Question

A company has three operational departments namely weaving, processing and packing with capacity to produce three different types of clothes namely suiting’s, shirting’s and woolens yielding a profit of rupees 2 ,4and 3 per meter respectively.1 meter of suiting requires 3 minutes in weaving, 2 minutes in processing and 1 minute in packing. Similarly 1 meter of shirting requires 4 minutes in weaving ,1 minute in processing and 3 minutes in packing.1 meter of woolen requires 3 minutes in each department .In a week total run time of each department is 60,40 and 80 hours for weaving, processing and packing respectively. Formulate this as LPP and find the solution.

Answer 1

Answer: Scroll down for solution

Step-by-step explanation: To formulate this problem as a Linear Programming Problem (LPP), we need to define the decision variables, objective function, and constraints.

1. Decision Variables:

Let's denote the number of meters of suiting, shirting, and woolen produced as:

- x1: Number of meters of suiting produced

- x2: Number of meters of shirting produced

- x3: Number of meters of woolen produced

2. Objective Function:

The objective is to maximize the profit, which can be calculated as follows:

Profit = 2x1 + 4x2 + 3x3

3. Constraints:

a) Weaving Department:

The total run time available for weaving is 60 hours per week. The time required to produce 1 meter of suiting, shirting, and woolen in the weaving department is given as 3 minutes, 4 minutes, and 3 minutes, respectively. Since there are 60 minutes in an hour, the constraint for the weaving department can be expressed as:

3x1 + 4x2 + 3x3 ≤ 60

b) Processing Department:

The total run time available for processing is 40 hours per week. The time required to produce 1 meter of suiting, shirting, and woolen in the processing department is given as 2 minutes, 1 minute, and 3 minutes, respectively. The constraint for the processing department can be expressed as:

2x1 + 1x2 + 3x3 ≤ 40

c) Packing Department:

The total run time available for packing is 80 hours per week. The time required to produce 1 meter of suiting, shirting, and woolen in the packing department is given as 1 minute, 3 minutes, and 3 minutes, respectively. The constraint for the packing department can be expressed as:

1x1 + 3x2 + 3x3 ≤ 80

d) Non-negativity constraint:

The number of meters produced cannot be negative, so we have the constraint:

x1, x2, x3 ≥ 0

Now, we have the LPP formulated with the decision variables, objective function, and constraints. To find the solution, we can use a method such as the Simplex method or graphical method to optimize the objective function while satisfying the constraints.