multiobjective analysis. an example adam miller is an independent consultant. two year’s ago he...

Multiobjective Analysis

An Example

Adam Miller is an independent consultant. Two year’s ago he signed a lease for office space. The lease is about to expire and he needs to decide whether to renew it or move to a new location. Adam defines five overriding objectives that he needs his office to fulfill: a short commute, good access to clients, good office services, sufficient space and low cost.

Consequence Table

Alternatives

Objectives Parkway Lombard Baranov Montana Pierpoint

Commute (min.)

45 25 20 25 30

Cust. Access (%)

50 80 70 85 75

Office Services

A B C A C

Office Size (sq. ft.)

800 700 500 950 700

Monthly Cost ($)

1850 1700 1500 1900 1750

Ranking Table

Alternatives

Objectives Parkway Lombard Baranov Montana Pierpoint

Commute (min.)

5 2 (tie) 1 2 (tie) 4

Cust. Access (%)

5 2 4 1 3

Office Services

1 (tie) 3 4 (tie) 1 (tie) 4 (tie)

Office Size (sq. ft.)

2 3 (tie) 5 1 3 (tie)

Monthly Cost ($)

4 2 1 5 3

Eliminating “Dominated” Alternatives

Dominance – If alternative A is better than alternative B on some objectives and no worse than B on all other objectives, B can be eliminated from consideration.

Example – Lombard Dominates Pierpoint

Eliminating “Dominated” Alternatives

Practical Dominance – If alternative A is better than alternative B on some objectives and no worse than B on all but one objective, B may be eliminated from consideration.

Example – Except for cost Montana dominates Parkway. Miller believes that the advantages of Montana justify the extra cost so that Montana dominates Parkway.

Updated Consequence Table

Alternatives

Objectives Lombard Baranov Montana

Commute (min.) 25 20 25

Cust. Access (%) 80 70 85

Office Services B C A

Office Size (sq. ft.) 700 500 950

Monthly Cost ($) 1700 1500 1900

“Even Swaps”

If every alternative for a given objective is rated equally you can eliminate that objective

Even Swaps is a way to adjust the values of different alternatives’ objectives in order to make them equivalent.

Even Swaps

First, determine the change necessary to cancel out an objective.

Second, assess what change in another objective would compensate for the needed change.

Third, make the even swap.

Even Swap

Alternatives

Objectives Lombard Baranov Montana

Commute (min.) 25 20 → 25 25

Cust. Access (%) 80 70 → 78 85

Office Services B C A

Office Size (sq. ft.) 700 500 950

Monthly Cost ($) 1700 1500 1900

Even Swap

Alternatives

Objectives Lombard Baranov Montana

Cust. Access (%) 80 78 85

Office Services B C → B A → B

Office Size (sq. ft.) 700 500 950

Monthly Cost ($) 1700 1500 → 1700 1900 → 1800

Dominance

Alternatives

Objectives Lombard Baranov Montana

Cust. Access (%) 80 78 85

Office Size (sq. ft.) 700 500 950

Monthly Cost ($) 1700 1700 1800

Even Swap

Objectives Lombard Montana

Cust. Access (%) 80 85

Office Size (sq. ft.) 700 → 950 950

Monthly Cost ($) 1700 → 1950 1800

Dominance

Objectives Lombard Montana

Cust. Access (%) 80 85

Monthly Cost ($) 1950 1800

Conclusion

Montana location is the final choice.

Multiobjective Value Analysis A procedure for ranking alternatives and

selecting the most preferred Appropriate for multiple conflicting

objectives and no uncertainty about the outcome of each alternative.

The Value Function Approach Specify decision alternatives and objectives Evaluate objectives for each alternative

A Multiobjective ExampleA prospective home buyer has visited four open houses in Medfield over the weekend. Some details on the four houses are presented in the following table.

A Multiobjective Example

Price

No. of bedrooms

No. of bathrms.

Style

$389,900 3 1.5 Ranch

$530,000 4 2 Colonial

$549,900 5 3

Garrison Colonial

$599,000 4 2.5 Colonial

The Value Function Approach Determine a value function which combines

the multiple objectives into a single measure of the overall value of each alternative.

The simplest form of this function is a simple weighted sum of functions over each individual objective.

The Value Function ApproachThis functional form:

requires specificatuion of

the objectives

... the weights

and the single objective value

functions

v x x x w v x w v x w v x

x x

w w

v x v x

m m m m

m

m

m m

( , ,... ) ( ) ( ) ... ( )

,...

( ),... ( )

1 2 1 1 1 2 2

1

1

1 1

The Value Function ApproachEstimating the single objective value functions Price - price ranges from roughly $300,000 to

$600,000 dollars with lower amounts being preferred.

Suppose that a decrease in price from $600,000 to $450,000 will increase value by the same amount as would a decrease in price from $450,000 to $300,000.

The Value Function Approach This implies that over the range $300,000 to

$600,00 the value function for price is linear and the value for each price alternative can be found by linear interpolation.

First set v1(389,900)=1 and v1(599,000)=0.

