Facility location problem

Facility Location Problems

You are an expert in discrete facility location, the open-sites-and-assign-customers structure that underlies supply chain design, public service siting, and clustering. This skill covers the uncapacitated and capacitated facility location problems (UFLP, CFLP), the p-median, and the p-center, with strong-vs-weak MIP formulations, Benders and Lagrangian solution paths, greedy and interchange heuristics, a VNS metaheuristic, and a stochastic-demand extension sketch. Use the framework below to identify the variant, choose the formulation whose LP relaxation is tight enough, and deliver a solution with an independently validated objective and a defensible optimality gap.

Initial Assessment

Establish the following before formulating or coding anything:

• Identify the variant. Are fixed opening costs present (UFLP/CFLP) or is the number of facilities fixed at p (p-median/p-center)? Is the objective total cost (sum) or worst-case distance (max)? Are capacities binding? These four answers select the model.
• Check the assignment regime. May a customer's demand be split across facilities (multi-source), or must each customer be served by exactly one facility (single-source)? Single-source CFLP embeds a generalized assignment problem and is much harder — even finding a feasible assignment for fixed open sites is NP-hard.
• Estimate size. Record n customers and m candidate sites. The strong formulation has n * m linking constraints; at n * m >= ~5e6 a direct MIP strains memory and you should plan for Benders, Lagrangian relaxation, or candidate reduction.
• Audit the cost data. Are assignment costs distances, distance × demand, or full freight quotes? Are fixed costs amortized to the same time horizon as the flow costs? Mixed units silently corrupt the trade-off the model is supposed to make.
• Check the distance structure. Euclidean/haversine from coordinates, or shortest paths on a road network? For p-center and covering arguments, confirm whether the triangle inequality holds — approximation guarantees depend on it.
• Clarify exact-vs-heuristic requirements. Strategic location decisions are solved rarely and justify exact MIP with a proven gap. Repeated tactical re-solves (e.g., location inside a larger loop) favor the interchange/VNS heuristics plus a Lagrangian bound.
• Confirm solver availability. Gurobi license present? If not, the models below port directly to HiGHS/SCIP via the same constraint families; the heuristics are pure numpy.
• Ask about uncertainty. Is demand known, or are there scenarios/forecast errors? Facilities are long-lived first-stage decisions; if demand is uncertain, a deterministic model with average demand can be badly wrong (see the stochastic extension sketch below).
• Agree on deliverables. Open-site list, assignment plan, objective split into fixed and service cost, capacity utilization, bound and gap, and an independent feasibility check.

Problem Variants and Formulations

Notation: customers j ∈ J with |J| = n, candidate sites i ∈ I with |I| = m, fixed opening cost f_i ≥ 0, assignment cost c_ji ≥ 0 for serving the full demand of j from i, demand d_j > 0, capacity q_i > 0, and p the number of facilities to open. Decision variables: y_i ∈ {0,1} (open site i), x_ji ∈ [0,1] (fraction of customer j served from i).

UFLP — uncapacitated facility location

$$ \min \sum_{i} f_i y_i + \sum_{j}\sum_{i} c_{ji} x_{ji} \quad \text{s.t.} \quad \sum_{i} x_{ji} = 1 ;; \forall j, \qquad x_{ji} \le y_i ;; \forall j,i, \qquad x \ge 0, ; y \in {0,1}^m. $$

Two classic formulations differ only in the linking constraints:

• Strong (disaggregated): x_ji <= y_i for every pair — n * m rows. The LP relaxation is tight; on many benchmark instances it is integral or nearly so (Cornuéjols, Nemhauser & Wolsey 1990, "The uncapacitated facility location problem", in Discrete Location Theory).
• Weak (aggregated): sum_j x_ji <= n * y_i — only m rows. Same integer optimum, but the LP bound is far weaker: the LP can open y_i = 1/n slivers of every facility, and branch-and-bound explodes.

Always start from the strong formulation. Modern solvers presolve and cut well, but no amount of cutting reliably recovers the strength you gave away by aggregating. With y integral, the assignment subproblem is trivial (serve each customer from the cheapest open site), so x may be declared continuous — the optimal x is automatically integral when assignment costs are nonnegative.

CFLP — capacitated facility location

Add capacity rows and keep the disaggregated links as redundant-for-the-IP tightening:

$$ \sum_{j} d_j x_{ji} \le q_i y_i ;; \forall i, \qquad x_{ji} \le y_i ;; \forall j,i. $$

The capacity row already links x to y, so the x_ji <= y_i rows are logically redundant — but they substantially tighten the LP relaxation and are standard practice. A useful valid inequality is the total-capacity cover sum_i q_i y_i >= sum_j d_j. Single-source CFLP declares x binary; the assignment subproblem for fixed y becomes a generalized assignment problem (strongly NP-hard), and instance difficulty jumps accordingly.

p-median and p-center

Both fix the number of open facilities and drop fixed costs (ReVelle & Swain 1970 for the p-median IP; Hakimi 1964, 1965 for the network versions):

$$ \text{p-median:} ;; \min \sum_{j}\sum_{i} d_j , t_{ji} , x_{ji} \qquad \text{p-center:} ;; \min z, ;; \sum_{i} t_{ji} x_{ji} \le z ;; \forall j, $$

both subject to sum_i x_ji = 1, x_ji <= y_i, sum_i y_i = p, with t_ji the distance. The p-median minimizes demand-weighted average distance (efficiency); the p-center minimizes the worst customer distance (equity/emergency coverage). The p-center MIP has notoriously weak LP bounds because a single z aggregates everything; the practical exact method is bisection on the radius with set-covering feasibility subproblems (Daskin 1995, Network and Discrete Location), implemented below.

