1. (My take on) Class Objectives Learn how to… –think about large, complex problems without much direction –make good assumptions –solve problems using a range of modeling tools –present and explain your solution This last part is the most important –if you do not clearly, concisely and convincingly present your findings, then your reader/TA/professor/boss has no reason to believe that you adequately modeled the problem and no way to understand it if you did

2. Formatting Your Homework Your reader ought to be able to recreate your solution given minimal information from you –The TA’s need to see… Your assumptions and a defense of them (your reasoning and/or a reference) Your solution and how you got there (ex: equations you used and not your entire excel spreadsheet) –When was the last time you read a published paper with MS Excel cells in it? There are ways to explain your solution without showing all of your cells.

3. Excel Example from HW 2 Question 2 (15 pts): A benefit-cost study of a proposed dam is conducted. The dam costs $75 million to construct. The study estimates a continuous stream of social benefits of $9.5 million per year (from avoided flood damage, hydroelectric power, etc.) and costs of $4 million ($2 million from operation and $2 million in environmental damages). a) [6 pts] Assuming a social marginal rate of time preference of 4% per year, how many years does it take for the dam to “break even” (i.e., the NPV of benefits just exceed the NPV of costs)? The following equation is used to estimate the continuous net present worth (NPV) of the dam: Where the discount rate, r, equals 4% and n is number of years. Can “hide” spreadsheet rows not needed

4. Theory Behind Minimal Information (very intelligent people have thought about this before us) Occam’s razor –“Entities should not be multiplied unnecessarily” Einstein’s principle –“Everything should be made as simple as possible, but not simpler” Definition of engineering elegance from Antoine de Saint-Exup'ery (aviator and author of The Little Prince) –"A designer knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away" Antoine de Saint-Exup'ery quote from http://dict.die.net/elegant/, Occam’s razor and Einstein quote from http://razorland55.free.fr/ockham.htm

5. Grading of the Homework Give just enough information - nothing more, nothing less –Solutions following this theory are tough to master (and to grade) and take practice Points will be taken off for… –Consistently giving too much information –Poor presentation including hard to follow logic, poor spelling, poor writing, or generally messy work Try to format your homework like a project

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

7. 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.

8. 12-706 and 73-359 Dominance Example #1  CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.

9. 12-706 and 73-359 But..  Need to be careful of “when” to eliminate dominated alternatives, as we’ll see.

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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.

16. 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

17. 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

18. 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?

19. 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.

20. 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?

21. 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

22. 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 )

23. 12-706 and 73-359 Recall: Choosing a Car Example  CarFuel Eff (mpg) Comfort  Index  Mercedes25 10  Chevrolet283  Toyota356  Volvo309

24. 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?

25. 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

26. 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?

27. 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

28. 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

29. 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.

30. 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)

31. 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)

32. 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)

33. 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

34. 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!

35. 12-706 and 73-359 Next time: Advanced Methods  More ways to combine tradeoffs and weights  Swing weights  Etc.