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:

Atomic Components

Fundamental elements that are not divisible into smaller parts within the context of a given framework.

Tautology

A statement that is always true by its logical structure, regardless of the truth of its components.

Equivalent

Having equal value or significance, or being virtually identical in effect or function.

Symbolic Notation

A system of symbols used to represent numbers, operations, relations, or concepts, especially in mathematical and logical expressions.