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M O D U L E
Analytic Hierarchy Process
TEACHING SUGGESTIONS
Factor evaluations:
Teaching Suggestion M1.1: Using Multifactor
Decision-Making Techniques.
Many decisions students make involve a number of factors. Thus,
multifactor decision-making techniques can be useful and practical. This section can be started by having students give examples
of decisions that require the analysis of multiple factors. Buying a
car or stereo and picking the best job offer are examples. Once students understand the principles of multiplying factor weights
times factor evaluations, they will be able to understand the use of
AHP.
Teaching Suggestion M1.2: Using AHP.
Have the students describe situations where AHP would be preferred over the multifactor evaluation process. You may want to
take one of these situations and show how pairwise comparisons
can be made. Students can then be asked to complete the AHP
problem and determine the best solution. This can lead to in-class
discussions on the AHP process.
SOLUTIONS TO QUESTIONS AND PROBLEMS
M1-1. Multifactor decision making is appropriate when a decision involves a number of factors. Deciding to buy a house, for example, can involve the price, location, taxes, utilities, and so forth.
M1-2. When using multifactor decision making, each factor receives an importance weight. These weights will sum to 1. Then
every alternative and factor combination will receive a factor evaluation. The factor weights are multiplied by the factor evaluations
to get a weighted evaluation for each alternative. The alternative
with the highest weighted evaluation is selected.
M1-3. The analytic hierarchy process should be used when it is
difficult or impossible to determine factor weights and factor evaluations subjectively. In this case, pairwise comparisons are performed to assist in the decision-making process and determine the
best alternative.
M1-4.
Here is an analysis of George’s decision.
Factor weights:
Factor
Importance (Weight)
Price
Color
Warranty
Size
Brand name
0.4
0.1
0.1
0.1
0.3
Factor
Sun
Hitek
Surgo
Price
Color
Warranty
Size
Brand name
0.7
0.9
0.8
0.8
0.9
0.6
0.9
0.9
0.8
0.9
0.8
0.4
0.4
0.2
0.6
Evaluation of SUN:
Factor
Name
Factor
Rating
Factor
Evaluation
Weighted
Evaluation
Price
Color
Warranty
Size
Brand name
Total
0.4
0.1
0.1
0.1
0.3
1.0
0.7
0.9
0.8
0.8
0.9
0.28
0.09
0.08
0.08
0.27
0.80
Factor
Name
Factor
Rating
Factor
Evaluation
Weighted
Evaluation
Price
Color
Warranty
Size
Brand name
Total
0.4
0.1
0.1
0.1
0.3
1.0
0.6
0.9
0.9
0.8
0.9
0.24
0.09
0.09
0.08
0.27
0.77
Evaluation of HITEK:
Evaluation of SURGO:
Factor
Name
Factor
Rating
Factor
Evaluation
Weighted
Evaluation
Price
Color
Warranty
Size
Brand name
Total
0.4
0.1
0.1
0.1
0.3
1.0
0.8
0.4
0.4
0.2
0.6
0.32
0.04
0.04
0.02
0.18
0.60
SUN is selected, with the highest total weighted evaluation of
0.80.
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ANALYTIC HIERARCHY PROCESS
Consistency information follows:
Linda’s problem can be analyzed as follows:
Price
Car 1
Car 1
Car 2
Car 3
Car 2
Car 3
2
7
4
Weighted sum vector ⫽ (0.7096 2.0468 0.2460)
Consistency vector ⫽ (3.0011 3.0031 3.0004)
Lambda ⫽ 3.0015
Value of CI ⫽ 0.0008
RI ⫽ 0.5800
The following will be the priorities for price:
CR ⫽ 0.0013
Priority for car 1 is 0.6025.
Priority for car 2 is 0.3151.
Priority for car 3 is 0.0824.
M1-8.
Factors
Consistency information follows:
Price
Price
Warranty
Style
Weighted sum vector ⫽ (1.8096 0.9460 0.2473)
Consistency vector ⫽ (3.0035 3.0019 3.0005)
Warranty
Style
2
9
6
Lambda ⫽ 3.0020
The following will be the priorities for the factors:
Value of CI ⫽ 0.0010
Priority for price is
0.6049.
Priority for warranty is 0.3337.
Priority for style is
0.0614.
