A Model to Determine Priority in AHP Using Coefﬁcient Correlation

Ella Silvana Ginting

, Devy Mathelinea

and Herman Mawengkang

Faculty of Technology Management and Business, Universiti Tun Hussein Onn Malaysia, Johor, Malaysia

Faculty of Mathematics, Universitas Sumatera Utara, Medan, Indonesia

Keywords:

Analytic Hierarchy Process (AHP), Correlation Coefﬁcient Maximization Approach (CCMA), Additive

Normalization (AN).

Abstract:

Analytic hierarchy process (AHP) is an effective decision-making method for making decisions on complex

issues. Through AHP unstructured complex issues are simpliﬁed into structured parts in a hierarchy.Various

methods in AHP are used to determine the priority of multi-criteria issues, but in this thesis a priority de-

termination model is proposed through the correlation coefﬁcient or Correlation Coefﬁcient Maximization

Approach (CCMA). Then compared with the addition normalization method or Additive normalization (AN).

As an application of this method, it is about the election of a school principal by a foundation. Of the three

alternatives speciﬁed, there are four criteria that must be possessed, namely knowledge, quality of work, re-

sponsibility, and work discipline. The results obtained show that the order of priority through the calculation

of CCMA and AN is the same.

1 INTRODUCTION

Basically every individual is a decision maker. Every-

thing that is done consciously or not is the result of a

decision. The information obtained helps to under-

stand events. In order to make the right and good de-

cisions, clear and accurate information is needed. Not

all information is useful for increasing understanding

and consideration, if you only make decisions intu-

itively then you tend to believe that all kinds of infor-

mation are useful and the better.

In making a decision, it is necessary to know the

problem, needs and objectives of the decision, the de-

cision criteria, the sub-criteria and the groups affected

and the alternative actions to be taken. Then trying to

determine the best alternative, for example in the case

of resource allocation, it takes priority for alternatives

to allocate the right resources.

Decision making, which gathers most of the infor-

mation, has become a science of mathematics today.

Decision making involving many criteria and many

sub-criteria, is used to rank the alternatives for a deci-

sion.

The Analytic Hierarchy Process (AHP) was de-

veloped in the early 1970s by Thomas L. Saaty, is

an effective decision-making method by using factors

of logic, intuition, experience, knowledge, emotion

and feeling to be optimized in a systematic process.

(Saaty, 1990), argued that AHP has been accepted as

the most superior multi-criteria decision model, both

among academics and among practitioners for mak-

ing decisions on complex issues. The rules in AHP re-

late to the workings of the human mind, because AHP

simply relies on intuition as its main input, but intu-

ition must come from a decision maker who is well

informed and understands the decision problem.

The preparation of the decision structure in pri-

oritizing a problem is carried out by decomposing,

namely breaking the whole problem into elements of

the problem, so that the inﬂuencing factors and alter-

native decisions will be described which will be de-

termined in the form of a hierarchy of all elements.

In setting the priority of the elements in a deci-

sion problem is to make a pairwise comparison of a

speciﬁed criterion, so that a very inﬂuential scale is

obtained to compare the two elements. The results of

this assessment are presented in the form of a matrix

called the Pairwise Comparison Matrix. The prepa-

ration of the pairwise comparison matrix is to deter-

mine the importance value of each element in the de-

cision structure. Pairwise comparison matrices are

made based on the levels of each factor.

240

Ginting, E., Mathelinea, D. and Mawengkang, H.

A Model to Determine Priority in AHP Using Coefﬁcient Correlation.

DOI: 10.5220/0012447600003848

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 3rd International Conference on Advanced Information Scientiﬁc Development (ICAISD 2023), pages 240-243

ISBN: 978-989-758-678-1

2 METHOD

Priority estimation from the pairwise comparison ma-

trix is a major part of AHP. Through the pairwise

comparison matrix, the priority level of each element

will be determined. By developing priority vectors

for all matrices in the hierarchies formed for a par-

ticular decision problem, it is possible to perform

aggregation and obtain the ﬁnal priority vector (to-

tal priority). There are different techniques for de-

termining priority vectors from comparison matrices

and much research effort has been directed towards

ﬁnding the best estimation method. The eigenvector

(EV) method, which was proposed for the ﬁrst time

by (Saaty, 1977), proved that the eigenvector principle

of the comparison matrix can be used as the required

priority vector, for consistent or non-consistent judg-

ments from decision makers. The standard procedure

for determining priority vectors with the EV method

is based on a square comparison matrix and normaliz-

ing the number of rows. Saaty also proposed several

simple approximation methods to obtain the required

vectors.

