BU.520.601 BU.520.601 Decision Models DecisionAnalysis1 Summer 2013

BU.520.601

DecisionAnalysis 1

BU.520.601 Decision Models

Decision Analysis

Summer 2013

BU.520.601DecisionAnalysis 2

Let us flip a fair coin 1000 (there is a fee).If you win a toss, I give you $102. If I win, you give me $100.How much fee will you pay for the playing the entire game?

Let us flip a fair coin once (there is a fee).If you win I give you $102. If I win, you give me $100.How much fee will you pay me for playing the game: $5, $2,

$1, $0? You can select any other amount.

Suppose you are getting ready to go the office in a crowded metro. Carrying an umbrella is a hassle; you will carry it only when you feel necessary. Forecast for today is 70% chance of rain and the sky is overcast. Should you carry an umbrella - Yes or no?

My decision would be “Yes” and it is a good decision.

However, there are two possible outcomes - it will rain or not.

If it does not rain, it does not mean I have made a bad decision.


Decision Analysis (DA)Decision Analysis (DA)

• DA is a methodology applicable to analyze a wide variety of problems.

• Although DA was used in the 1950s (at Du Pont) and early 1960s (at Pillsbury), major DA development took place in mid sixties. One of the earliest application (at GE) was to analyze whether a super heater should be added to the current power reactor.

• DA has been considered as a technology to assist (individuals and) organizations in decision making by quantifying the considerations (even though they may be subjective) to deduce logical actions.


Decision Analysis (DA)Decision Analysis (DA)One can discuss many topics listed below; we will look at a few.• Problem Formulation.• Decision Making with / without Probabilities.• Risk Analysis and Sensitivity Analysis.• Decision Analysis with Sample / Perfect Information.• Multistage decision making.

Tools and terminology• Basic statistics and probability• Influence diagram / payoff table /

decision tree• EMV: Expected Monetary Value • EVSI / EVPI : Expected Value of

Sample / Perfect Information

• Bayes’ rule• Decision vs. outcome• Risk management• Minimax / maximin /• Utility theory


Decision analysis without probabilitiesDecision analysis without probabilities

Alternatives Economic Condition

Recession Normal Boom

Project A 4075 5000 6100Project B 0 5250 12080Project C 2500 7000 10375Project D 1500 6000 9500

Example: There are four projects; I can select only one. The payoff table shows potential “payoff” depending upon likely economic conditions.

Concepts covered: Payoff table.Different approaches: Maximax, maximin, minimax regret

Since the payoff in project C is higher than the payoff for D for every economic condition, we say that project C is dominant.

We can eliminate project D from consideration.


MaximaxMaximax



Project A 4075 5000 6100Project B 0 5250 12080Project C 2500 7000 10375

If you are an optimist, you will decide on the basis of Maximax.

Step 1: Pick the max value for each alternative.

6100

12080

10375

Step 2:Then pick the alternative with max payoff.


MaximinMaximin




1: Pick the min value for each alternative.

4075

0

2500

If you are a conservative you will use Maximin.

2: Then pick the alternative with max payoff.


Alternatives Regret Table



Minimax RegretMinimax Regret




You are neither optimist nor conservative.

Step 1: Calculate the maximum for each outcome.

4075| 7000| 12080

Stet 2: Prepare “Regret Table” by subtracting each outcome cell value from its maximum.

At least one number for each regret table outcome is zero and there

are no negative numbers. Why?


Alternatives Regret Table



Minimax Regret..Minimax Regret..




Step 3: Pick the max value for each alternative.

5980

4075

1705

Step 4: Pick the alternative with minimum regret.

4075| 7000| 12080


General commentsGeneral comments

The above three approaches we used involved Decision Making without Probabilities.

Table columns show outcomes (also called

state of nature).

Payoff tableAlternatives Economic Condition



• The maximax payoff criterion seeks the largest of the maximum payoffs among the actions.

• The maximin payoff criterion seeks the largest of the minimum

payoffs among the actions. • The minimax regret criterion seeks the smallest of the

maximum regrets among the actions.


Decisionpoint

Decision analysis with probabilitiesDecision analysis with probabilities

Typically, we use a tree diagram for the decision analysis.1. A decision point is shown by a rectangle

20%55%

25% Chance events must be mutually exclusive and exhaustive (total probability = 1).

4. At the end of each branch is an endpoint shown as a triangle where a payoff will be identified.

2. Alternatives available at a decision point are shown as decision branches (DB).

3. At the end of each DB, there can be two or more chance events shown by a node and chance branches (CB).

CB

DB


Decision analysis with probabilitiesDecision analysis with probabilities

At the chance node, we calculate the average (i.e. expected) payoff. The terminology used is Expected Monetary Value (EMV)

Decision point: Chance event : End point:DB: Decision Branch CB: Chance Branch

If there is no chance event for a particular decision branch, it’s EMV is equal to the payoff. 20%

55%

25%

DB

CB

We select the decision with the highest EMV .

