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this is my very first post here. I think this might be a really stupid question (sorry), but I just need to guarantee it. I have the following equation:

$$F_i = \min_{i \ \leq \ j \ \leq \ q} \{\sum_{i \ < \ p \ \leq \ j} W_p \ + (\min\{l_j - B_j,\ L - P_j\})^+\} $$

Consider $i = 0$ and $q = 3$. Then $0 \leq j \leq 3$.

My doubt is: if $i < p \leq j$ ($p \in \mathbb{N})$ then in the first "iteration" the internal Sum is:

$$\sum_{0 \ < \ p \ \leq \ 0} W_p$$

I mean, this is clearly an invalid range. Mathematically, should this be considered equal to $0$ or equal to $W_0$? The reason I'm asking this is because I have been implementing an algorithm from the literature for my paper and this really makes all the difference to the problem. I get drastically different results depending in the chosen approach. Thanks in advance.

Anthartiky
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  • Related, possible duplicate: https://math.stackexchange.com/questions/2942128/simple-way-of-explaining-the-empty-product/2942841#2942841 https://math.stackexchange.com/questions/2310758/empty-set-and-empty-sum/2310774#2310774 – Ethan Bolker Jan 22 '19 at 22:38
  • An empty sum (adding nothing together) is equal to zero, which is at least intuitive. An empty product is equal to $1$, which is less intuitive to me, but which makes sense in reasonable applications (like $0!=1$, which makes sense if $1!=1\times 0!$) – Mark Bennet Jan 22 '19 at 22:54

1 Answers1

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When the lower bound is higher than the upper bound a sum is equal to $0$ and a product is equal to $1$.

lightxbulb
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