10 The Analytic Hierarchy Process Using Decision Lens

Decision TechnologyDecision TechnologyModeling, Software and ApplicationsModeling, Software and Applications

Matthew J. LiberatoreMatthew J. LiberatoreRobert L. NydickRobert L. Nydick

John Wiley & Sons, Inc. John Wiley & Sons, Inc.

Chapter 10Chapter 10

The Analytic Hierarchy Process The Analytic Hierarchy Process Using Decision LensUsing Decision Lens

CAR PURCHASE EXAMPLECAR PURCHASE EXAMPLE

We now consider an example to illustrate the We now consider an example to illustrate the basics of the AHP.basics of the AHP.

After completing this example, you will After completing this example, you will understand the application of the AHP through understand the application of the AHP through Decision Lens.Decision Lens.

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

After launching Decision Lens, begin to create a After launching Decision Lens, begin to create a model by following the five steps in order: 1. model by following the five steps in order: 1. Build Model, 2. Compare Criteria, 3. Evaluate Build Model, 2. Compare Criteria, 3. Evaluate Alternatives, 4. Allocate Resources, and 5. Alternatives, 4. Allocate Resources, and 5. Reporting.Reporting.

From the Build Model step, select “Create Tree-From the Build Model step, select “Create Tree-View” in the upper left hand corner and enter View” in the upper left hand corner and enter a description in “Goal Name,” such as, “Select a description in “Goal Name,” such as, “Select the best car” and click OK.the best car” and click OK.

DECISION LENS: FILE SETUPDECISION LENS: FILE SETUP

To enter the criteria, for example, cost, safety, and To enter the criteria, for example, cost, safety, and appearance, select “Add Child” and add the three appearance, select “Add Child” and add the three criteria. criteria.

Select cancel when all criteria has been entered.Select cancel when all criteria has been entered.

To add the alternative cars select the “Identify To add the alternative cars select the “Identify Alternatives” option in the upper left hand corner.Alternatives” option in the upper left hand corner.

Enter a criteria and alternative description in the Enter a criteria and alternative description in the Definition location. This is done using the Definition location. This is done using the notation button (looks like a piece of paper) to the notation button (looks like a piece of paper) to the right of the criteria and alternatives in the tree right of the criteria and alternatives in the tree view and alternatives section.view and alternatives section.



Select the “Add Alternative” option three times to enter Select the “Add Alternative” option three times to enter the alternatives: Honda, Mazda, Volvo. the alternatives: Honda, Mazda, Volvo.

Identify “musts” to limit alternatives. For example, the Identify “musts” to limit alternatives. For example, the car must get at least 20 mpg and must have at least car must get at least 20 mpg and must have at least dual airbags.dual airbags.

Select the “Identify Participants” and enter the wife and Select the “Identify Participants” and enter the wife and husband (Mary and Joe) as one decision maker. In husband (Mary and Joe) as one decision maker. In this example, we will assume that Mary and Joe will this example, we will assume that Mary and Joe will only enter one set of judgments. We will show later only enter one set of judgments. We will show later how to have each participant add their own how to have each participant add their own judgments.judgments.

ANALYZING THE HIERARCHYANALYZING THE HIERARCHY

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

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

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.

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.

DETERMINING PRIORITIESDETERMINING PRIORITIES

The couple begins by making The couple begins by making pairwise comparison pairwise comparison judgmentsjudgments between each pair of criteria. between each pair of criteria.

In our example, three judgments are needed: Cost In our example, three judgments are needed: Cost to Safety, Safety to Appearance, and Cost to to Safety, Safety to Appearance, and Cost to Appearance.Appearance.

The scale on the next page is the standard one to The scale on the next page is the standard one to nine developed by Saaty. nine developed by Saaty.

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. The form of pairwise comparison matrices. The computation of the weights are also shown.computation of the weights are also shown.

Consider the pairwise comparison matrix to Consider the pairwise comparison matrix to compare the criteria.compare the criteria.

If we compare the Cost to Cost, obviously they If we compare the Cost to Cost, obviously they are equal. Therefore, a 1 (equal preferred) is are equal. Therefore, a 1 (equal preferred) is placed in the first row, first column entry of placed in the first row, first column entry of the matrix.the matrix.

