Given the Following All-Integer Linear Program

question 16

Essay

Given the following all-integer linear program:
MAX 3x₁ + 2x₂
s.t.3x₁ + x₂ < 9
x₁ + 3x₂ < 7
-x₁ + x₂ < 1
x₁,x₂ > 0 and integer
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.

---

Definitions:

MCR Instruction

A Master Control Reset instruction, typically used in programmable logic controllers (PLCs) to control the operation of blocks of logic or functions.

Circuit

A closed loop or pathway that allows for the flow of electricity, consisting of various electrical components such as resistors, capacitors, and inductors.

Logic Continuity

The property of a logical scheme or electronic circuit to have a connected and coherent path for signal flow, ensuring intended operation.