Facility Location MILP Skill

Workflow 1 (Pyomo with Commercial/Open-Source Solver)

Modeling stage

Strategy Overview

This workflow uses Pyomo's abstract modeling syntax to create a portable MILP formulation, which can be interfaced with various solvers via SolverFactory.

Step 1 - Define Sets and Parameters

• Define index sets for facilities and customers as Python lists or sets.
• Organize parameters (fixed costs, capacities, demands, variable costs) as dictionaries keyed by these indices for clarity and scalability.
• For incomplete or synthetic data, generate deterministic approximations (e.g., base_cost + (i*factor + j) % mod) to ensure reproducibility.

Step 2 - Create Decision Variables

• Create binary variables y[i] for facility opening decisions (domain=pyo.Binary).
• Create continuous, non-negative variables x[i,j] for allocation amounts from facility i to customer j (domain=pyo.NonNegativeReals).

Step 3 - Formulate Objective and Constraints

• Formulate the objective to minimize total cost: sum of fixed_cost[i] * y[i] plus sum of variable_cost[i,j] * x[i,j].
• Add demand satisfaction constraints: for each customer j, sum of x[i,j] over all facilities must equal demand[j].
• Add capacity-linking constraints: for each facility i, sum of x[i,j] over all customers must be less than or equal to capacity[i] * y[i]. This enforces that no allocation occurs from a closed facility.

Formulation Template

{
  "sets": ["facilities", "customers"],
  "parameters": {
    "fixed_cost": {"index": "facilities"},
    "capacity": {"index": "facilities"},
    "demand": {"index": "customers"},
    "variable_cost": {"index": ["facilities", "customers"]}
  },
  "decision_variables": {
    "y": {"index": "facilities", "type": "binary"},
    "x": {"index": ["facilities", "customers"], "type": "continuous", "lb": 0}
  },
  "objective": {
    "sense": "min",
    "expression": "sum(fixed_cost[i] * y[i] for i in facilities) + sum(variable_cost[i,j] * x[i,j] for i in facilities for j in customers)"
  },
  "constraints": {
    "demand_satisfaction": {"index": "customers", "expression": "sum(x[i,j] for i in facilities) == demand[j]"},
    "capacity_linking": {"index": "facilities", "expression": "sum(x[i,j] for j in customers) <= capacity[i] * y[i]"}
  }
}