Network Flow Optimization

Network Flow Problem Framework

Problem Classification

1. Transportation Problem

m sources → n destinations
- Single commodity
- Direct shipments only
- Minimize total transportation cost

2. Transshipment Problem

Sources → Intermediate nodes → Destinations
- Allows intermediate stops
- More flexible routing
- Includes warehouses, hubs, cross-docks

3. Minimum Cost Flow Problem

General network with supplies, demands, costs, capacities
- Most general formulation
- Subsumes transportation and transshipment
- Linear programming problem

4. Maximum Flow Problem

Single source → Single sink
- Maximize total flow
- Subject to arc capacities
- Applications: network capacity, throughput

5. Multi-Commodity Flow

Multiple products/commodities sharing network
- Product-specific demands
- Shared arc capacities
- More complex but realistic

Mathematical Formulations

Minimum Cost Flow Problem

Network:

• G = (N, A): Directed graph with nodes N and arcs A

Parameters:

• b_i: Net supply at node i
  • b_i > 0: supply node
  • b_i < 0: demand node (demand = -b_i)
  • b_i = 0: transshipment node
• c_{ij}: Unit cost on arc (i,j)
• u_{ij}: Capacity on arc (i,j)
• l_{ij}: Lower bound on arc (i,j) (often 0)

Decision Variables:

• x_{ij}: Flow on arc (i,j)

Objective Function:

Minimize: Σ_{(i,j) ∈ A} c_{ij} × x_{ij}

Constraints:

1. Flow conservation at each node:
   Σ_{j:(i,j)∈A} x_{ij} - Σ_{j:(j,i)∈A} x_{ji} = b_i,  ∀i ∈ N

   (Outflow - Inflow = Net Supply)

2. Arc capacity constraints:
   l_{ij} ≤ x_{ij} ≤ u_{ij},  ∀(i,j) ∈ A

3. Non-negativity (if no lower bounds):
   x_{ij} ≥ 0,  ∀(i,j) ∈ A

4. Balanced network:
   Σ_{i∈N} b_i = 0
   (Total supply = Total demand)

Properties:

• Linear programming problem
• Polynomial-time solvable
• Network simplex very efficient

Transportation Problem

Simplified formulation:

• m sources with supplies s_i
• n destinations with demands d_j
• Cost c_{ij} to ship from source i to destination j

Variables:

• x_{ij}: Amount shipped from source i to destination j

Objective:

Minimize: Σ_i Σ_j c_{ij} × x_{ij}

Constraints:

1. Supply constraints:
   Σ_j x_{ij} ≤ s_i,  ∀i (sources)

2. Demand constraints:
   Σ_i x_{ij} ≥ d_j,  ∀j (destinations)

3. Non-negativity:
   x_{ij} ≥ 0,  ∀i,j

Multi-Commodity Flow

Additional notation:

• K: Set of commodities/products
• b_i^k: Supply/demand of commodity k at node i
• c_{ij}^k: Cost per unit of commodity k on arc (i,j)
• u_{ij}: Total capacity on arc (i,j) (shared)

Variables:

• x_{ij}^k: Flow of commodity k on arc (i,j)

Objective:

Minimize: Σ_k Σ_{(i,j)∈A} c_{ij}^k × x_{ij}^k

Constraints:

1. Flow conservation per commodity:
   Σ_j x_{ij}^k - Σ_j x_{ji}^k = b_i^k,  ∀i ∈ N, ∀k ∈ K

2. Shared arc capacity:
   Σ_k x_{ij}^k ≤ u_{ij},  ∀(i,j) ∈ A

3. Non-negativity:
   x_{ij}^k ≥ 0,  ∀(i,j) ∈ A, ∀k ∈ K

Solution Methods

1. Minimum Cost Flow with NetworkX

import networkx as nx
import matplotlib.pyplot as plt

def solve_min_cost_flow_nx(nodes, arcs, supplies, costs, capacities):
    """
    Solve minimum cost flow using NetworkX

