Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he...
-
Upload
diana-hart -
Category
Documents
-
view
213 -
download
0
Transcript of Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he...
![Page 1: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/1.jpg)
Multiobjective Analysis
![Page 2: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/2.jpg)
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.
![Page 3: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/3.jpg)
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 ($)
1850 1700 1500 1900 1750
![Page 4: Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he signed a lease for office space. The lease is about to.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/4.jpg)
Ranking Table
Alternatives
Objectives Parkway Lombard Baranov Montana Pierpoint
Commute (min.)
5 2 (tie) 1 2 (tie) 4
Cust. Access (%)
5 2 4 1 3
Office Services
1 (tie) 3 4 (tie) 1 (tie) 4 (tie)
Office Size (sq. ft.)
2 3 (tie) 5 1 3 (tie)
Monthly Cost ($)
4 2 1 5 3
![Page 5: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/5.jpg)
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
![Page 6: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/6.jpg)
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.
![Page 7: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/7.jpg)
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 ($) 1700 1500 1900
![Page 8: Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he signed a lease for office space. The lease is about to.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/8.jpg)
“Even Swaps”
If every alternative for a given objective is rated equally you can eliminate that objective
Even Swaps is a way to adjust the values of different alternatives’ objectives in order to make them equivalent.
![Page 9: Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he signed a lease for office space. The lease is about to.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/9.jpg)
Even Swaps
First, determine the change necessary to cancel out an objective.
Second, assess what change in another objective would compensate for the needed change.
Third, make the even swap.
![Page 10: Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he signed a lease for office space. The lease is about to.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/10.jpg)
Even Swap
Alternatives
Objectives Lombard Baranov Montana
Commute (min.) 25 20 → 25 25
Cust. Access (%) 80 70 → 78 85
Office Services B C A
Office Size (sq. ft.) 700 500 950
Monthly Cost ($) 1700 1500 1900
![Page 11: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/11.jpg)
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 ($) 1700 1500 → 1700 1900 → 1800
![Page 12: Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he signed a lease for office space. The lease is about to.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/12.jpg)
Dominance
Alternatives
Objectives Lombard Baranov Montana
Cust. Access (%) 80 78 85
Office Size (sq. ft.) 700 500 950
Monthly Cost ($) 1700 1700 1800
![Page 13: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/13.jpg)
Even Swap
Objectives Lombard Montana
Cust. Access (%) 80 85
Office Size (sq. ft.) 700 → 950 950
Monthly Cost ($) 1700 → 1950 1800
![Page 14: Multiobjective Analysis. An Example Adam Miller is an independent consultant. Two year’s ago he signed a lease for office space. The lease is about to.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/14.jpg)
Dominance
Objectives Lombard Montana
Cust. Access (%) 80 85
Monthly Cost ($) 1950 1800
![Page 15: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/15.jpg)
Conclusion
Montana location is the final choice.
![Page 16: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/16.jpg)
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.
![Page 17: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/17.jpg)
The Value Function Approach Specify decision alternatives and objectives Evaluate objectives for each alternative
![Page 18: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/18.jpg)
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.
![Page 19: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/19.jpg)
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
![Page 20: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/20.jpg)
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.
![Page 21: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/21.jpg)
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
![Page 22: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/22.jpg)
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.
![Page 23: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/23.jpg)
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
![Page 24: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/24.jpg)
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
![Page 25: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/25.jpg)
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
![Page 26: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/26.jpg)
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.
![Page 27: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/27.jpg)
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
![Page 28: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/28.jpg)
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.
![Page 29: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/29.jpg)
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
![Page 30: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/30.jpg)
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
![Page 31: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/31.jpg)
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)
![Page 32: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/32.jpg)
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..
![Page 33: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/33.jpg)
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.
![Page 34: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/34.jpg)
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.
![Page 35: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/35.jpg)
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.
![Page 36: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/36.jpg)
The Value Function ApproachThen for price
for bedrooms
for bathrooms
w w
w w
w w
1
2
3
25
2
.
![Page 37: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/37.jpg)
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
.
.
.
![Page 38: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/38.jpg)
The Value Function ApproachFinally
and
w w
w w
w w
w w
1
2
3
4
12
25 29
2 24
3 35
.
. .
.
.
![Page 39: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/39.jpg)
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.
![Page 40: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/40.jpg)
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
![Page 41: 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.](https://reader036.fdocuments.in/reader036/viewer/2022062713/56649ce35503460f949af897/html5/thumbnails/41.jpg)
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.