This problem can be solved using linear programming (LP). The goal is to determine how many of each type of desk should be produced in order to maximize profit, while satisfying the constraints on the available time in the cabinet, finishing, and crating shops.

Let:

• $x_1$ represent the number of Executive desks to produce,
• $x_2$ represent the number of Office desks to produce,
• $x_3$ represent the number of Student desks to produce.

The objective is to maximize profit given by:

$\text{Maximize } Z = 150x_1 + 125x_2 + 50x_3$

Constraints:

We are constrained by the number of available hours in the three shops (cabinet, finishing, and crating):

1. Cabinet shop constraint:

The time spent in the cabinet shop for each desk is given by: $2x_1 + 1x_2 + 1x_3 \leq 16 \quad \text{(hours available in cabinet shop)}$

2. Finishing shop constraint:

The time spent in the finishing shop for each desk is given by: $1x_1 + 2x_2 + 1x_3 \leq 16 \quad \text{(hours available in finishing shop)}$

3. Crating shop constraint:

The time spent in the crating shop for each desk is given by: $1x_1 + 1x_2 + 0.5x_3 \leq 10 \quad \text{(hours available in crating shop)}$

Non-negativity constraints:

Since the number of desks produced cannot be negative: $x_1, x_2, x_3 \geq 0$

Steps to Solve:

1. Set up the objective function and constraints: $\text{Maximize } Z = 150x_1 + 125x_2 + 50x_3$ Subject to: $2x_1 + 1x_2 + 1x_3 \leq 16 \quad \text{(Cabinet shop)}$ $1x_1 + 2x_2 + 1x_3 \leq 16 \quad \text{(Finishing shop)}$ $1x_1 + 1x_2 + 0.5x_3 \leq 10 \quad \text{(Crating shop)}$ $x_1, x_2, x_3 \geq 0$

2. Graphical method or Simplex Method: Since this is a linear programming problem with three variables, the best method is to use the Simplex Method (or use a solver like Excel, MATLAB, Python, etc.) to find the solution. A graphical approach would not be feasible for three variables.

3. Solve for the optimal values of $x_1$, $x_2$, and $x_3$ that maximize profit while satisfying all constraints.

If you'd like, I can use Python or another solver to solve this system for you. Would you like me to compute the solution, or would you prefer to solve it using a tool of your choice?

Let me know if you have any questions or need further explanation.

Questions:

1. What are the steps involved in the Simplex method?
2. How does the graphical method work for two-variable linear programming problems?
3. How do constraints impact the feasible region in linear programming?
4. Can the problem be solved using integer programming if only whole desks can be made?
5. How does changing the available shop hours impact the solution?

Tip: Understanding how the constraints affect the feasible region is crucial when solving optimization problems.