    Args:
        nodes: list of node IDs
        arcs: list of (source, target) tuples
        supplies: dict {node: supply} (negative for demand)
        costs: dict {(source, target): cost}
        capacities: dict {(source, target): capacity}

    Returns:
        optimal flow solution
    """

    # Create directed graph
    G = nx.DiGraph()

    # Add nodes with demands
    for node in nodes:
        demand = -supplies.get(node, 0)  # NetworkX uses demand (negative supply)
        G.add_node(node, demand=demand)

    # Add arcs with costs and capacities
    for (i, j) in arcs:
        G.add_edge(i, j,
                  weight=costs.get((i,j), 0),
                  capacity=capacities.get((i,j), float('inf')))

    # Solve min cost flow
    try:
        flow_dict = nx.min_cost_flow(G)

        # Extract flows
        flows = {}
        for i in flow_dict:
            for j in flow_dict[i]:
                if flow_dict[i][j] > 0:
                    flows[(i,j)] = flow_dict[i][j]

        # Calculate total cost
        total_cost = nx.cost_of_flow(G, flow_dict)

        return {
            'status': 'Optimal',
            'flows': flows,
            'total_cost': total_cost,
            'flow_dict': flow_dict
        }

    except nx.NetworkXUnfeasible:
        return {'status': 'Infeasible'}

    except Exception as e:
        return {'status': f'Error: {str(e)}'}


# Example usage
if __name__ == "__main__":
    # Example: Supply chain network
    # 2 plants → 2 DCs → 3 customers

    nodes = ['Plant1', 'Plant2', 'DC1', 'DC2',
             'Customer1', 'Customer2', 'Customer3']

    # Define arcs (directed edges)
    arcs = [
        # Plant to DC
        ('Plant1', 'DC1'), ('Plant1', 'DC2'),
        ('Plant2', 'DC1'), ('Plant2', 'DC2'),
        # DC to Customer
        ('DC1', 'Customer1'), ('DC1', 'Customer2'), ('DC1', 'Customer3'),
        ('DC2', 'Customer1'), ('DC2', 'Customer2'), ('DC2', 'Customer3')
    ]

    # Supplies (positive) and demands (negative)
    supplies = {
        'Plant1': 100,
        'Plant2': 150,
        'DC1': 0,  # Transshipment
        'DC2': 0,  # Transshipment
        'Customer1': -80,
        'Customer2': -90,
        'Customer3': -80
    }

    # Transportation costs
    costs = {
        ('Plant1', 'DC1'): 10, ('Plant1', 'DC2'): 12,
        ('Plant2', 'DC1'): 8, ('Plant2', 'DC2'): 11,
        ('DC1', 'Customer1'): 5, ('DC1', 'Customer2'): 7, ('DC1', 'Customer3'): 6,
        ('DC2', 'Customer1'): 6, ('DC2', 'Customer2'): 5, ('DC2', 'Customer3'): 8
    }

    # Arc capacities
    capacities = {
        ('Plant1', 'DC1'): 80, ('Plant1', 'DC2'): 70,
        ('Plant2', 'DC1'): 90, ('Plant2', 'DC2'): 100,
        ('DC1', 'Customer1'): 60, ('DC1', 'Customer2'): 70, ('DC1', 'Customer3'): 50,
        ('DC2', 'Customer1'): 70, ('DC2', 'Customer2'): 60, ('DC2', 'Customer3'): 80
    }

    print("="*70)
    print("MINIMUM COST FLOW PROBLEM")
    print("="*70)
    print(f"Nodes: {len(nodes)}")
    print(f"Arcs: {len(arcs)}")
    print(f"Total supply: {sum(v for v in supplies.values() if v > 0)}")
    print(f"Total demand: {-sum(v for v in supplies.values() if v < 0)}")

    result = solve_min_cost_flow_nx(nodes, arcs, supplies, costs, capacities)

    print(f"\n{'='*70}")
    print(f"OPTIMAL SOLUTION")
    print(f"{'='*70}")
    print(f"Status: {result['status']}")
    print(f"Total Cost: ${result['total_cost']:,.2f}")

    print(f"\nOptimal Flows:")
    for (i, j), flow in result['flows'].items():
        cost = costs.get((i,j), 0)
        capacity = capacities.get((i,j), 'inf')
        print(f"  {i:12} → {j:12}: {flow:6.1f} units "
              f"(cost=${cost}, capacity={capacity})")

