1.1 Title of Research
FEASIBILITY STUDY OF PRIORITISATION METHOD IN AHP (ANALYTIC
HIERARCHY PROCESS)
1
CHAPTER 1
INTRODUCTION
1.0
Introduction
The Analytical Hierarchy Process (AHP), developed by Saaty in 1970’s (Saaty,
1977; Cook, 1988; Bryson, 1995; Olson et al, 1995; Barzilai, 1997; Forman and Peniwati,
1998; Indrani, 2002) is a prevailing and widely used technique to solve complex
problems in multicriteria decision-making (Cook, 1988; Kumar and Ganesh, 1996; Zahir,
1999; Saaty, 2003). In AHP, the decision problem is structured as a hierarchic structure
with its elements, which are criterion on the top level and alternatives at the bottom level
(Srdjevic, 2003). Composite (final) priorities vector are derived from several alternatives
under some criteria through pairwise comparison of the alternatives at the bottom level of
the hierarchy. The AHP first determines the relative weights of all the criteria and
moving on to the alternatives in a top-down approach.
This paper ponders on the understanding of deriving priorities from a pairwise
comparison matrix, which is the principle of peewees comparison by using the
prioritization methods. In the next section, we explained three of the prioritization
methods in AHP, which are Additive Normalization Method, Eigenvector Method and
Geometric Mean Method. Section 3, discussed the example while section 4 concentrated
on the experiment and analysis. Finally in section 5, concluding remarks and future
works are presented.
1.2
Objective
i. To investigate state-of-the-art in prioritization methods within AHP.
ii. To identify the criteria that will be used to evaluate the prioritization methods.
iii. To compare each prioritization method based on selected criteria.
2
1.3
Scope
i. The research will focus on prioritization methods in AHP.
ii. The research will select the criteria will be used to evaluate the prioritization
methods.
iii. The research will evaluate each prioritization method based on selected criteria.
iv. This research will identify the best method based on selected criteria.
3
CHAPTER 2
LITERATURE REVIEW
2.0
Introduction
Multiple criteria decision making (MCDM) refers to making decisions in the
presence of multiple criteria. There are many methods available for solving MCDM
problems and one of the most popular methods is Analytic Hierarchy Process (AHP). The
next subtopic will discuss about the AHP.
2.1
AHP
The Analytic Hierarchy Process (AHP) is a flexible decision making method in
helping the decision makers to set priorities and make the best decision when both
qualitative and quantitative aspects of a decision need to be considered (Kumar and
Ganesh, 1996). AHP is usually used to solve a complex multi-criteria decision problem
(Cook, 1988; Kumar and Ganesh, 1996a; Sajjad, 1999; Saaty, 2003; Anderson et al.,
2003). AHP is a basic approach to decision making and has been acknowledged as an
important multi-criteria decision model (Sajjad, 1999).
Saaty is the one who responsible in introducing the AHP in 1970’s (Saaty, 1977;
Saaty et al., 1994; Cook, 1988; Mikhailov, 2003). Decision makers use the AHP to
resolve a problem because it allows the problem to be modeled in a hierarchical structure.
Decision makers must first understand and determine the goal, criteria and alternatives of
the problem before a hierarchic structure can be developed as show in Figure 1. This
structure essentially signifies the three level-hierarchy structures that indicate the
relationship of the goal, the criteria (A, B, C and D) and the alternatives (1, 2 and 3)
(Saaty, 1994).
4
Level 1
GOAL
Pairwise comparison
A
B
C
Level 2
D
Pairwise comparison
1
2
3
Level 3
Figure 1 show the three level-hierarchy model
Based on Figure 1, the AHP can be compartmentalize into three basic principles
which are decomposition of the problem, pairwise comparisons and composition of the
resulting priorities or synthesis of priorities (Kumar and Ganesh, 1996b; Liu et al., 1999).
The explanations about these three basic principles are as follow:
i. Decompose of the problem
Decomposition of the problem is done by structuring the problem in a hierarchical
structure in which is shown in Figure 1 (example of a three level hierarchy)
containing the goal or objective, criterion and alternatives. Level 1 which is the top
level for the hierarchy represents the main objective of the problem while Level 2
signifies the criteria of the problem. The lowest level which is Level 3 shows the
alternatives with respect to the criteria at the upper level. Essentially, the hierarchy
is usually in the form of three level hierarchy but Liu (1999) had denoted that the
level of hierarchy should divided into a four level-hierarchy which contains the goal
level, criterion level, sub criterion level and scheme (alternatives) level (Liu et al.,
1999).
