Muir Manufacturing produces two popular grades of commercial carpeting among its many other products. In the coming production period, Muir needs to decide, how many rolls of each grade should be produced in order to maximize profit? Each roll of Grade X carpet uses 50 units of synthetic-fiber, requires 25 hours of production time, and needs 20 units of foam backing. Each roll of Grade Y carpet uses 40 units of synthetic-fiber, requires 28 hours of production time, and needs 15 units of foam backing. The profit per roll of Grade X carpet is $200 and the profit per roll of Grade Y carpet is $160. In the coming production period, Muir has 3000 units of synthetic-fiber available for use. Workers have been scheduled to provide at least 1800 hours of production time (overtime is a possibility). The company has 1500 units of foam backing available for use. Develop a linear programming model for this problem.
1. Define the Decision Variables.
2. Write the Objective.
3. Write the Constraints.

Kindly check explanation

Explanation:

A) DECISION VARIABLES :

The decision variables in the question above lies in Muir Manufacturing Company's Production target in the coming production period:

The decision variables are thus:

1) Determining the number of rolls of Grade X carpet to manufacture.

11) Determining the number of Grade Y carpet to manufacture.

2) OBJECTIVES :

The objective is to maximize profit :

Profit per roll of Grade X carpet = $200

Profit per roll of Grade Y carpet = $160

The objective function is thus :

$200x + $160y

C.) THE CONSTRAINTS :

1) Synthetic fibre required:

Grade X = 50 units

Grade Y = 40 units

Total available units = 3000

50X + 40Y ≤ 3000

Production time:

Grade X = 25 hours of production time

Grade Y = 28 hours of production time

Scheduled production time to be provided by workers is atleast 1800 hours (since there is possibility for overtime).

25X + 28Y ≥ 1800

111) Foam backing required:

Grade X = 20 units

Grade Y = 10 units

Total available units = 1500

20X + 10Y ≤ 1500

Each roll of Grade X carpet uses 50 units of synthetic-fiber, requires 25 hours of production time, and needs 20 units of foam backing. Each roll of Grade Y carpet uses 40 units of synthetic-fiber, requires 28 hours of production time, and needs 15 units of foam backing.

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