THE ANALYTIC HIERARCHY PROCESS CAR PURCHASE EXAMPLE

1
THE ANALYTIC HIERARCHY PROCESSCAR PURCHASE
EXAMPLE
2
CAR PURCHASE EXAMPLE

• We now consider a motivating example.
• After completing this example, you will have an
  understanding of the basics of AHP and its
  application through Expert Choice
  (www.expertchoice.com).
• We want to apply the AHP to help a couple decide
  which car they should purchase.

3
CAR PURCHASE EXAMPLE

• The couple is considering three criteria cost,
  safety, and appearance.
• They have narrowed their alternatives to three
  specific cars Honda, Mazda, and Volvo.
• We demonstrate how to build the AHP hierarchy in
  Expert Choice.

4
EXPERT CHOICE FILE SETUP

• After launching Expert Choice, select the File,
  New option, and after selecting a destination
  folder, enter a file name such as CARS. (Expert
  Choice add the AHP file extension.)
• Next, enter a description for your goal, such as,
  Select the best car.

5
EXPERT CHOICE FILE SETUP

• To enter the criteria, for example, cost, safety,
  and appearance, use the Edit, and Insert Child of
  Current Node commands.
• Use the Esc key or hit an extra enter when
  finished entering the criteria.
• To add the alternative cars select the Edit,
  Alternative, and Insert commands.

6
EXPERT CHOICE FILE SETUP

• You can also use the Add Alternative button in
  the upper right hand corner of the model window.
• Repeat for all alternatives.
• Additional details can be found in the Expert
  Choice tutorial provided with the software.

7
ANALYZING THE HIERARCHY

• 1. Determine the weights of the alternatives for
  each criterion.
• 2. Determine the priorities or weights of the
  criteria in achieving the goal.
• 3. Determine the overall weight of each
  alternative in achieving the goal. This is
  accomplished by combining the results of the
  first two stages and is called synthesis.

8
ANALYZING THE HIERARCHY

• To complete the first stage, the couple can base
  their judgments on the following (hypothetical)
  performance information.
• All alternative pairwise comparisons should be
  based on data.

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HYPOTHETICAL DATA FOR CAR PURCHASE EXAMPLE
Car Cost Safety Appearance
Honda 22,000 28 Sporty Mazda
28,500 39 Slick Volvo 33,000 52 Dull
Safety Rating from a consumer testing service -
the higher the number, the safer the car.
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DETERMINING PRIORITIES

• The couple begins by making pairwise comparison
  judgments between each pair of cars for the cost
  criterion.
• In our example, three judgments are needed Honda
  to Mazda, Mazda to Volvo, and Honda to Volvo.
• The scale on the next page is the standard one.

11
STANDARD 1 - 9 MEASUREMENT SCALE

• Intensity of Importance Definition
  Explanation
• 1 Equal importance Two activities contribute
  equally
• 3 Moderate importance Experience and judgment
  slightly favor one
• activity over another
• 5 Strong importance Experience and judgment
  strongly favor one
• activity over another
• 7 Very strong An activity is favored very
  strongly over
• another
• 9 Extreme importance The evidence favoring one
  activity over
• another is of the highest possible order
• of affirmation
• 2, 4, 6, 8 For compromise Sometimes one needs
  to interpolate a
• values compromise between the above judgment
• numerically because there is no good
• word to describe it
• 1.1 - 1.9 For tied activities When elements
  are close and nearly
• indistinguishable moderate is 1.3 and
• extreme is 1.9
• Reciprocals of above If activity A has For
  example, if the pairwise comparison of

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COST PAIRWISE COMPARISONS

• The pairwise comparisons are represented in the
  form of pairwise comparison matrices.
• The computation of the weights are also shown.
• Consider the pairwise comparison matrix to
  compare the cars for the cost criterion.
• Remember that the costs of the three cars are
  22000, 28500, and 33000, respectively.

