Mutli-Attribute Decision Making

Scott MatthewsCourses: 12-706 / 19-702

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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.

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Dominance Example #1

CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.

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But..

Need to be careful of “when” to eliminate dominated alternatives, as we’ll see.

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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

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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 everythingAssumes all B/C quantifiable

BCA still important : economic efficiency

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Structuring Objectives

Choose a college

Max. Reputation Min. Cost Max Atmosphere

Academic Social Tuition Living Trans.Making this tree is useful for

Communication (for DM process) Creation of alternatives Evaluation of alternatives

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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

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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

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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.

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Choosing a Car

Car Fuel Eff (mpg) Comfort IndexMercedes 25 10Chevrolet 28 3Toyota 35 6Volvo 30 9Which dominated, non-dominated?

Dominated can be removed from decision BUT we’ll need to maintain their values for

ranking

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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

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Tradeoff of Car Problem

Fuel Eff

Comfort

10

5

0 10 20 30

MV

T

C

1) What is tradeoff between Mercedes and Volvo?

2) What can we see graphicallyabout dominated alternatives?

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Tradeoff of Car Problem

Fuel Eff

Comfort

10

5

0 10 20 30

M(25,10)V(30,9)

T

C

-15

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.

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Tradeoff of Car Problem

Fuel Eff

Comfort

10

5

0 10 20 30

M(25,10)V(30,9)

T (35,6)

-15

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?

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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

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Additive Utility

We motivated 2-attribute version already

Generally:U(x1,..,xm) = k1U1(x1) + … + kmUm(xm)

=ik iU i

(x )i=1

m

∑

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Recall: Choosing a Car Example

Car Fuel Eff (mpg) Comfort

IndexMercedes 25 10Chevrolet 28 3Toyota 35 6Volvo 30 9

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Tradeoff of Car Problem

Fuel Eff

Comfort

10

5

0 10 20 30

MV

T

C

1) What is tradeoff between Mercedes and Volvo?

2) What can we see graphicallyabout dominated alternatives?

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Proportional Scoring

Called proportional because scales linearlyComfort Index: Best = 10, Worst = 3

Uc(Mercedes) = 1; Uc(Chevrolet) = 0

Uc(V) = 9-3/10-3 = 6/7; Uc(T) = 6-3/10-3 = 3/7 i.e., Volvo is 1/7 away from best to worst

Ui (x) = x−WorstBest−Worst

Ui (x) =x −i−x

i+x −

i−x

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Prop Scoring (cont.)

Fuel Economy: Best = 35, Worst = 25 UF(Toyota) = 1; UF(Mercedes) = 0 UF(V) = 30-25/35-25 = 5/10 UF(C) = 28-25/35-25 = 3/10 i.e., Volvo is halfway between best/worst

See why we kept “dominated” options?

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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

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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

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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.

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“Pricing out”

Book uses $ / unit tradeoffOur example has no $ - but same idea“Pricing out” simply means knowing

your willingness to make tradeoffsAssume 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)

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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) kC(6/7) + kF(5/10) = kC(4/7) + kF(6/10) kC (2/7) = kF(1/10) <=> kF = kC (20/7) But kF + kC =1 <=> kC (20/7) + kC = 1 kC (27/7) = 1 ; kC = 7/27 = 0.26 (so kf=0.74)

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With these weights..

U(M) = 0.26*1 + 0.74*0 = 0.26U(V) = 0.26*(6/7) + 0.74*0.5 = 0.593U(T) = 0.26*(3/7) + 0.74*1 = 0.851U(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)

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Indifference - 2:1

Fuel Eff

Comfort

10

5

0 10 20 30

M

H

T

C

V

0.260.59 0.85

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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!

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Next time: Advanced Methods

More ways to combine tradeoffs and weights

Swing weightsEtc.