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Multiobjective Value Analysis. 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.
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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.
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 v1(389,900)=1 and v1(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 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
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.
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.
