THE ANALYTIC HIERARCHY PROCESS CAR PURCHASE EXAMPLE

THE ANALYTIC THE ANALYTIC HIERARCHY HIERARCHY

PROCESSPROCESS

CAR PURCHASE CAR PURCHASE EXAMPLEEXAMPLE

CAR PURCHASE EXAMPLECAR PURCHASE EXAMPLE

We now consider a motivating example. We now consider a motivating example.

After completing this example, you will have an After completing this example, you will have an understanding of the basics of AHP and its understanding of the basics of AHP and its application through Expert Choice application through Expert Choice (www.expertchoice.com).(www.expertchoice.com).

We want to apply the AHP to help a couple We want to apply the AHP to help a couple decide which car they should purchase. decide which car they should purchase.

CAR PURCHASE EXAMPLECAR PURCHASE EXAMPLE

The couple is considering three criteria: cost, The couple is considering three criteria: cost, safety, and appearance.safety, and appearance.

They have narrowed their alternatives to three They have narrowed their alternatives to three specific cars: Honda, Mazda, and Volvo.specific cars: Honda, Mazda, and Volvo.

We demonstrate how to build the AHP hierarchy We demonstrate how to build the AHP hierarchy in Expert Choice.in Expert Choice.

After launching Expert Choice, select the After launching Expert Choice, select the FFile, ile, NNew ew option, and after selecting a destination option, and after selecting a destination folder, enter a file name such as CARS. folder, enter a file name such as CARS. (Expert Choice add the AHP file extension.)(Expert Choice add the AHP file extension.)

Next, enter a description for your goal, such as, Next, enter a description for your goal, such as, “Select the best car.”“Select the best car.”

EXPERT CHOICE: FILE SETUPEXPERT CHOICE: FILE SETUP

To enter the criteria, for example, cost, safety, To enter the criteria, for example, cost, safety, and appearance, use the and appearance, use the EEdit, dit, and and Insert Insert CChild hild of Current Nodeof Current Node commands. commands.

Use the Esc key or hit an extra enter when Use the Esc key or hit an extra enter when finished entering the criteria.finished entering the criteria.

To add the alternative cars select the To add the alternative cars select the EEdit, dit, AAlternative, lternative, andand IInsert nsert commands. commands.


You can also use the “Add Alternative” button You can also use the “Add Alternative” button in the upper right hand corner of the model in the upper right hand corner of the model window. window.

Repeat for all alternatives.Repeat for all alternatives.

Additional details can be found in the Expert Additional details can be found in the Expert Choice tutorial provided with the software.Choice tutorial provided with the software.


ANALYZING THE HIERARCHYANALYZING THE HIERARCHY

1.1. Determine the weights of the alternatives for Determine the weights of the alternatives for each criterion.each criterion.

2.2. Determine the priorities or weights of the Determine the priorities or weights of the criteria in achieving the goal.criteria in achieving the goal.

3.3. Determine the overall weight of each Determine the overall weight of each alternative in achieving the goal. This is alternative in achieving the goal. This is accomplished by combining the results of the accomplished by combining the results of the first two stages and is called synthesis.first two stages and is called synthesis.

ANALYZING THE HIERARCHYANALYZING THE HIERARCHY

To complete the first stage, the couple can base To complete the first stage, the couple can base their judgments on the following their judgments on the following (hypothetical) performance information.(hypothetical) performance information.

All alternative pairwise comparisons should be All alternative pairwise comparisons should be based on data.based on data.

HYPOTHETICAL DATA FOR CAR HYPOTHETICAL DATA FOR CAR PURCHASE EXAMPLEPURCHASE EXAMPLE

CarCar CostCost Safety*Safety* AppearanceAppearance HondaHonda $22,000$22,000 2828 SportySporty Mazda Mazda 28,500 28,500 3939 SlickSlick Volvo Volvo 33,000 33,000 5252 DullDull

* Safety Rating from a consumer testing service - the * Safety Rating from a consumer testing service - the higher the number, the safer the car.higher the number, the safer the car.

DETERMINING PRIORITIESDETERMINING PRIORITIES

The couple begins by making The couple begins by making pairwise pairwise comparison judgmentscomparison judgments between each pair of between each pair of cars for the cost criterion.cars for the cost criterion.

In our example, three judgments are needed: In our example, three judgments are needed: Honda to Mazda, Mazda to Volvo, and Honda Honda to Mazda, Mazda to Volvo, and Honda to Volvo.to Volvo.

The scale on the next page is the standard one. The scale on the next page is the standard one.