A. ORIGINAL CRITERIA PAIRWISE COMPARISON MATRIX A. ORIGINAL CRITERIA PAIRWISE COMPARISON MATRIX

CostCost SafetySafety Appearance Appearance

CostCost 11

SafetySafety

AppearanceAppearance

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 important to itself.important to itself.




CostCost 11

SafetySafety 11

AppearanceAppearance 11


Suppose we believe that Safety is equally to Suppose we believe that Safety is equally to moderately more important than Cost. Place a moderately more important than Cost. Place a 1/2 in the row 1, column 2 entry and its 1/2 in the row 1, column 2 entry and its reciprocal value (2) in row 2, column 1 entry.reciprocal value (2) in row 2, column 1 entry.

This actually means that Cost is one half times This actually means that Cost is one half times as important as Safety and that Safety it twice as important as Safety and that Safety it twice as important as Cost.as important as Cost.




CostCost 11 1/21/2

SafetySafety 22 11

AppearanceAppearance 11


Suppose that we judge Safety to be strongly Suppose that we judge Safety to be strongly more important than Appearance. more important than Appearance.

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




CostCost 11 1/21/2

SafetySafety 22 11 55

AppearanceAppearance 1/51/5 11


The last set of judgments that are needed for this The last set of judgments that are needed for this problem are that Cost is moderately more problem are that Cost is moderately more important than Appearance.important than Appearance.

These judgments are also shown.These judgments are also shown.




CostCost 11 1/21/2 33


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


The matrix is now complete and the weights for each The matrix is now complete and the weights for each criterion can be computed.criterion can be computed.

The exact computational procedure is implemented in The exact computational procedure is implemented in Decision Lens, however, we show a simple three step Decision Lens, however, we show a simple three step procedure can be used to approximate the weights for procedure can be used to approximate the weights for each criterion.each criterion.

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


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.



CostCost 11 1/21/2 33


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

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

COLUMN TOTALSCOLUMN TOTALS


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 CRITERIA PAIRWISE COMPARISON MATRIX A. ORIGINAL CRITERIA PAIRWISE COMPARISON MATRIX


CostCost 11 1/21/2 33SafetySafety 22 11 55AppearanceAppearance 1/31/3 1/51/5 11

-------- ------- ------------- ------- -----COLUMN TOTALSCOLUMN TOTALS 10/310/3 17/1017/10 99


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 CRITERIA PAIRWISE COMPARISON MATRIX A. ORIGINAL CRITERIA PAIRWISE COMPARISON MATRIX


CostCost 11 1/21/2 33SafetySafety 22 11 55AppearanceAppearance 1/31/3 1/51/5 11





CostCost SafetySafety Appearance Appearance CostCost 11 1/21/2 33SafetySafety 22 11 55AppearanceAppearance 1/31/3 1/51/5 11


B. ADJUSTED CRITERIA PAIRWISE COMPARISON MATRIXB. ADJUSTED CRITERIA PAIRWISE COMPARISON MATRIX

CostCost SafetySafety AppearanceAppearanceCostCost 3/10*3/10* 5/175/17 3/93/9SafetySafety 6/106/10 10/1710/17 5/95/9AppearanceAppearance 1/101/10 2/172/17 1/91/9

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

the Cost column total (10/3).the Cost column total (10/3).


For the third column, judgments totaling 9 were For the third column, judgments totaling 9 were awarded. Cost received 3 of 9 (33.3%), awarded. Cost received 3 of 9 (33.3%), Safety 5 of 9 (55.6%), and Appearance 1 of 9 Safety 5 of 9 (55.6%), and Appearance 1 of 9 (11.1%) of the weight. (11.1%) of the weight.

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




CostCost SafetySafety Appearance Appearance CostCost 11 1/21/2 33SafetySafety 22 11 55AppearanceAppearance 1/31/3 1/51/5 11


B. ADJUSTED CRITERIA PAIRWISE COMPARISON MATRIXB. ADJUSTED CRITERIA PAIRWISE COMPARISON MATRIX WEIGHTS WEIGHTS

CostCost SafetySafety AppearanceAppearance Row Avg.Row Avg.

