Development of a Model based on Evaluation Considering Explicit and

Implicit Element in Multiple Criteria Decision Making

Rumiko Azuma

1

and Shinya Nozaki

2

1

Department of Social Informatics, Aoyama Gakuin University, Sagamihara, Kanagawa, Japan

2

Department of Electrical and Electronics Engineering, University of the Ryukyus, Okinawa, Japan

Keywords:

Decision-making Model, Implicit Evaluation, Principal Component Analysis, Analytic Hierarchy Process.

Abstract:

The Analytic Hierarchy Process (AHP) is a decision-making method for smoothly managing problems, cri-

teria, and alternatives. AHP can be used to respond to multiple criteria, and allows for the quantiﬁcation of

subjective human judgments, as well as objective evaluations. In a classical AHP, a decision-maker derives a

list of priorities by consciously comparing criteria and alternatives in order to deriving a comprehensive evalu-

ation. However, when the number of criteria increases, the problem also becomes complicated and the subjec-

tive judgment of the decision-maker tends to be clouded by ambiguity and inconsistency. As the solution, this

study proposes a method whereby latent elements are extracted from the data given by the decision-maker, and

an evaluation is made from a different aspect based on the extracted elements. This allows for the construction

of a model in which a decision is made from both explicit and implicit elements by making a ﬁnal synthesis

of the results obtained using the conventional method as well as the evaluation obtained using the method

proposed in this study. As a result, we can conclude that it is possible to make a decision that is not affected

by the ambiguity or inconsistency of the decision-maker.

1 INTRODUCTION

The analytic hierarchy process (AHP) (Saaty, 1980) is

well known as the procedure to solve multiple criteria

decision-making problems. AHP is the method which

quantiﬁes human’s subjective judgments, and makes

a decision by combining them and system approach

in the analysis of problem. AHP is used in a variety

of multiple-choice situations such as economic prob-

lems, management problems, medical issues, energy

problems, educational problems and city planning.

When making a decision, having a large number

of various criteria and alternatives tends to compli-

cate the problem and make it impossible to arrive at

the most appropriate decision. One problem is that

the hierarchal structure becomes complicated. When

creating a hierarchal structure, it is necessary to set

independent items in the criteria. If each criterion is

not independent, it is necessary to deﬁne a multi-level

hierarchy, such as AHP inner-dependence method

(Saaty and Takizawa, 1986) or dominant AHP (Ki-

noshita and Nakanishi, 1997)(Kinoshita and Nakan-

ishi, 1998). However, even in them it is impossible to

account for all of the implicit dependencies between

the criteria at a level beneath the decision-maker’s

awareness.

Another problem is that inconsistencies may oc-

cur in choices when criteria or alternatives must be

evaluated using subjective human judgment. Ambi-

guities and inconsistencies tend to occur more often

in human judgment when the number of criteria and

alternatives increase. The work involved in making

a pairwise comparison therefore becomes unmanage-

able and consistency consequently suffers. As a re-

sult, the reliability of the ﬁnal evaluation decreases,

and it is difﬁcult to make the best decision. In order to

resolve this problem, the absolute measure method on

AHP (Saaty, 1986) has been proposed. The method

is effective in case containing too many alternatives

and can avoid the rank reversal problem. However

using the the method, the results often lose reliability

because the comparison matrix does not always have

sufﬁcient consistency.

This study proposes a method whereby implicit

elements are extracted from the data of a decision-

maker’s judgments using principle component anal-

ysis (PCA) (Jolliffe, 2002), and new evaluation is

derived based on them. The conventional method

involves the decision-maker coming to a decision

based on explicit elements. The implicit elements ex-

271

Azuma R. and Nozaki S..

Development of a Model based on Evaluation Considering Explicit and Implicit Element in Multiple Criteria Decision Making.

DOI: 10.5220/0004905802710276

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 271-276

ISBN: 978-989-758-017-8

Copyright

c

2014 SCITEPRESS (Science and Technology Publications, Lda.)

tracted by PCA are as new criteria. Then, to utilize

new criteria enables to derive new evaluation from

a different aspect. Furthermore, a ﬁnal evaluation

can be made by synthesizing the explicit evaluation

and the implicit evaluation by the proposed method,

thereby constructing a decision-making model that is

not affected by ambiguity or inconsistencies of the

decision-maker.

There are researches to examine about best

method by comparing the evaluation by AHP with

the evaluation by PCA (Kim, 2006)(Wu et al., 2011).

On the other hand, there is research by which PCA

is applied to the decision-making method (Lee et al.,

2010). However, our approach is to integrate the eval-

uation by latent elements extracted by applying PCA

into AHP values which the decision-maker scored

subjectively. It is applied only to the process in which

the absolute measure method because PCA is an ef-

fective technique to normally-distributed data. Then,

our approach enables the decision-makers to achieve

clearer result.

