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8/21/2019 SYNDICATE 4 - YE51 - AHP Case.pptx
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Multi-criteria decisionmaking.
Analytic Hierarchy Process
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Case
2
A motorist is using the AHP to choose a new car from three possible
models: an Arrow, a bestmobile and a commuter. The choice will be
based on just two attributes: cost and style. The motorist considersthat cost is weakly more important than style.
hen asked to compare the costs of the cars, the motorist makes
the followig statements: on cost, the bestmobile is weakly preferred
to the arrow, but the arrow is weakly preferred to the commuter. Alsothe bestmobile is e!tremely preferred to the commuter.
"n style, the arrow is #ery strongly preferred to the bestmobile, but
the commuter is weakly preferred to the arrow. Also the commuter is
e!tremely preferred to the bestmobile.
a. $onstruct a hierarchy to represent the decision problem
b. $alculate the weight for each table in hierarchy and hence
determine which car should be purchased
c. $alculate the inconsistency ratios for the motorist comparisons of
the cars on %i& cost and %ii& style and interpret your results
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Analytic Hierarchy Process
'Step 1: Structure a hierarchy.
CostCost StyleStyle
COMMUTER
ARROWBESTMOBILE
COMMUTER
ARROWBESTMOBILE
COMMUTER
ARROWBESTMOBILE
COMMUTER
ARROWBESTMOBILE
Overall Goal
Criteria
DecisionAlternatives
To Select a new CarTo Select a new Car
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Analytic Hierarchy Process
Numericalratings
Verbal judgments
1 Equally important(preferred)
2 eakly prefered
! Strongly prefered" E#tremely prefered
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COST
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E#ample$ Synthesi%ing&rocedure - 1
Step 1$ Sum the 'alues in each column.
Cost BM AR CM
BM 1 2 6
AR 1/2 1 2
CM 1/6 1/3 1
Column totals10/6 10/3 9
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E#ample$ Synthesi%ing&rocedure - 2
Step 2$ i'ide each element of the matri# yits column total.
◦ *ll columns in the normali%ed pair+isecomparison matri# no+ ha'e a sum of 1.
Cost BM AR CM
BM 6/10 6/10 2/3
AR 3/10 3/10 2/9
CM 1/10 1/10 1/9
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E#ample$ Synthesi%ing&rocedure - ,
Step ,$ *'erage the elements in each ro+.◦ he 'alues in the normali%ed pair+ise
comparison matri# ha'e een con'ertedto decimal form.
◦ he result is usually represented as the(relati'e) priority 'ector (eigen 'ector).
Cost BM AR CM Row Avg.
BM 0.6 0.6 0.6670.622333
AR 0.3 0.4 0.222 0.307333
CM 0.1 0.1 0.111 0.103667
Total 1,033333
0.103667
0.307333
0.622333
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E#ample$ Consistency Checking- 1
Step 1$ Multiply each 'alue in the rst column of thepair+ise comparison matri# y the relati'epriority of the rst item considered. Sameprocedures for other items. Sum the 'aluesacross the ro+s to otain a 'ector of 'alueslaeled /+eighted sum.0
=
+
+
=
+
+
3.0
82.0
86.1
0.104
207.0
622.0
102.0
0.307
615.0
0.104
0.312
0.623
1
2
6
0.103667
31
1
2
0.307333
61
21
1
0.622333
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E#ample$ Consistency Checking- 2
Step 2$ i'ide the elements of the 'ector of+eighted sums y the correspondingpriority 'alue.
t!" 3# Com"ut! t$! av!%ag! o& t$! valu!s 'om"ut!( )n
st!" 2 *λma+.
=
2.981
2.691
2.990
104.0/31.0
307.0/826.0
622.0/86.1
887.23
981.2691.2990.2-
ma+ =
++=
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ST!"
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E#ample$ Synthesi%ing&rocedure - 1
Step 1$ Sum the 'alues in each column.
tl! BM AR CM
BM 1 1/4 1/6
AR 4 1 1/2
CM 6 2 1
Column totals11 13/4 10/6
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E#ample$ Synthesi%ing&rocedure - 2
Step 2$ i'ide each element of the matri# yits column total.
◦ *ll columns in the normali%ed pair+isecomparison matri# no+ ha'e a sum of 1.
tl! BM AR CM
BM 1/11 1/13 1/10
AR 4/11 4/13 3/10
CM 6/11 8/13 6/10
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E#ample$ Synthesi%ing&rocedure - ,
Step ,$ *'erage the elements in each ro+.◦ he 'alues in the normali%ed pair+ise
comparison matri# ha'e een con'ertedto decimal form.
◦ he result is usually represented as the(relati'e) priority 'ector (eigen 'ector).
tl! BM AR CM Row Avg.
BM 0.0909 0.0769 0.10.089
AR 0.3636 0.3076 0.3 0.324
CM 0.5454 0.6153 0.6 0.587
Total 1.000
0.587
0.324
0.089
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E#ample$ Consistency Checking- 1
Step 1$ Multiply each 'alue in the rst column of thepair+ise comparison matri# y the relati'epriority of the rst item considered. Sameprocedures for other items. Sum the 'aluesacross the ro+s to otain a 'ector of 'alueslaeled /+eighted sum.0
=
+
+
=
+
+
769.1
974.0
268.0
587.0
294.0
098.0
648.0
324.0
081.0
534.0
356.0
0.089
1
1/2
1/6
0.587
2
1
1/4
324.0
6
4
1
0.089
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E#ample$ Consistency Checking- 2
Step 2$ i'ide the elements of the 'ector of+eighted sums y the correspondingpriority 'alue.
t!" 3# Com"ut! t$! av!%ag! o& t$! valu!s 'om"ut!( )n
st!" 2 *λma+.
=
014.3
3.006
3.011
587.0/769.1
324.0/974.0
089.0/268.0
011.33
014.3006.3011.3-
ma+ =
++=
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