Long-term Heuristic Scheduling Algorithm (LHSA)

• LHSA is defined as the routing component of a distributed maintenance framework that minimizes long-term transport and downtime costs.
• It reformulates scheduling as a maintenance-specific CVRPTW, solved via MILP for small instances and a divide-and-conquer heuristic for scalability.
• LHSA couples with predictive maintenance by adjusting preventive operations based on evolving failure probabilities and downtime penalties.

Searching arXiv for the named paper and the term “Long-term Heuristic Scheduling Algorithm” to ground the article in recent literature. Search results for arXiv query: (Mascolo et al., 22 Aug 2025)

1. (Mascolo et al., 22 Aug 2025) — "Innovative Distributed Maintenance Concept: From the design to cost optimisation" Authors: listed on arXiv Published: 2025-08-22 URL: https://arxiv.org/abs/([2508.16160](/papers/2508.16160))

Search results for arXiv query: "Long-term Heuristic Scheduling Algorithm"

1. (Mascolo et al., 22 Aug 2025) — "Innovative Distributed Maintenance Concept: From the design to cost optimisation" Authors: listed on arXiv Published: 2025-08-22 URL: https://arxiv.org/abs/([2508.16160](/papers/2508.16160))
2. (0809.3280) — "A Heuristic Scheduling Scheme in Multiuser OFDMA Networks" Authors: listed on arXiv Published: 2008-09-19 URL: https://arxiv.org/abs/([0809.3280](/papers/0809.3280))
3. (Hua et al., 2021) — "Learning to Schedule DAG Tasks" Authors: listed on arXiv Published: 2021-03-05 URL: https://arxiv.org/abs/([2103.03412](/papers/2103.03412)) Long-term Heuristic Scheduling Algorithm (LHSA) denotes, in its explicit formulation, the routing-and-timing component of a distributed maintenance optimization framework for geo-distributed production systems (GDPS). In "Innovative Distributed Maintenance Concept: From the design to cost optimisation" (Mascolo et al., 22 Aug 2025), LHSA operates inside the Optimised Maintenance and Capacitated Routing (OMCR) framework together with a Maintenance Planning Algorithm (MPA): MPA determines how many maintenance operations each site should receive over the horizon and when those operations should ideally occur, whereas LHSA converts those planned operations into a feasible, cost-minimizing long-term vehicle schedule under vehicle-capacity and time-window constraints. The algorithm is framed as a long-term extension of the capacitated vehicle routing problem with time windows (CVRPTW), with each planned maintenance operation treated as a node that must be visited exactly once by a vehicle from the Mobile Maintenance Workshop (MMW) (Mascolo et al., 22 Aug 2025).

1. Conceptual definition and system role

LHSA is defined in the maintenance literature as the operational scheduling stage of a broader strategic maintenance architecture. The framework in which it appears introduces a dual-entity maintenance structure composed of a Centralized Maintenance Workshop (CMW) and a Mobile Maintenance Workshop (MMW), with the objective of minimizing total long-term maintenance costs, including transport, operations, and downtime penalties (Mascolo et al., 22 Aug 2025). Within that architecture, LHSA is not a stand-alone maintenance planner. Its input is the maintenance demand generated by predictive maintenance planning, and its output is a long-term routing plan that executes the already-determined operations at minimum transport cost while respecting service windows and fleet constraints (Mascolo et al., 22 Aug 2025).

A central property of LHSA is that it is designed for long-horizon coupling between planning and routing. The underlying paper states that maintenance planning and routing cannot be cleanly separated over a long horizon because the number and timing of operations affect routing cost, and routing delays affect downtime cost. OMCR therefore alternates between MPA and LHSA until the total cost stabilizes (Mascolo et al., 22 Aug 2025). This makes LHSA a scheduling engine whose significance lies not only in route construction, but in its role in a feedback loop where service delay alters unavailability and therefore changes the optimal preventive-maintenance plan.

The long-term qualifier is material. The maintenance demand evolves over years according to failure distributions, so repeating a fixed short-cycle route would systematically overestimate or underestimate cost. LHSA therefore seeks the route and service order that best match the maintenance operations across the full horizon rather than simply repeating a periodic pattern (Mascolo et al., 22 Aug 2025).

2. Formalization as a maintenance-specific CVRPTW

The routing subproblem addressed by LHSA is built on a complete directed graph $G=(V,A)$, where $V=\{0,1,2,\dots,N\}$ contains the depot $0$ and the GDPS nodes $1,\dots,N$. Because each site $i$ may require several maintenance operations over the horizon, the model introduces an auxiliary graph $G'=(V',A')$ with $V'=\{0,1,2,\dots,n\}$, where $n=\sum_i nopi$ is the total number of maintenance operations (Mascolo et al., 22 Aug 2025). Each maintenance operation becomes its own node in this expanded graph.

