The Analytic Hierarchy Process

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The Analytic Hierarchy Process

• ARE 511 Construction Maintenance Modeling

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• The analytic hierarchy process (AHP), developed
  by Thomas L. Saaty is designed to solve complex
  problems involving multiple criteria.
• The process requires the decision maker to
  provide judgments about the relative importance
  of each of the criteria and then to specify a
  preference for each decision alternative on each
  criterion.
• The output of the AHP is a prioritized ranking
  indicating the overall preference for each of the
  decision alternatives.
• In order to introduce the AHP, we consider the
  problem faced by Dave Payne. Dave is planning to
  purchase a new car. Alter a preliminary analysis
  of the makes and models available, Dave has
  narrowed the list of decision alternatives to
  three cars, which we will refer to as car A, car
  B, and car C. A summary of the information Dave
  has collected about the cars has been provided.

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Car A Car B Car C
Price 13,100 11,200 9,500
MPG 18 23 29
Interior Deluxe Above average Standard
Body 4-door midsize 2-door sport 2-door compact
Radio AM/FM, tape AM/FM AM
Engine 6-cylinder 4-cylinder turbo 4-cylinder

• Based on the information in the table - as well
  as his own personal feelings resulting from
  driving each carDave decided that there were
  several criteria that he needed to consider in
  making the purchase decision.
• After some thought, he selected purchase price ,
  miles per gallon ( MPG ), comfort, and style as
  the four criteria to be considered. Quantitative
  data regarding the purchase price and MPG
  criteria are provided directly in the table.

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• However, measures of comfort and style cannot be
  specified so easily. Dave will need to consider
  factors such as car interior, type of radio, ease
  of entry and exit, seat-adjustment features etc.,
  in order to determine the comfort level for each
  car.
• The style criterion will need to be measured in
  terms of Daves subjective evaluation of each
  car.
• Even when we deal with a criterion as easily
  measured as purchase price, however, subjectivity
  becomes an issue whenever a particular decision
  maker indicates his or her personal preferences.
  For instance, car A costs 3600 more than car C
  this difference might represent a great deal of
  money to one person, but not very much money to
  another person. Thus, whether car A is considered
  extremely more expensive than car C or only
  moderately more expensive than car C is a
  subjective judgment that will depend primarily on
  the financial status of the person making the
  comparison. AHPs advantage is that it can handle
  situations in which the subjective judgments of
  individuals constitute an important part of the
  decision process

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• Developing the Hierarchy
• The first step in the AHP is to develop a
  graphical representation of the problem in terms
  of the overall goal, the criteria, and the
  decision alternatives. Such a graph depicts the
  hierarchy for the problem.
• The following figure shows the hierarchy for the
  car-selection problem.

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• Note that the first level of the hierarchy shows
  that the overall goal is to select the best car.
  At the second level, we see that the four
  criteria (purchase price, MPG, comfort, and
  style) will contribute to the achievement of the
  overall goal. Finally, at the third level we see
  that each decision alternative (car A, car B, and
  ear C) can contribute to each criterion in a
  unique way.
• The approach AHP takes is to have the decision
  maker specify his or her judgments about the
  relative importance of each criterion in terms of
  its contribution to the achievement of the
  overall goal.
• At the next level, the AHP asks the decision
  maker to indicate a preference or priority for
  each decision alternative in terms of how it
  contributes to each criterion. For example, in
  the car-selection problem, Dave will need to
  specify his judgment about the relative
  importance of each of the four criteria. He will
  also need to indicate his preference for each of
  the three cars relative to each criterion. Given
  information on relative importance and
  preferences, a mathematical process is used to
  synthesize the information and provide priority
  ranking of the three cars in terms of their
  overall preference.

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• ESTABLISHING PRIORITIES USING THE AHP
• In this section we will show how the AHP utilizes
  pair wise comparisons to establish priority
  measures for both the criteria and the decision
  alternatives.
• The sets of priorities that need to be determined
  in the car-selection problem are as follows
• The priorities of the four criteria in terms of
  the overall goal
• The priorities of the three cars in terms of the
  purchase-price criterion
• The priorities of the three cars iii terms of the
  MPG criterion
• The priorities of the three cars in terms of the
  comfort criterion
• The priorities of the three cars in terms of the
  style criterion
• In the following discussion we will demonstrate
  how to establish priorities for the three cars in
  terms of the comfort criterion. The other sets of
  priorities can be determined in a similar fashion.

