The Linear Programming Problem Whose Output Follows Determines How Many

question 42

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
$$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\hline 1 & 100 & 108 & 123.75 \\2 & 57 & 120 & \text { none } \\3 & 8 & 25 & 58 \\4 & 41.5 & 50 & 54 \\\hline\end{array}$$
-By how much can the profit on green nail polish increase before the solution would change?

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Definitions:

Activity Variance

A metric used in cost accounting to measure the difference between planned activity and actual activity, often analyzed to improve budgeting and control processes.

Direct Labor

The labor costs that can be directly traced to the production of specific goods or services.

Activity Level

Activity level refers to the volume of production or operations within a specific period, influencing costs and budgeting in cost accounting.

Revenue And Spending Variance

The difference between the actual revenue and spending versus the budgeted amounts, indicating whether a business is over or under its budgeted figures.