UNIT 5 – DECISION MAKING UNDER UNCERTAINTY · • Decision trees are also used for displaying decision problems with uncertainty. • If probabilistic information regarding the

School of the Built Environment

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UNIT 5 – DECISION MAKING

UNDER UNCERTAINTY

• This unit:

– Discusses the techniques to deal with uncertainties


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INTRODUCTION

• Few decisions in construction industry are made with certainty.

• Need to look at:

– The magnitudes of uncertainties

– How uncertainties might affect decisions

– How uncertainties influence subsequent outcomes of decisions


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TRADITIONAL STRATEGY

• A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.

• The decision alternatives are the different possible strategies the decision maker can employ.

• The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.


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PAYOFF TABLES

• The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff.

• A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table.

• Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.


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EXPECTED VALUE APPROACH

• If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach.

• Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring.

• The decision yielding the best expected return is chosen.


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• The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative.

• The expected value (EV) of decision alternative di is defined as:

where: N = the number of states of nature

P(Sj ) = the probability of state of nature sj

Vij = the payoff corresponding to decision alternative di and state of nature sj


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Example – lets consider the example of a housing contractor who builds residential houses. The contractor is planning for the number of houses to be built for the coming year. Assume that each house costs $15,000 and sells for $0,000 and that the probability distribution of the market demand in this area for new houses is know as:

P0 = Prob (demand = 00) = 0.2

P1 = Prob (demand = 10) = 0.4

P2 = Prob (demand = 20) = 0.3

P3 = Prob (demand = 30) = 0.1

How many houses should the contractor build for the year?


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What are the states of nature? # of houses sold

What are the decision alternatives? # of houses built

The payoff table is shown below.

Decision

(houses

to be

built)

State of nature

(Sold houses)

0 10 20 30

0 0 0 0 0

10 -150 350 350 350

20 -300 200 700 700

30 -450 50 550 1050


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The probability distribution of market demand is added to the payoff table to calculate the EV of each decision

The maximum EV is when the contractor builds 20 houses.

Decision

(houses

to be

built)

State of nature

(Sold houses)

0 10 20 30 EV

0 0 0 0 0 0

10 -150 350 350 350 250

20 -300 200 700 700 300

30 -450 50 550 1050 200

Prob DD 0.2 0.4 0.3 0.1


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EXPECTED VALUE OF

PERFECT INFORMATION

• Frequently information is available which can improve the probability estimates for the states of nature.

• The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur.

• The EVPI is the price that one would be willing to pay in order to gain access to perfect information.

• The EVPI provides an upper bound on the expected value of any sample or survey information.


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EXPECTED VALUE OF

PERFECT INFORMATION

• EVPI Calculation

– Step 1:

Determine the optimal return corresponding to each state of nature.

– Step 2:

Compute the expected value of these optimal returns.

– Step 3:

Subtract the EV of the optimal decision from the amount determined in step (2).


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EXPECTED VALUE OF

PERFECT INFORMATION

Decision

(houses

to be

built)

State of nature

(Sold houses)

0 10 20 30 EV

0 0 0 0 0 0

10 -150 350 350 350 250

20 -300 200 700 700 300

30 -450 50 550 1050 200

Prob DD 0.2 0.4 0.3 0.1

Optimal 0 350 700 1050 455

EVPI = 455 – 300 = 155


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DECISION TREES

• A decision tree is a chronological representation of the decision problem.

• They provide a highly effective structure within which one can lay out options and investigate the possible outcomes of choosing those options.

• They also help you to form a balanced picture of the risks and rewards associated with each possible course of action.


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DECISION TREES

• A decision tree is read from left to right.

• Each decision tree has two types of nodes:

– round nodes correspond to the states of nature, outcome or chance event

– square nodes correspond to the decision alternatives

• Lines connecting the nodes show the direction of influence


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DECISION TREES

• The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.

• At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.


