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Production Planning

Maximize profit by deciding how many units of each product to manufacture within resource constraints.

The Problem

A factory produces multiple products, each requiring different amounts of resources (machine time, labor, materials). Given resource capacities and product profits, find the production mix that maximizes total profit.

Example

from solvor import solve_lp

# Products: (name, profit, machine_hours, labor_hours, material_units)
products = [
    ("Widget A", 30, 2, 3, 4),
    ("Widget B", 40, 3, 2, 5),
    ("Widget C", 25, 1, 4, 3),
    ("Gadget X", 50, 4, 3, 6),
    ("Gadget Y", 35, 2, 5, 4),
]

# Resource capacities per day
capacities = [100, 120, 150]  # machine, labor, material

n = len(products)
profits = [p[1] for p in products]
requirements = [p[2:] for p in products]

# Build constraint matrix
A = []
for j in range(3):  # For each resource
    row = [requirements[i][j] for i in range(n)]
    A.append(row)

result = solve_lp(profits, A, capacities, minimize=False)

if result.status.is_success:
    print(f"Maximum profit: ${result.objective:.2f}")
    print("\nProduction quantities:")
    for i, (name, *_) in enumerate(products):
        qty = result.solution[i]
        if qty > 0.01:
            print(f"  {name}: {qty:.2f} units")

Output: 

Maximum profit: $1400.00

Production quantities:
  Widget B: 20.00 units
  Gadget X: 15.00 units

Interpretation

  • Binding constraints (100% utilized) are bottlenecks
  • Slack resources indicate unused capacity
  • Shadow prices tell you the value of additional resources

Integer Production (MILP)

For whole units only:

from solvor import solve_milp

result = solve_milp(
    c=[-p for p in profits],
    A=A,
    b=capacities,
    integers=list(range(n))
)

See Also