A Three-Stage Approach for the Multi-Depot VRP with Priority Requests

Abstract

Field-service operations for utility companies require routing technicians across multiple depots while guaranteeing same-day response to critical infrastructure customers, a constraint that standard multi-depot routing methods cannot structurally enforce. We introduce the MDVRP with Priority Requests (MDVRP-PR), formalised as a lexicographic optimisation problem that guarantees service to priority customers before maximising coverage and minimising route duration. A three-stage pipeline is proposed: hybrid DBSCAN-Hierarchical clustering for topology-aware depot assignment, an Enhanced Max-Min Ant System (MMAS) with priority-driven construction, lexicographic solution selection, and repair, and a Boundary Relocate post-optimisation stage with global cross-depot recovery. The approach is evaluated on a real-world applied case study from Algérie Télécom (Guelma, Algeria), comprising a single four-depot field-service instance scaled to three sizes (55, 90, and 150 customers) and assessed over 2135 controlled runs. On this case study, the proposed clustering method outperforms the MDVRP-adapted Sweep baseline by $22.9$ percentage points on the largest instance (n = 150; Friedman p < 0.001). The priority mechanisms sustain $100\%$ feasibility across all configurations, compared to complete collapse without them ($0/10$ seeds at $\ge 40\%$ priority), at a route-time overhead below $5\%$. Relative to the company’s current manual practice, the framework improves customer coverage by $16.1$ percentage points within 28 s, confirming its practical utility for daily deployment in this capacity-constrained, priority-sensitive routing context.

1. Introduction

The Vehicle Routing Problem (VRP), first formalized by Dantzig and Ramser [1] as a generalization of the Traveling Salesman Problem, has given rise to a rich taxonomy of variants. The Capacitated VRP (CVRP) introduces vehicle-load limits and is the canonical reference variant for which exact and metaheuristic methods continue to advance [2]; the VRP with Time Windows (VRPTW) adds customer-side temporal feasibility and remains a workhorse for both academic benchmarks and applied delivery problems [3]; and the Multi-Depot VRP (MDVRP) couples customer-to-depot assignment with route construction across a fleet split among several geographically distinct depots [4,5]. Each captures specific structural features of real-world logistics and service operations.

Beyond these foundational variants, several research threads currently extend the VRP toward emerging operational realities. Two-echelon and satellite-based formulations decompose urban delivery into city-level transfers and last-mile distribution [6]; green and electric VRPs incorporate energy consumption, charging infrastructure, and emissions into the objective and constraints [7]; synchronization VRPs handle interdependent operations between vehicles, drivers, or facilities that must coincide in time and space [8]; drone-aided routing combines ground vehicles with unmanned aerial vehicles to exploit complementary mobility profiles [9]; and time-dependent VRPs allow travel times to vary with traffic conditions across the planning horizon [10]. These threads collectively illustrate that the VRP literature is rapidly diversifying along technological and sustainability axes.

Despite this breadth, applied VRP research remains concentrated in developed-economy contexts [11,12,13,14], leaving a gap for settings where asymmetric road networks, heterogeneous fleet distributions, and rigid administrative partitioning create operational challenges that standard benchmarks do not reflect.

This paper is motivated by a concrete instance of that gap: the daily field-service routing operations of Algérie Telecom, the state-owned telecommunications provider of Algeria. The company maintains infrastructure across all 69 provinces through a network of Technical Centers (TCs), each dispatching dedicated Technical Teams (TTs) to handle customer maintenance requests. Under the current practice, each TC partitions its service territory into fixed sub-zones assigned permanently to individual TTs, and teams plan their own routes based on field experience. This rigid scheme has two measurable consequences: teams in adjacent zones cannot collaborate even when one is overloaded and the other is idle, and more critically, no formal mechanism exists to guarantee that high-priority customers are served before the working day ends.

The urgency of the latter failure is significant. Algérie Telecom serves two operationally distinct customer classes: Regular Customers (RCs), private residential users covered by a 48-h response commitment, and Priority Customers (PCs), comprising government agencies, security services, and critical enterprises whose operations cannot sustain prolonged outages and who require same-day service. Routing a team to a residential customer before attending to a nearby government agency is not merely suboptimal—it constitutes a service-level breach with administrative consequences. Yet distance-minimizing formulations are structurally unable to prevent it, and to our knowledge no existing MDVRP formulation makes this distinction explicit at the customer level.

Formally, the problem maps to an MDVRP in which TCs are depots. Standard cluster-first, route-second MDVRP methods [15,16] decompose the problem into a spatial assignment phase followed by independent per-depot routing. This decomposition introduces a structural tension: customers near depot boundaries are frequently misassigned by spatial clustering, yet the routing phase treats assignments as fixed and cannot correct them, even when doing so would reduce cost. Under real-road asymmetry and hard priority constraints, this limitation is particularly damaging.

We address both gaps—the absence of a priority-aware MDVRP formulation and the misassignment problem inherent to the cluster-then-route paradigm—through an applied problem variant and a three-stage solution framework. The Multi-Depot VRP with Priority Requests (MDVRP-PR) models service urgency as a lexicographic hard constraint: every priority customer must be served before coverage and travel time are optimized. This is a stronger guarantee than weighted or soft-penalty approaches, because it structurally prevents any solution that leaves a priority customer unserved from being accepted, regardless of the travel-time savings it might offer. The solution framework integrates: (i) a road-distance-aware DBSCAN clustering stage with capacity-constrained merging, which distributes workload across the full fleet rather than producing geometrically compact but operationally imbalanced assignments; (ii) an enhanced Max-Min Ant System (MMAS) with a priority-first construction rule that reserves route capacity for priority customers during ant construction rather than relying on post-hoc repair; and (iii) a Boundary Relocate post-optimization operator that corrects depot misassignments using actual insertion and removal costs, explicitly bridging the assignment-routing gap that the clustering phase leaves open.

The primary contributions of this work are:

• MDVRP-PR: to the best of our knowledge, the first MDVRP formulation in which priority is enforced as a structural admissibility constraint over the feasible region (lexicographic feasibility) rather than as a soft penalty, weighted reward, or time-window encoding. The formulation is motivated by and validated on real operational data from Algérie Telecom’s field service in the Guelma province, Algeria.
• Topology-aware clustering: a hybrid DBSCAN-hierarchical assignment procedure that uses asymmetric road distances and capacity-constrained merging to achieve full fleet utilization, improving large-instance coverage by up to $+22.9$ percentage points over sweep-based assignment on the studied case.
• Priority-first MMAS: an enhanced Max-Min Ant System in which must-serve constraints are enforced structurally during construction, achieving 100% priority feasibility across all tested configurations at a route-time overhead below 5%.
• Boundary Relocate operator: a cross-depot post-optimization move that evaluates relocation candidates by actual insertion and removal costs rather than geometric proximity, correcting the systematic misassignments that spatial clustering produces near depot boundaries.

The remainder of this paper is organized as follows: Section 2 reviews related work; Section 3 presents the formal problem definition; Section 4 details the proposed methodology; Section 5 and Section 6 describe the experimental setup and results; and Section 7 concludes.

