In this case,

? max =

(2.0154+0.2648+0.7306) = 3.0108

Now consistency index

can be found out by above mentioned formula as

CI

= (? max-n)/ (n-1) = (3.0108-3)/(3-1) = 0.0054

RI = 1.98*(N-2)/N =

1.98*1/3 = 0.66

CR = CI/RI = 0.0082

As

we have discussed we accept consistency ratio value

up to 0.1. Here, CR value is 0.0082, which is under acceptable range. Therefore,

we can assume that our judgments matrix is reasonably consistent so we may

continue the process of decision-making using AHP.

Step-7 Local Priorities for Alternatives

We

have to find out local priorities of alternatives following the same procedure

as we did for local priorities of criteria’s.

We

have three criteria’s and two alternatives; hence, three matrixes will be

formed for both alternative to find out weightage of alternative A and B for

coal, water and market.

For

this purpose, we do a pair wise comparison, of all the alternatives, with

respect to each criterion.

In

our example,

1. Compare

Project A and B with respect to availability of coal criterion.

2. Compare

Project A and B with respect to availability of water criterion.

3. Compare

Project A and B with respect to availability of market criterion.

We

use the same Saaty’s pair wise comparison scale to give weightage to

alternatives.

Let’s

assume that we obtained following three responses-

Comparison

with respect to coal : X Coal

Alternatives

Project A

Project B

Project A

1

8

Project B

0.125

1

Comparison

with respect to water : X water

Alternatives

Project A

Project B

Project A

1

7

Project B

0.1429

1

Comparison

with respect to water : X Market

Alternatives

Project A

Project B

Project A

1

0.25

Project B

4.00

1

Step-8 Local weightage of Alternatives

To

find out the weightage of alternatives, same procedures can be followed and

repeated as done for finding out local weightage for criteria’s. Normalized

matrix can be written as –

X’

coal –

Alternatives

Project A

Project B

Project A

0.889

0.889

Project B

0.111

0.111

W Coal A – 0.889

W Coal B- 0.111

X’

Water-

Alternatives

Project A

Project B

Project A

0.875

0.875

Project B

0.125

0.125

W water A – 0.875

W water B- 0.125

X’

market-

Alternatives

Project A

Project B

Project A

0.200

0.200

Project B

0.800

0.800

W market A – 0.2

W market B- 0.8

As

we can see, the columns of all three consistency matrix is identical.

Therefore, we can say that all three matrix are consistent. There is no further

need of finding consistency ratios for validation.

Step-9 Global weights

In this step we find out the overall priorities of each

alternative.

0.3421

0.0882

0.6687

0.8

0.2

0.875

0.125

0.889

0.111

Global

weights can be found out my multiplying local weights for each alternatives.

Global

weights for project A = (0.6687*.889+ 0.0882*0.875+ 0.3421*0.2) = 0.72

Global weights for

project B = (0.6687*0.111+ 0.0882*0.125+ 0.3421*0.8) = 0.28

Step-9 Analysis of result and decision making

For

NTPC project A (0.72) is more feasible compared to project B (0.28) given the

preference of each criteria’s availability of coal, water and market.

EIGEN

VECTOR METHOD

There

is an alternate method to solve analytic hierarchy process problems called

Eigen vector method.

This

method is similar to the additive normalization method in the sense that all

initial steps are same in both the method. It differs only in the process of

calculation of eigen values or weightage of criteria’s and alternatives.

Followings

are the steps to solve analytic hierarchy process problem through eigen vector

method-

Step1:

Find out the comparison matrix

Step2:

Prepare N*N identity matrix, where N is the size of comparison matrix.

Step3:

Find out determinant of matrix (A-?I), where ? is eigen value. Find out the maximum value of eigen value ?

for which determinant of matrix (A-?I) is zero, using goal seek (what –if

analysis) feature of Excel.

Step-4: Use

value of ? max to check the consistency of matrix as we did in additive

normalization method.

To

elaborate and understand concepts behind each step, we’ll again consider one

example. Since we have taken one example of individual decision making. Lets

take example of group decision.

Group Decision

Here

unlike last example, priorities of criteria’s and that of alternatives are

finding out by putting the questions and options in front of group of

respondents.

Lets

take one example where goal is to take the decision to buy a steam generator

(boiler) for 500 MW NTPC plant. We have again two alternatives supplier for it

–

1. M/s

BHEL

2. M/s

Doosan

To

choose one alternative NTPC is considering various criteria’s such as

a) Safety

of boiler

b) Cost

of boiler

c) Efficiency

of boiler

To

take the decision of such type a better approach is to put the questionnaire in

front of subject experts and take the decisions.

Comparison

matrix can be prepared once we have answers of all questionnaires.

For

this project following questionnaires has been prepared-

Questionnaires-

A power company has

to purchase steam generator for its plant. Company is considering various

criterions to take decisions of choosing best supplier (Between M/s BHEL and

M/s Doosan). Comparison is being done for exactly same type and capacity of

boiler. Please fill the questionnaires as per your priorities of one over other

from scale 1 to 9.

1.

Preference of safety of boiler over cost

of boiler.

2.

Preference of safety of boiler over efficiency

of boiler.

3.

Preference of cost of boiler over

efficiency of boiler.

4.

If only efficiency is considered, How

much M/S BHEL is preferred over M/s Doosan ?

5.