2. Transportation Problem with PuLP

from pulp import *
import numpy as np

def solve_transportation_problem(sources, destinations, supplies,
                                demands, costs):
    """
    Solve Transportation Problem

    Args:
        sources: list of source IDs
        destinations: list of destination IDs
        supplies: dict {source: supply}
        demands: dict {destination: demand}
        costs: dict {(source, dest): unit_cost}

    Returns:
        optimal transportation plan
    """

    # Create problem
    prob = LpProblem("Transportation", LpMinimize)

    # Decision variables
    x = {}
    for i in sources:
        for j in destinations:
            x[i,j] = LpVariable(f"ship_{i}_{j}", lowBound=0, cat='Continuous')

    # Objective: Minimize total transportation cost
    prob += (
        lpSum([costs[i,j] * x[i,j] for i in sources for j in destinations]),
        "Total_Cost"
    )

    # Constraints

    # 1. Supply constraints (can't ship more than available)
    for i in sources:
        prob += (
            lpSum([x[i,j] for j in destinations]) <= supplies[i],
            f"Supply_{i}"
        )

    # 2. Demand constraints (must meet all demand)
    for j in destinations:
        prob += (
            lpSum([x[i,j] for i in sources]) >= demands[j],
            f"Demand_{j}"
        )

    # Solve
    import time
    start_time = time.time()
    prob.solve(PULP_CBC_CMD(msg=0))
    solve_time = time.time() - start_time

    # Extract solution
    if LpStatus[prob.status] in ['Optimal', 'Feasible']:
        shipments = {}
        for i in sources:
            for j in destinations:
                if x[i,j].varValue > 0.01:
                    shipments[i,j] = x[i,j].varValue

        # Calculate utilization
        source_utilization = {}
        for i in sources:
            total_shipped = sum(x[i,j].varValue for j in destinations)
            source_utilization[i] = (total_shipped / supplies[i]) * 100

        return {
            'status': LpStatus[prob.status],
            'total_cost': value(prob.objective),
            'shipments': shipments,
            'source_utilization': source_utilization,
            'solve_time': solve_time
        }
    else:
        return {
            'status': LpStatus[prob.status],
            'solve_time': solve_time
        }


# Example usage
sources = ['Factory_A', 'Factory_B', 'Factory_C']
destinations = ['Market_1', 'Market_2', 'Market_3', 'Market_4']

supplies = {
    'Factory_A': 200,
    'Factory_B': 300,
    'Factory_C': 250
}

demands = {
    'Market_1': 150,
    'Market_2': 200,
    'Market_3': 180,
    'Market_4': 170
}

# Randomly generate costs
np.random.seed(42)
costs = {}
for i in sources:
    for j in destinations:
        costs[i,j] = np.random.uniform(10, 50)

print("\n" + "="*70)
print("TRANSPORTATION PROBLEM")
print("="*70)
print(f"Sources: {len(sources)}")
print(f"Destinations: {len(destinations)}")
print(f"Total supply: {sum(supplies.values())}")
print(f"Total demand: {sum(demands.values())}")

result = solve_transportation_problem(sources, destinations, supplies,
                                     demands, costs)

print(f"\n{'='*70}")
print(f"OPTIMAL SOLUTION")
print(f"{'='*70}")
print(f"Status: {result['status']}")
print(f"Total Cost: ${result['total_cost']:,.2f}")

print(f"\nOptimal Shipments:")
for (i, j), quantity in result['shipments'].items():
    cost_per_unit = costs[i,j]
    total_arc_cost = quantity * cost_per_unit
    print(f"  {i} → {j}: {quantity:.1f} units "
          f"(@${cost_per_unit:.2f}/unit = ${total_arc_cost:,.2f})")

print(f"\nSource Utilization:")
for source, util in result['source_utilization'].items():
    print(f"  {source}: {util:.1f}% ({supplies[source]} available)")