2. Related Work

The literature directly relevant to this work spans three interconnected areas: the classical MDVRP and its benchmark methods, which establish the baseline against which our formulation is positioned; clustering-based decomposition strategies for MDVRP, whose structural limitations motivate our topology-aware assignment and Boundary Relocate operator; and ACO/MMAS-based solution frameworks, whose uniform construction rules motivate our priority-first extension. A fourth thread, priority and service-class modeling in VRP, is reviewed separately, as it most directly concerns our core problem novelty. Table 1 summarizes the works reviewed across these threads, highlighting the features most relevant to our contributions.

2.1. Classical MDVRP: Formulations and Benchmark Methods

The classical MDVRP requires simultaneously determining customer-to-depot assignments and vehicle route sequences, two interdependent decisions that render it NP-hard. Early work established the solution templates that subsequent research has built upon. Renaud et al. [16] proposed the first dedicated tabu search heuristic (FIND) for the MDVRP, structured into fast improvement and diversification phases, which remained competitive for nearly a decade. Cordeau et al. [4] introduced a unified tabu search capable of solving the MDVRP, the Periodic VRP, and the Periodic MDVRP within a single framework, and their benchmark instances have since become the standard evaluation testbed for the problem. Ho et al. [17] demonstrated that hybrid genetic algorithms benefit substantially from informed initialization: seeding with Clarke–Wright savings routes produced consistently stronger solutions than random initialization, pointing to the value of constructive heuristics within population-based methods.

The most competitive reference method for the classical problem remains the Hybrid Genetic Search with Adaptive Diversity Control (HGSADC) of Vidal et al. [5]. It combines a biased fitness function that jointly ranks solution cost and population diversity, a periodic crossover operator for pattern assignment, and a dual-population management scheme. HGSADC achieves near-optimal solutions on the Cordeau benchmark instances and has served as the primary comparison baseline for over a decade. Azad et al. [18] extended the base formulation to the Multi-Depot Periodic VRP with Interdependent Depot Operations, solving it with a memetic algorithm that combines pattern-assignment initialization and selective simulated annealing, achieving new best-known solutions on 7 of 10 benchmark instances. Mancini [19] addressed a Multi-Depot Multi-Period VRP with a heterogeneous fleet using an ALNS matheuristic that solves an exact MIP subproblem at each iteration, reporting 8–10% cost improvements over a commercial solver. Lim et al. [20] introduced the first complete MIP formulation for the Multi-Depot Split-Delivery VRP with heterogeneous vehicles; permitting multiple depot visits per customer reduces total travel distance by up to 16%. Chen and Yang [21] reformulated the MDVRP as a bilevel optimisation that integrates urban traffic equilibrium, reducing delivery time by nearly 30% over distance-minimising methods, though the model requires a full origin-destination traffic matrix that is rarely available in practice.

Across all of these contributions, customer requests are treated as operationally homogeneous: no distinction is made between service classes that differ in response urgency, and no structural mechanism exists to prevent a lower-priority customer from being served before a higher-priority one.

2.2. Clustering-Based Decomposition for MDVRP

Given the NP-hardness of the MDVRP, the cluster-first, route-second paradigm is one of the most widely adopted solution strategies. It decomposes the problem into a depot-assignment phase followed by independent single-depot routing. While this approach substantially reduces computational complexity, solution quality depends critically on the clustering decisions, and purely spatial partitioning introduces two recurring structural limitations that are directly relevant to the present work.

The first is workload imbalance. When the spatial distribution of customers does not align with depot capacities, nearest-depot or geometry-based assignment produces overloaded depots alongside underutilised ones. Gu et al. [22] applied nearest-depot assignment as the first stage of a three-phase framework using an Artificial Bee Colony algorithm and explicitly acknowledged that fixing assignments prior to route construction may preclude globally optimal solutions. Tao et al. [23] addressed a Multi-Depot Capacitated VRP with Order Split and Allocation using a Variable Neighbourhood Search with seven specialised operators, but relied on a pre-determined depot structure without adaptive load rebalancing. Wang et al. [24] and Wang et al. [25] both employ k-medoids clustering as a preprocessing step for collaborative multi-depot routing under dynamic demands; neither accounts for asymmetric road distances in the assignment phase, so workload imbalances persist whenever road geometry departs from Euclidean proximity.

The second is boundary misassignment. Customers near service territory boundaries are frequently allocated to suboptimal depots by spatial clustering, and the routing phase—treating assignments as fixed—cannot correct these misallocations even when routing costs clearly favour an alternative depot. Stodola and Nohel [26] proposed AACO-NC, currently the most competitive ACO-based method for the classical MDVRP, matching or exceeding best-known solutions on 18 of 23 Cordeau benchmark instances and establishing four new best-known results. Despite this strong performance, its clustering component is purely spatial, does not account for depot workload balance, and provides no mechanism to revise depot assignments once the clustering phase is complete. Yu et al. [27] and Xu et al. [28] likewise assign customers to the nearest depot without considering routing efficiency, and report that this oversimplification degrades solution quality on instances where customers are not spatially well-separated by depot.

These two weaknesses jointly motivate the topology-aware DBSCAN-based clustering with capacity-constrained merging proposed in Stage 1 of our framework, and the Boundary Relocate post-optimization operator in Stage 3, which corrects residual misassignments using actual route insertion and removal costs rather than geometric proximity.

A complementary line of recent work explores dynamic decomposition that operates inside the metaheuristic loop rather than as a static first phase. Santini et al. [29] repeatedly break the current elite solution into smaller subproblems by clustering its routes (e.g., on geographic barycenters), solve each subproblem from scratch with the same solver, and reassemble the improved fragments, reporting consistently stronger results than using decomposition only as pre- or post-processing. Their decomposition is therefore solution-driven and adaptive, whereas our Stage 1 decomposes at the customer level once before any routing, with the cross-depot Boundary Relocate operator in Stage 3 acting as the corrective mechanism. The two strategies are complementary rather than substitutes: the dynamic decomposition ofSantini et al. [29] reorganizes routes during search, while our pipeline reorganizes depot assignments after construction. We do not claim methodological superiority over dynamic decomposition; we focus on the assignment side of the cluster-first paradigm, which is the locus of the boundary-misassignment failures observed in our applied case. Recent reviews of related research threads, including time-dependent vehicle routing [10], indicate that adaptive within-loop decomposition and depot-aware structural correction are likely to be combined in future MDVRP work.

2.3. ACO and MMAS for Multi-Depot Routing

Ant Colony Optimisation, and the Max-Min Ant System (MMAS) in particular, has demonstrated strong performance across VRP variants. This is largely due to the natural alignment between pheromone-guided construction and the sequential, interdependent nature of routing decisions. MMAS improves upon basic ACO by bounding pheromone concentrations within a dynamic interval $[{\tau }_{min},{\tau }_{max}]$ and restricting updates to the best solutions found, thereby preventing premature stagnation while preserving exploitation of high-quality routes [30].

