Ahp calculations

1. 1
The Analytic Hierarchy Process (AHP)
Presented by
Jeff Kunz
to the
Eagle City Hall Location Options Task Force
February/March 2010

2. What is the Analytic Hierarchy Process (AHP)?
What is its purpose? How does it work?
`
“A structured technique for dealing with complex decisions. […] Based on
mathematics and psychology, it was developed by Thomas L. Saaty in the
1970s and has been extensively studied and refined since then. The AHP
provides a comprehensive and ration al framework for structuring a
decision problem, for representing and quantifying its elements, for relating
those elements to overall goals, and for evaluating alternative solutions.It
is used around the world in a wide variety of decision situations, in fields
such as government, business, indus try, healthcare and education.” - Wikipedia
`
“The purpose of the AHP is to assist people in organizing their thoughts
and judgments to make more effective decisions. […] The Analytic
Hierarchy Process (AHP) provides the objective mathematics to process
the inescapably subjective and personal preferences of an individual or
group in making decisions. […] Fundamentally, the AHP works by
developing priorities for alternatives and the criteria used to judge the
alternatives. […] First, priorities are derived for the criteria in terms of their
importance to achieve the goal, then priorities are derived for the
performance of the alternatives on each criterion. These priorities are
derived based on pair-wise assessments using judgments, or ratios of
measurements from a scale if one exists.[…] Finally, a weighting and
adding process is used to obtain overall priorities for the alternatives as to
how they contribute to the goal.”
- Saaty & Vargas, Models, Methods, Concepts & Applications of the Analytic Hierarchy
Process
, pp.12 & 27.
2

3. Different Forms of the
Analytic Hierarchy Process (AHP) Exist.
`
Different forms of the Analytic Hierarchy Process (AHP) exist. This
presentation is based on the “original” AHP version developed by Dr.
Thomas L. Saaty. - Saaty, Thomas L. “Priority Setting in Completing Problems.” IEEE Transactions on
Engineering Management
. Volume 30. Number 3. August 1983. Pages 140-155.
`
The “modified” AHP version normaliz es the pair-wise comparison values within each of the matrices and then averages the values in each row to
get the corresponding weights and ratings, whereas the “original” AHP
version calculates the nth root of the product of the pair-wise
comparison values in each row of the matrices and then normalizes the
aforementioned nth root of products to get the corresponding weights
and ratings.
3

4. AHP is a “Heuristic Algorithm.”
What is a “Heuristic Algorithm?”
`
The Analytic Hierarchy Process (AHP) is an example of a “heuristic
algorithm.”
`
A “heuristic” is an “intuitive rule of thumb for dealing with some aspect of
a model [or problem].” - Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page CD12-3.
`
An “algorithm” is “a procedure for solving a mathematical problem(as of
finding the greatest common divisor)in a finite number of steps that
frequently involves repetition of an operation; broadly: a step-by-step
procedure for solving a problem or accomplishing some end, especially by a
computer.”
- Merriam-Webster Online
`
A “heuristic algorithm” is one that “provides good approximate [but not
necessarily optimal] solutions to a given model [or problem]. Often (but by no
means always) in employing such an algorithm one may be able to
precisely measure the ‘goodness’ of the approximation.” - Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page CD12-2.
4

5. Basic AHP Procedure.
`
Step 1. Develop the weights for the criteria by
`
developing a single pair-wise comparison matrix for the criteria;
`
multiplying the values in each row together and calculating the nth root of
said product;
`
normalizing the aforementioned nth root of products to get the appropriate
weights; and
`
calculating and checking the Consistency Ratio (CR).
`
Step 2. Develop the ratings for each decision alternative for each criterion by
`
developing a pair-wise comparison matrix for each criterion, with each matrix
containing the pair-wise comparisons of the performance of decision
alternatives on each criterion;
`
multiplying the values in each row together and calculating the nth root of
said product;
`
normalizing the aforementioned nth root of product values to get the
corresponding ratings; and
`
calculating and checking the Consistency Ratio (CR).
`
Step 3. Calculate the weighted average rating for each decision alternative.
Choose the one with the highest score. - Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page CD12-26.
5

6. An AHP Example:
Judy Grim’s Computer Decision.
`
Judy Grim is looking for a new computer system for her small business.
`
She has determined that the most importa nt overall criteria (aka factors) are
hardware, software and vendor support.
`
Furthermore, she has narrowed down her decision alternatives to three possible
computer systems, labeled as SYSTEM-1, SYSTEM-2 and SYSTEM-3. - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
6