5
ii. Pair wise comparisons
The pairwise comparison is the second principle of the AHP. The pairwise
comparisons are constructed by comparing pairs of elements in each level of
hierarchy with respect to every element in the higher level. This pairwise is used to
establish priorities for each set of elements in each level of hierarchy. Comparing
the pairs of elements is generated by giving a comparative judgments of preferences
for each pair of elements in every level using the Saaty’s nine-point scale
(Mikhailov, 2003) (see Appendix A). This comparison process is carried out to
determine which of the element in a pair is more desirable or preferred compared to
the others. These comparisons are positioned into a positive reciprocal or pair wise
comparison matrix. The derivation of the priorities from pair wise comparisons
matrix is the main concept of the AHP (Mikhailov and Singh, 1999; Srdjevic, 2003).
The AHP allows decision makers to derive ratio scale priority or weights from the
pair wise comparisons matrices (Anderson et al., 2003). The priorities or the
priority vector for every set in a level is estimated by using the prioritization
method (i.e. eigenvector method, additive normalization method, geometric mean
method). The details explanation about the pairwise comparison concept is
elucidated in subtopic 2.2.
iii. Composition of the resulting priorities or synthesis of priorities
This principle is applied to attain the composite priority for the lowest level
elements, which are the alternatives based on the overall preferences expressed by
the decision makers. Every priority vector (priorities) in the lowest level is weighted
by the higher level priorities in order to derive the composite (final) priority (the
overall relative weights of the alternatives) that reflects the overall importance of
each alternative (Mikhailov, 2003; Indrani, 2002; Saaty, 2003; Srdjevic, 2003). The
prioritized ranking of the decision alternatives can be derived from the composite
priority. Different methods of prioritization may lead to different final values (Saaty,
2003).
6
The major constituent of the AHP is the estimation of priorities from pairwise
comparison matrices. The priority vector can be derived from comparison matrices using
prioritization methods. The original prioritization method in AHP is Eigenvector Method
(EVM), proposed by Saaty (1990, 1994). Nevertheless, a number of criticisms have been
launched at EVM and the major criticisms are rank reversal, inadequacy of the scale, and
number of pairwise comparisons (Kumar and Ganesh, 1996).
2.2
Pairwise Comparison Concept
The decision makers can compare any two elements and provide a numerical value
of the ratio of their relative importance in the pairwise comparison processes. Consider a
prioritization of n elements stated as T1, T2,…, Tn. The intensity of preference element Ti
over element Tj which represent a judgment can be indicated as aij for i, j = 1, 2,…, n
(Indrani, 2002) by comparing these two elements. If element Ti is preferred to Tj, then
a ij > 1 or otherwise a ij < 1 and a ij = 1 (for all i, j = 1, 2,…, n) when the two elements is of
the same importance. Hence, the reciprocal property a ji = 1/ a ij by assumption will always
holds, with aii = 1 (for all i = 1, 2,…, n) (Srdjevic, 2003; Mikhailov and Singh, 1999).
Finally, a positive reciprocal matrix of pairwise comparison with the property A = a ij is
constructed by having a dimension of n × n (Golany and Kress, 1993).
For further understanding, let assume that matrix A is a reciprocal matrix:
T1
T1 ⎡ a11
⎢
T
A = 2 ⎢a 21
Μ⎢Λ
⎢
Tn ⎣a n1
T2
Λ
a12 Λ
Λ Λ
Λ Λ
Λ
Λ
Tn
a1n ⎤
a 2 n ⎥⎥
Λ ⎥
⎥
a nn ⎦
with,
7
⎡ w1 ⎤
⎢ ⎥
weights = ⎢ w 2 ⎥ ,
⎢ Μ⎥
⎢ ⎥
⎣ wn ⎦
where n is the number of elements and T are the objects while w is the derive weights
from the reciprocal matrix. For the elements of the main diagonal in matrix A which are
aii ,...ann , the elements will always equal to 1. Judgments are only required to the upper
diagonal of the matrix and only need n(n – 1)/2 of the judgments to generate a matrix for
every prioritization while the symmetrical elements are communally reciprocal (Srdjevic,
2003). This means that the elements below the diagonal elements are satisfying the
equation which is a ji = 1/ aij . The weights for the reciprocal matrix are attained by using
either one of prioritization methods in AHP that will be discussed in the next section.
