As in any linear programming (LP) problem, the first and most important step is that of problem
formulation. There are three decision variables – that is, number of pizza slices, hot dogs, and
BBQ sandwiches Julia must sell to maximize her profit and help her decide if she should lease
the booth or not. Therefore,
let X1 = pizza slices,
X2 = hot dogs,
X3 = BBQ sandwiches
For the first game, we have the following model:
Maximize Z = $0.75X1 + $1.05X2 +$1.35X3
Subject to,
0.75X1 + 0.45X2 + 0.90X3 ≤ $1,500 (budget)
24X1 + 16X2 + 25X3 ≤ 55,296 in² (oven space)
Obs.: the calculation for the oven space is as follows:
Pizza slice: total space required for a 14 X 14 pizza = 196 in². Since there are 8 slices, we
divide 196 by 8, and this gives us approximately 24 in² per slice.
The total dimension of oven is the dimension of the oven shelf, 36 in X 48 in = 1,728 in²,
multiplied by 16 shelves = 27,648 in², which is multiplied by 2, before kickoff and during
the halftime, giving us a total space of 55,296 in².
X1 ≥ X2 + X3 or X1 – X2 – X3 ≥ 0 (pizza sales)
X2/X3 ≥ 2.0 or X2 – 2X3 ≥ 0 (hot dog sales vs. BBQ sandwiches)
Using QM for Windows, this problem can be easily solved and we get:
X1 = 1,250 slices of pizza
X2 = 1,250 hot dogs
X3 = 0 BBQ sandwiches
Profit Z = $2,250.00
Now, we can answer the four questions of this case problem. These are:
A)
Julia should receive a profit of $2,250 for the first game. With her costs of $,1000 for
the lease of booth and $100/game for the lease of oven, Julia will clear $1,150 per game.
So, Julia should lease the booth.
B)
From QM for window, we can see that the dual value is $1.50 for each additional dollar.
Therefore, Julia would increase her profit, if she borrows some money. However, the
upper limit of the sensitivity rage is $1,658.88, so she should only borrow $158.88 and
her additional profit would be $238.32 or a total profit of $2,488.32.
C)
Yes, she should hire her friend for $100.00 per game, for it is almost impossible for her to
prepare all the food in such a short time.
D)
The biggest uncertainty is the weather that could affect the sales projection.
Observe that all sensitivity analysis results can be easily obtained from QM for Window. From
this case problem, you will realize why QM for Window (and other similar LP computer
programs readily available in the market) is the most widely used optimization technique in
Management Science.