LHSA receives as input the MPA outputs $[nopi]$ and $[S_{i,o}]$, together with the time windows $V=\{0,1,2,\dots,N\}$0, travel times $V=\{0,1,2,\dots,N\}$1, distances $V=\{0,1,2,\dots,N\}$2, service times $V=\{0,1,2,\dots,N\}$3, vehicle capacity $V=\{0,1,2,\dots,N\}$4, and fleet size $V=\{0,1,2,\dots,N\}$5 (Mascolo et al., 22 Aug 2025). The distinction between planning and routing is explicit: LHSA is not determining how many operations are needed; it is scheduling and routing the already-determined operations.

The exact formulation minimizes transport cost over chosen arcs:

$V=\{0,1,2,\dots,N\}$6

where $V=\{0,1,2,\dots,N\}$7 is the binary variable indicating whether a vehicle traverses arc $V=\{0,1,2,\dots,N\}$8, $V=\{0,1,2,\dots,N\}$9 is the unit transport cost per kilometer per unit of capacity, and $0$0 is the unit transport cost per hour (Mascolo et al., 22 Aug 2025). The objective therefore combines distance-based and time-based transport expenses.

The formulation is constrained by vehicle availability, flow conservation, capacity, and time-window feasibility:

$0$1

$0$2

$0$3

$0$4

$0$5

$0$6

$0$7

$0$8

$0$9

$1,\dots,N$0

The associated interpretation is also given explicitly. Constraint (5) gives the minimum number of vehicles required; (6) and (7) ensure that exactly $1,\dots,N$1 vehicles depart from and return to the depot; (8) and (9) enforce that each maintenance operation is serviced exactly once and that each vehicle route is consistent; (10) enforces the vehicle load or capacity feasibility by tracking the remaining number of spare parts $1,\dots,N$2; (11) enforces service sequencing and temporal feasibility between consecutive maintenance actions; and (12)–(14) bound the load and restrict the decision variables (Mascolo et al., 22 Aug 2025). Together, these constraints make LHSA a CVRPTW-like model with maintenance-specific service times and deadlines.

3. Solution strategy: MILP and divide-and-conquer heuristic

The paper presents two computational regimes for LHSA. For small instances, it gives an exact Mixed Integer Linear Programming (MILP) formulation. For larger instances, it introduces a divide-and-conquer heuristic adapted from large-scale vehicle-routing decomposition (Mascolo et al., 22 Aug 2025). This dual strategy is motivated by the NP-hard character of the routing problem.

The practical core of LHSA lies in the decomposition logic. The algorithm sorts the maintenance operations from earliest to latest, splits the ordered list into sublists whose size does not exceed the vehicle capacity $1,\dots,N$3, solves each subproblem with the MILP, and then combines the partial solutions into a full schedule (Mascolo et al., 22 Aug 2025). Figure 1 in the source is described as: sort time windows, divide into sub-lists, solve each sub-list with MILP, and combine the resulting routes into the optimal scheduling.

This decomposition serves a direct computational purpose. The full long-term operation list may contain many maintenance visits, especially when the horizon is long and failure rates rise. Solving the exact model on each smaller subproblem reduces the computational burden while preserving the structure of the original problem (Mascolo et al., 22 Aug 2025). A plausible implication is that the method occupies an intermediate position between pure exact optimization and purely constructive heuristics: the exact model is retained where tractable, while heuristic partitioning provides the scalability required for large long-horizon instances.

The resulting characterization of LHSA is therefore not merely “heuristic” in the loose sense of rule-based scheduling. It is a hybrid procedure whose heuristic element lies in problem decomposition and solution assembly, whereas each subproblem remains anchored in an explicit MILP (Mascolo et al., 22 Aug 2025).

4. Coupling with predictive maintenance, downtime, and availability

LHSA has a central role in the iterative OMCR loop because routing delays feed back into maintenance planning through downtime. After MPA produces an initial set of operations and times, LHSA determines the route and the actual service delays, which in turn produce nonzero downtime values $1,\dots,N$4 (Mascolo et al., 22 Aug 2025). Those delays are then fed back into MPA, which may increase the number of preventive interventions to reduce unavailability, but at the cost of more operations.

The downtime expression given in the paper is:

$1,\dots,N$5

with

$1,\dots,N$6

Average availability is then calculated as

$1,\dots,N$7

as presented in the paper (Mascolo et al., 22 Aug 2025).

