Final answer:
The linear programming problem is to minimize the total cost of production, C = 60x + 80y, with x representing the number of units of product A and y the number of units of product B, subject to the constraints of y ≥ 200, x ≤ 400, and x + y ≤ 500, with both x and y being greater than or equal to zero.
Step-by-step explanation:
To formulate this problem as a linear programming problem, we need to define variables, the objective function, and the constraints based on the provided information. Let's define x as the number of units of product A and y as the number of units of product B. The objective function would be to minimize the total cost of production, which can be represented as C = 60x + 80y, where 60 and 80 represent the costs of producing one unit of product A and B respectively.
There are several constraints to consider:
The company has to supply at least 200 units of product B, so we have y ≥ 200.
The total machine hours for product A are limited to 400 hours, which gives us the constraint x ≤ 400 since one unit of product A requires one machine hour.
One unit of each product requires one labor hour each, with a total of 500 labor hours available, leading to the constraint x + y ≤ 500.
Also, we must consider that the number of products manufactured cannot be negative, thus x ≥ 0 and y ≥ 0.
The linear programming problem can therefore be summarized as follows:
Objective Function:
Minimize C = 60x + 80y
Subject to Constraints:
y ≥ 200
x ≤ 400
x + y ≤ 500
x ≥ 0
y ≥ 0