Analytical Hierarchy Process Approach An Application Of Engineering Education

Analytical Hierarchy Process approach An application
of engineering education
P. Kousalya
Vignana Bharathi Institute of Technology, Ghatkesar, Hyderabad, INDIA.
kousalya29@yahoo.com
G. Mahender Reddy
Gurunank Engineering College, Ibrahimpatnam, Hyderabad, INDIA
mahender1563@yahoo.co.in
S. Supraja
Vignan s Institute of Technology & Science, Nalgonda, INDIA.
sprj10@gmail.com
V. Shyam Prasad
Gurunank Engineering College, Ibrahimpatnam, Hyderabad, INDIA.
shyamnow4u@gmail.com
Abstract
This paper aims at giving an application of Analytical Hierarchy Process
(AHP, a Multi Criteria Decision Making method) . Here AHP is applied for
selection of a student from an Engineering college who is eligible for All
Round Excellence Award for the year 2004-05 by taking subjective judgments
of decision maker into consideration. Seven criteria were identified for getting
this award and the alternatives are the five Branches of an Engineering
college, in the state of Andhra Pradesh, INDIA. It is observed that a student of
ECE (Electronics and communications engineering) branch has received the
award.
Key words: AHP, Multi Criteria Decision Making, consistency,
priorities, Eigen vector method
Mathematica Aeterna, Vol. 2, 2012, no. 10, 861 - 878

1 Introduction
The Analytic Hierarchy Process (AHP) is a multi-criteria decision-
making approach and was introduced by Saaty .The AHP has attracted the
interest of many researchers mainly due to the nice mathematical properties
of the method and the fact that the required input data are rather easy to
obtain. The AHP is a decision support tool which can be used to solve
complex decision problems. It uses a multi-level hierarchical structure of
objectives, criteria, sub criteria, and alternatives. The data are derived by
using a set of pair wise comparisons. These comparisons are used to obtain
the weights of importance of the decision criteria, and the relative
performance measures of the alternatives in terms of each individual decision
criterion. If the comparisons are not perfectly consistent, then it provides a
mechanism for improving consistency.
The Analytical Hierarchy Process or AHP was first developed by Professor
Thomas L. Saaty in the 1970 s and since that time has received wide
application in a variety of areas. AHP has been applied for a vast number of
areas, but it was not applied for the problem that has been taken up in this
paper. Thomas.L.Saaty [4,5 ] has explained about Hierarchies, multiple
objectives and Fuzzy sets . Patric T.Harker [1] explained AHP in detail as an
art of Science and Decision-making. R.Ramanathan and L.S.Ganesh [2]
used group preference aggregation methods employed in AHP by deriving
members weightages. AHP has been applied to many areas but problem
chosen presently was not taken into consideration earlier in the literature.
862 P. Kousalya, G. Mahender Reddy, S. Supraja and V. Shyam Prasad

2 Methodology
A. Establishment of a structural Hierarchy
A complex decision is to be structured in to a hierarchy descending
from an overall objective to various criteria, sub criteria till the lowest level.
The overall goal of the decision is represented at the top level of the
hierarchy. The criteria and the sub criteria, which contribute to the decision,
are represented at the intermediate levels. Finally the decision alternatives
are laid down at the last level of the hierarchy. According to Saaty (2000), a
hierarchy can be constructed by creative thinking, recollection and using
people s perspectives.
B. Establishment of comparative judgments
Once the hierarchy has been structured, the next step is to determine the
priorities of elements at each level. A set of comparison matrices of all
elements in a level with to respect to an element of the immediately higher
level are constructed. The pair wise comparisons are given in terms of how
much element A is more important than element B. The preferences are
quantified using a nine point scale that is shown inTable1.
C. Synthesis of priorities and measurement of consistency
The pair wise comparisons generate the matrix of rankings for each level
of the hierarchy after all matrices are developed and all pair wise
comparisons are obtained, Eigen vectors (relative weights) are obtained.
Eigen Vector Method: Suppose we wish to compare a set of n objects in
pairs according to their relative weights. Denote the objects by A1,A2, ..An
and their weights by w1,w2, ..wn. The pair wise comparisons may be
represented by a matrix as given in Table 1a).
Analytical Hierarchy Process approach 863

