The Linear Programming Problem Whose Output Follows Is Used to Determine

question 44

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) By how much can the amount of space decrease before there is a change in the profit?
b) By how much can the amount of space decrease before there is a change in the product mix?
c) By how much can the amount of time available to set up the display can increase before the solution (product mix) would change?
d) What is the lowest value for the amount of time available to set up the display increase before the solution (product mix) would change?

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Definitions:

Pectoral Girdle

The skeletal structure that provides support for the upper limbs, including the clavicles and scapulae.

Clavicles

Clavicles, commonly known as collarbones, are paired bones that sit horizontally at the top of the chest, providing support and stability to the shoulders.

Scapula

A flat, triangular bone located in the upper back, which serves as an important part of the shoulder joint.

Osteoclasts

A type of bone cell that breaks down bone tissue, helping to reshape bones as they grow and repair.