The Linear Programming Problem Whose Output Follows Is Used to Determine

question 1

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

$$\begin{array} { c c c c } \text { Constraint } & \begin{array} { c } \text { Lower } \\\text { Limit }\end{array} & \begin{array} { c } \text { Current } \\\text { Value }\end{array} & \begin{array} { c } \text { Upper } \\\text { Limit }\end{array} \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$
-a) To what value can the per bottle profit on red nail polish drop before the solution (product mix) would change?
b) By how much can the per bottle profit on green nail polish increase before the solution (product mix) would change?

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Definitions:

Effective Communication

The conveyance and reception of ideas, messages, or information through appropriate channels, ensuring clarity and mutual understanding.

Intergroup Conflict

A situation where groups within an organization or society are in disagreement or conflict with each other, often due to competing interests or goals.

Conflict Sources

Factors or situations that can lead to disagreements or disputes within a group or organization.

Cultural Effects

The impact of a society's culture on the values, behaviors, and perceptions of its members.