13
COST PAIRWISE COMPARISONS

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

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COST PAIRWISE COMPARISONS

• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1
  
• 28.5K Mazda
• 33K Volvo

15
COST PAIRWISE COMPARISONS

• The other entries along the main diagonal of the
  matrix are also 1.
• This simply means that everything is equally
  preferred to itself.

16
COST PAIRWISE COMPARISONS

• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1
  
• 28.5K Mazda 1
  
• 33K Volvo 1

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COST PAIRWISE COMPARISONS

• Suppose we believe the Honda (22000) is equally
  to moderately preferred to the Mazda (28500).
  Place a 2 in the row 1, column 2 entry.
• Some might argue that the Honda should be 1.295
  times better than the Mazda (28,500/22,000).

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COST PAIRWISE COMPARISONS

• Do you agree?
• It depends!
• For some, 28,500 is significantly greater than
  22,000, implying a judgments greater than 1.295.
  
• Others with a lot of money may perceive virtually
  no difference between the two costs, implying a
  judgment somewhere between 1 and 1.295.

19
COST PAIRWISE COMPARISONS

• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2
  
• 28.5K Mazda 1
  
• 33K Volvo 1

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COST PAIRWISE COMPARISONS

• If the Honda is 2 times better than the Mazda,
  this implies that the Mazda (28500) is one half
  as good as the Honda (22000).
• The reciprocal judgment, (1/2), should be placed
  in the row 2, column 1 entry of the matrix.

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COST PAIRWISE COMPARISONS

• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2
  
• 28.5K Mazda 1/2 1
  
• 33K Volvo 1

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COST PAIRWISE COMPARISONS

• Suppose that we judge the Mazda (28500) to be
  equally to moderately preferred to the Volvo
  (33000).
• The following judgments would be entered in the
  matrix.

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COST PAIRWISE COMPARISONS

• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/2 1

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COST PAIRWISE COMPARISONS

• Assuming perfect consistency of judgments, we
  would expect that the Honda (22000) is 4 times
  (that is, moderately to strongly) preferred to
  the Volvo (33000).
• We will relax this assumption later.

25
COST PAIRWISE COMPARISONS

• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/4 1/2 1

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COST PAIRWISE COMPARISONS

• The matrix is now complete and the weights for
  each car (for the cost criterion) can be
  computed.
• The exact computational procedure is implemented
  in Expert Choice.
• For details see Expert Choice homepage and
  download AHPDEMO.EXE.

27
COST PAIRWISE COMPARISONS

• A simple three step procedure can be used to
  approximate the weights for each alternative.
• Essentially, this procedure normalizes the ratios
  of the judgments between any pair of alternatives.

28
COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• 
  
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/4 1/2 1
• ------- ------- -------
• COLUMN TOTALS
• 
  

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COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• 
  
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/4 1/2 1
• ------- ------- -------
• COLUMN TOTALS 7/4 7/2 7
  

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COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• 
  
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/4 1/2 1
• ------- ------- -------
• COLUMN TOTALS 7/4 7/2 7
  

31
COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• 
  
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/4 1/2 1
• ------- ------- -------
• COLUMN TOTALS 7/4 7/2 7
• 
  
• B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• Honda 4/7 4/7 4/7
  
• Mazda 2/7 2/7 2/7
• Volvo 1/7 1/7 1/7
• 
  

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COST PAIRWISE COMPARISONS

• Notice that no variation is seen across the rows
  because the judgments are perfectly consistent.
• For the third column, judgments totaling 7 were
  awarded. The Honda received 4 of 7 (57.1), the
  Mazda 2 of 7 (28.6), and the Volvo 1 of 7
  (14.3) of the weight.
• Similar comparisons can be made for the other two
  columns.