2 BASIC CONCEPT

2.1 Analytic Hierarchy Process using

Absolute Measurement

The AHP is a technique used for dealing with prob-

lems which involve the consideration of multiple cri-

teria simultaneously. It is based on the principles

of decomposition structures, comparative judgments,

and synthesis of priorities. Comparative judgments

are necessary to perform the pairwise comparisons of

criteria and alternatives. However, in case containing

too many alternatives, it is burdensome for decision-

maker to draw pairwise comparison in alternatives.

It produces sometimes bad consistency. Saaty pro-

posed an absolute measure method on AHP to solve

the problem (Saaty, 1986). The difference between

the method and the conventional relative measure-

ment is in the procedure of scoring the alternatives

corresponding to criteria. The method is adopted that

indirect comparison. A decision-maker evaluates al-

ternatives using absolute measurement by linguistic

scales as ”very good”, ”good” and etc. The evalua-

tion value of linguistic scale is acquired by pairwise

comparison of criteria as in Table 1. Table 2 is a ex-

ample of evaluation values for linguistic scales. They

are derived from eigenvector calculated by the ratio of

a linguistic scale. In our proposal model, the absolute

measurement method is adopted.

Table 1: A example of pairwise comparison of linguistic

scale.

very good good common bad

very good 1 2 5 7

good 1/2 1 3 5

common 1/5 1/3 1 2

bad 1/7 1/5 1/2 1

Table 2: Evaluation value about Table 1.

linguistic scale weight

bad 0.120

common 0.209

good 0.569

very good 1.000

2.2 Principal Component Analysis

(PCA)

PCA is a data representation method and is a kind of

multivariate analysis. It can extract new indexes with-

out correlation from each data and analyze weight of

data in each element. Moreover, new indexes can be

extracted from a few of data set. This is achieved by

transforming to a new set of variables, the principal

components, which are uncorrelated, and which are

ordered so that the ﬁrst few retain most of the varia-

tion present in all of the original data.

Suppose that matrix A is made from the result of

a questionnaire ﬁlled out by decision-maker in Figure

1, and that S

a

is a covariance matrix of A, and that

λ

k

(k = 1, 2, . .. , n) are a eigenvalue of S

a

. n is number

of questionnaire items. Suppose that λ

1

is the largest

eigenvalue, and v

1

is the corresponding eigenvector.

It can be shown that for the second,third, . . . , nth

principal component,the vectors of coefﬁcients v

2

,v

3

,

. . . ,v

n

are the eigenvectors corresponding to λ

2

,λ

3

, .

. . , λ

n

. The vectors v

k

are principal components and

each of them is new indicator uncorrelated.

v

T

k

= ⌊v

k1

v

k2

. . . v

kn

⌋ (1)

where v

k

corresponds to kth column of new indicators

in Figure 1.

The principal component score z

ki

of ith alterna-

tive corresponding new indicator v

k

is given as

z

ki

= v

k1

a

i1

+ v

k2

a

i2

+ · ·· + v

kn

a

in

(2)

where a

in

is score of alternative A

i

corresponding cri-

terion C

n

.

ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems

272

Figure 1: Example of a matrix which is made from ques-

tionnaire data and new indicators which is made from the

matrix by PCA.

3 PROPOSED MODEL

We propose a method by adopting PCA. The pro-

posed method can extract new criteria from the abso-

lute evaluation values given by a decision-maker and

evaluate based on them. We call a result of it the eval-

uation based on implicit elements. In contrast, we call

the conventional absolute measure method on AHP

the evaluation based on explicit elements. Moreover,

we develop a new approach to derive clearer priority

by synthesizing implicit and explicit evaluation.

Figure 2 shows the ﬂowchart of proposal model

for considering both evaluations. In Figure 2, through

Step 1 to Step 5 is the same process as the conven-

tional AHP. The evaluation X

AHP

in Step 5 is given

as

X

AHP

= Sw (3)

X

AHP

i

=

n

∑

k=1

s

ki

w

k

where matrix S consists of scores s

ki

of each alter-

native about criteria in step 4, and vector w is the

weights of criteria in Step 3. X

AHP

i

is regarded as

explicit evaluation of ith alternative..

We derive the implicit evaluation in Step 6. At

ﬁrst, PCA is applied to the matrix S in order to acquire

new indicators v

1

, v

2

, . . . , v

m

. Number of new indica-

tors becomes less than half that of original criteria by

adopting until 90% of contribution rate. Moreover,

in the process of implicit evaluation, the decision-

maker does not need to be conscious of dependency

among criteria because v

k

is independent component.

