Constrained Optimization MCP Server

""" Transportation/Logistics Optimization Example This module demonstrates transportation and logistics optimization for a supply chain network. The problem involves determining optimal flow quantities through a multi-stage network to minimize total transportation costs while satisfying supply constraints, demand requirements, and flow conservation at intermediate nodes. The linear programming problem minimizes total transportation cost subject to: - Supply capacity constraints at suppliers - Demand requirements at customers - Flow conservation at warehouses (intermediate nodes) - Non-negativity constraints on all flows This is a classic transportation/transshipment problem suitable for HiGHS. """ from returns.result import Failure, Success from usolver_mcp.solvers.highs_solver import simple_highs_solver def create_logistics_problem(): """ Create a transportation/logistics optimization problem. Network structure: - 2 Suppliers (S1, S2) with supply capacities - 2 Warehouses (W1, W2) acting as transshipment points - 2 Customers (C1, C2) with demand requirements Returns: dict: Problem parameters for the HiGHS solver """ # Decision variables (flows): # S1->W1, S1->W2, S2->W1, S2->W2 (supplier to warehouse) # W1->C1, W1->C2, W2->C1, W2->C2 (warehouse to customer) # Transportation costs per unit costs = [ 12.5, # S1 -> W1 14.2, # S1 -> W2 13.8, # S2 -> W1 11.9, # S2 -> W2 8.4, # W1 -> C1 9.1, # W1 -> C2 10.5, # W2 -> C1 6.2, # W2 -> C2 ] # Variable definitions (all continuous, non-negative flows) variables = [ {"name": "S1_W1", "lb": 0, "type": "cont"}, # Supplier 1 to Warehouse 1 {"name": "S1_W2", "lb": 0, "type": "cont"}, # Supplier 1 to Warehouse 2 {"name": "S2_W1", "lb": 0, "type": "cont"}, # Supplier 2 to Warehouse 1 {"name": "S2_W2", "lb": 0, "type": "cont"}, # Supplier 2 to Warehouse 2 {"name": "W1_C1", "lb": 0, "type": "cont"}, # Warehouse 1 to Customer 1 {"name": "W1_C2", "lb": 0, "type": "cont"}, # Warehouse 1 to Customer 2 {"name": "W2_C1", "lb": 0, "type": "cont"}, # Warehouse 2 to Customer 1 {"name": "W2_C2", "lb": 0, "type": "cont"}, # Warehouse 2 to Customer 2 ] # Constraint matrix # Rows: [S1_supply, S2_supply, W1_flow_conservation, W2_flow_conservation, C1_demand, C2_demand] # Cols: [S1_W1, S1_W2, S2_W1, S2_W2, W1_C1, W1_C2, W2_C1, W2_C2] constraint_matrix = [ # Supply constraints [1, 1, 0, 0, 0, 0, 0, 0], # S1 supply: S1_W1 + S1_W2 <= 50 [0, 0, 1, 1, 0, 0, 0, 0], # S2 supply: S2_W1 + S2_W2 <= 40 # Flow conservation at warehouses (inflow = outflow) [1, 0, 1, 0, -1, -1, 0, 0], # W1: S1_W1 + S2_W1 = W1_C1 + W1_C2 [0, 1, 0, 1, 0, 0, -1, -1], # W2: S1_W2 + S2_W2 = W2_C1 + W2_C2 # Demand constraints [0, 0, 0, 0, 1, 0, 1, 0], # C1 demand: W1_C1 + W2_C1 >= 30 [0, 0, 0, 0, 0, 1, 0, 1], # C2 demand: W1_C2 + W2_C2 >= 25 ] # Constraint senses and right-hand sides constraint_senses = ["<=", "<=", "=", "=", ">=", ">="] rhs_values = [ 50, 40, 0, 0, 30, 25, ] # Supply limits, flow conservation, demand requirements return { "sense": "minimize", "objective_coeffs": costs, "variables": variables, "constraint_matrix": constraint_matrix, "constraint_senses": constraint_senses, "rhs_values": rhs_values, "description": "Transportation network optimization to minimize shipping costs", } def solve_logistics_optimization(): """ Solve the logistics optimization problem and return results. Returns: dict: Solution results including optimal flows and total cost """ problem_params = create_logistics_problem() result = simple_highs_solver(**problem_params) # Parse the result from the HiGHS solver match result: case Success(solution): if solution.status.value == "optimal": # Extract solution values flows_list = solution.solution or [] # Map solution values to flow variables flow_vars = [ "S1_W1", "S1_W2", "S2_W1", "S2_W2", "W1_C1", "W1_C2", "W2_C1", "W2_C2", ] flows = {} for i, var_name in enumerate(flow_vars): if i < len(flows_list): flows[var_name] = flows_list[i] else: flows[var_name] = 0.0 return { "status": "optimal", "flows": flows, "total_cost": solution.objective_value or 0, } else: return { "status": solution.status.value, "error": f"Problem status: {solution.status.value}", } case Failure(error): return {"status": "error", "error": str(error)} case _: return {"status": "error", "error": "Unexpected result type"} def analyze_solution(results, problem_params): """Analyze the logistics solution and verify constraints.""" if results["status"] != "optimal": return None flows = results["flows"] # Calculate supply utilization s1_used = flows.get("S1_W1", 0) + flows.get("S1_W2", 