Formulate the following problem as a linear programming problem. Set up the prob

Question

Formulate the following problem as a linear programming problem. Set up the problem. A company which produces three kinds of spaghetti sauce has two plants. The East plant produces 3, 500 jars of plain sauce, 6, 500 jars of sauce with mushrooms, and 3,000 jars of hot spicy sauce per day. The West plant produces 2, 500 jars of plain sauce, 2,000 jars of sauce with mushrooms, and 1, 500 jars of hot spicy sauce per day. The cost to operate the East plant is $8, 500 per day and the cost to operate the West plant is $9, 500 per day. How many days should each plant operate to minimize cost and to fill an order for at least 8,000 jars of plain sauce, 9,000 jars of sauce with mushrooms, and 6,000 jars of hot spicy sauce? (Let x_1 equal the number of days East plant should operate and x_2 the number of days West plant should operate.)

Explanation / Answer

ANSWER :-

minimize 8500x + 9500x
subject to 3500x + 2500x 8000 (plain sauce)

6500x + 2000x 9000 (mushroom sauce)
3000x + 1500x 6000 (hot spicy sauce)

x 0 (nonnegative number of operating days for East plant)
x 0 (nonnegative number of operating days for West plant)

The East plant produces more of every sauce and has lower operations costs. Therefore, the West plant doesn't need to run at all.

WIth the given information ,the East plant has to run just long enough to satisfy the sauce that has the highest ratio of (amount required) to (amount produced).

In this case, that sauce is the plain sauce, which requires 8000/3500 = 16/7 = 2 and 2/7 days of production to achieve,

So the Optimal solution is : (x,x) = (16/7,0).

The operating cost associated with this solution is about $19,428.57

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