Facility-Location and Path Optimization

• FLPO is a coupled optimization framework that jointly determines facility locations and routing paths to minimize aggregate transportation costs.
• It employs diverse methodologies including mixed-integer programming, LP relaxations, and entropy-based models to address NP-hard challenges.
• FLPO applications span urban planning, supply-chain design, and network routing, with strategies tailored for online, dynamic, and robust settings.

Searching arXiv for recent and directly relevant FLPO papers and the cited facility-location/path-optimization works. Facility-Location and Path Optimization (FLPO) denotes a family of coupled optimization problems in which facility decisions and path, routing, or assignment decisions are optimized jointly. In one explicit formulation, FLPO is identified as a key subclass of Parameterized Sequential Decision-Making, where multiple agents jointly optimize discrete action policies and shared continuous parameters, with facility locations represented by $\mathcal{Y}=(y_1,\dots,y_M)$ and the objective of minimizing cumulative transportation cost within the network (Basiri et al., 30 Jul 2025). Closely related work uses the same label for simultaneous facility location with path optimization, constrained resource allocation, robust allocation under disruption risk, urban accessibility optimization driven by shortest-path-like costs, and facility placement under disjoint shortest-path requirements (2002.03505). This suggests that FLPO is best understood as a research umbrella rather than a single canonical mathematical program.

1. Scope and constitutive problem structures

A central explicit FLPO formulation models $N$ agents $v_i$, $M$ facilities $f_j$ at locations $y_j\in\mathcal{D}\subset\mathbb{R}^d$, start points $s_i$, destinations $\Delta_i$, and agent weights $\rho_i\in[0,1]$. For agent $i$, the admissible paths are sequences through facilities and the destination,

$N$0

with binary path-selection variables $N$1, one path per agent,

$N$2

and objective

$N$3

The resulting mixed-integer program jointly optimizes $N$4 and $N$5, and the paper states that this problem is NP-hard because of its mixed discrete-continuous structure and highly non-convex objective (Basiri et al., 30 Jul 2025).

Other FLPO formulations make the path component implicit rather than explicit. In large-scale urban facility location, accessibility is measured by demand-weighted travel cost

$N$6

where $N$7 is the distance between urban regions and residents go to their closest facility; in that setting, the path layer is represented by a precomputed origin-destination cost matrix rather than route variables (Su et al., 2024). In network design settings, the path component can be the shortest-path structure itself: a customer may be required to be served by two facilities reachable through disjoint shortest paths, either in a pathwise-disjoint or setwise-disjoint sense, yielding minimum-cardinality facility sets under routing-policy-dependent disjointness constraints (Johnson et al., 2016).

The same family also contains formulations in which path optimization is embedded in an entropy-based sequence model. In “simultaneous facility location with path optimization,” a node $N$8 connects to a destination $N$9 through a sequence of facilities, and the objective is the expected cost over paths under a maximum-entropy relaxation (2002.03505). Taken together, these models show that FLPO can involve explicit path sequences, shortest-path cost surrogates, disjoint-path requirements, or assignment structures that stand in for routing.

2. Core mathematical programs

The most basic facility-location component in this literature is the classical opening-plus-assignment structure: open facilities, assign demand, and minimize opening and connection cost. A robust online extension is Online Multi-Facility Location, where a client must be connected to at least $v_i$0 open facilities, with objective

$v_i$1

subject to

$v_i$2

This model generalizes classical online facility location by replacing single-service assignment with $v_i$3-service robustness (Markarian et al., 2020).

A second major class is two-stage robust bilevel facility location under disruption risk. Here the leader opens facilities $v_i$4, nature chooses a disruption vector $v_i$5 from a budgeted uncertainty set,

$v_i$6

and a follower allocates flow $v_i$7 and unmet demand $v_i$8. The upper-level objective is

$v_i$9

while the follower minimizes total unmet demand $M$0 subject to disrupted capacities and demand balance (Hu et al., 22 Mar 2026). This formulation couples strategic siting, operational allocation, and worst-case facility failures.

