C2 Shell- Decision Trees
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Jan 9, 2024
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6. Sequential Decision-Making Problems
Many real-life decision problems are slightly more complex than the real-estate developer problem that we have considered. One reason for the complexity is that there are many stages in the decision making process; that is, we have a sequential
decision-making problem
.
A decision tree is a useful approach to visually display these types of problems
– allowing for a better understanding of the problem. A decision tree consists of
branches and two types of nodes:
• A decision node (represented by a square node
) indicates that a decision needs to be made at that point in the process. The branches that emanate out of a decision node are the different alternatives from which the decision maker has to make a choice.
• A event (or chance) node (represented by a circle node ) indicates that a random event occurs at that point. The branches that emanate out of an event node are the different states of nature that may occur then, and the sum of the probabilities of the branches emanating from a event node must equal 1.
At each event node, we calculate its EMV using the respective probabilities and payoffs for each of the different states of nature. The EMV provides a valuation of the event node.
A decision tree always begins with a decision node. We initially construct the tree from left to right. At each node (decision or event), we record for each branch the partial cash flow associated with that branch. At each event node, we record for each branch the probability associated with that branch. When no more decisions or events are to be considered the branch is terminated. At the end of each terminated branch we record the complete cash flow or terminal value (TV) for that branch. Once we have completed the decision tree, we evaluate it from right to left to arrive at the optimal decision. At each event node we calculate the expected value (EMV) for that node and record it as a terminal value for the branch leading into that node. At each decision node we record the decision with the best EMV and record that EMV as TV for the branch leading into that node. 7
Example: A beverage company is considering replacing an existing product with a
new sugar-free product. Introduction would require a one-time outlay of $40,000. The market research manager feels that the levels of possible demand depend only upon counter-measure taken by competitive brands. During the time period stipulated by the board of directors, sales of the new drink will be either 1.2 million
or 1.6 million cans (the latter occurring if competition does not react rapidly). If introduced, each can will bring $0.10 profit, not including the fixed cost of introduction. Based on general industry studies the probabilities of “low” and “high” demand are 0.6 and 0.4, respectively. If the company does not introduce the
new product, it could continue focusing on its existing product with profit of $90,000.
What is the optimal strategy for the company?
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$160K
$90K
$-40K
$120K
Sequential Decision-Making Example
A computer company wants a new storage device for a proposed new computer system. Since the computer company does not have research people available to develop the new storage device, it will subcontract the development to an independent research firm. The computer company has offered a fee of $250,000 for the best proposal for developing the new storage device. The contract will go to
the firm with the best technical plan and the highest reputation for technical competence. DriveTek Research Institute wants to enter the competition. Management estimates
a cost of $50,000 to prepare a proposal with a fifty-fifty chance of winning the contract.
DriveTek’s engineers are not sure about how they will develop the storage device if they are awarded the contract. Three alternative approaches can be tried. The first approach is a mechanical method with a cost of $120,000, and the engineers are certain they can develop a successful model with this approach. A second approach involves electronic components. The engineers estimate that the electronic approach will cost only $50,000 to develop a model of the storage device, but with only a 50 percent chance of satisfactory results. A third approach uses magnetic components; this costs $80,000, with a 70 percent chance of success. DriveTek Research can work on only one approach at a time and has time to try only two approaches. If it tries either the magnetic or electronic method and the attempt fails, the second choice must be the mechanical method to guarantee a successful model. What is DriveTek’s optimal strategy?
Create the Decision Tree.
After we have drawn the tree we will use the software TreePlan to create the tree in Excel. Detailed instructions for TreePlan can be found in the course information folder as well as the file necessary for installation.
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$-80K
$-120K
$-50K
$250K
$-50K
$-120K
$-120K
Decision Tree results are highly dependent on the stated probabilities. Hence conducting sensitivity analysis on those probabilities is often critical.
Create a data table in which you vary the probability of success for the electrical approach from .1 to .9 in increments of .05. Record the branch decision result, the EMV of the branch, the final decision and the EMV of the final decision.
Refine the range of probabilities in increments of .01 to better identify the critical point at which a decision will change.
What can you conclude from the data table?
Create a two-way data table in which you vary the probability of success for both the electrical and the magnetic approaches. Note, unlike a one-way table,
two-way data tables can only record a single output, for example, the decision of the chosen method.
What can you conclude from this data table?
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Side Trip Performing Sensitivity Analysis using Data Tables
Data Tables allow you to repeated input a value into one cell of a spreadsheet and record the output from one or more calculated cells. One of the unverified assumptions was the probability of success in Electrical approach. We can input a range of probabilities and record the resulting branch decision, the EMV of the branch, the final decision and the EMV of the final decision.
Steps for creating a data table:
1.
Create a column (or row) that contains the desired input values. 2.
In the column to the immediate right and one row above the first input, enter “=cell reference” where cell reference is the cell containing the desired output value.
3.
Repeat step 2 for each desired output.