If the matrix A is constructed consistently, then elements aij have the perfect values
aij = wi w j (for all i and j) and aij = aik a kj (for all k other than i and j) (Saaty, 1977; Cook,
1988; Saaty, 1990; Kumar and Ganesh, 1996a; Kumar and Ganesh, 1996b; Mikhailov
and Singh, 1999; Saaty, 2003; Srdjevic, 2003). Conversely, the matrix is seldom
constructed consistently because it depends on the decision maker evaluation.
2.3
Prioritization Method
AHP uses Prioritization Methods as the methods to derive priorities vector or
weights from pairwise comparison matrices (Saaty, 1990). The common and well-known
prioritization method in AHP is Eigenvector method, which was proposed by Saaty
(Saaty, 1977). There are other methods such as Modified Eigenvectors, Additive
Normalization, Direct Least Squares, Weighted Least Squares, Logarithmic Least
Squares (Geometric Means), Logarithmic Goal Programming, and Fuzzy Preferences
Programming. In the next subtopic, the methods discussed are only Eigenvector,
Logarithmic Least Squares (Geometric Means), and Additive Normalization.
8
2.3.1 Eigenvector Method (EV)
The Eigenvector method is the fundamental of the prioritization method and also
the most appealing method compare to the others (Mikhailov and Singh, 1999). It is the
first original method proposed by Saaty’s to derive the priorities in AHP. Saaty proved
that the principal eigenvector of A can be used as the desired priority vector by using the
Frobenius Theorem or Perron-Frobenius theory (Saaty, 1977). The priority vector w can
be attained by solving the equation (1) (Saaty, 1977; Takeda et al., 1987; Saaty, 1990;
Kumar and Ganesh, 1996a; Kumar and Ganesh, 1996b; Mikhailov and Singh, 1999;
Srdjevis, 2003; Saaty, 2003):
Aw = λ max w ,
λ max ≥ n ,
(1)
where λmax is the principal eigenvalue of A. If matrix A is consistent, λmax = n or otherwise
λmax ≥ n (Saaty, 1977).
The decision makers must first square the pairwise comparison matrix before the
priority vector w is calculated. In deriving the priority vector, the total of each row in the
pairwise comparison matrix are divided by the total of all the values in the same matrix.
These two processes (squaring and deriving priority vector) will be iterated until the
difference between the sums in two consecutive calculations is smaller than a prescribe
value or sometimes equal to zero (Kumar and Ganesh, 1996a; Kumar and Ganesh,
1996b). The priority vector w attained in the process will correspond and satisfy the
equation (1). This method is quite easy to understand but the calculation is rather
complicated if the dimension of the matrix is expanding and decision makers may need to
have computational assistance from a software or program to help them. The steps for
this method are as follow:
9
i.
Square the matrix.
ii.
Calculate the first priority vector.
Sum every element in each row of the pairwise comparison matrix. Divide the
value of each row attained by the total sum of the values in order to obtain the
first normalize priority vector.
iii.
Repeat step i and ii iterated until the resulting priority vector is similar with the
result of the previous iteration or satisfy the equation (1).
2.3.2 Geometric Mean Method (EV)
Geometric mean method (GM) is also known as logarithmic least squares method
(Saaty, 1990; Golany and Kress, 1993; Genest and Zhang, 1996) or approximate
eigenvector method (Kumar and Ganesh, 1996a). It is one of the methods used in
deriving estimates of ratio-scales from positive reciprocal matrix and was also proposed
by Saaty. GM minimizes the objective function (2):
∑ (log a
n
i , j =1
ij
2
− log wi / w j ) .
(2)
The a ij is assumed can be replaced by log aij in (2) even the two values will not be
the same. GM solution is given by the relation (3) (Saaty, 1990):
wi =
n
∏a
j =1
ij
n
n
i =1
j =1
∑∏a
ij
,
i = 1,..., n .
(3)
In order to obtain the priority vector w, decision makers have to multiply every
value in each row of a pairwise comparison matrix and power the values by 1⁄n (number
of dimension). In deriving the priority vector, each value attained then is divided by the
sum of all values. The priority vector is the normalized vector derived after the process is
completed. The steps for this method are as follow:
10
i.
Multiply each element in every row and then power of 1/n.
ii.