The simplest method is the addition normalization

method or the additive normalization (AN) method.

This method generates priority by taking the sum of

the columns in the comparison matrix and by averag-

ing the values obtained in the rows. Although AN

is not widely accepted in the scientiﬁc community

which prefers more sophisticated methods, it is still

widely used because of its simplicity. The results of

the analysis show that this method is competitive with

other methods.

An interesting modiﬁcation of the EV method pro-

posed by (Cogger and Yu, 1985), is based on the

premise that the overall preference intensity informa-

tion is contained in the upper triangular matrix of

the comparison matrix. The calculation procedure is

recursive and simple, but the study by (Golany and

Kress, 1993), shows that this method is not effective

and can be excluded.

Used the EV method to investigate prioritization

with the addition of alternatives to the Analytic Hi-

erarchy Process. With the addition of alternatives

to the Analytic Hierarchy Process for certain cases,

the priority order of the previous alternatives can be

changed. But by modifying the way to normalize

the Eigen Vector (EV) it produces a procedure that is

able to maintain the order of priority (Schmidt et al.,

2015).

Most of the other methods of obtaining priority

from a comparison matrix are considered extreme be-

cause they are based on an optimization approach.

The prioritization problem is expressed as minimiz-

ing a certain objective function that measures the de-

viation between the ideal solution and the actual solu-

tion, subject to some additional constraints. As stated

by (Mikhailov and Singh, 1999), priority assessment

can be formulated as a non-linear optimization prob-

lem with constraints and solved by the direct least-

squares (DLS) method (Chu et al., 1979), stated that

although this method minimizes the Euclidean dis-

tance between the ideal solution and the actual solu-

tion, this method generally results in multiple solu-

tions which can be considered as a drawback from a

practical point of view. To eliminate it, several opti-

mization methods such as the weighted least-squares

method or the weighted least-squares (WLS) method

are proposed, using a modiﬁed Euclidean norm as the

objective function.

(Crawford and Williams, 1985) suggested the log-

arithmic least squares (LLS) method provides an ex-

plicit solution through an optimization procedure that

minimizes the logarithm of the objective function

by fulﬁlling the multiplication constraints. (Cook

and Kress, 1988), put forward the logarithmic least

absolute values (LLAV) method, a median-relation

method that is not biased toward determining extreme

values. (Wang et al., 2007), put forward an estimation

of priority in AHP through a correlation coefﬁcient,

known as the Correlation Coefﬁcient Maximization

Approach (CCMA). CCMA is able to maximize the

correlation coefﬁcient between its own priorities and

each column of the pairwise comparison matrix. Of

the various methods in determining priorities, the au-

thors put forward a model that will be discussed in

setting priorities, namely through the correlation co-

efﬁcient or the Correlation Coefﬁcient Maximization

Approach (CCMA).

3 RESULTS AND DISCUSSION

Correlation Coefﬁcient Maximization Approach

(CCMA) as an approach to maximizing the correla-

tion coefﬁcient used in determining the priority of the

Pairwise Comparison Judgment Matrices (PCJM).

According to (Wang et al., 2007), prioritization by

maximizing the correlation coefﬁcient can maximize

the correlation coefﬁcient between priorities and

each column of the pairwise comparison matrix.

Suppose A = (a

i j

)

nxn

is a pairwise comparison matrix

i j

= 1/a

i j

= 1 and a

i j

> 0 for i, j = 1,..,n

and W = (w

,...,w

)

as a priority vector with

∑

i=1

= 1 and w

≥ 0, for i = 1, ..., n . According

to (Saaty, 1988), if a

i j

= a

k j

for k = 1, ..., n then

A = (a

i j

)

nxn

it is called a perfectly consistent pairwise

comparison matrix. For a perfectly consistent com-

A Model to Determine Priority in AHP Using Coefﬁcient Correlation

241

parison matrix A = (a

i j

)

nxn

, it can be characterized

exactly by a priority vector W = (w

,...,w

)

follows:

i j

= w

,withi, j = 1,...,n (1)

From equation (1) obtained:

∑

j=1

∑

j=1

i j

∑

j=1

i j

, j = 1,...,n (2)

Based on equation 1, a consistent comparison ma-

trix A = (a

i j

)

nxn

) can be presented as an n column

vector as follows:







...













...







,...,







...







