A company manufactures x units of Product A and y units of Product B, on two machines, I and II. It has been determined that the company will realize a profit of $4/unit of Product A and a profit of $3/unit of Product B. To manufacture a unit of Product A requires 6 min on Machine I and 5 min on Machine II. To manufacture a unit of Product B requires 9 min on Machine I and 4 min on Machine II. There are 5 hr of machine time available on Machine I and 3 hr of machine time available on Machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?
A company manufactures x units of Product A and y units of Product B, on two machines, I and II. It has been determined that the company will realize a profit of $4/unit of Product A and a profit of $3/unit of Product B. To manufacture a unit of Product A requires 6 min on Machine I and 5 min on Machine II. To manufacture a unit of Product B requires 9 min on Machine I and 4 min on Machine II. There are 5 hr of machine time available on Machine I and 3 hr of machine time available on Machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit?
Sol:-
Let x be the number of units of Product A produced and y be the number of units of Product B produced. Then, the objective function we want to maximize is:
z = 4x + 3y
subject to the following constraints:
6x + 9y <= 300 (Machine I time)
5x + 4y <= 180 (Machine II time)
x >= 0
y >= 0
To put this in standard form for the simplex method, we introduce slack variables s1 and s2, which represent the unused machine time for machines I and II, respectively. The model becomes:
maximize: z = 4x + 3y
subject to: 6x + 9y + s1 = 300
5x + 4y + s2 = 180
x >= 0
y >= 0
s1 >= 0
s2 >= 0
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