To determine how many units of each product should be produced, we need to solve this as a linear programming problem. The goal is to maximize the contribution margin while satisfying constraints on machine availability and minimum demand.

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Let:

• $x$: the number of units of Product C produced.
• $y$: the number of units of Product B produced.

Constraints:

1. Steaming Machine Time Constraint: Each product uses time on the steaming machine: $15x + 24y \leq 4800 \text{ hours } \times 60 = 288,000 \text{ minutes.}$

2. Minimum Demand Constraint: $x \geq 3000, \quad y \geq 3000.$

3. Non-Negativity: $x \geq 0, \quad y \geq 0.$

Objective Function:

We want to maximize the total contribution margin: $\text{Maximize } Z = 16x + 32y.$

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Solving the Problem:

1. Convert the time constraint into a usable inequality: $15x + 24y \leq 288,000.$

2. Set the minimum demand constraints: $x \geq 3000, \quad y \geq 3000.$

3. Solve the system to maximize $Z = 16x + 32y$.

Let’s calculate the feasible solution and the optimal allocation.from scipy.optimize import linprog

Coefficients for the objective function (negative for maximization)

c = [-16, -32] # Maximize 16x + 32y

Coefficients for inequality constraints (Ax <= b)

A = [ [15, 24], # Time constraint [-1, 0], # x >= 3000 (converted to -x <= -3000) [0, -1] # y >= 3000 (converted to -y <= -3000) ]

b = [288000, -3000, -3000] # Right-hand side values

Bounds for x and y (x >= 3000, y >= 3000)

x_bounds = (3000, None) y_bounds = (3000, None)

Solve the linear programming problem

result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method="highs")

Extract results

result.x, result.fun(array([ 3000., 10125.]), -372000.0)(array([ 3000., 10125.]), -372000.0)successThe optimal production plan is:

• Produce 3,000 units of Product C.
• Produce 10,125 units of Product B.

This maximizes the contribution margin to a value of $372,000.

Explanation:

• The machine time constraint and minimum demand requirements are fully utilized to prioritize the product with a higher contribution margin (Product B).
• You can now allocate resources effectively to achieve the best profitability within the constraints.

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Would you like to review the detailed breakdown of the calculations or any adjustments? Here are some related questions to consider:

1. How does the production plan change if the available machine hours increase?
2. What happens if the minimum demand for Product C or B changes?
3. How can the business increase its overall profitability under these constraints?
4. What is the shadow price (marginal value) of an additional machine hour?
5. How does changing the contribution margin for either product affect the solution?

Tip: Analyze the sensitivity of constraints to understand how small changes in machine time or demand affect profitability.