Question

“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 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.
A. 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 graphically.
B. If Angela and Zooey increased the menu price on the fish dinner so that the profit for both
dinners was the same, what effect would that have on their solution? Suppose Angela and
Zooey reconsidered the demand for beef dinners and decided that at least 20% of their
customers would purchase beef dinners. What effect would this have on their meal
preparation plan?

Answer 1

(a) The solution is:

J20 A B C D X2 Used Available 0.25 20 15 0.5 S - -32 0.92 -0.1 8 Obj. Fun. 40 20 12 16 10 Max. 11 12 13 Z 14 800

x1 = 40

x2 = 20

Z = 800

The graphical solution is:

Beef Max 16 Fish+16Beef 1 Fish+1 Beef<=60 15 Fish+30Beef <=1200 O 2 Fish-3Beef >=0 0 - 1 Fish+9Beef>=0 O none Fish Beef Z D 40 20 960 960 34.28571 22.85714 914.2857 Fish

The next solution is:

M20 X for : B A C D E G F Available Used 60 1s 0.5 5 0.25 20 0 -0.1 0.92 8 Obj. Fun. 40 20 16 16 10 Max. 11 12 13 Z 960 14

x1 = 40

x2 = 20

Z = 960

(b) The solution is:

K17 x1 Used Available x 2 11s 0.25 0.55 20 -0.2 0.8 2 8 Obj. Fun. 40 20 10 Max. 12 16 12 13 Z 14 800

x1 = 40

x2 = 20

Z = 800