Optimised Maintenance & Capacitated Routing

• OMCR is an integrated heuristic framework that combines centralized maintenance with mobile service fleets to optimize preventive replacement scheduling and routing in geo-distributed systems.
• It couples predictive maintenance scheduling with capacitated vehicle routing and time windows, balancing transport, operations, and downtime costs to achieve up to 50% cost reductions.
• The framework employs a two-stage algorithm with innovative depot siting and fleet capacity selection to enable scalable, tractable routing across numerous service sites.

Optimised Maintenance and Capacitated Routing (OMCR) is an integrated heuristic framework for the strategic optimization of distributed maintenance operations in geo-distributed production systems (GDPS). It was introduced as a dual-entity maintenance concept combining a Centralized Maintenance Workshop (CMW) and a Mobile Maintenance Workshop (MMW), with the objective of minimizing total long-term maintenance costs through the joint treatment of transport, operations, and downtime penalties. The framework couples predictive maintenance scheduling with capacitated vehicle routing with time windows, and its reported experimental validation indicates lifecycle maintenance cost reductions of up to 50% together with increased scalability for systems exceeding 30 GDPS (Mascolo et al., 22 Aug 2025).

1. System model and maintenance architecture

OMCR is defined on a geo-distributed production system containing a fixed depot and multiple geographically dispersed service targets. The CMW is a fixed depot that monitors condition-monitoring data, holds an effectively unlimited spare-parts stock, and dispatches MMWs. In the routing formulation it acts as the root node, or depot, for all vehicle-routing models. The MMW is a fleet of $m$ homogeneous vehicles, each with capacity $Q$ spare parts and staffed by an operator; each vehicle departs the CMW, visits a subset of GDPS to perform planned maintenance in the form of equipment replacements, and returns to the CMW when stock is empty or routing dictates. Each GDPS contains one piece of equipment subject to stochastic failure and must be served within a tight time window $[a_i,b_i]$ (Mascolo et al., 22 Aug 2025).

Entity	Role	Formal characteristics
CMW	Fixed depot and monitoring center	Unlimited spare-parts stock; root node in routing
MMW	Mobile service fleet	$m$ homogeneous vehicles; capacity $Q$; operator-staffed
GDPS	Distributed service locations	$N$ sites; one equipment unit per site; stochastic failure

This architecture formalizes maintenance as neither purely local repair nor purely centralized dispatch. A common misunderstanding is to treat OMCR as only a routing formulation. In the reported framework, routing is downstream of a maintenance-planning stage, and the CMW–MMW split is a structural assumption used to coordinate long-term preventive replacement timing with fleet movement. This suggests that OMCR is designed for environments in which centralized servicing and high availability are both material constraints.

2. Long-term cost model

OMCR minimizes a three-term cost function:

$$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$

The transport term is

$$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$$

the operations term is

$$\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$$

and the downtime term is

$$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$$

Here $Q$0 indicates that vehicle $Q$1 travels from node $Q$2 to node $Q$3; $Q$4 and $Q$5 are distance and travel time; $Q$6 and $Q$7 are unit transport costs; $Q$8 is the number of preventive replacements at site $Q$9; $[a_i,b_i]$0 is per-operation cost; $[a_i,b_i]$1 is the cumulative failure probability; $[a_i,b_i]$2 is the per-hour downtime penalty; and $[a_i,b_i]$3 is mean repair time (Mascolo et al., 22 Aug 2025).

The formulation is significant because it makes explicit the trade-off between preventive intervention frequency and service logistics. Too-frequent replacements increase $[a_i,b_i]$4, while deferring replacement increases expected downtime exposure. In OMCR, transport is therefore not an auxiliary consideration but one of three co-equal cost drivers.

3. Two-stage optimization procedure

The framework is organized around a two-stage algorithmic approach: a Maintenance Planning Algorithm (MPA) and a Long-term Heuristic Scheduling Algorithm (LHSA). The MPA minimizes $[a_i,b_i]$5 for each site over a horizon $[a_i,b_i]$6. Its inputs are the failure law for equipment at site $[a_i,b_i]$7, the costs $[a_i,b_i]$8 and $[a_i,b_i]$9, $m$0, and the scheduling horizon. The failure law is specified as

$m$1

The decision variables are $m$2 and replacement epochs $m$3. The search logic uses golden-section search, or a similar one-dimensional search, on $m$4 and associated $m$5 to find the trade-off between excessive preventive replacement and long downtime. Its outputs are the optimal $m$6 and, for each replacement $m$7, a time window $m$8 centered on $m$9 (Mascolo et al., 22 Aug 2025).