Complexity and method selection

All four problems are NP-hard on general instances; the p-median is NP-hard even on planar graphs of maximum degree 3 (Kariv & Hakimi 1979). For metric UFLP the best known approximation factor is 1.488 (Li 2013) against a hardness floor of 1.463 (Guha & Khuller 1999); the p-center admits a tight 2-approximation (Gonzalez 1985; Hochbaum & Shmoys 1985). The p-median on trees is polynomial (Tamir 1996).

Setting	Recommended route
UFLP, n * m up to ~1e6 links	strong MIP directly (this skill's solve_uflp)
UFLP, larger	Benders with closed-form cuts (Fischetti, Ljubić & Sinnl 2017) or Lagrangian/dual ascent
CFLP, multi-source	MIP with capacity + disaggregated links + total-capacity cover
CFLP, single-source, large	Lagrangian relaxation + repair heuristic, or LNS over assignments
p-median, m <= ~1000	MIP, or Lagrangian (Beasley 1993) for bound + heuristic
p-median, large	VNS / fast interchange (Hansen & Mladenović 1997; Resende & Werneck 2007)
p-center	bisection over the radius with covering MIPs (below)
Stochastic demand	two-stage scenario model; see the sketch below and stochastic-optimization

Instance Generation and Validation

Standard benchmarks: the OR-Library cap* UFLP/CFLP sets (Beasley 1990), the large Körkel-Ghosh UFLP instances, and the OR-Library p-median set. For controlled experiments, generate Euclidean instances with seeds so every run is reproducible.

import numpy as np


def generate_instance(
    n_customers: int,
    n_facilities: int,
    seed: int,
    capacity_ratio: float | None = None,
    fixed_cost_range: tuple[float, float] = (100.0, 300.0),
) -> dict[str, np.ndarray]:
    """Generate a Euclidean facility-location instance with a fixed seed.

    Returns 'cust_xy', 'fac_xy', 'dist' (n x m Euclidean distances),
    'demand', 'cost' (demand-weighted distance, the UFLP/CFLP assignment
    cost), 'fixed_cost', and — when capacity_ratio is given — 'capacity'
    sized so total capacity = capacity_ratio * total demand.
    """
    rng = np.random.default_rng(seed)
    cust_xy = rng.uniform(0.0, 100.0, size=(n_customers, 2))
    fac_xy = rng.uniform(0.0, 100.0, size=(n_facilities, 2))
    dist = np.linalg.norm(cust_xy[:, None, :] - fac_xy[None, :, :], axis=2)
    demand = rng.integers(1, 20, size=n_customers).astype(np.float64)
    inst = {
        "cust_xy": cust_xy,
        "fac_xy": fac_xy,
        "dist": dist,
        "demand": demand,
        "cost": demand[:, None] * dist,
        "fixed_cost": rng.uniform(*fixed_cost_range, size=n_facilities),
    }
    if capacity_ratio is not None:
        total = float(demand.sum())
        inst["capacity"] = np.full(
            n_facilities, capacity_ratio * total / n_facilities
        )
    return inst


inst = generate_instance(n_customers=50, n_facilities=10, seed=42, capacity_ratio=2.0)
print(inst["dist"].shape, float(inst["demand"].sum()) > 0, inst["capacity"].shape)
# Expected: (50, 10) True (10,) — identical numbers on every rerun of seed 42.

Tune hardness through capacity_ratio (1.1–1.5 makes CFLP capacity-bound and hard; >= 3 makes it UFLP-like) and the ratio of fixed to assignment costs (high fixed costs → few facilities, long assignments, weaker LP bounds).

Every solution — exact or heuristic — must pass an independent validator that shares no code with the models. This is the second opinion that catches sign errors, transposed matrices, and constraints you forgot to add.

import numpy as np


def validate_solution(
    open_mask: np.ndarray,
    assign: np.ndarray,
    fixed_cost: np.ndarray,
    cost: np.ndarray,
    demand: np.ndarray | None = None,
    capacity: np.ndarray | None = None,
    p: int | None = None,
    objective: str = "sum",
    tol: float = 1e-6,
) -> tuple[bool, float, list[str]]:
    """Independent feasibility check and objective recomputation.

    open_mask: (m,) boolean. assign: (n, m) with assign[j, i] = fraction of
    customer j served from site i. cost: (n, m) cost of serving ALL of j
    from i; pass demand-weighted distance for the p-median, raw distance
    with objective='max' for the p-center. Returns (feasible, objective,
    violation messages).
    """
    violations: list[str] = []
    row_sum = assign.sum(axis=1)
    unserved = np.flatnonzero(np.abs(row_sum - 1.0) > tol)
    if unserved.size:
        violations.append(f"customers not fully assigned: {unserved[:5].tolist()}")
    if float(assign.min()) < -tol:
        violations.append("negative assignment fractions")
    closed = assign[:, ~open_mask]
    if closed.size and float(closed.max()) > tol:
        violations.append("assignment to closed facilities")
    if capacity is not None and demand is not None:
        load = (demand[:, None] * assign).sum(axis=0)
        over = np.flatnonzero(load > capacity * (1.0 + tol) + tol)
        if over.size:
            violations.append(f"capacity exceeded at sites: {over.tolist()}")
    if p is not None and int(open_mask.sum()) != p:
        violations.append(f"{int(open_mask.sum())} sites open, expected p={p}")
    service = (cost * assign).sum(axis=1)
    if objective == "max":
        obj = float(service.max())
    else:
        obj = float(fixed_cost[open_mask].sum() + service.sum())
    return (not violations), obj, violations


