Usually, linear-fractional programming problems are solved by transforming them into equivalent linear programming problems. That is done using Charnes-Cooper method which involves introducing a new variable t , only then, wikipedia say: this allows us to require that the denominator of the objective function $(\mathbf {d} ^{T}\mathbf {x} +\beta )$ equals 1.
But why we must add a new variable? Why can't we just add a constraint, the denominator to be equal to 1, and maximize the numerator, as we do using Charnes-Cooper method, but without the new variable. Since the denominator is constrained to be equal to 1, the original objective function is supposed to be equal to the numerator. So the problem becomes: $$ \begin{aligned} \text{maximize}\quad &{\mathbf {c} ^{T}\mathbf {x} +\alpha} \\ {\text{subject to}} \quad & A\mathbf {x} \leq \mathbf {b},\\ \quad & \mathbf {d} ^{T}\mathbf {x} +\beta=1 \end{aligned} $$
I know that it wrong. It doesn't work. I tried. But still, intuitively, I don't understand why.
Could you help me understand why my approach doesn't work, to sharpen my intuition?
Thank you!
The intuition is that forcing the original denominator to be $1$ imposes an additional constraint that further restricts the problem and possibly excludes all optimal solutions of the original problem. By introducing an additional variable, you are transforming the problem in a way that preserves optimality in the sense that every optimal solution to the original problem is also optimal to the transformed problem, and vice versa.