Consider the Following Linear Program Objective Function Value = 20

Essay

Consider the following linear program:
$$\begin{array} { c c } \text { Max } & 3 x _ { 1 } + 4 x _ { 2 } \text { (\$ Profit) } \\\text { s.t. } & x _ { 1 } + 3 x _ { 2 } \leq 12 \\& 2 x _ { 1 } + x _ { 2 } \leq 8 \\& x _ { 1 } \leq 3 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$$
Objective Function Value = 20.000 $$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { X1 } & 2.400 & 0.000 \\\text { X2 } & 3.200 & 0.000\end{array}$$ $$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\1 & 0.000 & 1.000 \\2 & 0.000 & 1.000 \\3 & 0.600 & 0.000\end{array}$$ OBJECTIVE COEFFICIENT RANGES $$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\text { X1 } & 1.333 & 3.000 & 8.000 \\\text { X2 } & 1.500 & 4.000 & 9.000\end{array}$$ RIGHT HAND SIDE RANGES $$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\1 & 9.000 & 12.000 & 24.000 \\2 & 4.000 & 8.000 & 9.000 \\3 & 2.400 & 3.000 & \text { No Upper Limit }\end{array}$$
a.What is the optimal solution including the optimal value of the objective function?
b.Suppose the profit on x₁ is increased to $7. Is the above solution still optimal? What is the value of the objective function when this unit profit is increased to $7?
c.If the unit profit on x₂ was $10 instead of $4, would the optimal solution change?
d.If simultaneously the profit on x₁ was raised to $5.5 and the profit on x₂ was reduced to $3, would the current solution still remain optimal?