Recap: How the Process Works

(1) Determine the weights. The weights can be absolute or relative. Weights encompass two parts -- the quantitative weight and the current evaluation of its importance towards explaining the objective.

(2) Once the weighting scheme is determined [shown in matrix A], solve the linear equation (or use approximation methods):

A w = max w that is, (A - max I) w = 0If this equation has a nonzero solution for w, then max [which is a scalar] is said

to be an eigenvalue or characteristic value of A [which is an n x n matrix of pairwise comparisons] and w [which is an n x 1 matrix] is said to be an

eigenvector belonging to . I is the identity matrix, which is a diagonal matrix with the main diagonal terms equal to 1 and zero elsewhere.

(3) The solution provides the answer to the most likely outcome, given your judgmental rankings of all the individual criteria.

The Problem Setup: Form Matrix of Ratio Comparisons and

Multiply by w

Approximation & Exact Methods to Derive a Solution

1. Normalize the geometric means of the rows. This result coincides with the eigenvector solution when n 3.

2. Normalize the elements (first) in each column of the judgment matrix and then average over each row. A simple way to obtain an

estimate of max , if the exact value of w [an n x 1 matrix] is available in normalized form, is to add the columns of A and

multiply the resulting vector by the priority vector w.

3. Use linear algebra to solve for the eigenvector and normalize the values for max .

Cell J9 is the geometric average ----

=(A9*B9*C9*D9*E9*F9*G9*H9)^(1/8)

A Short Cut for Determining Weights Comparison of the Geometric Average Approximation and Exact

Linear Algebra Calculation for an 8 x 8 Matrix

Another Method:Solving for the Weights by Successive

Squaring and Checking Differences

Second Iteration

Compare the Results Using a Statistical Package (RATS) to Calculate the

Eigenvector.

• How consistent are the inputs? In the application of AHP it is possible that inconsistent judgments of input desirability may have crept into the process. For example, the original matrix, A, may not have full transitivity -- that is, A(1) may be preferred to A(2) and A(2) to A(3), but A(3) may be preferred to A(1).

• To determine how much inconsistency is in the A matrix, Saaty defined a measure of deviation from consistency, called a consistency index, as: C.I. = (max - N)/(N-1), where N is the dimension of the matrix.

• Then, Saaty calculated a consistency ratio (C.R.) as the ratio of the C.I. to a random index (R.I.) which is the average C.I. of sets of judgments (from a 1 to 9 scale) for randomly generated

reciprocal matrices.

Saaty’s Calculated Random Index Measures for Various Sizes

of “N”

Average Random Consistency Index (R.I.)n 1 2 3 4 5 6 7 8 9 10

R.I. 0 0 .52 .89 1.11 1.25 1.35 1.40 1.45 1.49

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AHP Diagnostics

Example 1: Real GDP Growth Forecasting

From the Perspective of Late 1991

V ery S tron g(5 .5 -6 .5 % G row th )

S tron g(4 .5 -5 .5 % G row th )

C on ven tion a l A d ju s tm en t

M od era te(3 .0 -4 .5 % G row th )

W eak(2 .0 -3 .0 % G row th )

E con om ic R es tru c tu rin g

Th e S tren g th o f th e R ecovery

Based on Saaty and Vargas, “Incorporating Expert Judgment in Economic Forecasts: The Case of the U.S. Economy in 1992”

Definitions Used

• Conventional adjustment assumes a status quo with regard to the system of causes and consequences in the economy.

• Economic restructuring assumes a new environment.

Adding Another Level of Subfactors to the AHP Model

Subfactor Set 1

C on su m p tion E xp orts In ves tm en t F isca l P o lic y C on fid en ce

Subfactor Set 2

F in an c ia l S ec to r D efen se P os tu re G lob a l C om p etit ion

The Expanded AHP Model

Consumption Exports Investment Fiscal Policy Confidence

Very Strong Strong Moderate Weak

Conventional Adjustment

Financial Sector Defense Posture Global Competition

Very Strong Strong Moderate Weak

Economic Restructing

Strength of the Recovery

Once the Structure of the Model is Formulated, the Judgmental Weights Must be Assigned for a Given Forecast Horizon . . .

Which primary factor will be most influential in determining the strength of the recovery?

Conventional Adjustment (CA) Restructuring (R)Conventional Adjustment (CA) 1 1/5Restructuring (R) 5 1

Vector Weights0.1670.833

Step 1: Determining Weights Top Down

Determining the First Weight Using the Geometric Average

Step 1: 1 x (1/5) = 0.20

Step 2: 0.20 ^ (0.5) = 0.447

Step 3: Normalize Weights(0.447/2.683) =0.167

Note: In order to Normalize the Weight in Step 3, you need to know that the second row geometric average is 2.236.

Which subfactor is more important in influencing conventional adjustment?

Step 2-A: Determining Weights Top Down

C E I K F MConsumption (C) 1.0 7.000 3.000 1.000 7.000 3.000Exports (E) 0.143 1 0.200 0.200 1.000 0.200Investment (I) 0.333 5.000 1 0.333 0.333 0.200Confidence (K) 1.000 5.000 3.000 1 7.000 3.000Fiscal Policy (F) 0.143 1.000 3.000 0.143 1 0.143Monetary Policy (M) 0.333 7.000 5.000 0.333 7.000 1

Step 2-B: Determining Weights Top Down

Which subfactor is more important in influencing restructuring?

Follow the Same Process for Next Level . . .

FS DS GCFinancial Sector (FS) 1 0.200 0.333Defense Posture (DS) 5.000 1 3.000Global Competition (GC) 3.000 0.333 1

Vector Weights for Other AlternativesBased on Saaty’s Example

Relative likelihood of the strength of the recovery if consumption drives the expansion under conventional adjustment

Relative likelihood of the strength of the recovery if investment drives the expansion under conventional adjustment

Relative likelihood of the strength of the recovery if exports drives the expansion under conventional adjustment

Relative likelihood of the strength of the recovery if confidence drives the expansion under conventional adjustment

VS = 0.423

S = 0.423

M = 0.104

W = 0.051

VS = Very Strong Growth S = Strong Growth M = Moderate Growth W = Weak Growth

VS = 0.095

S = 0.095

M = 0.249

W = 0.560

VS = 0.182

S = 0.182

M = 0.545

W = 0.091

VS = 0.376

S = 0.376

M = 0.193

W = 0.054

Vector Weights for Other AlternativesBased on Saaty’s Example

Relative likelihood of the strength of the recovery if fiscal policy drives the expansion under conventional adjustment

Relative likelihood of the strength of the recovery if monetary policy drives the expansion under conventional adjustment

VS = 0.125

S = 0.125

M = 0.625

W = 0.125

VS = Very Strong Growth S = Strong Growth M = Moderate Growth W = Weak Growth

VS = 0.084

S = 0.084

M = 0.649

W = 0.183

Calculate the Weights for the Economic Restructuring Alternative . . .

Vector Weights for Other AlternativesBased on Saaty’s Example

Relative likelihood of the strength of the recovery if financial sector drives the expansion under restructuring

VS = 0.095

S = 0.095

M = 0.249

W = 0.560

VS = Very Strong Growth S = Strong Growth M = Moderate Growth W = Weak Growth

VS = 0.055

S = 0.118

M = 0.262

W = 0.565

Relative likelihood of the strength of the recovery if defense posture drives the expansion under restructuring

Relative likelihood of the strength of the recovery if global competition drives the expansion under restructuring

VS = 0.101

S = 0.101

M = 0.348

W = 0.449

GDP Forecasting Model

Conventional Adjustment0.167

Economic Restructing.833

Consumption0.311

Exports0.037

Investment0.075

Fiscal Policy0.063

Confidence0.300

Financial Sector0.105

Defense Sector0.637

Global Competition0.258

Very Strong Strong Moderate Weak

Strength of the Recovery

Once the Structure of the Model is Formulated and the Judgmental Weights Assigned, the Forecast Now Can Be Determined . . .

Overall Results of the ForecastBased on Mid-Points of Forecast Ranges

Using Matrix Multiplication Obtain the Forecast as Follows:

This Same Model Could Have “Alternatives” as a Time Horizon Instead of Strength of Recovery

• Three Months

• Six Months

• 12 Months

• 24 Months or More

Model Could Be Used for Various Forecast Horizons

Example 2:Forecasting Foreign Exchange Rates

Six Primary Indicators of Future Spot Exchange Rates: (1) Relative interest rates (INTRAT); (2) Forward exchange rate biases (FDBIAS); (3) Official exchange market intervention (EXCINT); (4) Relative degree of confidence in the U.S. economy (CONFUS); (5) the size and recent direction of the U.S. current account balance (CURBAL); (6) Past behavior of exchange rates (PASREC).

Source: Saaty and Vargas, Prediction, Projection and Forecasting, Kluwer Academic Publishers, Norwell, MA, 1991.

Yen/Dollar Exchange Rate Forecasting Hierarchy

GOAL Forex Value of the Yen in 90-Days

Three Levels of Criteria

Level 1: (A) Relative interest rates (INTRAT); (B) Forward exchange rate biases (FDBIAS); (C) Official exchange market intervention (EXCINT); (D) Relative degree of confidence in the U.S. economy (CONFUS); (E) the size and recent direction of the U.S. current account balance (CURBAL); (F) Past behavior of exchange rates (PASREC)

Level 2: (A) Fed Policy; (B) Size of Federal Deficit; (C) BoJ Policy; (D) Consistent/Erratic Forex Intervention; (E) Forward Rate Premium/Discount; (F) Size of Forward Rate Differential; (G) Relative Interest Rates; (H) Relative Growth; (I) Relative Political Stability; (J) Size of Current Account; (K) Expected Change in Current Account; (L) Past Exchange Rate Relative/Irrelative

Level 3: Generally of the form: (A) Higher/Tighter, (B) Neutral, and (C) Lower/Easier

Sharp Decline / Moderate Decline / No Change / Moderate Increase / Sharp IncreaseForecast

Some Other Applications of AHP to the Forecasting Problem

• To Address the Duration or Time to Event Question.

• To Assess the Likelihood of Strength or Weakness.

• To Determine Judgmental Probabilities.• To Judge the Likelihood of a Cyclical

Turning Point.• Could Be Used to Develop a Consensus Forecast

Based on Alternative Techniques or Inputs.