Given the Following All-Integer Linear Program

question 23

Essay

Given the following all-integer linear program:
​
Max 15x₁ + 2x₂
​ s. t. 7x₁ + x₂ < 23
3x₁ - x₂ < 5
x₁, x₂ > 0 and integer
​
a. Solve the problem as an LP, ignoring the integer constraints.
b. What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution?
c. What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain.
d. Show that the optimal objective function value for the ILP is lower than that for the optimal LP.
e. Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal? Comment on the MILP's optimal objective function compared to the corresponding LP & ILP.

---

Definitions:

Trailer Lighting Circuit

An electrical circuit dedicated to the operation of lights on a trailer, ensuring visibility and safety during towing.

LED Light Units

Electrical devices that produce light using light-emitting diodes, known for their efficiency and longevity.

Trailer Lighting Standards

Refers to the regulatory and technical specifications designed to ensure safety and uniformity in the lighting systems used on trailers, including tail lights, brake lights, and turn signals.

FMVSS-108

Federal Motor Vehicle Safety Standard 108, outlining requirements for automotive lighting, reflective devices, and associated equipment in the United States.