STANDARD 1 - 9 MEASUREMENT SCALESTANDARD 1 - 9 MEASUREMENT SCALEIntensity of ImportanceIntensity of Importance DefinitionDefinition ExplanationExplanation

11 Equal importanceEqual importance Two activities contribute equallyTwo activities contribute equally

33 Moderate importanceModerate importance Experience and judgment slightly favor oneExperience and judgment slightly favor one

activity over anotheractivity over another

55 Strong importanceStrong importance Experience and judgment strongly favor oneExperience and judgment strongly favor one

activity over anotheractivity over another

77 Very strongVery strong An activity is favored very strongly overAn activity is favored very strongly over

anotheranother

99 Extreme importanceExtreme importance The evidence favoring one activity overThe evidence favoring one activity over

another is of the highest possible orderanother is of the highest possible order

of affirmationof affirmation

2, 4, 6, 82, 4, 6, 8 For compromiseFor compromise Sometimes one needs to interpolate aSometimes one needs to interpolate a

valuesvalues compromise between the above judgmentcompromise between the above judgment

numerically because there is no goodnumerically because there is no good

word to describe itword to describe it

1.1 - 1.91.1 - 1.9 For tied activitiesFor tied activities When elements are close and nearlyWhen elements are close and nearly

indistinguishable; moderate is 1.3 andindistinguishable; moderate is 1.3 and

extreme is 1.9extreme is 1.9

Reciprocals of aboveReciprocals of above If activity A hasIf activity A has For example, if the pairwise comparison ofFor example, if the pairwise comparison of

one of the above one of the above A to B is 3.0, then the pairwise comparisonA to B is 3.0, then the pairwise comparison

numbers assignednumbers assigned of B to A is 1/3of B to A is 1/3

to it when compared to it when compared

with activity B, with activity B,

then B has the then B has the

reciprocal value reciprocal value

when compared to A.when compared to A.

COST PAIRWISE COST PAIRWISE COMPARISONSCOMPARISONS

The pairwise comparisons are represented in the The pairwise comparisons are represented in the form of pairwise comparison matrices. form of pairwise comparison matrices.

The computation of the weights are also shown.The computation of the weights are also shown.

Consider the pairwise comparison matrix to Consider the pairwise comparison matrix to compare the cars for the cost criterion. compare the cars for the cost criterion.

Remember that the costs of the three cars are: Remember that the costs of the three cars are: $22000, $28500, and $33000, respectively.$22000, $28500, and $33000, respectively.

If we compare the Honda to the Honda, If we compare the Honda to the Honda, obviously they are equal. obviously they are equal.

Therefore, a 1 (equal preferred) is placed in the Therefore, a 1 (equal preferred) is placed in the first row, first column entry of the matrix.first row, first column entry of the matrix.


A. ORIGINAL COST PAIRWISE COMPARISON MATRIX A. ORIGINAL COST PAIRWISE COMPARISON MATRIX

Honda Mazda Volvo Honda Mazda Volvo

22K22K Honda Honda 11

28.5K28.5K Mazda Mazda

33K33K VolvoVolvo

COST PAIRWISE COMPARISONSCOST PAIRWISE COMPARISONS

The other entries along the main diagonal of the The other entries along the main diagonal of the matrix are also 1. matrix are also 1.

This simply means that everything is equally This simply means that everything is equally preferred to itself.preferred to itself.




22K22K Honda 1 Honda 1

28.5K28.5K Mazda Mazda 1 1

33K33K VolvoVolvo 11


Suppose we believe the Honda ($22000) is Suppose we believe the Honda ($22000) is equally to moderately preferred to the Mazda equally to moderately preferred to the Mazda ($28500). Place a 2 in the row 1, column 2 ($28500). Place a 2 in the row 1, column 2 entry.entry.

Some might argue that the Honda should be Some might argue that the Honda should be 1.295 times better than the Mazda 1.295 times better than the Mazda (28,500/22,000). (28,500/22,000).


Do you agree?Do you agree?

It depends! It depends!

For some, $28,500 is significantly greater than For some, $28,500 is significantly greater than $22,000, implying a judgments greater than $22,000, implying a judgments greater than 1.295. 1.295.

Others with a lot of money may perceive virtually Others with a lot of money may perceive virtually no difference between the two costs, implying a no difference between the two costs, implying a judgment somewhere between 1 and 1.295.judgment somewhere between 1 and 1.295.




22K22K Honda 1 Honda 1 22

28.5K28.5K Mazda 1Mazda 1

33K33K VolvoVolvo 1 1


If the Honda is 2 times better than the Mazda, If the Honda is 2 times better than the Mazda, this implies that the Mazda ($28500) is one this implies that the Mazda ($28500) is one half as good as the Honda ($22000). half as good as the Honda ($22000).

The reciprocal judgment, (1/2), should be placed The reciprocal judgment, (1/2), should be placed in the row 2, column 1 entry of the matrix.in the row 2, column 1 entry of the matrix.




