Question 1 (10 points)
A general contractor has three options when bidding for three projects. He has the resources to undertake
only one of the projects and must select the most profitable option. The first option is to act as a
general contractor submitting a lump sum bid for a bridge project. The project has a 60% chance of
generating $400,000 profit and a 40% chance of $200,000 loss. The contractor estimates that it will cost
$5,000 to prepare the bid for the 1st option. The second project is a design and build scheme for a new
pumping station at a waterworks. Calculations show that the 2nd option has 90% chance of potential
profit $220,000, but a 10% chance of a $100,000 loss. There shall be $20,000 bidding cost for the
2nd option. The third project is a management contract for a building project. The project has a 95%
chance of generating $180,000 profit and a 5% chance of a $20,000 loss. Cost of bidding for the 3rd
option is $10,000. Which option is the most profitable? Show your calculations by drawing a decision
tree. EMV=Probability x Impacts
Question 2 (10 points)
A technology company must decide whether to bid for a government contract to develop a piece of
equipment, and if the company decides to bid, it must decide how much to bid. The government will
award the contract to the low bidder. However, there is 20% probability that there will not be any other
bidders. If there are other bidders, there is a 20% chance of underbid competitors if the bid price is
$160K; 60% chance of underbid competitors if the bid price is $170K; and 90% chance of underbid
competitors if the bid price is $180K. The company estimates that the cost of placing a bid is $7.5K and
the cost of developing the equipment, given that it wins the contract, is $150K. So, for example, if the
company bids $170K and wins the bid, its profit will be $20K less the cost of placing the bid.
Show your calculations by drawing a decision tree. EMV=Probability x Impacts
Single Criteria Decision Making: EMV Method
Single Criteria Decision Making: Decision Tree using the
EMV Method
Introduction
In day-to-day work, construction managers may face problems which involve
probability, that is, for any one particular action that a manager takes, there may be
several probable outcomes.
Let us consider a simple example. A contractor has to decide whether to rent
concrete pumping equipment in order to complete a foundation work tomorrow.
If he does not rent the equipment, work will be delayed and he will suffer a loss
of $10,000. If he rents the equipment, he anticipates two possible outcomes,
depending on the weather:
a. If the weather is fine, the concrete pumping equipment will be fully
utilized. The concrete will be completed and he will gain a profit of
$20,000 after deducting the rent for the equipment.
b. If there is rain, the equipment cannot function and the work will be
delayed. The contractor will have to pay $10,000 rent and he will also
suffer a loss of $10,000 for the delay of work.
In the above example, the two outcomes involve probability. But it is possible for the
contractor to estimate, by the guesswork or with the help of historical rainfall records
provided by the Meteorological Observatory, the probability of each outcome.
Having determined the probability of the two possible outcomes, he can make a
decision whether or not to rent the equipment. This chapter describes a mathematical
method called decision analysis which helps the contractor reach a decision.
Single Criteria Decision Making
Decision Tree Technique
The first step in decision analysis is to identify the various actions which can be taken
at a decision point and the probable outcomes of each action taken. In the example
given in Section 1, the contractor has arrived at a decision point where he can take
either of the two actions (that is, rent or not rent the equipment). There is only one
outcome of the first action (there is delay of work and he suffer a loss). The second
action has two probable outcomes (either the weather is fine or there is rain). Other
information needed for decision analysis is the probability of occurrence of each
outcome, together with its outcome value (a measure of the profit or loss associated
with the outcome). In this example, the probability of occurrence of each outcome has
not been further investigated, but it can be known after obtaining past rainfall records
from the Meteorological Observatory. The outcome value is the profit depending on
the weather conditions (for example, the outcome value if the weather is fine is
$20,000 and the value if it rains is -$20,000).
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Single Criteria Decision Making: EMV Method
The data is usually represented in a decision tree, which forms the decision model of
the problem. Another example is given below to demonstrate how a decision tree is
drawn.
Example 1
A contractor succeeds in bidding for a contract at a tender price of $20 million.
He estimates that there are three possible outcomes in completing the contract:
i
the actual construction cost is $15 million (i.e. profit $5 million) with a
probability of 0.2;
ii. the actual construction cost is $17 million (i.e. profit $3 million) with a
probability of 0.65; or
iii. the actual construction cost is $19 million (i.e. profit of $1 million) with a
probability of 0.15.
Our task is to express the above data in the form of a decision tree for the contractor.
Solution
The decision tree is shown below in Fig. 1:
$5m
$3m
$1m
Fig 1 Example of a very simple decision tree
A decision node is a branching point in a decision tree where alternative actions can
be taken. In this example, the contractor offers only one tender (one action). Hence,
there is only one branch from this node see Fig. 1.