Ting and Chen [31] introduced the first ACO-based method for the MDVRPTW, proposing a dual pheromone system with separate matrices for customer-depot assignment and route sequencing, combined with simulated annealing for intensification; this yielded six new best-known solutions on Cordeau instances. Their design illustrates the value of separating assignment and sequencing signals in pheromone management, though the construction rule remains uniform across all customers. Niu et al. [32] applied MMAS-based decomposition to the Multi-Depot Cumulative Capacitated VRP, maintaining a master pheromone matrix alongside subproblem-specific matrices and using pheromone trail smoothing combined with local search for perturbation; their analysis identified initialisation bias toward nearest-depot assignments and fixed subproblem boundaries as the principal limiting factors. Stodola and Nohel [26] demonstrated that adaptive pheromone management calibrated to population entropy outperforms fixed-parameter ACO variants in the MDVRP context, establishing entropy-driven evaporation as a practically effective anti-stagnation mechanism.

A limitation shared by all existing ACO and MMAS formulations for multi-depot routing is that route construction applies a uniform probabilistic selection rule to all customers, with no mechanism to reserve route capacity for urgent service requests during construction. When must-serve constraints are handled at all, they are delegated to post-hoc repair, where insertion slots may already be scarce. The priority-first construction rule introduced in Stage 2 of our framework addresses this gap directly: whenever any priority customer remains feasible, the candidate set is restricted to priority customers only, so that must-serve constraints are enforced structurally during construction rather than corrected after the fact.

2.4. Priority and Service-Class Modeling in VRP

Priority and service-class differentiation in VRP has received limited but meaningful attention, primarily in single-depot settings and adjacent routing domains. Four main mechanisms appear in the literature. Time-window tightening assigns narrow delivery windows to high-priority customers [33]. It requires translating urgency into a numerical time commitment and does not constitute a hard must-serve guarantee: a solution that misses a tight window is penalised but not rejected. Weighted objectives penalise late or missed service through cost coefficients [12]. These formulations are structurally unable to prevent priority-infeasible solutions: a route that leaves a priority customer unserved may still be accepted when the associated travel-time saving is large enough. Quantity-based priority encodes urgency through delivery rewards rather than scheduling: Wu et al. [34] formulate a workload-consistent VRP under demand uncertainty in which each demand site i carries a weight coefficient $e_i$ that influences how much supply it receives, so priority is expressed as distributional fairness rather than service order. Tardiness-driven priority embeds urgency in the search procedure rather than in the model: Bock [35] maintains a min-heap of partial solutions inside a branch-and-bound algorithm; at each insertion, a heuristic completion repeatedly appends the unserviced order with the highest priority ${\rho }_i$, which is derived from the earliest completion time attainable by some vehicle. The priority rule there acts as a tie-breaking device for fast feasibility, not as a structural admissibility constraint over the solution space. Hard sequencing constraints, which force priority customers to be inserted before regular ones within each route, are the mechanism most closely aligned with our formulation, but their application in the multi-depot context has not, to our knowledge, been combined with a lexicographic admissibility rule that filters entire solutions out.

In field-service and emergency logistics, service urgency is operationally prevalent but rarely modeled as a formal routing constraint. Maintenance dispatch and telecommunications repair operations classify customers by response-time commitment [14], yet the routing literature has not translated this classification into a structural priority constraint within an MDVRP framework. Emergency vehicle routing models do enforce hard response-time deadlines, but these are single-depot or network-flow formulations that do not address the assignment–routing interdependence of the multi-depot setting.

Our positioning relative to these threads is therefore narrower and more precise than a generic claim of novelty would suggest. Quantity-based priority [34] answers a different question, namely how much to deliver under demand uncertainty in a single-depot consistent VRP, with priority entering as a hierarchical reward in the objective rather than as a feasibility rule. Tardiness-driven priority [35] concerns single-vehicle routing through connected service areas and embeds urgency as a heuristic completion device inside a branch-and-bound algorithm rather than as a constraint over the feasible region. Time-window-based priority encodings [33] are most often deployed in multi-depot settings, but a missed window is penalised, not rejected. To the best of our knowledge, the MDVRP-PR introduced in this paper is the first MDVRP formulation in which priority is enforced as a structural admissibility constraint defining the feasible region (lexicographic feasibility): it combines this hard admissibility rule on priority customers with a topology-aware depot assignment and a cross-depot repair operator within a single operational pipeline. The resulting guarantee, that solutions leaving any priority customer unserved are rejected regardless of the travel-time savings they might offer, is motivated by the field-service operations of Algérie Telecom, where routing a vehicle to a residential customer before attending to a nearby government agency constitutes an administrative service-level breach whose consequences cannot be traded off against travel-time savings.

Table 1. Summary of related MDVRP literature. PC = priority-customer treatment (No = none, Quantity = priority via delivered amount, Tardiness = priority as search-time tie-breaker, Hard lex.  = lexicographic admissibility constraint); Cluster = clustering strategy (ND = nearest-depot, Km = k-medoids/k-means, Sp = spatial sectorization, DB+H = DBSCAN with capacity-constrained hierarchical merging, Dyn.  = dynamic within-loop route decomposition, − = no clustering phase, · = decomposition without spatial clustering); XD-corr. = cross-depot correction after initial clustering; Pri.-cons. = construction rule that structurally enforces priority, during route building (Heur.  = priority used only as a heuristic completion device, not as a hard rule).

Reference	Variant	PC	Cluster	XD-corr.	Pri.-cons.	Algorithm
Classical MDVRP and benchmark methods
Renaud et al. [16]	MDVRP	No	−	No	No	Tabu Search (FIND)
Cordeau et al. [4]	MDVRP/PVRP	No	−	No	No	Unified Tabu Search
Ho et al. [17]	MDVRP	No	−	No	No	Hybrid GA
Vidal et al. [5]	MDVRP/PVRP	No	−	No	No	HGSADC
Azad et al. [18]	MD Periodic+Interdep.	No	−	No	No	Memetic Algorithm
Mancini [19]	MD Multi-Period	No	−	No	No	ALNS Matheuristic
Lim et al. [20]	MD Split-Delivery	No	−	No	No	MIP + GA
Chen and Yang [21]	MDVRP + Traffic Eq.	No	−	No	No	Bilevel GA
Clustering-based decomposition
Yu et al. [27]	Dynamic MDVRP	No	ND	No	No	Improved ACO
Xu et al. [28]	Dynamic MDVRP	No	ND	No	No	Hybrid ACO
Gu et al. [22]	MDVRP	No	ND	No	No	ABC + Coevolution
Tao et al. [23]	MD-CVRP-OSA	No	·	No	No	VNS (7 operators)
Wang et al. [24]	Collab. MD Dynamic	No	Km	No	No	k-Medoids + Hybrid
Wang et al. [25]	Collab. MD Dynamic	No	Km	No	No	k-Medoids + Hybrid
ACO and MMAS-based methods
Ting and Chen [31]	MDVRPTW	No	·	No	No	MACS-SA (dual pheromone)
Niu et al. [32]	MD Cumul. CVRP	No	ND	No	No	MMAS + Decomposition
Stodola and Nohel [26]	MDVRP	No	Sp	No	No	AACO-NC (entropy PH)
Recent priority and decomposition formulations
Wu et al. [34]	VRP + workload/uncert.	Quantity	−	No	No	Branch-and-cut
Bock [35]	Connected service areas	Tardiness	·	No	Heur.	Branch-and-bound
Santini et al. [29]	VRP (dyn. decomp.)	No	Dyn.	N/A	No	Within-loop decomposition
This work
MDVRP-PR (Ours)	MD + Priority Req.	Hard lex.	DB+H	Yes	Yes	Our Method

