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 ObjectivesParkwayLombardBaranovMontanaPierpoint Commute (min.) Cust. Access (%) Office Services ABCAC Office Size (sq. ft.) Monthly Cost ($)

Ranking Table Alternatives ObjectivesParkwayLombardBaranovMontanaPierpoint Commute (min.) 52 (tie)1 4 Cust. Access (%) Office Services 1 (tie)34 (tie)1 (tie)4 (tie) Office Size (sq. ft.) 23 (tie)51 Monthly Cost ($) 42153

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 ObjectivesLombardBaranovMontana Commute (min.) Cust. Access (%) Office ServicesBCA Office Size (sq. ft.) Monthly Cost ($)

“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 ObjectivesLombardBaranovMontana Commute (min.)2520 → 2525 Cust. Access (%)8070 → 7885 Office ServicesBCA Office Size (sq. ft.) Monthly Cost ($)

Even Swap Alternatives ObjectivesLombardBaranovMontana Cust. Access (%) Office ServicesBC → BA → B Office Size (sq. ft.) Monthly Cost ($) → → 1800

Dominance Alternatives ObjectivesLombardBaranovMontana Cust. Access (%) Office Size (sq. ft.) Monthly Cost ($)

Even Swap ObjectivesLombardMontana Cust. Access (%)8085 Office Size (sq. ft.)700 → Monthly Cost ($)1700 →

Dominance ObjectivesLombardMontana Cust. Access (%)8085 Monthly Cost ($)

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 Example A 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

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 Approach

Estimating 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 v 1 (389,900)=1 and v 1 (599,000)=0.  Then

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 v 2 (5)=1 and v 2 (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 v 2 (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 v 3 (3)=1 and v 3 (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, v 3 (2)=0+x=0+1/4=.25, and v 3 (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 v 4 (Colonial)=1, v 4 (Garrison Colonial)=.5 and v 4 (Ranch)=0

A Multiobjective Example

The Value Function Approach Determine 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 Approach

Determine the overall value of each alternative Compute the weighted sum of the single objective values for each alternative.  Rank the alternatives from high to low.

A Multiobjective Example

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.