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:

Site Navigation

Site navigation refers to the system that allows users to browse and find content on a website efficiently, typically involving menus, links, and search functions.

Content Skimming

The practice of quickly reading through content to get an overview or the gist without focusing on details.

Inverted Pyramid

A writing style that presents the most important information first, followed by supporting details, commonly used in journalism and news reporting.

Planning Phase

The initial stage in a project or process where goals are defined, strategies are developed, and tasks are outlined.