THE ANALYTIC
HIERARCHY
PROCESS
CAR PURCHASE
EXAMPLE
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.
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.
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.”
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.
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.
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.
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.
HYPOTHETICAL DATA FOR CAR
PURCHASE EXAMPLE
Car
Honda
Mazda
Volvo
Cost
$22,000
28,500
33,000
Safety*
28
39
52
Appearance
Sporty
Slick
Dull
* Safety Rating from a consumer testing service - the
higher the number, the safer the car.
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.
STANDARD 1 - 9 MEASUREMENT SCALE
Intensity of Importance
1
3
Definition
Equal importance
Moderate importance
5
Strong importance
7
Very strong
9
Extreme importance
2, 4, 6, 8
1.1 - 1.9
Reciprocals of above
For compromise
values
For tied activities
If activity A has
one of the above
numbers assigned
to it when compared
with activity B,
then B has the
reciprocal value
when compared to A.
Explanation
Two activities contribute equally
Experience and judgment slightly favor one
activity over another
Experience and judgment strongly favor one
activity over another
An activity is favored very strongly over
another
The evidence favoring one activity over
another is of the highest possible order
of affirmation
Sometimes one needs to interpolate a
compromise between the above judgment
numerically because there is no good
word to describe it
When elements are close and nearly
indistinguishable; moderate is 1.3 and
extreme is 1.9
For example, if the pairwise comparison of
A to B is 3.0, then the pairwise comparison
of B to A is 1/3
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.
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.
COST PAIRWISE COMPARISONS
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
28.5K
Mazda
33K
Volvo
Volvo
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.
COST PAIRWISE COMPARISONS
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
28.5K
Mazda
1
33K
Volvo
Volvo
1
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).
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.
COST PAIRWISE COMPARISONS
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1
33K
Volvo
Volvo
1
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.
COST PAIRWISE COMPARISONS
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
Volvo
1
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.
COST PAIRWISE COMPARISONS
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/2
Volvo
2
1
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.
COST PAIRWISE COMPARISONS
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/2
Volvo
4
2
1
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.
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.
COST PAIRWISE
COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/2
------------COLUMN TOTALS
Volvo
4
2
1
-------
COST PAIRWISE
COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/2
------------COLUMN TOTALS
7/4
7/2
Volvo
4
2
1
------7
COST PAIRWISE
COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/2
------------COLUMN TOTALS
7/4
7/2
Volvo
4
2
1
------7
COST PAIRWISE COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/2
------------COLUMN TOTALS
7/4
7/2
Volvo
4
2
1
------7
B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
Honda
4/7*
4/7
Mazda
2/7
2/7
Volvo
1/7
1/7
Volvo
4/7
2/7
1/7
* This entry is obtained by dividing the Honda entry in the original matrix (1)
by the Honda column total (7/4).
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.
COST PAIRWISE COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/2
------------COLUMN TOTALS
7/4
7/2
Volvo
4
2
1
------7
B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
Honda
4/7*
4/7
Mazda
2/7
2/7
Volvo
1/7
1/7
Volvo
4/7
2/7
1/7
* This entry is obtained by dividing the Honda entry in the original matrix (1)
by the Honda column total (7/4).
COST PAIRWISE COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/2
------------COLUMN TOTALS
7/4
7/2
B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
Honda
4/7*
4/7
Mazda
2/7
2/7
Volvo
1/7
1/7
Volvo
4
2
1
------7
Volvo
4/7
2/7
1/7
TOTAL
WEIGHTS
(ROW AVG.)
0.571
0.286
0.143
--------1.000
* This entry is obtained by dividing the Honda entry in the original matrix (1)
by the Honda column total (7/4).
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 (3:1).
This button appears on the top left-hand side of
the toolbar to the right of the model view
button.
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.
EXPERT CHOICE:
Entering Judgments
For our problem, Expert Choice only displays
three judgments.
1’s along the main diagonal and reciprocal
judgments do not appear.
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.
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.
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.
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.
INCONSISTENCY OF
JUDGMENTS
Inconsistency of judgments may result from:
problems of estimation;
errors between the comparisons;
or, the comparisons may be naturally
inconsistent.
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.
INCONSISTENCY OF
JUDGMENTS
The point is not to stop inconsistency from
occurring.
Make sure that the level of inconsistency
remains within some reasonable limit.
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.
