| Course | BU 407 Quantitative Decision-Making |
|---|---|
| Unit | Unit 2 |
| Paper type | Expected value comparison with decision tree |
| Length | About 1,050 words, 6 pages |
| Format | APA 7 student paper |
| School | Herzing University |
| Program | BS in Business Analytics |
| Updated | September 2026 |
Free sample paper for BU 407 Unit 2
Automated Line or Smaller Filler, and What a Perfect Forecast Is Worth: An Expected Value Comparison With a Decision Tree for a Composite Dairy Cooperative
Student Name
Business, Herzing University
BU 407: Quantitative Decision-Making
Instructor Name
February 4, 2026
Automated Line or Smaller Filler, and What a Perfect Forecast Is Worth: An Expected Value Comparison With a Decision Tree for a Composite Dairy Cooperative
From Table to Tree
The payoff table in Unit 1 narrowed Northwoods Dairy Cooperative's yogurt capacity decision to two finalists: a new automated cup-filling line and a second, smaller filler on the existing line. Northwoods is the composite Wisconsin cooperative used throughout this course. The criteria that need no probabilities disagreed about which finalist to choose, so this unit brings in probabilities and compares the two on expected value. It also keeps doing nothing on the tree as a reference branch. Because both investments would run for several years, this comparison uses the net present value of each option over five years in each market state, supplied by the cooperative's finance office, rather than the average annual contributions used in the payoff table.
The planning committee assigned probabilities to the three market states after reviewing regional grocery data and the chain's store-brand discussions: 0.5 that the market holds at recent growth, 0.3 that it softens and 0.2 that it grows strongly. The committee's estimates are judgments, and the sensitivity of the result to them is discussed below.
The Tree
The decision tree begins with a square decision node offering three branches: install the automated line, add the smaller filler or do nothing. Each investment branch leads to a round chance node with three branches, one for each market state, labeled with its probability and ending in the five-year net present value of that outcome. The table below repeats the tree's figures for readers who cannot follow a diagram; values are in millions of dollars and belong to the composite case.
| Decision branch | Market holds (p = 0.5) | Market softens (p = 0.3) | Market grows (p = 0.2) | Expected value |
|---|---|---|---|---|
| Automated line | $1.10 | minus $0.40 | $2.60 | $0.95 |
| Smaller filler | $0.80 | $0.10 | $1.40 | $0.71 |
| Do nothing | $0 | $0 | $0 | $0 |
Rolling Back the Tree
Rollback works from the right side of the tree to the left, replacing each chance node with its expected value and each decision node with its best branch. For the automated line, the expected value is 0.5 times $1.10 million, plus 0.3 times minus $0.40 million, plus 0.2 times $2.60 million: $0.55 million minus $0.12 million plus $0.52 million, which is $0.95 million. For the smaller filler, it is 0.5 times $0.80 million, plus 0.3 times $0.10 million, plus 0.2 times $1.40 million: $0.40 million plus $0.03 million plus $0.28 million, which is $0.71 million. Doing nothing is worth $0. At the decision node, the automated line has the highest expected value, $0.24 million more than the smaller filler, so rollback chooses it.
Render et al. (2018) note that expected value is the right criterion for a decision maker who is risk neutral, one who would accept a fair gamble. A cooperative with limited reserves may not be. The automated line carries a 30 percent chance of losing $0.40 million, while the smaller filler never loses money in any state. That difference does not appear in the expected values and has to be weighed separately.
One way to weigh that downside is to ask how large a loss the cooperative can absorb in a bad year. Its members are paid for their milk from the same revenue that would fund the line, so a $0.40 million loss would reduce member payments or require borrowing. A board that sets a rule, such as no investment with more than a 25 percent chance of losing money, would rule out the automated line on the committee's probabilities, since the soft market carries a 30 percent chance. Stating such a rule before looking at the results keeps the choice from being bent to fit a preferred option.
What a Perfect Forecast Would Be Worth
Before recommending an investment, it is worth asking how much it would be worth to know the market state in advance. With perfect information, the cooperative would choose the best option for each state: the automated line if the market holds, earning $1.10 million; the smaller filler if the market softens, earning $0.10 million; and the automated line if the market grows, earning $2.60 million. Weighting these by the probabilities gives the expected value with perfect information: 0.5 times $1.10 million, plus 0.3 times $0.10 million, plus 0.2 times $2.60 million, which is $1.10 million. The expected value of perfect information is that figure minus the best expected value without information, $1.10 million minus $0.95 million, or $0.15 million.
The result sets a ceiling on what any market research is worth. No study can be worth more than $150,000, and a real study, which would be imperfect, is worth considerably less. If the chain's store-brand discussions could be settled with a signed letter of intent at little cost, that information would be valuable; a $200,000 consulting study would not be.
When the Options Come Out Close
The two options differ by $0.24 million in expected value, which is meaningful but not large compared with the spread of outcomes. A simple check shows how sensitive the choice is to the probabilities. If the probability that the market softens rose from 0.3 to 0.45, with the other states reduced proportionally, the automated line's expected value would fall faster than the filler's, because its soft-market outcome is a loss. Working the algebra with the other two probabilities kept in their 5 to 2 ratio, the options break even when the chance of a soft market reaches about 0.53; below that the automated line leads. That breakeven deserves a line in the board memo, since the committee's probabilities are judgments. Raiffa (1968) recommended exactly this kind of sensitivity check whenever probabilities are subjective, so that a decision maker can see how firmly the recommendation rests on them.
