Consider the Following Linear Program The Management Scientist Provided the Following Solution Output:
OPTIMAL SOLUTION

question 50

Essay

Consider the following linear program:
$$\begin{array} { c r } \text { Min } & 6 x _ { 1 } + 9 x _ { 2 } ( \$ \text { cost } ) \\\text { s.t. } & x _ { 1 } + 2 x _ { 2 } \leq 8 \\& 10 x _ { 1 } + 7.5 x _ { 2 } \geq 30 \\&x _ { 2 } \geq 2 \\&x _ { 1 } , x _ { 2 } \geq 0\end{array}$$
The Management Scientist provided the following solution output:
OPTIMAL SOLUTION
Objective Function Value = 27.000 $$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { X1 } & 1.500 & 0.000 \\\text { X2 } & 2.000 & 0.000\end{array}$$ $$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\1 & 2.500 & 0.000 \\2 & 0.000 & - 0.600 \\3 & 0.000 & - 4.500\end{array}$$ OBJECTIVE COEFFICIENT RANGES $$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\text { X1 } & 0.000 & 6.000 & 12.000 \\\text { X2 } & 4.500 & 9.000 & \text { No Upper Limit }\end{array}$$ RIGHT HAND SIDE RANGES $$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\1 & 5.500 & 8.000 & \text { No Upper Limit } \\2 & 15.000 & 30.000 & 55.000 \\3 & 0.000 & 2.000 & 4.000\end{array}$$
a.What is the optimal solution including the optimal value of the objective function?
b.Suppose the unit cost of x₁ is decreased to $4. Is the above solution still optimal? What is the value of the objective function when this unit cost is decreased to $4?
c.How much can the unit cost of x₂ be decreased without concern for the optimal solution changing?
d.If simultaneously the cost of x₁ was raised to $7.5 and the cost of x₂ was reduced to $6, would the current solution still remain optimal?
e.If the right-hand side of constraint 3 is increased by 1, what will be the effect on the optimal solution?

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Definitions:

Percentage Analysis

A financial analysis tool that expresses each item within a financial statement as a percent of a base item, facilitating comparison.

Working Capital

A financial metric representing the difference between a company's current assets and current liabilities, indicating its operational liquidity.

Current Liabilities

Short-term financial obligations that are due within one year or within a business's operating cycle, whichever is longer.

Company Finance

An area of finance dealing with the sources of funding and the capital structure of corporations, the actions that managers take to increase the value of the firm to the shareholders, and the tools and analysis used to allocate financial resources.