A third class is dynamic facility location, in which the metric changes over time and the objective balances facility opening, connection, and switching costs. At each time $M$1, one selects open facilities $M$2 and assignments $M$3, paying a switching cost whenever $M$4 for client $M$5. The dynamic model is therefore not merely repeated static facility location; it is a time-coupled location-allocation problem with stability as an explicit decision criterion (An et al., 2014).

A fourth class arises in capacitated multi-channel systems. In the extended multi-channel facility location problem, facilities have capacities $M$6, each fulfilment channel $M$7 has capacity $M$8, and the objective combines opening cost, channel-specific shipment cost, and penalty for unfulfilled demand under a cardinality bound $M$9 (Agarwal et al., 2023). This is still facility location, but the path or service layer is now mode- or channel-specific.

These formulations are not interchangeable. Some are continuous-space location-allocation models, some are graph problems with shortest-path constraints, some are online covering problems, and some are bilevel robust programs. This suggests that FLPO is defined more by the coupling of facility decisions with path, service, or network-transport structure than by any single shared constraint set.

3. Robustness, redundancy, and temporal change

Robustness enters FLPO through several distinct mechanisms. In Online Multi-Facility Location, robustness is explicit redundancy: each client must be served by at least $f_j$0 different facilities. The paper motivates this by fault tolerance, facility failure, link failure, and replication, and notes that when $f_j$1 the model reduces exactly to known online facility location problems (Markarian et al., 2020). In graph terms, the requirement becomes purchasing $f_j$2 edge-disjoint paths from a root to a client in a bipartite root-facility-client graph, so the facility-location and path-design interpretations coincide.

In shortest-path service design, robustness is encoded through disjointness rather than multiplicity. The set-disjoint and path-disjoint variants require that each customer be covered by a pair of facilities whose shortest paths are disjoint, with the set-disjoint formulation motivated by OSPF and the path-disjoint formulation by MPLS (Johnson et al., 2016). Here, the same customer is served twice, but feasibility depends on routing-policy semantics, not only on the number of facilities.

In robust bilevel location, robustness is adversarial and capacity-based. Facilities may be disrupted, with at most $f_j$3 facilities unavailable, and the follower minimizes unmet demand after the disruption. The model admits decision-dependent uncertainty by imposing $f_j$4, so only opened facilities can be disrupted, and Proposition 7 shows that this reduces the search space without changing the worst-case value (Hu et al., 22 Mar 2026). In this setting, robustness is neither path multiplicity nor disjointness; it is the ability of the opened network to maintain service under worst-case capacity loss.

Dynamic facility location introduces a different notion of resilience: stability over time. If the distance metric changes between time steps, repeatedly recomputing a static optimum may induce many client switches. The switching cost therefore penalizes instability and creates a trade-off between per-period efficiency and temporal continuity (An et al., 2014). A common misconception is that all robust FLPO models enforce the same form of redundancy. The literature instead distinguishes among $f_j$5-service redundancy, disjoint-route protection, disruption resilience, and temporal stability.

4. Algorithmic paradigms

Several algorithmic paradigms dominate this literature. Online competitive analysis is central in the multi-facility setting. For the non-metric case, Online Non-metric Multi-Facility Location is reformulated as a path-purchasing problem on a bipartite graph with a root, and the paper gives an online $f_j$6-competitive randomized algorithm based on a fractional phase, threshold rounding, and explicit fallback path purchases, together with an $f_j$7 lower bound under $f_j$8 (Markarian et al., 2020). In the metric case, the paper gives a reduction from metric online facility location and obtains an $f_j$9-competitive algorithm whenever an $y_j\in\mathcal{D}\subset\mathbb{R}^d$0-competitive online metric facility-location algorithm is available (Markarian et al., 2020).