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COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• 
  
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
  
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/4 1/2 1
• ------- ------- -------
• COLUMN TOTALS 7/4 7/2 7
• 
  
• B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• Honda 4/7 4/7 4/7
  
• Mazda 2/7 2/7 2/7
• Volvo 1/7 1/7 1/7

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COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• 
  
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  2
• 33K Volvo 1/4 1/2 1
• ------- ------- -------
• COLUMN TOTALS 7/4 7/2 7
• 
  
• B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
  WEIGHTS
• Honda Mazda Volvo
  (ROW AVG.)
• Honda 4/7 4/7 4/7
  0.571
• Mazda 2/7 2/7 2/7
  0.286
• Volvo 1/7 1/7 1/7
  0.143
• 
  ---------

35
EXPERT CHOICE Entering Judgments

• Expert Choice offers a variety of modes for
  entering the judgments.
• Highlight the cost node and select the Pairwise
  Numerical comparison button (31).
• This button appears on the top left-hand side of
  the toolbar to the right of the model view
  button.

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EXPERT CHOICE Entering Judgments

• Sliding the bar between Honda and Mazda to the
  left so that it rests on the 2 means that the
  Honda is two times better than the Mazda when
  considering cost.
• If the Mazda were 2 times better than the Honda,
  the bar would be slid to the 2 on the right.
• The other comparisons are entered in a similar
  fashion.

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EXPERT CHOICE Entering Judgments

• For our problem, Expert Choice only displays
  three judgments.
• 1s along the main diagonal and reciprocal
  judgments do not appear.

38
EXPERT CHOICE Entering Judgments

• There are different modes for entering judgments.
• The Pairwise Verbal Comparisons (ABC) and the
  Pairwise Graphical Comparisons (the button that
  looks like a bar graph) are available.
• The only difference between these modes is how
  the pairwise comparison questions are displayed.

39
EXPERT CHOICE Entering Judgments

• A 1-9 scale is used for numerical comparisons.
• The verbal comparisons are equal, moderate,
  strong, very strong, and extreme.
• The graphical mode makes comparisons based on the
  length of two bars.
• The user selects the desired mode.

40
EXPERT CHOICE Entering Judgments

• After entering all pairwise comparisons, record
  judgments by clicking Yes.
• The model view will be displayed with alternative
  weights for the cost criterion now appearing.

41
INCONSISTENCY OF JUDGMENTS

• Since our pairwise comparisons were perfectly
  consistent, Expert Choice reports Incon 0.00.
• If this ratio is greater than 0.1 some revision
  of judgments is required.
• Select Inconsistency (within any Pairwise
  Comparison mode) to identify the most
  inconsistent judgments.

42
INCONSISTENCY OF JUDGMENTS

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

43
INCONSISTENCY OF JUDGMENTS

• One example of natural inconsistency is in a
  sporting contest.
• 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 necessarily imply that team A is
  six times as likely to beat team C.
• This inconsistency may result because of the way
  that the teams match-up overall.

44
INCONSISTENCY OF JUDGMENTS

• The point is not to stop inconsistency from
  occurring.
• Make sure that the level of inconsistency remains
  within some reasonable limit.

45
INCONSISTENCY OF JUDGMENTS

• How does a judgment change affect the car
  weights?
• Suppose the Mazda to Volvo changes from 2 to 3.
• This obviously changes the comparison for Volvo
  to Mazda from (1/2) to (1/3).
• The judgments are now somewhat inconsistent.

46
COST PAIRWISE COMPARISONS

• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  3
• 33K Volvo 1/4 1/3 1

47
COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  3
• 33K Volvo 1/4 1/3 1
• ------- ------- -------
• COLUMN TOTALS 7/4 10/3 8

48
COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  3
• 33K Volvo 1/4 1/3 1
• ------- ------- -------
• COLUMN TOTALS 7/4 10/3 8
• 
  
• B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• Honda 4/7 6/10 4/8
  
• Mazda 2/7 3/10 3/8
• Volvo 1/7 1/10 1/8
• 
  

49
COST PAIRWISE COMPARISONS

• 1. SUM THE ELEMENTS IN EACH COLUMN OF THE
  ORIGINAL MATRIX.
• 2. DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY
  ITS COLUMN SUM.
• THIS RESULTS IN THE ADJUSTED MATRIX.
• 3. COMPUTE THE ROW AVERAGES - THESE ARE THE
  WEIGHTS.
• A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
  