7. An AHP Example:
Judy Grim’s Decision Hierarchy.
`
For illustration purposes, Judy Grim places her criteria and alternative computer
systems into a decision hierarchy (as shown below).
`
The decision hierarchy for her computer selection has three different levels.
`
The top level of the hierarchy describes the overall decision, which is to
select the best computer system for her small business.
`
The middle level of the hierarchy describes the decision criteria that are to
be considered: hardware, software and vendor support.
`
The lower level of the hierarchy reveals the alternative computer systems:
SYSTEM-1, SYSTEM-2 and SYSTEM-3.
- Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
7

8. The Key To AHP Usage: Pairwise Comparisons.
`
As the decision-maker, Judy Grim, will use pair-wise comparisons to establish
the relative priority of each criterion against every other criterion as well as the
relative priority of each system against every other system for each criterion.
`
The pair-wise comparisons use a scale that ranges from equally preferred to
extremely preferred. Here is the scale: (IMPORTANT NOTE: Reciprocal
relationships are also possible – the reciprocal of an integer nis equal to 1/n.)
Examples:
Suppose that Judy decides that software is “very to extremely strongly
preferred” over hardware. According to the scale above, she should use a value
of “8” to denote the preference of this pair-wise comparison.
Suppose that Judy decides that hardware is “moderately non-preferred” over
vendor support. According to the scale above, she should use a value of “1/3” to
denote the non-preference of this pair-wise comparison. - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
1−Equally preferred
2−Equally to moderately preferred
3−Moderately preferred
4−Moderately to strongly preferred
5−Strongly preferred
6−Strongly to very strongly preferred
7−Very strongly preferred
8−Very to extremely strongly preferred
9−Extremely preferred 8

9. Step 1: Develop the Weights for the Criteria.
`
1. Develop the weights for the criteria by
`
developing a single pair-wise comparison matrix for the criteria ;
`
multiplying the values in each row together and calculating the nth root
of said product;
`
normalizing the aforementioned nth root of products to get the
appropriate weights; and
`
calculating and checking the Consistency Ratio (CR).
- Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page
CD12-26.
`
In comparing the three criteria – hardware, software and vendor support, Judy
has determined that software is the most important. Specifically, `
Software is “very to extremely strongly preferred” over hardware (number 8).
`
Software is “moderately preferred” over vendor support (number 3).
`
In comparing vendor support to hardware, she has determined that vendor
support is more important. Specifically, `
Vendor support is “moderately preferred” over hardware (number 3).
`
With the aforementioned pair-wise comparison values, Judy can now construct
the pair-wise comparison matrix and then compute the weights for hardware,
software and vendor support. (Refer to next slide.) - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
9

10. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
The 3x3 matrix above contains all of the pair-wise comparisons for the criteria.
(Since there are three criteria, th e matrix must be of size 3x3.)
`
Remember: `
Software is “very to extremely strongly preferred” over hardware (number 8),
as shown in the second row, first column of the matrix.
`
Software is “moderately preferred” over vendor support (number 3), as shown
in the second row, third column of the matrix.
`
Vendor support is “moderately preferred” over hardware (number 3), as
shown in the third row, first column of the matrix.
`
The “equally preferred” values shown along the upper-left to lower-right
diagonal are comparing each criteria to itself and so, by definition, must be
equal to one.
-
Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
Hardware Software
Vendor
support
Hardware 1.0000.125 0.333
Software 8.000 1.0003.000
Vendor support3.000 0.333 1.000
10

11. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
The remaining values shown in the matrix represent the reciprocalpair-wise
comparisons of relationships previously mentioned. Specifically, `
Hardware is “very to extremely strongly non-preferred” over software (fraction
1/8, or 0.125), as shown in the first row, second column of the matrix.
(Because software is “very to extremely strongly preferred” over hardware
(number 8), consistency requires the reciprocal relationship be expressed as
hardware is “very to extremely strongly non-preferred” over software (fraction
1/8, 0r 0.125).)
`
Hardware is “moderately non-preferred” over vendor support (fraction 1/3, or
0.333), as shown in the first row, third column of the matrix.
`
Vendor support is “moderately non-preferred” over software (fraction 1/3, or
0.333), as shown in the third row, second column of the matrix.
`
The values shown in blue (in the upper right-hand corner of the matrix) are the
only values the decision-maker must enter into the Excel spreadsheet. All other
values are automatically calculated and correctly placed into the spreadsheet.
(Doing so reduces the number of pair-wise comparisons the decision-maker
must manually enter while also reducing human error related to data entry.) - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
Hardware Software
Vendor
support
Hardware 1.0000.125 0.333
Software 8.000 1.0003.000
Vendor support3.000 0.333 1.000
11

12. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
1. Develop the weights for the criteria by
`
developing a single pair-wise comparison matrix for the criteria;
`
multiplying the values in each row together and calculating the nth
root of said product;
`
normalizing the aforementioned nth root of products to get the
appropriate weights; and
`
calculating and checking the Consistency Ratio (CR).
- Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page
CD12-26.
`
Note the column labeled “3
rd
root of product” in the matrix shown above. The
third-root-of-product values in each row are calculated as follows: (Again, the
third-root-of-product is calculated because there are three criteria.)
`
Hardware: (1.000 x 0.125 x 0.333) = (0.042)
(1/3)
= 0.347;
`
Software: (8.000 x 1.000 x 3.000) = (24.000)
(1/3)
= 2.884;
`
Vendor Support: (3.000 x 0.333 x 1.000) = (1.000)
(1/3)
= 1.000.
`
Each of the aforementioned third-root-of-product values are then added
together to equal 4.231. - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
Hardware Software
Vendor
support
3rd root of
product
Hardware 1.0000.125 0.3330.347
Software 8.000 1.0003.0002.884
Vendor support3.000 0.333 1.000 1.000
4.231
12

13. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
1. Develop the weights for the criteria by
`
developing a single pair-wise comparison matrix for the criteria;
`
multiplying the values in each row together and calculating the nth root
of said product;
`
normalizing the aforementioned nth root of products to get the
appropriate weights; and
`
calculating and checking the Consistency Ratio (CR).
- Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page
CD12-26.
`
Note the column labeled “Priority vector” in the matrix above. The third-root-of-
product values (and total) from the previous step will be normalized to get the
appropriate weights for each criteria. The weights for each criteria are calculated
as follows: (The “Priority vector” values are the criteria weights.) `
Hardware: (0.347 / 4.231) = 0.082;
`
Software: (2.884 / 4.231) = 0.682;
`
Vendor Support: (1.000 / 4.231) = 0.236.
`
Note when calculated correctly, the weights for each criteria must sum to one. - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
Hardware Software
Vendor
support
3rd root of
product
Priority
vec t or
Hardware 1.0000.125 0.3330.347 0.082
Software 8.000 1.0003.0002.884 0.682
Vendor support3.000 0.333 1.000 1.000 0.236
4.231 1.000
13

14. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
1. Develop the weights for the criteria by
`
developing a single pair-wise comparison matrix for the criteria;
`
multiplying the values in each row together and calculating the nth root of said
product;
`
normalizing the aforementioned nth root of products to get the appropriate
weights; and
`
calculating and checking the Consistency Ratio (CR).
- Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page CD12-26.
`
The Consistency Ratio (CR) tells the decision-maker how consistent he/she has been
when making the pair-wise comparisons. Calculating the Consistency Ratio (CR) is a
four-step process. `
First, the pair-wise comparison values in each column are added together (as the
“Sum” values) and each sum is then multiplied by the respective weight (from the
“Priority vector” column) for that criteria. Specifically,
`
Hardware: (1.000 + 8.000 + 3.000) = 12.000 x 0.082 = 0.983;
`
Software: (0.125 + 1.000 + 0.333) = 1.458 x 0.682 = 0.994;
`
Vendor Support: (0.333 + 3.000 + 1.000) = 4.333 x 0.236 = 1.024.
`
- Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
Hardware Software
Vendor
support
3rd root of
product
Priority
vec t or
Hardware 1.0000.125 0.3330.347 0.082
Software 8.000 1.0003.0002.884 0.682
Vendor support3.000 0.333 1.000 1.000 0.236
Sum 12.000 1.458 4.333 4.231 1.000
Sum*PV 0.983 0.994 1.024 3.002
14

15. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
Calculating and checking the Consistency Ratio (CR) (Cont’d): `
Note the row labeled “Sum*PV” shown in the matrix above. Each value in this
row shows the result of multiplying the respective sum (shown in the row
immediately above) by the respective weight for that criteria (shown in the
column labeled “Priority vector”).
`
Second, the aforementioned values (shown in the row labeled “Sum*PV”) are
added together to yield a total of 3.002 (i.e., 0.983 + 0.994 + 1.024 = 3.002). This
value is known Lambda-max. IMPORTANT NOTE: Unlike the weights for the
criteria which must sum to one, Lambda-max will not necessarily equal one.
`
Third, the Consistency Index (CI) is calculated. The formula is:
CI = (Lambda-max –n) / (n–1)
where nis the number of criteria or systems being compared. In this case, n= 3,
for the three different criteria being compared. For this particular case, the
calculation is:
CI = (3.002 – 3) / (3 – 1) = (0.002) / (2) = 0.001
- Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
15
Hardware Software
Vendor
support
3rd root of
product
Priority
vec t or
Hardware 1.0000.125 0.3330.347 0.082
Software 8.000 1.0003.0002.884 0.682
Vendor support3.000 0.333 1.000 1.000 0.236
Sum 12.000 1.458 4.333 4.231 1.000
Sum*PV 0.983 0.994 1.024 3.002
Lambda max= 3.002
CI = 0.001

16. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
Calculating and checking the Consistency Ratio (CR) (Cont’d): `
Fourth (and lastly), the Consistency Ratio (CR) is calculated by dividing the
Consistency Index (CI) (from the previous step) by a Random Index (RI), which is
determined from a lookup table. The Random Index (RI) is a direct function of
the number of criteria or systems being considered.
`
The table of Random Indices (RI) is shown below:
- Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
Hardware Software
Vendor
support
3rd root of
product
Priority
vec t or
Hardware 1.0000.125 0.3330.347 0.082
Software 8.000 1.0003.0002.884 0.682
Vendor support3.000 0.333 1.000 1.000 0.236
Sum 12.000 1.458 4.333 4.231 1.000
Sum*PV 0.983 0.994 1.024 3.002
Lambda max= 3.002
CI = 0.001
CR = 0.001
16
nRandom Index (RI)
10.00
20.00
30.58
40.90
51.12
61.24
71.32
81.41
91.45

17. Step 1: Develop the Weights for the Criteria.
(Cont’d)
`
Calculating and checking the Consistency Ratio (CR) (Cont’d): `
In general, the Consistency Ratio (CR) is calculated as:
Consistency Ratio (CR) = Consistency Index (CI) / Random Index (RI)
`
In this case, n= 3 because three criteria are being compared, and so the
Random Index (RI) equal to 0.58 (from the table) must be used. Therefore,
CR = CI / RI = 0.001 / 0.58 = 0.001
`
The Consistency Ratio (CR) tells the decision-maker how consistent he/she has
been when making the pair-wise comparisons. A higher number means the
decision-maker has been less consistent, whereas a lower number means the
decision-maker has been more consistent.
`
If the Consistency Ratio (CR) <
0.10, the decision-maker’s pair-wise comparisons
are relatively consistent. (In Judy’s case, the CR equals 0.001, meaning her pair-
wise comparisons are relatively consistent and no corrective action is necessary.)
`
If the Consistency Ratio (CR) > 0.10, the decision-maker should seriously
consider re-evaluating his/her pair-wise comparisons – the source(s) of
inconsistency must be identified and resolved and the analysis re-done.
Ad t d f R d & St i Qtitti Ali f M t 7
th
diti P
Hardware Software
Vendor
support
3rd root of
product
Priority
vec t or
Hardware 1.0000.125 0.3330.347 0.082
Software 8.000 1.0003.0002.884 0.682
Vendor support3.000 0.333 1.000 1.000 0.236
Sum 12.000 1.458 4.333 4.231 1.000
Sum*PV 0.983 0.994 1.024 3.002
Lambda max= 3.002
CI = 0.001
CR = 0.001
17

18. Step 2: Develop the Ratings for Each Decision
Alternative for Each Criterion.
`
2. Develop the ratings for each decision alternative for each criterion by
`
developing a pair-wise comparison matrix for each criterion, with each
matrix containing the pair-wise co mparisons of the performance of
decision alternatives on each criterion;
`
multiplying the values in each row together and calculating the nth root of said
product;
`
normalizing the aforementioned nth root of product values to get the
corresponding ratings; and
`
calculating and checking the Consistency Ratio (CR).
- Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page CD12-26.
`
During the second step of the AHP process, the decision-maker determines the
ratings for each decision alterative for each criterion. There will be one pair-wise
comparison matrix for each criterion. And within each matrix, the pair-wise
comparisons will rate each system relative to every other system. `
For example, Judy has identified three crit eria applicable to her computer decision
– hardware, software and vendor support. Accordingly, she will develop three
separate matrices – one matrix for the hardware, one matrix for the software and
one matrix for the vendor support.
`
Within each of the aforementioned three matrices, there will be pair-wise
comparisons for each system against every other system relative to that criterion.
Since there are three computer systems under evaluation, each matrix must be of
size 3x3.
- Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
18

