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)
Formulate the linear programming problem (LPP)
Answers
GPT-4o
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.
### 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.