Recommendation
On expected value, the cooperative should install the automated line, which is worth about $0.95 million against $0.71 million for the smaller filler. Two conditions accompany the recommendation. First, the board should decide whether it can absorb the 30 percent chance of a $0.40 million loss; if it cannot, the smaller filler is the prudent choice at a cost of $0.24 million in expected value. Second, the cooperative should spend no more than a small fraction of the $150,000 value of perfect information on further market research before deciding, such as seeking a letter of intent from the grocery chain. Northwoods Dairy Cooperative, its probabilities and values are composites created for this course (Hillier & Lieberman, 2015).
References
Hillier, F. S., & Lieberman, G. J. (2015). Introduction to operations research (10th ed.). McGraw-Hill Education.
Raiffa, H. (1968). Decision analysis: Introductory lectures on choices under uncertainty. Addison-Wesley.
Render, B., Stair, R. M., Jr., Hanna, M. E., & Hale, T. S. (2018). Quantitative analysis for management (13th ed.). Pearson.
The Herzing BU 407 Unit 2 instructions
For Unit 2, students compare alternatives using expected value, usually with a decision tree. Most versions want the tree drawn with decision and chance nodes, probabilities and payoffs on every branch, a rollback calculation showing each node's expected value and a recommendation. Many rubrics also ask for the expected value of perfect information and for a discussion of whether expected value is the right criterion given the decision maker's attitude toward risk. Instructors appreciate a table repeating the tree's figures, since hand-drawn diagrams can be hard to read. The probabilities should have a stated source. Expect a few pages with a diagram or table, citing decision analysis sources in APA 7, and include a check of how sensitive the choice is to the probabilities.
Inside the BU 407 Unit 2 example
The sample explains how the payoff table narrowed the choice and why five-year net present values are used on the tree. It states the probabilities and their source. The tree is described and its figures repeated in a table with the expected value of each branch. The rollback section shows the arithmetic for each chance node and the choice at the decision node, then notes the downside the averages hide and offers a loss-tolerance rule the board could apply. The next section calculates the expected value with perfect information and the value of perfect information, and explains what that means for research spending. A sensitivity section finds the breakeven probability, and the recommendation states two conditions.
Where the points sit in the BU 407 Unit 2 rubric
The rubric for the expected value unit generally assesses the structure of the tree, the accuracy of the rollback, the calculation of the value of information and the quality of the recommendation. Structure earns full points when decision and chance nodes are correctly placed and every branch carries a probability and a payoff. Rollback accuracy is judged by correct expected values at each node, shown with their arithmetic. The value of information scores highest when it is calculated correctly and interpreted as a ceiling on research spending, as the sample does. The recommendation rewards discussion of risk beyond the expected value and a sensitivity check on the probabilities. Clear tables and citations round out the score, and a breakeven point that is actually computed rather than guessed stands out.
BU 407 Unit 2 help: mistakes that cost points
The most frequent Unit 2 mistake is rolling back the tree from left to right instead of from the outcomes back to the decision. Another is forgetting that probabilities at each chance node must sum to one. Students often calculate the value of perfect information incorrectly, subtracting the wrong expected value or using the worst outcome in each state instead of the best. Some papers recommend the highest expected value without noting a large chance of loss. Others use probabilities with no stated source. Finally, check how sensitive the answer is; if a small change in one probability would reverse the choice, the recommendation should say so plainly.
Get a BU 407 Unit 2 example written to your instructions
Send us the Unit 2 instructions and your BU 407 rubric, together with any probabilities the scenario fixes. We write a custom example that draws the tree before folding it, keeps probabilities attached to states, rolls back from the right and prices perfect information. First custom sample free, returned in 24 to 48 hours.
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BU 407 Unit 2 questions, answered
Where do the probabilities come from?
From the scenario in most sections, and where the assignment leaves them open, from a stated assumption you can defend. Say which it is. A likelihood drawn from a published market report belongs with a citation; one you chose belongs with a sentence of reasoning and a note that the case is composite. What no assignment wants is a figure presented as fact with no origin.
Does the tree have to be drawn by hand?
Sections differ, and either a drawing tool or a neat hand sketch is usually accepted so long as it reproduces cleanly. What matters is that a marker can read the labels: state, probability and payoff on every branch, with the node types distinguishable. If your file compresses the image, put the table version underneath it so nothing is lost in the submission.
What if the two options come out almost equal?
Then the interesting work begins. Near ties are where sensitivity belongs: say how far a probability would have to move before the winner changes, and how far a payoff would. A recommendation admitting the margin is thin and naming the number to watch reads as stronger than one asserting a clear victory the arithmetic does not support.
Where can I find a free BU 407 Unit 2 sample paper?
The complete BU 407 Unit 2 comparison, tree table through recommendation, is free to read above. For a decision tree built on your own options and probabilities, share the prompt and your figures, and the desk will draft a free custom comparison, commonly inside 48 hours.
Does the BU 407 Unit 2 tree have to be drawn by hand?
Not necessarily; software drawings are usually fine. What matters is that every node and branch is labeled clearly. The sample also repeats the tree's figures in a table, which protects the work if a diagram is hard to read.