## INTRODUCTION: models. Higher model with more than 2

INTRODUCTION:            Analytical hierarchy process (AHP) was introduced by
Thomas saaty in the year 1970. This method is vastly used in many industries
and various government organizations in several countries. This is a
mathematical based model used to analyze various complex problems. It can be
used even in psychological decision making. If there are only two decision
alternatives, the decision is quite simple. Choose one between the two.             But if there are more
than 2 alternatives and several criteria for selecting them, then the problem
starts becoming complicated. For instance, if we have to buy one a mobile phone
(from A, B and C) and we has four criterion: colour, memory space, cost and
service availability then it becomes quite tedious without some scientific
tools. This problem is even more blatant in big organizations where there are
hundreds of options and hundreds of criterion. Such complex decisions can be
analysed and processed by AHP. The alternatives need not be tangible. They can
be intangible also.            AHP works by giving
proper weights and by setting priorities to each of the alternative and then
the decision is left to the decision makers. These weights represent the
priorities of the alternatives. Thus the more the weight, the more the
priority. The decision can be based on the responses of several people also.              Throughout the
discussion, the mobile phone example stated above will be used for discussion.
In this project I have considered a two level hierarchy model instead of
starting with a simple one level hierarchy model and then proceeding to the two
level models. Higher model with more than 2 levels are more practical but this
paper is limited to 2 levels and with only one example

A

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B

C

D

Criteria  Alternatives                                                                                                    Procedure for assigning relative ranks:The AHP works by comparing two criteria or two
alternatives at a time. This process is done in a all combinations such that
all the criteria or alternatives available are compared with each other. The
respondent is asked to rate the criteria and the given alternatives as
suggested in the questioner. The procedure and the process of filling the
questioner are as given below.

q-1

Colour

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

memory

q-2

Colour

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

price

q-3

Colour

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

service

q-4

memory

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

price

q-5

memory

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

service

q-6

price

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

service

Explanation:Any AHP questioner can be filled up in the ranking
from 1 through 9. This scale is developed by researchers though their vast
research. 1—both priorities are equal 3—moderately important. 5—strong 7—very strong9—extreme In the above questioner, the respondent chose 9 on the
right side (in question 1). This means his priority of memory over colour is 9
times. Similarly his priority of memory over service (q-5) is 7. The respondent is also asked the rate the various
alternatives with respective to each criteria available in a similar manner. The
ranking in between those explained above indicate that the importance is in
between them. Such as 2 indicates that the importance is between equal and
moderate. For example the below questioner is asked based on the
colour. That is if all other criteria are ignored and only colour is considered
for the selection of alternative then the rankings given by the respondent to
the mobiles are:

q-1

Mobile A

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

Mobile B

q-2

Mobile A

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

Mobile C

q-3

Mobile A

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

Mobile D

q-4

Mobile B

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

Mobile C

q-5

Mobile B

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

Mobile D

q-6

Mobile C

9

8

7

6

5

4

3

2

1

2

3

4

5

6

7

8

9

Mobile D

These responses are to be filled
in any software (MS excel in our case) in form of a matrix as described below.

Mobile A

Mobile B

Mobile C

Mobile D

Mobile A

1

Q1

Q2

Q3

Mobile B

1/Q1

1

Q4

Q5

Mobile C

1/Q2

1/Q4

1

Q6

Mobile D

1/Q3

1/Q5

1/Q6

1

1.     The rating of mobile A with mobile B as entered by the respondent in
questions-1, 2,3 …6 are to be entered as shown in the table above.( upper half
of the principle diagonal).2.     For example in the question-1, (row1, col-2) 1/5 has to be entered as
the priority of Mobile B is greater than the mobile A by 5 times. 7 has to be
entered in the question-3 as the priority of mobile A is 7 times than mobile D.
3.     The inverse of the responses of the questions 1 through 6 are to be
entered in the lower half of the principle diagonal. This is because, if the
priority of alternative or criteria 1 over 2 is given as in Q-1, the priority
of criteria-2 over 1 is given by its inverse. ( logical)4.     The elements of principle diagonal are always 1. This is because; we are
comparing each option with itself. So the weights would invariably 1. 5.     The matrix so formed is called original matrix.6.     The original matrix for the criteria and alternatives are given below:

ORIGINAL MATRIX FOR CRITERIA

COLOUR

MEMORY

PRICE

SERVICE

COLOUR

1.00

0.11

0.17

8.00

MEMORY

9.00

1.00

0.14

7.00

PRICE

6.00

7.00

1.00

8.00

SERVICE

0.13

0.14

0.13

1.00

ORIGINAL MATRIX FOR ALTERNATIVES (wrt colour)

MOBILE
A

MOBILE
B

MOBILE
C

MOBILE
D

MOBILE
A

1.00

5.00

6.00

0.14

MOBILE
B

0.20

1.00

4.00

0.13

MOBILE
C

0.17

0.25

1.00

0.14

MOBILE
D

7.00

8.00

7.00

1.00

Some things to be noted:
1.
The above tables
represent the exact priorities as set by the respondent. This does not in any
way depend on the discretion of the researcher.
2.
The comparison of
the alternatives is taking place pair wise.
3.
The number of
questions to be framed by the researcher is (n2-n)/2, where n is
the number of alternatives.
4.
Any change, if
needed is to be done in the upper half of the principle diagonal.
5.
This is a simple
example with only one set of criterion and one set of alternatives. In many
practical real life conditions, there will be a number of levels of hierarchy
and so the number of matrices.
6.
In case the decision
is a group decision and a number of people participate in the rankings of the
criterion and the alternatives, then the geometric mean of all the responses
corresponding to each of the question is to be taken and that is to be used
as the response of the group as a whole.