Table 1a) Matrix containing weights
A1 A2 An
A1 w1/w1 w1/w2 w1/wn
A2 w2/w1 w2/w1 w2/w1
.
.
.
An wn/w1 wn/w1 wn/w1
The matrix shown in Table 1a) has positive entries everywhere and
satisfies the reciprocal property aji = 1/aij. It is called a reciprocal matrix. If
we multiply this matrix by the transpose of the vector wT = ( w1,w2, ..wn) we
obtain the vector nw.
Table1: Saaty s Ratio scale for pair wise comparison of importance
of weights of criteria/ alternatives
Intensity of
Importance
Definition Explanation
1 Equal importance Two elements contribute equally to
the property
3 Moderate importance
of one over another
Experience and judgment slightly
favor one over the other
5 Essential or strong
importance
Experience and judgment strongly
favor one over another
7 Very strong
importance
An element is strongly favored and
its dominance is demonstrated in
practice.
9 Extreme importance The evidence favoring one element
over another is one of the highest
possible order of affirmation
2,4,6,8 Intermediate values
between two adjacent
judgments
Comprise is needed between two
judgments
Reciprocals When activity i compared to j is assigned one of the above
numbers, the activity j compared to i is assigned its reciprocal
Rational Ratios arising from forcing consistency of judgments

Our problem takes the form Aw= nw. We started with the assumption that
w was given.But if we only had A and wanted to recover w, we would have to
solve the system (A- nI) w = 0 in the unknown w. This has a nonzero
solution if n is an eigenvalue of A, i.e., it is a root of the characteristic
equation of A. But A has unit rank since every row is a constant multiple of
the first row. Thus all the eigenvalue i ,i=1,2, ..n of A are zero except one.
Also it is known that


n
i 1
 i = tr (A) = n, and i = 0, i  max . The solution w of this problem is any
column of A. These solutions differ by a multiplicative constant. However, this
solution is normalized so that its components sum to unity. The result is unique
solution no matter which column is used. The matrix A satisfies the cardinal
consistency property
The consistency ratio is calculated as per the following steps i) Calculate the
Eigen vector or the relative weights and max for each matrix of order n.ii)
Compute the consistency index for each matrix of order n by the formulae CI
= (max n ) / (n 1) iii) The consistency ratio is then calculated using the
formulae CR = CI / RI ,where RI is a known random consistency index
obtained from a large number of simulation runs and varies depending upon
the order of the matrix .

Table 2: Average random index (RI) based on matrix size (adapted from
Saaty, 2000)
Size of
matrix (n)
Random consistency
index (RI)
1 0
2 0
3 0.52
4 0.89
5 1.11
6 1.25
7 1.35
8 1.40
9 1.45
10 1.49
The acceptable CR range varies according to the size of the
matrix i.e. 0.05 for a 3 by 3 matrix, 0.08 for a 4 by 4 matrix and 0.1 for all
larger matrices, for n  5 (Saaty, 2000) if the value of CR is equal to, or less
than that value it implies that the evaluation within the matrix is acceptable
or indicates a good level of consistency in the comparative judgments
represented in that matrix. If CR is more than that acceptable value,
inconsistency of the judgments within the matrix has occurred and the
evaluation process should be reviewed.
2.1. The Use of Pairwise Comparisons
One of the most crucial steps in many decision-making methods is the
accurate estimation of the pertinent data. This is a problem not bound in the
AHP method only, but it is crucial in many other methods which need to
elicit qualitative information from the decision-maker. Very often qualitative
data cannot be known in terms of absolute values. For instance, "With
respect to Academics Criterion, what is the relative performance of EEE
over ECE? " Although information about questions like the previous one are

