Bus Rental Optimization: Minimize Transportation Costs

What This Problem Teaches

Let's Set Up the Problem

This is a constrained optimization problem where we need to visualize the constraints and available options. Let's organize what we know:

Key insight: While small buses are more cost-efficient per seat, we're limited by having only 9 drivers total. This constraint forces us to find the optimal balance.

Solution: Method 1 — Constraint Analysis

Step 1 — Define variables and objective function

Let L = number of large buses and S = number of small buses.

We want to minimize the total cost: C = 500L + 300S

Step 2 — Write all constraints as inequalities

Step 3 — Find feasible integer combinations

Since we can't use partial buses, we need integer solutions. Let's test combinations systematically, starting from the driver limit and working down:

Step 4 — Identify the minimum cost

The combination (4, 5) gives exactly 400 seats using all 9 drivers at the lowest cost of $3500.

Solution: Method 2 — Linear Programming with Graphical Analysis

Step 1 — Graph the feasible region

Let's visualize this problem on a coordinate system where L (large buses) is the x-axis and S (small buses) is the y-axis:

Step 2 — Evaluate corner points of feasible region

In linear programming, the optimal solution occurs at a vertex of the feasible region. We already identified these in Method 1, but graphically we can see that (4,5) is indeed at the intersection of our key constraints.

Step 3 — Confirm optimality

The cost function C = 500L + 300S has a slope that favors smaller buses. The point (4,5) represents the maximum use of cheaper small buses while still meeting the capacity requirement.

The Answer

Verification

Let's verify our solution satisfies all constraints:

Does This Make Sense?

Our answer passes several sanity checks:

• Capacity: We use exactly the minimum capacity needed (400 seats) with no waste
• Resource utilization: We use all 9 available drivers efficiently
• Cost efficiency: We maximize the use of cheaper small buses (cost $7.50/seat) while using the minimum necessary large buses (cost $10.00/seat)
• Boundary solution: This is a corner point solution, which is expected in linear programming

If we used fewer total buses, we couldn't seat all 400 students. If we used more buses, we'd exceed our driver limit or increase costs unnecessarily.

Common Pitfalls

Recognizing This Problem Type

Watch for these keywords that signal a linear programming optimization problem:

• "Minimize cost" or "maximize profit" — indicates an objective function to optimize
• "Subject to" or "limited by" — introduces constraints that bound your solution
• Multiple resource limits — drivers, buses, capacity all create boundaries
• "How many of each type" — suggests you're choosing between alternatives

Similar problems appear in manufacturing (minimize production cost subject to demand), nutrition (minimize cost subject to dietary requirements), and scheduling (minimize staff while meeting coverage needs).

The Math Behind the Optimization

This problem demonstrates linear programming, where we optimize a linear objective function subject to linear constraints. The general form is:

In our bus problem:

• Variables: L and S (number of each bus type)
• Objective function: Minimize 500L + 300S
• Constraints: Capacity, drivers, and availability limits

The Fundamental Theorem of Linear Programming guarantees that if an optimal solution exists, it occurs at a vertex of the feasible region. That's why we only needed to test the corner points.

Real-World Applications

This exact type of optimization appears frequently in:

• Transportation logistics: Airlines optimizing aircraft assignment to routes
• Manufacturing: Factories determining product mix to maximize profit within capacity constraints
• Resource allocation: Hospitals scheduling staff to minimize costs while meeting patient care requirements
• Investment portfolio: Balancing different asset types to minimize risk while achieving target returns

What If?

2026-07-27