22K22K Honda 1 2Honda 1 2

28.5K28.5K Mazda Mazda 1/21/2 1 1

33K33K VolvoVolvo 1 1


Suppose that we judge the Mazda ($28500) to be Suppose that we judge the Mazda ($28500) to be equally to moderately preferred to the Volvo equally to moderately preferred to the Volvo ($33000). ($33000).

The following judgments would be entered in The following judgments would be entered in the matrix.the matrix.




22K22K Honda 1 2Honda 1 2

28.5K28.5K Mazda 1/2 1Mazda 1/2 1 22

33K33K VolvoVolvo 1/21/2 1 1


Assuming perfect consistency of judgments, we Assuming perfect consistency of judgments, we would expect that the Honda ($22000) is 4 would expect that the Honda ($22000) is 4 times (that is, moderately to strongly) times (that is, moderately to strongly) preferred to the Volvo ($33000). preferred to the Volvo ($33000).

We will relax this assumption later.We will relax this assumption later.




22K22K Honda 1 2Honda 1 2 44

28.5K28.5K Mazda 1/2 1Mazda 1/2 1 2 2

33K33K VolvoVolvo 1/41/4 1/2 11/2 1


The matrix is now complete and the weights for The matrix is now complete and the weights for each car (for the cost criterion) can be each car (for the cost criterion) can be computed.computed.

The exact computational procedure is The exact computational procedure is implemented in Expert Choice. implemented in Expert Choice.

For details see Expert Choice homepage and For details see Expert Choice homepage and download AHPDEMO.EXE.download AHPDEMO.EXE.


A simple three step procedure can be used to A simple three step procedure can be used to approximate the weights for each alternative.approximate the weights for each alternative.

Essentially, this procedure normalizes the ratios Essentially, this procedure normalizes the ratios of the judgments between any pair of of the judgments between any pair of alternatives.alternatives.


1.1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.

2.2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.

THIS RESULTS IN THE ADJUSTED MATRIX.THIS RESULTS IN THE ADJUSTED MATRIX.

3.3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.



22K22K Honda 1 2 4 Honda 1 2 4

28.5K28.5K Mazda 1/2 1Mazda 1/2 1 2 2

33K33K Volvo 1/4 1/2 1Volvo 1/4 1/2 1

------- ------- -------------- ------- -------

COLUMN TOTALSCOLUMN TOTALS









28.5K28.5K Mazda 1/2 1Mazda 1/2 1 2 2

33K33K Volvo 1/4 1/2 1Volvo 1/4 1/2 1

------- ------- -------------- ------- -------

COLUMN TOTALSCOLUMN TOTALS 7/4 7/2 7 7/4 7/2 7


1.1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.







28.5K28.5K Mazda 1/2 1Mazda 1/2 1 2 2

33K33K Volvo 1/4 1/2 1Volvo 1/4 1/2 1

------- ------- -------------- ------- -------

COLUMN TOTALS 7/4 7/2 7 COLUMN TOTALS 7/4 7/2 7


1.1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. 22.. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.

THIS RESULTS IN THE ADJUSTED MATRIX.THIS RESULTS IN THE ADJUSTED MATRIX.3.3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX A. ORIGINAL COST PAIRWISE COMPARISON MATRIX

Honda Mazda Volvo Honda Mazda Volvo 22K22K Honda 1 2 4 Honda 1 2 4 28.5K28.5K Mazda 1/2 1Mazda 1/2 1 2 2 33K33K Volvo 1/4 1/2 1Volvo 1/4 1/2 1

------- ------- -------------- ------- -------COLUMN TOTALS 7/4 7/2 7COLUMN TOTALS 7/4 7/2 7

B. ADJUSTED COST PAIRWISE COMPARISON MATRIX B. ADJUSTED COST PAIRWISE COMPARISON MATRIX

Honda Mazda VolvoHonda Mazda VolvoHondaHonda 4/7* 4/7 4/74/7* 4/7 4/7 MazdaMazda 2/7 2/7 2/72/7 2/7 2/7VolvoVolvo 1/7 1/7 1/71/7 1/7 1/7

* This entry is obtained by dividing the Honda entry in the original matrix (1) by * This entry is obtained by dividing the Honda entry in the original matrix (1) by

the Honda column total (7/4).the Honda column total (7/4).


Notice that no variation is seen across the rows Notice that no variation is seen across the rows because the judgments are perfectly consistent.because the judgments are perfectly consistent.

For the third column, judgments totaling 7 were For the third column, judgments totaling 7 were awarded. The Honda received 4 of 7 (57.1%), the awarded. The Honda received 4 of 7 (57.1%), the Mazda 2 of 7 (28.6%), and the Volvo 1 of 7 Mazda 2 of 7 (28.6%), and the Volvo 1 of 7 (14.3%) of the weight. (14.3%) of the weight.

Similar comparisons can be made for the other two Similar comparisons can be made for the other two columns.columns.


1.1. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX. 2.2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.

