Advanced Engineering Project

Management

Dr. Nabil I. El SawalhiAssistant professor of Construction

Management

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Decision trees

Decision trees are tools for classification and prediction.

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Decision Trees

The Payoff Table approach is useful for a non-sequential or single stage.

Many real-world decision problems consists of a sequence of dependent

decisions.

Decision Trees are useful in analyzing multi-stage decision processes.

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DECISION TREES

Used when:

Single stage decision-making is required;

Multi-stage decision-making is required;

Schematic representation is useful.

Consists of: Nodes; commonly represented by squares

Branches; represented by lines

Chances; represented by circles

Probability estimates;

Payoffs.

End nodes - represented by triangles

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Decision nodes require a conscious decision on which branch to choose, typically shown as a square.

Chance nodes show different possible events that can confront a chosen strategy, typically shown as a circle.

Decision Branches represent a strategy or course of action, sometimes shown as two parallel lines.

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Chance Branches represent a chance-determined event, sometimes shown as a

single line.

Terminal Branches mark the end of the decision tree.

Decision trees can be deterministic or probabilistic (stochastic).

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DETERMINISTIC DECISION TREE

Example 1.

Excavator replacement decision

The site manager for Droflas Construction has three alternative choices relating to

the replacement of a mechanical

excavator. They are shown in the payoff

matrix:

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Profit or Payoff ()

Strategy Year 1 Year 2 Total

S1 : Replace Now 4000 6000 10000

S2 : Replace after

1 year

5000 4000 9000

S3 : Do not

Replace

5000 3000 8000

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Draw the appropriate decision tree and

identify the appropriate solution

10000

DN#1

DN#2

First year Second year

Repla

cenow

Do not replace Repl

ace

Do not replace

4000

6000

5000 4000

3000

Decision Tree

9000

8000

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Example 2

A manager has developed a table that shows ($000) for future store. The payoffs depend on the size of the store and the strength of demand:

Small 30 50 Large 10 80 The manager estimate that the probability of low

demand is equal to the probability of high demand. The manager could request that a local research firm conduct a survey (cost $2000) that would better indicate wither demand will be low or high. In discussion with the research firm the manager has learned the following about the reliability of survey conducted by the firm.

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Actual results

Low high

Survey showed low 0.9 0.3

high 0.1 0.7

a. if the manager should decide to use the survey, what would the revised probabilities be demand and what

probabilities should be used for survey results (i.e.

survey shows high demand)

B. construct a tree diagram

C. determine the EMV

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A. the following are revised probabilities if survey shows low demand

Actual

demand

Conditio

nal p

Prior p Joint p Revised p

low 0.9 x 0.5 = .45 .45/.6=.75

high 0.3 x 0.5 = .15 .15/.6=.25

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A. the following are revised probabilities if

survey shows low demand

Actual demand

Condition

al p

Prior p Joint p Revised p

low 0.1 x 0.5 = .05 .45/.4=.125

high 0.7 x 0.5 = .35 .15/.4=.875

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d1

d2

d3

No Survey

Survey

Low p .6

HD p .4

Large

Small.25

.7530

50

10

80

30

50

10

80

.25

.75

.125

.875

.125

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d1

d4

Large

Small

. 5

. 5 30

50

10

80

. 5

. 5

No Survey

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Example 3

The Metal Discovery Group (MDG) is a company set up to conduct geological explorations of parcels of land in

order to ascertain whether significant metal deposits

(worthy of further commercial exploitation) are present or

not. Current MDG has an option to purchase outright a

parcel of land for 3m.

If MDG purchases this parcel of land then it will conduct a geological exploration of the land. Past experience

indicates that for the type of parcel of land under

consideration geological explorations cost approximately

1m and yield significant metal deposits as follows:

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manganese 1% chance

gold 0.05% chance

silver 0.2% chance

Only one of these three metals is ever found (if at all), i.e. there is no chance of finding two or more of these

metals and no chance of finding any other metal.

If manganese is found then the parcel of land can be sold for 30m, if gold is found then the parcel of land can be sold for 250m and if silver is found the parcel of land can be sold for 150m.

