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 Safety rating Price 0.20.40.60.81.0 0.2 0.4 0.6 0.8 1.0

11. Example: Constraint between safety rating and price Safety rating Price 0.20.40.60.81.0 0.2 0.4 0.6 0.8 1.0

12. Example: Constraint between safety rating and price Safety rating Price 0.20.40.60.81.0 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 Safety rating Price 0.20.40.60.81.0 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