THIS RESULTS IN THE ADJUSTED MATRIX.THIS RESULTS IN THE ADJUSTED MATRIX.3.3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS. A. ORIGINAL COST PAIRWISE COMPARISON MATRIX A. ORIGINAL COST PAIRWISE COMPARISON MATRIX












B. ADJUSTED COST PAIRWISE COMPARISON MATRIX WEIGHTS B. ADJUSTED COST PAIRWISE COMPARISON MATRIX WEIGHTS

Honda Mazda Volvo (ROW AVG.)Honda Mazda Volvo (ROW AVG.)Honda Honda 4/7* 4/7 4/7 0.5714/7* 4/7 4/7 0.571 MazdaMazda 2/7 2/7 2/7 0.2862/7 2/7 2/7 0.286VolvoVolvo 1/7 1/7 1/7 0.1431/7 1/7 1/7 0.143

--------- --------- TOTALTOTAL 1.0001.000




Expert Choice offers a variety of modes for Expert Choice offers a variety of modes for entering the judgments. entering the judgments.

Highlight the cost node and select the Pairwise Highlight the cost node and select the Pairwise Numerical comparison button (3:1).Numerical comparison button (3:1).

This button appears on the top left-hand side of This button appears on the top left-hand side of the toolbar to the right of the model view the toolbar to the right of the model view button.button.

EXPERT CHOICE: EXPERT CHOICE: Entering JudgmentsEntering Judgments

Sliding the bar between Honda and Mazda to the Sliding the bar between Honda and Mazda to the left so that it rests on the 2 means that the left so that it rests on the 2 means that the Honda is two times better than the Mazda Honda is two times better than the Mazda when considering cost.when considering cost.

If the Mazda were 2 times better than the Honda, If the Mazda were 2 times better than the Honda, the bar would be slid to the 2 on the right.the bar would be slid to the 2 on the right.

The other comparisons are entered in a similar The other comparisons are entered in a similar fashion.fashion.


For our problem, Expert Choice only displays For our problem, Expert Choice only displays three judgments. three judgments.

1’s along the main diagonal and reciprocal 1’s along the main diagonal and reciprocal judgments do not appear.judgments do not appear.


There are different modes for entering There are different modes for entering judgments.judgments.

The Pairwise Verbal Comparisons (ABC) and The Pairwise Verbal Comparisons (ABC) and the Pairwise Graphical Comparisons (the the Pairwise Graphical Comparisons (the button that looks like a bar graph) are button that looks like a bar graph) are available.available.

The only difference between these modes is how The only difference between these modes is how the pairwise comparison questions are the pairwise comparison questions are displayed.displayed.


A 1-9 scale is used for numerical comparisons.A 1-9 scale is used for numerical comparisons.

The verbal comparisons are: equal, moderate, The verbal comparisons are: equal, moderate, strong, very strong, and extreme.strong, very strong, and extreme.

The graphical mode makes comparisons based The graphical mode makes comparisons based on the length of two bars.on the length of two bars.

The user selects the desired mode.The user selects the desired mode.


After entering all pairwise comparisons, record After entering all pairwise comparisons, record judgments by clicking Yes.judgments by clicking Yes.

The model view will be displayed with The model view will be displayed with alternative weights for the cost criterion now alternative weights for the cost criterion now appearing.appearing.


INCONSISTENCY OF INCONSISTENCY OF JUDGMENTSJUDGMENTS

Since our pairwise comparisons were perfectly Since our pairwise comparisons were perfectly consistent, Expert Choice reports Incon: 0.00.consistent, Expert Choice reports Incon: 0.00.

If this ratio is greater than 0.1 some revision of If this ratio is greater than 0.1 some revision of judgments is required.judgments is required.

Select Select InconsisInconsisttencyency (within any Pairwise (within any Pairwise Comparison mode) to identify the most Comparison mode) to identify the most inconsistent judgments.inconsistent judgments.


Inconsistency of judgments may result from:Inconsistency of judgments may result from:

problems of estimation;problems of estimation;

errors between the comparisons;errors between the comparisons;

or, the comparisons may be naturally or, the comparisons may be naturally inconsistent. inconsistent.


One example of natural inconsistency is in a One example of natural inconsistency is in a sporting contest. sporting contest.

If team A is twice as likely to beat team B, and if If team A is twice as likely to beat team B, and if team B is three times as likely to beat team C, team B is three times as likely to beat team C, this does not necessarily imply that team A is this does not necessarily imply that team A is six times as likely to beat team C. six times as likely to beat team C.

This inconsistency may result because of the This inconsistency may result because of the way that the teams “match-up” overall.way that the teams “match-up” overall.


The point is not to stop inconsistency from The point is not to stop inconsistency from occurring. occurring.

Make sure that the level of inconsistency Make sure that the level of inconsistency remains within some reasonable limit. remains within some reasonable limit.