Figure 2: The ﬂowchart of proposed decision-making

model.

Next, the weight w

′

k

of each indicator v

k

as new crite-

ria is acquired by pairwise comparison. Here, vector

w

′

= (w

′

1

, w

′

2

, . . . , w

′

m

) is a weight vector of new crite-

ria. Based on Eq.(2), a vector z

k

= (z

k1

, z

k2

, . . . , z

kn

)

made up of kth principal scores is attained as

z

k

= S · v

k

(k = 1, 2, . . . , m). (4)

Finally, the priority of alternatives based on im-

plicit elements is acquired on the following Eq.(6).

X

PCA

= Zw

′

(5)

where matrix Z = (z

1

, z

2

, . . . , z

m

). The implicit eval-

uation value of ith alternative is given as

X

PCA

i

=

m

∑

k=1

z

ki

w

′

k

(6)

Final evaluation is obtained in Step 7. We deﬁne

ﬁnal evaluation of ith alternative to synthesize Eq.(3)

and Eq.(6) as

X

i

= X

AHP

i

· X

PCA

i

(7)

4 APPLICATION

Suppose that a family is looking for the new house.

After visiting much real estate, eight houses remained

as possible houses for new life. We shall call them

A, B, C, D, E, F and G. A decision-maker has to

decide which house is the best. The decision-maker

DevelopmentofaModelbasedonEvaluationConsideringExplicitandImplicitElementinMultipleCriteriaDecision

Making

273

has identiﬁed the following decision criteria. Access,

Price, Safety, Comfort, Location, Width, Equipment

and Appearance (hereinafter referred to as ”Ac”, ”Pr”,

”Sa”, ”Co”, ”Lo”, ”Wi”, ”Eq” and ”Ap”). We apply

our procedure to the above mentioned example, fol-

lowing the steps in ﬂowchart in Figure 2.

4.1 Evaluation based on Explicit

Elements

The decision-maker constructs a evaluation matrix

with respect to decision criteria in Step 2. It is per-

formed through a pairwise comparison shown in Ta-

ble 3. The values for pairwise comparison in Table 3

is scored on 9-point measurement at the same as the

conventional AHP.

In Step 3, a weight vector w of criteria is acquired

as an eigenvector for a maximum eigenvalue of the

matrix composed of Table 3, given as

w = (0.156, 0.233, 0.269, 0.138, 0.081, (8)

0.067, 0.031, 0.025)

T

.

In Step 4, the decision-maker evaluates each alter-

native about criteria. The alternatives are scored by

evaluation values acquired according to the linguistic

scales provided for each criterion as Table 4 to 7, and

not by pairwise comparison. The result is described

in Table 8.

Table 8 is regarded as matrix S and the evalua-

tion X

AHP

based on explicit elements is derived from

Eq.(3) in Step 5.

X

AHP

= Sw (9)

= (0.370, 0.458, 0.451, 0.458,

0.459, 0.327, 0.472)

T

There is little difference among priorities of B, C, D

and E in the result of explicit evaluation. Then, in

addition to the conventional method, it is necessary to

evaluate alternatives from another perspective.

4.2 Evaluation based on Implicit

Elements

According to the Step 6, we obtain an evaluation

based on implicit element by adopting PCA. The re-

sult of Table 9 is obtained by applying PCA to the data

in Table 8. The result gives us new indicators consist-

ing of principal component score. The ﬁrst PC (prin-

cipal component) can be interpreted as being highly

positively related to the abundances of Ap (0.531) and

Lo (0.432), and negatively related to the abundance of

Sa (-0.455), that is, it expresses the beautiful urbane

Table 3: Pairwise comparison between each criterion.

Ac Pr Sa Co Lo Wi Eq Ap

Ac 1 1 1/3 1 3 3 5 5

Pr 1 1 1 3 5 3 5 7

Sa 3 1 1 3 5 3 5 7

Co 1 1/3 1/3 1 3 3 5 5

Lo 1/3 1/5 1/5 1/3 1 3 3 5

Wi 1/5 1/5 1/5 1/5 1/3 1/3 1 1

Ap 1/5 1/7 1/7 1/5 1/5 1/5 1 1

λ

max

= 8.606 C.I. = 0.087

Table 4: Evaluation values of criterion ”Ac”.

linguistic scale value

inconvenience 0.188

moderate 0.354

convenience 1.000

Table 5: Evaluation values of criterion ”Pr”.

linguistic scale value

low 1.000

moderate 0.464

expensive 0.208

very expensive 0.098

Table 6: Evaluation values of criteria ”Sa”,”Co”, ”Lo”,”Eq”

and ”Ap”.

linguistic scale value

very good 1.000

good 0.464

moderate 0.208

bad 0.098

Table 7: Evaluation values of criterion ”Wi”.

linguistic scale value

very large 1.000

large 0.538

moderate 0.274

narrow 0.129

very narrow 0.068

Table 8: Absolute evaluation of alternatives about each cri-

terion.