Sum all the values attained in i.
iii.
Divide the value of each row attained by the total sum of the values in order to
obtain the priority vector w as shown in relation (3).
2.3.3 Additive Normalization Method (AN)
This method is widely used and popular because of its simplicity and very easy to
understand (Srdjevic, 2003). Every element of matrix A is normalized by dividing each
element in a column by the sum of the elements in the same column to create a
normalized pairwise comparison matrix A ′ . The procedure to acquire matrix A ′ is as
given by the relations (4) (Srdjevic, 2003):
a ij′ = a ij
n
∑a
i =1
ij
,
i , j = 1, 2 ,..., n .
(4)
The priority vector w is attained by dividing the total sum of elements in each row
of matrix A ′ by the number of the matrix dimension, n. The formula is given by the
relations (5) (Srdjevic, 2003):
n
w i = (1 / n )∑ a ij′ ,
i = 1, 2 ,..., n .
(5)
j =1
The AN method has three basic steps of procedure in order to attain the priorities or
the priority vector (Anderson et al., 2003). Firstly, the decision makers have to total all
the values in each column of a pairwise comparisons matrix before dividing each of the
values in the matrix by its column total. This will result a matrix refer to as a normalized
pairwise comparison matrix. Finally the priority vector can be derived by computing the
average of the elements in each row of the normalized pairwise comparison matrix. In
other words, the number of the matrix dimension divides bthe total of each row of the
normalized pairwise comparison matrix in order to establish the priority vector. This
11
method is stated as an easy and simple method because it does not need any complex
calculation procedure to gain the priority vector (Srdjevic, 2003). The steps for this
method are as follow:
i.
Sum the values of every element in each column of the pairwise comparison
matrix.
ii.
Divide each element in a column of matrix pairwise comparison matrix by the
total sum of that column (values attained in process i). A normalized matrix is
generated at the end of this step as related to relation (4).
iii.
Compute the average of all the elements in each row of the normalized matrix to
acquire the priority vector w. Priority vector w is attained by using the relation (5).
2.4
Evaluation Criteria
The evaluation criteria are the criteria needed to measure the performance of each
prioritization methods in AHP. These criteria will compare and determine the best
method among all the prioritization methods in AHP.
2.4.1 Euclidean Distance (ED)
The ED is used to estimate the overall distance between all the judgment elements
in the comparison matrix and associated ratios of the priorities from the derived vector w
(Srdjevic, 2003). The best method is determined by the least ED value. The ED is
measured in the following way (Srdjevic, 2003):
⎡ n n
2⎤
ED = ⎢ ∑ ∑ (a ij − wi w j ) ⎥
⎣ i =1 j =1
⎦
1/ 2
i , j = 1, 2 ,..., n .
12
2.4.2 Total Deviation (TD)
The total of the deviations between ratios of weights and their equivalent entry in
each matrix is evaluated by using the TD (Golany and Kress, 1993). TD concludes the
best method with the TD value which used the same concept with ED criteria. The TD is
indicated as (Mikhailov and Singh, 1999):
2
⎡ n ⎡w
⎤
⎤
TD = ∑ ⎢ ∑ ⎢ i − aij ⎥ ⎥
w
i =1 ⎢ j =1 ⎣
⎦⎥ ⎥⎦
⎣ ⎢ j
1/ 2
n
i , j = 1, 2 ,..., n .
2.4.3 Minimum Violation (MV)
The MV total up all the violations related to the priority vector w for any techniques
used and is articulated as (Golany and Kress, 1993; Mikhailov and Sing, 1999; Srdjevic,
2003):
MV =
n
n
∑∑I
i =1 j =1
ij
,
where the rule of the ‘violation’, Iij is based on (Golany and Kress, 1993; Mikhailov and
Sing, 1999; Srdjevic, 2003),
⎧ 1
⎪0 .5
⎪
I ij = ⎨
⎪0 .5
⎪⎩ 0
if w i > w j and a ij > 1,
if w i = w j and a ij ≠ 1,
if w i ≠ w j and a ij = 1,
otherwise .
i , j = 1, 2 ,..., n .
If an alternative j is preferred to alternative i in every pairwise comparison even
though alternative i obtain a larger weight in the final priority vector derived (Golany and
Kress, 1993; Mikhailov and Sing, 1999; Srdjevic, 2003) (i.e. a ji > 1 but the priority vector
shows that wi>wj), this will be called a ‘violation’. The MV values obtained is divided by
13
n2 (n is the size of a matrix) in the purpose of conserving the constant comparisons of the
prioritization methods at the global hierarchy scale (Srdjevic, 2003).
