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I’m working on a transportation network optimization problem and could use some insights on modeling it effectively using Gurobi. The network includes both fixed and variable costs, and I’m tasked with comparing two different system configurations.

The network has nodes connected by edges, each with an associated variable cost proportional to the flow across that edge. Additionally, there’s a fixed cost for each node whenever it sends items through any of its outgoing edges. Here’s the basic setup:

• Each edge (i, j) has a variable cost c_{ij}.
• Each node i has a fixed cost f_i incurred once it sends out any items.
• Some nodes are designated as supply nodes, while others are demand nodes. The total supply matches or exceeds total demand.

Consider a simple network:

• Edges:(1,2), (2,3), (1,3) with variable costs.
• Nodes: Node 1 has a fixed cost f_1, and similarly for other nodes.
• Flows: If node 1 sends x_{12} items to node 2, the cost is (f_1 + c_{12} times x_{12}

1. Find the minimum total cost to satisfy all demands from the available supplies using any available paths.
2. The network is divided into k parts:
   • Part 1: Nodes 1, 2, 3
   • Part 2: Nodes 4, 5, 6, 7
   • Part 3: Nodes 8, 9, 10
   • Each part can only send items to other parts via one designated “gateway” node (e.g., nodes 2, 7, and 9).

I need to compare the total costs under these two systems to determine how much more expensive the partitioned system is compared to the optimal setup.

1. How can I model these two systems in Gurobi to ensure I’m accurately capturing both the fixed and variable costs?
2. What is the most efficient way to implement constraints that limit inter-part communication to specific nodes?
3. Any examples or similar Gurobi models that can guide the setup for this complex scenario?

I’ve started by defining the nodes, edges, and basic constraints for flow conservation in Gurobi, but I’m struggling with how to efficiently model the partitioned system’s restrictions and compare the two systems effectively.