RI ⫽ 0.5800
CR ⫽ 0.0017
M1-6.
Consistency information follows:
Warranty
Car 1
Car 1
Car 2
Car 3
1
3
1
8
Weighted sum vector ⫽ (1.8246 1.0044 0.1842)
Consistency vector ⫽ (3.0163 3.0097 3.0016)
Value of CI ⫽ 0.0046
1
5
Car 2
RI ⫽ 0.5800
Car 3
CR ⫽ 0.0079
The following are the final rankings—Car 1 is selected.
The following will be the priorities for warranty:
Priority for car 1 is 0.0768.
Priority for car 2 is 0.1863.
Priority for car 3 is 0.7370.
Consistency information follows:
Weighted sum vector ⫽ (0.2310 0.5640 2.2825)
Consistency vector ⫽ (3.0088 3.0276 3.0972)
Lambda ⫽ 3.0445
RI ⫽ 0.5800
CR ⫽ 0.0384
M1-7.
Car 1
Car 1
Car 2
Car 2
Car 3
1
3
3
8
Car 3
The following will be the priorities for style:
Priority for car 1 is 0.2364.
Priority for car 2 is 0.6816.
Priority for car 3 is 0.0820.
Ranking
0.4045
0.2946
0.3008
M1-9. The weighted averages of these scores are shown in the
table. Gina should choose Univesity B.
Value of CI ⫽ 0.0223
Style
Item
Car 1
Car 2
Car 3
Weight
Cost
0.6
Reputation
0.2
A
B
C
4
8
7
9
5
6
M1-10.
Quality of life
0.2
Weighted
Average
7
7
3
5.6
7.2
6.0
Using AHP, we have the following matrices.
Cost
A
B
C
Column Total
A
1
5
3
9
Normalized
A
B
C
Factor Evaluation
(Row Average)
0.1111
0.5556
0.3333
0.1304
0.6522
0.2174
0.0769
0.6923
0.2308
0.1062
0.6333
0.2605
A
B
C
B
C
0.2
0.333333
1
3
0.333333
1
1.533333 4.333333
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Reputation
A
A
1
B
0.142857
C
0.2
Column Total 1.342857
Normalized
A
B
C
A
0.7447
0.1064
0.1489
B
7
1
3
11
C
0.7895
0.0526
0.1579
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The following will be the priorities for price:
C
5
0.333333
1
6.333333
B
0.6364
0.0909
0.2727
ANALYTICAL HIERARCHY PROCESS
Priority for system 1 (S-1) is 0.6039.
Priority for system 2 (S-2) is 0.3258.
Priority for system 3 (S-3) is 0.0703.
Consistency information follows:
Factor Evaluation
(Row Average)
0.7235
0.0833
0.1932
Weighted sum vector ⫽ (1.8178 0.9792 0.2109)
Consistency vector ⫽ (3.0099 3.0056 3.0011)
Lambda ⫽ 3.0055
Value of CI ⫽ 0.0028
RI ⫽ 0.5800
CR ⫽ 0.0048
Quality of Life
A
B
C
Column Total
Normalized
A
B
C
A
1
1
0.2
2.2
B
1
1
0.142857
2.142857
C
5
7
1
13
A
B
C
0.4545
0.4545
0.0909
0.4667
0.4667
0.0667
0.3846
0.5385
0.0769
Brand Name
S-1
S-1
S-2
S-3
Factor Evaluation
(Row Average)
0.4353
0.4866
0.0782
S-2
S-3
1
6
4
The following will be the priorities for brand name:
Priority for system 1 (S-1) is 0.4838.
Priority for system 2 (S-2) is 0.4232.
Priority for system 3 (S-3) is 0.0930.
Consistency information follows:
Weighted sum vector ⫽ (1.4649 1.2789 0.2794)
Factors
Cost
Reputation
Quality of life
Column Total
Cost
1
0.333333
0.142857
1.47619
Reputation
3
1
0.5
4.5
Consistency vector ⫽ (3.0278 3.0220 3.0051)
Quality of life
7
2
1
10
Lambda ⫽ 3.0183
Value of CI ⫽ 0.0092
RI ⫽ 0.5800
CR ⫽ 0.0158
Normalized
Cost
Reputation
Quality of life
Quality
Cost Reputation of life
0.6774
0.6667
0.7000
0.2258
0.2222
0.2000
0.0968
0.1111
0.1000
Factor Evaluation
(Row Average)
0.6814
0.2160
0.1026
Using the factor weights, we find the following weighted averages
for each university.