2.4.4 Computational Time
The number of CPU seconds required by each of the prioritization method that has
been programmed to resolve any given problem (in this research is by using C language)
is referred to as the computational time. In most cases, the complexity level of each
prioritization method determines the time it takes to complete the problem; which means
the more complex prioritization method takes more time when compared to a lesser
method (Golany & Kress). In certain cases, the computational time is affected by the size
of a matrix which more time is allocated to a bigger matrix size.
2.4.5 Ratio Differences
The ratio differences occur when the sum of the entire priority vector derived is not
equivalence to 1. The exact value for the total sum of the final priority vector should
equal to 1 as the equation given below (Golany and Kress):
n
∑w
i =1
i
= 1.
The difference occur specify a slightly error in the implementation of the
computational by using the programming.
14
CHAPTER 3
METHODOLOGY
3.0
Introduction
The methodology used in this research is indicated in Figure 2. The figure shows all
the processes involve in determining the best method in prioritization method.
Proposal
Literature Review
Analysis Prioritization
Method
Develop Prioritization
Method
Fail
Experimental
Success
Result
Figure 2 shows the processes involve in this research.
As shown in Figure 2, there are six main processes in this research. The processes
implied are writing the proposal, prepare the literature review, analysis the prioritization
15
method, develop a program based on the prioritization methods chose, run the
experimental and lastly attain the results. These processes are explicated in details in the
next subtopic.
3.1
Proposal
The first thing that is needed to be done is writing the proposal for the research. In
this proposal, the main criteria needed are such as the title of the research need to be
conducted, objective, scope and a brief review about the research. These criteria had been
conversed in chapter 1.
3.2
Literature Review
The next process is about preparing the literature review. In this process we studied
about the AHP, prioritization methods in AHP and also the evaluation criteria. In this
state, we try to understand various prioritization methods explained by different
researcher. There is also a variety of evaluation criteria that we require to comprehend in
order to get as much knowledge as we could. All of this information is gathered by
journals of different researcher that studied in the same scope of work. We also used
books written by expertise in this field (AHP) to guide our review as it is very important
to address true facts in this research.
3.3
Analysis Prioritization Method
This process will take place after the preparation for the literature review is finished.
Based on the knowledge and information gathered in the preparation for the literature
review process, we determined which prioritization methods and evaluation criteria are
the best required and suitable to be studied in this research. After all the reviews done, we
16
concludes to concentrate on only three prioritization methods which are Eigenvector
method, Geometric Mean method, and Additive Normalization method. These three
methods had been conferred before in chapter 2. The reasons for Eigenvector method is
chose in this research are:
i.
It is the first and fundamental prioritization method introduced in AHP technique.
ii.
This prioritization method had been the main method studied for many past
researchers in their research.
The Geometric Mean method is selected as one of the prioritization method studied
because this method is frequently used and also stated as a defended method by Saaty
himself (Saaty, 1990). While for the last method which is Additive Normalization method
is elected because of it simplicity in deriving priority vectors and it is also latest method
in AHP (Srdjevic, 2003). For the evaluation criteria, we wind up to choose only five
criteria which are Euclidean Distance, Total Deviation, Minimum Violation,
Computational Time and Ratio Differences. The details for these evaluation criteria are
revealed in chapter 2.
3.4
Develop Prioritization Method
A programming that signifies the three prioritization methods selected is developed
by using C language in this process. The programming will do the same process of
deriving the priority vectors as the manual based on the given formula for each method. It
also includes the evaluation criteria that are used to determine which of the prioritization
methods studied is superior compare to the others. Users can choose whether to generate
the matrix required interactively or randomly by using the programming.
17
3.5
Experimental
The experimental process engages the experiment done by using the programming
developed in the previous process. In this research, the experimental studied are done by
using different dimension or size of matrices in order to establish the best prioritization
method. The experimental process also includes the experiment based on real scenario of
a majoring selection for the students of Science Computer and Information System
faculty in Universiti Teknologi Malaysia (UTM).
3.6
Result
This is the last process involve in the methodology. This process will give the
outcome or result for this research. This topic is discussed in details in chapter 4.
