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Using Preference Constraints to Solve Multi-criteria Decision Making Problems

Using Preference Constraints to Solve Multi-criteria Decision Making Problems. Tanja Magoč, Martine Ceberio, and François Modave Computer Science Department, The University of Texas at El Paso. Outline. Multi-criteria decision making (MCDM) Traditional techniques to solve MCDM problem

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Using Preference Constraints to Solve Multi-criteria Decision Making Problems

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  1. Using Preference Constraints to Solve Multi-criteria Decision Making Problems Tanja Magoč, Martine Ceberio, and François Modave Computer Science Department, The University of Texas at El Paso

  2. Outline • Multi-criteria decision making (MCDM) • Traditional techniques to solve MCDM problem • A novel approach • Representing preferences over criteria in terms of constraints • Narrowing down the search space

  3. Multi-criteria decision making • Comparison of multidimensional alternatives to select the optimal one • Pick the “best” car to buy • Elements of a MCDM setting: • a set of alternatives • cars (finitely many) • a set of criteria • color, safety rating, price • a set of values of the criterion • color={red, black, grey}, safety rating={1,2,3,4,5}, price=[11000, 53000] • a preference relation for each criterion: • red>black>grey, lower price is more preferred than higher price, higher safety rating is more preferred than lower safety rating • Challenge: Combine partial preferences into a global preference

  4. Traditional techniques to solve MCDM problems • Utility based approaches: • Maximax strategy • Maximin strategy • Weighted sum approach • Non-additive approaches • The Choquet integral w.r.t. a non-additive measure

  5. Narrow down the search space • Narrow down the search space from all possible values that criteria can take to a smaller space that contains the best alternative based on the preferences of an individual. • Assumption: the decision maker is able to express his/her preference of a criterion over another criterion by means of how much of a criterion he/she would sacrifice in order to obtain higher value of the other criterion. • E.g. the individual knows how much more he/she is willing to pay for an increase in one star of safety rating. • Different tradeoff at different value.

  6. Process of narrowing down the search space • Map the domain of each criterion into an interval • Convert each of the intervals into the interval [0,1] • higher preference is given to values closer to 1 • Original search space=cross product of these intervals • Constraints on the search space: preferences provided by decision maker • Input how much the individual is willing to “pay” using one criterion to increase the value of other criterion by 0.2 • Use standard techniques for solving problems with constraints to narrow the search space

  7. Example: Map domains into intervals • Map domains of criteria into intervals: • Color={red, black, gray}[1,3] with gray<black<red • Safety rating={1,2,3,4,5}[1,5] • Price=[11000,53000] • Convert all interval domains onto a common scale [0,1], where higher number means a higher preference: • Color: u(c)=(c-1)/2 • Safety rating: u(s)=(s-1)/5 • Price: u(p)=(53000-p)/42000

  8. Example: Importance of an increase • The decision maker gives an importance value from 0-10 for each increase in 0.2 value in a criterion relative to another criterion • 0 means the individual is not willing to sacrifice other particular criterion at all for increase in the value of the criteria • 10 means the individual is willing to sacrifice lot for increase in the value of the criteria

  9. Example: Importance of an increase • Safety rating relative to price: • 0.00.2: 10 • 0.20.4: 9 • 0.40.6: 7 • 0.60.8: 5 • 0.81.0: 2 • Sum of values = 33 • Increase in safety rating from 0.0 to 0.2 is worth decreasing value of price by (10/33)*100% • Increase in safety rating from 0.8 to 1.0 is worth decreasing value of price by (2/33)*100%

  10. Example: Constraint based on safety rating and price Price 1.0 0.8 0.6 0.4 0.2 Safety rating 0.2 0.4 0.6 0.8 1.0

  11. Example: Constraint between safety rating and price Price 1.0 0.8 0.6 0.4 0.2 Safety rating 0.2 0.4 0.6 0.8 1.0

  12. Example: Constraint between safety rating and price Price 1.0 0.8 0.6 0.4 0.2 Safety rating 0.2 0.4 0.6 0.8 1.0

  13. Example: All constraints together • Constraints: • safety rating – price • price – color • safety rating – color • Each constraint itself narrows down the search space • Use standard techniques used to narrow down the search space even more

  14. Example: Constraint between safety rating and price Price 1.0 0.8 0.6 0.4 0.2 Safety rating 0.2 0.4 0.6 0.8 1.0

  15. Conclusion • What have we done? • Represent multi-criteria decision making problem in terms of preference constraints defined by the decision maker • Reduce the initial search space by using standard continuous constraint solving techniques • Why is this approach better? • Speeds up the process of finding the best solution by fast elimination of domains that certainly do not contain the best solution • What is the next step? • Define more complex/general preferences • Combine importance and interactions (Choquet) with preferences

  16. Thank you QUESTIONS???

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