1. Ice Cream Production (25 points)
Chin’s Ice Cream Co. is producing three ﬂavors of ice cream: vanilla, mint chip, and chocolate.
These products are manufactured using common ingredients: milk, eggs, and pre-made powder with
unit prices $5, $3, and $10, respectively. The three products also rely on common manufacturing
processes: mixing and freezing. The cost is $1.5 per hour per gallon for mixing, and $1 per hour
per gallon for freezing.
Production of a gallon of vanilla ice cream requires 3 units of milk, 2 units of eggs, 1 unit of pre-
made powder, 2 hours of mixing, and 5 hours of freezing. Production of a gallon of mint chip ice
cream requires 4 units of milk, 1.5 units of eggs, 0.5 units of pre-made power, 2 hours of mixing,
and 4 hours of freezing. Production of a gallon of chocolate ice cream requires 3.5 units of milk, 1
unit of eggs, 1.5 units of pre-made powder, 1 hour of mixing, and 3 hours of freezing. Each of the
three ﬂavors sells for $80 per gallon.
A total of 1000 units of milk, 500 units of eggs, and 250 units of pre-made powder are delivered to
the warehouse each day. The factory is equipped to mix up to 20 gallons per hour and freeze up
to 40 gallons per hour, for up to 24 hours per day. A contractual arrangement with a restaurant
chain requires that Chin’s produces at least 50 gallons of vanilla ice cream per day. The company’s
objective is to maximize proﬁt.
a) Formulate a linear program that can be used to determine how much vanilla, mint chip, and
chocolate ice cream Chin’s should produce each day. Solve the linear program and provide
the numerical quantities.
b) Formulate the dual of the linear program formulated in part (a).
c) Provide the optimal dual objective value. Explain in no more than sixty words why this can
be done by inspection, without solving the dual.
d) Identify dual variables that are equal to zero at an optimal solution to the dual. Explain in
no more than sixty words why this can be done by inspection, without solving the dual.