LP relaxation solution is the solution obtained by relaxing the integer constraints in an Integer Programming (IP) problem, allowing fractional values for the decision variables.
Rounding the LP relaxation solution up or down to the nearest integer can have various effects on the solution of an IP problem. It is important to carefully consider the implications of rounding to obtain feasible and optimal solutions. There are four possible outcomes:
a. Produce an infeasible solution: Rounding the LP relaxation solution can lead to constraint violations, resulting in an infeasible solution. This can occur when rounding a fractional LP solution to an integer value leads to a solution that does not satisfy the original constraints of the IP problem.
b. Simplify the IP solution procedure: Rounding the LP relaxation solution does not necessarily simplify the IP solution procedure, as it can result in suboptimal or infeasible solutions. The additional step of rounding may not always lead to a simpler or more efficient solution process.
c. Eliminate the need for Branch and Bound (B&B): Rounding alone cannot eliminate the need for B&B in IP problems. B&B is a common technique used to find optimal integer solutions, and rounding may not be sufficient to guarantee the optimality of the solution.
d. Reduce the risk of infeasibility: Rounding does not guarantee a reduction in the risk of infeasibility; in fact, it can increase the likelihood of obtaining an infeasible solution due to constraint violations. Rounding should be done with caution to avoid introducing infeasibilities into the solution.
In summary, rounding the LP relaxation solution up or down to the nearest integer may produce an infeasible solution and does not necessarily simplify the IP solution procedure, eliminate the need for B&B, or reduce the risk of infeasibility. Careful consideration of the implications of rounding is important to obtain feasible and optimal solutions to IP problems.