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MDG can, if they wish, pay 750,000 for the right to conduct a three-day test exploration before deciding

whether to purchase the parcel of land or not. Such

three-day test explorations can only give a preliminary

indication of whether significant metal deposits are

present or not and past experience indicates that three-

day test explorations cost 250,000 and indicate that significant metal deposits are present 50% of the time.

If the three-day test exploration indicates significant metal deposits then the chances of finding manganese,

gold and silver increase to 3%, 2% and 1% respectively.

If the three-day test exploration fails to indicate

significant metal deposits then the chances of finding

manganese, gold and silver decrease to 0.75%, 0.04%

and 0.175% respectively.

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What would you recommend MDG should do and why?

A company working in a related field to MDG is prepared to pay half of all costs associated with this parcel of land

in return for half of all revenues. Under these

circumstances what would you recommend MDG should

do and why?

Below we carry out step 1 of the decision tree solution procedure which (for this example) involves working out

the total profit for each of the paths from the initial node

to the terminal node (all figures in '000000).

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Step 1

path to terminal node 8, abandon the project - profit zero

path to terminal node 9, we purchase (cost 3m), explore (cost 1m) and find manganese (revenue 30m), total profit 26 (m)

path to terminal node 10, we purchase (cost 3m), explore (cost 1m) and find gold (revenue 250m), total profit 246 (m)

path to terminal node 11, we purchase (cost 3m), explore (cost 1m) and find silver (revenue 150m), total profit 146 (m)

path to terminal node 12, we purchase (cost 3m), explore (cost 1m) and find nothing, total profit -4 (m)

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path to terminal node 13, we conduct the three-day test (cost 0.75m + 0.25m), find we have an enhanced chance of significant metal deposits, purchase and

explore (cost 4m) and find manganese (revenue 30m), total profit 25 (m)

path to terminal node 14, we conduct the three-day test (cost 0.75m + 0.25m), find we have an enhanced chance of significant metal deposits, purchase and

explore (cost 4m) and find gold (revenue 250m), total profit 245 (m)

path to terminal node 15, we conduct the three-day test (cost 0.75m + 0.25m), find we have an enhanced chance of significant metal deposits, purchase and

explore (cost 4m) and find silver (revenue 150m), total profit 145 (m)

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path to terminal node 16, we conduct the three-day test (cost 0.75m + 0.25m), find we have an enhanced chance of significant metal deposits, purchase and

explore (cost 4m) and find nothing, total profit -5 (m)

path to terminal node 17, we conduct the three-day test (cost 0.75m + 0.25m), find we have an enhanced chance of significant metal deposits, decide to abandon,

total profit -1 (m)

path to terminal node 18, we conduct the three-day test (cost 0.75m + 0.25m), find we have an reduced chance of significant metal deposits, purchase and

explore (cost 4m) and find manganese (revenue 30m), total profit 25 (m)

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path to terminal node 19, we conduct the three-day test (cost 0.75m + 0.25m), find we have an reduced chance of significant metal deposits, purchase and

explore (cost 4m) and find gold (revenue 250m), total profit 245 (m)

path to terminal node 20, we conduct the three-day test (cost 0.75m + 0.25m), find we have an reduced chance of significant metal deposits, purchase and

explore (cost 4m) and find silver (revenue 150m), total profit 145 (m)

path to terminal node 21, we conduct the three-day test (cost 0.75m + 0.25m), find we have an reduced chance of significant metal deposits, purchase and

explore (cost 4m) and find nothing, total profit -5 (m)

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path to terminal node 22, we conduct the three-day test (cost 0.75m + 0.25m), find we have an reduced chance of

significant metal deposits, decide to

abandon, total profit -1 (m)

Hence we can arrive at the table below indicating for each branch the total profit

involved in that branch from the initial

node to the terminal node.

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Terminal node Total profit

8 0

9 26

10 246

11 146

12 -4

13 25

14 245

15 145

16 -5

17 -1

18 25

19 245

20 145

21 -5

22 -1 AEPM L11 27

We can now carry out the second step of the decision tree solution procedure where we work from the right-

hand side of the diagram back to the left-hand side.

Step 2

Consider chance node 7 with branches to terminal nodes 15-21 emanating from it. The expected monetary value

for this chance node is given by

0.0075(25) + 0.0004(245) + 0.00175(145) + 0.99035(-5) = -4.4125

Hence the best decision at decision node 5 is to abandon (EMV=-1).