The process of AHP to
assign weights can be done by two main methods. 1. Additive identity method
and 2. Eigen Matrix method.
We will discuss the Eigen
matrix method as it is more prominent and more precise.

Theory for finding the
weights :

Procedure for finding
weights:
Let I be the identity
matrix and A be the original matrix as discussed earlier.
I=

indentity
matrix

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

Let lamda (?) be a constant and delta, the
determinanat of the matrix such that
Det= determinant of (A-
?*I).
Since I is a 4×4 matrix
and ? is a constant, (A- ?*I) is also a 4×4 matrix. A is also a 4×4 matrix.
Therefore (A- ?*I) is also a 4×4 matrix. We have to choose ? such that
determinant is equal to zero. Procedure to be followed for finding the
weights using excel is explained below.

lamda

5.1971888

det

9.607E-07

A-LAMDA*I

W1

W2

W3

W4

LHS

SIGN

RHS

WEIGHTS

0.09

0.26

0.62

0.03

-4.20

0.11

0.17

8.00

0.00

=

0

9.00

-4.20

0.14

7.00

0.00

=

0

6.00

7.00

-4.20

8.00

0.00

=

0

0.13

0.14

0.13

-4.20

0.00

=

0

Z

1

=

1

1.
Assign some
arbitrary values to the ? (more than 6 or7 because while solving the Excel
comes down from that value to the right value of ?. If we choose a value very
less, then we may not find the right value. The arbitrary value should always
be greater than the number of constraints, 4 in this case).
2.
Now find the
determinant using the formula MDETERM and selecting the whole matrix. This
will produce some arbitrary value in the determinant cell.
3.
Use the goal seek
from the ‘what-if analysis’ of Excel and find the values of ? (make the
determinant equal to zero by changing the values of ?). This will give us the
value of ? , when the determinant is zero.
4.
Now we need to find the
weights of each criterion (w1, w2, w3, w4). Let the matrix W (w1, w2, w3, w4)
be a column matrix (4×1).
5.
Matrix
multiplication of (A- ?*I) 4×4 and W4x1 gives a 4×1 matrix. The
elements of this matrix must be equal to zero for a matrix to be consistent.
6.
The sum of all
weights should be equal to one.
7.
We can solve for the
weights (matrix W) using the solver of Excel using the above two conditions.
8.
In our example, we
got the weights of criteria as: 0.09, 0.26, 0.62, and 0.03 for color, memory,
price, and service availability.
9.
These weights are
called global weights.

Consistency of the
weights:
Consistency means rational
judgment of the decision maker. Mathematically we can say that the matrix is
consistent if
aij*ajk
= aik for all i,j,k >0
eg.  a12 – ratio of color to memory.
(color/memory)
a23—ratio of memory to
price. ( memory/price)
a13—ratio of color to
price. (Color/price). This should be equal to a12 *a23
logically.
If this is not equal, that
means that the respondent has not given the weights correctly. Or he may not
have a clear cut idea of what exactly he wants. In such cases the respondent
is advised to fill his choices again.
If there are only 2
choices, the weights are always consistent. However in case of higher order
comparisons, it generally not consistent.
In order to decide if this inconsistency is tolerable or not, we need
a quantifiable measure of consistency for the matrix A.
If the matrix A is
consistent, the values of ?, which we obtained from the procedure as described
earlier must be equal to the number of criterion (4 in this case). If the
value is greater than this, then there is some inconsistency.
The consistency index is
given by the formula below. ?max, which we obtained from solving
the matrix above ( 5.19 in this example).
CI= (?max-n)/(n-1) ; where n is the number of criterion ( 4 in this case)
Solving we get the CI as 0.399.
As the ?max moves
closer to n, the consistency improves and the matrix A is more consistent.
Thus the constancy ratio is given by
CR= CI/ RI,
Where RI- random
consistency and is given by 1.98*(n-2)/n. this value is empirically obtained
after several millions of experiments. For n=2, the RI= 0.99
CR= 0.399/ 0.99 =
0.403   or 40.3 %.
The industry standards of
acceptable limit in 10% or 0.1.
Therefore in our example,
since the CR is 40.3% , the Reponses filled by the respondent are quite
inconsistent and are thus not acceptable.

