Given the following all-integer linear program:
$\begin{array} { l l } \text { Max } & 3 x _ { 1 } + 2 x _ { 2 } \\\\\text { s.t. } & 3 x _ { 1 } + x _ { 2 } \leq 9 \\& x _ { 1 } + 3 x _ { 2 } \leq 7 \\& - x _ { 1 } + x _ { 2 } \leq 1 \\& x _ { 1 } , x _ { 2 } \geq 0 \text { and integer }\end{array}$ a.Solve the problem as a linear program ignoring the integer constraints.Show that the optimal solution to the linear program gives fractional values for both x₁ and x₂.
b.What is the solution obtained by rounding fractions greater than of equal to 1/2 to the next larger number? Show that this solution is not a feasible solution.
c.What is the solution obtained by rounding down all fractions? Is it feasible?
d.Enumerate all points in the linear programming feasible region in which both x₁ and x₂ are integers, and show that the feasible solution obtained in (c) is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

Question

Given the following all-integer linear program:

\begin{array} { l l } \text { Max } & 3 x _ { 1 } + 2 x _ { 2 } \\\\\text { s.t. } & 3 x _ { 1 } + x _ { 2 } \leq 9 \\& x _ { 1 } + 3 x _ { 2 } \leq 7 \\& - x _ { 1 } + x _ { 2 } \leq 1 \\& x _ { 1 } , x _ { 2 } \geq 0 \text { and integer }\end{array}

a.Solve the problem as a linear program ignoring the integer constraints.Show that the optimal solution to the linear program gives fractional values for both x₁ and x₂.
b.What is the solution obtained by rounding fractions greater than of equal to 1/2 to the next larger number? Show that this solution is not a feasible solution.
c.What is the solution obtained by rounding down all fractions? Is it feasible?
d.Enumerate all points in the linear programming feasible region in which both x₁ and x₂ are integers, and show that the feasible solution obtained in (c) is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

Quizplus · Accepted Answer

The Answer of Given the following all-integer linear program: $$\begin{array} { l l } \text { Max } & 3 x _ { 1 } + 2 x _ { 2 } \\ \text { s.t. } & 3 x _ { 1 } + x _ { 2 } \leq 9 \ & x _ { 1 } + 3 x _ { 2 } \leq 7 \ & - x _ { 1 } + x _ { 2 } \leq 1 \ & x _ { 1 } , x _ { 2 } \geq 0 \text { and integer } \end{array}$$ a.Solve the problem as a linear program ignoring the integer constraints.Show that the optimal solution to the linear program gives fractional values for both x₁ and x₂. b.What is the solution obtained by rounding fractions greater than of equal to 1/2 to the next larger number? Show that this solution is not a feasible solution. c.What is the solution obtained by rounding down all fractions? Is it feasible? d.Enumerate all points in the linear programming feasible region in which both x₁ and x₂ are integers, and show that the feasible solution obtained in (c) is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

Given the Following All-Integer Linear Program ASolve the Problem as a Linear Program Ignoring the Integer