0) s2_used = flows.get("S2_W1", 0) + flows.get("S2_W2", 0) # Calculate demand satisfaction c1_satisfied = flows.get("W1_C1", 0) + flows.get("W2_C1", 0) c2_satisfied = flows.get("W1_C2", 0) + flows.get("W2_C2", 0) # Calculate warehouse throughput w1_inflow = flows.get("S1_W1", 0) + flows.get("S2_W1", 0) w1_outflow = flows.get("W1_C1", 0) + flows.get("W1_C2", 0) w2_inflow = flows.get("S1_W2", 0) + flows.get("S2_W2", 0) w2_outflow = flows.get("W2_C1", 0) + flows.get("W2_C2", 0) return { "supply_utilization": { "S1": {"used": s1_used, "capacity": 50, "utilization": s1_used / 50}, "S2": {"used": s2_used, "capacity": 40, "utilization": s2_used / 40}, }, "demand_satisfaction": { "C1": {"satisfied": c1_satisfied, "required": 30}, "C2": {"satisfied": c2_satisfied, "required": 25}, }, "warehouse_flows": { "W1": { "inflow": w1_inflow, "outflow": w1_outflow, "balanced": abs(w1_inflow - w1_outflow) < 1e-6, }, "W2": { "inflow": w2_inflow, "outflow": w2_outflow, "balanced": abs(w2_inflow - w2_outflow) < 1e-6, }, }, } def print_results(results, problem_params) -> None: """Print logistics optimization results in a formatted way.""" if results["status"] != "optimal": print(f"Problem Status: {results['status']}") if "error" in results: print(f"Error: {results['error']}") return flows = results["flows"] print("Transportation/Logistics Optimization Results") print("=" * 60) print(f"\nMinimum Total Transportation Cost: ${results['total_cost']:,.2f}") # Print optimal flows print("\nOptimal Flow Distribution:") print("-" * 40) print("Supplier to Warehouse flows:") print(f" S1 -> W1: {flows.get('S1_W1', 0):6.2f} units") print(f" S1 -> W2: {flows.get('S1_W2', 0):6.2f} units") print(f" S2 -> W1: {flows.get('S2_W1', 0):6.2f} units") print(f" S2 -> W2: {flows.get('S2_W2', 0):6.2f} units") print("\nWarehouse to Customer flows:") print(f" W1 -> C1: {flows.get('W1_C1', 0):6.2f} units") print(f" W1 -> C2: {flows.get('W1_C2', 0):6.2f} units") print(f" W2 -> C1: {flows.get('W2_C1', 0):6.2f} units") print(f" W2 -> C2: {flows.get('W2_C2', 0):6.2f} units") # Analysis analysis = analyze_solution(results, problem_params) if analysis: print("\nSupply Chain Analysis:") print("-" * 30) supply_util = analysis["supply_utilization"] print("Supply Utilization:") for supplier, data in supply_util.items(): print( f" {supplier}: {data['used']:6.2f} / {data['capacity']:6.2f} ({data['utilization']:6.1%})" ) demand_sat = analysis["demand_satisfaction"] print("\nDemand Satisfaction:") for customer, data in demand_sat.items(): status = "✓" if data["satisfied"] >= data["required"] else "✗" print( f" {customer}: {data['satisfied']:6.2f} / {data['required']:6.2f} {status}" ) warehouse_flows = analysis["warehouse_flows"] print("\nWarehouse Flow Balance:") for warehouse, data in warehouse_flows.items(): status = "✓" if data["balanced"] else "✗" print( f" {warehouse}: In={data['inflow']:6.2f}, Out={data['outflow']:6.2f} {status}" ) def main() -> None: """Main function to run the logistics optimization example.""" print(__doc__) problem_params = create_logistics_problem() results = solve_logistics_optimization() print_results(results, problem_params) def test_logistics() -> None: """Test function for pytest.""" create_logistics_problem() results = solve_logistics_optimization() # Test that we get an optimal solution assert results["status"] == "optimal" # Test that all flows are non-negative flows = results["flows"] flow_vars = ["S1_W1", "S1_W2", "S2_W1", "S2_W2", "W1_C1", "W1_C2", "W2_C1", "W2_C2"] for var in flow_vars: if var in flows: assert flows[var] >= -1e-6 # Allow small numerical errors # Test supply constraints s1_total = flows.get("S1_W1", 0) + flows.get("S1_W2", 0) s2_total = flows.get("S2_W1", 0) + flows.get("S2_W2", 0) assert s1_total <= 50 + 1e-6 # S1 supply constraint assert s2_total <= 40 + 1e-6 # S2 supply constraint # Test demand constraints c1_total = flows.get("W1_C1", 0) + flows.get("W2_C1", 0) c2_total = flows.get("W1_C2", 0) + flows.get("W2_C2", 0) assert c1_total >= 30 - 1e-6 # C1 demand constraint assert c2_total >= 25 - 1e-6 # C2 demand constraint # Test flow conservation at warehouses w1_in = flows.get("S1_W1", 0) + flows.get("S2_W1", 0) w1_out = flows.get("W1_C1", 0) + flows.get("W1_C2", 0) w2_in = flows.get("S1_W2", 0) + flows.get("S2_W2", 0) w2_out = flows.get("W2_C1", 0) + flows.get("W2_C2", 0) assert abs(w1_in - w1_out) < 1e-6 # W1 flow conservation assert abs(w2_in - w2_out) < 1e-6 # W2 flow conservation # Test that total cost is positive assert results["total_cost"] > 0 if __name__ == "__main__": main()