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• Pair-wise Comparisons
• Pair-wise comparisons are fundamental building
  blocks of the AHP.
• In establishing the priorities for the three cars
  in terms of comfort, we will ask Dave to state a
  preference for the comfort of the cars when the
  cars are considered two at a time (pair wise).
  That is Dave will be asked to compare the comfort
  of car A to car B, car A to car C, and car B to
  car C in three separate comparisons.
• The AHP employs an underlying scale with values
  from 1 to 9 to rate the relative preferences for
  two items.
• Researchers and experience have confirmed the
  9-unit scale as a reasonable basis for
  discriminating between the preferences for two
  items.

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Verbal Judgment of Preference Numerical Rating
Extremely preferred 9
Very strongly to extremely 8
Very strongly preferred 7
Strongly to very strongly 6
Strongly preferred 5
Moderately to strongly 4
Moderately preferred 3
Equally to moderately 2
Equally preferred 1
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• In the car-selection example, suppose that Dave
  has compared the comforts of car A with those of
  car B and is convinced that car A is more
  comfortable.
• Dave is then asked to state his preference for
  the comfort of car A compared to that of car B
  using one of the verbal descriptions shown in
  earlier table.
• If he believes that car A is moderately preferred
  to car B, a value of 3 is utilized in the AHP if
  he believes that car A is strongly preferred, a
  value of 5 is utilized if he believes that car A
  is very strongly preferred, a value of 7 is
  utilized if he believes that car A is extremely
  preferred, a value of 9 is utilized. Values of 2,
  4, 6, and 8 are the intermediate values for the
  scale. A value of 1 is reserved for the case
  where the two items are judged to be equally
  preferred.

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• Suppose that when asked his preference between
  cars A and B with respect to the comfort
  criterion, Dave states that car A is between
  equally and moderately more preferred than car B
  the numerical measure that reflects this judgment
  is 2.
• Dave is then asked to provide his preference
  between car A and car C. Suppose in this case he
  states that car A is very strongly to extremely
  more preferred than car C this corresponds to a
  numerical rating of 8.
• Finally, Dave is asked to state his preference
  for car B compared to car C. Suppose in this case
  he indicates that car B is strongly to very
  strongly preferred to car C the AHP would assign
  a numerical rating of 6

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• The Pairwise Comparison Matrix
• In order to develop the priorities for the three
  cars in terms of the comfort criterion, we need
  to develop a matrix of the pairwise comparison
  ratings.
• Since three cars are being considered, the
  pairwise comparison matrix will consist of three
  rows and three columns. As shown below

Comfort Car A Car B Car C
Car A 2 8
Car B 6
Car C
Note In the pairwise comparison matrix, the
value in row i and column j is the measure of
preference of the car in row i when compared to
the car in column j.
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• We see that the value in the matrix that
  corresponds to comparing car A with car B is 2,
  the value that corresponds to comparing car A
  with cur C is 8 and the value that corresponds
  to comparing car B with car C is 6.
• In order to determine the remaining entries in
  the pairwise comparison matrix, first note that
  when we compare any car against itself , the
  judgment must be that they are equally preferred.
  Hence, the AHP assigns a 1 to all elements on the
  diagonal of the pairwise comparison matrix.
• Given these entries, all that remains is to
  determine the rating for car B compared to car A,
  car C compared to car A, and car C compared to
  car B.
• Obviously, we could follow the same procedure and
  ask Dave to provide his preferences for these
  pairwise comparisons. However, since we already
  know that Dave has rated his preference for car A
  compared to car B as 2, there is no need for him
  to make another pairwise comparison with these
  two cars.

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• In fact, we will conclude that the preference
  rating for car B when compared to car A is simply
  the reciprocal of the preference rating for car A
  when compared to car B 1/2.
• Using this logic, the AHP obtains the preference
  rating of car B compared to car A by computing
  the reciprocal of the rating of car A compared to
  car B.
• Using this inverse, or reciprocal, relationship,
  we find that the rating of car C compared to car
  A is ¼ and the rating of car C compared to car B
  is ¼. Using these numerical values of preference,
  the complete pairwise comparison matrix for the
  comfort criterion is shown in completed table.

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Comfort Car A Car B Car C
Car A 1 2 8
Car B 1/2 1 6
Car C 1/8 1/6 1
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• Synthesis
• Once the matrix of pairwise comparisons has been
  developed, we can calculate what is called the
  priority of each of the elements being compared.
  For example, we would now like to use the
  pairwise comparison information to estimate the
  relative priority for each of the cars in terms
  of the comfort criterion. This part of the AHP is
  referred to as synthesization.
• The exact mathematical procedure required to
  perform this synthesization involves the
  computation of eigenvalues and eigenvectors and
  is beyond the scope of this text. However, the
  following three-step procedure provides a good
  approximation of the synthesized priorities.