Branching out from an action is a chance node. It is a branching point in a decision
tree where various possible outcomes occur. In this example, there are three possible
outcomes of the chance node. The probability of each outcome is written on the
branch line.
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Single Criteria Decision Making: EMV Method
Corresponding to each branch from a chance node is an outcome node. In Fig. 1, the
possible outcome values, or profits of each outcome ($5 million, $3 million and $1
million), have been written next to each outcome node.
Let us now consider a more realistic example.
Example 2
A contractor is performing some work in a coastal area which is subject to
destructive hurricane conditions. During the course of the construction work he
has to provide storage for his plant for a week. The contractor can either leave
the plant on site or he can move it to a hurricane-safe storage place.
If he keeps the plant idle on site, he can either build a protective shelter at a cost
of $30,000 which will protect the plant against minor (but not major) hurricanes,
or he can leave the plant unprotected on site with no cost incurred. However, he
then risks losing the plant if a hurricane, both major or minor, occurs.
On the other hand, he can take the plant off the site and move it to a safe place,
store it there for a week, and then move it back to the site at a total cost of
$42,500.
The plant, which cost $400,000, will be destroyed by either a minor or major
hurricane if it is left unprotected on site. A major hurricane but not a minor
hurricane will also destroy it if a protective shelter is built on site. Hurricanes
cannot damage the plant if it is stored in the safe place away from the site. The
probabilities of being hit by a major or a minor hurricane during the week, as
estimated from the past records of the Meteorological Observatory, are 0.01 and
0.09 respectively.
The task is to draw a decision tree for the contractor.
Solution
(i) The decision tree if the plant is left unprotected is given below.
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Single Criteria Decision Making: EMV Method
Fig 2
Decision tree for unprotected plant
(ii) The decision tree if the plant is left on site protected by a shelter is given below:
Fig. 3a Decision tree for plant protected by a shelter
It can also be presented as shown in Fig. 3b. In fact it is a better presentation than
Fig. 3a.
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Single Criteria Decision Making: EMV Method
Fig. 3b Decision tree for plant protected by a shelter (using alternative
presentation)
(iii) The whole problem can be presented by the following decision tree:
Fig. 4a Decision tree representing the whole problem
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Single Criteria Decision Making: EMV Method
It can also be represented as shown in Fig. 4b. As mentioned before, it is a better
form of presentation than Fig. 4a. In certain cases, such as decision trees for finding
the value of information, this form of presentation must be used.
Fig. 4b Decision tree representing the whole problem (using alternative
presentation)
Expected Monetary Value (EMV)
Before going on to discuss the procedure for finding the optimal decision, we have to
know what an expected monetary value (EMV) attached to a particular action is.
The EMV of the outcome branching from a chance node is defined as the sum of the
products of the outcome values and their respective probabilities. Mathematically, it
can be written as follows:
n
EMV =
px
i =1
i
i
where n = number of outcomes branching from a chance node
xi = value of outcome i, and
pi = probability of occurrence of outcome i.
The EMV of the outcomes for Example 1 in Section 2 is therefore calculated as 0.2 x
5,000,000 + 0.65 x 3,000,000 + 0.15 x 1,000,000 = 3,100,000.
The EMV is $3,100,000. This is the expected profit of the contract. It must be noted,
however, that the profit arising from this contract may not be exactly $3,100,000. This
profit is only an average figure. This means that if a considerable number of similar
contracts were to be taken, then the profit for such contracts would be $3,100,000 on
average. It is unlikely that this particular contract will bring a profit of exactly
$3,100,000 if it is done just once.
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Single Criteria Decision Making: EMV Method
The EMV definition is in fact obtained from the definition of average (or mean, or
expected) value. In statistics, the definition of an average is given by the following
expression:
i=n
f x
Average =
i =1
i=n
i
f
i =1
i
i
where xi = value of outcome i (i=1, …n), and
fi = frequency of occurrence of outcome i.
The EMV of a set of outcomes is actually the average of the outcome values. The
proof is given below:
Average of the outcome values = (f1x1 + f2x2 + … + fnxn) / fi
= (f1/fi)xi + (f2/fi)x2 + … + (fn/fi)xn
= p1 x1 + p2 x2 + … + pn xn
(where pi = probability of event i)
i=n
Therefore, average =
px
i =1
i
i
= EMV
Using EMV to Solve a Decision Problem
Section 2 shows how to represent a decision problem in the form of a decision tree,
and Section 3 explains the EMV criterion. The next step is to solve a decision
problem.
