The optimal objective value of the LP relaxation model of an integer programming (IP) model always gives an upper-bound to that of the IP.

The statement is true. Here's why:

1. Integer Programming (IP) is a mathematical optimization program in which all the variables are constrained to be integers. This can be a complex problem due to the discrete nature of integer variables.

2. Linear Programming (LP) is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. LP relaxation is a method of simplifying an IP problem by allowing the variables to take on non-integer values.

3. The optimal objective value of the LP relaxation model is the best possible outcome when the variables are allowed to be non-integer. This is typically easier to solve than the original IP problem.

4. However, because the original IP problem requires the variables to be integers, the optimal solution to the LP relaxation model may not be feasible for the IP problem.

5. Therefore, the optimal objective value of the LP relaxation model provides an upper bound to that of the IP model. This means that the best possible outcome of the IP problem cannot be better than the optimal objective value of the LP relaxation model.

6. In other words, the LP relaxation provides a best-case scenario for the IP problem. If the optimal solution to the LP relaxation model is an integer, it is also the optimal solution to the IP problem. If it is not an integer, the optimal solution to the IP problem will be less than or equal to the optimal solution of the LP relaxation model.