The problem involves setting up a linear programming (LP) model based on the provided information. Let's break down the information and set up the LP:

Problem Breakdown:

• Products: Two products, $A$ and $B$, are manufactured.
• Profit per unit:
  • Profit from product $A$ = Rs 30
  • Profit from product $B$ = Rs 40
• Processing requirements:
  • Product $A$ requires 1 hour on machine $M1$ and 2 hours on machine $M2$.
  • Product $B$ requires 2 hours on machine $M1$ and 1 hour on machine $M2$.
• Machine availability:
  • Machine $M1$ is available for 7 hours.
  • Machine $M2$ is available for 10 hours.

Variables:

Let:

• $x$ be the number of units of product $A$ produced.
• $y$ be the number of units of product $B$ produced.

Objective Function:

Maximize the profit $Z$: $Z = 30x + 40y$

Constraints:

1. Machine $M1$ time constraint: $1x + 2y \leq 7$
2. Machine $M2$ time constraint: $2x + 1y \leq 10$
3. Non-negativity constraint: $x \geq 0, \quad y \geq 0$

LP Model:

The LP problem can be formulated as follows:

$\text{Maximize } Z = 30x + 40y$

$\text{Subject to:}$ $x + 2y \leq 7$ $2x + y \leq 10$ $x \geq 0, \quad y \geq 0$

This is the complete formulation of the linear programming problem based on the given information.

Would you like to solve this LP problem graphically or using any other method? Let me know if you have any questions or if you would like to dive into the solution process.

Related Questions:

1. How can we graphically represent the feasible region of this LP problem?
2. What are the corner points of the feasible region, and how do they help in finding the optimal solution?
3. How does the simplex method solve this LP problem?
4. What would happen if the availability of machine $M1$ increased to 8 hours?
5. How would the objective function change if the profit per unit of product $A$ was reduced to Rs 20?

Tip:

Always ensure to check the feasibility of the constraints and the non-negativity condition when solving any linear programming problem.