Let us use the decision model in Example 2 as an example. The problem may be
solved using a number of steps.
Step 1
Set up the decision tree for the problem.
The model is reproduced below for easy reference. For convenience, the decision
nodes are labeled A and B and the chance nodes are numbered 1 and 2.
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Single Criteria Decision Making: EMV Method
Fig. 5 Decision tree representing the problem in Example 2
Step 2
Find the EMVs at the various chance nodes and enter the values onto the decision tree.
The EMV at chance node 1 (EMV1) is -$40,000 and the EMV at chance node 2
(EMV2) is -$34,000. The calculations are left to the readers
Step 3
At decision node B, compare the EMVs and select the one with a higher value (i.e.
maximum profit or minimum loss). In this case, EMV2 is selected as it represents the
minimum loss. Write
on the route which has not been selected.
At decision node A, compare the EMVs of the two routes which are left uncompared
and select the one with the smaller loss. The EMVs here are -$34,000 (EMV2) and
-$42,500 and so EMV2 is selected and an
Choose the path without an
is put on the unselected route.
on the route. This path represents the optimal decision.
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Single Criteria Decision Making: EMV Method
Fig. 6 Finding the optimal path
We can now reach a decision. The plant should not be moved, but a shelter should be
built to protect it from minor hurricanes. Such a decision has an expected monetary
value of -$34,000. The negative sign indicates that it is a loss rather than a profit.
The same answer can be obtained if Fig 4b is used. This exercise is left to the readers.
It is here reiterated that Fig. 4b is a better form and should be encouraged to use,
although the above example has not used Fig 4b to illustrate the calculation.
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Single Criteria Decision Making: EMV Method
5 Summary
The following is a summary of the steps for making optimal decision based on the
EMV criterion.
1. Represent a decision problem by a decision tree. The tree is drawn after
alternative actions are identified and the values of the outcomes and their
respective probabilities are known.
2. Use the EMV criterion as the basis for decision making. Calculate the EMVs at
the chance nodes. Start with the chance nodes nearest to the tips of the tree (at the
most right-hand side of the tree).
3. Beginning with tip of the decision tree, compare the EMVs of the alternative
actions at each decision node. Select the alternative action with the highest EMV
(i.e. the maximum expected profit or minimum expected loss) and assign this
EMV to that decision node.
4. Proceed node by node from right to left. Put an
been selected.
on the routes which have not
5. The overall decision is given by the route (from left to right) with no
BSCI 5460/6460: Planning and Decision Making in Construction
signs.
Page 10 of 10
Go to the website ~ www.projectrisk.com
Use Decision Trees to Make
Important Project Decisions1
By David T. Hulett, Ph.D.
Introduction
A large part of the risk management process involves looking into the future, trying to
understand what might happen and whether it matters. An important quantitative technique
which has been neglected in recent years is enjoying something of a revival – decision trees.
Decision tree analysis is included in the PMBOK® Guide as one of the techniques of
Quantitative Risk Analysis.
At heart the decision tree technique for making decisions in the presence of uncertainty is really
quite simple, and can be applied to many different uncertain situations. For instance: Should we
use the low-price bidder? Should we adopt a state-of-the-art technology? While making many
decisions is difficult, the particular difficulty of making these decisions is that the results of
choosing the alternatives available may be variable, ambiguous, unknown or unknowable.
While it may be easy to make a decision for which the results are known, we need a rule to make
decisions in an uncertain world. That rule is based on probability, the language most useful for
describing and analyzing the future. If the future were certain we would probably decide to take
the path that promises the highest value or lowest cost. With uncertainty, we will generally take
the path which has the highest expected monetary value or lowest expected cost. These concepts
combine the probability that an event will occur with the impact if it does; in other words,
expected monetary value and expected cost follow the definition of project risk in the PMBOK®
Guide (namely “an uncertain event or condition that, if it occurs, has a positive or negative effect
on at least one project objective”).
Many decisions are like this in risky projects, and we often need to make a decision even if we
do not know for sure how it will turn out. These can be very important decisions for the project,
and making them correctly increases the possibility of project success.
Simple Decision – One Decision Node and Two Chance Nodes
We can illustrate decision tree analysis by considering a common decision faced on a project.
We are the prime contractor and there is a penalty in our contract with the main client for every
day we deliver late. We need to decide which sub-contractor to use for a critical activity. Our
aim is to minimize our expected cost. It is often difficult to argue for using the higher-priced subcontractor, even if that one is known to be reliable. The lower-bidding sub-contractor also
promises a successful delivery, although we suspect that he cannot do so reliably. A rigorous
analysis of this decision using a simplified decision tree structure that minimizes our expected
cost is shown below:
•
•
One sub-contractor is lower-cost ($110,000 bid). We estimate however that there is a
50% chance that this contractor will be 90 days late and our contract with the main client
specifies that we must pay a delay penalty of $1,000 per calendar day for every day we
deliver late.
