Question 1

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?

Accepted Answer

The answer of Consider the following linear program:
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Question 2

Given the following linear program: $$\begin{array} { l r } \operatorname { Max } & 5 x _ { 1 } + 7 x _ { 2 } \ \text { s.t. }& x _ { 1 } \leq 6 \ & 2 x _ { 1 } + 3 x _ { 2 } \leq 19 \ & x _ { 1 } + x _ { 2 } \leq 8 \ & x _ { 1 } , x _ { 2 } \geq 0 \end{array}$$ The graphical solution to the problem is shown below.From the graph we see that the optimal solution occurs at x₁ = 5,x₂ = 3,and obj.func.= 46. a.Calculate the range of optimality for each objective function coefficient. b.Calculate the dual price for each resource.

Accepted Answer

The answer of Given the following linear program:
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Question 3

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?

Accepted Answer

The answer of Consider the following linear program:
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Question 4

Portions of a Management Scientist output are shown below.Use what you know about the solution of linear programs to fill in the ten blanks.
LINEAR PROGRAMMING PROBLEM
MAX 12X1+9X2+7X3
S.T.
1)3X1+5X2+4X3<150
2)2X1+1X2+1X3<64
3)1X1+2X2+1X3<80
4)2X1+4X2+3X3>116
OPTIMAL SOLUTION
Objective Function Value = 336.000 $$\begin{array} { c c c } 
\text { Variable } & \text { Value } & \text { Reduced Cost } \
\text { X1 } &-& 0.000 \
\text { X2 } &  24.000& -\
\text { X3 } & -& 3.500 \
\end{array}$$ $$\begin{array} { c c c } 
\text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \
1 & 0.000 & 15.000 \
2 & -& 0.000 \
3 & -& 0.000 \
4 & 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 } & 5.400 & 12.000 & \text { No Upper Limit } \
\text { X2 } & 2.000 & 9.000 & 20.000 \
\text { X3 } & \text { No Lower Limit } & 7.000 & 10.500
\end{array}$$ RIGHT HAND SIDE RANGES $$\begin{array} { c c c r } 
\text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \
1 & 145.000 & 150.000 & 156.667 \
2 & -& -& 64.000 \
3 & -& - & 80.000 \
4 & 110.286& 116.000 & 120.000
\end{array}$$

Accepted Answer

The answer of Portions of a Management Scientist output are...

Quiz 8: Linear Programming: Sensitivity Analysis and Interpretation of Solution