How does a judgment change affect the car How does a judgment change affect the car weights?weights?

Suppose the Mazda to Volvo changes from 2 to Suppose the Mazda to Volvo changes from 2 to 3. 3.

This obviously changes the comparison for This obviously changes the comparison for Volvo to Mazda from (1/2) to (1/3). Volvo to Mazda from (1/2) to (1/3).

The judgments are now somewhat inconsistent. The judgments are now somewhat inconsistent.




28.5K28.5K Mazda 1/2 1Mazda 1/2 1 3 3

33K33K Volvo 1/4Volvo 1/4 1/31/3 11









28.5K28.5K Mazda 1/2 1Mazda 1/2 1 3 3

33K33K Volvo 1/4 1/3 1Volvo 1/4 1/3 1

------- ------- -------------- ------- -------

COLUMN TOTALSCOLUMN TOTALS 7/4 10/3 87/4 10/3 8





------- ------- -------------- ------- -------COLUMN TOTALS COLUMN TOTALS 7/4 10/3 87/4 10/3 8







THIS RESULTS IN THE ADJUSTED MATRIX.THIS RESULTS IN THE ADJUSTED MATRIX.3.3. COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.

A. ORIGINAL COST PAIRWISE COMPARISON MATRIX A. ORIGINAL COST PAIRWISE COMPARISON MATRIX Honda Mazda Volvo Honda Mazda Volvo

22K22K Honda 1 2 4 Honda 1 2 4 28.5K28.5K Mazda 1/2 1Mazda 1/2 1 3 3 33K33K Volvo 1/4 1/3 1Volvo 1/4 1/3 1


B. ADJUSTED COST PAIRWISE COMPARISON MATRIX WEIGHTSB. ADJUSTED COST PAIRWISE COMPARISON MATRIX WEIGHTS

Honda Mazda Volvo (ROW AVG.)Honda Mazda Volvo (ROW AVG.)Honda 4/7* 6/10 4/8Honda 4/7* 6/10 4/8 0.5570.557

Mazda 2/7 3/10 3/8Mazda 2/7 3/10 3/8 0.3200.320Volvo 1/7 1/10 1/8Volvo 1/7 1/10 1/8 0.1230.123

---------------- TOTAL TOTAL 1.0001.000 * This entry is obtained by dividing the Honda entry in the original matrix (1) by * This entry is obtained by dividing the Honda entry in the original matrix (1) by




The new weights are: 0.557, 0.320, and 0.123. The new weights are: 0.557, 0.320, and 0.123. The inconsistency resulted in some change in The inconsistency resulted in some change in the original weights of 0.571, 0.286, and the original weights of 0.571, 0.286, and 0.143. 0.143.

As expected, the weight for the Mazda increased As expected, the weight for the Mazda increased while the weight for the Volvo decreased.while the weight for the Volvo decreased.

The weights now vary across each row. The weights now vary across each row. Essentially, inconsistency measures the Essentially, inconsistency measures the degree of variation across the rows.degree of variation across the rows.

To make this change in Expert Choice, highlight cost To make this change in Expert Choice, highlight cost node and select any Pairwise Comparison modenode and select any Pairwise Comparison mode..

Within the numerical mode, slide the comparison bar Within the numerical mode, slide the comparison bar to the left from 2 to 3, select the Model View, and to the left from 2 to 3, select the Model View, and record the judgments to see the new weights.record the judgments to see the new weights.

The weights of 0.558, 0.320, and 0.122 are slightly The weights of 0.558, 0.320, and 0.122 are slightly different from the three-step procedure weights.different from the three-step procedure weights.

This is not due to rounding -- Expert Choice gives This is not due to rounding -- Expert Choice gives the exact results.the exact results.

EXPERT CHOICE: EXPERT CHOICE: Revising JudgmentsRevising Judgments


The inconsistency ratio is now 0.02.The inconsistency ratio is now 0.02.

The weights can also be used to measure the The weights can also be used to measure the effectiveness of the alternatives. effectiveness of the alternatives.

For example, based on all comparisons, the For example, based on all comparisons, the Honda is 1.74 (0.558/0.320) times better than Honda is 1.74 (0.558/0.320) times better than the Mazda.the Mazda.


We knew that a $22,000 car is better than a We knew that a $22,000 car is better than a $28,500 car, but now we know how much $28,500 car, but now we know how much better.better.

Why is this ratio 1.74 and not the pairwise Why is this ratio 1.74 and not the pairwise comparison of 2?comparison of 2?

Inconsistency in the judgments!Inconsistency in the judgments!

REMAINING COMPUTATIONSREMAINING COMPUTATIONSNext, the cars must be pairwise compared for the Next, the cars must be pairwise compared for the

safety criterion and then for the appearance safety criterion and then for the appearance criterion. criterion.