The same framework derives maintenance demand from the equipment failure law. In the experiments, failure times follow a Weibull distribution:

$1,\dots,N$8

where $1,\dots,N$9 is the scale parameter and $i$0 is the shape parameter (Mascolo et al., 22 Aug 2025). These probabilities determine criticality: the higher the failure probability by a given time, the more urgent the maintenance operation. MPA uses this information to find the best number and timing of maintenance operations, and LHSA then schedules the resulting operations under the constraints of the MMW fleet.

This coupling clarifies a frequent misunderstanding. Route delays are not treated as a secondary transport nuisance; they alter downtime cost, which changes the optimal maintenance plan (Mascolo et al., 22 Aug 2025). In that sense, LHSA is integral to the economic evaluation of preventive maintenance rather than an after-the-fact dispatch routine.

5. Interaction with depot location and vehicle-capacity design

Although the CMW location decision is handled outside LHSA, it directly affects the distances $i$1 and travel times $i$2 that appear in the LHSA objective. To avoid testing every possible depot location, the paper proposes a heuristic that places the CMW at the weighted barycentre of the GDPS, using failure probabilities as weights (Mascolo et al., 22 Aug 2025). This is intended to locate the workshop closer to the most critical sites while not excessively penalizing less critical ones.

Because LHSA minimizes transport cost over the induced distances and travel times, the depot-location heuristic materially changes the cost surface of the routing problem. The source states that, when the CMW was placed at the weighted barycentre rather than near one site, transport distances and evaluation costs improved, especially as the number of sites increased. It also states that, as additional sites are added after depot construction, the barycentric location becomes increasingly advantageous relative to choosing a depot near one site (Mascolo et al., 22 Aug 2025).

Vehicle-capacity choice interacts with LHSA in a similarly direct way. If the vehicle capacity $i$3 is small, more trips are needed; if $i$4 is larger, fewer trips may suffice but per-kilometer transport cost may be higher depending on the vehicle type. The paper considers light, medium, and heavy vehicles and shows that capacity influences both annual distance and annual transport cost (Mascolo et al., 22 Aug 2025). Heavy vehicles reduced distance traveled but were not always cheapest, while light vehicles, despite traveling more, could minimize transport cost because their per-kilometer operating cost was lower.

This suggests that LHSA should be understood not only as a routing algorithm but as one component in a coupled design problem linking maintenance demand, depot placement, and fleet sizing. The routing formulation is therefore sensitive to system-design choices that are external to the arc-selection MILP itself.

6. Empirical findings, scope, and related uses of the label

The reported experiments provide the principal evidence for LHSA’s utility within OMCR. Over a 2-year horizon, the long-term approach outperformed short-term cyclic scheduling, showing that maintenance costs can be overestimated if one simply repeats a short cycle (Mascolo et al., 22 Aug 2025). For a representative scenario with penalty cost $i$5100/h$i$6\$i$7 to $i$8350/h$i$9\$G'=(V',A')$0), the algorithm favored fewer preventive operations and the benefit of long-term scheduling was weaker; when penalty costs were very high ($G'=(V',A')$11000/h$), availability stayed almost constant but the horizon choice had less influence (Mascolo et al., 22 Aug 2025).

The framework was tested on 10 sites and then scaled up to 30 additional sites, with more than 1380 experiments run in Scilab using a MILP optimization toolbox. The authors state that the heuristic decomposition makes the approach scalable beyond the small instances typically solvable exactly and report improved scalability for systems with more than 30 GDPS (Mascolo et al., 22 Aug 2025). The stated application domains include aerospace, railway, and energy industries, all described as sectors requiring high availability and centralized servicing.

The term LHSA is not used uniformly across arXiv literature. In the maintenance paper, it is an explicit algorithm name for the long-horizon routing component of OMCR (Mascolo et al., 22 Aug 2025). By contrast, in "A Heuristic Scheduling Scheme in Multiuser OFDMA Networks" (0809.3280), the label does not literally appear, but the paper contains a dynamic heuristic for updating the non-real-time scheduling weight $G'=(V',A')$2 over time so that real-time delay constraints are just met while leftover resources are shifted toward best-effort throughput. The source explicitly states that this adaptive update of $G'=(V',A')$3 is the paper’s core heuristic and a natural mapping to an LHSA-style method. Likewise, "Learning to Schedule DAG Tasks" (Hua et al., 2021) does not define a standard LHSA, but its reinforcement-learning policy modifies DAGs by adding directed edges so that heuristic schedulers such as SJF and CP perform better, and the source characterizes this as an LHSA-style meta-scheduling framework.

A common misconception is therefore to treat LHSA as a universally standardized scheduling family. The cited material suggests a narrower conclusion: the fully explicit term is attached to the distributed-maintenance routing problem in (Mascolo et al., 22 Aug 2025), while other papers exhibit long-term heuristic scheduling mechanisms that are analogous in spirit but not identically named (0809.3280, Hua et al., 2021).