fixed = np.array([4.0, 3.0, 4.0])
cost = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
open_mask = np.array([True, False, True])
assign = np.zeros((4, 3))
assign[[0, 1, 2, 3], [0, 0, 2, 2]] = 1.0
print(validate_solution(open_mask, assign, fixed, cost))
# Expected: (True, 20.0, []) — feasible, objective matches the UFLP optimum below.

Exact MIP Models in gurobipy

State each model, then build it from named constraint-builder functions so every constraint family is testable in isolation. The 4-customer, 3-site instance used in the demos has known optima: UFLP 20.0, CFLP 20.0, p-median 12.0, p-center 5.0, all opening sites {0, 2}.

UFLP with selectable formulation strength

import gurobipy as gp
import numpy as np
from gurobipy import GRB


def add_assignment_constraints(
    model: gp.Model, x: gp.tupledict, n: int, m: int
) -> None:
    """Each customer is fully assigned: sum_i x[j,i] == 1."""
    for j in range(n):
        model.addConstr(
            gp.quicksum(x[j, i] for i in range(m)) == 1, name=f"assign[{j}]"
        )


def add_strong_linking_constraints(
    model: gp.Model, x: gp.tupledict, y: gp.tupledict, n: int, m: int
) -> None:
    """Disaggregated x[j,i] <= y[i]: n*m rows, tight LP relaxation."""
    for j in range(n):
        for i in range(m):
            model.addConstr(x[j, i] <= y[i], name=f"link[{j},{i}]")


def add_weak_linking_constraints(
    model: gp.Model, x: gp.tupledict, y: gp.tupledict, n: int, m: int
) -> None:
    """Aggregated sum_j x[j,i] <= n*y[i]: m rows, loose LP relaxation."""
    for i in range(m):
        model.addConstr(
            gp.quicksum(x[j, i] for j in range(n)) <= n * y[i],
            name=f"aglink[{i}]",
        )


def solve_uflp(
    fixed_cost: np.ndarray,
    cost: np.ndarray,
    strong: bool = True,
    time_limit: float = 120.0,
) -> tuple[float, np.ndarray, np.ndarray]:
    """Solve the UFLP. Returns (objective, open mask, assignment matrix)."""
    n, m = cost.shape
    model = gp.Model("uflp")
    model.Params.OutputFlag = 0
    model.Params.TimeLimit = time_limit
    y = model.addVars(m, vtype=GRB.BINARY, name="y")
    x = model.addVars(n, m, vtype=GRB.CONTINUOUS, lb=0.0, ub=1.0, name="x")
    add_assignment_constraints(model, x, n, m)
    if strong:
        add_strong_linking_constraints(model, x, y, n, m)
    else:
        add_weak_linking_constraints(model, x, y, n, m)
    model.setObjective(
        gp.quicksum(float(fixed_cost[i]) * y[i] for i in range(m))
        + gp.quicksum(
            float(cost[j, i]) * x[j, i] for j in range(n) for i in range(m)
        ),
        GRB.MINIMIZE,
    )
    model.optimize()
    if model.Status not in (GRB.OPTIMAL, GRB.TIME_LIMIT) or model.SolCount == 0:
        raise RuntimeError(f"no solution found (status {model.Status})")
    open_mask = np.array([y[i].X > 0.5 for i in range(m)])
    assign = np.array([[x[j, i].X for i in range(m)] for j in range(n)])
    return float(model.ObjVal), open_mask, assign


fixed = np.array([4.0, 3.0, 4.0])
cost = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
obj, open_mask, assign = solve_uflp(fixed, cost, strong=True)
print(round(obj, 2), np.flatnonzero(open_mask).tolist())
# Expected: 20.0 [0, 2] — open sites 0 and 2; customers 0,1 -> 0 and 2,3 -> 2.

To see the formulation gap, solve the LP relaxations of both variants on a generated instance: the weak LP bound is typically far below the strong one, while the strong LP is often integral. This comparison is the single most instructive experiment in MIP formulation strength.

CFLP with capacities and single-source option

import gurobipy as gp
import numpy as np
from gurobipy import GRB


def add_demand_constraints(
    model: gp.Model, x: gp.tupledict, n: int, m: int
) -> None:
    """Each customer's demand is fully served: sum_i x[j,i] == 1."""
    for j in range(n):
        model.addConstr(
            gp.quicksum(x[j, i] for i in range(m)) == 1, name=f"demand[{j}]"
        )


def add_capacity_constraints(
    model: gp.Model,
    x: gp.tupledict,
    y: gp.tupledict,
    demand: np.ndarray,
    capacity: np.ndarray,
) -> None:
    """Capacity with opening link: sum_j d_j x[j,i] <= q_i y[i]."""
    n, m = len(demand), len(capacity)
    for i in range(m):
        model.addConstr(
            gp.quicksum(float(demand[j]) * x[j, i] for j in range(n))
            <= float(capacity[i]) * y[i],
            name=f"capacity[{i}]",
        )