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Procedure for Synthesizing Judgments Step 1 Sum
the values in each column of the pairwise
comparison matrix. Step 2 Divide each element
in the pairwise comparison matrix by its column
total the resulting matrix is referred to as the
normalized pairwise column. Step 3 Compute the
average of the elements in each row of the
normalized matrix these averages provide an
estimate of the relative priorities of the
elements being compared. To see how the
synthesization process works for our example
problem, we carry out the procedure using the
pairwise comparison matrix shown in table.
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Step 1 Sum the values in each column.
Comfort Car A Car B Car C
Car A 1 2 8
Car B 1/2 1 6
Car C 1/8 1/6 1
Column totals 13/8 19/6 15
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Step 2 Divide each element of the matrix by its
column total.
Comfort Car A Car B Car C
Car A 8/13 12/19 8/15
Car B 4/13 6/19 6/15
Car C 1/13 1/19 1/15
Note that all columns in the normalized pairwise
comparison matrix now have a sum of 1.
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Step 3 Average the elements in each row. (The
values in the normalized pairwise comparison
matrix have been converted to decimal form.)
Comfort Car A Car B Car B Car C
Car A 0.615 0.632 0.533 0.593
Car B 0.508 0.316 0.400 0.341
Car C 0.677 0.053 0.067 0.066
Total 1.000
This synthesis provides the relative priorities
for the three cars with respect to the comfort
criterion. Thus, we see that, considering
comfort, the must preferred car is car A (with a
priority of 0.593). Car B (with a priority of
0.341) is second, followed by car C (with a
priority of 0.066).
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Consistency A key step in the AHP is the
establishment of priorities through the use of
the pairwise comparison procedure. An important
consideration in terms of the quality of the
ultimate decision relates to the consistency of
judgments that the decision maker demonstrated
during the series of pairwise comparisons. For
example, consider a situation involving the
comparison of three job offers with respect to
the salary criterion. Suppose that the following
pairwise comparison matrix was developed.
Salary Job 1 Job 2 Job 3
Job 1 1 2 8
Job 2 1/2 1 3
Job 3 1/8 1/3 1
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• The interpretation of the preference scores is
  that the preference for job 1 is twice the
  preference for job 2, and the preference for job
  2 is three limes the preference for job 3.
• Using these two pieces of information, we would
  logically concIude that the preference for job 1
  should be 2 x 3 6 times the preference for job
  3.
• The fact that the pairwise comparison matrix
  showed a preference of instead of 6 indicates
  that some lack of consistency exists in the
  pairwise comparisons.
• However, it has to be realized that perfect
  consistency is very difficult to achieve and that
  some lack of consistency is expected to exist in
  almost any set of pairwise comparisons.
• To handle the consistency question, the AHP
  provides a method for measuring the degree of
  consistency among the pairwise judgments provided
  by the decision maker, If the degree of
  consistency is acceptable, the decision process
  can continue. However, if the degree of
  consistency is unacceptable, the decision maker
  should reconsider and possibly revise the
  pairwise comparison judgments before proceeding
  with the analysis.

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• The AHP provides a measure of the consistency of
  pairwise comparison judgments by computing a
  consistency ratio.
• This ratio is designed in such a way that values
  of the ratio exceeding 0.10 are indicative of
  inconsistent judgments in such cases the
  decision maker would probably want to reconsider
  and revise the original values in the pairwise
  comparison matrix.
• Values of the consistently ratio of 0.10 or less
  are considered to indicate a reasonable level of
  consistency in the pairwise comparisons.
• Although the exact mathematical computation of
  the consistency ratio is beyond the scope of this
  text, an approximation of the ratio can be
  obtained. We will illustrate this computational
  procedure for the car-selection problem by
  considering Daves pairwise comparison for the
  comfort criterion.

