Taco Loco Is Considering a New Addition to Their Menu $\text { Variable Cells }$

Short Answer

Taco Loco is considering a new addition to their menu. They have test marketed a number of possibilities and narrowed them down to three new products, X, Y, and Z. Each of these products is made from a different combination of beef, beans, and cheese, and each product has a price point. Taco Loco feels they can sell an X for $17, a Y for $13, and a Z for $14. The company's management science consultant formulates the following linear programming model for company management.
Max R = 14Z + 13Y + 17X
subject to:
Beef 2Z + 3Y + 4X ≤ 28
Cheese 9Z + 8Y + 11X ≤ 80
Beans 4Z + 4Y + 2X ≤ 68
X,Y,Z ≥ 0
The sensitivity report from the computer model reads as follows:
$\text { Variable Cells }$
$\begin{array}{llccccc}\hline \text { Cell } & \text { Name } & \begin{array}{c}\text { Final } \\\text { Value }\end{array} & \begin{array}{c}\text { Reduced } \\\text { Cost }\end{array} & \begin{array}{c}\text { Objective } \\\text { Coefficient }\end{array} & \begin{array}{c}\text { Allowable } \\\text { Increase }\end{array} & \begin{array}{c}\text { Allowable } \\\text { Decrease }\end{array} \\\hline \text { \$C\$4 } & \mathrm{Z} & 1.45 & 0 & 14 & 0.63 & 5.33 \\\hline \text { \$D\$4 } & \text { Y } & 8.36 & 0 & 13 & 8 & 0.56 \\\hline \text { \$E\$4 } & \mathrm{X} & 0 & -0.818 & 17 & 0.818 & 1 \mathrm{E}+30 \\\hline\end{array}$

$\text { Constraints }$
$\begin{array}{llccccc}\hline {\text { Cell }} & {\text { Name }} & \begin{array}{c}\text { Final } \\\text { Value }\end{array} & \begin{array}{c}\text { Shadow } \\\text { Price }\end{array} & \begin{array}{c}\text { Constraint } \\\text { R.H. Side }\end{array} & \begin{array}{c}\text { Allowable } \\\text { Increase }\end{array} & \begin{array}{c}\text { Allowable } \\\text { Decrease }\end{array} \\\hline \text { \$F\$6 } & \text { Beef } & 28 & 0.45 & 28 & 2 & 10.22 \\\hline \text { \$F\$7 } & \text { Cheese } & 80 & 1.45 & 80 & 46 & 5.33 \\\hline \text { \$F\$8 } & \text { Beans } & 39.27 & 0 & 68 & 1 \mathrm{E}+30 & 28.73 \\\hline\end{array}$
-If Taco Loco reduces the price of the X product by about 82 cents, then their optimal product mix will contain ________ X.