A builder needs to transport sand and gravel. A truck

Solution

Each truck carries 4 tons of sand and 3 tons of gravel. Each van carries 2 tons of sand and 2 tons of gravel. The requirement is:

• At least 16 tons of sand: 4x + 2y ≥ 16
• At least 12 tons of gravel: 3x + 2y ≥ 12
• Non-negativity: x ≥ 0, y ≥ 0

Objective: Minimize cost C = 800x + 600y Solve the constraints as equations (1) 4x + 2y = 16 ⇒ 2x + y = 8 (2) 3x + 2y = 12 From (1): y = 8 − 2x Substitute into (2): 3x + 2(8 − 2x) = 12 3x + 16 − 4x = 12 ⇒ −x = −4 ⇒ x = 4 Then y = 8 − 2(4) = 0 One point: (4, 0) Try other feasible combinations Try x = 2: Sand: 4×2 + 2y ≥ 16 ⇒ 8 + 2y ≥ 16 ⇒ y ≥ 4 Gravel: 3×2 + 2y ≥ 12 ⇒ 6 + 2y ≥ 12 ⇒ y ≥ 3
So y = 4 works ⇒ Point: (2, 4) Try x = 0: Sand: 4×0 + 2y ≥ 16 ⇒ y ≥ 8 Gravel: 3×0 + 2y ≥ 12 ⇒ y ≥ 6 y = 8 satisfies ⇒ Point: (0, 8) Calculate cost at feasible points

• At (4, 0): C = 800×4 + 600×0 = ₹3200
• At (2, 4): C = 800×2 + 600×4 = ₹4000
• At (0, 8): C = 800×0 + 600×8 = ₹4800

Minimum cost = ₹3200 when 4 truck trips and 0 van trips are used.