def add_linking_constraints(
    model: gp.Model, x: gp.tupledict, y: gp.tupledict, n: int, m: int
) -> None:
    """x[j,i] <= y[i]: redundant for the IP, tightens the LP substantially."""
    for j in range(n):
        for i in range(m):
            model.addConstr(x[j, i] <= y[i], name=f"link[{j},{i}]")


def solve_cflp(
    fixed_cost: np.ndarray,
    cost: np.ndarray,
    demand: np.ndarray,
    capacity: np.ndarray,
    single_source: bool = False,
    time_limit: float = 120.0,
) -> tuple[float, np.ndarray, np.ndarray]:
    """Solve the CFLP; x becomes binary when single_source=True."""
    n, m = cost.shape
    model = gp.Model("cflp")
    model.Params.OutputFlag = 0
    model.Params.TimeLimit = time_limit
    y = model.addVars(m, vtype=GRB.BINARY, name="y")
    xtype = GRB.BINARY if single_source else GRB.CONTINUOUS
    x = model.addVars(n, m, vtype=xtype, lb=0.0, ub=1.0, name="x")
    add_demand_constraints(model, x, n, m)
    add_capacity_constraints(model, x, y, demand, capacity)
    add_linking_constraints(model, x, y, n, m)
    model.addConstr(
        gp.quicksum(float(capacity[i]) * y[i] for i in range(m))
        >= float(demand.sum()),
        name="total_capacity",
    )
    model.setObjective(
        gp.quicksum(float(fixed_cost[i]) * y[i] for i in range(m))
        + gp.quicksum(
            float(cost[j, i]) * x[j, i] for j in range(n) for i in range(m)
        ),
        GRB.MINIMIZE,
    )
    model.optimize()
    if model.Status not in (GRB.OPTIMAL, GRB.TIME_LIMIT) or model.SolCount == 0:
        raise RuntimeError(f"no solution found (status {model.Status})")
    open_mask = np.array([y[i].X > 0.5 for i in range(m)])
    assign = np.array([[x[j, i].X for i in range(m)] for j in range(n)])
    return float(model.ObjVal), open_mask, assign


fixed = np.array([4.0, 3.0, 4.0])
cost = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
demand = np.array([4.0, 3.0, 5.0, 4.0])
capacity = np.array([9.0, 8.0, 9.0])
obj, open_mask, assign = solve_cflp(fixed, cost, demand, capacity)
print(round(obj, 2), np.flatnonzero(open_mask).tolist())
# Expected: 20.0 [0, 2] — capacities 9+9 cover total demand 16 with no split.

p-median

import gurobipy as gp
import numpy as np
from gurobipy import GRB


def add_assignment_constraints(
    model: gp.Model, x: gp.tupledict, n: int, m: int
) -> None:
    """Each customer is assigned: sum_i x[j,i] == 1."""
    for j in range(n):
        model.addConstr(
            gp.quicksum(x[j, i] for i in range(m)) == 1, name=f"assign[{j}]"
        )


def add_linking_constraints(
    model: gp.Model, x: gp.tupledict, y: gp.tupledict, n: int, m: int
) -> None:
    """Customers may only use open medians: x[j,i] <= y[i]."""
    for j in range(n):
        for i in range(m):
            model.addConstr(x[j, i] <= y[i], name=f"link[{j},{i}]")


def add_cardinality_constraint(model: gp.Model, y: gp.tupledict, p: int) -> None:
    """Open exactly p facilities: sum_i y[i] == p."""
    model.addConstr(y.sum() == p, name="cardinality")


def solve_pmedian(
    dist: np.ndarray, demand: np.ndarray, p: int, time_limit: float = 120.0
) -> tuple[float, np.ndarray, np.ndarray]:
    """Solve the p-median; objective is demand-weighted total distance."""
    n, m = dist.shape
    model = gp.Model("pmedian")
    model.Params.OutputFlag = 0
    model.Params.TimeLimit = time_limit
    y = model.addVars(m, vtype=GRB.BINARY, name="y")
    x = model.addVars(n, m, vtype=GRB.CONTINUOUS, lb=0.0, ub=1.0, name="x")
    add_assignment_constraints(model, x, n, m)
    add_linking_constraints(model, x, y, n, m)
    add_cardinality_constraint(model, y, p)
    model.setObjective(
        gp.quicksum(
            float(demand[j] * dist[j, i]) * x[j, i]
            for j in range(n)
            for i in range(m)
        ),
        GRB.MINIMIZE,
    )
    model.optimize()
    if model.Status not in (GRB.OPTIMAL, GRB.TIME_LIMIT) or model.SolCount == 0:
        raise RuntimeError(f"no solution found (status {model.Status})")
    open_mask = np.array([y[i].X > 0.5 for i in range(m)])
    assign = np.array([[x[j, i].X for i in range(m)] for j in range(n)])
    return float(model.ObjVal), open_mask, assign


dist = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
obj, open_mask, assign = solve_pmedian(dist, np.ones(4), p=2)
print(round(obj, 2), np.flatnonzero(open_mask).tolist())
# Expected: 12.0 [0, 2] — best pair of medians for unit demands.

p-center: direct MIP and radius bisection

The direct MIP works but bounds poorly. The bisection method exploits that the optimal radius is one of the n * m distance values: binary-search the sorted unique distances, and at each candidate radius r ask the set-covering feasibility question "can p sites cover every customer within r?".

import gurobipy as gp
import numpy as np
from gurobipy import GRB


def add_radius_constraints(
    model: gp.Model, x: gp.tupledict, z: gp.Var, dist: np.ndarray
) -> None:
    """Worst-case distance bound: sum_i t[j,i] x[j,i] <= z for every j."""
    n, m = dist.shape
    for j in range(n):
        model.addConstr(
            gp.quicksum(float(dist[j, i]) * x[j, i] for i in range(m)) <= z,
            name=f"radius[{j}]",
        )


def solve_pcenter_mip(
    dist: np.ndarray, p: int, time_limit: float = 120.0
) -> tuple[float, np.ndarray]:
    """Direct p-center MIP. Weak LP bound; fine for small instances."""
    n, m = dist.shape
    model = gp.Model("pcenter")
    model.Params.OutputFlag = 0
    model.Params.TimeLimit = time_limit
    y = model.addVars(m, vtype=GRB.BINARY, name="y")
    x = model.addVars(n, m, vtype=GRB.CONTINUOUS, lb=0.0, ub=1.0, name="x")
    z = model.addVar(lb=0.0, name="z")
    for j in range(n):
        model.addConstr(
            gp.quicksum(x[j, i] for i in range(m)) == 1, name=f"assign[{j}]"
        )
        for i in range(m):
            model.addConstr(x[j, i] <= y[i], name=f"link[{j},{i}]")
    model.addConstr(y.sum() == p, name="cardinality")
    add_radius_constraints(model, x, z, dist)
    model.setObjective(z, GRB.MINIMIZE)
    model.optimize()
    if model.Status not in (GRB.OPTIMAL, GRB.TIME_LIMIT) or model.SolCount == 0:
        raise RuntimeError(f"no solution found (status {model.Status})")
    return float(model.ObjVal), np.array([y[i].X > 0.5 for i in range(m)])


def solve_pcenter_bisection(dist: np.ndarray, p: int) -> tuple[float, np.ndarray]:
    """Exact p-center via binary search on the radius over unique distances.

    Each test solves a small covering MIP: choose <= p sites such that every
    customer has an open site within radius r (Daskin 1995).
    """
    n, m = dist.shape
    radii = np.unique(dist)
    lo, hi = 0, len(radii) - 1
    best_r, best_mask = float(radii[-1]), np.zeros(m, dtype=bool)
    while lo <= hi:
        mid = (lo + hi) // 2
        r = float(radii[mid])
        cover = dist <= r + 1e-9
        model = gp.Model("cover")
        model.Params.OutputFlag = 0
        y = model.addVars(m, vtype=GRB.BINARY, name="y")
        for j in range(n):
            model.addConstr(
                gp.quicksum(y[i] for i in range(m) if cover[j, i]) >= 1,
                name=f"cover[{j}]",
            )
        model.addConstr(y.sum() <= p, name="cardinality")
        model.setObjective(0, GRB.MINIMIZE)
        model.optimize()
        if model.Status == GRB.OPTIMAL:
            best_r = r
            best_mask = np.array([y[i].X > 0.5 for i in range(m)])
            hi = mid - 1
        else:
            lo = mid + 1
    return best_r, best_mask


dist = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
r, mask = solve_pcenter_bisection(dist, p=2)
print(round(r, 2), np.flatnonzero(mask).tolist())
# Expected: 5.0 [0, 2] — same optimum as solve_pcenter_mip on this instance.

The bisection solves O(log(n*m)) covering MIPs, each of which is far easier than the monolithic p-center model. On instances where the direct MIP stalls with a 50% gap, the bisection typically finishes in seconds.

Construction and Improvement Heuristics

The classic UFLP heuristics remain the right first move on large instances: greedy ADD (Kuehn & Hamburger 1963) for construction, then interchange (open/close/swap) improvement. With zero fixed costs and swap-only moves, the same interchange is the Teitz & Bart (1968) vertex-substitution heuristic for the p-median. The implementation below evaluates every move by full recomputation — O(m^2) trials of O(n*m) each per pass — which is simple and correct; production codes use Whitaker's (1983) fast-swap data structures, refined by Resende & Werneck (2007).

import numpy as np


def uflp_total_cost(
    open_mask: np.ndarray, fixed_cost: np.ndarray, cost: np.ndarray
) -> float:
    """Fixed cost of the open set plus closest-open-site assignment cost."""
    if not open_mask.any():
        return float("inf")
    return float(fixed_cost[open_mask].sum() + cost[:, open_mask].min(axis=1).sum())


def greedy_add_uflp(fixed_cost: np.ndarray, cost: np.ndarray) -> np.ndarray:
    """Greedy ADD: repeatedly open the site with the largest total-cost drop."""
    n, m = cost.shape
    open_mask = np.zeros(m, dtype=bool)
    best_serve = np.full(n, np.inf)
    total = float("inf")
    while True:
        cand_assign = np.minimum(best_serve[:, None], cost).sum(axis=0)
        cand_total = fixed_cost[open_mask].sum() + fixed_cost + cand_assign
        cand_total[open_mask] = np.inf
        i = int(np.argmin(cand_total))
        if cand_total[i] >= total - 1e-9:
            return open_mask
        total = float(cand_total[i])
        open_mask[i] = True
        best_serve = np.minimum(best_serve, cost[:, i])


def interchange(
    open_mask: np.ndarray,
    fixed_cost: np.ndarray,
    cost: np.ndarray,
    swap_only: bool = False,
) -> tuple[np.ndarray, float]:
    """Best-improvement open/close/swap descent. With swap_only=True and zero
    fixed costs this is Teitz-Bart vertex substitution for the p-median."""
    mask = open_mask.copy()
    total = uflp_total_cost(mask, fixed_cost, cost)
    improved = True
    while improved:
        improved = False
        best_total, best_mask = total, mask
        open_idx = np.flatnonzero(mask)
        closed_idx = np.flatnonzero(~mask)
        moves = [(a, b) for a in open_idx for b in closed_idx]
        if not swap_only:
            moves += [(a, -1) for a in open_idx if len(open_idx) > 1]
            moves += [(-1, b) for b in closed_idx]
        for a, b in moves:
            trial = mask.copy()
            if a >= 0:
                trial[a] = False
            if b >= 0:
                trial[b] = True
            t = uflp_total_cost(trial, fixed_cost, cost)
            if t < best_total - 1e-9:
                best_total, best_mask, improved = t, trial, True
        mask, total = best_mask, best_total
    return mask, total


fixed = np.array([4.0, 3.0, 4.0])
cost = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
g = greedy_add_uflp(fixed, cost)
print(np.flatnonzero(g).tolist(), uflp_total_cost(g, fixed, cost))
final, total = interchange(g, fixed, cost)
print(np.flatnonzero(final).tolist(), total)
# Expected: greedy opens [1, 2] at 21.0; interchange swaps to [0, 2] at 20.0.

Note the instructive failure: greedy stops at {1, 2} (cost 21) because no single addition improves, yet the swap 1 -> 0 reaches the optimum 20. Greedy for UFLP has a worst-case guarantee only for the related maximization form (Cornuéjols, Fisher & Nemhauser 1977); always follow it with interchange.

Metaheuristic: VNS for the p-median

Variable neighborhood search is the method of choice for large p-median instances (Hansen & Mladenović 1997): shake by replacing k medians at random, descend with vertex substitution, and grow k only when stuck. The local-search step below vectorizes the insertion scan over all candidate sites at once. For VNS design choices — neighborhood ordering, skewed acceptance, reduced VNS — see variable-neighborhood-search; this block keeps the algorithm minimal.

import numpy as np


def pmedian_cost(dist: np.ndarray, demand: np.ndarray, medians: np.ndarray) -> float:
    """Demand-weighted cost of serving every customer from its closest median."""
    return float((demand * dist[:, medians].min(axis=1)).sum())


def interchange_step(
    dist: np.ndarray, demand: np.ndarray, medians: np.ndarray
) -> tuple[np.ndarray, float, bool]:
    """One best-improvement vertex-substitution pass, vectorized over insertions."""
    n, m = dist.shape
    cand = np.setdiff1d(np.arange(m), medians)
    cur = pmedian_cost(dist, demand, medians)
    best_cost, best_sol = cur, medians
    d_sel = dist[:, medians]
    for r in range(len(medians)):
        rest = np.delete(d_sel, r, axis=1)
        rest_min = rest.min(axis=1) if rest.shape[1] else np.full(n, np.inf)
        new_min = np.minimum(rest_min[:, None], dist[:, cand])
        costs = (demand[:, None] * new_min).sum(axis=0)
        k = int(np.argmin(costs))
        if costs[k] < best_cost - 1e-9:
            best_cost = float(costs[k])
            best_sol = medians.copy()
            best_sol[r] = cand[k]
    return best_sol, best_cost, best_cost < cur - 1e-9


def vns_pmedian(
    dist: np.ndarray,
    demand: np.ndarray,
    p: int,
    seed: int = 0,
    k_max: int = 3,
    max_no_improve: int = 30,
) -> tuple[np.ndarray, float]:
    """Basic VNS for the p-median: shake k medians, descend, move or grow k."""
    rng = np.random.default_rng(seed)
    m = dist.shape[1]
    sol = rng.choice(m, size=p, replace=False)
    improved = True
    while improved:
        sol, _, improved = interchange_step(dist, demand, sol)
    best_sol, best_cost = sol, pmedian_cost(dist, demand, sol)
    stall, k = 0, 1
    while stall < max_no_improve and m > p:
        kk = min(k, p, m - p)
        out = rng.choice(p, size=kk, replace=False)
        pool = np.setdiff1d(np.arange(m), best_sol)
        trial = best_sol.copy()
        trial[out] = rng.choice(pool, size=kk, replace=False)
        improved = True
        while improved:
            trial, _, improved = interchange_step(dist, demand, trial)
        cost = pmedian_cost(dist, demand, trial)
        if cost < best_cost - 1e-9:
            best_sol, best_cost, k, stall = trial, cost, 1, 0
        else:
            k, stall = k % k_max + 1, stall + 1
    return np.sort(best_sol), best_cost


dist = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
sol, cost = vns_pmedian(dist, np.ones(4), p=2, seed=7)
print(sol.tolist(), cost)
# Expected: [0, 2] 12.0 — matches the p-median MIP optimum on this instance.

On instances with m <= ~2000, VNS with these defaults reliably lands within 0–1% of the OR-Library optima in seconds. Always report the gap against the Lagrangian or LP bound, not just the best heuristic value.

Advanced Techniques

Lagrangian relaxation of the assignment constraints

Dualize sum_i x_ji = 1 with multipliers λ_j (free sign). The relaxed problem separates by facility: each site i has reduced profit ρ_i = f_i + sum_j min(0, c_ji - λ_j) and opens exactly when ρ_i < 0, giving the bound

$$ L(\lambda) = \sum_j \lambda_j + \sum_i \min(0, \rho_i). $$

The subproblem has the integrality property, so max_λ L(λ) equals the strong-formulation LP bound (Geoffrion 1974, "Lagrangean relaxation for integer programming") — you get the strong bound without ever forming the n*m constraint matrix. Maximize L by subgradient steps on g_j = 1 - sum_i x_ji; full subgradient loop, step-size rules, and the primal-recovery heuristic are in lagrangian-relaxation (its worked example is exactly this UFLP path).

import numpy as np


def uflp_lagrangian_bound(
    fixed_cost: np.ndarray, cost: np.ndarray, lam: np.ndarray
) -> tuple[float, np.ndarray]:
    """Lagrangian lower bound for UFLP and the subgradient at lam."""
    slack = cost - lam[:, None]
    rho = fixed_cost + np.minimum(slack, 0.0).sum(axis=0)
    open_i = rho < 0.0
    x = (slack < 0.0) & open_i[None, :]
    bound = float(lam.sum() + rho[open_i].sum())
    return bound, 1.0 - x.sum(axis=1)


fixed = np.array([4.0, 3.0, 4.0])
cost = np.array([[2.0, 6.0, 9.0], [5.0, 3.0, 8.0], [9.0, 7.0, 2.0], [8.0, 6.0, 3.0]])
bound, subgrad = uflp_lagrangian_bound(fixed, cost, np.array([6.0, 5.0, 7.0, 6.0]))
print(bound)
# Expected: 20.0 — at these multipliers the bound already equals the optimum.

Benders decomposition with closed-form cuts

Project out x: keep y and one epigraph variable η_j per customer in the master. For a master point ŷ, the customer-j subproblem is min_i { c_ji : ŷ_i = 1 }, whose dual solution is available in closed form. With i* the cheapest open site for j, the optimality cut is

$$ \eta_j \ge c_{j i^} - \sum_{i:, c_{ji} < c_{j i^}} (c_{j i^*} - c_{ji}) , y_i , $$

separable per customer and added lazily at integer master solutions. This is the engine behind the state-of-the-art exact UFLP results of Fischetti, Ljubić & Sinnl (2017), "Redesigning Benders decomposition for large-scale facility location". The full callback implementation (lazy constraints, root-node cut loop, normalization) is the UFLP worked example in benders-decomposition — use it when n * m links no longer fit in memory.

Dual ascent and DUALOC

Erlenkotter (1978), "A dual-based procedure for uncapacitated facility location", ascends the strong-LP dual greedily: raise each customer's dual v_j until it hits the next assignment cost c_ji, keep facility slack complementary, and read off a primal solution from the tight constraints. DUALOC routinely closes UFLP instances without any LP solver and remains a strong warm-start generator: run dual ascent, take its primal solution as a MIP start, and its bound as the initial cutoff.

Stochastic-demand extension (sketch)

Facilities are first-stage decisions; assignments can adapt after demand is revealed. With scenarios s of probability π_s and demands d_j^s, the two-stage model is

$$ \min ; \sum_i f_i y_i + \sum_s \pi_s \sum_j \sum_i c_{ji}^s , x_{ji}^s $$

subject to per-scenario assignment and capacity constraints, with the single y shared across scenarios. The extensive form is just the CFLP model above with x indexed by scenario — build it by calling the same constraint builders once per scenario. Uncertainty matters most when capacities bind: the deterministic average-demand solution can be infeasible in high-demand scenarios. For scenario generation, sample average approximation, EVPI/VSS, and the L-shaped method (which is exactly Benders applied to this structure), see stochastic-optimization; the survey of Snyder (2006), "Facility location under uncertainty: a review", maps the modeling choices.

Candidate and variable reduction

For continental-scale instances, most x_ji pairs are absurd (customer in one region, site in another). Restrict each customer to its k nearest candidate sites (k = 20–50) before building the model; this shrinks the strong formulation from n*m to n*k links. The reduction is heuristic — verify it by checking that no customer's chosen site is its k-th nearest (if it is, increase k and re-solve), or make it exact by pricing the deleted variables' reduced costs after solving.

Practical Challenges

The strong formulation exhausts memory before the solver even starts. At n = m = 3000, the disaggregated UFLP has 9M linking rows. Options in order of preference: candidate reduction to k nearest sites; Benders with closed-form cuts (no x in the master at all); Lagrangian relaxation (no constraint matrix at all). Do not fall back to the weak formulation — its LP bound makes branch-and-bound hopeless at this size.

The CFLP LP relaxation is much weaker than the UFLP one. Capacity rows let the LP open fractional facilities to "rent" capacity cheaply. Keep the disaggregated x_ji <= y_i links even though the capacity row already links x to y, add the total-capacity cover sum q_i y_i >= sum d_j, and consider flow-cover cuts (Gurobi separates these automatically when the model is written with explicit capacity structure).

Single-source CFLP feasibility is itself hard. With x binary, fixing y leaves a generalized assignment problem, so a heuristic that picks good sites can still die on the assignment. Solve the multi-source LP first to choose sites, then solve the single-source assignment as a separate MIP over the open sites only; or use Lagrangian relaxation of the assignment constraints with a knapsack-based repair.

The p-center MIP stalls with a huge gap. A single max-variable z gives the LP almost no information. Switch to the radius-bisection method above: O(log(n*m)) covering problems, each tiny. If even those are slow, the 2-approximation of Gonzalez (1985) (iteratively pick the farthest customer as the next center) gives an instant upper bound and a starting interval for the bisection.

The heuristic answer has no quality certificate. Stakeholders deciding on warehouses will ask how far from optimal the plan is. Always pair a heuristic incumbent with a bound: the Lagrangian bound above costs milliseconds per evaluation, and 200 subgradient iterations typically land within 1–2% of the strong LP value. Report gap = (UB - LB) / UB.

Identical candidate sites create symmetry. Multiple sites with equal (f_i, c_·i) columns (e.g., generic lots in the same zone) make branch-and-bound enumerate permutations. Aggregate them into one site with an integer opening variable y_i ∈ {0, ..., u} and capacity u * q, or add lexicographic ordering constraints y_i >= y_{i'} within each identical group.

Fixed and variable costs live on different time scales. Fixed costs are capital expenditures; assignment costs are per-shipment. Amortize the fixed cost to the planning horizon (e.g., annualized capex against annual flow cost) before modeling. If the answer flips when the amortization period changes from 5 to 10 years, report both solutions — the model is telling you the decision is finance-driven, not logistics-driven.

Distance ties make heuristic results unstable across seeds. Grid-like coordinates produce many equal distances; argmin tie-breaking then differs across runs and platforms. Break ties deterministically (lowest index), run the metaheuristic over >= 10 seeds, and report best/mean/spread rather than a single run.

Tools & Libraries

Library	When to use	Note
gurobipy	exact UFLP/CFLP/p-median/p-center MIPs	the models in this skill; callbacks enable the Benders path
HiGHS / SCIP (via PuLP, Pyomo, python-mip)	no commercial license	same constraint families port directly; expect 2-10x slower on hard CFLP
OR-Tools CP-SAT	single-source CFLP with side constraints	strong on assignment-feasibility-dominated instances
numpy	heuristics, bounds, validators	greedy/interchange/VNS and the Lagrangian bound above
scipy.spatial.distance.cdist	building distance matrices	one call replaces nested loops; supports many metrics
networkx	road-network distances	multi-source Dijkstra from candidate sites, then matrix assembly
spopt (PySAL)	quick GIS-flavored location models	ready-made p-median/p-center/LSCP/MCLP wrappers over PuLP

Output Format

A complete facility-location deliverable contains:

1. Model summary — one table stating what was solved:

Item	Value
Variant	CFLP, multi-source
Formulation	strong links + total-capacity cover
Size	n=500 customers, m=60 sites, 30,060 binaries/continuous
Solver, settings	Gurobi 12, TimeLimit=600, MIPGap=0.5%

2. Solution report — open sites with their fixed costs; objective split into fixed vs. service cost; per-site utilization load_i / q_i for capacitated variants; the assignment plan as a CSV (customer, site, fraction, cost).
3. Quality certificate — best bound, incumbent, gap, and solve time for exact runs; for heuristics, the Lagrangian/LP bound, the gap against it, and best/mean/std over seeds.
4. Validation line — output of validate_solution on the reported plan: feasibility flag, recomputed objective, and confirmation that it matches the solver objective to tolerance. A deliverable without this line is not done.
5. Sensitivity notes — how the open set changes when fixed costs are amortized differently, when p varies by ±1, or across demand scenarios; location decisions are strategic, and the stability of the open set matters more than the third decimal of the objective.
6. Artifacts — instance file (or generator call with seed), solution CSV/JSON, and the exact solver log or heuristic convergence trace.

Questions to Ask

• Are fixed opening costs known, or is the number of facilities fixed by policy (p-median/p-center)?
• Should the model minimize total cost or the worst customer's distance? Who is the worst-case constraint protecting?
• Can a customer's demand be split across facilities, or is single sourcing contractual?
• Do facilities have hard capacities? What happens to unmet demand — lost, penalized, or outsourced?
• How many customers and candidate sites? Is there a shortlist of candidates, or is every customer location also a candidate?
• Are distances Euclidean, road-network, or tariff-based? Do you have the matrix, or coordinates only?
• Is demand deterministic, or should the plan hedge across forecast scenarios?
• What time horizon amortizes the fixed costs, and is the demand figure for the same horizon?
• Is a proven optimum required for the investment decision, or is a bounded-gap heuristic acceptable?

Related Skills

• benders-decomposition — when the strong UFLP/CFLP formulation no longer fits in memory; its worked example is UFLP with closed-form lazy cuts.
• lagrangian-relaxation — when you need the strong-LP bound and a primal heuristic without building the full model; its worked example dualizes the UFLP assignment constraints.
• variable-neighborhood-search — when instances outgrow exact methods; design details for the shaking and descent used in the p-median VNS above.
• stochastic-optimization — when demand is uncertain and facilities must be sited before it is revealed; two-stage scenario models and SAA.
• milp-modeling-gurobi — when you need the general gurobipy modeling workflow behind the constraint-builder pattern used in every model here.