LP-based relaxation and rounding reappear in hard-capacitated settings. “LP-Based Algorithms for Capacitated Facility Location” introduces MFN-LP, a multi-commodity-flow-based relaxation with constant integrality gap. The associated algorithm produces a semi-integral solution and then rounds it to an integral one, yielding a 288-approximation relative to the LP optimum while respecting capacities exactly (An et al., 2014). This is noteworthy because it uses multi-commodity flows and matchings, not only the standard assignment LP, to encode residual feasibility under partial assignments.

Graph-theoretic repair methods arise when facility decisions interact with combinatorial parity or path constraints. For parity-constrained facility location, the paper first solves an unconstrained metric facility location instance and then corrects parity violations by constructing an auxiliary graph and computing a minimum-cost $y_j\in\mathcal{D}\subset\mathbb{R}^d$1-join, obtaining the first constant-factor approximation for the parity-constrained problem (Kim et al., 2019). For disjoint-path facility location, the facility-location problem is reduced to Set Cover by Pairs; worst-case approximability is poor, but the paper develops fast heuristics and a lower-bounding integer-programming formulation for the set-disjoint variant that equals the optimal solution value for all instances in its testbed (Johnson et al., 2016).

Dynamic settings require a different rounding philosophy. “Dynamic Facility Location via Exponential Clocks” extends the standard LP relaxation across time steps and introduces a clustering-free LP-rounding method based on competing exponential clocks on facilities and clients. Clients connect by following the smallest clock in their neighborhood, and the resulting algorithm gives the first constant approximation for dynamic facility location (An et al., 2014). The absence of clustering is emphasized as crucial because traditional cluster-based rounding is highly sensitive to small metric changes and can trigger frequent switches.

5. Learning-based, entropy-based, and control-theoretic methods

A major recent strand replaces exact combinatorial search by structured surrogates. In explicit FLPO, the Maximum Entropy Principle introduces a free-energy objective

$y_j\in\mathcal{D}\subset\mathbb{R}^d$2

with Gibbs-optimal path distributions for fixed $y_j\in\mathcal{D}\subset\mathbb{R}^d$3. “Parametrized Multi-Agent Routing via Deep Attention Models” then approximates these Gibbs policies using the Shortest Path Network, a permutation-invariant encoder-decoder. The paper reports “up to 100$y_j\in\mathcal{D}\subset\mathbb{R}^d$4 speedup in policy inference and gradient computation compared to MEP baselines,” an “average optimality gap of approximately 6%,” and, on a small FLPO instance, a match to Gurobi’s optimal cost with annealing at a “1500$y_j\in\mathcal{D}\subset\mathbb{R}^d$5 speedup” (Basiri et al., 30 Jul 2025).

A related but older entropy-based direction addresses inequality constraints directly. “Inequality Constraints in Facility Location and Other Similar Optimization Problems: An Entropy Based Approach” treats facility location, simultaneous facility location with path optimization, and last-mile delivery using deterministic annealing with auxiliary exponential penalties on violated constraints. The method permits temporary violation during early search and gradually reduces it as annealing proceeds, so feasibility is approached continuously rather than imposed by hard combinatorial branching (2002.03505).

Control-theoretic variants tighten this idea further. “A Control Barrier Function Approach to Constrained Resource Allocation Problems in a Maximum Entropy Principle Framework” reformulates capacitated facility location as a dynamic control design problem. The shifted free energy serves as a Control Lyapunov Function, while capacity and probability-domain constraints are encoded as Control Barrier Functions; numerical experiments report that the method is on average $y_j\in\mathcal{D}\subset\mathbb{R}^d$6 faster than SGF and $y_j\in\mathcal{D}\subset\mathbb{R}^d$7 faster than SLSQP, with “negligible growth in computation time with problem size” relative to those baselines (Bayati et al., 2 Apr 2025).