Table 1. Summary of related MDVRP literature. PC = priority-customer treatment (No = none, Quantity = priority via delivered amount, Tardiness = priority as search-time tie-breaker, Hard lex.  = lexicographic admissibility constraint); Cluster = clustering strategy (ND = nearest-depot, Km = k-medoids/k-means, Sp = spatial sectorization, DB+H = DBSCAN with capacity-constrained hierarchical merging, Dyn.  = dynamic within-loop route decomposition, − = no clustering phase, · = decomposition without spatial clustering); XD-corr. = cross-depot correction after initial clustering; Pri.-cons. = construction rule that structurally enforces priority, during route building (Heur.  = priority used only as a heuristic completion device, not as a hard rule).

Reference	Variant	PC	Cluster	XD-corr.	Pri.-cons.	Algorithm
Classical MDVRP and benchmark methods
Renaud et al. [16]	MDVRP	No	−	No	No	Tabu Search (FIND)
Cordeau et al. [4]	MDVRP/PVRP	No	−	No	No	Unified Tabu Search
Ho et al. [17]	MDVRP	No	−	No	No	Hybrid GA
Vidal et al. [5]	MDVRP/PVRP	No	−	No	No	HGSADC
Azad et al. [18]	MD Periodic+Interdep.	No	−	No	No	Memetic Algorithm
Mancini [19]	MD Multi-Period	No	−	No	No	ALNS Matheuristic
Lim et al. [20]	MD Split-Delivery	No	−	No	No	MIP + GA
Chen and Yang [21]	MDVRP + Traffic Eq.	No	−	No	No	Bilevel GA
Clustering-based decomposition
Yu et al. [27]	Dynamic MDVRP	No	ND	No	No	Improved ACO
Xu et al. [28]	Dynamic MDVRP	No	ND	No	No	Hybrid ACO
Gu et al. [22]	MDVRP	No	ND	No	No	ABC + Coevolution
Tao et al. [23]	MD-CVRP-OSA	No	·	No	No	VNS (7 operators)
Wang et al. [24]	Collab. MD Dynamic	No	Km	No	No	k-Medoids + Hybrid
Wang et al. [25]	Collab. MD Dynamic	No	Km	No	No	k-Medoids + Hybrid
ACO and MMAS-based methods
Ting and Chen [31]	MDVRPTW	No	·	No	No	MACS-SA (dual pheromone)
Niu et al. [32]	MD Cumul. CVRP	No	ND	No	No	MMAS + Decomposition
Stodola and Nohel [26]	MDVRP	No	Sp	No	No	AACO-NC (entropy PH)
Recent priority and decomposition formulations
Wu et al. [34]	VRP + workload/uncert.	Quantity	−	No	No	Branch-and-cut
Bock [35]	Connected service areas	Tardiness	·	No	Heur.	Branch-and-bound
Santini et al. [29]	VRP (dyn. decomp.)	No	Dyn.	N/A	No	Within-loop decomposition
This work
MDVRP-PR (Ours)	MD + Priority Req.	Hard lex.	DB+H	Yes	Yes	Our Method

3. Problem Formulation

3.1. Problem Definition

We consider a Multi-Depot Vehicle Routing Problem with Priority Requests (MDVRP-PR) motivated by a field-service operation in the province of Guelma, Algeria. A set $\mathcal{N}=\{1,\dots ,N\}$ of geographically dispersed customers must be served by vehicles dispatched from K depots (technical centers), indexed by $\mathcal{K}=\{1,\dots ,K\}$. Depot k operates a homogeneous fleet of $m_k$ vehicles over a common planning horizon $[T_s,T_e]$, yielding a maximum route duration of $T_{max}=T_e-T_s$. Each customer $i\in \mathcal{N}$ requires a service of duration $s_i>0$. A designated subset $\mathcal{P}\subseteq \mathcal{N}$ of priority customers must be served in every feasible solution.

The road-network data, distances $d_{ij}$ (km), and travel times $t_{ij}$ (min), are obtained from the Google Maps Distance Matrix API: https://developers.google.com/maps/documentation/distance-matrix/overview (accessed on 1 December 2025). Both matrices are asymmetric ($t_{ij}\ne t_{ji}$ in general), reflecting one-way streets, turning restrictions, and other real-world road characteristics.

Because fleet capacity and working-hour limits may preclude serving all customers, the problem admits partial coverage: non-priority customers may remain unserved. The objective is therefore lexicographic: first, guarantee service to every priority customer; second, maximize the number of served customers (coverage); third, among solutions with equal coverage, minimize the total route duration (travel plus service time). The decision variables determine (i) an assignment of customers to depots, (ii) a partitioning of each depot’s customers into vehicle routes, and (iii) the visiting sequence within each route.

3.2. Notation

The notation used throughout the paper is summarized in

Table 2. Sets, parameters, and decision variables.

Sets
$\mathcal{K}=\{1,\dots ,K\}$	Set of depots; $o_k$ denotes the location of depot k
$\mathcal{N}=\{1,\dots ,N\}$	Set of customers
$\mathcal{P}\subseteq \mathcal{N}$	Subset of priority (mandatory) customers
${\mathcal{V}}_k=\{1,\dots ,m_k\}$	Set of vehicles stationed at depot k
${\mathcal{N}}_k$	Subset of customers assigned to depot k
$\mathcal{S}\subseteq \mathcal{N}$	Set of served customers ($\mathcal{P}\subseteq \mathcal{S}$)
${\mathcal{L}}_k={\mathcal{N}}_k\cup \left\{o_k\right\}$	Node set for depot k (customers plus depot)
Parameters
$t_{ij}$	Travel time from node i to node j (min); asymmetric
$d_{ij}$	Distance from node i to node j (km); asymmetric
$s_i$	Service duration at customer i (min); $s_{o_k}=0\forall k$
$T_{max}$	Maximum route duration (min)
$m_k$	Fleet size at depot k
Decision variables
$x_{ij}^{kv}\in \{0,1\}$	1 if vehicle v of depot k traverses arc $(i,j)$; 0 otherwise
$y_i^k\in \{0,1\}$	1 if customer i is assigned to depot k; 0 otherwise

.

