The Linear Programming Problem
Max
6x₁ + 2x₂ + 3x₃ $\le$

question 6

Essay

The linear programming problem
Max
6x₁ + 2x₂ + 3x₃ + 4x₄
s.t.
x₁ + x₂ + x₃ + x₄ $\le$ 100
4x₁ + x₂ + x₃ + x₄ $\le$ 160
3x₁ + x₂ + 2x₃ + 3x₄ $\le$ 240
x₁, x₂, x ₃, x₄ $\ge$ 0
has the final tableau $$\begin{array} { c c | c c c c c c c | c } & & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm { x } _ { 4 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \\\text { Basis } & \mathrm { c } _ { \mathrm { B } } & 6 & 2 & 3 & 4 & 0 & 0 & 0 & \\\hline \mathrm { x } _ { 2 } & 2 & 0 & 1 & 1 / 2 & 0 & 3 / 2 & 0 & - 1 / 2 & 30 \\\mathrm { x } _ { 1 } & 6 & 1 & 0 & 0 & 0 & - 1 / 3 & 1 / 3 & 0 & 20 \\\mathrm { x } _ { 4 } & 4 & 0 & 0 & 1 / 2 & 1 & - 1 / 6 & - 1 / 3 & 1 / 2 & 50 \\\hline & \mathrm { z } _ { \mathrm { j } } & 6 & 2 & 3 & 4 & 2.33 & .67 & 1 & 380 \\& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & 0 & 0 & 0 & - 2.33 & - .67 & - 1 &\end{array}$$ Fill in the table below to show what you would have found if you had used The Management Scientist to solve this problem.
LINEAR PROGRAMMING PROBLEM
MAX
6X1+2X2+3X3+4X4
S.T.
1) 1X1 + 1X2 + 1X3 + 1X4 < 100
2) 4X1 + 1X2 + 1X3 + 1X4 < 160
3) 3X1 + 1X2 + 2X3 + 3X4 < 240
OPTIMAL SOLUTION Objective Function Value $=$
$\begin{array}{ccc}\text { Variable } & \text { Value } & \text { Reduced Cost } \\X 1 & - &- \\X 2 & - &- \\X 3 & - &- \\X 4 & - &-\end{array}$

$\begin{array}{ccc}\text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\1 & - & - \\2 & - & - \\3 & - & -\end{array}$
$\text { OBJECTIVE COEFFICIENT RANGES }$

$\begin{array}{cccc}\text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\mathrm{X} 1 & - & -&- \\\mathrm{X} 2 & - & -&- \\\mathrm{X} 3 & - & -&- \\\mathrm{X} 4 & - & -&-\end{array}$
$\text { RIGHT HAND SIDE RANGES }$

$\begin{array}{cccc}\text { Constraint }& \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\1 & - & -&-\\2 & - & -&-\\3 & - & -&-\\\end{array}$

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