3. Multi-Commodity Flow

def solve_multi_commodity_flow(nodes, arcs, products, supplies, costs,
                               arc_capacities):
    """
    Solve Multi-Commodity Flow Problem

    Args:
        nodes: list of nodes
        arcs: list of (source, target) tuples
        products: list of product IDs
        supplies: dict {(node, product): supply} (negative for demand)
        costs: dict {(source, target, product): cost}
        arc_capacities: dict {(source, target): shared capacity}

    Returns:
        optimal multi-commodity flow
    """

    prob = LpProblem("Multi_Commodity_Flow", LpMinimize)

    # Decision variables: x[i,j,k] = flow of product k on arc (i,j)
    x = {}
    for (i, j) in arcs:
        for k in products:
            x[i,j,k] = LpVariable(f"flow_{i}_{j}_{k}",
                                 lowBound=0, cat='Continuous')

    # Objective: Minimize total cost across all products
    prob += (
        lpSum([costs.get((i,j,k), 0) * x[i,j,k]
               for (i,j) in arcs for k in products]),
        "Total_Cost"
    )

    # Constraints

    # 1. Flow conservation for each product at each node
    for node in nodes:
        for k in products:
            # Outflow - Inflow = Supply
            outflow = lpSum([x[i,j,k] for (i,j) in arcs if i == node])
            inflow = lpSum([x[i,j,k] for (i,j) in arcs if j == node])

            prob += (
                outflow - inflow == supplies.get((node, k), 0),
                f"Flow_Conservation_{node}_{k}"
            )

    # 2. Shared arc capacity constraints
    for (i, j) in arcs:
        prob += (
            lpSum([x[i,j,k] for k in products]) <= arc_capacities.get((i,j), float('inf')),
            f"Capacity_{i}_{j}"
        )

    # Solve
    import time
    start_time = time.time()
    prob.solve(PULP_CBC_CMD(msg=1, timeLimit=600))
    solve_time = time.time() - start_time

    # Extract solution
    if LpStatus[prob.status] in ['Optimal', 'Feasible']:
        flows = {}
        for (i,j) in arcs:
            for k in products:
                if x[i,j,k].varValue > 0.01:
                    flows[i,j,k] = x[i,j,k].varValue

        # Calculate arc utilization
        arc_utilization = {}
        for (i,j) in arcs:
            total_flow = sum(x[i,j,k].varValue for k in products)
            capacity = arc_capacities.get((i,j), float('inf'))
            if capacity != float('inf'):
                arc_utilization[i,j] = (total_flow / capacity) * 100

        return {
            'status': LpStatus[prob.status],
            'total_cost': value(prob.objective),
            'flows': flows,
            'arc_utilization': arc_utilization,
            'solve_time': solve_time
        }
    else:
        return {
            'status': LpStatus[prob.status],
            'solve_time': solve_time
        }


# Example usage
nodes = ['Plant', 'DC1', 'DC2', 'Customer1', 'Customer2']
arcs = [
    ('Plant', 'DC1'), ('Plant', 'DC2'),
    ('DC1', 'Customer1'), ('DC1', 'Customer2'),
    ('DC2', 'Customer1'), ('DC2', 'Customer2')
]
products = ['ProductA', 'ProductB', 'ProductC']

# Product-specific supplies/demands
supplies = {
    ('Plant', 'ProductA'): 100,
    ('Plant', 'ProductB'): 150,
    ('Plant', 'ProductC'): 120,
    ('Customer1', 'ProductA'): -40,
    ('Customer1', 'ProductB'): -60,
    ('Customer1', 'ProductC'): -50,
    ('Customer2', 'ProductA'): -60,
    ('Customer2', 'ProductB'): -90,
    ('Customer2', 'ProductC'): -70
}