These judgments are shown on the next page.These judgments are shown on the next page.

The safety comparisons are all inverted, that is, The safety comparisons are all inverted, that is, for each comparison, the top bar was moved to for each comparison, the top bar was moved to the left.the left.

This means that the Mazda is 2 times more This means that the Mazda is 2 times more preferred than the Honda, with respect to safety.preferred than the Honda, with respect to safety.

SAFETY & APPEARANCE JUDGMENTSSAFETY & APPEARANCE JUDGMENTS

Safety Pairwise Comparison Matrix Safety Pairwise Comparison Matrix

HondaHonda MazdaMazda Volvo Volvo

2828 Honda Honda 11 1/21/2 1/51/5

3939 Mazda Mazda 22 11 1/41/4

5252 Volvo Volvo 55 44 11

Appearance Pairwise Comparison Matrix Appearance Pairwise Comparison Matrix

HondaHonda MazdaMazda Volvo Volvo

SportySportyHondaHonda 11 55 99

SlickSlick MazdaMazda 1/51/5 11 22

DullDull VolvoVolvo 1/91/9 1/21/2 1 1

REMAINING COMPUTATIONSREMAINING COMPUTATIONS

Next, the criteria must be pairwise compared. Next, the criteria must be pairwise compared.

These judgments are shown on the next page. These judgments are shown on the next page.

There are no data to support these judgments There are no data to support these judgments since they are purely a reflection of your since they are purely a reflection of your preferences.preferences.

CRITERIA JUDGMENTSCRITERIA JUDGMENTS

Original Criteria Pairwise Comparison MatrixOriginal Criteria Pairwise Comparison Matrix

CostCost SafetySafetyAppearanceAppearance

CostCost 11 1/21/2 33

SafetySafety 22 11 55

AppearanceAppearance 1/31/3 1/51/5 11

REMAINING COMPUTATIONSREMAINING COMPUTATIONS

The last stage computes the final weights for The last stage computes the final weights for each car. each car.

Multiply the criteria weight by the car weight for Multiply the criteria weight by the car weight for each criterion and then sum over all criteria. each criterion and then sum over all criteria.

This is nothing more than a weighted average.This is nothing more than a weighted average.

The computational results are shown next.The computational results are shown next.

FINAL CAR WEIGHTSFINAL CAR WEIGHTSCRITERIA WEIGHTSCRITERIA WEIGHTS

COST SAFETY APPEARANCECOST SAFETY APPEARANCE

0.309 0.582 0.1090.309 0.582 0.109

CARS CARS FINAL WEIGHTS FINAL WEIGHTS

Honda 0.558 0.117 0.761Honda 0.558 0.117 0.761

Mazda 0.320 0.200 0.158Mazda 0.320 0.200 0.158

Volvo 0.122 0.683 0.082Volvo 0.122 0.683 0.082



0.309 0.582 0.1090.309 0.582 0.109


Honda 0.558 0.117 0.761 Honda 0.558 0.117 0.761 0.3240.324

Mazda 0.320 0.200 0.158Mazda 0.320 0.200 0.158

Volvo 0.122 0.683 0.082Volvo 0.122 0.683 0.082

Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324

0.1730.173 0.068 0.068 0.083 0.083



0.309 0.582 0.1090.309 0.582 0.109


Honda 0.558 0.117 0.761 0.324Honda 0.558 0.117 0.761 0.324

Mazda 0.320 0.200 0.158 Mazda 0.320 0.200 0.158 0.2320.232

Volvo 0.122 0.683 0.082Volvo 0.122 0.683 0.082

Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324

0.1730.173 0.068 0.068 0.083 0.083

Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232

0.0990.099 0.116 0.116 0.017 0.017



0.309 0.582 0.1090.309 0.582 0.109


Honda 0.558 0.117 0.761 0.324Honda 0.558 0.117 0.761 0.324

Mazda 0.320 0.200 0.158 0.232Mazda 0.320 0.200 0.158 0.232

Volvo 0.122 0.683 0.082Volvo 0.122 0.683 0.082 0.4440.444

Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324Honda: (0.558)(0.309) + (0.117)(0.582) + (0.761)(0.109) = 0.324

0.1730.173 0.068 0.068 0.083 0.083

Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232Mazda: (0.320)(0.309) + (0.200)(0.582) + (0.158)(0.109) = 0.232

0.0990.099 0.116 0.116 0.017 0.017

Volvo: (0.122)(0.309) + (0.683)(0.582) + (0.082)(0.109) = 0.444Volvo: (0.122)(0.309) + (0.683)(0.582) + (0.082)(0.109) = 0.444

0.0380.038 0.397 0.397 0.009 0.009

LOCAL VS GLOBAL WEIGHTSLOCAL VS GLOBAL WEIGHTS

For cost, the local weights for the cars are 0.558, For cost, the local weights for the cars are 0.558, 0.320, and 0.122 and sum to 1.000.0.320, and 0.122 and sum to 1.000.

