Dr. Y. İlker TOPCU

www.ilkertopcu.net

www.isl.itu.edu.tr/ya/

I would like to acknowledge Prof. T.L. Saaty's “ANP_Slideshow_July_2001.ppt” slides submitted by www.creativedecisions.net which greatly influence these notes...

The Analytic Network Process

The Analytic Network Process (ANP)for Decision Making and Forecasting

with Dependence and Feedback

• The alternatives depend on the criteria as in a hierarchy but may also depend on each other.

• The criteria themselves can depend on the alternatives and on each other as well.

• Feedback improves the priorities derived from judgments and makes prediction much more accurate.

Feedback Network with components having Inner and Outer Dependence among Their Elements

Loop in a component indicates inner dependence of the elements in that componentwith respect to a common property.

Arc from componentC4 to C2 indicates theouter dependence of the elements in C2 on theelements in C4 with respectto a common property.

C4

C1

C2

C3

Feedback

The Management of a Water Reservoir

Flood Recreation Hydro-Control Electric

Power

Low Intermediate HighLevel Level Level

A Feedback System with Two Components

Inner and Outer Dependence

The elements in a component may influence other elements in the same component (inner dependence) and those in other components (outer dependence) with respect to each of several properties. We want to determine the overall influence of all the elements.

In that case we must organize the properties or criteria and prioritize them in the framework of a network, perform comparisons and synthesize to obtain the priorities of these properties. We then derive the influence of elements in the feedback system with respect to each of these properties. Finally, we weight the resulting influences by the importance of the properties and add to obtain the overall influence of each element.

Weighting The Components

In the ANP one often needs to prioritize the influence of the components themselves on each other component to which the elements belong. This influence is assessed through paired comparisons with respect to a control criterion.

The priority of each component is used to weight the priorities of all the elements in that component. The reason for doing this is to enable us to perform feedback multiplication of priorities by other priorities in a cycle, an infinite number of times. The process would not converge unless the resulting matrix of priorities is column stochastic (each of its columns adds to one).

Why ANP?

• The power of the Analytic Network Process (ANP) lies in its use of ratio scales to capture all kinds of interactions and make accurate predictions, and, even further, to make better decisions.

• So far, it has proven itself to be a success when expert knowledge is used with it to predict sports outcomes, economic turns, business, social and political decision outcomes.

The Pairwise Comparison Questions

Two kinds of questions encountered in the ANP:

“of the sub-elements, which one influences the parent element more and how much more?”

or

“of the dependent factors, which one influnces the common factor more and how much more?”

• Each element is represented at one row and one respective column

• The computed eigenvector of the sub-elements with respect to their parent element is placed to the column representing the parent element and the rows representing the sub-elements.

The Supermatrix

Supermatrix of a Hierarchy

0 0 0 0 0 0

W21 0 0 0 0 0W =

Wn-1, n-2 0 00 0 0 0 Wn, n-1 I

0 W32 0 0 0 0

0 0 0

Supermatrix of a Network

c1 c2 cN

e11e12 e1n1e21e22 e2n2

eN1eN2 eNnN

W11 W12 W1N

W21 W22 W2N

WN1 WN2 WNN

W =

c1

c2

cN

e11

e12

e1n1e21

e22

e2n2eN1

eN2

eNuN

where

Wi1 Wi1 Wi1

Wij =

(j1) (j2) (jnj)

(j1) (j2) (jnj)Wi2 Wi2 Wi2

Wini Wini

Wini

(j1) (j2) (jnj)

• If the column sum of any column in the composed supermatrix is greater than 1 (there are more than one eigenvector), that column will be normalized.

• Such a supermatrix is called as weighted supermatrix.

The Weighted Supermatrix

• The weighted supermatrix is then raised to a significantly large power in order to have the converged or stable values.

• The values of this limit matrix are the desired priorities of the elements of the decision network with respect to the goal.

The Limiting Supermatrix

Goal Learning Friends School life Vocational trainingCollege preparation Music classes A B CGoal 0 0 0 0 0 0 0 0 0 0

Learning 0 0 0 0 0 0 0 0 0 0Friends 0 0 0 0 0 0 0 0 0 0

School life 0 0 0 0 0 0 0 0 0 0Vocational training 0 0 0 0 0 0 0 0 0 0

College preparation 0 0 0 0 0 0 0 0 0 0Music classes 0 0 0 0 0 0 0 0 0 0Alternative A 0.3676 0.16 0.33 0.45 0.77 0.25 0.69 1 0 0Alternative B 0.3781 0.59 0.33 0.09 0.06 0.5 0.09 0 1 0Alternative C 0.2543 0.25 0.34 0.46 0.17 0.25 0.22 0 0 1

