Why don't we allow a linear programming problem to have strictly '<' or '>' constraints?

Question

I am new to linear programming and I have been asked this question "Why don't we allow a linear programming problem to have strictly '<' or '>' constraints?" But unable to answer it.

Kindly provide me an explanation on this.

Answer 1

Consider the linear program on $\mathbb R$ consisting of one constraint: $x < 1$, with the function to be optimized being $f(x) = x$. What's the optimum? At what point is it achieved?

Answer: There's no optimum. Normally, it'd be at $x = 1$, but that just barely fails to meet the constraint. But for any $x$ less than $1$, there's a better solution, namely $(1+x)/2$.

Answer 2

Because of The Divisibility Assumption.

The Divisibility Assumption requires that each decision variable is allowed to assume fractional values. For example, the Divisibility Assumption implies that it is acceptable to produce $1.5$ or $1.63$ of a product or service.