EMGT 269 - Elements of Problem Solving and Decision Making

EMGT 269 - Elements of Problem Solving and Decision Making

15. Conflicting Objectives

Utility Functions:

·  Captures Risk Attitudes towards Monetary-Payoff

Utility Functions can be applied to model risk attitude with respect to other attributes as well e.g.:

·  Market Share (in %)

·  Death Toll in Transportation Accidents (in # Deaths)

·  Etc.

1.  If you feel one unit of the attribute of a fundamental objective is always worth the same to you in terms of dollars, you can establish such a utility function by:

STEP 1: Establish Utility Function for Monetary-Payoff

STEP 2: Establish Monetary equivalent of one unit on the measurement scale of the fundamental objective.

STEP 3:Transform Utility Function for Monetary-Payoff to Utility Function for other fundamental objective.

2.  If you feel the above does not apply, you need to asses the utility function directly in terms of the attribute of the fundamental objective.

EXAMPLE: MONETARY VALUE LIFE-SAVING

Life Valuation for Purposes of Cost-Benefit Analyses

(Source: Henley Kumamoto. 1981)

Approaches / Typical Values / Some Limitations
(1) Implicit Value / $9,000-$9,000,000 / Assumes past decisions are optimal
(2) Human Capital / $100,000-$400,000 / Based solely on life-time income. Ignores individual Preferences. Discriminates against unproductive members of society
(3) Insurance Premiums / Wide Range / Does not take into account individuals's interest in protecting his own life
(4) Court Awards / $250,000 / Based on lost income
(5) Willingness to Pay / $180,000-$1,000,000 / Difficult to estimate. Depends on Risk Situation

Summary: All measures depend to some extent on the lifetime earning potential of the individuals at risk and ignore perception of seriousness.

Conclusion: Cannot be rigorously determined. Choose value (say $300,000) according to personal values, third party interests and psychological factors.

STEP 1: Utility for Monetary-Payoff: U(X) = a*Ln(X) + b, a>0

STEP 2: Y : Measured in Lives, X : Measured in Dollars

"1 Life saved is equivalent to (say) $300,000"

X=300000*Y

STEP 3: U(X) = U(300000*Y)= a*Ln(300000*Y) + b

= a*Ln(Y) + Ln(300000) + b

What if more than risk attitude and monetary pay-off

in the potential outcome is important?

A doctor prescribing medical treatment must consider a variety of issues:

·  Potential Health complications for the patient (perhaps death)

·  Money cost to the patients

·  Patient's time spent being treated

·  Cost to insurance companies

·  Payments to doctor.

·  Utilization of resources (nurses, hospital space, equipment)

·  Information gained in treating this patient (may be helpful in treating others.

In choosing a site for a new airport near Mexico City, the head of the ministry of public works had to balance such objective as:

·  Minimize the costs to the federal government.

·  Raise the capacity of airport facilities.

·  Improve the safety of the system.

·  Reduce Noise Levels.

·  Reduce access time to users.

·  Minimize Displacement of people for expansion.

·  Improve regional development (roads, for instance).

·  Achieve Political Aims.

Identifying objectives is Creative Process

·  STEP 1: Establish Fundamental Objective Hierarchy

·  STEP 2: Classify how to measure (operationalize) Fundamental Objectives

OBJECTIVE / ATTRIBUTE
Maximize profit / Money ( for example dollars)
Maximize Revenue / Money ( for example dollars)
Maximize Savings / Money ( for example dollars)
Minimize Cost / Money ( for example dollars)
Maximize Market Share / Percentage
Maximize Rate of Return / Percentage
Maximize proximity / Miles, minutes
Maximize Safety / # Deaths

When is an attribute operational?

·  Can you explain to someone what to measure and why?

·  Does it take a reasonable amount of effort to measure?

·  If the measurement were given to you by someone else, could you tell how well the objective was achieved. (Usually this means identifiying a worst and best case).

·  STEP 3: Establish an individual utility function for each attribute associated with a fundamental objective.

·  STEP 4: Establish a MULTI ATTRIBUTE UTILITY FUNCTION by combining the individual utility functions.