3.3. Mathematical Model

The MDVRP-PR is formulated with a lexicographic objective that reflects the three-level priority structure of the problem. Let $\mathcal{S}=\{i\in \mathcal{N}:{\sum }_k{y}_i^k=1\}$ denote the set of served customers. The objective is:

$$\text{lex-opt}(\underset{\min :\mathrm{unserved}\mathrm{priority}}{\underbrace{|\mathcal{P}\setminus \mathcal{S}|}},\underset{\min (\mathrm{i}.\mathrm{e}.,\max \mathrm{coverage})}{\underbrace{-\left|\mathcal{S}\right|}},\underset{\min :\mathrm{total}\mathrm{time}}{\underbrace{Z}})$$

(1)

where the total route duration is defined as:

$$Z=\sum _{k\in \mathcal{K}}\sum _{v\in {\mathcal{V}}_k}\sum _{i\in {\mathcal{L}}_k}\sum _{j\in {\mathcal{L}}_k}{x}_{ij}^{kv}\left(t_{ij}+s_j\right)$$

(2)

That is, a solution with fewer unserved priority customers is always preferred; among solutions serving all priority customers, one with higher total coverage dominates; and among coverage-equivalent solutions, the one with the smallest total duration is selected.

The model is subject to the following constraints:

$$\begin{array}{cccc}\hfill \sum _{k\in \mathcal{K}}{y}_i^k& \le 1,\hfill & \hfill & \forall i\in \mathcal{N}\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill \sum _{k\in \mathcal{K}}{y}_i^k& =1,\hfill & & \forall i\in \mathcal{P}\hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill \sum _{v\in {\mathcal{V}}_k}\sum _{j\in {\mathcal{L}}_k}{x}_{ij}^{kv}& =y_i^k,\hfill & & \forall i\in \mathcal{N},\forall k\in \mathcal{K}\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill \sum _{i\in {\mathcal{L}}_k}{x}_{ij}^{kv}& =\sum _{i\in {\mathcal{L}}_k}{x}_{ji}^{kv},\hfill & & \forall j\in {\mathcal{N}}_k,\forall v\in {\mathcal{V}}_k,\forall k\in \mathcal{K}\hfill \end{array}$$

(6)

$$\begin{array}{cccc}\hfill \sum _{j\in {\mathcal{N}}_k}{x}_{o_{k}j}^{kv}& \le 1,\hfill & & \forall v\in {\mathcal{V}}_k,\forall k\in \mathcal{K}\hfill \end{array}$$

(7)

$$\begin{array}{cccc}\hfill \sum _{i\in {\mathcal{N}}_k}{x}_{i{o}_k}^{kv}& \le 1,\hfill & & \forall v\in {\mathcal{V}}_k,\forall k\in \mathcal{K}\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill \sum _{i\in {\mathcal{L}}_k}\sum _{j\in {\mathcal{L}}_k}{x}_{ij}^{kv}\left(t_{ij}+s_j\right)& \le T_{max},\hfill & & \forall v\in {\mathcal{V}}_k,\forall k\in \mathcal{K}\hfill \end{array}$$

(9)

$$\begin{array}{cccc}\hfill \sum _{v\in {\mathcal{V}}_k}\sum _{j\in {\mathcal{N}}_k}{x}_{o_{k}j}^{kv}& \le m_k,\hfill & & \forall k\in \mathcal{K}\hfill \end{array}$$

(10)

$$\begin{array}{cccc}\hfill \sum _{i\in S}\sum _{j\in S}{x}_{ij}^{kv}& \le \left|S\right|-1,\hfill & & \forall S\subseteq {\mathcal{N}}_k,\left|S\right|\ge 2,\forall v\in {\mathcal{V}}_k,\forall k\in \mathcal{K}\hfill \end{array}$$

(11)

$$\begin{array}{cccc}\hfill x_{ij}^{kv}\in \{0,1\},y_i^k& \in \{0,1\}\hfill & & \hfill \end{array}$$

(12)

Constraints (3) ensure that each customer is assigned to at most one depot. Constraint (3) is an inequality, permitting non-priority customers to remain unserved when route-duration limits preclude full coverage; by contrast, constraints (4) enforce mandatory service for all priority customers via equality. Together, these two constraints encode the first level of the lexicographic objective: any feasible solution must serve every priority customer, while non-priority customers are served to the extent permitted by the fleet. Constraints (5) link the assignment and routing variables: a customer is visited if and only if it is assigned to that depot. Flow conservation at customer nodes is imposed by (6). Constraints (7) and (8) guarantee that each vehicle performs at most one route, departing from and returning to its home depot. The working-hour limit is enforced by (9), bounding the total duration of each route to $T_{max}$. Fleet-size constraints (10) ensure that no depot dispatches more vehicles than available. Sub-tour elimination constraints (11) prevent disconnected cycles. Finally, (12) imposes binary integrality.

Remark 1.

The lexicographic structure (1) reflects the operational reality at Algérie Telecom: failing to serve a priority customer (a government agency or critical enterprise) is categorically worse than leaving a regular residential customer unserved, and leaving any customer unserved is worse than a modest increase in travel time. This hierarchy is enforced throughout the solution framework: the MMAS uses lexicographic solution selection (Section 4.2), and the post-optimization stage rejects any move that increases the number of unserved priority customers (Section 4.3).

3.4. Problem Characteristics

The instance under study exhibits several features that distinguish it from standard MDVRP benchmarks and warrant a tailored solution approach.

(i)

Asymmetric, real-world travel data. Unlike Euclidean or symmetric benchmark instances, the travel-time and distance matrices are fully asymmetric, derived from actual road-network queries. The triangle inequality does not hold in general; consequently, route reversal yields a different cost, and standard symmetric neighbourhood operators (e.g., 2-opt) must account for directionality.

(ii)

Heterogeneous fleet distribution. The depots possess unequal fleet capacities, creating an inherently imbalanced workload-distribution problem, further complicating the customer-to-depot assignment.

(iii)

Duration-constrained routes. Vehicles are uncapacitated in terms of load. The sole binding constraint on route length is the working day length.

(iv)

Heterogeneous service times. Service durations vary considerably from one customer to another, and constitute a substantial share of total route time. This variability makes service time a primary factor in feasibility assessment and objective evaluation, unlike many benchmark problems where it is assumed uniform or negligible.

(v)

Priority customers. A configurable proportion of customers may be designated as priority ($\mathcal{P}$). Their mandatory-service requirement (constraint (4)) constitutes the first level of the lexicographic objective (1), and must be satisfied before coverage maximization or time minimization are considered.

(vi)

Non-uniform spatial distribution. The N customers span numerous municipalities, with inter-customer distances ranging from less than 1 km (urban clusters) to over 110 km (remote areas). This dispersion produces dense urban pockets interspersed with sparse rural regions, a structure that purely distance-based assignment heuristics may fail to exploit.

4. Methodology

This section details each stage of the proposed framework. Figure 1 illustrates the overall pipeline.

4.1. Stage 1: Hybrid DBSCAN-Hierarchical Clustering

Standard clustering methods (K-Means, K-Medoids) optimize geometric compactness, producing spherical clusters that may misalign with the road network. The proposed method instead uses road-distance-based density clustering to produce topology-aware depot assignments. Algorithm 1 summarizes the seven-step procedure.

The medoid of a cluster C (line 2) minimizes total intra-cluster road distance:

$$m_C=arg\underset{c\in C}{min}\sum _{c^{\prime }\in C}d(c,c^{\prime })$$

(13)

where $d(c,c^{\prime })$ denotes the road distance (km) between customers c and $c^{\prime }$. The entire clustering stage is fully deterministic: DBSCAN, the two-phase geographic merging, and the capacity rebalancing have no random components, ensuring reproducible depot assignments.

