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# Linear Programming Model for Maximization

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[IV](20) The Golden Gate Cranberry Company purchases cranberries from local growers and
makes cranberry sauce and cranberry juice. It costs \$0.70 to produce a can of cranberry sauce
and \$0.95 to produce a bottle of cranberry juice. In order to present a representative marketing mix
to its customers the company has made it a policy that at least thirty percent, but not more than
sixty percent, of the items it produces be cranberry sauce.
The company wants to produce up to, but no more than, the demand for each product. The
company marketing manager estimates that the demand for cranberry sauce is a maximum of
5,000 cans plus an additional 3 cans for each \$1 spent on advertising. The maximum demand
for cranberry juice is estimated to be 4,000 bottles plus an additional 5 bottles for every \$1 spent
to promote cranberry juice. The company has \$16,000 to spend on producing and advertising
cranberry sauce and cranberry juice. Cranberry sauce sells for \$1.45 per can while cranberry juice
sells for \$1.75 per bottle.
The company wants to know how many units of each cranberry product to produce and how
much to spend on advertising for each product in order to maximize profit. Formulate a linear
programming model for this problem. Make sure you clearly define decision variables. Do not
attempt to solve this problem.

https://brainmass.com/math/linear-programming/linear-programming-model-maximization-350265

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Decision Variables:
x = The number of cans of sauce produced
y = The number of bottles of juice produced
z = Amount in dollars spent on advertising
Objective Function:
Maximize Profit, P = 1.45x + 1.75y
Constraints:
0.70x + 0.95y + z â‰¤ 16000 [Cost]
0.3(x + y) â‰¤ x â‰¤ 0.6(x + y), that is (a) -0.7x + 0.3y â‰¤ 0 and (b) 0.4x - 0.6y â‰¤ 0 [Representative Mix]
x â‰¤ 5000 + 3z, that is x - 3z â‰¤ 5000 [Demand for Sauce]
y â‰¤ 4000 + 5z, that is y - 5z â‰¤ 4000 [Demand for Juice]
x, y â‰¥ 0 and Integers, and z â‰¥ 0 [Non-negativity]

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