A fundamental objective hierarchy should be:

1.  Complete: "Do you feel something is missing"

2.  Small: Only those objectives should be contained in the fundamental objective hierarchy that are appropriate for the decision context.

3.  Not redundant: Every objective should appear once in the hierarchy and the objectives should not be closely related. (e.g. at the same level in the hierarchy it should not be the case that the achievement of two objectives, implies the achievement of a third objective)

4.  Decomposable: Can you think about each objective seperately.

5.  Attribute Scales must be operational.

6.  AN APPROACH TO TRADING OFF OBJECTIVES:

THE MULTI ATTRIBUTE UTILITY FUNCTION

Trading Off Conflicting Objectives:

How much worth is a unit increase in one objective worth to you in terms of units of the other objective?

·  The assessment above is subjective.

TWO OBJECTIVE EXAMPLE: BUYING A CAR

BUY THE MOST RELIABLE CAR

FOR THE LEAST AMOUNT OF MONEY

·  Establish a price range i.e. a minimum and maximum price

·  Establish a lifespan range i.e. a minimum and maximum desirable life span.

·  Select all cars that fit within the above ranges.

PORTALO / A / NORUSHI / STANDAR M.
PRICE ($1000) / 17 / 15 / 10 / 8
LIFESPAN (Years) / 12 / 7 / 9 / 6

STEP 1: Make Dominance Considerations

Conclusion: Car A is dominated by the Norushi

STEP 2: Choose most important objective and price out other objective.

·  Standard ® Norushi Û Price: $2000 ­ , Life: 3 Years­

In other words: Am I willing to pay $2000/3 = $666.67 for an extra year gained in life span?

If the answer is no, you choose the standard motors. If the answer is yes, you continue.

·  Norushi ® Portalo Û Price: $7000 ­ , Life: 3 Years­

In other words: Am I willing to pay $7000/3 = $2333.33 for an extra year gained in life span?

If the answer is no, you choose the Norushi. If the answer is yes, you choose the Portalo.

Comments:

·  Procedure work in case of two objectives. Three objectives would already be more difficult

·  One of the objectives is measured in $ which allows pricing out.

A more general approach is required

ADDITIVE MULTI-ATTRIBUTE UTILITY FUNCTION

STEP 1: Establish a range for each objective i: (Ai, Bi)

STEP 2: Establish a utility function for each objective i:

Ui (Xi)

STEP 3: Establish an importance weight ki for each

objective.

STEP 4: Calculate the combined utility for all objectives

,

SUGGESTED FIRST CUT APPROACH IN STEP 2:

Assume Risk Neutrality in STEP 2 above by using PROPORTIONAL SCORES, i.e.

,

Note:

·  Above formula implies Risk Neutrality in terms of attribute scale associated with , because function is linear function.

·  ,

CAR EXAMPLE CONTINUED:

PORTALO / NORUSHI / STANDAR M.
PRICE ($1000) / 17 / 10 / 8
LIFESPAN (Years) / 12 / 9 / 6

STEP 1:

·  PRICE: Worst = 17, Best = 8

·  LIFESPAN: Worst = 6, Best =12

STEP 2:

INDIVIDUAL UTILITY SCORES

PORTALO / NORUSHI / STANDAR M.
PRICE ($1000) / / /
LIFESPAN (Years) / / /

or

PORTALO / NORUSHI / STANDAR M.
PRICE ($1000) / 0 / 0.778 / 1.0
LIFESPAN (Years) / 1.0 / 0.50 / 0.0

CASE 1: Price is equally important as Life Span Þ

kp=0.50, kL=0.50

CASE 2: Price is twice as important as Life Span Þ

CASE 3:

·  Price is your most important objective

·  You are willing to pay $600 dollars for an extra year of lifespan.

Thus: U(8000,6) = U(8600,7)

· 

·  Up(8600) = =0.933

·  UL(7) ==0.167

· 

· 

·  Weights must sum up to 1:

INDIFFERENCE CURVES

Definition:

An indifference curve with value C of a Multi-Atribute Utility Function is a set of values for the individual attributes of the objectives for which the combined utility is constant and equals C.

