1. Mutli-Attribute Decision Making Scott Matthews Courses: 12-706 / 19-702

2. 12-706 and 73-359 Dominance  To pick between strategies, it is useful to have rules by which to eliminate options  Let’s construct an example - assume minimum “court award” expected is $2.5B (instead of $0). Now there are no “zero endpoints” in the decision tree.

3. 12-706 and 73-359 Dominance Example #1  CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.

4. 12-706 and 73-359 But..  Need to be careful of “when” to eliminate dominated alternatives, as we’ll see.

5. 12-706 and 73-359 Multi-objective Methods  Multiobjective programming  Mult. criteria decision making (MCDM)  Is both an analytical philosophy and a set of specific analytical techniques  Deals explicitly with multi-criteria DM  Provides mechanism incorporating values  Promotes inclusive DM processes  Encourages interdisciplinary approaches

6. 12-706 and 73-359 Decision Making  Real decision making problems are MC in nature  Most decisions require tradeoffs  E.g. college-selection problem  BCA does not handle MC decisions well  It needs dollar values for everything  Assumes all B/C quantifiable  BCA still important : economic efficiency

7. 12-706 and 73-359 Structuring Objectives Choose a college Max. ReputationMin. CostMax Atmosphere AcademicSocial TuitionLivingTrans.  Making this tree is useful for  Communication (for DM process)  Creation of alternatives  Evaluation of alternatives

8. 12-706 and 73-359 Desirable Properties of Obj’s  Completeness (reflects overall objs)  Operational (supports choice)  Decomposable (preference for one is not a function of another)  Non-redundant (avoid double count)  Minimize size

9. 12-706 and 73-359 MCDM Terminology  Non-dominance (aka Pareto Optimal)  Alternative is non-dominated if there is no other feasible alternative that would improve one criterion without making at least one other criterion worse  Non-dominated set: set of all alternatives of non-dominance

10. 12-706 and 73-359 More Defs  Measures (or attributes)  Indicate degree to which objective is achieved or advanced  Of course its ideal when these are in the same order of magnitude. If not, should adjust them to do so.  Goal: level of achievement of an objective to strive for  Note objectives often have sub-objectives, etc.

11. 12-706 and 73-359 Choosing a Car  CarFuel Eff (mpg) Comfort  Index  Mercedes2510  Chevrolet283  Toyota356  Volvo309  Which dominated, non-dominated?  Dominated can be removed from decision  BUT we’ll need to maintain their values for ranking

12. 12-706 and 73-359 Conflicting Criteria  Two criteria ‘conflict’ if the alternative which is best in one criteria is not the best in the other  Do fuel eff and comfort conflict? Usual.  Typically have lots of conflicts.  Tradeoff: the amount of one criterion which must be given up to attain an increase of one unit in another criteria

13. 12-706 and 73-359 Tradeoff of Car Problem Fuel Eff Comfort 10 5 0 2030 M V T C 1) What is tradeoff between Mercedes and Volvo? 2) What can we see graphically about dominated alternatives?

14. 12-706 and 73-359 Tradeoff of Car Problem Fuel Eff Comfort 10 5 0 2030 M(25,10) V(30,9) T C 5 The slope of the line between M and V is -1/5, i.e., you must trade one unit less of comfort for 5 units more of fuel efficiency.

15. 12-706 and 73-359 Tradeoff of Car Problem Fuel Eff Comfort 10 5 0 2030 M(25,10) V(30,9) T (35,6) 5 Would you give up one unit of comfort for 5 more fuel economy? -3 5 THEN Would you give up 3 units of comfort for 5 more fuel economy?

16. 12-706 and 73-359 Multi-attribute utility theory  To solve, we need 2 parts:  Attribute scales for each objective  Weights for each objective  Our weights should respect the “Range of the attribute scales”  This gets to the point of 0-1, 0-100, etc scales  Does not matter whether we have “consistent” scales as long as weights are context-specific (e.g. 100x different if 0-1, 0-100)  However we often use consistent scales to make the weighting assessment process easier

17. 12-706 and 73-359 Additive Utility  We motivated 2-attribute version already  Generally:  U(x 1,..,x m ) = k 1 U 1 (x 1 ) + … + k m U m (x m )

18. 12-706 and 73-359 Recall: Choosing a Car Example  CarFuel Eff (mpg) Comfort  Index  Mercedes25 10  Chevrolet283  Toyota356  Volvo309

19. 12-706 and 73-359 Tradeoff of Car Problem Fuel Eff Comfort 10 5 0 2030 M V T C 1) What is tradeoff between Mercedes and Volvo? 2) What can we see graphically about dominated alternatives?

20. 12-706 and 73-359 Proportional Scoring  Called proportional because scales linearly  Comfort Index: Best = 10, Worst = 3  U c (Mercedes) = 1; U c (Chevrolet) = 0  U c (V) = 9-3/10-3 = 6/7; U c (T) = 6-3/10-3 = 3/7  i.e., Volvo is 1/7 away from best to worst

21. 12-706 and 73-359 Prop Scoring (cont.)  Fuel Economy: Best = 35, Worst = 25  U F (Toyota) = 1; U F (Mercedes) = 0  U F (V) = 30-25/35-25 = 5/10  U F (C) = 28-25/35-25 = 3/10  i.e., Volvo is halfway between best/worst  See why we kept “dominated” options?

22. 12-706 and 73-359 Next Step: Weights  Need weights between 2 criteria  Don’t forget they are based on whole scale  e.g., you value “improving salary on scale 0-100 at 3x what you value fun going from 0-100”. Not just “salary vs. fun”  If choosing a college, 3 choices, all roughly $30k/year, but other amenities different.. Cost should have low weight in that example  In Texaco case, fact that settlement varies across so large a range implies it likely has near 100% weight

23. 12-706 and 73-359 Weights - Car Example  Start with equal weights (0.5, 0.5) for C,F  U(M) = 0.5*1 + 0.5*0 = 0.5  U(V) = 0.5*(6/7) + 0.5*0.5 = 0.678  U(T) = 0.5*(3/7) + 0.5*1 = 0.714  U(C) = 0.5*0 + 0.5*0.3 = 0.15  As expected, Chevrolet is worst (dominated)  Given 50-50 weights, Toyota has highest utility

24. 12-706 and 73-359 What does this tell us?  With equal weights, as before, we’d be in favor of trading 10 units of fuel economy for 7 units of comfort.  Or 1.43 units F per unit of C  Question is: is that right?  If it is, weights are right, else need to change them.

25. 12-706 and 73-359 “Pricing out”  Book uses $ / unit tradeoff  Our example has no $ - but same idea  “Pricing out” simply means knowing your willingness to make tradeoffs  Assume you’ve thought hard about the car tradeoff and would trade 2 units of C for a unit of F (maybe because you’re a student and need to save money)

26. 12-706 and 73-359 2:1 Tradeoff Example  Find an existing point (any) and consider a hypothetical point you would trade for.  You would be indifferent in this trade  E.g., V(30,9) -> H(31,7)  H would get Uf = 6/10 and Uc = 4/7  Since we’re indifferent, U(V) must = U(H)  k C (6/7) + k F (5/10) = k C (4/7) + k F (6/10)  k C (2/7) = k F (1/10) k F = k C (20/7)  But k F + k C =1 k C (20/7) + k C = 1  k C (27/7) = 1 ; k C = 7/27 = 0.26 (so k f =0.74)

27. 12-706 and 73-359 With these weights..  U(M) = 0.26*1 + 0.74*0 = 0.26  U(V) = 0.26*(6/7) + 0.74*0.5 = 0.593  U(T) = 0.26*(3/7) + 0.74*1 = 0.851  U(H) = 0.26*(4/7) + 0.74*0.6 = 0.593  Note H isnt really an option - just “checking” that we get same U as for Volvo (as expected)

28. 12-706 and 73-359 Indifference - 2:1 Fuel Eff Comfort 10 5 0 2030 M H T C V 0.26 0.59 0.85

29. 12-706 and 73-359 Notes  Make sure you look at tutorial at end of Chapter 4 on how to simplify with plug-ins  Read Chap 15 Eugene library example!

30. 12-706 and 73-359 Next time: Advanced Methods  More ways to combine tradeoffs and weights  Swing weights  Etc.