision-maker ”Which is more important a sense

of responsibility or an inventive idea for person per-

ception ?”. In negative, ”Which doesn’t need a sense

of responsibility or an inventive idea for person per-

ception ?”. Then, he gives the ratio as pairwise com-

parison value. Table 1 and Table 2 are results given

by a decision-maker through a pairwise comparison.

Values in square are calculated by given comparison

value. Both tables consisted of n − 1 values are satis-

ﬁed in Saaty’s C.I..

Table 1: Positive pairwise comparison by personality.

C

1

C

2

C

3

C

1

1 8/2 24/4

C

2

1 6/4

C

3

1

Table 2: Negative pairwise comparison by personality.

C

1

C

2

C

3

C

1

1 3/7 2/8

C

2

1 7/12

C

3

1

In next step, we check the dual consistency. Two

adjacency matrixes are created from Table 1 and Table

2 by Eq. (7).

M

P

=

1 1 1

0 1 1

0 0 1

,M

N

=

1 0 0

1 1 0

1 1 1

.

Then, reachability matrixes R

P

and R

N

are ob-

tained as

R

P

=

1 1 1

0 1 1

0 0 1

,R

N

=

1 0 0

1 1 0

1 1 1

.

Finally, the dual consistency index C can be calcu-

lated by Eq. (9).

C = O

In the calculation, C is zero matrix. It shows that

the dual consistency is satisﬁed under criteria. More-

over, both eigen vectors of maximum eigen values are

weight vector on criteria, givens as

W

T

P

= (0.706,0.176,0.118)

W

T

N

= (0.136,0.318,0.546).

By the above result, the order relation isC

1

≥ C

2

≥ C

3

in positive and negative. It shows that they are duality

in order relation.

5 CONCLUSIONS

This paper suggested a new decision-making model

to analyze the human’s double psychology in fraud

prevention.

We proposed the dual hierarchy, having posi-

tive matrix and negative matrix to evaluate two-sided

question. We discussed about the relation that both

matrixes are duality. Further, the technique to judge

a dual consistency of both matrixes, provides consis-

tency index based on a concept of reachability matrix

through the order relation of both matrixes. As a re-

sult, we are able to check any inconsistency between

positive and negative mentals.

In our future work, we plan to develop a method

for fraud detection with our decision-making model.

REFERENCES

Ataka, H. (1966). Boolean algebra. Kyouritsu, Japan.

Azuma, R. and Miyagi, H. (2009). Ahp for risk manage-

ment based on expected utility theory. In IEEJ C,

EISS. IEEJ.

RIAN, H. S. and Takashi, S. (1995). Decision-making of

person perception by the observed quantities : Prob-

lem of the person perception and personnel choice by

the extended fuzzy relation equation. Japan Society

for Fuzzy Theory.

T. L. Saaty (1980). The Analytic Hierarchy Process. Mc-

GrawHill.

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

688