School Fundraiser Model
A class sells candles to fund a trip. Each candle costs $4 to buy and sells for $9. Unsold candles can be returned for $2 each. Fixed costs for advertising and booth rental are $300. The class must raise at least $1,200 net. Demand is uncertain: there is a 50% chance they sell 150 candles, a 30% chance they sell 250, and a 20% chance they sell 350. The class can order any whole number of candles but wants to avoid ordering more than 300 because storage is limited. They must decide an order quantity before knowing demand.
1. Build a net-income model for an order quantity Q and demand D (with D ≤ Q), and compute net income for Q = 300 under each demand scenario.
Read the worked answer
Model. If D ≤ Q: revenue = 9D, returns = 2(Q − D), purchase cost = 4Q, fixed cost = 300. Net = 9D + 2(Q − D) − 4Q − 300 = 7D − 2Q − 300.
Q = 300.
- D = 150: 7(150) − 600 − 300 = 1,050 − 900 = $150.
- D = 250: 7(250) − 900 = 1,750 − 900 = $850.
- D = 350: demand exceeds Q, so all 300 sell, no returns: net = 9(300) − 4(300) − 300 = 2,700 − 1,200 − 300 = $1,200.
Check the D ≤ Q formula for D = 350. It would give 7(350) − 900 = $1,550, but that assumes 350 candles were available, which they were not. The cap matters.
2. Compute the expected net income for Q = 300 and determine whether it meets the $1,200 goal on average. Then find the order quantity (whole number, ≤ 300) that maximizes expected net income.
Read the worked answer
Expected net for Q = 300. 0.5(150) + 0.3(850) + 0.2(1,200) = 75 + 255 + 240 = $570. This is well below $1,200; the goal is not met on average.
General expected net. For Q ≤ 250, all scenarios have D ≥ Q except D = 150 when Q > 150. Consider Q = 250: D = 150 gives 7(150) − 500 − 300 = 250; D = 250 gives 7(250) − 500 − 300 = 950; D = 350 gives 9(250) − 1,000 − 300 = 950. Expected = 0.5(250) + 0.3(950) + 0.2(950) = 125 + 285 + 190 = $600.
Try Q = 200. D = 150: 7(150) − 400 − 300 = 350; D = 250 and 350: 9(200) − 800 − 300 = 700 each. Expected = 0.5(350) + 0.5(700) = $525.
Try Q = 150. All demands ≥ 150, so all sell: 9(150) − 600 − 300 = 450. Expected = $450.
Try Q = 300 gave $570. So the maximum among these is Q = 250 at $600. A finer check near 250 (e.g., Q = 240) gives similar or lower values because the marginal candle sells with probability 0.5 at D = 150, where the return loss ($2) exceeds the gain. The best order is about 250 candles, expected net $600.
3. The class wants at least a 70% chance of net income of $1,200 or more. Using the best order quantity, assess whether this is achievable and recommend an action.
Read the worked answer
Best order, Q = 250. Net incomes: D = 150 → $250; D = 250 → $950; D = 350 → $950. Only the top two scenarios reach $950, still below $1,200.
Probability of ≥ $1,200. With Q = 250, no scenario reaches $1,200, so the probability is 0. Even Q = 300 reaches $1,200 only in the D = 350 case, probability 0.20, far below 70%.
Maximum possible net. Selling 350 candles would need Q = 350, but storage caps Q at 300. At Q = 300, the best case is $1,200, so $1,200 is the ceiling and can only be hit in one scenario.
Recommendation. The 70% goal is infeasible under the storage cap and price structure. The class should raise the selling price, cut the $300 fixed cost, or negotiate extra storage to order 350. Otherwise it should lower its target to about $600 (the expected value at Q = 250) and plan accordingly.