Training budget with tiered fees
A fictional EU agency has €84,000 to train 60 staff for a certification. The platform fee is €12,000, payable once. An in-person course costs €1,450 per participant, but a 10% discount applies if total in-person enrolment is 25 or more. Travel is reimbursed at €180 per in-person participant. An online course costs €620 per participant and has no travel cost. In-person cohorts must have between 12 and 30 participants; online capacity is unlimited. All 60 staff must complete one course. A colleague proposes 35 in-person and 25 online and claims the total cost is €79,750. The head of unit wants the maximum number of in-person places that keeps the programme within budget, while respecting cohort rules. The budget covers fees and travel only; staff time is not charged.
1. Calculate the maximum number of in-person places that fits the budget. Show the constraint.
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Subtract the fixed platform fee: €84,000 − €12,000 = €72,000 for variable costs. For x in-person places at the discounted rate, the per-person cost is €1,450 × 0.9 + €180 = €1,305 + €180 = €1,485. Online costs €620 per person. Total variable cost = 1,485x + 620(60 − x) = 37,200 + 865x. Set 37,200 + 865x ≤ 72,000, so 865x ≤ 34,800, giving x ≤ 40.23. The maximum whole number is 40. Check cohort rules: 40 can be split into two cohorts of 20, both between 12 and 30. Cost check: 40 × 1,485 = €59,400; 20 × 620 = €12,400; variable total €71,800; plus €12,000 = €83,800. At 41 in-person, variable cost is 41 × 1,485 + 19 × 620 = €72,665, total €84,665, over budget.
2. Is the colleague's proposal feasible, and by how much is their cost estimate wrong?
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It is feasible. Thirty-five in-person participants cannot form one cohort because the maximum is 30, but they can be split into two cohorts of 18 and 17, both above the minimum of 12. The actual cost is 35 × €1,485 = €51,975 for in-person fees and travel, plus 25 × €620 = €15,500 online, giving variable costs of €67,475. Adding the €12,000 platform fee gives €79,475. The colleague's figure of €79,750 is therefore €275 too high. The proposal is within budget, but it is not the maximum: 40 in-person places are affordable.
3. The agency later learns that the discount applies per cohort, not to total enrolment. How does that change the maximum?
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If the discount is per cohort, each cohort of 25–30 pays €1,485 per person including travel; cohorts of 12–24 pay €1,630. To maximise in-person places, use one discounted cohort of 25 or more and one smaller cohort. For 39 in-person, the best split is 27 + 12: 27 × €1,485 = €40,095; 12 × €1,630 = €19,560; in-person total €59,655. Online for 21 people is €13,020, so variable cost is €72,675 and total is €84,675, over budget. For 38 in-person, split 26 + 12: €38,610 + €19,560 = €58,170; online for 22 is €13,640; variable total €71,810; total €83,810, under budget. The maximum is therefore 38, not 40.