Cumulative review: Decisions under uncertainty
Decide whether a print shop should pay for a demand report when waiting also changes what its regular customer will pay.
Combine probability, expected value, spreadsheet scenarios, and opportunity cost to choose between acting now and buying more information.
One press, two customers
You own a print shop with one press slot left for Friday. A regular customer offers a job worth $600 in profit if you confirm it now. An event organizer might need a rush run worth $1,600. To take that order, you have to leave the slot open. If the order never comes, the press sits idle.
You can buy an $80 update on event registrations before deciding which job to take. Waiting has a consequence: the regular customer would have to arrange its own finishing work, so its revised job would leave you only $450. How much should you pay for the update when part of the price of waiting never appears on an invoice?
The shop, customers and figures are fictional. Each dollar amount is additional profit for Friday after all costs that change, except the separately stated report fee. You can fund any option. Choose the plan with the highest expected profit. Allow three minutes to read, twelve to solve, and ten to check your reasoning.
What the update tells you
The update says either “busy” or “quiet.” The organizer can still place an order after a quiet report or decline after a busy one. Use these 100 comparable past Fridays as the probability model for this exercise. The shop verified whether the rush order was needed on every date, even when its press was unavailable. Reading the report does not change demand.
| Registration report | Rush order needed | No rush order | Total Fridays |
|---|---|---|---|
| Busy | 30 | 10 | 40 |
| Quiet | 10 | 50 | 60 |
| Total | 40 | 60 | 100 |
Before the update, the regular job and the rush opportunity are mutually exclusive: either uses the entire slot. If you buy the update, the regular customer guarantees the revised $450 offer until you receive the result. You can still reserve the slot for the rush job on the original terms. There is no third use for the press.
Recall yesterday’s quality alert: when Meridian knew the screen had flagged a batch, which batches belonged in the denominator of its defect estimate? Apply the same reasoning when the shop sees a quiet report.
Review questions
- Calculate the chance of a rush order before the report, after a busy report, and after a quiet one. Is a rush order independent of what the report says? Give numerical evidence.
- If you act now, compare the sure job with holding the slot open. What profit do you give up by reserving the press? How much better or worse is that choice on average?
- If you wait, choose an action after each possible report. Work backward to calculate expected profit before the fee, the most you would pay for the update, and your decision at $80.
- Make a small spreadsheet or paper model with two scenarios. Keep the $450 fallback offer in the first; reduce it to $350 in the second. Change no other input. Does the recommendation survive? Name a check that would catch using the original $600 offer after waiting.
- Write a two-sentence instruction for the person scheduling the press. Include an action for each report and one assumption to verify before spending the money.
Try the questions before reading the worked answers.
Worked answers
1. Choose the denominator. Before seeing the report, 40 of 100 Fridays required a rush run, so the chance is 40%. After a busy report, it is 30 ÷ 40 = 75%. After a quiet one, it is 10 ÷ 60 = 1 in 6, about 16.7%. The report and demand are dependent: knowing the report changes the estimated chance. As a check, the chance of both a busy report and a rush order is 30%, whereas multiplying the separate 40% rates gives only 16%.
Recall answer: Meridian counted defective batches among all flagged batches. Here the quiet-Friday denominator is 60. Dividing the same ten orders by 40 rush orders would tell you how often a rush order followed a quiet report, which answers a different question.
2. Decide what you would do now. Holding the slot has expected profit of (0.40 × $1,600) + (0.60 × $0) = $640. Accepting the regular job gives $600. Reserve the press if you must act now. The $600 is the opportunity cost of reserving: it is the profit from the best alternative you give up. Your expected advantage is $640 − $600 = $40. Do not deduct the $600 from the rush payoff and then compare the result with $600 again.
On this one Friday, reserving pays either $1,600 or nothing. The expected $40 advantage is small compared with the $600 you forgo if the order never comes.
3. Price the report. On a busy report, reserving earns 0.75 × $1,600 = $1,200 in expectation, more than $450. On a quiet report, reserving earns (1 ÷ 6) × $1,600 ≈ $266.67, so take the regular job for $450. Work backward through these two decisions. Before you know which report will arrive, the plan is worth (0.40 × $1,200) + (0.60 × $450) = $750 before the fee.
The most you should pay is $750 − $640 = $110. Compare with the best choice available now, not the sure $600 job you would already decline. At $80, buy the update: expected profit is $670, or $30 more than acting now. The fee is paid once, whichever report arrives. This is the practical test of information value taught in MIT’s decision analysis course: what improvement can you make by acting on the result? MIT OpenCourseWare: Value of Test.
4. Test the fallback offer. Keep one input cell for the profit on the regular job after waiting. Each scenario uses the same rule: on each report, take the larger of the rush job’s expected profit and the fallback offer. Weight the chosen amounts by the report probabilities, then subtract the fee.
| Regular job after waiting | Busy: reserve | Quiet: take regular job | Expected profit after $80 fee | Best plan |
|---|---|---|---|---|
| $450 | $1,200 | $450 | $670 | Buy update |
| $350 | $1,200 | $350 | $610 | Reserve now for expected $640 |
The busy column shows expected profit given a busy report. It does not mean the organizer pays $1,200. In the second scenario, $350 still beats $266.67 on quiet days. Losing another $100 on the fallback therefore costs 0.60 × $100 = $60 on average, enough to reverse the decision.
For a hand check, imagine 100 Fridays under the original report plan: 30 busy Fridays bring $1,600 each, ten busy Fridays bring nothing, and 60 quiet Fridays bring $450 each. Subtract 100 report fees of $80. That leaves $67,000, or $670 per Friday. A formula that quietly keeps the original $600 offer would overstate the result by 0.60 × ($600 − $450) = $90 per Friday. That $90 is the expected cost of waiting on the customer. It is already reflected in the $670 result.
5. Give the instruction. “Confirm that the regular customer will hold the $450 offer until the report arrives, then buy the $80 update. Reserve the slot on a busy report and take the regular job on a quiet one; if the customer offers only $350 for waiting, skip the report and reserve now.”
Finish the week’s decision tree
Keep the two choices, the report probabilities, and the Friday profits together on one page. Mark which offer expires while you wait. That detail explains why the same report at the same price can be worth buying in one scenario and worth skipping in the other.
Next week turns to the data behind estimates like these. You’ll start by looking at how business results are distributed and what an average can hide.
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