# Product-specific costs
costs = {}
for (i,j) in arcs:
    for k in products:
        costs[i,j,k] = np.random.uniform(5, 25)

# Shared arc capacities
arc_capacities = {
    ('Plant', 'DC1'): 200,
    ('Plant', 'DC2'): 180,
    ('DC1', 'Customer1'): 100,
    ('DC1', 'Customer2'): 120,
    ('DC2', 'Customer1'): 110,
    ('DC2', 'Customer2'): 130
}

print("\n" + "="*70)
print("MULTI-COMMODITY FLOW PROBLEM")
print("="*70)
print(f"Nodes: {len(nodes)}")
print(f"Arcs: {len(arcs)}")
print(f"Products: {len(products)}")

result = solve_multi_commodity_flow(nodes, arcs, products, supplies,
                                   costs, arc_capacities)

print(f"\n{'='*70}")
print(f"OPTIMAL SOLUTION")
print(f"{'='*70}")
print(f"Status: {result['status']}")
print(f"Total Cost: ${result['total_cost']:,.2f}")

print(f"\nOptimal Flows (sample - first 15):")
count = 0
for (i,j,k), flow in result['flows'].items():
    if count >= 15:
        break
    cost_per_unit = costs[i,j,k]
    print(f"  {i} → {j} ({k}): {flow:.1f} units @${cost_per_unit:.2f}/unit")
    count += 1

print(f"\nArc Utilization:")
for (i,j), util in result['arc_utilization'].items():
    capacity = arc_capacities[i,j]
    print(f"  {i} → {j}: {util:.1f}% (capacity={capacity})")

Advanced Algorithms

1. Maximum Flow (Ford-Fulkerson)

def max_flow_ford_fulkerson(graph, source, sink):
    """
    Maximum flow using Ford-Fulkerson algorithm with BFS (Edmonds-Karp)

    Args:
        graph: dict {node: {neighbor: capacity}}
        source: source node
        sink: sink node

    Returns:
        maximum flow value and flow assignment
    """
    from collections import deque, defaultdict

    # Create residual graph
    residual = defaultdict(lambda: defaultdict(int))
    for u in graph:
        for v in graph[u]:
            residual[u][v] = graph[u][v]

    def bfs_find_path():
        """Find augmenting path using BFS"""
        visited = {source}
        queue = deque([(source, [source])])

        while queue:
            node, path = queue.popleft()

            if node == sink:
                return path

            for neighbor in residual[node]:
                if neighbor not in visited and residual[node][neighbor] > 0:
                    visited.add(neighbor)
                    queue.append((neighbor, path + [neighbor]))

        return None

    max_flow_value = 0

    # Find augmenting paths
    while True:
        path = bfs_find_path()

        if path is None:
            break

        # Find bottleneck capacity
        flow = min(residual[path[i]][path[i+1]]
                  for i in range(len(path)-1))

        # Update residual graph
        for i in range(len(path)-1):
            u, v = path[i], path[i+1]
            residual[u][v] -= flow
            residual[v][u] += flow

        max_flow_value += flow

    # Extract final flow
    flow_assignment = {}
    for u in graph:
        for v in graph[u]:
            flow_on_edge = graph[u][v] - residual[u][v]
            if flow_on_edge > 0:
                flow_assignment[u,v] = flow_on_edge

    return {
        'max_flow': max_flow_value,
        'flow_assignment': flow_assignment
    }


# Example
graph = {
    'S': {'A': 10, 'B': 5},
    'A': {'B': 15, 'C': 10},
    'B': {'D': 10},
    'C': {'D': 10, 'T': 5},
    'D': {'T': 10},
    'T': {}
}

result = max_flow_ford_fulkerson(graph, 'S', 'T')

print("\n" + "="*70)
print("MAXIMUM FLOW PROBLEM")
print("="*70)
print(f"Maximum Flow: {result['max_flow']}")
print(f"\nFlow Assignment:")
for (u,v), flow in result['flow_assignment'].items():
    print(f"  {u} → {v}: {flow}")