CostCost 3/10*3/10* 5/175/17 3/93/9 .309.309SafetySafety 6/106/10 10/1710/17 5/95/9 .581.581AppearanceAppearance 1/101/10 2/172/17 1/91/9 .110.110

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

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

the Cost column total (10/3).the Cost column total (10/3).


Entering judgments in Decision Lens is simple. Choose Entering judgments in Decision Lens is simple. Choose “Compare Criteria” (step 2) and the “Pairwise “Compare Criteria” (step 2) and the “Pairwise Comparison” option (upper left hand corner) to enter the Comparison” option (upper left hand corner) to enter the criteria comparisons. criteria comparisons.

Since Safety is judged to be twice as important as Cost, Mary Since Safety is judged to be twice as important as Cost, Mary and Joe would click on the 2 option on the right hand side and Joe would click on the 2 option on the right hand side of the bar. Since there is only one decision maker, once of the bar. Since there is only one decision maker, once Mary and Joe enter their judgment the same value appears Mary and Joe enter their judgment the same value appears in the average row. If Cost was judged to be twice as in the average row. If Cost was judged to be twice as important to Safety, the 2 on the left would be selected.important to Safety, the 2 on the left would be selected.

To enter a decimal judgment right click on the bar and select To enter a decimal judgment right click on the bar and select Decimal Vote.Decimal Vote.

Entering JudgmentsEntering Judgments

Choosing the “Next Vote” button allows the other Choosing the “Next Vote” button allows the other comparisons to be entered.comparisons to be entered.

Make sure you notice which judgment is being entered Make sure you notice which judgment is being entered next. For example, after hitting the “Next Vote” next. For example, after hitting the “Next Vote” button once the Safety to Appearance comparison is button once the Safety to Appearance comparison is entered (5).entered (5).

In this example, Decision Lens only requires three In this example, Decision Lens only requires three judgments. judgments.

1’s along the main diagonal and reciprocal judgments 1’s along the main diagonal and reciprocal judgments are not entered.are not entered.

DECISION LENS: DECISION LENS: Entering JudgmentsEntering Judgments

After entering all pairwise comparisons, select the After entering all pairwise comparisons, select the “Compute” button to display the criteria weights.“Compute” button to display the criteria weights.

Decision Lens reports that the weights for Cost, Safety, Decision Lens reports that the weights for Cost, Safety, and Appearance are 0.309, 0.582, and 0.109, and Appearance are 0.309, 0.582, and 0.109, respectively.respectively.

Remember that the 3-step approximation produced Remember that the 3-step approximation produced weights of 0.309, 0.581, and 0.110, respectively.weights of 0.309, 0.581, and 0.110, respectively.

This difference is not due to rounding – Decision Lens This difference is not due to rounding – Decision Lens gives the exact weights.gives the exact weights.

DECISION LENS: DECISION LENS: Entering JudgmentsEntering Judgments

INCONSISTENCY OF INCONSISTENCY OF JUDGMENTSJUDGMENTS

Decision Lens also reports the inconsistency at Decision Lens also reports the inconsistency at 0.004. 0.004.

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

Select “Inconsistency Analysis” in the upper left Select “Inconsistency Analysis” in the upper left hand corner and then “Matrix View” for hand corner and then “Matrix View” for information about inconsistent judgments.information about inconsistent judgments.


Inconsistency of judgments may result from:Inconsistency of judgments may result from:problems of estimation; errors between the comparisons; problems of estimation; errors between the comparisons; or, the comparisons may be naturally inconsistent. or, the comparisons may be naturally inconsistent.

One example of natural inconsistency is in sports. One example of natural inconsistency is in sports.

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

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


We knew that Safety was more important than We knew that Safety was more important than Cost, but now we know how much more Cost, but now we know how much more important.important.

We judged Safety to be 2 times more important We judged Safety to be 2 times more important than Cost. The weights tell us that Safety is than Cost. The weights tell us that Safety is 1.88 times more important than Cost 1.88 times more important than Cost (0.582/0.309). Which is “correct” and why is (0.582/0.309). Which is “correct” and why is there a difference?there a difference?

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

three criteria: cost, safety and appearance. three criteria: cost, safety and appearance.