19. Step 2: Develop the Ratings for Each Decision
Alternative for Each Criterion. (Cont’d)
`
Here are the three matrices to determine the ratings for each decision alternative
(system) for each criterion:
`
Concerning hardware, `
Judy determines that the hardware for computer SYSTEM-1 is “moderately
preferred” to computer SYSTEM-2 (number 3).
`
She determines that the hardware for computer SYSTEM-1 is “extremely
preferred” to computer SYSTEM-3 (number 9).
`
And, finally, she determines the hardware for computer SYSTEM-2 is “strongly
to very strongly preferred” to computer SYSTEM-3 (number 6).
`
The “equally preferred” values shown along the upper-left to lower-right diagonal
are comparing each system to itself and so, by definition, must be equal to one.
`
The remaining values shown in the matrix represent the reciprocalpair-wise
comparisons of relationships previously mentioned.
`
The values shown in blue (in the upper right-hand corner of the matrix) are the
only values the decision-maker must enter into the Excel spreadsheet. All other
values are automatically calculated and correctly placed into the spreadsheet.
19
SYSTEM-1 SYSTEM-2 SYSTEM-3
3rd root of
product
Priority
vec t or
1.0003.000 9.0003.000 0.663
0.333 1.0006.0001.260 0.278
0.111 0.167 1.000 0.265 0.058
Sum 1.444 4.167 16.000 4.524 1.000
Sum*PV 0.958 1.160 0.936 3.054
Lambda max= 3.054
CI = 0.027
CR = 0.046
SYSTEM-1
SYSTEM-2
SYSTEM-3
Hardware

20. Step 2: Develop the Ratings for Each Decision
Alternative for Each Criterion. (Cont’d)
`
Concerning software, `
Judy determines that the software for computer SYSTEM-2 is “equally to
moderately preferred” to computer SYSTEM1 (number 2).
`
She determines that the software for computer SYSTEM-3 is “very to extremely
strongly preferred” to computer SYSTEM-1 (number 8).
`
And, finally, she determines that the software for computer SYSTEM-3 is
“strongly preferred” to SYSTEM 2 (number 5).
`
The “equally preferred” values shown along the upper-left to lower-right diagonal
are comparing each system to itself and so, by definition, must be equal to one.
`
The remaining values shown in the matrix represent the reciprocalpair-wise
comparisons of relationships previously mentioned.
`
The values shown in blue (in the upper right-hand corner of the matrix) are the
only values the decision-maker must enter into the Excel spreadsheet. All other
values are automatically calculated and correctly placed into the spreadsheet. - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
20
SYSTEM-1 SYSTEM-2 SYSTEM-3
3rd root of
product
Priority
vec t or
1.0000.500 0.1250.397 0.087
2.000 1.0000.2000.737 0.162
8.000 5.000 1.000 3.420 0.751
Sum 11.000 6.500 1.325 4.554 1.000
Sum*PV 0.959 1.052 0.995 3.006
Lambda max= 3.006
CI = 0.003
CR = 0.005
SYSTEM-1
SYSTEM-2
SYSTEM-3
Software

21. Step 2: Develop the Ratings for Each Decision
Alternative for Each Criterion. (Cont’d)
`
Concerning vendor support, `
Judy determines that the vendor support for computer SYSTEM-1 is “equally
preferred” to computer SYSTEM-2 (number 1).
`
She determines that the vendor support for computer SYSTEM-1 is “strongly to
very strongly preferred” to computer SYSTEM-3 (number 6).
`
And, finally, she determines that the vendor support for computer SYSTEM-2 is
“moderately preferred” to computer SYSTEM-3 (number 3).
`
The “equally preferred” values shown along the upper-left to lower-right diagonal
are comparing each system to itself and so, by definition, must be equal to one.
`
The remaining values shown in the matrix represent the reciprocalpair-wise
comparisons of relationships previously mentioned.
`
The values shown in blue (in the upper right-hand corner of the matrix) are the
only values the decision-maker must enter into the Excel spreadsheet. All other
values are automatically calculated and correctly placed into the spreadsheet. - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
21
SYSTEM-1 SYSTEM-2 SYSTEM-3
3rd root of
product
Priority
vec t or
1.0001.000 6.0001.817 0.499
1.000 1.0003.0001.442 0.396
0.167 0.333 1.000 0.382 0.105
Sum 2.167 2.333 10.000 3.641 1.000
Sum*PV 1.081 0.924 1.048 3.054
Lambda max= 3.054
CI = 0.027
CR = 0.046
SYSTEM-1
SYSTEM-2
SYSTEM-3
Vendor support

