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I am reading lecture slides for a logistics course and for one of the Linear Programming contrainsts, the summation is written as follows:

$$ \sum_{i \in I} X_{ij} $$

and

$$ \sum_{c \in C} Y_{jc} $$

This is about shipping from plant (i) to warehouse (j) and from warehouse (j) to customer base (c)

What seems weird to me is that the summation index only contains part of the variables in the sumnmation. Would it make any difference if I wrote the following instead?

$$ \sum_{i \in I, j\in J} X_{ij} $$

and

$$ \sum_{j\in J, c \in C} Y_{jc} $$

Thanks for your time! Much appreciated!

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The first summation indicates a sum from all plants to one (unspecified) warehouse, whereas your first summation indicates a sum across all possible combinations of warehouses and plants.

Suppose $X_{ij}$ represents the quantity of a product being shipped monthly from plant $i$ to warehouse $j$. Then the total incoming shipping for warehouse $j$ is $\sum_{i \in I}X_{ij}$, and the total amount of incoming shipping to all warehouses is $\sum_{i \in I, j \in J}X_{ij}$.

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  • $\begingroup$ Each of the summations results in a quantity that still has a subscript $j$. That indicates you have a number of them, in this case one for each warehouse. You may want to sum over $j$ to get a total, but that is not indicated here. $\endgroup$ – Ross Millikan Nov 30 '13 at 20:02

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