Analytic Hierarchy Process

To accompany Quantitative Analysis for Management, 9e \by Render/Stair/Hanna

M1-1 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Analytic Hierarchy Analytic Hierarchy ProcessProcess



Introduction Introduction

AHP was developed by Thomas L. Saaty and published in his 1980 book, The Analytic

Hierarchy Process.

Analytic hierarchy process (AHP) is an approach designed to quantify the preferences for various factors and alternatives.

This process involves pairwise comparisons. The decision maker starts by laying out the

overall hierarchy of the decision. This hierarchy reveals the factors to be

considered as well as the various alternatives in the decision. Then, a number of pairwise comparisons are done, which result in the determination of factor weights and factor evaluations.



Analytic Hierarchy ProcessAnalytic Hierarchy Process

Break decision into stages or levels.

Starting at the lowest level, for each level, make pairwise comparison of the factors.

9-step scale:1. equally preferred2. equally to moderately preferred3. moderately preferred4. moderately to strongly preferred5. strongly preferred6. strongly to very strongly preferred7. very strongly preferred8. very to extremely preferred9. extremely preferred



Analytic Hierarchy ProcessAnalytic Hierarchy Process

Develop the matrix representation: Comparison matrix Normalized matrix Priority matrix

Develop and the consistency ratio.

Determine factor weights. Perform a multifactor

evaluation.



Judy Grim's Computer Judy Grim's Computer DecisionDecision

• As an example of this process, we take the case of Judy Grim, who is looking for a new computer systems for her small business.

• She has determined that the most important overall factors hardware, software, and vendor support.

• Furthermore, Judy has narrowed down her alternatives to three possible computer systems. She has labeled these SYSTEM-1, SYSTEM-2, and SYSTEM-3.

• To begin, Judy has placed these factors and alternatives into a decision hierarchy (see Figure 1).



Decision Hierarchy for Decision Hierarchy for Computer System SelectionComputer System Selection

Select Computer System

Hardware Software Vendor Support

System: System: System:1 2 3 1 2 3 1 2 3

Figure (1)

The key to using AHP is pairwise comparisons. The decision maker, Judy

Grim, needs to compare two different alternatives using a scale that ranges from equally preferred to extremely preferred.



Using Pairwise Using Pairwise ComparisonsComparisons

• Judy begins by looking at the hardware factor and by comparing computer SYSTEM-1 with computer SYSTEM-2. Using the 9-step scale.

• Judy determines that the hardware for computer SYSTEM-1 is moderately preferred to computer SYSTEM-2. Thus, Judy uses the number 3, representing moderately preferred.

• She believes that the hardware for computer SYSTEM-1 is extremely preferred to computer SYSTEM-3. This is a numerical score of 9.

• She believes that the hardware for computer SYSTEM-2 is strongly to very strongly preferred to the hardware for computer SYSTEM-3, a score of 6.



Beginning Comparison Beginning Comparison MatrixMatrix

Judy Grim has used the 9-point scale for pairwise comparison to evaluate each system on hardware capabilities

Hardware

Syst

em-1

Sys t

em-3

System-1

System-2

System-3

Syst

em-2

3 9

6

1

1



Comparison Matrix Comparison Matrix (continued)(continued)

HardwareSy

stem

-1

Sys t

em-3

System-1

System-2

System-3

Syst

em-2

3 9

6

1

1

1

1/3

1/9 1/6



HardwareSy

stem

-1

Sys t

em-3

System-1

System-2

System-3

Syst

em-2

3 9

6

1

1

1

1/3

1/9 1/6

1.444 4.167 16.0Column Totals

Normalizing the Normalizing the MatrixMatrix

The totals are used to create a normalized matrix



HardwareSy

stem

-1

Sys t

em-3

System-1

System-2

System-3

Syst

em-2

0.6923 0.7200

0.2300 0.2400 0.3750

0.0769 0.0400 0.0625

Normalized MatrixNormalized Matrix

0.5625

= 1/ 1.444 = .333/ 1.444



Final Matrix for Final Matrix for HardwareHardware

3/)0625.00400.00769.0(3/)3750.02400.02300.0(3/)5625.07200.06923.0(

0.05980.28190.6583

Averages Row

Factor System-1 System-2 System-3

Hardware 0.6583 0.2819 0.0598

To determine the priorities for hardware for the three computer systems, we simply find the

average of the various rows from the matrix of numbers as follows:



