Question

A company has three operational departments (weaving, processing, and packing) with the capacity to produce three different types of clothes namely suiting, shirting, and woolens yielding a profit of Birr 2, 4, and 3 respectively. One meter of suiting requires 3 minutes in weaving, 2 minutes in processing, and 1 minute in packing. Similarly, one meter of shirting requires 4 minutes in weaving, 1 minute in processing, and 3 minutes in packing. One meter of woolen requires 3 minutes in each department. In a week, the total run time of each department is 60, 40, and 80 hours for weaving, processing, and packing respectively.
Formulate the linear programming problem (LPP)

Answer 1

To formulate this linear programming problem (LPP), we need to define the decision variables, the objective function, and the constraints based on the given information.

### Decision Variables:
Let's denote:
- \( x_1 \) as the number of meters of suiting produced per week.
- \( x_2 \) as the number of meters of shirting produced per week.
- \( x_3 \) as the number of meters of woolen produced per week.

### Objective Function:
The company aims to maximize the profit, where the profit for each meter of suiting, shirting, and woolen is Birr 2, 4, and 3 respectively. Therefore, the objective function is:
\[ \text{Maximize } Z = 2x_1 + 4x_2 + 3x_3 \]

### Constraints:
1. **Weaving Department:** The total time required for weaving is limited to 60 hours per week. In minutes, this is \( 60 \times 60 = 3600 \) minutes.
\[ 3x_1 + 4x_2 + 3x_3 \leq 3600 \]

2. **Processing Department:** The total time required for processing is limited to 40 hours per week. In minutes, this is \( 40 \times 60 = 2400 \) minutes.
\[ 2x_1 + 1x_2 + 3x_3 \leq 2400 \]

3. **Packing Department:** The total time required for packing is limited to 80 hours per week. In minutes, this is \( 80 \times 60 = 4800 \) minutes.
\[ 1x_1 + 3x_2 + 3x_3 \leq 4800 \]

4. **Non-negativity Constraints:** The production volumes cannot be negative.
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]
\[ x_3 \geq 0 \]

### Formulated LPP:
Putting it all together, the linear programming problem can be formulated as follows:

**Maximize:**
\[ Z = 2x_1 + 4x_2 + 3x_3 \]

**Subject to:**

1. Weaving constraint:
\[ 3x_1 + 4x_2 + 3x_3 \leq 3600 \]

2. Processing constraint:
\[ 2x_1 + 1x_2 + 3x_3 \leq 2400 \]

3. Packing constraint:
\[ 1x_1 + 3x_2 + 3x_3 \leq 4800 \]

4. Non-negativity constraints:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]
\[ x_3 \geq 0 \]

This formulation provides the framework to use linear programming techniques to determine the optimal production levels of suiting, shirting, and woolen to maximize the profit.