The couple can base their judgments on the The couple can base their judgments on the following (hypothetical) performance following (hypothetical) performance information.information.

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

We need to compare Honda to Mazda, Mazda to We need to compare Honda to Mazda, Mazda to Volvo, and Honda to Volvo for each criterion. Volvo, and Honda to Volvo for each criterion.

HYPOTHETICAL DATA FOR CAR HYPOTHETICAL DATA FOR CAR PURCHASE EXAMPLEPURCHASE EXAMPLE

CarCar CostCost Safety*Safety* AppearanceAppearance HondaHonda $22,000$22,000 AverageAverage SportySporty Mazda Mazda 28,500 28,500 Above AverageAbove Average SlickSlick Volvo Volvo 33,000 33,000 ExcellentExcellent DullDull

* Safety Rating from a consumer testing service.* Safety Rating from a consumer testing service.

COST JUDGMENTSCOST JUDGMENTS

Cost Pairwise Comparison Matrix Cost Pairwise Comparison Matrix

HondaHonda MazdaMazdaVolvo Volvo

$22k$22k Honda Honda 11 22 44

$28.5k$28.5kMazdaMazda 1/21/2 11 33

$33k$33k Volvo Volvo 1/41/4 1/31/3 11

SAFETY & APPEARANCE JUDGMENTSSAFETY & APPEARANCE JUDGMENTS

Safety Pairwise Comparison Matrix Safety Pairwise Comparison Matrix

HondaHonda MazdaMazda Volvo Volvo

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

AAAA Mazda Mazda 22 11 1/41/4

ExcExc 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

ENTERING JUDGMENTS IN DECISION LENSENTERING JUDGMENTS IN DECISION LENS

These judgments now need to be entered in Decision These judgments now need to be entered in Decision Lens.Lens.

Select the third step: “Evaluate Alternatives,” click Select the third step: “Evaluate Alternatives,” click on the “Pairwise Alternatives” button, Yes, and on the “Pairwise Alternatives” button, Yes, and “Next.”“Next.”

You will then be taken through screens that will You will then be taken through screens that will allow you to enter the 9 pairwise comparison allow you to enter the 9 pairwise comparison judgments (3 each for the three criteria). Select judgments (3 each for the three criteria). Select “Next Vote” to enter all of the judgments. “Next Vote” to enter all of the judgments.

ENTERING JUDGMENTS IN DECISION LENSENTERING JUDGMENTS IN DECISION LENS

Again, be careful to make sure that you know which Again, be careful to make sure that you know which judgments are being entered at each step.judgments are being entered at each step.

After all 9 judgments have been entered select “Next After all 9 judgments have been entered select “Next Vote” again to display the final weights.Vote” again to display the final weights.

REMAINING COMPUTATIONSREMAINING COMPUTATIONS

The alternative weights are computed just like the The alternative weights are computed just like the criteria weights and are based on the alternative criteria weights and are based on the alternative pairwise comparisons for each criterion.pairwise comparisons for each criterion.

To compute the final weights, multiply the criteria To compute the final weights, multiply the criteria weight by the car weight for each criterion and weight by the car weight for each criterion and then sum over all criteria. This is nothing more then sum over all criteria. This is nothing more than a weighted average.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.

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 the couple buy the Volvo?Should the couple 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. .

Distributive synthesis should be used when you Distributive synthesis should be used when you are concerned about the priorities of all the are concerned about the priorities of all the alternatives or when you want to allocate a alternatives or when you want to allocate a fixed amount of resources across all fixed amount of resources across all alternatives.alternatives.

Ideal synthesis should be used when you are Ideal synthesis should be used when you are choosing one alternative and don’t really care choosing one alternative and don’t really care about the alternatives you do not choose. For about the alternatives you do not choose. For example, you are buying one car.example, you are buying one car.


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

At this time Decision Lens does not report ideal At this time Decision Lens does not report ideal weights. This feature will be added in the next weights. This feature will be added in the next release.release.

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.

DECISION LENS: DECISION LENS: Sensitivity AnalysisSensitivity Analysis

In Decision Lens sensitivity analysis is performed In Decision Lens sensitivity analysis is performed by selecting “Analysis/Sensitivity” within the by selecting “Analysis/Sensitivity” within the Evaluate Alternatives step. We use two options Evaluate Alternatives step. We use two options to display the sensitivity results: Scoresheet to display the sensitivity results: Scoresheet Sensitivity and Barchart Sensitivity. Sensitivity and Barchart Sensitivity.