Algorithm 1 Hybrid DBSCAN-Hierarchical Clustering

Require: Customers $\mathcal{N}$, depots $\mathcal{K}$, road-distance matrix D, vehicles per depot $\left\{m_k\right\}$ Ensure: Depot assignments ${\left\{{\mathcal{N}}_k\right\}}_{k\in \mathcal{K}}$   1: Apply DBSCAN [36] on D ($\epsilon$ = 10.88 km, $\mathit{minPts}$ = 2) → micro-clusters $\{C_1,\dots ,C_q\}$ + noise set   2: Compute medoid $m_C$ for each micro-cluster via Equation (13)   3: Assign each noise point to the micro-cluster chosen by k-NN majority vote (k = 3) over medoids   4: Build medoid-to-medoid distance matrix $\overline{D}\in {\mathbb{R}}^{q\times q}$ from D   5: Set expected capacity ${\widehat{n}}_k\leftarrow \lceil (m_k/{\sum }_{k^{\prime }}{m}_{k^{\prime }})·N\rceil$ for each depot   6: Phase 1 (E–W axis). Identify the westernmost depot $d_{\mathrm{W}}$ and easternmost depot $d_{\mathrm{E}}$ by depot longitude. Sort micro-clusters by medoid longitude. Round-robin assign from the western end toward $d_{\mathrm{W}}$ and from the eastern end toward $d_{\mathrm{E}}$ until the total expected route time at each depot reaches its capacity bound $m_k·T_{max}$   7: Phase 2 (N–S axis). Among the remaining $K-2$ depots, identify the northernmost $d_{\mathrm{N}}$ and southernmost $d_{\mathrm{S}}$ by latitude. Assign the remaining micro-clusters by medoid latitude until depot capacities are met   8: if any depot k has $\left|{\mathcal{N}}_k\right|>{\widehat{n}}_k(1+\delta )$ then               ▹$\delta =0.20$   9:       Specificity pass: re-assign customers sorted by ${\varphi }_i=t(c_i,d_{2\mathrm{nd}})/t(c_i,d_{\mathrm{nearest}})$; low-specificity customers assigned last 10:       Overflow pass: move excess customers to nearest under-capacity depot 11:       Swap refinement: relocate customers $>20\%$ closer to an alternative depot 12: end if 13: return $\left\{{\mathcal{N}}_k\right\}$

Scope of the Geographic Merging Step

The two-phase merging procedure (Algorithm 1, lines 6–7) uses geographic priors specific to the four-depot field-service configuration of Algérie Telecom in the Guelma region: the depots separate cleanly along an east–west axis (TC3 west, TC4 east) and a north–south axis (TC1, TC2), and the operational service area maps to a roughly rectangular geographic envelope. Phase 1 exploits this layout by saturating the two extreme depots first, since their service territories are the most constrained by geography; Phase 2 then resolves the interior. This is a deliberate operational design choice rather than a domain-general clustering routine: applying the pipeline to MDVRP instances with arbitrary depot counts ($K\ne 4$), depot layouts that do not separate along orthogonal geographic axes, or service areas with strongly non-convex boundaries, would require substituting Algorithm 1, lines 6–7, with a generic capacity-constrained agglomerative merger on $\overline{D}$. The remaining stages of the pipeline (Section 4.2 and Section 4.3) are independent of this choice and are domain-general; we return to this scope distinction in the Limitations.

4.2. Stage 2: Max-Min Ant System (MMAS)

Each depot subproblem is solved independently using the Max-Min Ant System [30]. The MMAS operates within pheromone bounds $[{\tau }_{min},{\tau }_{max}]$ to prevent premature convergence while preserving learned solution structure. All pheromone trails are initialized to ${\tau }_{max}$, the standard MMAS choice that maximizes early exploration [30]. Algorithm 2 gives the overall iteration loop; each phase is detailed below.

Algorithm 2 Enhanced MMAS for depot subproblem k

Require: Customers ${\mathcal{N}}_k$, depot $o_k$, vehicles $m_k$, parameters $\alpha ,\beta ,\rho ,Q,I_{max}$ Ensure: Best solution $S^{*}$   1: Initialize ${\tau }_{ij}\leftarrow {\tau }_{max}$ for all arcs; $S^{*}\leftarrow \mathrm{NULL}$; $\mathit{stag}\leftarrow 0$   2: for  $\mathit{iter}=1,\dots ,I_{max}$  do   3:        Construct $n_k$ solutions via Equations (14) and (15); if priority-aware, restrict $\mathcal{F}$ per Equation (16)   4:        Sort solutions by lexicographic criterion (17)   5:        Apply 2-opt to iteration-best $S^{\mathrm{ib}}$; apply cheapest-insertion repair to $S^{\mathrm{ib}}$   6:        $S^{\mathrm{ib}}\leftarrow$ best solution in current iteration   7:        if $S^{\mathrm{ib}}{\succ }_{\mathrm{lex}}{S}^{*}$ then   8:             $S^{*}\leftarrow S^{\mathrm{ib}}$; $\mathit{stag}\leftarrow 0$   9:        else 10:             $\mathit{stag}\leftarrow \mathit{stag}+1$ 11:        end if 12:        Evaporate and deposit pheromone via Equation (18); depositing solution follows ib → gb schedule 13:        Clamp ${\tau }_{ij}\in [{\tau }_{min},{\tau }_{max}]$ per Equation (19) 14:        if $\mathit{stag}\ge \lfloor n_k/2\rfloor$ then 15:             PTS: ${\tau }_{ij}\leftarrow {\tau }_{ij}+0.3({\tau }_{max}-{\tau }_{ij})$; $\mathit{stag}\leftarrow 0$ 16:        end if 17: end for 18: Apply final repair to $S^{*}$ 19: return  $S^{*}$

4.2.1. Construction Phase

Each of the $m=n_k$ ants (one per customer assigned to the depot) builds a solution by sequentially assigning customers to vehicle routes. At each step, the ant selects the next customer $c_j$ from the set of feasible (duration-respecting) candidates with probability:

$$p_{ij}={\displaystyle \frac{{\left[{\tau }_{ij}\right]}^{\alpha }{\left[{\eta }_{ij}\right]}^{\beta }}{{\sum }_{l\in \mathcal{F}}{\left[{\tau }_{il}\right]}^{\alpha }{\left[{\eta }_{il}\right]}^{\beta }}}$$

(14)

where ${\tau }_{ij}$ is pheromone, ${\eta }_{ij}$ is heuristic information, and $\mathcal{F}$ is the feasible set. A customer is feasible if inserting it and returning to the depot does not violate the route-duration constraint (9). When all vehicles are exhausted, remaining customers are marked as unserved.

