Let's break down the problem for part (b) systematically.

We need to maximize profit given the following updated scenario:

Profit per set:

• Bargain Set Profit: Now $430 instead of $360.
• Deluxe Set Profit: Still $520.

Production constraints:

• Assembly Line: 3600 hours available.
• Cabinet Shop: 1200 hours available.
• Testing and Packing Department: 2800 hours available.

Resource usage:

• Bargain Set:

  • 5 hours on the assembly line
  • 1 hour in the cabinet shop
  • 4 hours in testing and packing

• Deluxe Set:

  • 7 hours on the assembly line
  • 3 hours in the cabinet shop
  • 4 hours in testing and packing

Defining Variables:

Let:

• $x$ be the number of bargain sets produced.
• $y$ be the number of deluxe sets produced.

Objective Function (Profit):

The profit function is: $P = 430x + 520y$ We want to maximize this profit.

Constraints:

We have the following resource constraints:

1. Assembly Line: $5x + 7y \leq 3600$
2. Cabinet Shop: $x + 3y \leq 1200$
3. Testing and Packing: $4x + 4y \leq 2800$
4. Non-negativity: $x \geq 0, \quad y \geq 0$

Solving the Linear Programming Problem:

To find the optimal solution, we need to solve this linear programming problem, typically using the graphical method, the simplex method, or optimization software.

I will use the constraints to set up and solve the system.

Let me calculate that for you! scipy.optimize import linprog

Coefficients for the objective function (maximize profit)

c = [-430, -520] # We negate the coefficients to maximize (as linprog minimizes)

Coefficients for the inequality constraints (Ax <= b)

A = [ [5, 7], # Assembly line constraint [1, 3], # Cabinet shop constraint [4, 4] # Testing and packing constraint ]

Right-hand side of the inequalities

b = [3600, 1200, 2800]

Bounds for the variables (non-negativity constraints)

x_bounds = (0, None) y_bounds = (0, None)

Solving the linear program

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

Extracting the optimal values

optimal_x = result.x[0] optimal_y = result.x[1] max_profit = -result.fun # We negated the coefficients, so we need to negate the result

optimal_x, optimal_y, max_profitTo maximize profit, the company should produce:

• 650 deluxe sets
• 50 bargain sets

The maximum profit will be $305,500.

Would you like a detailed explanation of how this was calculated? Or do you have any questions about this solution?

Here are some follow-up questions to consider:

1. How would the maximum profit change if the available work-hours for the cabinet shop were reduced by 100 hours?
2. What would happen if the company could increase the available work-hours for the assembly line by 500 hours?
3. How would the optimal number of sets change if the profit for the deluxe sets dropped to $490?
4. What methods can be used to solve such linear programming problems without software?
5. How does the concept of "constraints" affect the optimal production decision?

Tip: In linear programming, when maximizing profit, always make sure to reverse the signs of the profit function's coefficients because many solvers like linprog minimize functions by default.