• Honda Mazda Volvo
• 22K Honda 1 2 4
  
• 28.5K Mazda 1/2 1
  3
• 33K Volvo 1/4 1/3 1
• ------- ------- -------
• COLUMN TOTALS 7/4 10/3 8
• 
  
• B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
  WEIGHTS
• Honda Mazda Volvo
  (ROW AVG.)
• Honda 4/7 6/10 4/8
  0.557
• Mazda 2/7 3/10 3/8
  0.320
• Volvo 1/7 1/10 1/8
  0.123
• 
  --------

50
INCONSISTENCY OF JUDGMENTS

• The new weights are 0.557, 0.320, and 0.123.
  The inconsistency resulted in some change in the
  original weights of 0.571, 0.286, and 0.143.
• As expected, the weight for the Mazda increased
  while the weight for the Volvo decreased.
• The weights now vary across each row.
  Essentially, inconsistency measures the degree of
  variation across the rows.

51
EXPERT CHOICE Revising Judgments

• To make this change in Expert Choice, highlight
  cost node and select any Pairwise Comparison
  mode.
• Within the numerical mode, slide the comparison
  bar to the left from 2 to 3, select the Model
  View, and record the judgments to see the new
  weights.
• The weights of 0.558, 0.320, and 0.122 are
  slightly different from the three-step procedure
  weights.
• This is not due to rounding -- Expert Choice
  gives the exact results.

52
INCONSISTENCY OF JUDGMENTS

• The inconsistency ratio is now 0.02.
• The weights can also be used to measure the
  effectiveness of the alternatives.
• For example, based on all comparisons, the Honda
  is 1.74 (0.558/0.320) times better than the Mazda.

53
INCONSISTENCY OF JUDGMENTS

• We knew that a 22,000 car is better than a
  28,500 car, but now we know how much better.
• Why is this ratio 1.74 and not the pairwise
  comparison of 2?
• Inconsistency in the judgments!

54
REMAINING COMPUTATIONS

• Next, the cars must be pairwise compared for the
  safety criterion and then for the appearance
  criterion.
• These judgments are shown on the next page.
• The safety comparisons are all inverted, that is,
  for each comparison, the top bar was moved to the
  left.
• This means that the Mazda is 2 times more
  preferred than the Honda, with respect to safety.

55
SAFETY APPEARANCE JUDGMENTS

• Safety Pairwise Comparison Matrix
  
• Honda Mazda Volvo
• 28 Honda 1 1/2 1/5
• 39 Mazda 2 1 1/4
• 52 Volvo 5 4 1
• Appearance Pairwise Comparison Matrix
  
• Honda Mazda Volvo
• SportyHonda 1 5 9
• Slick Mazda 1/5 1 2
• Dull Volvo 1/9 1/2 1

56
REMAINING COMPUTATIONS

• Next, the criteria must be pairwise compared.
• These judgments are shown on the next page.
• There are no data to support these judgments
  since they are purely a reflection of your
  preferences.

57
CRITERIA JUDGMENTS

• Original Criteria Pairwise Comparison Matrix
• Cost Safety Appearance
• Cost 1 1/2 3
• Safety 2 1 5
• Appearance 1/3 1/5 1

58
REMAINING COMPUTATIONS

• The last stage computes the final weights for
  each car.
• Multiply the criteria weight by the car weight
  for each criterion and then sum over all
  criteria.
• This is nothing more than a weighted average.
• The computational results are shown next.