22. Step 2: Determine the Ratings for Each Decision
Alternative For Each Criterion.
`
2. Develop the ratings for each decision alternative for each criterion by
`
developing a pair-wise comparison matrix for each criterion, with each matrix
containing the pair-wise comparisons of the performance of decision
alternatives on each criterion;
`
multiplying the values in each row together and calculating the nth root
of said product;
`
normalizing the aforementioned nth root of product values to get the
corresponding ratings; and
`
calculating and checking the Consistency Ratio (CR).
- Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page
CD12-26.
`
Multiplying the values in each row together and calculating the nth root of said product, normalizing the aforementioned nth root of product values to get the
corresponding ratings, and calculating and checking the Consistency Ratio (CR) are
identical to Step 1. Accordingly, these sub-steps of the second step are omitted.
(Refer to the preceding three slides fo r the aforementioned calculations.)
`
Note that the Consistency Ratios (CRs) for all three of the aforementioned matrices
depicting the ratings for each decision alternative for each criterion are less-than-
or-equal-to 0.10, meaning that Judy’s pair-wise comparisons are relatively
consistent. Therefore, no correction actions are necessary.
- Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
22

23. Step 3: Calculate the Weighted Average Rating
for Each Decision Alternative; Choose the One
With the Highest Score.
`
3. Calculate the weighted average rating for each decision alternative. Choose
the one with the highest score. - Adapted from: Moore & Weatherford, Decision Modeling With Microsoft Excel
, 6
th
edition. Page
CD12-26.
`
In the third and final phase, Judy determines the final scores for each system by (a)
multiplying the criteria weights (from Step 1) by the ratings for the decision
alternatives for each criteria (from Step 2); and (b) summing the respective
products. (This is known as a sum-of-products mathematical operation and AHP
refers to this matrix as the “Principle of Composition of Priorities.”) Here are the
specific calculations:
SYSTEM-1: (0.082 x 0.663) + (0.682 x 0.087) + (0.236 x 0.499) = 0.232;
SYSTEM-2: (0.082 x 0.278) + (0.682 x 0.162) + (0.236 x 0.396) = 0.227; and
SYSTEM-3: (0.082 x 0.058) + (0.682 x 0.751) + (0.236 x 0.105) = 0.542.
`
Judy should select SYSTEM-3 as the “best” computer system for her new business
because it possesses the highest score (at 0.542). - Adapted from: Render & Stair, Quantitative Analysis for Management
, 7
th
edition. Pp. 522-529.
23
Hardware Software
Vendor
supportScore
0.082 0.682 0.236 1.000
SYSTEM-1 0.663 0.087 0.499 0.232
SYSTEM-2 0.278 0.162 0.396 0.227
SYSTEM-3 0.0580.751 0.105 0.542Winne
r
1.000 1.000 1.000 1.000
Criteria
Options

24. References.
[1] Wikipedia, “Analytic Hierarchy Process (AHP).” (Available at
http://en.wikipedia.org/wiki/Analytic_hierarchy_process
.)
[2] Merriam-Webster Online. “Algorithm.” (Available at http://www.merriam-
webster.com/dictionary/algorithm
.)
[3] Saaty, Thomas L. Vargas, Luis G. Models, Methods, Concepts & Applications of the
Analytic Hierarchy Process.
Kluwer Academic Publishers: Norwell, Mass. 2001.
[4] Saaty, Thomas L. “Priority Setting in Completing Problems.” IEEE Transactions on
Engineering Management
. Volume 30. Number 3. August 1983. Pages 140-155.
[5] Moore, Jeffrey H. Weatherford, Larry R. Decision Modeling with Microsoft Excel
, 6
th
edition. Prentice-Hall, Inc.: Upper Saddle River, NJ. 2001.
[6] Render, Barry. Stair Jr, Ralph M. Quantitative Analysis for Management
, 7
th
edition. Prentice-Hall, Inc: Upper Saddle River, NJ. 2000.
24

25. Q&A.
`
Questions?
25