The higher-cost sub-contractor bids $140,000. We know this contractor and assess that it
poses a low 10% chance of being late, and only 30 days late at that. Of course, our
customer will impose on us the same $1,000 delay penalty per day for late delivery.
We need to know if there is any benefit to using the higher-cost sub-contractor, and we suspect it
may lie in the greater reliability of performance we expect. Of course, both we and our customer
need to be convinced of the benefit. A formal analysis using decision trees will ascertain if there
is a benefit, and will also document it for the customer.
The steps we need to implement are as follows:
•
•
Identify the major decisions to be made (decision nodes) and the major uncertainties
(event nodes) that relate to the consequences.
Construct the structure of the decision and all of its (main) consequences. Because each
decision or event node has at least two alternatives, the structure of the decision looks
like a tree, typically placed on its side with the root on the left and the branches on the
right, with potentially many branches.
Figure 1: Contractor Decision - Basic Decision Tree Structure
•
Estimate the costs and benefits of each alternative decision. This is usually a task of
some importance since the final result of the decision tree analysis will depend largely on
the accuracy of these comparative estimates.
•
Calculate the value of the project for each path, beginning on the left-hand side with the
first decision and cumulating the values to the final branch tip on the right as if each of
the decisions were taken and each event occurred. This action is called “rolling
forward.” For example, taking the top-most branch, the lower bidder bids $110,000 and if
we are 90 days late because he is late, $90,000 in penalties will be added by our customer
for a total cost to us of $200,000. This rolling forward calculation of the four possible
path values is shown in Figure 2 below.
Figure 2: Contractor Decision – Rolling Forward to find the Path Values
•
To identify the correct decision and its value (to “solve the tree”) we need to estimate
some additional data, namely the probability of each possible uncertain outcome.
Estimating these probabilities is not as easy as it might appear, since there are often no
useful databases from which to extract the data. Expert judgment is usually required and
that judgment may be poorly-informed or biased.
•
To solve the tree we must calculate the value of each node – including both chance nodes
and decision nodes. We start with the path values at the far right-hand end of the tree and
then moving from the right to the left calculate the value of each node as it is
encountered. This process is called “folding back” the tree.
The rules for finding the values of the chance and decision nodes are:
The value of each chance node is found by multiplying the values of the uncertain
alternatives by their probabilities of occurring and sum the results. This value is
known as Expected Monetary Value (EMV).
o The value of a decision node is the highest value of the succeeding branches
leading from that node.
o
This procedure is illustrated in Figure 3 below.
Figure 3: Contractor Decision - Folding back to find the EMV
The “folding back” calculations for our simple example are as follows:
•
•
For the “Low Bidder but Risky” alternative
1. Multiply -$200,000 by 50% for -$100,000
2. Multiply -$110,000 by 50% for -$55,000
3. Add these numbers (- $100,000 - $55,000) for - $155,000. This is the Expected
Cost of this option, since it is 50% likely that the low bidder will be late.
For the Reliable High Bidder
1. Multiply -$170,000 x 10% for - $17,000
2. Multiply -$140,000 x 90% = - $126,000
3. Add the results (- $17,000 and - $126,000) for - $143,000
In the Contractor Decision case there is only one decision node, the original one. The
alternatives are:
1. Low Bidder but Risky at -$155,000
2. Reliable High Bidder at -$143,000.
Clearly, the reliable high bidder has the edge here and is actually expected to cost less for the
project because of their greater on-time reliability. The value of the process is to represent the
values corrected for, or incorporating in, their uncertainty.
Tree with Embedded Decision Nodes – The Value of “Folding Back”
For simple decision trees with just one decision and chance nodes like the one in our earlier
example, the full value of the folding back technique is not evident. However, many decision
trees on real projects contain embedded decision nodes. The only way to solve such decision
trees is to use the folding back technique from right to left. This is because the value of each
node depends on the values of those nodes to its right (in the standard left-to-right orientation of
decision trees depicted here).
In folding back, the right-hand-most nodes must be valued before those next in right-to-left
order, as shown in the following example using a technology decision.