18
CHAPTER 4
RESULT AND DISCUSSION
4.0
Introduction
This chapter discussed the result and discussion about the research been made. In
order to obtain the result for the best prioritization method among the three methods
being studied, the program developed is applied in a case study of selecting the major
course for the students of Science Computer and Information System faculty in Universiti
Teknologi Malaysia (UTM). The case study is conferred in the next topic.
4.1
Case Study of Majoring Selection
The case study for the simulation is about a majoring selection for the students of
Science Computer and Information System faculty in Universiti Teknologi Malaysia
(UTM). A survey is conducted to distinguish the preference of the student in choosing
their majoring. The case study deals with 3 level-hierarchies as illustrated in Figure 2.
The hierarchy in Figure 2 is structured by following the first principle of AHP.
Majoring
Selection
Level 1
Pairwise comparison
Difficul
ty
CPA
Interest
Workload
Level 2
Pairwise comparison
SE
MIS
GMM
CMI
CSC
Level 3
Figure 2 show the three level-hierarchy of the majoring selection
19
The hierarchy in Figure 2 contains of 4 × 4 matrix at the Level 2 where all the
criteria are compared by importance with respect to the goal at Level 1. There are four
matrices of size 5 × 5 containing alternatives at Level 3. The judgment alternatives at
Level 3 are based on the criteria at Level 2. Based from Figure 2, the element of criteria
selected are CPA, Interest and Complexity while for the alternatives are Software
Engineering (SE), Information System (IS), Graphic Multimedia (GMM), Computer
Modeling Industry (CMI) and Computer System and Communication (CSC). The
students’ preferences or judgment of each element in each level with respect to the
element of the higher level is based on the Saaty’s nine-point scale (see Appendix A).
The pairwise judgments matrices for the problem are presented in Appendixes B.
The pairwise judgments are then entered into the programming that has been
developed to obtain the priority vectors. The priority vectors are derived by using
different prioritization methods (EV, GM and AN) for all judgment matrices and
represented in Table 1 to Table 5. These priority vectors are also called as local priority
vectors. The priority vectors are derived with respect to the second principles of AHP.
The weights derived in the priority vectors are ranked to determine which of the criteria
or alternatives is more preferred (most important) compare to the others.
Table 1 denotes the priority vector derived for the criteria (referred as M1). Based
on the Table 1, the CPA element is more important compare to the others and followed
by Complexity, Skills and the least important is Interest.
Table 1: Priority vectors for criteria (M1)
Priority Vectors
Criteria
Rank
EV
GM
AN
Difficulty
0.18501
0.18959
0.18593
3
CPA
0.53210
0.52766
0.52347
1
Interest
0.05921
0.05728
0.06173
4
Workload
0.22368
0.22546
0.22887
2
20
The priority vectors obtain for alternatives with respect to the criteria of Difficulty
(referred as M2) is represented in Table 2. The table shows that SE is preferred for every
method and closely followed by CSC. IS is ranked in 3 and trail by CMI and lastly by
GMM.
Table 2: Priority vectors for alternatives based on Difficulty (M2)
Alternative
Priority Vectors
Rank
EV
GM
AN
SE
0.34131
0.34322
0.33882
1
IS
0.24345
0.24121
0.24370
3
GMM
0.04067
0.04099
0.04160
5
CMI
0.06541
0.06470
0.06855
4
CSC
0.30916
0.30988
0.30733
2
Table 3 indicates the priority vector derived for alternatives with respect to the
criteria of CPA (referred as M3). From this table, it is clear that IS obtain the highest
value for every method and is ranked in 1. In the 2nd rank is CSC and GMM is in rank 3
according to the importance. The least important alternatives is CMI and followed by SE.
Table 3: Priority vectors for alternatives based on CPA (M3)
Alternative
Priority Vectors
Rank
EV
GM
AN
SE
0.03592
0.03548
0.03737
5
IS
0.46503
0.45799
0.45378
1
GMM
0.20082
0.20130
0.20408
3
CMI
0.08888
0.08904
0.09260
4
CSC
0.20936
0.21619
0.21217
2
21
The CSC alternative is chosen to be the most important alternative for each of the
prioritization method in the priority vectors for performance matrix of alternatives with
respect to the criteria of Interest (referred as M4). The comparison result of the priority
vectors obtained is obvious with CSC alternative is having the highest folowed by SE,
CMI, IS and GMM. This result can be seen in Table 4 as the table highlight the priority
vectors acquired for matrix M4.
Table 4: Priority vectors for alternatives based on Interest (M4)
Alternative
Priority Vectors
Rank
EV
GM
AN
SE
0.30916
0.29922
0.29957
2
IS
0.11709
0.11580
0.12571
4
GMM
0.04106
0.04122
0.04363
5
CMI
0.21975
0.20910
0.21700
3
CSC
0.31293
0.33466
0.31410
1
The priority vectors attained for the last performance matrix of alternatives with
respect to the criteria of Workload (referred as M5). is represented in Table 5. The most
important alternative for this performance matrix is all the same for every method which
is SE. Nevertheless, the second most important alternative is different with CMI is the
second most important alternative for EV method while CSC is the second important
alternative for GM and AN method. This case is still the same for the third important
alternative (which is rank 3) where CSC is the result for EV method while for GM and
AN method, the result is CMI. The least important alternative is the same for each
method which is GMM alternative and lastly is IS.
22
Table 5: Priority vectors for alternatives based on Workload (M5)
Alternative
Rank
Priority Vectors
EV
GM
AN
SE
0.39822
0.40170
0.39122
1
IS
0.05238
0.05292
0.05399
5
GMM
0.07126
0.07373
0.07524
4
CMI
0.24009
0.23371
0.23191
2(EV) / 3
(others)
CSC
0.23805
0.23794
0.24764
3(EV) / 2
(others)
4.2
Result
The result for each evaluation criteria are as below:
•
Euclidean Distance (ED)
The ED values computed by using the programming developed for each method
based on the hierarchical matrices are shown in Table 6. Based on Table 6, the AN
method is the best method for all matrices except for matrix M2 with GM method as
the best method. The overall result indicates that the AN method is superior than the
other two methods based on the ED criteria.
23
Table 6: Results of the ED evaluation for each method
Method /
Matrices
ED Value
M1
M2
M3
M4
M5
EV
4.833248
4.119511
5.784161
5.107903
4.970120
GM
4.949018
4.051668
5.765705
5.063570
4.948545
AN
4.515245
4.117686
5.169671
4.758167
4.652029
AN
GM
AN
AN
AN
Result
•
Total Deviation (TD)
The TD values of each method for the hierarchical matrices are presented in Table 7.
From the table, it is clear that AN method had overcome the other two methods for
every matrix in the hierarchy except for matrix M2 that determines GM method is
superior to the other two methods. In this case, it is clear that AN method is
superior to the other methods.
Table 7: Results of the TD evaluation for each method
Method /
Matrices
TD Value
M1
M2
M3
M4
M5
EV
7.469370
8.101945
9.589574
9.457125
9.276966
GM
7.361627
8.002236
9.634067
9.687864
9.195814
AN
7.217233
8.026870
8.933754
8.951055
8.719060
AN
GM
AN
AN
AN
Result
•
Minimum Violation (MV)
Table 8 shows the MV values of each method for the hierarchical matrices. The MV
values of each method are all the same for matrix M1, M2, M3 and M4 which
indicates that there is no method is superior to the others. In the other hand, it shows
that EV method for matrix M5 is superior than the other two methods because it
24
does not have any MV values compare to the others. The overall results indicate
that each of the method can be selected as a good prioritization method.
Table 8: Results of the MV evaluation for each method
Method /
Matrices
MV Value
M1
M2
M3
M4
M5
EV
0.06250
0.04000
0.04000
0.04000
0.00000
GM
0.06250
0.04000
0.04000
0.04000
0.04000
AN
0.06250
0.04000
0.04000
0.04000
0.04000
Equal
Equal
Equal
Equal
EV
Result
•
Computational Time
Table 9 represents the computational time for each method based on each of the
performance matrices only in order to derived local priority vector and not the
overall process of determining the final priority vector.. From the table, all the
methods have equal performance for each matrix M1, M2, M3 and M4, M5. As a
conclusion, in this case, AN method seems to be more superior to the other two
methods in all the performance matrices.
Table 9: Results of the Computational Time evaluation for each method
Method /
Matrices
Computational Time Value
M1
M2
M3
M4
M5
EV
0.000000
0.000000
0.000000
0.000000
0.000000
GM
0.000000
0.000000
0.000000
0.000000
0.000000
AN
0.000000
0.000000
0.000000
0.000000
0.000000
Equal
Equal
Equal
Equal
Equal
Result
25
•
Ratio Differences
The difference occur specify a slightly error in the implementation of the
computational by using the programming. Table 10 shows the ratio differences
values for the matrices (M1 – M5). Every method is equal to each other for matrix
M2 and for matrix M4 the method selected is AN. For the matrix M1, EV and AN
method are selected because there are no ratio differences between the two methods.
