Analytic Hierarchy

Process

Zheng-Wen Shen

2006/04/11

Outline

1. Introduction of AHP

2. How the AHP works

3. Example

1. Introduction of AHP

Salary is

important

..

Location

is

important..

Long term

prospect is

important..

Interest is

important..

Is job

1 best ?

Is Job

2 best ?

Is Job

3 best ?

Is Job

4 best ?

Crystal is looking for job…

AHP Features

AHP is a powerful tool that may be used to

make decisions when

multiple and conflicting objectives/criteria are

present,

and both qualitative and quantitative aspects

of a decision need to be considered.

AHP reduces complex decisions to a

series of pairwise comparisons.

2. How the AHP works

1. Computing the vector of objective

weights

2. Computing the matrix of scenario scores

3. Ranking the scenarios

4. Checking the consistency

consider m evaluation criteria and n scenarios.

AHP Steps

1. Computing the vector of objective

weights

2. Computing the matrix of scenario scores

3. Ranking the scenarios

4. Checking the consistency

Step 1: Computing the vector of

objective weights

Pairwise comparison matrix A [m × m].

Each entry ajk of A represents the

importance of criterion j relative to criterion

k:

If ajk > 1, j is more important than k

if ajk < 1, j is less important than k

if ajk = 1, same importance

ajk and akj must satisfy ajkakj = 1.

Step 1: Computing the vector of

objective weights

The relative importance between two criteria is

measured according to a numerical scale from 1

to 9.

A Anorm (Normalized)

Step 1: Computing the vector of

objective weights Preferences on Objectives

Weights on Objectives

AHP Steps

1. Computing the vector of objective

weights

2. Computing the matrix of scenario scores

3. Ranking the scenarios

4. Checking the consistency

Step 2: Computing the matrix of

scenario scores The matrix of scenario scores S [n × m]

Each entry sij of S represents the score of the scenario i with respect to the criterion j

The score matrix S is obtained by the columns sj calculated as follows: A pairwise comparison matrix Bj is built for each

criterion j.

Each entry bjih represents the evaluation of the

scenario i compared to the scenario h with respect to the criterion j according to the DM’s evaluations.

From each matrix Bj a score vectors sj is obtained (as in Step 1).

Step 2: Computing the matrix of

scenario scores

Location scores Relative Location scores

Relative scores for each objective

AHP Steps

1. Computing the vector of objective

weights

2. Computing the matrix of scenario scores

3. Ranking the scenarios

4. Checking the consistency

Step 3: Ranking the scenarios

Once the weight vector w and the score matrix S

have been computed, the AHP obtains a vector

v of global scores by multiplying S and w

v = S · w.

The i-th entry vi of v represents the global score

assigned by the AHP to the scenario i

The scenario ranking is accomplished by

ordering the global scores in decreasing order.

Step 3: Ranking the scenarios

Relative scores for each objective

Weights on Objectives

A

B

C: .335 D: .238

AHP Steps

1. Computing the vector of objective

weights

2. Computing the matrix of scenario scores

3. Ranking the scenarios

4. Checking the consistency

Step 4: Checking the consistency

When many pairwise comparisons are

performed, inconsistencies may arise.

criterion 1 is slightly more important than

criterion 2

criterion 2 is slightly more important than

criterion 3

inconsistency arises if criterion 3 is more

important than criterion 1

Step 4: Checking the consistency

The Consistency Index (CI) is obtained:

x is the ratio of the j-th element of the vector A · w to the corresponding element of the vector w

CI is the average of the x

A perfectly consistent DM should always obtain CI = 0

but inconsistencies smaller than a given threshold are tolerated.

3. Example (1/7)

Small example, m = 3 criteria and n = 3

scenarios.

Criterion 1

0 S3 S2 S1

Criterion 2

0 S3 S2 S1

Criterion 3

0 S3 S2 S1

Example (2/7)

pairwise comparison matrix A for the 3

criteria

Weight Vector

Example (3/7)

pairwise scenario comparison matrices for

the first criterion:

Score Vector

Example (4/7)

pairwise scenario comparison matrices for

the first criterion:

Score Vector

Example (5/7)

pairwise scenario comparison matrices for

the first criterion:

Score Vector

Example (6/7)

Score Matrix S is :

Global Score Vector

Example (7/7)

The rank is:

Scenario 1: 0.523

Scenario 2: 0.385

Scenario 3: 0.092