Ac Pr Sa Co Lo Wi Eq Ap

A 0.354 0.208 0.208 1.000 0.208 0.538 0.464 0.208

B 1.000 0.098 0.208 0.464 0.464 0.274 0.464 0.464

C 0.354 1.000 0.098 0.208 0.208 0.538 0.098 0.464

D 0.188 0.464 0.464 1.000 0.464 0.274 0.098 0.098

E 1.000 0.208 0.464 0.464 0.464 1.000 1.000 0.208

F 1.000 0.208 0.098 0.464 1.000 0.129 0.208 1.000

G 0.188 1.000 0.464 0.208 0.464 0.538 0.098 0.098

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274

Table 9: Principal component which obtained from absolute

evaluation.

Principal Component : PC

1st PC 2nd PC 3rd PC 4th PC

Ac 0.349 0.466 0.202 -0.015

Pr -0.168 -0.541 0.343 0.156

Sa -0.455 0.167 -0.220 0.543

Co -0.162 0.182 -0.705 -0.213

Lo 0.432 0.086 -0.118 0.776

Wi -0.351 0.274 0.476 0.077

Eq -0.159 0.589 0.231 -0.080

Ap 0.531 0.010 0.052 -0.144

cumulative con-

tribution ratio

41.3% 71.2% 90.7% 97.7%

Table 10: Pairwise comparison between new criterion.

C

1

C

2

C

3

C

4

C

1

1 1/5 1/3 1/5

C

2

5 1 3 1/3

C

3

3 1/3 1 1/5

C

4

5 3 5 1

λ

max

= 4.198 C.I. = 0.066

house which is not located in safety area. The sec-

ond PC, on the other hand, is positively related to the

abundance of Eq (0.589) and Ac (0.466), and nega-

tively related to the abundance of Pr (-0.541). There-

fore, the second index means that a house is prized

convenience more than price. Similarly, the third in-

dicator is interpreted in terms of Pr, Wi and Co, as

a house which is affordable and large but inconve-

niently located. The forth indicator is characterized

by Lo (0.776) and Sa (0.543), as a house that is at

safe place and good environment. Four indicators are

identiﬁed as the following decision criteria.

• the beautiful urbane house (C

′

1

)

• the house at more convenient place (C

′

2

)

• the affordable house (C

′

3

)

• the safety house in a good environment (C

′

4

)

The decision-maker constructs a evaluation matrix

with respect to four criteria by using 9-point measure-

ment. It is shown in Table 10. The weight vector w

′

of new criteria C’ is acquired as an eigenvector for a

maximum eigenvalue of the matrix composed of Ta-

ble 10, given as

w

′

= (0.064, 0.271, 0.122, 0 .544)

T

(10)

By applying Eq.(4), the vector z

k

of principal

component scores is acquired regarding the new cri-

terion C

′

k

. Then, we obtain the implicit evaluation by

normalization of Eq.(6).

X

PCA

= (z

1

z

2

z

3

z

4

)w

′

(11)

= (0.126, 0.231, 0.052, 0.006,

0.366, 0.196, 0.022)

T

As in Step 7, the ﬁnal evaluation X is obtained to

synthesize X

AHP

and X

PCA

, given as

X = (0.110, 0.251, 0.056, 0.007, (12)

0.399, 0.152, 0.025)

T

The priority of each alternative is E > B > F > A >

C > G > D.

5 CONCLUSIONS

In this paper, a decision-making model for consider-

ing both explicit and implicit element was presented.

We utilized the absolute measure method on AHP for

determining the evaluation based on explicit element.

On the other hand, we proposed the method adopting

PCA for determining the evaluation based on implicit

element.

The increase in criteria or alternatives becomes

frequently the cause of making vagueness in the

decision-maker’s judgment. In the case, we think that

our proposed procedure is effective. In the conven-

tional method, the priority of alternatives is obtained

using directly the score given by the decision-maker.

However, utilizing the score accompanied by vague

judgments directly makes the reliability of evaluation

lower. Therefore, we proposed the method which en-

ables decision-makers to evaluate based on implicit

elements extracted from the score accompanied by

vague judgments. Further, we tried to develop the

model for decision making by synthesizing explicit

evaluation and implicit evaluation. Our approach en-

ables decision-makers to achieve clearer result in de-

cision making.

ACKNOWLEDGEMENTS

This work was supported by The Ministry of Edu-

cation,Culture,Sports,Science and Technology under

Grant-in-Aid for Young Scientists (B), 24700909.

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