Compute break-even and simulate diaper inventory
Overview
This question evaluates quantitative modeling skills including break-even and contribution-margin analysis, inventory simulation, and operational levers for supply sustainability in a service context, and is categorized under Statistics & Math for a Data Scientist role.
You operate a cloth-diaper service with these weekly parameters: 350 babies served; charge 0.05 per diaper cleaned; delivery service cost is 1,040 total. Inventory rules and timing: On Monday each week the route runs. First Monday there is no pickup. From the second Monday onward, for each baby you (a) pick up 72 used diapers from the prior week, (b) clean them the same day and return them to clean inventory, then (c) deliver 80 clean diapers for the coming week. “Inventory” means clean, ready-to-deliver diapers on hand at the facility immediately after that Monday’s cleanings and deliveries. Start with 70,000 clean diapers on Monday, January 1 (assume January 1 is a Monday). Answer the following:
- Compute the weekly break-even number of babies (round up to a whole baby). Show contribution margin, formula, and result.
- With 350 babies, compute weekly profit (revenue − variable costs − fixed costs). Show all steps.
- After the third Monday’s cycle completes, how many clean diapers remain in inventory?
- If you never manufacture or purchase new diapers (rely only on cleaning returns), on which calendar Monday will inventory first go negative? Provide the exact date.
- Propose three operational levers to prevent stock-outs (e.g., increase pickup frequency, adjust per-baby drop quantity, change pricing). For one lever, quantify the minimum weekly change required to keep inventory non-negative for the next 8 Mondays while serving 350 babies. State and justify any assumptions you add.