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Estimating the Consistency Ratio Step 1
Multiply each value in the first column of the
pairwise comparison matrix by the relative
priority of the first item considered multiply
each value in the second column of the matrix by
the relative priority of the second item
considered multiply each value in the third
column of the matrix by the relative priority of
the third item considered. Sum the values across
the rows to obtain a vector of values labeled
weighted sum. This computation for the
car-selection example is
Weighted Sum Vector
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Step 2 Divide the elements of the vector of
weighted sums obtained in 1 by the corresponding
priority value. For the car-selection example, we
obtain
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Step 3 Compute the average of the values
computed in step 2 this average is denoted by
?max. For the car-selection example, we obtain
Step 4 Compute the consistency index (CI), which
is defined us follows
Where n the number of items being compared.
For the car-selection example with n 3, we
obtain
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Step 5 Compute the consistency ratio (CR), which
is defined as follows
where RI, the random index, is the consistency
index of a randomly generated pairwise comparison
matrix, It can be shown that RI depends on the
number of elements being compared and takes on
the following values
n 3 4 5 6 7 8
RI 0.58 0.9 1.12 1.24 1.32 1.41
Thus, for our car- example with n 3 and RI
0.5 we obtain the following consistency ratio
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Other Pairwise Comparisons for the Car-Selection
Example In continuing with the AHP analysis of
the car-selection problem, we need to use the
pairwise comparison procedure to determine the
priorities of the three cars in terms of the
purchase price, MPG, and style criteria. This
requires that Dave express pairwise comparison
preferences for the cars, considering each of
these criteria one at a time. Daves preferences
are summarized in the pairwise comparison
matrices shown.
Price Car A Car B Car C
Car A 1 1/3 1/4
Car B 3 1 1/2
Car C 4 2 1
MPG Car A Car B Car C
Car A 1 1/4 1/6
Car B 4 1 1/3
Car C 6 3 1
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Style Car A Car B Car C
Car A 1 1/3 4
Car B 3 1 7
Car C ¼ 1/7 1

• The interpretation of the numerical values in the
  earlier tables is the same as the interrelation
  of the preference values we observed for the
  comfort criterion. For example, consider the
  comparison of car A and car B in terms of the
  purchase price criterion. Car B (11,200) is
  considered more preferable than car A (13,100).
• In fact, the pairwise comparison matrix shows
  Daves preference for car B is three times
  greater than his preference for car A in terms of
  purchase price. Similarly, car A is only ¼ as
  preferred as car B. Recall that the pairwise
  comparison matrix is set up to show the
  preference of the item in row i when compared to
  the item in column j

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• Following the same synthesis procedure that we
  used for the comfort criterion, the priority
  vectors for these criteria can be computed. The
  result of this synthesis is shown below.

Price MPG Style

• In interpreting these priorities we see that car
  C is the most preferable in terms of purchase
  price (0.557) and miles per gallon (0.639). Car B
  is the most preferable in terms of style (0.655).
  
• No car is the most preferred with respect to all
  criteria. Thus, before a final decision can be
  made, we must assess the relative importance of
  the criteria.

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• In addition to the pairwise comparisons for the
  decision alternatives, we must use the same
  pairwise comparison procedure to set priorities
  for all four criteria in terms of the importance
  of each in contributing toward the overall goal
  of selecting time best car.
• To develop this final pairwise comparison matrix,
  Dave would have to specify how important he
  thought each criterion was compared to each of
  the other criteria.
• In order to do this, six pairwise judgments have
  to be made purchase price compared to MPG
  purchase compared to comfort purchase price
  compared to style MPG compared to comfort MPG
  compared to style and comfort compared to style.
  
• For example, in the pairwise comparison of the
  purchase price and MPG criteria, Dave indicated
  that purchase price was moderately more important
  than MPG. Using the AHP 9-point numerical rating
  scale, a value of 3 was recorded to show the
  higher importance of the purchase-price
  criterion.

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The summary of the pairwise comparison matrix
preferences for the four criteria is shown in
table below.
Criterion Price MPG Comfort Style
Price 1 3 2 2
MPG 1/3 1 ¼ ¼
Comfort ½ 4 1 ½
Style ½ 4 2 1
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• The synthesization process described earlier in
  this section can now be used to convert the
  pairwise comparison information into the
  priorities for the four criteria. The results
  obtained are as follows

Criteria Priorities
Price 0.398
MPG 0.085
Comfort 0.218
Style 0.299

• We see that the purchase price (0.398) has been
  identified as the highest-priority or most
  important criterion in the car-selection
  decision. Style (0.299) and comfort (0.218) rank
  next in importance. MPG (0.085) is a relatively
  unimportant criterion in terms of the overall
  goal of selecting the best car.

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• Using The AHP To Develop An Overall Priority
  Ranking
• A matrix that summarizes the priorities for each
  car in terms of each criterion is given below.
  This matrix is referred to as the priority matrix.

Price MPG Comfort Style
Car A 0.123 0.087 0.593 0.265
Car B 0.320 0.274 0.341 0.655
Car C 0.557 0.639 0.066 0.080

• The overall priority for each decision
  alternative is obtained by summing the product of
  the criterion priority times the priority of the
  decision alternative with respect to that
  criterion. Recall that the criterion priorities
  were found to be 0.398 for purchase price, 0.085
  for MPG, 0.218 for comfort, and 0.299 for style.