The global weights are computed by multiplying The global weights are computed by multiplying the cost criterion weight by the local car the cost criterion weight by the local car weights.weights.

The global weights are 0.173, 0.099, and 0.038 The global weights are 0.173, 0.099, and 0.038 and sum to the cost criterion weight of 0.309.and sum to the cost criterion weight of 0.309.

The final weights are shown in Expert Choice The final weights are shown in Expert Choice after all comparisons are entered and when the after all comparisons are entered and when the Model View is displayed and the goal is Model View is displayed and the goal is highlighted.highlighted.

Choose Choose DisDisttributiveributive Mode. Mode.

The difference between the Distributive and The difference between the Distributive and Ideal modes will be discussed later.Ideal modes will be discussed later.

EXPERT CHOICE: SynthesisEXPERT CHOICE: Synthesis

INTERPRETING THE RESULTSINTERPRETING THE RESULTS

The final weights provide a measure of the The final weights provide a measure of the relative performance of each alternative. relative performance of each alternative.

It is important to properly interpret the meaning It is important to properly interpret the meaning of these numbers.of these numbers.

The Volvo is ranked first, the Honda second, The Volvo is ranked first, the Honda second, and Mazda third.and Mazda third.

The Volvo is preferred 1.37 (0.444/0.324) times The Volvo is preferred 1.37 (0.444/0.324) times more than the Honda.more than the Honda.

INTERPRETING THE RESULTSINTERPRETING THE RESULTSShould we buy the Volvo?Should we buy the Volvo?The output is a decision-making aid and cannot The output is a decision-making aid and cannot

replace the decision-maker.replace the decision-maker.

The results can be used to support discussion The results can be used to support discussion and possibly the judgments will be revised. and possibly the judgments will be revised.

This iterative process is quite normal. This iterative process is quite normal.

AHP can help to facilitate communication and AHP can help to facilitate communication and generate consensus between different groups. generate consensus between different groups.

SYNTHESIS MODESSYNTHESIS MODES

The process used to compute the final weights is The process used to compute the final weights is called called distributive synthesisdistributive synthesis. .

This method works well when there is a fixed This method works well when there is a fixed amount of resources that must be distributed amount of resources that must be distributed to a fixed set of alternatives.to a fixed set of alternatives.


In some cases after completing an AHP analysis, In some cases after completing an AHP analysis, an additional alternative may need to be an additional alternative may need to be considered. considered.

It is possible that a It is possible that a rank reversalrank reversal could occur. could occur.

Our rankings are: Volvo, Honda, and Mazda. Our rankings are: Volvo, Honda, and Mazda.

If another Volvo is added that is similar to the If another Volvo is added that is similar to the original Volvo, it is possible that the Honda original Volvo, it is possible that the Honda will be ranked higher than the original Volvo.will be ranked higher than the original Volvo.


In some cases this is acceptable, in others it is In some cases this is acceptable, in others it is not. not.

Distributive synthesis should not be used if Distributive synthesis should not be used if preservation of rank is important.preservation of rank is important.

Ideal SynthesisIdeal Synthesis should be used to prevent rank should be used to prevent rank reversal.reversal.

IDEAL MODEIDEAL MODE

The ideal mode gives the full weight of the The ideal mode gives the full weight of the criterion to the alternative that ranks highest criterion to the alternative that ranks highest under that criterion. under that criterion.

The other alternatives are given a portion of the The other alternatives are given a portion of the criterion weight based on their local weight.criterion weight based on their local weight.


The local weights for the three cars with respect to The local weights for the three cars with respect to cost are: 0.558, 0.320, and 0.122, respectively. cost are: 0.558, 0.320, and 0.122, respectively. The cost criterion weight is 0.309. The cost criterion weight is 0.309.

Since the Honda has the highest cost weight it is Since the Honda has the highest cost weight it is initially assigned the full cost weight of 0.309. initially assigned the full cost weight of 0.309.

Mazda would be (0.320 / 0.558)*(0.309) = 0.177. Mazda would be (0.320 / 0.558)*(0.309) = 0.177.

Volvo would be (0.122 / 0.558)*(0.309) = 0.068.Volvo would be (0.122 / 0.558)*(0.309) = 0.068.


Using the same approach, the weights for the Using the same approach, the weights for the three cars with respect to safety are: 0.100, three cars with respect to safety are: 0.100, 0.170, and 0.582, respectively.0.170, and 0.582, respectively.

The weights for the three cars with respect to The weights for the three cars with respect to appearance are: 0.109, 0.023, and 0.012, appearance are: 0.109, 0.023, and 0.012, respectively.respectively.


