A bit of a beginner question, but I'm currently working on routing and travelling salesman problem and a bit confused. Currently I have these two constraints in my model, but what is the mathemathical difference between these two?
- $$\sum_{i \in V}x_i,_j = 1 \ for \ all \ {j \in V}$$
- $$\sum_{j \in V}x_i,_j = 1 \ for \ all \ {i \in V}$$
Which constraint sums up the rows and which constraint sums up the columns?
Consider the matrix $X = (x_{i,j})_{i,j \in V}$. The first expression says that, for every column (which are indexed by $j \in V$), the sum of its elements is 1. In other words, the $j$-column is fixed and the row index $i$ runs: so it's a sum over a column. The second expression is the same but for every row.