Complete Network Flow Solver

class NetworkFlowSolver:
    """
    Comprehensive Network Flow Optimization Solver
    """

    def __init__(self):
        self.problem_type = None
        self.loaded = False

    def load_min_cost_flow(self, nodes, arcs, supplies, costs, capacities):
        """Load minimum cost flow problem"""
        self.nodes = nodes
        self.arcs = arcs
        self.supplies = supplies
        self.costs = costs
        self.capacities = capacities
        self.problem_type = 'min_cost_flow'
        self.loaded = True

        print(f"Loaded Minimum Cost Flow Problem:")
        print(f"  Nodes: {len(nodes)}")
        print(f"  Arcs: {len(arcs)}")
        total_supply = sum(v for v in supplies.values() if v > 0)
        total_demand = -sum(v for v in supplies.values() if v < 0)
        print(f"  Total supply: {total_supply}")
        print(f"  Total demand: {total_demand}")
        print(f"  Balanced: {abs(total_supply - total_demand) < 0.01}")

    def load_transportation(self, sources, destinations, supplies,
                          demands, costs):
        """Load transportation problem"""
        self.sources = sources
        self.destinations = destinations
        self.supplies_trans = supplies
        self.demands_trans = demands
        self.costs_trans = costs
        self.problem_type = 'transportation'
        self.loaded = True

        print(f"Loaded Transportation Problem:")
        print(f"  Sources: {len(sources)}")
        print(f"  Destinations: {len(destinations)}")
        print(f"  Total supply: {sum(supplies.values())}")
        print(f"  Total demand: {sum(demands.values())}")

    def solve_exact(self):
        """Solve with exact method"""
        if not self.loaded:
            raise ValueError("Problem not loaded")

        if self.problem_type == 'min_cost_flow':
            return solve_min_cost_flow_nx(
                self.nodes, self.arcs, self.supplies,
                self.costs, self.capacities
            )

        elif self.problem_type == 'transportation':
            return solve_transportation_problem(
                self.sources, self.destinations,
                self.supplies_trans, self.demands_trans,
                self.costs_trans
            )

    def visualize_network(self, solution=None):
        """Visualize network and flows"""
        if self.problem_type != 'min_cost_flow':
            print("Visualization only for min cost flow currently")
            return

        import matplotlib.pyplot as plt
        import networkx as nx

        G = nx.DiGraph()

        # Add nodes
        for node in self.nodes:
            supply = self.supplies.get(node, 0)
            if supply > 0:
                G.add_node(node, node_type='supply')
            elif supply < 0:
                G.add_node(node, node_type='demand')
            else:
                G.add_node(node, node_type='transshipment')

        # Add arcs
        for (i, j) in self.arcs:
            G.add_edge(i, j)

        # Layout
        pos = nx.spring_layout(G, k=2, iterations=50)

        plt.figure(figsize=(14, 10))

        # Draw nodes by type
        supply_nodes = [n for n in G.nodes() if G.nodes[n].get('node_type') == 'supply']
        demand_nodes = [n for n in G.nodes() if G.nodes[n].get('node_type') == 'demand']
        trans_nodes = [n for n in G.nodes() if G.nodes[n].get('node_type') == 'transshipment']

        nx.draw_networkx_nodes(G, pos, nodelist=supply_nodes,
                             node_color='lightgreen', node_size=800,
                             label='Supply Nodes')
        nx.draw_networkx_nodes(G, pos, nodelist=demand_nodes,
                             node_color='lightcoral', node_size=800,
                             label='Demand Nodes')
        nx.draw_networkx_nodes(G, pos, nodelist=trans_nodes,
                             node_color='lightblue', node_size=800,
                             label='Transshipment')

        # Draw edges
        nx.draw_networkx_edges(G, pos, alpha=0.5, arrows=True,
                             arrowsize=20, width=2)