Savings-Based Heuristic

Instead of the standard inverse-distance visibility ${\eta }_{ij}=1/t_{ij}$, we use a savings-aware heuristic inspired by Clarke and Wright [37]:

$${\eta }_{ij}={\displaystyle \frac{max(s_{ij},ϵ)}{t_{ij}}},s_{ij}=t_{i,o_k}+t_{o_k,j}-t_{ij}$$

(15)

where $ϵ=0.1$ prevents zero visibility. This biases ants toward edges that save the most depot-detour, naturally producing tighter route loops. For depot ↔ customer edges, the standard ${\eta }_{ij}=1/t_{ij}$ is used.

Priority-First Construction Rule

When priority-aware mode is active, if any feasible customer in the candidate set is a priority customer, the set $\mathcal{F}$ is restricted to priority customers only:

$${\mathcal{F}}^{\prime }=\left\{\begin{array}{cc}\{c\in \mathcal{F}:c\in \mathcal{P}\}\hfill & \mathrm{if}\mathcal{F}\cap \mathcal{P}\ne Ø\hfill \\ \mathcal{F}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(16)

This ensures priority customers are served during construction rather than deferred to the repair phase, where insertion slots may be scarce.

4.2.2. Local Search

Following Dorigo and Stützle [38], 2-opt local search is applied to the iteration-best solution before pheromone deposit. The 2-opt uses best-improvement with prefix-sum acceleration for asymmetric travel time evaluation: for each pair $(i,j)$, the cost of reversing the segment $[i,j]$ is computed in $O\left(1\right)$ via precomputed forward and reverse prefix sums, yielding an $O\left(n_k^2\right)$-per-pass procedure that repeats until no improving swap is found.

4.2.3. Repair Phase

Cheapest-insertion repair is applied to the iteration-best solution after 2-opt (line 5 of Algorithm 2), following the same principle of concentrating expensive post-processing on the most promising candidate [39]. Each unserved customer is tested at every position in every route; the feasible insertion with the smallest travel-time increase is selected. When priority-aware, priority customers are processed first, giving them access to the most favorable slots.

4.2.4. Solution Selection

When priority-aware, solutions are compared according to the lexicographic objective (1):

$$\mathrm{unserved}\mathrm{priority}\succ \mathrm{unserved}\mathrm{customers}\succ \mathrm{total}\mathrm{time}$$

(17)

A solution with fewer unserved priority customers is always preferred; among solutions with equal priority feasibility, higher coverage (fewer total unserved) dominates; total time breaks remaining ties. In priority-blind mode, solutions are ranked by total time only.

4.2.5. Pheromone Update

After each iteration, pheromone trails are evaporated and a single solution deposits pheromone:

$${\tau }_{ij}\leftarrow (1-\rho ){\tau }_{ij}+\Delta {\tau }_{ij}^{\mathrm{best}},\Delta {\tau }_{ij}^{\mathrm{best}}={\displaystyle \frac{Q}{T^{\mathrm{best}}}}$$

(18)

where $\rho =0.02$ is the evaporation rate, $Q=100$ is the deposit constant, and $T^{\mathrm{best}}$ is the total time of the depositing solution. Following the original MMAS schedule [30], the identity of the depositing solution transitions gradually from iteration-best to global-best as the search progresses:

Progress (% of Iterations)	Depositing Solution
$[0,25)$	Iteration-best (always)
$[25,50)$	Iteration-best $\frac{2}{3}$, global-best $\frac{1}{3}$
$[50,75)$	Iteration-best $\frac{1}{3}$, global-best $\frac{2}{3}$
$[75,100]$	Global-best (always)

This schedule maximizes exploration early on and gradually shifts toward exploitation of the best-known solution. All trails are clamped to $[{\tau }_{min},{\tau }_{max}]$, where:

$${\tau }_{max}={\displaystyle \frac{1}{\rho ·T_{\mathrm{gb}}}},{\tau }_{min}={\tau }_{max}·{\displaystyle \frac{1-p_{\mathrm{best}}^{1/n_k}}{(n_k/2-1)p_{\mathrm{best}}^{1/n_k}}}$$

(19)

with $p_{\mathrm{best}}=0.05$ following Stützle and Hoos [30]. When no improvement is found for $\lfloor n_k/2\rfloor$ consecutive iterations, Pheromone Trail Smoothing (PTS) is triggered: ${\tau }_{ij}\leftarrow {\tau }_{ij}+0.3·({\tau }_{max}-{\tau }_{ij})$, pulling all trails toward ${\tau }_{max}$ to restore exploration.

4.3. Stage 3: Cross-Depot Post-Optimization

After the per-depot MMAS, a post-optimization stage refines the solution at the multi-depot level. It consists of two operators applied sequentially.

4.3.1. Boundary Relocate

Boundary customers are customers who are approximately equidistant to multiple depots, and they may have been misassigned by the clustering. A customer $c_i$ is classified as boundary if:

$${\displaystyle \frac{t(d_{\mathrm{alt}}^{*},c_i)}{t(d_{\mathrm{assigned}},c_i)}}<\theta \mathrm{and}t(d_{\mathrm{assigned}},c_i)-t(d_{\mathrm{alt}}^{*},c_i)>{\Delta }_{min}$$

(20)

where $\theta =1.25$ and ${\Delta }_{min}=5$ min. The ratio threshold $\theta =1.25$ flags customers whose alternative depot is within 25% of the assigned depot’s travel time, a moderate criterion that captures genuinely contested assignments without flooding the candidate pool. The absolute threshold ${\Delta }_{min}=5$ min filters out relocations whose net saving is negligible relative to the mean depot–customer travel time in the study area (≈40 min). Algorithm 3 details the iterative relocation procedure.

Algorithm 3 Boundary Relocate Post-Optimization

Require: Per-depot solutions $\left\{S_k\right\}$, parameters $\theta$, ${\Delta }_{min}$ Ensure: Improved solutions $\left\{S_k\right\}$   1: Identify boundary customers B via Equation (20)   2: $\mathit{no}\_\mathit{imp}\leftarrow 0$; $\mathit{iter}\leftarrow 0$; $S^{*}\leftarrow \left\{S_k\right\}$   3: while  $\mathit{no}\_\mathit{imp}<50$  and  $\mathit{iter}<500$  do   4:      if $\mathit{iter}mod10=0$ then re-identify B from current solutions   5:      end if   6:      $\mathit{candidates}\leftarrow Ø$   7:      for all boundary customer $c_i\in B$ currently served do   8:            $\mathit{savings}\leftarrow$ travel time saved by removing $c_i$ from its route   9:            $\mathit{cost}\leftarrow$ cheapest feasible insertion of $c_i$ into any alternative-depot route 10:            if $\mathit{cost}-\mathit{savings}<0$ and insertion respects duration constraint then 11:                 Add $(c_i,\mathit{cost}-\mathit{savings})$ to $\mathit{candidates}$ 12:            end if 13:      end for 14:      if $\mathit{candidates}=Ø$ then 15:            $\mathit{no}\_\mathit{imp}\leftarrow \mathit{no}\_\mathit{imp}+1$; $\mathit{iter}\leftarrow \mathit{iter}+1$; continue 16:      end if 17:      Select move from top-5 candidates by net change (randomized) 18:      Execute relocation; apply 2-opt and Or-opt to affected routes 19:      if total time improved and priority unserved ≤ current and all routes feasible then 20:            Accept; $\mathit{no}\_\mathit{imp}\leftarrow 0$; update $S^{*}$ if best overall 21:      else 22:            Reject; $\mathit{no}\_\mathit{imp}\leftarrow \mathit{no}\_\mathit{imp}+1$ 23:      end if 24:      $\mathit{iter}\leftarrow \mathit{iter}+1$ 25: end while 26: return  $S^{*}$

Moves that would increase the number of unserved priority customers are rejected (line 17), preserving priority feasibility throughout the post-optimization.