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DECISION TREES

Decision node

Decision alternatives:1. Develop temp sensor2. Develop pressure sensor3. Do neither

State of nature nodes

States of nature:1. Success 2. Failure

Payoffs


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DECISION TREES

WITHOUT PROBABILITIES

• Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are:

– the optimistic (Maximax) approach

– the conservative (Maximin) approach

– the minimax regret approach


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OPTIMISTIC APPROACH

MAXIMAX

• Used by an optimistic decision maker.

• The decision with the largest possible payoff is chosen.

• If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.


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CONSERVATIVE APPROACH

MAXIMIN

• Used by a conservative decision maker.

• For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.)

• If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)


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MINIMAX REGRET APPROACH

• Requires the construction of a regret table or an opportunity loss table.

• This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature.

• Then, using this regret table, the maximum regret for each possible decision is listed.

• The decision chosen is the one corresponding to the minimum of the maximum regrets


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EXAMPLE 1

• Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits


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EXAMPLE 1

d1

d2

d3

S1

S1

S1

S2

S2

S2

S3

S3

S3

4

4

-2

0

3

-1

1

5

-3


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EXAMPLE 1 - MAXIMAX

• An optimistic decision maker would use the optimistic (maximax) approach. We choose the decision that has the largest single value in the payoff table.

Maximax decision Maximax

payoff


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EXAMPLE 1 - MAXIMAX

d1

d2

d3

S1

S1

S1

S2

S2

S2

S3

S3

S3

4

4

-2

0

3

-1

1

5

-3

5

3

4


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EXAMPLE 1 - MAXIMIN

• A conservative decision maker would use the conservative (maximin) approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs

Maximin decision

Maximin payoff

Minimum Payoff

-2

-1

-3


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EXAMPLE 1 - MAXIMIN

d1

d2

d3

S1

S1

S1

S2

S2

S2

S3

S3

S3

4

4

-2

0

3

-1

1

5

-3

-3

-1

-2


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EXAMPLE 1 – MINIMAX REGRET

• For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column.

States of nature

Decision S1 S2 S3

d1 4 4 -2

d2 0 3 -1

d3 1 5 -3

Column Max 4 5 -1

Regret table S1 S2 S3

d1 0 1 1

d2 4 2 0

d3 3 0 2

=-1-(-2)= 4 - 0


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EXAMPLE 1 – MINIMAX REGRET

• For each decision list the maximum regret. Choose the decision with the minimum of these values.

.

Regret table S1 S2 S3 Max regretd1 0 1 1 1d2 4 2 0 4d3 3 0 2 3

Minimax regret

Minimax regret

decision


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EXAMPLE 1 - MINIMAX REGRET

d1

d2

d3

S1

S1

S1

S2

S2

S2

S3

S3

S3

4

4

-2

0

3

-1

1

5

-3

3

4

1


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DECISION TREES

WITH PROBABILITIES

• Decision trees are also used for displaying decision problems with uncertainty.

• If probabilistic information regarding the states of nature is available, use the expected value (EV) approach.

• The EV for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring.

• The decision yielding the best EV is chosen.


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EXAMPLE 2

Company ABC has developed a new line of products. Top managementis attempting to decide on the appropriate marketing and productionstrategy. 3 strategies are being considered: A (Aggressive), B (basic) andC (Cautious). The market conditions under study are denoted by S(Strong) or W (Weak). Management's best estimate of the net profits (in$, mil) in each case is given in the following payoff table.

Management's estimates of the probabilities of a strong or a weak market are 0.45 and 0.55 respectively. Which strategy should be chosen.

DecisionState of nature

S W

A 30 -8

B 20 7

C 15 10


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SOLUTION 2

A

B

C

P=0.4530

-8

20

7

15

10

P=0.45

P=0.45

P=0.55

P=0.55

P=0.55

EV=9.1

EV=12.85

EV=12.25


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SENSITIVITY ANALYSIS

• Other critical points of the studied decision can be concluded from the decision tree analysis.