The LHSA then models the routing layer as a capacitated vehicle routing problem with time windows (CVRPTW) over all $Q$0 maintenance-stop nodes. It receives from the MPA the time windows $Q$1 and service times $Q$2 for each stop. The mixed-integer linear programming formulation is single-commodity and single-depot, with objective

$Q$3

subject to depot dispatch, single visit per stop, flow conservation, capacity propagation, time-feasibility, and window-respecting arrival constraints:

$Q$4

$Q$5

$Q$6

$Q$7

$Q$8

$Q$9

Because $N$0 can be large, LHSA uses a divide-and-conquer heuristic: all stops are sorted by earliest latest-start time, partitioned into sub-lists of at most $N$1 stops, each sub-list is solved by MILP, and the resulting tours are concatenated. LHSA yields an approximate routing solution and the actual downtime intervals $N$2 for feedback to the MPA (Mascolo et al., 22 Aug 2025).

This iterative coupling is central to OMCR. A common misconception is that maintenance timing is fixed before routing and unaffected by actual service realizability. In the reported formulation, LHSA feeds realized downtime intervals back to MPA, so the long-term schedule is refined against route feasibility.

4. Depot siting and fleet-capacity choice

A novel component of OMCR is its heuristic for CMW location determination. Site weights are defined by criticality over the planning horizon through total failure probability:

$N$3

The depot coordinates are then set to the weighted barycentre,

$N$4

The reported empirical comparison against a rule that chooses the closest existing site shows that, as $N$5 grows, barycentre-based placement yields lower aggregate transport cost and a smaller candidate set, since it reduces the search to one position rather than $N$6 (Mascolo et al., 22 Aug 2025).

OMCR also performs a discrete selection of MMW capacity. Vehicle capacity $N$7 is chosen from a small predefined set, exemplified by $N$8. The framework loops over candidate values, runs OMCR with iterative MPA–LHSA convergence for each, records $N$9, and selects the minimizing value $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$0. The stated trade-off is that larger $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$1 reduces distance but incurs higher per-hour or per-trip costs, and the framework seeks the empirically favorable compromise (Mascolo et al., 22 Aug 2025).

These two heuristics are structurally distinct. CMW placement alters the geometry of all routes, whereas MMW capacity changes the partitioning of work across routes. Their joint inclusion indicates that OMCR is not limited to schedule optimization after infrastructure has already been fixed.

5. Integrated formulation, tractability, and computational logic

The underlying problem admits an all-in-one MILP that would jointly decide the preventive-replacement counts $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$2, replacement epochs $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$3, and routing variables $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$4. The reported study states that such a model is NP-hard with worst-case time complexity $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$5. OMCR avoids this monolithic integration by iterating two smaller problems: MPA, with complexity $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$6 per site, and LHSA, whose subproblems are exponential only in sub-list size bounded by $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$7 (Mascolo et al., 22 Aug 2025).

The divide-and-conquer strategy reduces the largest routing instance from size $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$8 to chunks of size at most $$\mathrm{Total\_Cost}=\mathrm{Transport\_Cost}+\mathrm{Operations\_Cost}+\mathrm{Downtime\_Cost}.$$9, making each MILP tractable. The study further reports practical overall complexity of roughly $$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$$0, compared with $$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$$1 for monolithic solution. This computational decomposition is one of the framework’s defining characteristics.

A second common misunderstanding is to interpret OMCR as a fully integrated exact optimizer. The reported formulation does not claim that. Instead, it combines MILP subproblems, heuristic partitioning, and iteration between planning and scheduling. This suggests that OMCR prioritizes tractable long-horizon decision support over exact global optimality for the full joint problem.