The Weighted Sum VectorThe Weighted Sum Vector

333322311233222211133122111

VectorSum

WeightedW

Matrix

ComparisonPairwise Original

P

3f2f1fVector FinalF

pfpfpfpfpfpfpfpfpf

F = [ 0.6583 0.2819 0.0598]

3 90.33 1 60.11 0.167 1

1

(0.6583)(1) + (0.2819)(3) +(0.0598)(9) = 2.04230.6583)(0.33) + (0.2819)(1) + (0.0598)(6) = 0.8602(0.6583)(0.11) + (0.2819)(0.167) + (0.0598)(1) = 0.1799



The Consistency VectorThe Consistency Vector

33

22

11

///

fwfwfw

VectoryConsistenc

C

2.0423 / 0.6583 3.1025= 0.8602 0.2819 = 3.0512 0.1799/ 0.0598 3.0086



Computing LambdaComputing Lambda

n

i 1 ic

Lambda is the average value of the consistency vectors.

= 3.1025 + 3.0512 + 3.0086 3

= 3.0541



The Consistency IndexThe Consistency Index

esalternativ ofnumber n where

1Index

yConsistencCI

n

n

The consistency index is:

CI = 3.0541 – 3 3 – 1

= 0.0270



Consistency RatioConsistency Ratio

The consistency ratio (CR) tells how consistent the decision maker is with her answers. A higher number means less consistency. In general, a number of 0.10 or greater suggests the decision maker should reevaluate her responses during the pairwise comparison.

CR =CIRI (random index)

= 0.0270 0.58

= 0.0466

This is a table value

Is Judy consistent in her answers regarding hardware??



Random Index TableRandom Index Table



Achieving a Final Achieving a Final RankingRanking

We must now perform a second pairwise comparison regarding the relative importance of each of the remaining two factors.

Factor Evaluation

System 1 System 2 System 3

Hardware

Software

Vendor Support

0.6583 0.2819 0.0598

0.0874 0.1622 0.7504

0.4967 0.3967 0.1066

Table (1): Factor Evaluations



Achieving a Final Rank Achieving a Final Rank (continued)(continued)

Determining Factor Weights

Next, we need to determine the factor weights. In comparing the three factors, Judy determines that software is the most important. Software is very to extremely strongly preferred over hardware (number 8). Software is moderately preferred over vendor support (number 3). In comparing vendor support to hardware, we decide that the vendor support is more important. Vendor support is moderately preferred to hard ware (number 3).

With these values, we can construct the pairwise comparison matrix and then compute the weights for hardware, software, and support.



Achieving a Final Rank Achieving a Final Rank (continued)(continued)

• After making the appropriate calculations, the factor weights for hardware, software, and vendor support are shown in the next table:

Table (2): Factor weights



Judy Grim’s Final Judy Grim’s Final DecisionDecision

Overall Ranking• After the factor weights have been

determined, we can multiply the factor evaluations in table (1) times the factor weights in table (2). It will give us the overall ranking for the three computer systems, which is shown in next table.



ExampleExample

Select the Best CarSelect the Best Car

CostCost SafetySafety AppearanceAppearance

HondaHondaMazdaMazda VolvoVolvo



Overall GoalOverall Goal

CriteriaCriteria

DecisionDecisionAlternativesAlternatives



Example Example (continued)(continued)

Cost H

onda

V

olvo

Honda

Mazda

Volvo

Maz

da2 4

3

1

1

1

1/2

1/4 1/3




Safety H

onda

V

olvo

Honda

Mazda

Volvo

Maz

da1/2 1/5

1/4

1

1

1

2

5 4




Appearance H

onda

V

olvo

Honda

Mazda

Volvo

Maz

da5 9

2

1

1

1

1/5

1/9 1/2




Criteria C

ost

App

ear.

Cost

Safety

Appear.

Saf

ety

1/2 3

5

1

1

1

2

1/3 1/5



Factor Evaluation

Honda Mazda Volvo

Cost 0.5570.557 0.3200.320 0.1230.123

Safety 0.1170.117 0.2000.200 0.6830.683

Appear 0.7610.761 0.1580.158 0.0820.082



Factor Factor WeightCost 0.3090.309

SAFETYSAFETY 0.5820.582

APPEARANCEAPPEARANCE 0.1090.109



Overall RankingOverall Ranking

System or Alternative

Total Weighted Evaluation

Honda 0.3240.324

Mazda 0.2320.232

Volvo 0.4440.444

Best Decision!!

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