• Using the same facts from example 2:

EVA = 30*P(S) – 8*P(W); since P(S) + P(W) =1, then

EVA = 30*P(S) – 8*[1-P(S)]; which simplifies into

EVA = 38P(S) – 8; which is the equation of a straight line


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• Similarly solving for EVB & EVC gives:

EVB = 7 + 13P(S); and

EVC = 10 + 5P(S)

• All 3 equations can be plotted with EV as the y-axis and P(S) as the x-axis.

• The resultant graph is shown in next slide


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EV

-10

-5

0

5

10

15

20

25

30

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

EVA EVB EVC

P(S)

C strategyP(S) < 0.38

B strategyA strategyP(S) > 0.6


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CLASS EXERCISE 1

An investment performance is significantly affected by twovariables: the economic environment and whether acompeting building is developed. The cash flow under thevarious scenarios is as follows:


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CLASS EXERCISE 1

Through an assessment of the economic environment, theprobability estimates of the likelihood of each of the 3 economicsituations under consideration is:

The conditional probability of whether the competing building willbe built under the 3 economic conditions is estimated as follows:

What is the expected value of the investment?


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CLASS EXERCISE 2

Burger Prince Restaurant is considering opening a new

restaurant on Main Street. It has three different models, each

with a different seating capacity. Burger Prince estimates that the

average number of customers per hour will be 80, 100, or 120.

The payoff table for the three models is:

Given that the probability of S1, S2 and S3 is 0.4, 0.2 and 0.4

respectively, calculate the expected value for each decision & the

expected value of perfect information.


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CLASS EXERCISE 3

A contractor is concerned about the capacity of the existing

dewatering system to be used in a new project to keep the site

dry in order to prevent progress delay. Additional pumps will

add costs to the budget, and progress delay will cause

penalties to the contractor. Three scenarios (S1, S2 and S3)

are expected to make the water level in the construction site

high. S1 (the rain will be < 6 inch in 12 hour period) has the

probability of 0.5, S2 (the rain will reach 6 inch only one time in

12 hour period) has the probability of 0.3, and S3 (the rain will

reach 6 inch many times in 12 hour period) has the probability

of 0.2. The cost of pumps installation and the penalties for any

delay are shown in the following table.


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CLASS EXERCISE 3

A. Draw a decision tree to represent this problem

B. Which alternative is the best choice? And Why?

Alternative S1 S2 S3

Install 15,000 15,000 65,000

Do nothing 0 20,00 100,000


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CLASS EXERCISE 4

Two pumping systems “A” and ”B” are proposed for supplying

water to a residential area. The construction costs for system

“A” is $250,000 and for system “B” is $750,000. if partial failure

occurs, it is expected that the damage cost for system “A” is

$80,000 and for system “B is $15,000. If complete failure

occurs, it is expected that the damage cost for system “A” is

$450,000 and for system “B” is $400,000. the probabilities of

partial and complete failures are 5% and 1% respectively.

a. Draw a decision tree to represent this problem

b. Which alternative is the best choice? And Why?


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CLASS EXERCISE 5

Adrian is a developer and must build factories for the coming

year. Construction for the factories must be in quantities of 20.

The cost per factory is $70 if they build 20, $67 if they build 40,

$65 if they build 60, and $64 if they build 80. The factory will

be sold for $100 each. Any factory left over at the end of the

season can be sold (for certain) at $45 each. If Adrian run out

of factories during the year, then they will suffer a loss of

"goodwill" among their customers. They estimate this goodwill

loss to be $5 per customer who was unable to buy a factory.

Adrian estimate that the demand for factory this season will be

10, 30, 50, or 70 factory with probabilities of 0.2, 0.4, 0.3, and

0.1 respectively.


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CLASS EXERCISE 5

a. What are the decisions facing Adrian?

b. What are the states of nature?

c. Construct the payoff table.

d. What would be Adrian’s decision based on Maximax,

Maximin and Minimax Regret criterion?


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THE END

Any questions?

Documents

UNIT 5 – DECISION MAKING UNDER UNCERTAINTY · • Decision trees are also used for displaying decision problems with uncertainty. • If probabilistic information regarding the