Learning is also used when the path layer is implicit. In large-scale urban location selection, a knowledge-informed reinforcement-learning method uses swap operations guided by a graph neural network and achieves “comparable performance to commercial solvers with less than 5\% accessibility loss, while displaying up to 1000 times speedup” (Su et al., 2024). In multi-objective facility location, dual graph neural networks predict the distribution probability of the entire Pareto set and sample non-dominated solutions non-autoregressively, achieving performance comparable to a multi-objective evolutionary algorithm with significantly reduced search cost (Liu et al., 2022). Under parameter perturbations, supervised learning can also predict how much of a reference facility solution should remain fixed, and those predictions can be embedded as additional constraints in the MILP for the new instance (Lodi et al., 2019).

These methods do not eliminate the underlying combinatorics. Rather, they replace exhaustive search by free-energy continuation, learned surrogates, policy sampling, or predictive warm starts. This suggests that the current methodological frontier in FLPO lies in preserving enough problem structure to retain meaningful guarantees while exploiting repeated-instance structure and differentiable approximations.

6. Applications, empirical behavior, and open directions

The application range is broad. In supply-chain design under disruption risk, the robust bilevel model studies a two-echelon network with capacitated supply facilities and customers, and numerical results show that, compared to the centralized two-stage robust model, the bilevel model “typically result[s] in higher service efficiency and better utilization of supply capacity.” Under a small-scale disruption, it “delivers better service performance,” whereas under a large-scale disruption “the centralized model performs more effectively” (Hu et al., 22 Mar 2026). In large-scale multi-channel supply chains, a submodular-plus-optimal-transport algorithm for the extended multi-channel facility location problem obtains a “100-fold speedup in computation, while the difference in objective values lies within a narrow range of 3%” (Agarwal et al., 2023).

Urban and networked applications reveal different structural emphases. Urban FLPO emphasizes accessibility, demand heterogeneity, and geospatial scale; network-monitoring and content-distribution applications emphasize shortest-path routing policy, pairwise coverage, and fault tolerance (Su et al., 2024). In dynamic or online settings, the operative concern is not only cost but also how rapidly assignments should react to changing metrics or arrivals (An et al., 2014). In probabilistic metric models, even very simple heuristics can be asymptotically strong: on random shortest path metrics, opening the $y_j\in\mathcal{D}\subset\mathbb{R}^d$8 cheapest facilities is asymptotically optimal in expectation when $y_j\in\mathcal{D}\subset\mathbb{R}^d$9, and in the equal-opening-cost case the expected approximation ratio reduces to $s_i$0, $s_i$1, or $s_i$2 depending on the magnitude of the opening costs (Klootwijk et al., 2019).

Several open directions are explicit in the literature. Online multi-facility location raises the question of competitive ratios independent of $s_i$3 or $s_i$4, and of removing the metric dependence on $s_i$5 and $s_i$6 (Markarian et al., 2020). Robust bilevel location points to larger uncertainty classes, richer disruption models, multi-period settings, and direct integration with routing (Hu et al., 22 Mar 2026). Learning-based FLPO raises questions about distribution shift, sparse or road-network graphs, explicit capacity and congestion modeling, and scaling beyond hundreds of nodes (Basiri et al., 30 Jul 2025). Urban reinforcement learning points toward dynamic traffic assignment, multimodal accessibility, and end-to-end location-plus-network-design pipelines (Su et al., 2024).

A recurring misconception is that FLPO is already a fully standardized benchmark family. The present literature instead supports a narrower conclusion: there is a stable core idea—joint optimization of facility placement and path-, routing-, or service-allocation structure—but the dominant formulations still differ markedly in whether paths are explicit or implicit, whether uncertainty is online or robust, whether redundancy is modeled by multiplicity or disjointness, and whether optimization is exact, approximate, entropy-based, or learned. This suggests that FLPO is an active synthesis area whose unifying principle is coupling, not uniformity.