COST PAIRWISE COMPARISONS
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/3
Volvo
4
3
1
COST PAIRWISE COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/3
------------COLUMN TOTALS
7/4
10/3
Volvo
4
3
1
------8
COST PAIRWISE COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/3
------------COLUMN TOTALS
7/4
10/3
Volvo
4
3
1
------8
B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
Honda
4/7*
6/10
Mazda
2/7
3/10
Volvo
1/7
1/10
Volvo
4/8
3/8
1/8
* This entry is obtained by dividing the Honda entry in the original matrix (1)
by the Honda column total (7/4).
COST PAIRWISE COMPARISONS
1.
2.
3.
SUM THE ELEMENTS IN EACH COLUMN OF THE ORIGINAL MATRIX.
DIVIDE EACH ELEMENT IN THE ORIGINAL MATRIX BY ITS COLUMN SUM.
THIS RESULTS IN THE ADJUSTED MATRIX.
COMPUTE THE ROW AVERAGES - THESE ARE THE WEIGHTS.
A. ORIGINAL COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
22K
Honda
1
2
28.5K
Mazda
1/2
1
33K
Volvo
1/4
1/3
------------COLUMN TOTALS
7/4
10/3
B. ADJUSTED COST PAIRWISE COMPARISON MATRIX
Honda
Mazda
Honda
4/7*
6/10
Mazda
2/7
3/10
Volvo
1/7
1/10
Volvo
4
3
1
------8
Volvo
4/8
3/8
1/8
TOTAL
WEIGHTS
(ROW AVG.)
0.557
0.320
0.123
-------1.000
* This entry is obtained by dividing the Honda entry in the original matrix (1)
by the Honda column total (7/4).
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.
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.
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.
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!
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.
SAFETY & APPEARANCE JUDGMENTS
Safety Pairwise Comparison Matrix
Honda
Mazda
28
Honda
1
1/2
39
Mazda
2
1
52
Volvo
5
4
Appearance Pairwise Comparison Matrix
Honda
Mazda
SportyHonda
1
5
Slick Mazda
1/5
1
Dull Volvo
1/9
1/2
Volvo
1/5
1/4
1
Volvo
9
2
1
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.
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
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.
FINAL CAR WEIGHTS
COST
0.309
CARS
Honda
Mazda
Volvo
0.558
0.320
0.122
CRITERIA WEIGHTS
SAFETY
APPEARANCE
0.582
0.109
FINAL WEIGHTS
0.117
0.761
0.200
0.158
0.683
0.082
FINAL CAR WEIGHTS
COST
0.309
CARS
Honda
Mazda
Volvo
0.558
0.320
0.122
CRITERIA WEIGHTS
SAFETY
APPEARANCE
0.582
0.109
FINAL WEIGHTS
0.117
0.761
0.324
0.200
0.158
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
FINAL CAR WEIGHTS
COST
0.309
CARS
Honda
Mazda
Volvo
0.558
0.320
0.122
CRITERIA WEIGHTS
SAFETY
APPEARANCE
0.582
0.109
FINAL WEIGHTS
0.117
0.761
0.324
0.200
0.158
0.232
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
FINAL CAR WEIGHTS
COST
0.309
CARS
Honda
Mazda
Volvo
0.558
0.320
0.122
CRITERIA WEIGHTS
SAFETY
APPEARANCE
0.582
0.109
FINAL
0.117
0.761
0.200
0.158
0.683
0.082
WEIGHTS
0.324
0.232
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
IDEAL MODE
For each car, add the three criteria weights:
Honda
Cost
0.309
Safety
0.100
Appearance 0.109
Total
0.518
Mazda
0.177
0.170
0.023
0.370
Volvo
0.068
0.582
0.012
0.662
IDEAL MODE
For each car, add the three criteria weights:
Honda
Cost
0.309
Safety
0.100
Appearance 0.109
Total
0.518
Since the sum
Mazda Volvo
of the three
0.177 0.068 weights is
0.170 0.582 1.550, we
0.023 0.012 divide each
0.370 0.662 weight by 1.550
to normalize the
results.
IDEAL MODE
For each car, add the three criteria weights:
Since the sum
Honda Mazda Volvo
of the three
Cost
0.309
0.177 0.068 weights is
Safety
0.100
0.170 0.582 1.550, we
Appearance 0.109
0.023 0.012 divide each
Total
0.518
0.370 0.662 weight by 1.550
to normalize the
Total/1.550 0.335
0.239 0.427
results.
These are the ideal weights reported in
Expert Choice.
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.
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.
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.
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.
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.
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.