Least-Cost Bus Mix Under Capacity and Parking Limits
A middle school is planning a day trip for 212 students and 18 chaperones, 230 people in all. The district contracts two vehicle types: full-size coaches that seat 58 and cost $850 each, and minibuses that seat 24 and cost $400 each. District policy requires at least one certified staff member on every vehicle, and only 8 teachers have volunteered, so no more than 8 vehicles can be used. The destination's drop-off loop can accept at most 3 full-size coaches at one time; minibuses are unlimited. The principal's draft plan uses 3 coaches and 2 minibuses because “that keeps the groups large,” and the business office has flagged it as both short on seats and possibly not the cheapest option. The trip coordinator must find the least expensive feasible mix, first ignoring the drop-off limit and then applying it, and must also respond to a suggestion that the school simply charter 10 minibuses.
1. Ignoring the drop-off limit, find the least-cost vehicle mix that seats at least 230 people. Show the comparison, not just the answer.
Read the worked answer
Capacity check: 10 minibuses seat 240 but use 10 vehicles, exceeding the 8-vehicle cap, so that suggestion is infeasible before cost is even considered.
Feasible candidates with 8 or fewer vehicles:
- 4 coaches: 232 seats, 4 vehicles, 4 × 850 = $3,400
- 3 coaches + 3 minibuses: 174 + 72 = 246 seats, 6 vehicles, 2,550 + 1,200 = $3,750
- 2 coaches + 5 minibuses: 116 + 120 = 236 seats, 7 vehicles, 1,700 + 2,000 = $3,700
- 1 coach + 8 minibuses: 9 vehicles, over the cap; with 7 minibuses only 226 seats, short
- 3 coaches + 2 minibuses (the draft): 174 + 48 = 222 seats, short by 8 - infeasible
Least cost is 4 coaches at $3,400. Per-seat cost supports this: coaches run $850/58 ≈ $14.66 per seat versus $400/24 ≈ $16.67 for minibuses, so filling coaches first is efficient - but only while the seats are actually used, which is why the 232-seat total (2 spare) is acceptable while a nearly empty extra coach would not be.
2. Now apply the limit of at most 3 full-size coaches and find the least-cost feasible plan. Explain why the answer is not obtained by maximizing coaches.
Read the worked answer
With coaches capped at 3:
- 3 coaches + 3 minibuses: 246 seats, 6 vehicles, $3,750
- 3 coaches + 2 minibuses: 222 seats - short by 8, infeasible
- 2 coaches + 5 minibuses: 236 seats, 7 vehicles, $3,700
- 1 coach + 7 minibuses: 226 seats - short; 1 coach + 8 minibuses violates the 8-vehicle cap
- 0 coaches: 10 minibuses needed for 240 seats, over the vehicle cap
The least-cost feasible plan is 2 coaches + 5 minibuses at $3,700, using 7 of the 8 permitted vehicles and 8 teachers (6 staff needed, so staffing is fine).
Maximizing coaches is not the objective; minimizing cost subject to the seat floor is. Three coaches force a third minibus to cover the remaining 56 people, and 3 coaches + 3 minibuses costs $50 more while carrying 10 unnecessary seats. The binding constraints here are the coach cap and the 230-seat requirement, and the optimum sits at a point where dropping a coach saves $850 but forces two more minibuses at $800 - a net $50 gain.
3. The principal proposes 3 coaches + 3 minibuses to “keep everyone together.” Evaluate this proposal against the constraints and against the cost-minimizing plan.
Read the worked answer
Feasibility: 246 seats for 230 people, 6 vehicles (within the 8-vehicle cap), 3 coaches (at the drop-off limit), and 6 staff required against 8 available. It is feasible.
Cost: $3,750, which is $50 more than the $3,700 optimum, and it carries 16 spare seats rather than 6.
“Keeping everyone together” is not a stated constraint, so it cannot justify the extra $50 on cost grounds alone. It does deliver a real, if unquantified, benefit - one fewer vehicle to load, count, and supervise - so the honest framing is that the school is buying convenience for $50, not that the plan is cheaper or required. If the school prefers it, the decision should be recorded as a discretionary $50 trade-off; if cost is the governing criterion, 2 coaches + 5 minibuses is the defensible choice.