The Linear Programming Problem Whose Output Follows Is Used to Determine

question 66

Short Answer

The linear programming problem whose output follows is used to determine how many bottles of red nail polish (x1), blue nail polish (x2), green nail polish (x3), and pink nail polish (x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand for blue, green, and pink nail polish bottles combined is at least 50 bottles.
MAX $100 x _ { 1 } + 120 x _ { 2 } + 150 x _ { 3 } + 125 x _ { 4 }$
Subject to 1 . $x _ { 1 } + 2 x _ { 2 } + 2 x _ { 3 } + 2 x _ { 4 } \leq 108$
   2. $3 x _ { 1 } + 5 x _ { 2 } + x _ { 4 } \leq 120$
   3. $x _ { 1 } + x _ { 3 } \leq 25$
   4. $x _ { 2 } + x _ { 3 } + x _ { 4 } \geq 50$
    $x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \geq 0$
Optimal Solution:
Objective Function Value = 7475.000

$$\begin{array} { c | c | c } \text { Variable } & \text { Value } & \begin{array} { c } \text { Reduced } \\\text { Costs }\end{array} \\\hline x _ { 1 } & 8 & 0 \\x _ { 2 } & 0 & 5 \\x _ { 3 } & 17 & 0 \\x _ { 4 } & 33 & 0\end{array}$$

$$\begin{array} { c c c } \text { Constraint } & \begin{array} { c } \text { Slack/ } \\\text { Surplus }\end{array} & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges

$\begin{array}{c|c|c|c}\text { Variable } & \begin{array}{c}\text { Lower } \\\text { Limit }\end{array} & \begin{array}{c}\text { Current } \\\text { Value }\end{array} & \begin{array}{c}\text { Upper } \\\text { Limit }\end{array} \\\hline x_{1} & 87.5 & 100 & \text { none } \\x_{2} & \text { none } & 120 & 125 \\x_{3} & 125 & 150 & 162 \\x_{4} & 120 & 125 & 150\end{array}$ Right Hand Side Ranges

$$\begin{array} { c c c c } \text { Constraint } & \begin{array} { c } \text { Lower } \\\text { Limit }\end{array} & \begin{array} { c } \text { Current } \\\text { Value }\end{array} & \begin{array} { c } \text { Upper } \\\text { Limit }\end{array} \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$
-How much space will be left unused? How many minutes of idle time remain for setting up the display?