The EMV for chance node 6 is given by 0.03(25) + 0.02(245) + 0.01(145) + 0.94(-5) = 2.4

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Hence the best decision at decision node 4 is to purchase (EMV=2.4).

The EMV for chance node 3 is given by 0.5(2.4) + 0.5(-1) = 0.7

The EMV for chance node 2 is given by 0.01(26) + 0.0005(246) + 0.002(146) + 0.9875(-4) = -3.275

Hence at decision node 1 have three alternatives:

abandon EMV=0

purchase and explore EMV=-3.275

3-day test EMV=0.7

Hence the best decision is the 3-day test as it has the highest expected monetary value of 0.7 (m).

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Sharing the costs and revenues on a 50:50 basis merely halves all the monetary

figures in the above calculations and so

the optimal EMV decision is exactly as before. However in a wider context by

accepting to share costs and revenues the

company is spreading its risk and from

that point of view may well be a wise offer

to accept.

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STOCHASTIC DECISION TREES

Example 4

Based upon the recommendations of their strategic planning group, Droflas Associates has decided to expand their present organisation. Having considered several alternatives, the following strategies were considered to be viable options:

Strategy A: Build a large office with an estimated cost of 2M.

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This alternative can face two states of nature (market conditions), high demand for

surveying services with a probability of 0.7 or

low demand with a probability of 0.3. If the

demand is high, the company can expect to

receive an annual cash flow of 500000 for 7 years.

If the demand is low, the annual cash flow would be only 100000 because of the large fixed costs and inefficiencies caused by the

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Strategy B: Build a small office with an estimated cost of 1M.

This alternative also faces two states of nature, high demand with a probability of 0.7 and low

demand with a probability of 0.3. The company

expects to receive an annual cash flow of

300000 or 150000 if demand is high or low respectively. If the demand is low and remains

low for 2 years the office will certainly not be

expanded.

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However, if initial demand is high and remains high for 2 years they will face another decision of whether or not to expand the office. It is assumed that the cost of expanding the office at that time will be 1.5M. Further, it is assumed that after this second decision, the probabilities of high and low demand will remain the same.

If the decision to expand is made, the company then expects to receive an annual cash flow of 600000 or 100000 if the demand is high or low respectively.

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Which is the optimal strategy?

Elements needed to construct a decision tree:

All decision and chance nodes; Branches that connect various decision and

chance nodes;

Payoff (reward or cost), if any, associated with branches emanating from decision nodes;

Probability value associated with branches emanating from chance nodes;

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Payoffs associated with each chance node;

Payoffs associated with each terminal branch at the conclusion of each path that can be traced through various combinations that form the tree;

Position values of chance and decision nodes;

The process of rollback.

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Some possible refinements:

The sequence of decisions can involve a larger number of decisions;

At each decision node, consider a larger number of strategies;

At each chance node, consider a larger number of chance branches, or assume a continuous probability distribution at each chance node;

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More sophisticated and more detailed projections of cash flows can be introduced;

Discounted cash flows can be introduced; The quality of risk can be explicated by

estimating the range or standard deviation of the payoff distribution for each path;

Sensitivity testing and sensitivity analysis can be introduced.

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DN#2

DN#1

CN#1

CN#2

CN#4

CN#3

2 YEARS 5 YEARS

A1

A2

B1

B2

B3

B4

B5

HD .7, cash .5m

LD .3 cash .1m

Small

office,

1m

Large

office

2m

HD .7 ,

cash 0.3

LD .3 , 0.15 m

Expand

1.5m

Not

expand

HD .7,

cash .6

LD .3 ,

0.1

HD .7,

cash .3

LD .3 ,

0.15

2.25

1.275

2.66m

1.402

1.275

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Large office

EMV1=-2+0.7x.5x7+ 0.3x0.1x7 = 0.66m(best strategy)

Small office expand after 2y

DN #2 =.7x.3x5 + .3x.15x5=1.275m

Small office Not Expand

EMV2 =-1+ 1.275x.7 + .7x.3x2 + .3x .15x 2=0.402m

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