Given a smooth Fano variety $X$ over $\mathbb{C}$, we can define the index, $I(X)$, as the divisibility of $-K\_X$ inside of $Pic(X)$. There is a theorem which states that $I(X) \leq n+1$, where $n$ is the dimension of the variety. Moreover varieties of index $n$ are quadrics and varieties of index $n+1$ are projective spaces. I'm curious about the following modified definition of the index $I'(X)$. Namely, define $I'(X)$ to be the maximal number $d$ such that 
$-K\_X$ can be decomposed as a sum: 


$$-K\_X= D\_1 + D\_2 + \cdots + D\_d $$


where $D\_i$ are all ample. For Picard rank one, this of course doesn't do anything. Are there interesting examples of $I'(X) \geq n$ in higher Picard rank? 



Question about a variant of the index of a Fano manifold
Keyword: "pseudo-index". See, for instance, the work of Jiung-Cheng Chen and Cho - Miyaoka - Shepherd-Barron.