If only safety is considered, How much

M/S BHEL is preferred over M/s Doosan ?

6.

If only cost is considered, How much M/S

BHEL is preferred over M/s Doosan ?

Question 1,2,3: What

is relative importance of one criterion over other for selection of steam

generator?

Please compare the LHS and RHS criteria and circle your

answer using the scale below:

1-Equal

3- Moderate 5-Strong 7-Very Strong 9- Extreme

Safety of Boiler

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

Cost of Boiler

Safety of Boiler

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

Efficiency of Boiler

Cost of Boiler

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

Efficiency of Boiler

Question 4: If only

efficiency of boiler is considered, which supplier is preferable?

1-Equal

3- Moderate 5-Strong 7-Very Strong 9- Extreme

BHEL

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

DOOSAN

Question 5: If only

safety of boiler is considered, which supplier is preferable?

1-Equal

3- Moderate 5-Strong 7-Very Strong 9- Extreme

BHEL

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

DOOSAN

Question 6: If only

cost of boiler is considered, which supplier is preferable?

1-Equal

3- Moderate 5-Strong 7-Very Strong 9- Extreme

BHEL

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

DOOSAN

Explanation:

1-

If you

choose 1, when comparing BHEL with DOOSAN that means you give equal importance

to both.

2-

If you choose 9 towards right side (Towards

Doosan) that means you prefer Doosan extremely higher than BHEL on given

questions.

3-

If you choose 9 towards left side (Towards BHEL)

that means you prefer BHEL extremely higher than DOOSAN on given questions.

Following is the outcome of the survey received from 10

industry expert in matrix form.

Q1

Q2

Q3

Q4

Q5

Q6

RES1

9

7

0.5

1

1

5

RES2

8

8

0.2

0.33

3

4

RES3

9

5

0.5

0.25

2

2

RES4

4

6

1

0.13

0.25

6

RES5

8

5

2

2

0.2

3

RES6

9

7

0.5

8

0.5

0.25

RES7

6

8

0.25

4

0.33

0.5

RES8

8

7

3

3

0.5

1

RES9

7

9

4

2

4

1

RES10

9

9

1

0.11

8

0.5

To analyze further through Eigen value method, we’ll have to

take geometric means of responses.

Q1

Q2

Q3

Q4

Q5

Q6

7.502

6.960

0.827

0.922

0.978

1.463

Using these results now we can form Comparison matrix for 3

criteria’s as following-

Step1: Comparison Matrix:

Safety of Boiler

Cost of Boiler

Efficiency of Boiler

Safety of Boiler

1

7.5

6.96

Cost of Boiler

0.133

1

0.83

Efficiency of Boiler

0.144

1.205

1

Step2: Prepare Identity matrix

Identity matrix I (3*3) is given as –

1

0

0

0

1

0

0

0

1

Step3: (A-?I) Matrix

(A-?I) Matrix is

prepared for a variable unknown value of ?.

The value of ? is calculated

such that determinant of the matrix becomes zero. We use goal use tool in excel

to calculate value of ?.

(A-?I) Matrix

-2.00

7.50

6.96

? max

3.001385

0.13

-2.00

0.83

Deter

0.00

0.14

1.20

-2.001385353

Step4: Consistency check

Similar to additive normalization method, we can check for

the consistency.

Here Consistency Index (CI)=

0.000693

Random Consistency Index (RI) = 0.66

Consistency ratio = CI/RI = 0.00105

Since consistency ratio is much lower than acceptable range

of 0.1, we can use it for further analysis.

Step4: Weights using linear programming

Safety of Boiler

Cost of Boiler

Efficiency of Boiler

W1

W2

W3

CONSTRAINTS

0.7827

0.1006

0.1167

LHS

SIGN

RHS

(A-? maxI) Matrix

-2.00

7.50

6.96

0.00

=

0

0.13

-2.00

0.83

0.00

=

0

0.14

1.20

-2.00

0.00

=

0

Objective Function

1

Weights are found out using linear programming.

Objective function is taken-

w1+w2+w3 =1

Constraints are – (A-? maxI)*W

=0

Weights obtained clearly show that the experts give

maximum priority to safety than efficiency of boiler than cost of boiler.

Note- Similar procedure can be done to find out

local weights of alternatives. Global weights can be easily found out by

multiplying local weights following the same procedure as already explained in

additive normalization method.

Evaluation

of Error in AHP

If Wi, where i= 1,2,3,….n are actual

weights of priorities then element of comparison matrix (Aij) can be written as

Aij = (Wi / Wj )

If error ? is present, then (Wi / Wj ) = A ij* ?ij

?ij

= (Wi / Wj )* A ji

Steps to find out error in the AHP.

Step-1: Prepare

the comparison matrix from respondent’s priorities. For example- following table

is being prepared from the people response for the priorities of four

criteria’s for purchase of new car.

Brand

Cost

Efficiency

service

Brand

1

0.2

0.167

0.125

Cost

5

1

3

0.5

Efficiency

6

0.333

1

2

service

8

2

0.5

1

Step-2: Find out

transpose matrix Aji

Aji =

Brand

Cost

Efficiency

service

Brand

1

5

6

8

Cost

0.2

1

0.333

2

Efficiency

0.167

3

1

0.5

service

0.125

0.5

2

1

Weights can be found out by using eigen value method.

Calculated weights are as follows-

W1

W2

W3

W4

0.044

0.328

0.290

0.337