manufacturing company has three types of machines, A, B, and C, available for production. They need to produce at least 500 units every 8-hour workday. Machine type A can produce 60 units per hour with a quality rate of 98%. Machine type B can produce 50 units per hour with a quality rate of 96%, and Machine type C can produce 45 units per hour with a quality rate of 94%. The company has a limited number of machines and operators available. There are 8 Machine A units, 10 Machine B units, and 12 Machine C units available. The hourly operating costs for these machines are ₹400 for Machine A, ₹350 for Machine B, and ₹300 for Machine C. Additionally, each defective product costs the company ₹75 in rework and other expenses. The company wants to minimize the total production and quality control costs while ensuring that they produce at least 500 units per day

This is a classic optimization problem that can be solved using linear programming. The goal is to minimize the total cost while ensuring that the company produces at least 500 quality units per day. We will need to consider both the production and quality control costs.

### Variables:
Let:
- \( x_A \) = Number of Machine A units in use.
- \( x_B \) = Number of Machine B units in use.
- \( x_C \) = Number of Machine C units in use.

### Objective:
We want to minimize the total cost, which consists of both the operating costs and the costs due to defective products.

- **Total production cost** is the sum of the hourly costs for running each machine:
  \[
  \text{Production cost} = 400x_A + 350x_B + 300x_C
  \]

- **Effective production rates** (i.e., quality-adjusted production):
  - Machine A produces \( 60 \times 8 \times 0.98 = 470.4 \) units per machine per day.
  - Machine B produces \( 50 \times 8 \times 0.96 = 384 \) units per machine per day.
  - Machine C produces \( 45 \times 8 \times 0.94 = 338.4 \) units per machine per day.

- **Defective units**:
  - Machine A has \( 60 \times 8 \times 0.02 = 9.6 \) defective units per day.
  - Machine B has \( 50 \times 8 \times 0.04 = 16 \) defective units per day.
  - Machine C has \( 45 \times 8 \times 0.06 = 21.6 \) defective units per day.

The cost of reworking defective units is ₹75 per defective unit:
- Defective cost for Machine A: \( 9.6 \times 75 = 720 \) per machine per day.
- Defective cost for Machine B: \( 16 \times 75 = 1200 \) per machine per day.
- Defective cost for Machine C: \( 21.6 \times 75 = 1620 \) per machine per day.

The total defective cost is:
\[
\text{Defective cost} = 720x_A + 1200x_B + 1620x_C
\]

Thus, the total cost (production + defective) is:
\[
\text{Total cost} = (400x_A + 350x_B + 300x_C) + (720x_A + 1200x_B + 1620x_C)
\]
\[
\text{Total cost} = 1120x_A + 1550x_B + 1920x_C
\]

### Constraints:
1. **Production constraint**: The company must produce at least 500 quality units per day:
   \[
   470.4x_A + 384x_B + 338.4x_C \geq 500
   \]
   
2. **Machine availability constraints**:
   \[
   x_A \leq 8 \quad (Machine\ A\ units\ available)
   \]
   \[
   x_B \leq 10 \quad (Machine\ B\ units\ available)
   \]
   \[
   x_C \leq 12 \quad (Machine\ C\ units\ available)
   \]

3. **Non-negativity constraint**: \( x_A, x_B, x_C \geq 0 \).

### Problem formulation:
Minimize:
\[
1120x_A + 1550x_B + 1920x_C
\]
Subject to:
\[
470.4x_A + 384x_B + 338.4x_C \geq 500
\]
\[
x_A \leq 8,\ x_B \leq 10,\ x_C \leq 12
\]
\[
x_A, x_B, x_C \geq 0
\]

This can be solved using linear programming techniques such as the simplex method to find the optimal number of machines (A, B, and C) to minimize costs while meeting production requirements.

Would you like a detailed solution using specific values or a tool like Excel or Python to solve this optimization problem?

### Five follow-up questions:
1. Would you prefer a step-by-step calculation of this optimization?
2. Are there any other constraints that need to be considered, such as limited work shifts?
3. Would you like an explanation of how linear programming works for this type of problem?
4. Do you need help setting up this problem in a specific software?
5. How would you like to see the sensitivity analysis for the machine choices?

### Tip:
Using linear programming software like Excel Solver or Python's `scipy.optimize` can help automate complex cost minimization problems like this one!

Explore how to minimize production and quality control costs in a manufacturing company using linear programming. This example involves machines A, B, and C with specific production rates, costs, and constraints. Learn the steps to formulate and solve the optimization problem to meet production requirements efficiently.

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