Memory
S-1
S-1
S-2
S-3
1
2
1
7
1
6
S-2
S-3
The following will be the priorities for memory:
A
B
C
Weights
Cost Reputation Quality
of life
0.1062
0.7235
0.4353
0.6333
0.0833
0.4866
0.2605
0.1932
0.0782
0.6814
0.2160
0.1026
Weighted
Average
0.2733
0.4995
0.2272
Priority for system 1 (S-1) is 0.0919.
Priority for system 2 (S-2) is 0.1535.
Priority for system 3 (S-3) is 0.7545.
Consistency information follows:
Weighted sum vector ⫽ (0.2765 0.4631 2.3192)
Consistency vector ⫽ (3.0078 3.0164 3.0736)
Therefore, Gina should choose University B.
Lambda ⫽ 3.0326
Value of CI ⫽ 0.0163
RI ⫽ 0.5800
M1-11. The analysis to determine which computer system is to
be selected is as follows:
Price
S-1
S-2
S-3
S-1
S-2
2
S-3
8
5
CR ⫽ 0.0281
Speed
S-1
S-2
S-3
S-1
S-2
S-3
1
3
2
5
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Table for Factors for Problem M1-11
Factors
Price
Brand
Name
Memory
Speed
Flexibility
PC
Compatibility
9
4
5
1
3
2
1
4
1
5
2
1
2
1
6
1
3
1
6
Price
Brand name
1
2
Memory
Speed
1
2
Flexibility
PC compatible
Consistency information follows:
The following will be the priorities for speed:
Weighted sum vector ⫽ (2.1717 0.2371 0.6218)
Priority for system 1 (S-1) is 0.2299.
Priority for system 2 (S-2) is 0.6479.
Priority for system 3 (S-3) is 0.1222.
Consistency vector ⫽ (3.0389 3.0040 3.0122)
Lambda ⫽ 3.0184
Consistency information:
Value of CI ⫽ 0.0092
Weighted sum vector ⫽ (0.6902 1.9485 0.3667)
RI ⫽ 0.5800
Consistency vector ⫽ (3.0026 3.0071 3.0013)
CR ⫽ 0.0158
Lambda ⫽ 3.0037
The following will be the weights for the factors:
Value of CI ⫽ 0.0018
RI ⫽ 0.5800
CR ⫽ 0.0032
Flexibility
S-1
S-1
S-2
S-3
1
2
1
8
1
4
S-2
Weight for price is
Weight for brand name is
Weight for memory is
Weight for speed is
Weight for flexibility is
Weight for PC compatibility is
0.3849
0.0447
0.0816
0.0514
0.149
0.288
See the table for factors for Problem M1-11.
Consistency information follows:
冢
S-3
Weighted sum vector ⫽ 2.39 0.275
0.312 0.918
The following will be the priorities for flexibility:
Priority for system 1 (S-1) is 0.0909.
Priority for system 2 (S-2) is 0.1818.
Priority for system 3 (S-3) is 0.7273.
Consistency vector =
6.2208
冢6.0592
CR ⫽ 0.0232
Consistency vector ⫽ (3.0000 3.0000 3.0000)
The following are the final rankings—system 1 (S-1) is
selected.
Lambda ⫽ 3.0000
Value of CI ⫽ 0.0000
RI ⫽ 0.5800
Item
Ranking
CR ⫽ 0.0000
System 1 (S-1)
System 2 (S-2)
System 3 (S-3)
0.4928
0.2400
0.2671
S-1
S-2
S-3
8
S-2
S-3
The following will be the priorities for PC compatibility:
Priority for system 1 (S-1) is 0.7146.
Priority for system 2 (S-2) is 0.0789.
Priority for system 3 (S-3) is 0.2064.
冣
6.1480 6.0362
6.1485 6.2518
RI ⫽ 1.2400
Weighted sum vector ⫽ (0.2727 0.5455 2.1818)
S-1
冣
Value of CI ⫽ 0.0288
Consistency information follows:
PC Compatibility
0.493
1.801
4
1
3