6. Experimental evidence and sectoral applicability

The experimental setup uses a network of 10 GDPS randomly placed within a 50 km radius, with additional tests adding up to 30 more sites to assess scalability. Failure data follow Weibull distributions with $$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$$2 to reflect aging equipment. Cost scenarios vary the downtime penalty $$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$$3 against a replacement cost $$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$$4100{,}000$$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$$5\$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$6, $\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$7100/\mathrm{h}$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$8\$\mathrm{Transport\_Cost} =\sum_{k=1}^m \sum_{i,j\in V} \bigl(C_D\,d_{ij} + C_T\,t_{ij}\bigr)\,x_{ij}^k,$9. Vehicles are assigned nominal speed $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$0, capacities $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$1, and transport costs $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$23/\mathrm{km}$\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$3C_T=\$\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$4. The planning horizon $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$5 is swept from 2 months to 24 months. Reported metrics are average site availability $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$6, $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$7, $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$8, $\mathrm{Operations\_Cost} =\sum_{i=1}^N C_{R,i}\,n_i,$9, $\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$0, annual distance traveled, and CPU time. Implementation is in Scilab 5.5.2 with the FOSSEE optimization toolbox on an Intel i7-10850H with 32 GB RAM (Mascolo et al., 22 Aug 2025).

The principal results reported for OMCR are specific. In the medium-penalty case of $\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$1100/\mathrm{h}$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$2\mathrm{Total_Cost}$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$3Q=4$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$4\mathrm{Transport_Cost}$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$5\sum n_i\approx 120$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$6\sum n_i\approx 20$ (Mascolo et al., 22 Aug 2025).

The stated applicability covers aerospace, railway, and energy sectors, particularly where assets are geographically clustered and unavailability penalties are high. The reported implication is that long-term planning, barycentre-based CMW positioning, and a right-sized MMW fleet can reduce lifecycle maintenance costs by up to 50%. This suggests that OMCR is oriented toward settings in which centralized maintenance governance coexists with geographically dispersed operational assets.

7. Relation to dynamic capacitated routing research

OMCR is formulated around planned maintenance in a GDPS with CVRPTW scheduling, but it also sits near a broader routing literature concerned with dynamic capacitated routing. Tong et al.’s “A Novel Generalised Meta-Heuristic Framework for Dynamic Capacitated Arc Routing Problems” develops a framework for dynamic capacitated arc routing problems (DCARP) in which rescheduling occurs while service is already partially executed. The summary provided for that work states that, although the authors do not use the acronym “OMCR,” their framework can be seen as an “Optimised Maintenance and Capacitated Routing” approach in a dynamic environment (Tong et al., 2021).

The DCARP formulation is defined on a directed graph $$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$$7 with depot $$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$$8, task set $$\mathrm{Downtime\_Cost} =\sum_{i=1}^N \sum_{o=1}^{n_i} C_{P,i}\,F_i\bigl(TT_{i,o}\le S_{i,o}\bigr)\,\bigl(S_{i,o}-S_{i,o-1}-\mathrm{MTTR}_i\bigr).$$9, outside vehicles at stops $Q$00 with remaining capacities $Q$01, and up to $Q$02 fresh vehicles at the depot, each with capacity $Q$03. The objective minimizes total route cost subject to exact task coverage and capacity constraints. Dynamic changes are generated by a service simulator that executes a current solution, chooses a random stopping point, removes served tasks, records outside-vehicle states, simulates one breakdown event per vehicle, and then triggers dynamic events on arcs with probability $Q$04. The nine event types include vehicle breakdown, road closure, congestion, recovery from closure or congestion, worsening or easing congestion, demand increase, and new task creation (Tong et al., 2021).

The core contribution of that framework is a virtual-task transformation that converts a DCARP instance with outside vehicles into a static CARP instance in which all vehicles appear at the depot with capacity $Q$05. For each outside vehicle, a virtual task is added with head $Q$06, tail $Q$07, infinite deadheading cost, serving cost $Q$08, and demand $Q$09. After solving the transformed static instance, the virtual tasks are stripped out, the artificial serving costs are subtracted, and routes are re-anchored to outside-vehicle locations. Two dynamic-response strategies are used: a restart strategy and a sequence-transfer strategy that inherits and repairs the previous solution (Tong et al., 2021).

The empirical results reported for that dynamic framework are distinct from OMCR’s maintenance-cost results but relevant as neighboring methodology. Direct rescheduling through virtual tasks outperforms a return-first baseline when outside vehicles retain moderate or large remaining capacity; embedding the virtual-task idea into MASDC yields strictly better results on all 72 tested instances; and among the compared plugged-in static-CARP methods, memetic MAENS attains the best average ranking. A plausible implication is that OMCR’s long-term maintenance-routing framework could be related, in future extensions, to dynamic rescheduling mechanisms when maintenance execution is disrupted by breakdowns, road closures, or changing service demands.