01 May MAT 540 Julia’s Food Booth Case Problem
MAT 540 Julia’s Food Booth Case Problem
. Linear Programming Model:
a. Decision Variables: The problem is to determine the number of slices of pizza,
number of hot dogs, and the number of barbecue sandwiches that have to be sold in
order to maximize profit. Let x1 = the number of slices of pizza, x2 = the number of
hot dogs, and x3 = the number of BBQ sandwiches.
b. Objective function: The objective is to maximize profit. The price per unit and cost
per unit of each item is given in the problem. For example the cost of pizza is given to
be $6 for 8 slices. Therefore the cost of one slice of pizza will be 6/8 = $0.75. The
price of one slice of pizza is given to be $1.50. The net profit from selling x1 slices of
pizza will then be ($1.50 – $0.75)*x1 = $0.75*x1. Similarly you will find the net
profit from selling x2 hot dogs and x3 BBQ sandwiches. The total will be the net
profit and that will be the objective function.
c. Constraints: There are 4 constraints:
(i)
There is a restriction on budget. Julia has only got $1500. The total of costs of
x1 slices of pizza, x2 hot dogs, and x3 BBQ sandwiches cannot exceed $1500.
The cost per unit of each item is given in the problem.
(ii)
There is a restriction on the oven space. There are 16 shelves and each shelf is
3ft by 4ft (36 in by 48 in. note that 1ft = 12 in). The oven is going to be filled
twice, one before the game (kickoff) and one during half time. The size of one
big round (circular) pizza is given to be 14*14 = 196 square inches. Each big
pizza includes 8 slices of pizza and therefore the space needed for a slice of
pizza should be 196/8 = 24.5 square inches (you can take it to be
approximately 24 square inches). Then the space needed for x1 slices of pizza
will be 24*x1 square inches. The size of each hot dog and each BBQ
sandwich is also given. Determine the space needed for x2 hot dogs and x3
BBQ sandwiches in a similar way. The total will then give you the total space
needed for x1 slices of pizza, x2 hot dogs, and x3 BBQ sandwiches. The total
space available is the dimension of the shelf and that is 16*36*48 = 27648
square inches. The oven space is used twice, one before kickoff and one at
half time. Therefore the total space required will be 2*27648 = 55296 square
inches. Thus the total space needed for x1 slices of pizza, x2 hot dogs, and x3
sandwiches should not exceed 55296 square inches.
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