vital in making the correct decision, it is very difficult, if not impossible, to
quantify them correctly. Therefore, many decision-making methods attempt
to determine the relative importance, or weight, of the alternatives in terms
of each criterion involved in a given decision-making problem. Pairwise
comparisons are used to determine the relative importance of each
alternative in terms of each criterion. In this approach the decision-maker
has to express his opinion about the value of one single pairwise comparison
at a time. Usually, the decision-maker has to choose his answer among 10-17
discrete choices. Each choice is a linguistic phrase. Some examples of such
linguistic phrases are: "A is more important than B", or "A is of the same
importance as B", or "A is a little more important than B", and so on .The
main problem with the pairwise comparisons is how to quantify the
linguistic choices selected by the decision maker during their evaluation. All
the methods which use the pairwise comparisons approach eventually
express the qualitative answers of a decision maker into some numbers
which, most of the time, are ratios of integers. Since pairwise comparisons
are the keystone of these decision-making processes, correctly quantifying
them is the most crucial step in multi-criteria decision-making methods
which use qualitative data. Pairwise comparisons are quantified by using a
scale. Such a scale is a one-to-one mapping between the set of discrete
linguistic choices available to the decision maker and a discrete set of
numbers which represent the importance, or weight, of the previous
linguistic choices. The scale proposed by Saaty is depicted in table 1. Other
scales have also been proposed by others.
3 Physical Significance of all the Criteria
Attendance(C1): The students attendance has to be above 75% throughout
four-year period in all the semesters.

Academics(C2): The students academic record should be consistently above
70% in all the Semesters throughout four-year period.
Co-curricular activities(C3): A student has to participate in co-curricular
activities like Paper presentation, debates, Group Discussions or quizzes etc,
either in inter college or intra- college and need to win some prizes.
Extra curricular activities(C4): A student has to participate in extra-
curricular activities like Indoor games, Outdoor games which are held in
intra college or inter college and need to win some prizes.
Cultural activities(C5): A student has to participate in cultural activities like
Singing or choreography which are held in Intra College or Inter College and
need to win some prizes.
General behavior(C6): A student is required to be honest and need to
maintain good relationship with his / her peers and with teachers.
Departmental activities(C7): A student need to participate in the activities
conducted by the department and need to posses certain managerial skills
and need to coordinate different activities/events held in the department.
The matrix of pair wise comparisons of the criteria as given by the decision
maker is shown in Table 3, along with the resulting vector of priorities. The
vector of priorities is the principal Eigen vector of the matrix .It gives the
relative priority of the criteria measured on a ratio scale given in Table 1.Next
we move to pair wise comparisons of the lower level and lastly to the pair
wise comparisons of the lowest level .The elements to be compared pair wise
are the engineering branches with respect to how much better one is than the
other in satisfying each criterion in level 2.Thus there will be fifteen 5 x 5
matrices of judgments. To understand these judgments, a brief description of
the engineering branches is follows.

EEE: This branch consists of students who are good at academics,
attendance and Co-Curricular activities. Their participation is comparatively
less in Extra Curricular activities when compared to other branch students.
ECE: The students of this branch are highly motivated and hence have good
academic records and attendance. Their general behavior is good. The
departmental activities are conducted well. Though their participation in
Extra curricular activities and Cultural activities is less, compared to other
branch students, they are good at Co Curricular activities.
ICE: This branch consists of students who are less motivated and hence poor
in academics and attendance. Their relation ship with teachers and peers is
not good when compared with other branch students. They are good at Extra
curricular activities , cultural activities and are able to manage events well in
their departments.
CSE: The students of this branch are good in academics and attendance as
students of EEE.The relationship with peers is not good. They are good in
Extra Curricular activities and Co curricular activities. They manage events
well as students of EEE and ECE.
MECH: The students of this branch are less motivated and hence are not
good in academics and attendance. They are good at Extra Curricular
activities and Cultural activities, but not good at Co Curricular activities.
Their relationship with teachers and peers is not good.
Table 3 gives the opinions of the decision maker regarding the alternatives
with respect to each of the criteria/ sub criteria. The opinions given are
converted to numbers using Saaty s ratio scale. The consistency index and
Consistency ratio of each matrix are calculated using the formulae given in
section II. Table 4 gives the weights of the criteria calculated using Eigen
vector method shown in section II. Table 5 shows the local and global

