Python Gurobipy Maximum Flow: How to model "source" and "sink" in constraints

Question

I am attempting to capture the nature of a "source" node as well as a "sink" node where the net flow for each node aside from the source and sink is zero. Here is my code so far:

import gurobipy as gp
from gurobipy import GRB
import pandas
import os


# writes constraints and objective to model file for easier inspection
def write_to_file(model, filestring):
    directory = os.getcwd()
    filename = directory+filestring
    if model.write(filename):
        return 0
    else:
        print("failed to write to file")
        return -1

# nodes
# for my example, SOURCE s = 2 and SINK t = 7
vertices = [1,2,3,4,5,6,7]

# arcs = edges, each arc has a fixed capacity
arcs, capacity = gp.multidict({
    (2,1) : 100,
    (2,3) : 50,
    (3,4) : 25,
    (4,7) : 25,
    (3,6) : 200,
    (1,6) : 500,
    (3,7) : 25,
    (6,5) : 500,
    (6,7) : 50,
    (5,7) : 200
})

# create model
m = gp.Model("Maximum Flow Problem")

# define variables
flow = m.addVars(arcs, name = "flow")

# flow capacity
m.addConstrs(
    (flow.sum(i,j) <= capacity[i,j] for i,j in arcs), "cap")

# flow conservation ISSUE HERE!
m.addConstrs(
    (flow.sum('*',j) - flow.sum(j,'*') == 0 for j in vertices), "node")

obj = (flow.sum('*', 7) - flow.sum(7,'*'))
m.setObjective(obj, GRB.MAXIMIZE)

#write_to_file(m, "\\rec2_model_simple.lp")

m.optimize()

The issue has to do with the flow conservation constraint. I want to ensure all flow going in and out of all nodes that aren't s=2 and t=7 to equal zero; however, I do not know how to capture this in my constraint.

I would like to have something like:

m.addConstrs(
    (flow.sum('*',j) - flow.sum(j,'*') == 0 for j != 2 or j != 7 in vertices), "node")

This throws an error, but I am curious if there is something similar in nature to capture this notion of a "source" and a "sink" in my flow conservation constraint. This may be a question more geared towards the python language instead of gurobipy, but since it is a question specific to modeling, I felt as though it is appropriate here.

Answer 1

One way to model this is to add a dummy arc from the sink to the source and impose flow balance of 0 at every node, including the source and sink.

But if you prefer the conditional constraint, I think the proper syntax is:

m.addConstrs(
    (flow.sum('*',j) - flow.sum(j,'*') == 0 for j in vertices if j != 2 and j != 7), "node")