59
FINAL CAR WEIGHTS

• CRITERIA WEIGHTS
• COST SAFETY
  APPEARANCE
• 0.309 0.582 0.109
• CARS
  FINAL WEIGHTS
• Honda 0.558 0.117 0.761
• Mazda 0.320 0.200 0.158
• Volvo 0.122 0.683 0.082

60
FINAL CAR WEIGHTS

• CRITERIA WEIGHTS
• COST SAFETY
  APPEARANCE
• 0.309 0.582 0.109
• CARS
  FINAL WEIGHTS
• Honda 0.558 0.117 0.761
  0.324
• Mazda 0.320 0.200 0.158
• Volvo 0.122 0.683 0.082
• Honda (0.558)(0.309) (0.117)(0.582)
  (0.761)(0.109) 0.324
• 0.173 0.068 0.083

61
FINAL CAR WEIGHTS

• CRITERIA WEIGHTS
• COST SAFETY
  APPEARANCE
• 0.309 0.582 0.109
• CARS
  FINAL WEIGHTS
• Honda 0.558 0.117 0.761
  0.324
• Mazda 0.320 0.200 0.158
  0.232
• Volvo 0.122 0.683 0.082
• Honda (0.558)(0.309) (0.117)(0.582)
  (0.761)(0.109) 0.324
• 0.173 0.068 0.083
• Mazda (0.320)(0.309) (0.200)(0.582)
  (0.158)(0.109) 0.232
• 0.099 0.116 0.017

62
FINAL CAR WEIGHTS

• CRITERIA WEIGHTS
• COST SAFETY
  APPEARANCE
• 0.309 0.582 0.109
• CARS
  FINAL WEIGHTS
• Honda 0.558 0.117 0.761
  0.324
• Mazda 0.320 0.200 0.158
  0.232
• Volvo 0.122 0.683 0.082
  0.444
• Honda (0.558)(0.309) (0.117)(0.582)
  (0.761)(0.109) 0.324
• 0.173 0.068 0.083
• Mazda (0.320)(0.309) (0.200)(0.582)
  (0.158)(0.109) 0.232
• 0.099 0.116 0.017
• Volvo (0.122)(0.309) (0.683)(0.582)
  (0.082)(0.109) 0.444
• 0.038 0.397 0.009

63
LOCAL VS GLOBAL WEIGHTS

• For cost, the local weights for the cars are
  0.558, 0.320, and 0.122 and sum to 1.000.
• The global weights are computed by multiplying
  the cost criterion weight by the local car
  weights.
• The global weights are 0.173, 0.099, and 0.038
  and sum to the cost criterion weight of 0.309.

64
EXPERT CHOICE Synthesis

• The final weights are shown in Expert Choice
  after all comparisons are entered and when the
  Model View is displayed and the goal is
  highlighted.
• Choose Distributive Mode.
• The difference between the Distributive and Ideal
  modes will be discussed later.

65
INTERPRETING THE RESULTS

• The final weights provide a measure of the
  relative performance of each alternative.
• It is important to properly interpret the meaning
  of these numbers.
• The Volvo is ranked first, the Honda second, and
  Mazda third.
• The Volvo is preferred 1.37 (0.444/0.324) times
  more than the Honda.

66
INTERPRETING THE RESULTS

• Should we buy the Volvo?
• The output is a decision-making aid and cannot
  replace the decision-maker.
• The results can be used to support discussion and
  possibly the judgments will be revised.
• This iterative process is quite normal.
• AHP can help to facilitate communication and
  generate consensus between different groups.

67
SYNTHESIS MODES

• The process used to compute the final weights is
  called distributive synthesis.
• This method works well when there is a fixed
  amount of resources that must be distributed to a
  fixed set of alternatives.

68
SYNTHESIS MODES

• In some cases after completing an AHP analysis,
  an additional alternative may need to be
  considered.
• It is possible that a rank reversal could occur.
  
• Our rankings are Volvo, Honda, and Mazda.
• If another Volvo is added that is similar to the
  original Volvo, it is possible that the Honda
  will be ranked higher than the original Volvo.

69
SYNTHESIS MODES

• In some cases this is acceptable, in others it is
  not.
• Distributive synthesis should not be used if
  preservation of rank is important.
• Ideal Synthesis should be used to prevent rank
  reversal.