What if we are dealing with costs?


A larger tree diagramA larger tree diagram


You bought 500 units of X @$10 each.

Demand: X 300 400 500 600Pr(X) 0.30 0.45 0.20 0.05

Obviously, if demand exceeds 500, you will sell all 500. On the other hand, if demand is under 500, you will have leftover units. These leftover items can disposed off for $7 each ($3 loss, the dealer will no longer buy these leftover units from you).

You can sell these yourself for $16 each ($6/unit profit) but the demand is uncertain. The demand distribution is shown in the table.

A dealer has offered to buy these from you @$14 each ( you can make $4/unit profit).

What’s your decision?

Example 1Example 1


Suppose you have 500 units of X in stock, purchased for $10 each. Dealer sales price:$14, self sale price:$16 with salvage value:$7.

Demand: X 300 400 500 600Pr(X) 0.30 0.45 0.20 0.05

DealerSale

Self sale

Example 1 ..Example 1 ..

Start with the tree having 2 branches (DB) at the decision point. There are no chance events in the dealer sale branch,

500, 20%

600, 5%

400, 45%300, 30%

For the self sale, there are 4 mutually exclusive possibilities.


Suppose you have 500 units of X in stock, purchased for $10 each. Dealer sales price:$14, self sale price:$16 with salvage value:$7.

Demand: X 300 400 500 600Pr(X) 0.30 0.45 0.20 0.05

Example 1 ...Example 1 ...

DealerSale

Self sale500, 20%

600, 5%

400, 45%300, 30%

Payoff = 500*4 = 2000 EMV = 2000

Payoff = 300*6 – 200*3 = 1200

Payoff = 400*6 – 100*3 = 2100

Payoff = 500*6 = 3000

Payoff = 500*6 = 3000 EMV = 0.3*1200 + 0.45*2100 + 0.2* 3000 + 0.05*3000 = 2055

Your decision?


Risk ProfileRisk Profile

Payoff = 1200

Payoff = 2100

Payoff = 3000

Payoff = 3000

Self Sale300

400

500

600

20%

5%

45%

30%

Risk profile is the probability distribution for the payoff associated with a particular action.

The risk profile shows all the possible economic outcomes and provides the probability of each: it is a probability distribution for the principal output of the model.


Example 3Example 3

We have received RFP (Request For Proposal).• We may not want to bid at all (our cost: 0)• If we bid, we will have to spend $5k for proposal preparation.

Based on the information provided in the RFP, a quick decision is to bid either $115k or $120k or $125k.

We must select among 4 alternatives (including no bid).

• A quick estimate of the cost of the project (in addition to the preparation cost) is $95k.

• Looks like we may have a competitor. • If we bid the same amount as the competitor, we will get the

project because of our reputation with the client.• We have gathered some probabilities based on past experience.


Our bid (OB) must be 0 (no bid), 115, 120 or 125.

Competitor’s bid (CB): 0, under 115, 115 to under 120, 120 to under 125, 125 and over.

Assumption: If bids are equal, we get the contract.

Information : Preparation cost: $5 + Cost of work : $95 = $100 total

Profit for our bid

0 115 120 125

All numbers in thousand dollars

Use mini-max, maxi-max, etc?

There are probabilities involved.

Example 3..Example 3..

Competitor’s bid

1. No bid

2a. Under $115

2b. $115 to under $120

2c. $120 to under $125

2d. Over $125

00000

15-5151515

20-5-52020

25-5-5-525


1. There is a 30% probability that the competitor will not bid.

2. If the competitor does bid, there is

(a) 20% probability of bid under $115.

(b) 40% probability of bid $115 to under $120.

(c) 30% probability of bid under $120 to under $125.

(d) 10% probability of bid over $125.

Example 3…Example 3…

Prob.

-

20%

40%

30%

10%

Prob.

30%

70%

Profit for our bid

Competitor’s bid 0 115 120 1251. No bid 0 15 20 25

2a. Under $115 0 -5 -5 -5

2b. $115 to under $120 0 15 -5 -5

2c. $120 to under $125 0 15 20 -5

2d. Over $125 0 15 20 25

Actual Prob.

30%

14%

28%

21%

7%


Example 3:Example 3: Profit for our bid

Competitor 0 115 120 1251. No bid 0 15 20 25

2a. < $115 0 -5 -5 -5

2b. $115 to < $120 0 15 -5 -5

2c. $120 to < $125 0 15 20 -5

2d. > $125 0 15 20 25

Actual Prob.