These show how a change in a criterion weight These show how a change in a criterion weight affects the final weights of the alternatives.affects the final weights of the alternatives.

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 Decision Lens.done in Decision Lens.


Drag the cost criterion bar .309 to approximately .458, and Drag the cost criterion bar .309 to approximately .458, and see that the Volvo and Honda have the same highest final see that the Volvo and Honda have the same highest final weight. weight.

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

Sensitivity analysis could also be performed for the other Sensitivity analysis could also be performed for the other criteria.criteria.

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


Step 4 allows you to use Decision Lens to allocate Step 4 allows you to use Decision Lens to allocate limited resources. limited resources.

The purpose of the optimizer is the maximize the The purpose of the optimizer is the maximize the overall priority within the defined constraints on overall priority within the defined constraints on limited resources using the ratio of Benefit/Cost, limited resources using the ratio of Benefit/Cost, where Benefit is the AHP final weight.where Benefit is the AHP final weight.

We will use the file called Resource Allocation We will use the file called Resource Allocation Example.ANP. This model is already setup with Example.ANP. This model is already setup with five criteria, ten alternatives, three participants, five criteria, ten alternatives, three participants, and all necessary pairwise comparisons. Go to and all necessary pairwise comparisons. Go to Step 4: Allocate Resources.Step 4: Allocate Resources.

ALLOCATING RESOURCESALLOCATING RESOURCES

The define constraints screen is a summary of the The define constraints screen is a summary of the basic information entered for this problem. The basic information entered for this problem. The three dropdowns across the top are used to enter three dropdowns across the top are used to enter different resource types, different scenarios, and different resource types, different scenarios, and different years. This information can be changed different years. This information can be changed by selecting the edit button.by selecting the edit button.

To edit the funding pools: double click on Default To edit the funding pools: double click on Default Pool and rename as New Systems, enter 150000 as Pool and rename as New Systems, enter 150000 as the pool budget, and then click OK.the pool budget, and then click OK.

To add a new pool: type Infrastructure under Pool To add a new pool: type Infrastructure under Pool Name, 75000 under Pool Budget, and then Add Name, 75000 under Pool Budget, and then Add Funding Pool. Funding Pool.


This means that we have two resource pools and the This means that we have two resource pools and the different projects will compete for the resources.different projects will compete for the resources.

To specify a budget for a project 1: click the cell in To specify a budget for a project 1: click the cell in the Project 1 row under Requested Budget, type the Project 1 row under Requested Budget, type 30000 and Enter.30000 and Enter.

To specify that project 5 requires funding from both To specify that project 5 requires funding from both resource pools: click on the dropdown under the resource pools: click on the dropdown under the header Funding Pool for this project, select mixed header Funding Pool for this project, select mixed pools, type 10000 under Request for New Systems pools, type 10000 under Request for New Systems and 9000 for Infrastructure, click the F button once and 9000 for Infrastructure, click the F button once to change to H (discussed later), and then Update.to change to H (discussed later), and then Update.


To assign funding pools: for Projects 1-4, click on To assign funding pools: for Projects 1-4, click on the dropdown under the header Funding Pool and the dropdown under the header Funding Pool and select Infrastructure for each. This means that select Infrastructure for each. This means that these projects will compete for funding from the these projects will compete for funding from the Infrastructure budget while projects 6-10 will Infrastructure budget while projects 6-10 will compete for resources from the New Systems compete for resources from the New Systems budget.budget.

To add funding minimums: click in the row for To add funding minimums: click in the row for Project 1 under Minimums and enter 10000. We Project 1 under Minimums and enter 10000. We have now set a Hard Minimum of $10,000 for have now set a Hard Minimum of $10,000 for this project meaning that Project 1 cannot receive this project meaning that Project 1 cannot receive less than $10,000.less than $10,000.


To add a Fixed Amount Funding Minimum: click in To add a Fixed Amount Funding Minimum: click in the row for Project 2 under the Minimums and the row for Project 2 under the Minimums and enter 15000, and then click on the H twice to enter 15000, and then click on the H twice to make it an F. We now specify that Project 2 will make it an F. We now specify that Project 2 will receive exactly $15,000. receive exactly $15,000.

If you click the F twice to make it an S you have If you click the F twice to make it an S you have now changed Project 2 to a Soft Minimum now changed Project 2 to a Soft Minimum meaning that it must get at least $15,000 if it gets meaning that it must get at least $15,000 if it gets any money at all, otherwise it gets $0.any money at all, otherwise it gets $0.

For our problem, change this setting back to F for For our problem, change this setting back to F for Project 2.Project 2.


To add Project Dependencies: click Edit Dependencies, To add Project Dependencies: click Edit Dependencies, then click the upper left green cell twice so that an up then click the upper left green cell twice so that an up arrow appears and then click X to close window. This arrow appears and then click X to close window. This means that Project 2 cannot receive resources unless means that Project 2 cannot receive resources unless Project 1 gets funded. You can also enter other Project 1 gets funded. You can also enter other dependencies as needed.dependencies as needed.

We are now ready to optimize by selecting the We are now ready to optimize by selecting the Optimization step in the upper left hand corner and Optimization step in the upper left hand corner and then clicking on the yellow Optimize button.then clicking on the yellow Optimize button.

The results show the amount funded for each project. The results show the amount funded for each project. Notice that project 3 is not funded due to its low Notice that project 3 is not funded due to its low priority (0.690) and its high requested budget priority (0.690) and its high requested budget ($65,000).($65,000).


The Optimizer tries to get the highest possible The Optimizer tries to get the highest possible priority while staying within budget.priority while staying within budget.

Sensitivity analysis can be performed by sliding the Sensitivity analysis can be performed by sliding the arrow at the end of the criteria weight bar to arrow at the end of the criteria weight bar to another level to see how project funding changes.another level to see how project funding changes.

Proceeding to step 5, “Reporting,” will allow you to Proceeding to step 5, “Reporting,” will allow you to customize a report with selected output in RTF customize a report with selected output in RTF format.format.


This chapter discussed the application of the AHP This chapter discussed the application of the AHP and Decision Lens.and Decision Lens.

The example was a couple trying to decide which The example was a couple trying to decide which car to purchase.car to purchase.

We showed how to build and analyze the hierarchy, We showed how to build and analyze the hierarchy, enter and process all pairwise comparison enter and process all pairwise comparison judgments, and compute final weights. judgments, and compute final weights.

SummarySummary

The differences between the ideal and distributive The differences between the ideal and distributive modes are discussed.modes are discussed.

Sensitivity analysis is used to study how changes to Sensitivity analysis is used to study how changes to criteria weights impact the final alternative criteria weights impact the final alternative weights.weights.

Resource allocation and printing were also Resource allocation and printing were also presented.presented.

SummarySummary

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Copyright Copyright Matthew J. Liberatore and Robert L. Nydick. All rights Matthew J. Liberatore and Robert L. Nydick. All rights reserved. Reproduction or translation of this work beyond that named in reserved. Reproduction or translation of this work beyond that named in Section 117 of the United States Copyright Act without the express written Section 117 of the United States Copyright Act without the express written consent of the copyright owners is unlawful. Requests for further information consent of the copyright owners is unlawful. Requests for further information should be addressed to Matthew J. Liberatore and Robert L. Nydick. Adopters should be addressed to Matthew J. Liberatore and Robert L. Nydick. Adopters of the textbook are granted permission to make back-up copies for their own of the textbook are granted permission to make back-up copies for their own use only, to make copies for distribution to students of the course the textbook use only, to make copies for distribution to students of the course the textbook is used in, and to modify this material to best suit their instructional needs. is used in, and to modify this material to best suit their instructional needs. Under no circumstances can copies be made for resale. Matthew J. Liberatore Under no circumstances can copies be made for resale. Matthew J. Liberatore and Robert L. Nydick assume no responsibility for errors, omissions, or and Robert L. Nydick assume no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the damages, caused by the use of these programs or from the use of the information contained herein.information contained herein.

Automotive

10 The Analytic Hierarchy Process Using Decision Lens