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M
1
O
D
U
L
E
Analytic Hierarchy Process
TEACHING SUGGESTIONS
Factor evaluations:
Teaching Suggestion M1.1: Using Multifactor Decision-Making 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 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.
M1-1.
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.
M1-2.
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 decision-making process and determine the best alternative. M1-3.
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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Linda’s problem can be analyzed as follows:
M1-5.
Price
Car 1
Car 1 Car 2 Car 3
Consistency information follows:
Car 2
Car 3
2
7 4
Weighted sum vector
ϭ
(0.7096 2.0468 0.2460)
Consistency vector
ϭ
(3.0011 3.0031 3.0004)
Lambda
ϭ
3.0015
Value of CI
ϭ
0.0008
RI
ϭ
0.5800
CR
ϭ
0.0013
The following will be the priorities for price: 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: Weighted sum vector
ϭ
(1.8096 0.9460 0.2473)
Consistency vector
ϭ
(3.0035 3.0019 3.0005)
Lambda
ϭ
3.0020
Value of CI
ϭ
0.0010
RI
ϭ
0.5800
CR
ϭ
0.0017
Price
Price Warranty Style
Style
2
9 6
The following will be the priorities for the factors: Priority for price is 0.6049. Priority for warranty is 0.3337. Priority for style is 0.0614.
M1-6.
Consistency information follows:
Warranty
Car 1
Car 1
Car 2
Car 3
Weighted sum vector
ϭ
(1.8246 1.0044 0.1842)
1 3
1 8
Consistency vector
ϭ
(3.0163 3.0097 3.0016)
1 5
Value of CI
ϭ
0.0046
RI
ϭ
0.5800
CR
ϭ
0.0079
Car 2 Car 3
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
ϭ
(0.2310 0.5640 2.2825)
Consistency vector
ϭ
(3.0088 3.0276 3.0972)
Lambda
ϭ
3.0445
Value of CI
ϭ
0.0223
RI
ϭ
0.5800
CR
ϭ
0.0384
Car 1
Car 1 Car 2
Car 3
1 3
3 8
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
Car 1 Car 2 Car 3
0.4045 0.2946 0.3008
The weighted averages of these scores are shown in the table. Gina should choose Univesity B.
Car 2
Car 3
Item
M1-9.
M1-7.
Style
Warranty
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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MODULE 1
Reputation A A 1 B 0.142857 C 0.2 Column Total 1.342857
Normalized A B C
Quality of Life A B C Column Total Normalized A B C Factors Cost Reputation Quality of life Column Total
Normalized Cost Reputation Quality of life
B 7 1 3 11
B 0.6364 0.0909 0.2727
C 0.7895 0.0526 0.1579
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
Cost 1 0.333333 0.142857 1.47619
The following will be the priorities for price:
C 5 0.333333 1 6.333333
A 0.7447 0.1064 0.1489
Reputation 3 1 0.5 4.5
Quality Cost Reputation of life 0.6774 0.6667 0.7000 0.2258 0 .2222 0.2000 0.0968 0.1111 0.1000
271
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: Weighted sum vector
Using the factor weights, we find the following weighted averages for each university.
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
ϭ
(1.4649 1.2789 0.2794)
Consistency vector
ϭ
(3.0278 3.0220 3.0051)
Quality of life 7 2 1 10 Factor Evaluation (Row Average) 0.6814 0.2160 0.1026
S-2
Memory
Lambda
ϭ
3.0183
Value of CI
ϭ
0.0092
RI
ϭ
0.5800
CR
ϭ
0.0158
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
ϭ
(0.2765 0.4631 2.3192)
Consistency vector
ϭ
(3.0078 3.0164 3.0736)
Therefore, Gina should choose University B.
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
Speed
8 5
S-1 S-2 S-3
Lambda
ϭ
3.0326
Value of CI
ϭ
0.0163
RI
ϭ
0.5800
CR
ϭ
0.0281
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
The following will be the priorities for speed:
Consistency information follows:
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.
Weighted sum vector
ϭ
(2.1717 0.2371 0.6218)
Consistency vector
ϭ
(3.0389 3.0040 3.0122)
Lambda
ϭ
3.0184
Value of CI
ϭ
0.0092
RI
ϭ
0.5800
CR
ϭ
0.0158
Consistency information: Weighted sum vector
ϭ
(0.6902 1.9485 0.3667)
Consistency vector
ϭ
(3.0026 3.0071 3.0013)
Lambda
ϭ
3.0037
Value of CI
ϭ
0.0018
RI
ϭ
0.5800
CR
ϭ
0.0032
Flexibility
S-1
S-1
The following will be the weights for the factors:
S-2
S-3
1 2
1 8 1 4
S-2 S-3
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
See the table for factors for Problem M1-11. Consistency information follows: Weighted sum vector
ϭ
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 =
Consistency information follows: Weighted sum vector
ϭ
0.3849 0.0447 0.0816 0.0514 0.149 0.288
Value of CI
ϭ
0.0288
RI
ϭ
1.2400
CR
ϭ
0.0232
(0.2727 0.5455 2.1818)
2.39 0.312
0.275 0.918
ϭ
(3.0000 3.0000 3.0000)
Lambda
ϭ
3.0000
Value of CI
ϭ
0.0000
RI
ϭ
0.5800
Item
Ranking
CR
ϭ
0.0000
System 1 (S-1) System 2 (S-2) System 3 (S-3)
0.4928 0.2400 0.2671
S-1
S-1
S-2
The following are the final rankings—system 1 (S-1) is selected.
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.