        # Draw labels
        nx.draw_networkx_labels(G, pos, font_size=10)

        # If solution provided, highlight flows
        if solution and 'flows' in solution:
            edge_labels = {}
            for (i, j), flow in solution['flows'].items():
                cost = self.costs.get((i,j), 0)
                edge_labels[(i,j)] = f"{flow:.0f}\n${cost}"

            nx.draw_networkx_edge_labels(G, pos, edge_labels,
                                        font_size=8)

        plt.title("Network Flow Visualization")
        plt.legend()
        plt.axis('off')
        plt.tight_layout()
        plt.show()


# Complete example
if __name__ == "__main__":
    print("="*70)
    print("NETWORK FLOW OPTIMIZATION - COMPREHENSIVE EXAMPLE")
    print("="*70)

    # Create sample network
    nodes = ['S1', 'S2', 'DC1', 'DC2', 'DC3', 'D1', 'D2', 'D3']

    arcs = [
        ('S1', 'DC1'), ('S1', 'DC2'),
        ('S2', 'DC2'), ('S2', 'DC3'),
        ('DC1', 'D1'), ('DC1', 'D2'),
        ('DC2', 'D1'), ('DC2', 'D2'), ('DC2', 'D3'),
        ('DC3', 'D2'), ('DC3', 'D3')
    ]

    supplies = {
        'S1': 150, 'S2': 200,
        'DC1': 0, 'DC2': 0, 'DC3': 0,
        'D1': -100, 'D2': -120, 'D3': -130
    }

    costs = {}
    for (i,j) in arcs:
        costs[i,j] = np.random.uniform(8, 30)

    capacities = {}
    for (i,j) in arcs:
        capacities[i,j] = np.random.uniform(60, 150)

    # Create solver
    solver = NetworkFlowSolver()
    solver.load_min_cost_flow(nodes, arcs, supplies, costs, capacities)

    # Solve
    print("\n" + "="*70)
    print("SOLVING...")
    print("="*70)

    solution = solver.solve_exact()

    print(f"\n{'='*70}")
    print(f"OPTIMAL SOLUTION")
    print(f"{'='*70}")
    print(f"Status: {solution['status']}")
    print(f"Total Cost: ${solution['total_cost']:,.2f}")

    print(f"\nOptimal Flows:")
    for (i,j), flow in sorted(solution['flows'].items()):
        cost = costs[i,j]
        capacity = capacities[i,j]
        util = (flow / capacity) * 100
        print(f"  {i:4} → {j:4}: {flow:6.1f}/{capacity:6.1f} "
              f"({util:5.1f}%) @${cost:.2f}/unit")

    # Visualize
    solver.visualize_network(solution)

Tools & Libraries

Python Libraries

• NetworkX: Graph algorithms, flow optimization
• PuLP/Pyomo: MIP formulation
• OR-Tools: Google network optimization
• SciPy: Sparse matrix operations
• igraph: Fast network analysis

Commercial Software

• CPLEX/Gurobi: High-performance solvers
• AIMMS: Optimization modeling
• AMPL: Mathematical modeling

Common Challenges & Solutions

Large Networks: Use specialized algorithms (network simplex), decomposition

Integer Flows: Add integrality constraints, use branch-and-bound

Time-Varying Demands: Dynamic network flows, time-expanded networks

Uncertainty: Stochastic optimization, robust optimization

Multiple Objectives: Multi-objective optimization, weighted objectives

Output Format

Network Flow Solution:

• Total Cost: $X
• Maximum Flow: Y units
• Arc Utilization: Z%
• Flow Pattern: [detailed routing]

Questions to Ask

1. Network structure? (nodes, arcs, capacities)
2. Single or multi-commodity?
3. Supplies and demands?
4. Cost structure?
5. Capacity constraints?
6. Time dimension?
7. Optimization objective?

Related Skills

• facility-location-problem: Location decisions with flows
• distribution-center-network: Multi-echelon networks
• vehicle-routing-problem: Routing after flow allocation
• inventory-routing-problem: Integrated inventory-flow
• optimization-modeling: MIP formulation
• hub-location-problem: Hub-based flow networks