Assessing the exact
response which is inconsistent:
We need to form an error
matrix to determine the exact element where there is inconsistency. Let
·
Wi – the
weight matrix.  4×1 matrix
·
Wj- the
transpose of weight matrix. 1×4 matrix.
If there were no error and
the matrix consistent, then aij = (wi/ wj).
But since error is
present, each element of the matrix can be written as
aij = (wi/
wj) * ?ij.  Where
?ij is the error associated with each of the element of the
original matrix.
Therefore   ?ij =  (wi/ wj) *(1/ aij
)
Implies        ?ij =  (wi/ wj)* aji   ( taking transpose of the matrix A).
Where i, j =1, 2…n.

TRANSPOSE
OF ORIGINAL MATRIX

1.00

9.00

6.00

0.13

0.11

1.00

7.00

0.14

0.17

0.14

1.00

0.13

8

7

8

1

ERROR
MATRIX

TRANSPOSED
WEIGHTS

0.09

0.26

0.62

0.03

Original
weights

0.09

1.00

3.05

0.86

0.37

0.26

0.33

1.00

2.95

1.24

0.62

1.16

0.34

1.00

2.58

0.03

2.71

0.80

0.39

1.00

From the formula above the error
matrix can be calculated as shown in the error matrix table above. Note that
the principle diagonal elements of this matrix are also all unity since wi=
wj when i=j and the aij = 1 when i= j.  Significance of error
matrix and correction of error: In the error matrix, if the
elements below the principal diagonal are high, that means that it is the
element which is inconsistent. We are considering the elements below the
principle diagonal only because the error matrix is formed from the transpose
of the original matrix and in order to calculate the error in the responses are
indicated from the lower elements of the error matrix. In our example the
elements 3×1 and 4×1 of the error matrix are quite high compared to the other elements.
They correspond to the questions 2 and 3 of the questioner. It is the relative
rankings of colour to price and colour to service.   Therefore we may ask the
respondent to consider these particular responses once again and give new
rankings.  We can also guess what the
correct rankings could be. As discussed earlier the element aij
= (wi/ wj) if there is no error. Thus if we assume that
there is no error the response to that particular question has to be wi/
wj.  Therefore in our example, the response
to 1.     Question 2 has to be 0.09/0.62 = 0.142.     Question 3 has to be 0.09/ 0.03 = 3 By changing these obtained values
in the questions 2 and 3 and calculating the CR as explained above, we get  ?=     4.72CR = 24.3. As we can observe there is a
considerable change in the ? and the CR values. Similarly we can reduce the CR
to less than the recommended 10% by suggesting the decision maker to reconsider
his rankings.  The final weights of the criteria
are:Colour         :0.09Memory       :0.26Price            :0.62Service        :0.03   Finding the weights
of alternatives: In the exact similar way as
discussed above, the weights and error table of the alternatives (wrt colour)
are calculated in the Excel. The results are: 1.     CI = 0.202.      RI= 0.993.     CR=0.20 Weights of alternatives ( wrt
colour):1.     Mobile A – 0.212.     Mobile B – 0.083.     Mobile C – 0.044.     Mobile D – 0.67 The major inconstancies were found in:1.     Q4 – mobile B vs mobile C2.     Q1– mobile A vs mobile B. By re- ranking these two responses, we can get a CR value of below 10%.
Similarly the priorities wrt memory, price and service can be put in a matrix
form as below (questioner not shown in the report as the procedure is exactly
similar and so only the final matrix is formed):            The weights of the mobiles with respect to memory, price and service are
directly shown in the table format, even though they were calculated in the
similar way. Generally this matrix form of representation of the rankings given
by the respondent is more common.          To find global rankings: Now that we got all the weighted rankings of each of the criteria and
each of the alternatives w.r.t the each individual criterion, we can get the
global rankings (final weighted rankings considering all the criterion and
alternatives) as per the procedure below. :Global Weight of alternate =Where
i= 1 or 2 or 3 or 4 (mobile A, B, C or D)           j= 1 to 4 (colour, memory, price and
service respectively)eg:
for mobile A : (0.09*0.21)+(0.26*0.19)+(0.62*0.21)+(0.03*0.25)= 0.21similarly
global values for all the alternates are shown in the darker blocks.

Service (0.03)

Price (0.62)

Memory(0.26)

Colour(0.09)

Criteria

A (0.21)

A (0.19)

A (0.21)

A (0.25)

Alternatives

B(0.08)

B(0.08)

B(0.09)

B(0.08)

C(0.04)

C(0.04)

C(0.04)

C(0.04)

D(0.67)

D(0.68)

D(0.66)

D(0.63)

Clearly the priority
is the mobile D followed by mobile A. mobile B and mobile C is negligible in
priority.  Therefore the decision maker
AHP:·
Hierarchical
structuring of decision problem.·
Consensus
of a group is possible.·
Easier
representation and ease of decision taking.·
More
scientific method·
Ease
of locating the errors. ·
Many
easy to use software available in the market, especially Excel.·
Management
of any organisation generally agrees to the decision based on AHP.·
Data
is given by the decision makers themselves.
Sometime
if the CI is very high, it can only be asked to reconsider.

·
Generally
the ranking of the options available is quite cumbersome in case of many levels
of hierarchy.

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