The Linear Programming Problem Whose Output Follows Determines How Many

question 89

Short Answer

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes 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. Constraints 3 and 4 are marketing restrictions.
MAX    100x1 + 120x2 + 150x3 + 125x4
Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108
      2. 3x1 + 5x2 + x4 ? 120
      3. x1 + x3 ? 25
      4. x2 + x3 + x4 > 50
      x1, x2, x3, x4 ? 0
Optimal Solution:
Objective Function Value = 7475.000
$$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Costs } \\\hline \text { X1 } & 8 & 0 \\\text { X2 } & 0 & 5 \\\text { X3 } & 17 & 0 \\\text { X4 } & 33 & 0 \\\hline\end{array}$$
$$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Prices } \\\hline 1 & 0 & 75 \\2 & 63 & 0 \\3 & 0 & 25 \\4 & 0 & - 25 \\\hline\end{array}$$ Objective Coefficient Ranges
$$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline \text { X1 } & 87.5 & 100 & \text { none } \\\text { X2 } & \text { none } & 120 & 125 \\\text { X3 } & 125 & 150 & 162 \\\text { X4 } & 120 & 125 & 150\end{array}$$ Right Hand Side Ranges
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$
-To what value can the profit on red nail polish drop before the solution would change?

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Definitions:

Lateral Line Organs

Sensory organs in fish and some amphibians that detect water movements and vibrations, aiding in navigation and the detection of prey or predators.

Vibrations in Water

Oscillatory movements that propagate through water, typically caused by external forces like wind, seismic activity, or objects entering the water.

Invertebrates

Animals that lack a vertebral column, or backbone, including species such as insects, arachnids, mollusks, and crustaceans.

Statoliths

Granules of loose sand or calcium carbonate found in statocysts.