priorities of the alternatives. Figure 1 shows the hierarchical decomposition
of criteria, sub criteria and alternatives. Level 0 shows the overall goal of All
round Excellence award in the zeroth level, shown in blue color. The next
level, namely level 1 shows the criteria as its elements, which are shown in
orange colored cells. Its next level namely level 2 shows the sub criteria in the
lavender colored cells. It can be observed that not all the criteria have sub
criteria. The criteria like Academics and attendance do not have any criteria
as identified by the decision maker. The next level namely level 3 is the
highest level given by alternatives shown in green colored cells.
Figures 2 16 shows the weights of alternatives with respect to each of the
criteria or sub criteria. Table 6 shows the weights and ranks of alternatives.
Figure 17 shows the ranks of alternatives through bar diagrams.
4 Conclusions & Scope:
The above problem of selecting a student for all round Excellence award is an
application of Analytic hierarchy Process, as this uses qualitative opinions.
AHP has been used for a wide number of areas, but AHP waste not applied to
the problem studied here. The student of ECE gets the All Round Excellence
Award as he/she is good in academics and general behavior which are highest
priority criteria, had the largest priority from figure 17 and Table 6. The
student of CSE branch is equivalently good who performed better than EEE
students. It can be observed that the student of ICE branch is relatively very
poor in everything.
The Fuzzy AHP method could be applied to the above and the results could
be compared. Sensitivity analysis could be applied to check how sensitive the
alternatives and criteria weights are, to small changes.

Figure: 1 Hierarchical decomposition of criteria, sub criteria and alternatives
All Round Excellence Award
Academics(C1)
) Attendance(C2) Extra curricular
Activities(C4)
Co-curricular
Activities(C3)
Cultural
Activities(C5)
General
Behavior(C6)
Departmental
Activities(C7)
Paper Presentation(C31) Singing(C51)
Choreography(C52)
Honesty(C61)
Group Discussion(C32)
Debate(C33)
Quiz(C34)
In-door games(C41)
Out-door games(C42)
Peer Relationship
(C62)
Teacher
Relationship(C63)
Event Management
(C71)
Coordinating (C72)
EEE(A1) ECE(A2) ICE(A3) CSE(A4) MECH(A5)

Table 3 :Pair wise comparison matrices for level 2
Academics Attendance
EEE ECE ICE CSE MECH
priority
vector EEE ECE ICE CSE MECH
priority
vector
EEE 1 0.33 5 3 5 0.28428 EEE 1 0.33 0.2 0.33 5 0.1032
ECE 3 1 5 3 5 0.44204 ECE 3 1 5 3 7 0.508
ICE 0.2 0.2 1 0.33 1 0.0634 ICE 5 0.2 1 1 1 0.1608
CSE 0.33 0.33 3 1 3 0.15004 CSE 3 0.33 1 1 3 0.1999
MECH 0.2 0.2 1 0.3 1 0.0622 MECH 0.2 0.143 1 0.33 1 0.0633
max = 5.1892, C.I =0.0473,C.R=0.0426 max = 5.9761, C.I =0.244,C.R=0.2198
Paper presentation Debate
priority
priority
vector
EEE 1 0.33 5 1 5 0.23164 EEE 1 1 3 1 3 0.2818
ECE 3 1 5 3 5 0.4487 ECE 1 1 1 3 3 0.2818
ICE 0.2 0.2 1 0.33 0.33 0.05121 ICE 0.33 1 1 0.33 0.33 0.0934
CSE 1 0.33 3 1 3 0.18883 CSE 1 0.33 3 1 3 0.2258
MECH 0.2 0.2 3 0.33 1 0.07963 MECH 0.33 0.33 3 0.33 1 0.1163
max = 5.2914, C.I =0.0729,C.R=0.0656 max = 5.6874, C.I =0.1719,C.R=0.1548
Group discussion Quiz
priority
priority
vector
EEE 1 0.33 3 1 3 0.19448 EEE 1 1 3 1 0.33 0.1776
ECE 3 1 5 3 5 0.46212 ECE 1 1 3 3 5 0.381
ICE 0.33 0.2 1 0.33 0.33 0.05829 ICE 0.33 0.33 1 0.33 1 0.0915
CSE 1 0.33 3 1 3 0.19448 CSE 1 0.33 3 1 3 0.2212
MECH 0.33 0.2 3 0.33 1 0.09065 MECH 3 0.2 1 0.33 1 0.1287
max =5.1952, C.I =0.0488,C.R=0.044 max =5.9064, C.I =0.2266,C.R=0.2041
Indoor games Outdoor games
priority
priority
vector
EEE 1 1 3 0.33 0.33 0.13526 EEE 1 3 1 1 0.33 0.183
ECE 1 1 0.33 0.33 0.33 0.08698 ECE 0.33 1 0.33 0.33 0.33 0.0755
ICE 0.33 3 1 0.33 0.33 0.10836 ICE 1 3 1 1 1 0.2284
CSE 3 3 3 1 0.33 0.262 CSE 1 3 1 1 1 0.2284
MECH 3 3 3 3 1 0.40741 MECH 3 3 1 1 1 0.2846
max =5.547, C.I =0.1368,C.R=0.1232 max =5.1416, C.I =0.0354,C.R=0.0319
Singing Choreography
priority
priority
vector
EEE 1 3 3 1 1 0.27188 EEE 1 0.33 3 1 1 0.1811
ECE 0.33 1 3 0.33 0.33 0.11222 ECE 3 1 3 0.33 0.33 0.1808

