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:

Fundamental Shifts

Significant changes in the core or essential parts of a society, economy, or industry that alter its direction or methods.

Evolution

The process of change and development over time, often referring to the gradual adaptation of organisms or ideas.

Bottom-up Change

A change management approach where initiatives and ideas originate from lower levels of the organizational hierarchy, engaging employees and encouraging innovation and collaboration.

Radical Change

Significant, fundamental alterations to the existing state of an organization, industry, or market.