Max’s Bakery specializes in sourdough bread. A loaf of sourdough costs $2.00 to make and sells for a price of $4.00 in the store that day. All loaves not sold on the day that they are made are sold on the next day at $1.00 each.

Suppose that daily sourdough bread demand is either 20, 30, 40, or 50 loaves of bread, and Max also chooses between making one of those quantities. Suppose the probabilities corresponding to the demands of 20, 30, 40, and 50 loaves are 0.2, 0.3, 0.3, and 0.2, respectively

Which of the following statements are true about Max’s decision?

Question

Max’s Bakery specializes in sourdough bread. A loaf of sourdough costs $2.00 to make and sells for a price of $4.00 in the store that day.  All loaves not sold on the day that they are made are sold on the next day at $1.00 each.

Suppose that daily sourdough bread demand is either 20, 30, 40, or 50 loaves of bread, and Max also chooses between making one of those quantities.  Suppose the probabilities corresponding to the demands of 20, 30, 40, and 50 loaves are 0.2, 0.3, 0.3, and 0.2, respectively

Which of the following statements are true about Max’s decision?

BlackTom AI · Accepted Answer

We start by restating the problem setup and each answer option to be analyzed.

Question: Max’s Bakery must decide how many loaves to bake each day, given possible demand D in {20, 30, 40, 50} with probabilities {0.2, 0.3, 0.3, 0.2}, cost per loaf 2, sale price 4, and salvage value 1 for any unsold loaf the next day. If they bake q loaves, profit in a day with demand D is: profit = revenue from sold loaves (4 per loaf) plus salvage for leftovers (1 per loaf) minus cost (2 per loaf). This simplifies to profit = 3*min(q, D) − q. The statements to evaluate are:
- (A) The most that one would pay for perfect information about the demand is $12.
- (B) The EV with PI is $70.
- (C) Each of the four decision alternatives is optimal for one of the four decision criteria.
- (D) The expected daily demand is 36 loaves.
- (E) Exactly two answers are correct.
- (F) None of the answers are correct.

Now we analyze each option in turn, with explicit calculations where needed.

Option A: The most that one would pay for perfect information about the demand is $12.
- First compute the expected value with perfect information (EVPI). To do this, determine, for each possible demand D, which quantity q would maximize profit after observing D, then average those best profits using the demand probabilities.
- For D = 20: evaluate profits for q in {20,30,40,50} using profit = 3*min(q, D) − q. With D = 20, min(q, D) = 20 when q ≥ 20, so profits are: q=20 → 3*20−20=40; q=30 → 60−30=30; q=40 → 60−40=20; q=50 → 60−50=10. The best is q=20 with profit 40.
- For D = 30: min(q, D) is 20, 30, or 30 depending on q. Profits: q=20 → 40−20=20; q=30 → 90−30=60; q=40 → 90−40=50; q=50 → 90−50=40. Best is q=30 with profit 60.
- For D = 40: Profits: q=20 → 40−20=20; q=30 → 60−30=30; q=40 → 120−40=80; q=50 → 120−50=70. Best is q=40 with profit 80.
- For D = 50: Profits: q=20 → 40−20=20; q=30 → 60−30=30; q=40 → 120−40=80; q=50 → 150−50=100. Best is q=50 with profit 100.
- EVPI = 0.2*40 + 0.3*60 + 0.3*80 + 0.2*100 = 8 + 18 + 24 + 20 = 70. This is the value of perfect information, i.e., the maximum one would pay for it.
- Since the computed EVPI is 70, the claim that the most one would pay is 12 is incorrect.

Option B: The EV with PI is $70.
- From the calculation above, the EV with perfect information equals the average of the best profits per demand, which works out to 70.
- This directly supports the statement that EV with PI is $70, so this option is correct.

Option C: Each of the four decision alternatives is optimal for one of the four decision criteria.
- The four decision criteria in classical decision analysis could include: expected value (EV), expected value with perfect information (EVPI), maximax, maximin, or other criteria. Here, we have found that the EV without PI is maximized by q = 40 (EV = 59). The EV with PI chooses q = 20 for D=20, q=30 for D=30, q=40 for D=40, q=50 for D=50, resulting in the EVPI calculation. It is unlikely that all four q choices are each uniquely optimal for a distinct standard criterion, especially since some criteria would yield the same option across multiple states or would not align with the per-state optimal choices identified for PI. Without a precise definition of the four criteria, this claim is dubious and seems incorrect based on typical decision-analysis patterns.
- Therefore, this option is likely false.

Option D: The expected daily bread demand is 36 loaves.
- Compute E[D] = 20*(0.2) + 30*(0.3) + 40*(0.3) + 50*(0.2) = 4 + 9 + 12 + 10 = 35.
- The expected demand is 35, not 36, so this statement is false.

Option E: Exactly two answers are correct.
- From the analyses above, likely only Option B is correct; Option A is false, Option C false, Option D false. Therefore, exactly two answers being correct is not supported. This statement is false.

Option F: None of the answers are correct.
- Since Option B is correct, this blanket statement is false.

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