Figure 4: Technology Decision with Embedded Decision Nodes showing the benefit of folding
back
In this decision tree, we are faced with the choice of using an experimental technology or a
commercial-off-the-shelf (COTS) technology. If it is successful, the experimental technology
promises greater rewards. Of course both solutions could encounter “Minor Problems”, which
we are prepared to accept. But unfortunately, we need to acknowledge that both alternative
solutions carry a possibility of “Major Problems”. This is quite likely with the Experimental
technology for which we estimate an 80% probability of such problems, but even COTS have a
10% chance of major problems. If major problems occur, we must decide whether to fix the
problem or to “limp along.” Because of this decision there is an embedded decision node in each
branch after a chance node in this more realistic decision tree.
The folding back process starts at the right-hand-side of the tree, at the end of the branches. On
each branch we encounter a decision node first. We have estimated rewards to the “Limp
Along” scenario and the “Fix the Problem” scenario. Each of these is probably a net present
value (NPV) of future income streams with the latter, Fix the Problem, netting out the cost of the
actions necessary to perform the fix.
With both the Experimental and COTS technologies, the decision would be to FIX the Problem
if major problems occur since to “Limp Along” provides small rewards. The value (not the
“Expected Monetary Value” at this point since it is a decision rather than a chance node) of the
decision node is found by folding back from the right toward the left using the path values. That
value is found by selecting the highest path value offered:
1. $800,000 in the case of the COTS decision
2. $1,000,000 for the experimental technology.
There is no other way to discover this value than starting from the right-hand side of the tree and
folding back.
The values of the embedded decision nodes, $1,000,000 and $800,000, are then carried to the left
and used in the Expected Monetary Value (EMV) calculations that provide the value of the two
event nodes. The calculations are as follows for the COTS branch:
1. Multiply the value of the “major problems” node by its probability ($800,000 x 10%) for
the value $80,000.
2. We multiply the path value of the “minor problems by its probability ($1,000,000 x 90%)
for a value of $900,000.
3. Add the two values together for $980,000. This is the EMV of the COTS decision
To find the value of using the experimental technology:
1. Multiply the value of the major problems branch by its probability ($1,000,000 x 80%)
for a total of $800,000
2. Multiply the path value of the minor problems branch by its probability ($1,400,000 x
20%) for the total of $280,000.
3. Add these two values together to derive the EMV of the branch, $1,080,000.
The ultimate decision in the technology case is found by comparing the EMVs of two
technologies. This comparison indicates that we should choose the experimental technology
because, even though there is an 80% probability of major problems, the technology’s EMV is
higher than that for the COTS choice. The closeness of these values, $1,080,000 vs. $980,000,
indicates that this decision is close. The decision is quite sensitive to the accuracy of the
estimates and we are encouraged to both (1) make these estimates as accurate as possible, and (2)
evaluate factors that are not included in the decision tree, for instance whether proving the
experimental technology on this project might lead to future licensing revenues, which may
affect our decision.
The Risk Averse Organization
In the examples above we have assumed that the organization wants to choose whichever
decision maximizes its expected monetary value or minimizes its expected cost. This behavior,
which could be called “risk-neutral,” may represent an organization that has many projects and
can thrive if it succeeds “on the average.”
Many, if not most, organizations are cautious in situations where they think they might be
vulnerable to large losses. These organizations may shy away from project decisions which, if
they were to fail, would expose the organization to the probability of large losses, even if such
project decisions might also offer a possibility of large gains associated with success. This
behavior might be called “risk-averse”. Decisions made by risk-averse organizations’ tend to
maximize their expected utility rather than expected value, and that utility may give serious
(negative) weight to the possibility of large losses. Most decision tree software allows the user
to design a utility function that reflects the organization’s degree of aversion to large losses.
Conclusion
Project decisions, even quite simple ones, can be difficult to make because their implications are
often not certain. This is a fact of life for most project managers, who often face situations like
those explored above: the choice of alternative contractors and of alternative technologies. Each
of these decisions poses clear alternatives but murky consequences. Uncertain consequences are
best described and analyzed using probability concepts as part of a decision tree analysis to
maximize Expected Monetary Value or minimize expected costs to the organization.
Decision trees allow project managers to distinguish between decisions where we have control
and chance events that may or may not happen. It takes account of the costs and rewards of
decision options as well as the probabilities and impacts of associated risks. Structured analysis
using “rolling forward” and “folding back” allows the best decision option to be taken based on
calculation of Expected Monetary Value, although this may be influenced by the risk appetite of
the organization. The decision tree technique offers a powerful way of describing, understanding
and analyzing uncertainty, and can be a valuable part of the toolkit for any project manager who
needs to make decisions where the outcome is uncertain.
1
A version of this article was published in PM Network, May 2006 (© PMI), co-authored by
David Hulett and David Hilson.
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David T. Hulett, Ph.D.
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