The Linear Programming Problem Whose Output Follows Determines How Many

question 25

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
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$
-How much space will be left unused?

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Definitions:

Marketing Manager

A professional responsible for developing strategies to promote a product, service, or brand to increase sales.

Advertising Budget

An estimate of a company's promotional expenditures over a period. It includes costs related to advertising through various media channels.

Net Operating Income

The profit generated from a company's normal business operations, excluding expenses and revenues that are outside its core business activities.

Contribution Format

is a profit reporting format that separates fixed and variable costs, highlighting the contribution margin which helps in decision-making.