Given the Following All-Integer Linear Program:
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Max $3 x _ { 1 } + 2 x _ { 2 }$

question 51

Essay

Given the following all-integer linear program:
​
Max $3 x _ { 1 } + 2 x _ { 2 }$
s.t.
$\begin{array} { l l } \text { M.t. } & 3 x _ { 1 } + 2 x _ { 2 } \\ & 3 x _ { 1 } + x _ { 2 } \leq 9 \\ & x _ { 1 } + 3 x _ { 2 } \leq 7 \\ & - x _ { 1 } + x _ { 2 } \leq 1 \\ & x _ { 1 } , x _ { 2 } \geq 0 \text { and integer } \end{array}$

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.​
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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.

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Definitions:

Autonomic Nervous System

The part of the nervous system that controls involuntary actions of the body, such as heart rate and digestion.

Autonomic Nervous System

The part of the nervous system that controls involuntary actions of the body, such as heart rate and digestion, and is divided into the sympathetic and parasympathetic nervous systems.

Spinal Cord

A long, thin, tubular structure made up of nervous tissue, which extends from the brain down through the spinal column, playing a critical role in transmitting nerve signals between the brain and the rest of the body.

Stomach Muscles

The muscles associated with the stomach, primarily responsible for the mechanical digestion of food through churning movements, and part of the broader group of muscles in the abdominal region.