Goal Learning Friends School life Vocational trainingCollege preparation Music classes A B CGoal 0 0 0 0 0 0 0 0 0 0

Learning 0.32 0 0 0 0 0 0 0 0 0Friends 0.14 0 0 0 0 0 0 0 0 0

School life 0.03 0 0 0 0 0 0 0 0 0Vocational training 0.13 0 0 0 0 0 0 0 0 0

College preparation 0.24 0 0 0 0 0 0 0 0 0Music classes 0.14 0 0 0 0 0 0 0 0 0Alternative A 0 0.16 0.33 0.45 0.77 0.25 0.69 1 0 0Alternative B 0 0.59 0.33 0.09 0.06 0.5 0.09 0 1 0Alternative C 0 0.25 0.34 0.46 0.17 0.25 0.22 0 0 1

The School Hierarchy as Supermatrix

Limiting Supermatrix & Hierarchic Composition

The Management of a Water Reservoir

Here we are faced with the decision to choose one of the possibilities of maintaining the water level in a dam at:

Low (L), Medium (M) or High (H) depending on the relative importance of Flood Control (F), Recreation (R) and the generation of Hydroelectric Power (E) respectively for the three levels.

The first set of three matrices gives the prioritization of the alternatives with respect to the criteria and the second set, those of the criteria in terms of the alternatives.

A Feedback System with Two Components

Flood Recreation Hydro-Control Electric

Power

Low Intermediate HighLevel Level Level

1) Which level is best for flood control?

3) Which level is best for power generation?

2) Which level is best for recreation?

Flood Control

Low Med HighLowMediumHigh

Eigenvector

Consistency Ratio = .107

1 5 7 .722 1/5 1 4 .205 1/7 1/4 1 .073

Low Med HighLowMediumHigh

Eigenvector

Consistency Ratio = .056

1 1/7 1/5 .072 7 1 3 .649 5 1/3 1 .279

Recreation

Low Med HighLowMediumHigh

Eigenvector

Consistency Ratio = .101

1 1/5 1/9 .058 5 1 1/5 .207 9 5 1 .735

Power Generation

1) At Low Level, which attribute is satisfied best?

2) At Intermediate Level, which attribute is satisfied best?

3) At High Level, which attribute is satisfied best?

Low Level DamF R E Eigenvector

Flood Control 1 3 5 .637Recreation 1/3 1 3 .258Hydro-Electric 1/5 1/3 1 .105 Power

Consistency Ratio = .033

Intermediate Level DamF R E Eigenvector

Flood Control 1 1/3 1 .200Recreation 3 1 3 .600Hydro-Electric 1 1/3 1 .200 Power

Consistency Ratio = .000

High Level DamF R E Eigenvector

Flood Control 1 1/5 1/9 .060Recreation 5 1 1/4 .231Hydro-Electric 9 4 1 .709 Power

Consistency Ratio = .061

The six eigenvectors were then introduced as columns of the following stochastic supermatrix.

F R E L M H

0 0 0 .637 .200 .060 0 0 0 .258 .600 .231 0 0 0 .105 .200 .709.722 .072 .058 0 0 0.205 .649 .207 0 0 0.073 .279 .735 0 0 0

One must ensure that all columns sum to unity exactly.

FRELMH

The final priorities for both, the height of the dam and importance for the criteria were obtained from the limiting power of the supermatrix.

F R E L M H

0 0 0 .241 .241 .241 0 0 0 .374 .374 .374 0 0 0 .385 .385 .385.223 .223 .223 0 0 0.372 .372 .372 0 0 0.405 .405 .405 0 0 0

FRELMH

Limiting Supermatrix

• SCI is used to find how close the estimations revealed using ANP model are to actual data (i.e. market shares)

Saaty Compatibility Index (SCI)

SCI Calculation

A1 A1/A1 A1/A2 A1/A3A2 A2/A1 A2/A2 A2/A3A3 A3/A1 A3/A2 A3/A3

B1 B1/B1 B2/B1 B3/B1B2 B1/B2 B2/B2 B3/B2B3 B1/B3 B2/B3 B3/B3

Vector of actual data Pairwise comparison matrix

Vector of estimations Transpose of comparison matrix

A1/A1 * B1/B1 A1/A2 * B2/B1 A1/A3 * B3/B1A2/A1 * B1/B2 A2/A2 * B2/B2 A2/A3 * B3/B2A3/A1 * B1/B3 A3/A2 * B2/B3 A3/A3 * B3/B3

Hadamard multiplication

SCI =(A1/A1 * B1/B1 + A1/A2 * B2/B1 + A1/A3 * B3/B1 + A2/A1 * B1/B2 + A2/A2 * B2/B2 + A2/A3 * B3/B2 + A3/A1 * B1/B3 + A3/A2 * B2/B3 + A3/A3 * B3/B3) / 9