SYNDICATE 4 - YE51 - AHP Case.pptx

8/21/2019 SYNDICATE 4 - YE51 - AHP Case.pptx

1/16

Multi-criteria decisionmaking.

Analytic Hierarchy Process


2/16

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


3/16


'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


4/16


Numericalratings

Verbal judgments

1 Equally important(preferred)

2 eakly prefered

! Strongly prefered" E#tremely prefered


5/16

COST


6/16

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


7/16


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


8/16

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


9/16

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


10/16


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+ =

++=


11/16

ST!"


12/16


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


13/16


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


14/16

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


15/16


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


16/16


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+ =

++=

Documents

SYNDICATE 4 - YE51 - AHP Case.pptx