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

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?