Consider the Following Linear Program:
the Management Scientist Provided the Following

question 43

Essay

Consider the following linear program:
The Management Scientist provided the following solution output:
OPTIMAL SOLUTION
Objective Function Value = 27.000
MIN $6 x _ { 1 } + 9 x _ { 2 } ( \$$ cost $)$

s.t.
$\begin{aligned}x_{1}+2 x_{2} & \leq 8 \\10 x_{1}+7.5 x_{2} & \geq 30 \\x_{2} & \geq 2 \\x_{1}, x_{2} & \geq 0\end{aligned}$ ​
$$\begin{array} { l l l } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\mathrm { X } 1 & 1.500 & 0.000 \\\mathrm { X } 2 & 2.000 & 0.000\end{array}$$ OBJECTIVE COEFFICIENT RANGES
$$\begin{array} { l l c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\1 & 2.500 & 0.000 \\2 & 0.000 & - 0.600 \\3 & 0.000 & - 4.500\end{array}$$ RIGHT HAND SIDE RANGES
$$\begin{array} { l l l l } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\mathrm { X } 1 & 0.000 & 6.000 & 12.000 \\\mathrm { X } 2 & 4.500 & 9.000 & \text { No Upper Limit }\end{array}$$ ​
$$\begin{array} { l c c l } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\1 & 5.500 & 8.000 & \text { No Upper Limit } \\2 & 15.000 & 30.000 & 55.000 \\3 & 0.000 & 2.000 & 4.000\end{array}$$
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?

---

Definitions:

Hans Selye

A pioneering Hungarian-Canadian endocrinologist who is known for his research on the stress response and for introducing the concept of General Adaptation Syndrome.

Meyer Friedman

A cardiologist known for co-developing the theory that the Type A behavior pattern increases the risk of heart disease.

Walter Cannon

A physiologist known for his work on the fight or flight response and contributions to understanding the autonomic nervous system.

General Adaptation Syndrome

A three-stage response (alarm, resistance, exhaustion) that the body goes through when exposed to stress.