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Thus, the computation of the overall priority for
car A is as follows Overall car A priority
0.398(0.123) 0.085(0.087) 0.218(0.593)
0.299(0.265) 0.265 Repeating this
calculation for cars B and C provides their
overall priorities as follows Overall car B
priority 0.398(0.320) 0.085(0.274)
0.218(0.341) 0.299(0.655)
0.421 Overall car C priority 0.398(0.557)
0.085(0.639) 0.218(0.066)
0.299(0.080) 0.314
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• Ranking these priority values, we have the
  following AHP ranking of the decision
  alternatives

Alternative Priority
Car B 0.421
Car C 0.314
Car A 0.265
Total 1.000

• These results provide a basis for Dave to make a
  decision regarding the purchase of a car. Based
  on the AHP priorities, Dave should select car B.
• If Dave believes that the judgments that he has
  made regarding the importance of the criteria and
  his preferences for the cars in terms of the
  criteria are valid, then the AHP priorities show
  that car B is the preferred car.

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• USING EXPERT CHOICE TO IMPLEMENT THE AHP
• Expert Choice (EC), a software package marketed
  by Decision Support Software, provides a
  user-friendly procedure for implementing the AHP
  on a microcomputer. We now provide an
  introduction to this software package by showing
  how it cart be used to compute the priorities for
  the car-selection problem.
• Expert Choice enables the user to simply
  construct a graphical representation of the
  hierarchy. For example, to create the hierarchy
  for the car-selection example, the user selects
  the option to develop a new application what
  appears on the computers monitor is a request to
  define the overall goal.
• After the user defines the overall goal, a
  rectangular box, or node, appears on the screen,
  with the goal description written directly above
  it.

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The user selects the EDIT command and then the
INSERT option another rectangular box or node
appears below the goal node, and the user now
types the name of a criterion, which will be
entered inside the box. This process continues
until all four criterion nodes have been
specified. The figure given shows the partial
hierarchy appearing on the computer screen after
the four criteria have been specified.
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• In the figure we see that in addition to the
  names of each criterion , the criterion nodes
  also contain the decimal value of 0.250.
• This value represents the initial weight, or
  priority, given to each criterion at the start of
  the EC session.
• The user can now continue the process of using
  the EDIT command with the INSERT option to define
  the decision alternative nodes associated with
  each of the criterion nodes.
• In the following figure we show the result of
  defining the decision alternative nodes for the
  price criterion now that since there are three
  alternatives, the initial priorities are set at
  0.333.

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• A similar set of decision alternatives is then
  identified for each of the other three criteria.
• Once the user has developed the complete
  hierarchy for the problem, he or she can focus on
  any particular part of time hierarchy through
  time use of the REDRAW command.

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• In fact, to show the detail displayed in the
  earlier figure, all we did was to point to the
  price node (using the arrows on the keyboards
  numeric key pad) and then type R for redraw.
• Our intent here is not to attempt to show you how
  to use EC but merely to let you develop some
  appreciation for the ease with which the analysis
  can be performed using this software package.
• Now that the hierarchy has been input to EC, we
  are ready to begin developing the pairwise
  comparisons needed to establish priorities for
  the decision alternatives.
• In order to illustrate the type of approach used,
  we moved back to the goal node with EC and then
  selected time COMPARE command by typing C.
• After selecting the option to make comparisons
  based on the importance of the decision criteria,
  the EC system begins to go through the pairwise
  comparison analysis.

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• One portion of this analysis, which shows the
  approach used by EC to establish the comparative
  importance between the purchase price and MPG
  criteria, is shown in following figure.
• Note that this figure indicates to time EC system
  that price is moderately more important than MPG.
  This process continues until all the entries in
  the pairwise comparison matrix for criteria have
  been developed.
• The synthesization process is then performed to
  compute the priorities for the criteria.
• The process of entering pairwise preferences for
  the cars relative to each of the criteria was
  then performed in a similar manner.
• The overall decision was then arrived at by
  entering the command S which is an abbreviation
  for synthesizing this command is used only when
  we have entered all the data for the pairwise
  comparison matrices and want to obtain an overall
  prioritization of the decision alternatives.

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The priorities that were obtained after
synthesization

• The following figure shows the results obtained.
  Note that the results indicate that the final
  priority for car B, the most preferable, is
  0.422.
• The EC system is a very helpful software package
  in performing the multiple-criteria decision
  analysis of the AHP.
• In addition to providing the overall priorities
  for the decision alternatives, EC has the
  capability of doing what if types of analyses,
  where the decision maker can begin to learn how
  the overall priorities for the decision
  alternative are affected by changes in the
  preference input data.

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