CAR EXAMPLE CONTINUED:

Marginal Rate of Return in utilities:

Interpretation: The amount of utility in price your are willing to give up for one unit increase in utility off lifespan.

Denote:

Note: Cannot be easily understood.

Marginal Rate of Return in original attributes when using proportional scores for individual utility functions:

Interpretation: The amount of money in price your are willing to give up for an increase of one year in lifespan.

SUMMARY:

Developed a general approach for two objectives

that can easily be extended to more objectives.

CAR EXAMPLE CONTINUED:

Third Objective: Maximize Color Satisfaction

Three Colors: Blue Portalo, Red Norushi and Yellow Standard Motors

STEP 2: Assessing Individual Utility Function for Color Attribute.

Note that color does not have a natural attribute scale. Hence, we have to use a constructed scale. Through Expert Judgment we assess that on a 0%-100% scale, where 100% means complete color satisfaction, that the color red scores 30%. We also assess that Blue is twice as desirable as Red, and Yellow is 2.5 times desirable as red.

Conclusion:

Red = 30% Color Satisfaction

Blue = 60% Color Satisfaction

Yellow=75% Color Satisfaction

Assuming Risk Neutrality in Color Satisfaction Attribute:

·  U(Blue)=

·  U(Red)=

·  U(Yellow) =

ASSESSING WEIGHTS: MORE THAN TWO OBJECTIVES

1.  Swing Weights Approach

STEP 1: Consider worst conceivable scenario, i.e. all objectives at their worst values

STEP 2: Consider objectives separately increasing from worst to best and indicate the one that gives you the highest satisfaction.

STEP 3: For other objectives swing from worst to best and ask percentage satisfaction compared to most preferred objective.

STEP 4: Calculate the weights

CAR EXAMPLE CONTINUED:

STEP 2: If you could trade the car with individual utility scores (0,0,0) for the car with individual utility score (1,0,0), (0,1,0) or (0,0,1). Which one would you choose?

Answer: (1,0,0)

Conclusion: Changing from Worst to Best in the price objective is most important to you.

STEP 3:

·  Assess in % Satisfaction getting car (0,1,0) in STEP 2 in stead of car (1,0,0). Answer: 75%

·  Assess in % Satisfaction getting car (0,0,1) in STEP 2 in stead of car (1,0,0). Answer: 10%

STEP 4:

·  Combined Utility of Car (0,0,0): 0

·  Combined Utility of Car (1,0,0): kp

·  Combined Utility of Car (0,1,0): kL=0.75 * kp

·  Combined Utility of Car (0,0,1): kC=0.10 * kp

NOTE: Swing Weights Method is sensitive to range of individual attributes.

2.  Reference Lottery Approach

Indifference Þ ki = p

CAR EXAMPLE CONTINUED:

Answer: p=0.55 Þ kp = 0.55

Repeat the above procedure for other attributes and check whether weights sum up to 1. If not, re-scale such that sum equals 1.

Making Decisions with Multiple Objectives

Influence Diagram

Two Objectives:

·  Making Money (Measured in $)

·  Having Fun (Constructed attribute scale, see page 128)

Best, Good, Middle, Bad, Worst

Decision Tree

Analysis based on Two Objectives:

1.  Before overall score can be calculated the scales of each alternative need to be the same i.e.

·  Transform to 0-1 scale or 0%-100% scale

·  Set Best=100%, Worst=0%

·  Determine intermediate values

Having Fun objective:

Best(100%), Good(90%), Middle(60%), Bad(25%), Worst (0%)

Making Money Objective:

·  $2730.00=100%, $2047.50=0%

·  Intermediate dollar amount X:

2.  Assess Trade-off weights

= Weight for Salary = Weight for Fun Level

Using Expert Judgment:

Going from worst to best in salary objective is

1.5 times more important than going from

worst to best in having fun objective

·  . With follows that:

Optimal Decision Strategy: Forest-Job

"Forest-Job" stochastically dominates

"In-Town Job" alternative in terms of

the overall score

Lecture notes by: Dr. J. Rene van Dorp Session 12 - Page 208

Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen