MPCLP: Probabilistic Covering Location Problem

• MPCLP is a discrete optimization problem that selects a fixed number of facilities to maximize expected customer demand coverage using probabilistic measures that account for spatial decay and co-location.
• The formulation employs compact MINLP models and innovative valid inequalities, including submodular and outer-approximation cuts, to linearize non-linear constraints effectively.
• Empirical studies demonstrate that the advanced branch-and-cut algorithm, enhanced with EOA and LS cuts, achieves notable speedups and reduced LP gaps compared to previous methods.

The Multiple Probabilistic Covering Location Problem (MPCLP) is a discrete optimization problem that generalizes classical covering location formulations by incorporating joint probabilistic coverage under facility co-location and multiple coverage scenarios. MPCLP aims to select a fixed number of facilities to maximize the expected total covered demand of customers, where each facility-customer pair is associated with a coverage probability—frequently dependent on spatial factors such as distance. Recent advances in algorithmic formulations, most notably by Wang, Chen & Ljubić (2024), have introduced compact mixed integer nonlinear programming (MINLP) models and specialized branch-and-cut (B&C) algorithms leveraging state-of-the-art valid inequalities, significantly improving computational tractability and solution performance (Wang et al., 21 Nov 2025).

1. Formal Problem Definition and Notation

Consider a finite set $I$ of potential facility sites and a set $J$ of customer locations. For each customer $j \in J$, a nonnegative demand $d_j$ is specified. Coverage is probabilistic: $p_{ij} \in [0,1]$ denotes the probability that facility $i \in I$ covers customer $j$, typically with $p_{ij}$ decaying with distance or other attenuation factors. The total number of facilities to be opened is restricted to an integer $K \geq 1$, and co-location (multiple facilities at the same site) is allowed.

Decision variables are:

• $x_i \in \{0,1\}$: 1 if at least one facility is opened at $i$, 0 otherwise
• $z_j \in [0,1]$: joint probabilistic coverage “level” of customer $j$

With a joint coverage model encompassing both totally correlated and independent coverage paradigms, the probability that customer $j$ is not covered is the product of the events that each potential covering facility fails to cover $j$. This leads to the constraint:

$$z_j \leq 1 - \prod_{i \in I} (1 - p_{ij} x_i), \quad \forall j \in J$$

The objective is to maximize the total covered demand:

$\text{maximize } \sum_{j \in J} d_j z_j$

subject to the above constraints plus the facility count constraint $\sum_{i \in I} x_i = K$ and integrality/box constraints on variables. This MINLP formulation encapsulates the coverage dependencies induced by probabilistic effects and co-location.

2. Linearization via Submodular and Outer-Approximation Cuts

The nonlinear constraints in MPCLP, especially the product term $\prod_{i \in I} (1 - p_{ij} x_i)$, necessitate efficient linearization for tractable solution by modern MILP solvers.

Submodular Inequalities:

For the totally correlated coverage component, the function $f(z) = \max_{i \in I} p_i z_i$ is submodular over binary vectors $z \in \{0,1\}^{|I|}$. Leveraging the Nemhauser–Wolsey approach:

$$\zeta \leq p_\ell + \sum_{i \in I} (p_i - p_\ell)^+ z_i, \quad \forall \ell \in I \cup \{0\},\, p_0 := 0$$

This admits efficient separation in $O(|I|)$ time per customer.

Outer-Approximation (OA) Inequalities:

For the independent coverage part, outer-approximation cuts are constructed from supporting hyperplanes to the concave function $\eta_j = 1 - \prod_{i \in I}(1 - p_{ij})^{y_i}$, where $y_i$ counts facilities at $i$:

$$\eta \leq c(y^*) + \sum_{i \in I_P} a_i(y^*) y_i + \sum_{i \in I_F} y_i$$

with $I_P = \{i: p_i < 1\}$, $I_F = \{i: p_i = 1\}$, and explicit formulas for $c(y^*)$ and $a_i(y^*)$ per (Wang et al., 21 Nov 2025). OA cuts may be extended to fractional $y^*$ via appropriate rounding.

Both cut families critically reduce the search space size and maintain LP tractability.

3. Advanced Valid Inequalities: Enhanced OA and Lifted Subadditive Cuts

Two strong classes of valid inequalities have been developed to further tighten LP relaxations:

Enhanced Outer-Approximation (EOA) Inequalities:

By analyzing the relations $z_i \leq y_i \leq K z_i$, the OA cuts are strengthened by replacing coefficients $a_i(y^*) \geq 1 - c(y^*)$ with $1 - c(y^*)$ and substituting $y_i$ with $z_i$ for such indices. Letting $L = \{i \in I_P : a_i(y^*) \geq 1 - c(y^*)\}$, the EOA cut is:

$$\eta \leq c(y^*) + \sum_{i \in I_P \setminus L} a_i(y^*) y_i + (1 - c(y^*)) \sum_{i \in I_F \cup L} z_i$$

Lifted Subadditive (LS) Inequalities:

By expressing $1 - \prod_{i}(1-p_i)^{y_i}$ as a subadditive composition and applying coordinate-wise linear upper bounds, LS inequalities are:

$$\eta \leq 1 - p_C + p_C \left[ \sum_{i\in I_P\setminus C} h_{i,k_i}(y_i,z_i) + \sum_{i\in C} p_i(y_i - z_i) + \sum_{i\in I_F} z_i \right]$$

for any $C \subseteq I_P$ and integer breakpoints $k_i \in [1, K-1]$, where $p_C = \prod_{i \in C} (1-p_i)$ and $h_{i,k}(y_i,z_i)$ is as defined above. Under mild conditions, LS cuts are facet-defining; separation is heuristic due to NP-hardness.

4. Structure and Workflow of the Branch-and-Cut Algorithm

The branch-and-cut (B&C) algorithm exploits the compactness and tightness of the above formulations:

• Variables: $y_i$ (integer facility count), $z_i$ (binary), $\zeta_j$ and $\eta_j$ (continuous), with total variable count $2|I|+2|J|$
• Master MILP: Initialized with basic linking and box constraints.
• Node Processing Workflow:
  • Solve LP relaxation at each search node (best-bound first).
  • For each $j$, separate submodular cuts (for $\zeta_j$), OA and EOA cuts (for $\eta_j$).
  • Optionally, separate LS cuts using local search at fractional or integer solutions.
  • Iterate until no violated cuts remain.
  • If solution is integral, branch on $y_i$ or $z_i$ variables; otherwise, process further.
• Termination: Occurs when all nodes are pruned or specified time/optimality gap limits are attained.

This framework introduces only one $y_i, z_i$ pair per site and one $\zeta_j, \eta_j$ pair per customer, yielding $O(|I|+|J|)$ variables—an order of magnitude smaller than previous approaches relying on $O(K|I|)$ binaries. This results in smaller LPs and more manageable search trees.

5. Empirical Performance and Computational Study

Extensive computational evaluation considered 240 benchmark MPCLP instances with $|I| = |J|$ from 100 to 900, $K$ from 5 to 200, and multiple parameter settings (joint coverage weight $\theta \in \{0.2,0.5,0.8\}$, two decay scenarios). The algorithm was compared with a state-of-the-art B&C method (Álvarez-Miranda & Sinnl 2019), highlighting substantial advances:

Metric	Proposed B&C	Prior B&C	Note
Instances solved (opt.)	205/240	148/240	57 instances solved only by new method
Average CPU time (solved)	124.2 s	979.1 s	~8× speedup
Average B&B nodes	2995	8526
Root LP gap	0.60%	1.73%

The incremental impact of EOA and LS cuts was quantified:

• Vanilla (no EOA/LS): root gap ≈1.10%, 188 solved, average 406 s.
• EOA only: root gap 0.92%, 194 solved, avg. 311 s.
• LS only: root gap 0.67%, 201 solved, avg. 219 s.
• EOA + LS: root gap 0.60%, 205 solved, avg. 124 s.

Almost 70% of the testbed was solved within 100 s by the proposed algorithm, compared to ~55% by the prior approach; end-of-run optimality gaps always remained below 1% when EOA and LS cuts were active.

6. Significance and Future Directions

MPCLP encapsulates practical scenarios where coverage is inherently stochastic and redundant facility location (through co-location) is feasible. The compact formulation and advanced B&C techniques demonstrate dramatic reductions in model size, node counts, and computational times, resolving dozens of previously open instances to proven optimality (Wang et al., 21 Nov 2025). The methodologies, particularly the use of submodular, EOA, and LS inequalities, represent a generalizable framework for other probabilistic location and coverage models with joint and independent effects.

A plausible implication is the applicability of these techniques to broader classes of probabilistic combinatorial optimization problems, especially those with complex nonlinear dependencies structurable via submodularity and concavity. The strength of the valid inequalities and the separation procedures introduced may inform future optimization algorithm development beyond the specific context of probabilistic covering.