Then

The Value Function ApproachPrice = $530,000

530 000 389 900

599 000 389 900

530 000 1

0 1

67530 000 1

133 530 000

1

1

1

, ,

, ,

( , )

.( , )

. ( , )

v

v

v

The Value Function ApproachPrice = $549,900

549 900 389 900

599 000 389 900

549 900 1

0 1

77549 900 1

123 549 900

1

1

1

, ,

, ,

( , )

.( , )

. ( , )

v

v

v

The Value Function Approach Number of bedrooms - the number of bedrooms

for the four alternatives is 3, 4 or 5 with more bedrooms preferred to fewer.

Thus v2(5)=1 and v2(3)=0.

Suppose the increase in value in going from 3 to 4 bedrooms is twice the increase in value in going from 4 to 5 bedrooms.

The Value Function Approach Then if the value increase in going from 4 to 5

bedrooms is x, the value increase in going from 3 bedrooms to 4 is 2x.

And since the value increase in going from 3 bedrooms to 5 is 1, 2x+x=1.

Thus x=1/3 and finally the v2(4)=0+2(1/3) =.67

The Value Function Approach Number of bathrooms - The number of bathrooms for the

four alternatives are 1.5, 2, 2.5, and 3 with more bathrooms being preferred to fewer bathrooms.

Thus v3(3)=1 and v3(1.5)=0.

Suppose that the increase in value in going from 1.5 to 2 bathrooms is small and about equal to the increase in value in going from 2.5 to 3 bathrooms. The increase in value in going from 2 to 2.5 bathrooms is more significant and is about twice this value.

The Value Function Approach Then, the value increase in going from 1.5 to 2

bathrooms is x. The value increase in going from 2 to 2.5 bathrooms is 2x. And the value increase in going from 2.5 to 3 bathrooms is also x.

The sum of the value increases x+2x+x=1 and x=1/4.

So, v3(2)=0+x=0+1/4=.25, and v3(2.5)=0+x+2x=0+1/4+2/4=.75

The Value Function Approach Style - there are three house styles available:

Ranch, Colonial and Garrison Colonial. Suppose that Colonial, is most preferred, Ranch is

least preferred and the value of Garrison Colonial is about mid-value.

Then v4(Colonial)=1, v4(Garrison Colonial)=.5 and v4(Ranch)=0

A Multiobjective Example

Price

No. of bedrooms

No. of bathrms.

Style

$389,900 (1)

3 (0)

1.5 (0)

Ranch (0)

$530,000 (.33)

4 (.67)

2 (.25)

Colonial (1)

$549,900 (.23)

5 (1)

3 (1)

Garrison (.5)

$599,000 (0)

4 (.67)

2.5 (.75)

Colonial (1)

The Value Function ApproachDetermine the weights Consider the value increase that would result

from swinging each alternative (one at a time) from its worst value to its best value (e.g.. the value increase from swinging price from $599,000 to $389,900).

Determine which swing results in the largest value increase, the next largest, etc..

The Value Function Approach Suppose going from a Ranch to a Colonial results

in the largest value increase, going from 3 to 5 bedrooms the second largest, going from 1.5 bathrooms to 3 bathrooms the next largest and swinging price from $599,000 to $389,900 results in the smallest value increase.

The Value Function Approach Set the smallest value increase equal to w and set

each other value increase as a multiple of w. Suppose the bathroom swing is twice as valuable

as the price swing, the style swing is 3 times as valuable as the price swing and the bedroom swing falls about half way in between these two.

The Value Function Approach Since the single objective value functions are

scaled from 0 to 1 the weight for any objective is equal to its value increase for swinging from worst to best.

And because we would like the multiobjective value function to be scaled from 0 to 1, the weights should sum to 1.

The Value Function ApproachThen for price

for bedrooms

for bathrooms

w w

w w

w w

1

2

3

25

2

.

The Value Function Approachand for style

Now set

or

and

w w

w w w w

w

w

4 3

25 2 3 1

85 1

12

.

.

.

The Value Function ApproachFinally

and

w w

w w

w w

w w

1

2

3

4

12

25 29

2 24

3 35

.

. .

.

.

The Value Function ApproachDetermine the overall value of each alternativeCompute the weighted sum of the single objective

values for each alternative. Rank the alternatives from high to low.

A Multiobjective Example

Price

No. of Bedrms

No. of Bathrms

Style

WeightedSum

$389900 (1)

3 (0)

1.5 (0)

Ranch (0)

(.12)

$530000 (.33)

4 (.67)

2 (.25)

Colonial (1)

(.64)

$549900 (.23)

5 (1)

3 (1)

Garrison (.5)

(.73)

$599000 (0)

4 (.67)

2.5 (.75)

Colonial (1)

.(72)

.12 .29 .24 .35

The Value Function Approach The weighted sums provide a ranking of the

alternatives. The most preferred alternative has the highest sum.

The “ideal“ alternative would have a value of 1. The value for any alternative tell us how close it is to the theoretical ideal.