Let's start solving part (a), where we need to formulate the problem as a Linear Programming (LP) model.

Problem:

Reddy Mikks produces two types of paint:

• Exterior paint
• Interior paint

It uses two raw materials, M1 and M2, and we are given the following:

• Exterior paint requires:

  • 6 tons of M1
  • 1 ton of M2
  • Yields a profit of $5,000 per ton

• Interior paint requires:

  • 4 tons of M1
  • 2 tons of M2
  • Yields a profit of $4,000 per ton

The availability of the raw materials is:

• M1: 24 tons per day
• M2: 6 tons per day

Goal:

Maximize the total daily profit by determining the optimal product mix of interior and exterior paints.

Decision Variables:

Let:

• $x_1$ = tons of exterior paint produced per day
• $x_2$ = tons of interior paint produced per day

Objective Function:

Maximize the profit: $Z = 5000x_1 + 4000x_2$

Constraints:

We need to ensure the use of raw materials does not exceed the daily availability:

• M1 constraint (for both types of paint): $6x_1 + 4x_2 \leq 24$ (since 6 tons of M1 is used for each ton of exterior paint and 4 tons of M1 for each ton of interior paint)

• M2 constraint: $x_1 + 2x_2 \leq 6$ (since 1 ton of M2 is used for each ton of exterior paint and 2 tons for each ton of interior paint)

• Non-negativity constraints: $x_1 \geq 0, \quad x_2 \geq 0$

Linear Programming Model:

$\text{Maximize } Z = 5000x_1 + 4000x_2$ Subject to: $6x_1 + 4x_2 \leq 24$ $x_1 + 2x_2 \leq 6$ $x_1 \geq 0, \quad x_2 \geq 0$

This formulation represents the problem mathematically.

Let me know if you'd like me to proceed with part (b) or provide further clarification.

Here are some related questions:

1. What does the graphical solution look like for this LP problem?
2. How can we use the Simplex method to solve this LP problem?
3. What is the significance of dual variables in linear programming?
4. How does the availability of raw materials affect the product mix?
5. How can post-optimal analysis be used in this case?

Tip: Linear programming helps in optimizing resources and finding the best possible outcome (maximum profit or minimum cost) subject to given constraints.