What is the constraint for node 2 in the following shortest path problem?[Fill in the blank]

Question

BlackTom AI · Accepted Answer

Question: What is the constraint for node 2 in the following shortest path problem? [Fill in the blank]

Answer options:
 a. -X12--X13= 0
 b. -X12-X24= 1
 c. X12+ X13= 0
 d. -X12+ X24= 0

Now let's analyze each option one by one in the context of a typical shortest path flow/conservation constraint at node 2.

Option a: -X12--X13= 0. This would read as the flow from 1 to 2 (X12) plus the flow from 1 to 3 (X13) balancing to zero with a double negative sign in front of X13. The intended meaning of a node constraint is usually to balance inflows and outflows at a node. Having -X13 in this position would imply a subtraction rather than a proper balancing of flows entering and leaving node 2. It does not reflect the standard conservation form for node 2 where the sum of flows into the node should equal the sum of flows out of the node. Therefore, this option is inconsistent with typical node-conservation constraints.

Option b: -X12-X24= 1. This expression subtracts both X12 and X24 and sets the result to 1. In a conservation constraint, the right-hand side is typically 0 (net balance) or the supply/demand term for the node, not a fixed unrelated constant like 1. Moreover, both X12 and X24 are flows connected to node 2, but this equation does not represent a proper balance between inflows and outflows at node 2. Thus this choice is incorrect.

Option c: X12+ X13= 0. This states that the sum of X12 and X13 is zero. X12 is the flow from node 1 to 2, while X13 is from node 1 to 3; neither directly represents an outflow or inflow at node 2. The node 2 constraint should involve flows entering and leaving node 2 (e.g., X12 and X24), not X13 which connects to node 1–3 pathway. Therefore, this option misplaces the variables and fails to express node 2’s conservation.

Option d: -X12+ X24= 0. This option reads as the negative of the inflow from 1 to 2 plus the outflow from 2 to 4 equals zero, i.e., the inflow to node 2 from node 1 is balanced by the outflow from node 2 to node 4. This is a standard node-conservation form: sum of inflows to node 2 minus sum of outflows from node 2 equals zero (or equivalently, inflows = outflows). It correctly captures the balance at node 2 given the two incident arcs you can see in the diagram (1→2 and 2→4). Therefore, this option correctly expresses the node-2 constraint.

In summary, the reasoning shows why the first three options do not reflect the proper flow balance at node 2, while option d accurately enforces that the inflow to node 2 equals the outflow from node 2 through the available edges.

What is the constraint for node 2 in the following shortest path problem?[Fill in the blank]单项选择题

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