(3)

This shows that the n column vector is perfectly

correlated with the priority vector W = (w

,...,w

)

For example, R

is the correlation coefﬁcient be-

tween the priority vector W and the j-column vector

of the pairwise comparison matrix A = (a

i j

)

nxn

, then

it is obtained

∑

i=1

i j

− ¯a

)(w

− ¯w)

(

∑

i=1

i j

− ¯a

)

)(

∑

i=1

− ¯w)

)

(4)

with j = 1, ...n, ¯a

∑

i=1

i j

and

¯w =

∑

i=1

For a comparison matrix A = (a

i j

)

nxn

,asa

i j

, with (i, j = 1, 2, ...,n) and ¯a

= ¯w/w

for

( j = 1, ..., n), Equation 4 can be written as :

∑

i=1

− ¯w)(w

− ¯w)

(

∑

i=1

− ¯w)

)(

∑

i=1

− ¯w)

)

= 1, f or j = 1, ...,n

(5)

Based on the expression above, the optimization

model can be formulatee as follows.

MaximizeR =

∑

j=1

∑

j=1

∑

i=1

i j

− ¯a)

∑

i=1

i j

− ¯a

)

− ¯w)

∑

i=1

− ¯w)

(6)

Consider that

∑

i=1

= 1, and w

≥ 0, f ori =

1,...,n, and let

ˆw

∗

− ¯w)

∑

i=1

− ¯w)

(7)

i j

− ¯a

)

∑

i=1

i j

− ¯a

)

,i, j = 1,...,n (8)

We obtain:

∑

i=1

ˆw

= 1 dan

∑

i=1

i j

= 1,(i, j = 1,...,n)

Due to the Equations 5 - 7, the optimization model

can be transformed into the following expression.

MaximizeR =

∑

j=1

∑

i=1

i j

ˆw

∑

i=1

(

∑

j=1

i j

) ¯w)

(9)

We can derive Theorems from that optimization

model.

Theorem 1.

Let ˆw

∗

be the solution of optimization model and R

∗

is the value of objective function, then:

ˆw

∗

∑

j=1

i j

∑

i=1

(

∑

j=1

i j

)

(10)

∗

∑

i=1

(

∑

j=1

i j

)

(11)

From Equation 6, we get

= ¯w +

∑

i=1

− ¯w)

∗

,(i = 1, ..., n) (12)

Let β =

∑

i=1

− ¯w))

≥ 0, Equation 11 becomes:

= ¯w + β ˆw

+ β ˆw

,(i = 1, ..., n) (13)

(The differences of value will give different

priority vector).

To get the value of parameter, we could solve two

optimization model, as follows:

J =

∑

i=1

(

∑

j=1

− a

i j

)

∑

i=1

(

∑

j=1



β( ˆw

− a

i j

ˆw

∗

) −

i, j

− 1)



, f orβ ≥ 0

(14)

J =

∑

i=1

(

∑

j=1

i j

−

)

) =

∑

i=1

(

∑

j=1

i j

−

1/n + β ˆw

∗

1/n + β ˆw

∗

, f orβ ≥ 0

(15)

ICAISD 2023 - International Conference on Advanced Information Scientiﬁc Development

242

Teorema 2.

∗

∑

i=1

∑

j=1

i j

− 1)( ˆw

∗

− a

i j

ˆw

∗

)

∑

i=1

∑

j=1

( ˆw

∗

− a

i j

ˆw

∗

)

(16)

To determine the priority of a consistent pair com-

parison matrix, the CCMA determines the following

steps:

• Step 1. Normalize the pairwise comparison ma-

trix using Equation 8.

• Step 2. Calculate the weight of the transformation

using Equation 10 and maximizing the sum on the

correlation coefﬁcient with Equation 11,

• Step 3. Determine the value of the coefﬁcient with

Equation 16.

• Step 4. Calculate the ﬁnal priority (i = 1,2,...,n),

with the Equation 13.

4 CONCLUSIONS

This study proposes a model for determining priori-

ties in decision-making using the Analytic Hierarchy

Process (AHP) and the Correlation Coefﬁcient Max-

imization Approach (CCMA) to be applied to the se-

lection of school principals by a foundation, where

three alternatives and four criteria must be deter-

mined, namely knowledge, quality of work, respon-

sibility and work discipline. The results show that the

priority order through CCMA and Additive Normal-

ization (AN) calculations is the same. The proposed

model provides a competitive method for ranking al-

ternatives in decision-making and can be used to de-

termine priorities in complex decision scenarios.

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