To solve this problem step-by-step, let's start by defining variables, setting up the constraints based on the available hours in each shop, and then formulate the objective function for profit maximization.

Step 1: Define Variables

Let:

• $x$ = number of executive desks
• $y$ = number of office desks
• $z$ = number of student desks

Step 2: Define Constraints

From the table provided, each type of desk requires different hours in the cabinet shop, finishing shop, and crating shop, with available hours as follows:

1. Cabinet Shop Constraint:

   • Executive desk: 2 hours
   • Office desk: 1 hour
   • Student desk: 1 hour
   • Total available hours: 16 $2x + y + z \leq 16$

2. Finishing Shop Constraint:

   • Executive desk: 1 hour
   • Office desk: 2 hours
   • Student desk: 1 hour
   • Total available hours: 16 $x + 2y + z \leq 16$

3. Crating Shop Constraint:

   • Executive desk: 1 hour
   • Office desk: 1 hour
   • Student desk: 0.5 hours
   • Total available hours: 10 $x + y + 0.5z \leq 10$

4. Non-negativity Constraints: $x \geq 0, \quad y \geq 0, \quad z \geq 0$

Step 3: Define the Objective Function

The objective is to maximize the profit:

• Profit from executive desk: $150
• Profit from office desk: $125
• Profit from student desk: $50

The total profit $P$ can be expressed as: $P = 150x + 125y + 50z$

Step 4: Formulate the Linear Programming Problem

We have the following linear programming problem:

Maximize: $P = 150x + 125y + 50z$

Subject to: $2x + y + z \leq 16$ $x + 2y + z \leq 16$ $x + y + 0.5z \leq 10$ $x \geq 0, \quad y \geq 0, \quad z \geq 0$

Step 5: Solve the Problem Using Simplex Method or Graphical Analysis

Since this is a multi-variable linear programming problem, it is best solved using the Simplex method or a linear programming solver.

Would you like me to proceed with the calculation using a linear programming tool, or do you prefer a detailed explanation on how to solve this manually?

Let me know if you have questions or need further clarification on any part of the solution so far. Here are some related questions that could deepen your understanding:

1. How is the Simplex method applied in linear programming?
2. Why do we use inequalities as constraints in linear programming problems?
3. What are the real-life applications of linear programming in production optimization?
4. How does changing one of the available hours impact the maximum profit?
5. What would be the effect on the solution if the profit values were different?

Tip: In optimization problems, setting up constraints accurately is crucial as they define the feasible region for the solution.