1. Case Problem: “The Possibility” Restaurant Angela Fox and Zooey
Caulfield were food and nutrition majors at State University, as well as
close friends and roommates. Upon graduation Angela and Zooey
decided to open a French restaurant in Draperton, the small town where
the university was located. There were no other French restaurants in
Draperton, and the possibility of doing something new and somewhat
risky intrigued the two friends. They purchased an old Victorian home
just off Main Street for their new restaurant, which they named “The
Possibility.” Angela and Zooey knew in advance that at least initially they
could not offer a full, varied menu of dishes. They had no idea what their
local customers’ tastes in French cuisine would be, so they decided to
serve only two full-course meals each night, one with beef and the other
with fish. Their chef, Pierre, was confident he could make each dish so
exciting and unique that two meals would be sufficient, at least until
they could assess which menu items were most popular. Pierre indicated
that with each meal he could experiment with different appetizers,
soups, salads, vegetable dishes, and desserts until they were able to
identify a full selection of menu items. The next problem for Angela and
Zooey was to determine how many meals to prepare for each night so
they could shop for ingredients and set up the work schedule. They could
not afford too much waste. They estimated that they would sell a
maximum of 60 meals each night. Each fish dinner, including all
accompaniments, requires 15 minutes to prepare, and each beef dinner
takes twice as long. There is a total of 20 hours of kitchen staff labor
available each day. Angela and Zooey believe that because of the health
consciousness of their potential clientele, they will sell at least three fish
dinners for every two beef dinners. However, they also believe that at
least 10% of their customers will order beef dinners. The profit from each
fish dinner will be approximately $12, and the profit from a beef dinner
will be about $16. Formulate a linear programming model for Angela and
Zooey that will help them estimate the number of meals they should
prepare each night and solve this model using graphical or simplex
method.
2. Solve the following problem using graphical linear programming.
Minimize Z = 8x1 + 12x2
Subject to
5x1 + 2x2 ≥ 20
4x1 + 3x2 ≥ 24
x2 ≥ 2
x1, x2 ≥ 0
3. A small candy shop is preparing for the holiday season. The owner must decide
how many bags of deluxe mix and how many bags of standard mix of
Peanut/Raisin Delite to put up. The deluxe mix has 2⁄3 pound raisins and 1⁄3
pound peanuts, and the standard mix has 1⁄2 pound raisins and 1⁄2 pound
peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts
to work with. Peanuts cost $.60 per pound and raisins cost $1.50 per pound. The
deluxe mix will sell for $2.90 for a one-pound bag, and the standard mix will sell
for $2.55 for a one-pound bag. The owner estimates that no more than 110 bags
of one type can be sold.
A. If the goal is to maximize profits, how many bags of each type should be
prepared?
B. What is the expected profit?
4. Solve the following LPP.
Max Z = x1+2x2+3x3- x4
Subject to:
x1+2x2+3x3=15
2x1+x2+5x3=20
x1+2x2+x3+ x4 =10
x1, x2, x3, x4 ≥0