Sensitivity Ananlysis
Prob of Success
Decision
EMV
Final
Final
Electrical Method
Method
Method
Decision
EMV
Key
Method
2
90000
1
20000
1 mechanical
0.1
3
84,000
$ 1
17,000
$ 2 electrical
0.15
3
84,000
$ 1
17,000
$ 3 magnetic
0.2
3
84,000
$ 1
17,000
$ 0.25
3
84,000
$ 1
17,000
$ Final Decision
0.3
3
84,000
$ 1
17,000
$ 1 bid
0.35
3
84,000
$ 1
17,000
$ 2 no bid
0.4
3
84,000
$ 1
17,000
$ 0.45
2
84,000
$ 1
17,000
$ 0.5
2
90,000
$ 1
20,000
$ 0.55
2
96,000
$ 1
23,000
$ 0.6
2
102,000
$ 1
26,000
$ 0.65
2
108,000
$ 1
29,000
$ 0.7
2
114,000
$ 1
32,000
$ 0.75
2
120,000
$ 1
35,000
$ 0.8
2
126,000
$ 1
38,000
$ 0.85
2
132,000
$ 1
41,000
$ 0.9
2
138,000
$ 1
44,000
$ Cells
containing output information
Cells where changes in output will be recorded
Cells
containing input information
4.
Highlight the entire data table (but not the labels)
12
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5.
click on the Data tab, select What-if Analysis, and click on data table.
The following pop-up asks for the input cell:
Since our input data is arranged in a column, we use the column input cell to record the location in our spreadsheet of the cell containing the probability of success.
The resulting data table provides the desired comparisons. Prob of Success
Decision
EMV
Final
Final
Electrical Method
Method
Method
Decision
EMV
2
90000
1
20000
0.1
3
84,000
$ 1
17,000
$ 0.15
3
84,000
$ 1
17,000
$ 0.2
3
84,000
$ 1
17,000
$ 0.25
3
84,000
$ 1
17,000
$ 0.3
3
84,000
$ 1
17,000
$ 0.35
3
84,000
$ 1
17,000
$ 0.4
3
84,000
$ 1
17,000
$ 0.45
2
84,000
$ 1
17,000
$ 0.5
2
90,000
$ 1
20,000
$ 0.55
2
96,000
$ 1
23,000
$ 0.6
2
102,000
$ 1
26,000
$ 0.65
2
108,000
$ 1
29,000
$ 0.7
2
114,000
$ 1
32,000
$ 0.75
2
120,000
$ 1
35,000
$ 0.8
2
126,000
$ 1
38,000
$ 0.85
2
132,000
$ 1
41,000
$ 0.9
2
138,000
$ 1
44,000
$ 13
Note: the cells containing the location of the output calculation can be hidden by filling them in with black or changing the color of the font to white. Using Two Way Data Tables for Sensitivity Analysis
We can have our data tables include two inputs by using both options simultaneously. For example, the probability of electrical success using the column input and the probability of magnetic success (between .5 to .85) using the row input.
When using the two-way data table we can keep track of only one output. The location of that output is indicated in the upper left hand corner of the data table.
prob Mag Success
2
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Key
0.1
1
1
1
1
3
3
3
3
Mechanical
0.15
1
1
1
1
3
3
3
3
Electrical
0.2
1
1
1
1
3
3
3
3
Magnetic
prob
0.25
1
1
1
1
3
3
3
3
Elect
0.3
1
1
1
1
3
3
3
3
Success
0.35
1
1
1
1
3
3
3
3
0.4
1
1
1
1
3
3
3
3
0.45
2
2
2
2
2
3
3
3
0.5
2
2
2
2
2
2
3
3
0.55
2
2
2
2
2
2
2
3
0.6
2
2
2
2
2
2
2
2
0.65
2
2
2
2
2
2
2
2
0.7
2
2
2
2
2
2
2
2
0.75
2
2
2
2
2
2
2
2
0.8
2
2
2
2
2
2
2
2
0.85
2
2
2
2
2
2
2
2
0.9
2
2
2
2
2
2
2
2
Cells containing the output calculation
Cells containing the row input
Cells containing
the column input
In the resulting popup we enter the appropriate location of both the row and the column inputs. The location of the prob of success of electrical method is entered for the
“column input cell” and the location of the prob of success of magnetic method is entered in the “row input cell”.
14
Instructions: Please prepare a decision tree by hand Wynn Corporation is considering developing a new product to replace a current one. If they elect to stick
with the current product they can anticipate a profit of $1,200,000. It will cost $200,000 to attempt to develop the new product. If the development is highly successful (25% chance) they will introduce the new product to the market with an anticipated profit of $1,800,000 (not counting the development cost). If
the development is not successful (30% chance) Wynn would return to the current product. A third outcome is that the development could be moderately successful (with probability equal to 1- the other two probabilities). This would result in uncertain demand for the new product. In this scenario Wynn would have to decide whether to introduce the new product to the market or keep the current one. If Wynn introduces the new product there is a 60% chance of the high demand that would result in the same
$1,800,000 profit described above. But it is also possible (with probability of 1- the probability of high demand)) that, following only moderately successful development, the demand for the new product will be low and generate a profit of only $600,000
1. Create a decision tree that will help determine whether Wynn should develop the new product and assuming they decide to develop the product, and whether they should introduce it to the market. 15
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