ICE 0.33 0.33 1 0.33 0.33 0.07217 ICE 0.33 0.33 1 0.33 0.33 0.0748
CSE 1 3 3 1 1 0.27188 CSE 1 3 3 1 1 0.2817
MECH 1 3 3 1 1 0.27188 MECH 1 3 3 1 1 0.2817
max =5.1372, C.I =0.0343,C.R=0.0309 max =5.5737, C.I =0.1434,C.R=0.1292
Honesty Peer relationship
priority
priority
vector
EEE 1 1 3 1 3 0.27303 EEE 1 1 1 1 1 0.2
ECE 1 1 3 1 3 0.27303 ECE 1 1 1 1 1 0.2
ICE 0.33 0.33 1 0.33 1 0.09046 ICE 1 1 1 1 1 0.2
CSE 1 1 3 1 3 0.27303 CSE 1 1 1 1 1 0.2
MECH 0.33 0.33 1 0.33 1 0.09046 MECH 1 1 1 1 1 0.2
max =4.988, C.I =-0.003,C.R=-0.0027 max =5.000, C.I =0.0,C.R=0.0
Teacher relationship Event management
priority
priority
vector
EEE 1 1 3 1 3 0.27303 EEE 1 1 3 1 1 0.2284
ECE 1 1 3 1 3 0.27303 ECE 1 1 3 1 3 0.2846
ICE 0.33 0.33 1 0.33 1 0.09046 ICE 0.33 0.33 1 0.33 0.33 0.0755
CSE 1 1 3 1 3 0.27303 CSE 1 1 3 1 1 0.2284
MECH 0.33 0.33 1 0.33 1 0.09046 MECH 1 0.33 3 1 1 0.183
max =4.988, C.I =-0.003,C.R=-0.0027 max =5.1416, C.I =0.0354,C.R=0.0319
Table: 4 weight vector of criteria
Coordinating
priority
vector
EEE 1 3 3 1 1 0.29227
ECE 0.33 1 3 1 1 0.18796
ICE 0.33 0.33 1 0.33 1 0.09684
CSE 1 1 3 1 1 0.23461
MECH 1 1 1 1 1 0.18833
max =5.2931, C.I =0.0733,C.R=0.066
C1 C2 C3 C4 C5 C6 C7
priority
vector
C1 1 5 3 5 5 1 3 0.316851
C2 0.2 1 0.33 0.33 0.33 0.2 1 0.045595
C3 0.33 3 1 3 3 1 1 0.158735
C4 0.2 3 0.33 1 1 0.33 0.33 0.067133
C5 0.2 3 0.33 1 1 0.33 1 0.078653
C6 1 5 1 3 3 1 3 0.234051
C7 0.33 1 1 3 1 0.33 1 0.098985
max = 7.4524, C.I = 0.0754,C.R=0.0559