30%

14%

28%

21%

7%

$0

No bid

bid

$115 Win

Lose Payoff = (-5), Probability 14%

Payoff = 15, Probability 86%

(-5)*(0.14) + 15 * (0.86) = $12.2


Profit for our bid

Competitor 0 115 120 1251. No bid 0 15 20 25

2a. < $115 0 -5 -5 -5

2b. $115 to < $120 0 15 -5 -5

2c. $120 to < $125 0 15 20 -5

2d. > $125 0 15 20 25

Example 3:Example 3: Actual Prob.

30%

14%

28%

21%

7%

20, 58%

-5, 42%

25, 37%

-5, 63%

15, 86%

-5, 14%

Bid $120

Bid $115

Bid= $125

L

W

L

W

L

W

No bid

$0

$9.5

$6.1

$12.2 Our decision?

We will now use Excel to solve the problem.


Ex. 3: ExcelEx. 3: Excel =SUMPRODUCT(Profit_bid_115,Probabilities)

=MAX(D9:G9)INDEX+MATCH

HLOOKUP ?

Value we are looking (12.2) is not in the ascending order in the table.


Example 3: Sensitivity analysisExample 3: Sensitivity analysis

What if 30% probability of no bid from competitor is incorrect?

We can build a one variable data table. Variable: Competitor’s no bid probability.

We select two outputs: bid and (corresponding maximum) profit.


Profit for our bid

Competitor’s bid 0 115 120 1251. No bid 0 15 20 25

2a. Under $115 0 -5 -5 -5

2b. $115 to under $120 0 15 -5 -5

2c. $120 to under $125 0 15 20 -5

2d. Over $125 0 15 20 25

Ex. 3: DA and value of informationEx. 3: DA and value of information

Our decision was to bid $115 and EMV was $12.2. Suppose we get competitor’s bid information. Can we improve our profit?

What is the probability?

Earned Value of Perfect Information (EVPI) = $17.65 – $12.2 = $5.45

EMV = 0.3*25+0.14*0+0.28*15+0.21*20+0.07*25 = 17.65

0.300.7 * 0.2 = 0.140.7 * 0.4 = 0.280.7 * 0.3 = 0.210.7 * 0.1 = 0.07

Sometimes we may have partial information.


$0

No bid

OB= $115

$15-$5

$15 $15 $15

OB= $120

OB= $125

CB=0

CB

30%

70%

15(.3)+11(.7) = $12.2

30%

70%

$20

$5$9.5

30%

70%

$25

-$2$6.1

<115115 to <120

120 to < 125>125

30%10%

40%20%

EMV Payoff

-5(.2)+15(.4+.3+.1) = $11

bid$12.2Our

decision

Example 3: Alternate methodExample 3: Alternate method


Values 12.2, 9.5 and 6.1 represent Expected Monetary Values (EMV).

This line indicates the decision made.

This is called folding back the decision tree.

$0OB= $115

OB= $125

$12.2

$9.5

$6.1

bid$12.2

OB= $120

No bid

Example 3…..Example 3…..


Utility theoryUtility theoryConsider the gambling

problems again.– Let us flip a fair coin once.– If you win I give you $102– If I win, you give me $100– How much will you pay me

to play this game: $5, $2, $1, $0 ?

Consider another gamble– Let us flip the same coin

(500 times) with the same payoffs

– How much will you pay me to play this game?

• Different people will pay different amounts to play the first game

Expected payoff in the first game is $1 but most people do not want to play the game at all.

Why? Losing $100 is a bigger event than winning $102

• Most people will play the second game.

Still differ in how much they will pay.

• For most people a gain that is twice as big is not twice as good.

• A loss of twice as much is more than twice as bad.

• People’s attitude towards risk can be categorized as: risk averse, risk seeker and risk neutral.

• A common way to express it is through the decision-maker’s utility function.


0 100 0 100 0 100

U(100)

U(0)

U(100)

U(0)

U(100)

U(0)

0 100 0 100 0 100

U(100)

U(0)

U(100)

U(0)

U(100)

U(0)

Risk seeker Risk averse Risk neutral

Utility is a measure of relative satisfaction. We can plot a graph of amount of money spent vs. “utility” on a 0 to 100 scale. Typical shapes for different types of risk takers generally follow the patterns shown below.

Graphs above show that to achieve 50% utility, risk seekers will pay maximum, risk averse will pay minimum and risk neutral will pay an average amount.

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BU.520.601 BU.520.601 Decision Models DecisionAnalysis1 Summer 2013