AN method is selected for matrix M3 and for matrix M5, either EV or AN method
can be choose. The conclusion for the ratio differences criteia is still AN method is
the best compare to the others.
Table 10: Results of the Ratio Differences evaluation for each method
Method /
Matrices
Ratio Differences Value
M1
M2
M3
M4
M5
EV
0.000000
0.000000
-0.000010
0.000010
0.000000
GM
0.000010
0.000000
0.000000
0.000000
0.000010
AN
0.000000
0.000000
0.000000
-0.000010
0.000000
Result
EV & AN
Equal
AN
GM
EV & AN
Table 11 shows the final evaluation criteria values for every method. The final
evaluation criteria values are obtain by totaling the values of each evaluation criteria for
each method. The total time for the computational time is calculated for the overall
process needed by each of the prioritization method in order to derive the final priority
vector. From the result in Table 11, it is clear that AN method is superior to the other
methods.
26
Table 11: Final evaluation values for each method
Evaluation Criteria
Results
Methods
EV
GM
AN
Euclidean Distance
24.814943
24.778506
23.212798
AN
Total Deviation
43.894980
43.882608
41.847972
AN
Minimum Violation
0.182500
0.222500
0.222500
EV
Computational Time
0.078000
0.063000
0.015000
AN
All the priority vectors derived (for all the comparison matrices) are synthesized in
order to derive the composite (final) priority for the lowest level elements (alternatives).
The deriving of final priority is the third principle of AHP. Table 12 represents the final
priority vectors for each method used. From the table, the most recommended alternative
for every method is all the same which is absolutely alternative IS.
Table 12: Final priority vectors values for each method
Final Priority Vectors
Alternative / Methods
Rank
EV
GM
AN
SE
0.18964
0.19150
0.19059
3
IS
0.31113
0.30596
0.30297
1
GMM
0.13275
0.13297
0.13448
4
CMI
0.12611
0.12392
0.12769
5
CSC
0.24037
0.24564
0.24427
2
27
4.3
Discussion
The AN method is indeed to be the best method based on the performance matrices
(M1 – M5) but after the third principle of AHP is applied, every method gives the same
final result (final vector. This situation indicates that all the method is at equal
performance in deriving final priority vector. This result proved the statement of Golany
and Kress (1993) who stated that there is not a single prioritization method which is
superior to the others in all cases.
28
CHAPTER 5
CONCLUSION AND FUTURE WORKS
5.0
Conclusion and Future Works
Through research and testing based on the case study, it has been found in entirely
that the GM method is better than the EV method. This result is also supported by
previous a researcher, Takeda (1987). He conducted research on prioritization method
(GM, EV and Modified EV). However, the AN method is a new method that is not been
used by many researchers in their research. Hence, it is quite uncertain to state any proof
that AN method is more superior compare to the other two methods. Yet the studied
made by Srdjevic stated that AN method is superior to GM and EV method (Srdjevic,
2003). In this research, we have considered the degree of inconsistency for every
matrices developed which is very important to determine the best method.
The result acquired in this research may be influence by evaluation criteria used. In
this research, we have combined a few of the evaluation criteria from past researcher. For
the computational time evaluation, the times needed to derive the local priority vectors
are null for each of the method because of the simplicity of each the performance
matrices. Nevertheless, the time needed to obtain the final priority vector (the overall
process) is different for each method with EV method having the longest time to finish
the whole calculation followed by GM and AN method. This is due the complexity of the
calculation for each method. However, in this research, the matrices developed are simple
and less complex. Therefore the computer managed to calculate and obtain the priority
vectors without any computational time.
29
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32
Paper Publication.
Type of
Title
Level
conference
Seminar
Comparison Analysis of
Kebangsaan Sains
prioritization method in
Pemutusan 2004
Analytic Hierarchy
National
Process (AHP)
Synthesis
Prosiding Seminar
Perbandingan Kaedah
National
Kebangsaan Sosio- Eigenvektor dan Purata
Ekonomi dan IT
Geometrik di dalam
ke-2
Proses Hirarki Analitik
33