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

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

should buy the mobile D. Advantages of

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.

Disadvantages:·

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.