70
IDEAL MODE

• The ideal mode gives the full weight of the
  criterion to the alternative that ranks highest
  under that criterion.
• The other alternatives are given a portion of the
  criterion weight based on their local weight.

71
IDEAL MODE

• The local weights for the three cars with respect
  to cost are 0.558, 0.320, and 0.122,
  respectively. The cost criterion weight is
  0.309.
• Since the Honda has the highest cost weight it is
  initially assigned the full cost weight of 0.309.
  
• Mazda would be (0.320 / 0.558)(0.309) 0.177.
• Volvo would be (0.122 / 0.558)(0.309) 0.068.

72
IDEAL MODE

• Using the same approach, the weights for the
  three cars with respect to safety are 0.100,
  0.170, and 0.582, respectively.
• The weights for the three cars with respect to
  appearance are 0.109, 0.023, and 0.012,
  respectively.

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IDEAL MODE

• For each car, add the three criteria weights
• Honda Mazda Volvo
• Cost 0.309 0.177 0.068
• Safety 0.100 0.170 0.582
• Appearance 0.109 0.023 0.012
• Total 0.518 0.370 0.662

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IDEAL MODE

• For each car, add the three criteria weights
• Honda Mazda Volvo
• Cost 0.309 0.177 0.068
• Safety 0.100 0.170 0.582
• Appearance 0.109 0.023 0.012
• Total 0.518 0.370 0.662

Since the sum of the three weights is 1.550, we
divide each weight by 1.550 to normalize the
results.
75
IDEAL MODE

• For each car, add the three criteria weights
• Honda Mazda Volvo
• Cost 0.309 0.177 0.068
• Safety 0.100 0.170 0.582
• Appearance 0.109 0.023 0.012
• Total 0.518 0.370 0.662
• Total/1.550 0.335 0.239 0.427
• These are the ideal weights reported in
• Expert Choice.

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

• Sensitivity analysis is an important aspect of
  any decision-making process.
• Sensitivity analysis determines whether small
  changes in judgments affects the final weights
  and rankings of the alternatives.
• If so, the decision-maker may want to review the
  sensitive judgments.

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EXPERT CHOICE Sensitivity Analysis

• In Expert Choice sensitivity analysis from the
  GOAL shows how the weights and the rankings of
  the alternatives change if some or all of the
  criteria weights change.
• There are five graphical sensitivity analysis
  modes available Performance, Dynamic, Gradient,
  Two-Dimensional, and Difference.
• The first three show how a change in a criterion
  weight affects the final weights of the
  alternatives.

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EXPERT CHOICE Sensitivity Analysis

• The last two show how the alternatives perform
  with respect to any two criteria.
• Performance places all sensitivity information
  on a single chart with horizontal line graphs for
  the alternatives linked to vertical bars for the
  criteria.
• Dynamic two sets of dynamically linked
  horizontal bar graphs one for criteria and one
  for alternatives.

79
EXPERT CHOICE Sensitivity Analysis

• Gradient a line graph that shows how the weights
  of the alternatives vary according to the weight
  assigned to a specific criterion. (Use the
  X-Axis to change the selected criterion.)
• Two-Dimensional shows how well the alternatives
  perform with respect to any two criteria.
• Difference a graph that shows the differences
  between any two alternatives for any criterion.

80
EXPERT CHOICE Sensitivity Analysis

• An important use of sensitivity analysis is to
  determine how much a given criterion weight must
  change before there is a change in the rankings
  of the two highest alternatives.
• This type of breakeven analysis can be easily
  done in Expert Choice.

81
EXPERT CHOICE Sensitivity Analysis

• Choose Dynamic from the Sensitivity-Graphs
  option.
• Drag the cost criterion bar 30.9 to
  approximately 45.9, and see that the Volvo and
  Honda have the same highest final weight.
• The final rankings are relatively insensitive to
  a change in the cost weight since it had to be
  increased by almost 50 to get a change in the
  final rankings.
• The sensitivity results are different for the
  ideal mode.