# Linear Programming Model for Maximization

[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

## SOLUTION This solution is **FREE** courtesy of BrainMass!

The solution file is attached.

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]

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