Table: 5 Local and global priorities
C1
(0.
3)
C2
(0.0
4)
C3
(0.
2)
C4
(0.
2)
C5
(0.2
)
C6
(0.
2)
C7
(0.5
)
C8
(0.
5)
C9
(0.5
)
C10
(0.5
)
C11
(0.3
3)
C12
(0.3
3)
C13
(0.
33)
C14
(0.5)
C15
(0.5)
A1 0.2
8
0.1 0.23 0.2
8
0.19 0.1
8
0.14 0.1
8
0.27
2
0.18
1
0.27
3
0.2 0.27 0.23 0.29 0.22
5
A2 0.44
2
0.51 0.45 0.2
8
0.46 0.3
8
0.09 0.0
8
0.112 0.18
1
0.27
3
0.2 0.27 0.28 0.19 0.23
6
A3 0.0
63
0.01 0.0
5
0.0
9
0.06 0.0
9
0.11 0.2
3
0.07
2
0.07
5
0.09 0.2 0.0
9
0.08 0.1 0.10
2
A4 0.15 0.02 0.19 0.22 0.19 0.2
2
0.26 0.2
3
0.27
2
0.28
2
0.27
3
0.2 0.27 0.23 0.23 0.23
4
A5 0.0
6
0.01 0.0 0.11 0.09 0.1 0.4 0.2 0.27 0.28 0.09 0.2 0.0
9
0.18 0.1 0.19
7
Figure 4: weights of alternatives w.r.t Paper Presentation Figure 5: Weights of alternatives w.r.t Group Discussion
Figure2: weights of alternatives w.r.t Academics Figure 3: Weights of alternatives w.r.t Attendance

Figure 10: weights of alternatives w.r.t Singing Figure 11: Weights of alternatives w.r.t Choreography
Figure 8: weights of alternatives w.r.t Indoor Games Figure 9: Weights of alternatives w.r.t Outdoor Games
Figure 6: weights of alternatives w.r.t Debate Figure 7: Weights of alternatives w.r.t Quiz

Figure 16: weights of alternatives w.r.t Coordinating
Figure 14: weights of alternatives w.r.t Teacher relationship Figure 15: Weights of alternatives w.r.t Event management
Figure12: weights of alternatives w.r.t Honesty Figure 13: Weights of alternatives w.r.t Peer relationship

Table:6 Weights and Ranks of alternatives
S.No Alternatives Weights of
alternatives
Ranks of
alternatives
1. EEE 0.225 3
2. ECE 0.236 1
3. ICE 0.102 5
4. CSE 0.234 2
5. MECH 0.197 4
Figure: 17 Ranks of alternatives (five engineering branches)
References:
[1] Patric T.Harker, The art of Science and Decision-making: The Analytic
Hierarchy Process , Springer Verlag, 1989.
[2] R.Ramanathan and L.S.Ganesh, Group preference aggregation
methods employed in AHP: An evaluation and an intrinsic process for
deriving members weightages , European Journal of Operational Research
79(1994)249-265.
0.225 0.236
0.102
0.234
0.197
0
0.05
0.1
0.15
0.2
0.25
Ranks of branches for All round Excellence
Award
ALTERNATIVES

[3] T.Ramsha Prabhu and K.Vizaya Kumar, Fuzzy Hierarchical Decision
Making (FHDM): A Methodology for Technology choice (1996),
International Journal of Computer Applications in Technology, Vol.9 No.5
&6.
[4] Thomas.L.Saaty Exploring interface between Hierarchies, multiple
objectives and Fuzzy sets , Fuzzy sets and systems 1(1978), 57-68.
[5] Thomas, L.Saaty (1980), The Analytical Hierarchy Process , Tata
McGraw Hill, New York.
Proceedings of Conferences:
[6] Analytical Hierarchy Process for selection of a student for All Round
Excellence Award , National conference on Mathematics and computational
sciences, Adikavi Nannaya University, Andhra Pradesh, India(2012).
[7] Selection of a student for All Round Excellence Award A mathematical
model of a Multi-criteria Decision Making approach , 4th International
conference, Bengal Institute of Technology and Management, West Bengal
(2012).
Received: November, 2012

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