For each car, add the three criteria weights:For each car, add the three criteria weights:

HondaHonda MazdaMazda VolvoVolvo

CostCost 0.3090.309 0.1770.177 0.0680.068

SafetySafety 0.1000.100 0.1700.170 0.5820.582

AppearanceAppearance0.1090.109 0.0230.023 0.0120.012

TotalTotal 0.5180.518 0.3700.370 0.6620.662




CostCost 0.3090.309 0.1770.177 0.0680.068

SafetySafety 0.1000.100 0.1700.170 0.5820.582


TotalTotal 0.5180.518 0.3700.370 0.6620.662

Since the sum Since the sum of the three of the three weights is weights is 1.550, we 1.550, we divide each divide each weight by 1.550 weight by 1.550 to normalize the to normalize the results.results.




CostCost 0.3090.309 0.1770.177 0.0680.068

SafetySafety 0.1000.100 0.1700.170 0.5820.582


TotalTotal 0.5180.518 0.3700.370 0.6620.662

Total/1.550Total/1.550 0.3350.335 0.2390.239 0.4270.427

These are the ideal weights reported inThese are the ideal weights reported in

Expert Choice.Expert Choice.

Since the sum Since the sum of the three of the three weights is weights is 1.550, we 1.550, we divide each divide each weight by 1.550 weight by 1.550 to normalize the to normalize the results.results.

SENSITIVITY ANALYSISSENSITIVITY ANALYSIS

Sensitivity analysis is an important aspect of Sensitivity analysis is an important aspect of any decision-making process.any decision-making process.

Sensitivity analysis determines whether small Sensitivity analysis determines whether small changes in judgments affects the final changes in judgments affects the final weights and rankings of the alternatives. weights and rankings of the alternatives.

If so, the decision-maker may want to review If so, the decision-maker may want to review the sensitive judgments.the sensitive judgments.

EXPERT CHOICE: EXPERT CHOICE: Sensitivity AnalysisSensitivity Analysis

In Expert Choice sensitivity analysis from the In Expert Choice sensitivity analysis from the GOAL shows how the weights and the rankings GOAL shows how the weights and the rankings of the alternatives change if some or all of the of the alternatives change if some or all of the criteria weights change.criteria weights change.

There are five graphical sensitivity analysis modes There are five graphical sensitivity analysis modes available: Performance, Dynamic, Gradient, available: Performance, Dynamic, Gradient, Two-Dimensional, and Difference. Two-Dimensional, and Difference.

The first three show how a change in a criterion The first three show how a change in a criterion weight affects the final weights of the weight affects the final weights of the alternatives.alternatives.

The last two show how the alternatives perform The last two show how the alternatives perform with respect to any two criteria.with respect to any two criteria.

PerformancePerformance: places all sensitivity information : places all sensitivity information on a single chart with horizontal line graphs on a single chart with horizontal line graphs for the alternatives linked to vertical bars for for the alternatives linked to vertical bars for the criteria.the criteria.

DynamicDynamic: two sets of dynamically linked : two sets of dynamically linked horizontal bar graphs: one for criteria and one horizontal bar graphs: one for criteria and one for alternatives.for alternatives.


GradientGradient: a line graph that shows how the weights : a line graph that shows how the weights of the alternatives vary according to the weight of the alternatives vary according to the weight assigned to a specific criterion. (Use the assigned to a specific criterion. (Use the XX-Axis-Axis to change the selected criterion.)to change the selected criterion.)

Two-DimensionalTwo-Dimensional: shows how well the alternatives : shows how well the alternatives perform with respect to any two criteria.perform with respect to any two criteria.

DifferenceDifference: a graph that shows the differences : a graph that shows the differences between any two alternatives for any criterion.between any two alternatives for any criterion.


An important use of sensitivity analysis is to An important use of sensitivity analysis is to determine how much a given criterion weight determine how much a given criterion weight must change before there is a change in the must change before there is a change in the rankings of the two highest alternatives. rankings of the two highest alternatives.

This type of breakeven analysis can be easily This type of breakeven analysis can be easily done in Expert Choice.done in Expert Choice.


Choose Choose DDynamicynamic from the from the SensitivitSensitivityy-Graphs-Graphs option. option.

Drag the cost criterion bar 30.9% to approximately Drag the cost criterion bar 30.9% to approximately 45.9%, and see that the Volvo and Honda have the 45.9%, and see that the Volvo and Honda have the same highest final weight. same highest final weight.

The final rankings are relatively insensitive to a change The final rankings are relatively insensitive to a change in the cost weight since it had to be increased by in the cost weight since it had to be increased by almost 50% to get a change in the final rankings.almost 50% to get a change in the final rankings.

The sensitivity results are different for the ideal mode.The sensitivity results are different for the ideal mode.


THE ANALYTIC HIERARCHY PROCESS CAR PURCHASE EXAMPLE

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