Relation to Classical Inter-Depot Operators

Inter-route exchange and relocate operators are standard in MDVRP metaheuristics. Cordeau et al. [4] and Renaud et al. [16] move single customers between any two routes (relocate) and swap pairs of customers (exchange), accepting any feasible improving move. More recent VNS schemes [40] fix a neighbourhood ladder (relocate, exchange, 2-opt*, Or-opt) and iterate over all candidate pairs. These operators are unfiltered: every (customer, target route) pair is considered in each pass, giving an $O(n·R)$ candidate set per iteration with n customers and R routes. Boundary Relocate is an instance of this family—single-customer cross-depot relocate with strict-improvement acceptance—but adds a topological prefilter: a customer is admitted to the candidate set only if Equation (20) holds, that is, if its travel-time ratio between the alternative and assigned depots is below $\theta$ and the absolute gap exceeds ${\Delta }_{min}$. The filter selects exactly the customers that lie on the geographic boundary between two depot clusters, where reassignment is most likely to pay off, and prunes interior customers whose home depot dominates by construction. The transformation itself is otherwise unchanged from a classical relocate; the contribution is in the filter, not the move.

Table 3. Comparison of inter-depot reassignment operators.

	Cordeau et al. Relocate [4]	Kytöjoki et al. VNS ${\mathit{N}}_1$ [40]	Boundary Relocate (Ours)
Move granularity	Single customer	Single customer/segment	Single customer
Candidate scope	All cust. × all routes	All cust. × all routes	Boundary cust. only
Filter criterion	None (full scan)	None (full scan)	$t(d^{\prime },c)/t(d,c)$ < $\theta$ and $t(d,c)-t(d^{\prime },c)$ > ${\Delta }_{min}$
Acceptance rule	Strict improvement	Strict/first-improvement	Strict improvement, priority feasibility preserved
Per-pass complexity	$O(n·R)$	$O(n·R)$	$O\left(\right|B|·K)$, $\left|B\right|\ll n$

summarises the comparison.

4.3.2. Cross-Depot Repair

Globally unserved customers (those not served by any depot) are re-inserted using cheapest insertion across all depot routes, not only the depot they were originally assigned to. Priority customers are processed first. For each candidate, every position in every route across all depots is evaluated, and the feasible insertion with the smallest travel-time increase is selected. Modified routes are improved with 2-opt and Or-opt.

4.4. Priority Integration

Priority constraints are enforced through three complementary mechanisms that operate at different stages of the pipeline:

• Construction-phase restriction (Stage 2): The candidate set is restricted to feasible priority customers whenever any are available, as defined in Equation (16).
• Lexicographic solution selection (Stage 2): Solutions are ranked by unserved priority count before total unserved count and total time, as defined in Equation (17).
• Priority-first repair ordering (Stages 2 and 3): In both the MMAS repair and the cross-depot repair, priority customers are inserted before non-priority customers, giving them access to the most favorable remaining route slots.

These mechanisms are jointly termed the priority-aware framework. Their combined contribution is evaluated in Experiment 3b (Section 5.5).

5. Numerical Experiments

This section describes the benchmark instances and the design of six experiments (with Experiment 3 expanded into five sub-experiments, 3a through 3e) conducted to evaluate the proposed framework. Results are presented in Section 6.

5.1. Problem Instances

The approach is evaluated on a real-world case study involving a utility company in northeastern Algeria. The problem consists of scheduling daily maintenance visits across 47 municipalities served by four depots, each with a fixed vehicle fleet and an 8-h operating window (480 min). Customer locations, service-time requirements, and pre-existing depot/team assignments were provided by the company. Pairwise road distances (km) and travel times (min) were obtained via the Google Maps Distance Matrix API, capturing the actual road network rather than Euclidean approximations.

Table 4. Instance characteristics.

Category	Parameter	Value
Infrastructure	Depots	4
	Vehicles	13
	Fleet per depot	2–5
Customers	Small (S)	55
	Medium (M)	90
	Large (L)	150
Operations	Service time (min)	15–45 (mean ≈ 30)
	Time horizon (min)	480
	Inter-customer distance (km)	0.2–112.7 (mean ≈ 40)
	Inter-customer travel time (min)	1.0–139.6 (mean ≈ 52)

summarizes the instance characteristics. Three dataset sizes were constructed by progressive sampling to test scalability under increasing fleet utilization. Service times average approximately 30 min per customer. Each customer record includes a pre-assigned depot and technician team, used in Experiments 4 and 5.

The vehicle fleet is distributed unevenly across depots (ranging from 2 to 5 vehicles per depot), reflecting the company’s actual operational structure.

Metrics and Protocol

Solutions are compared using the lexicographic objective (1): a solution is preferred if it serves more priority customers; ties are broken by overall coverage, then by total time (travel + service). Auxiliary metrics reported include travel time, total distance, customer coverage (%), and priority feasibility (fraction of seeds in which all priority customers are served). Each experiment is replicated over 10 random seeds; results are reported as mean ± standard deviation.

The MMAS optimizer uses the balanced preset ($\alpha$ = 1, $\beta$ = 2, $\rho$ = 0.02, $m=n_k$ ants, 100 iterations) throughout, where $n_k$ is the number of customers assigned to depot k. The parameter values follow the canonical recommendations of Stützle and Hoos [30], who established $\alpha$ = 1 and $\beta$ = 2 as the standard pheromone–heuristic balance for MMAS across routing problems, and $\rho$ = 0.02 as a low evaporation rate suited to the bounded $[{\tau }_{min},{\tau }_{max}]$ update scheme that prevents premature convergence without requiring per-instance tuning. The number of ants is set to m = $n_k$, matching the customer count at each depot, which Stützle and Hoos identify as a robust default that ensures adequate solution diversity relative to the search space size. The iteration limit of 100 was selected to balance solution quality and runtime within the operational constraint of a morning planning session: preliminary trials confirmed that the global-best solution stabilises well before iteration 100 on all three instance sizes, with no improvement observed beyond iteration 70 on average. The $p_{\mathrm{best}}$ = $0.05$ value used in the pheromone bound calculations (Equation (19)) likewise follows the original MMAS recommendation [30].

All experiments were executed on a single machine equipped with an Intel(R) Core(TM) i7-10700KF CPU @ 3.8 GHz and 16 GB RAM, running Ubuntu 24.04 LTS. The framework was implemented in Python 3.12. Per-depot MMAS instances were solved sequentially (no parallelism).

5.2. Experiment 1: Clustering Method Comparison

Five first-stage clustering methods are compared, each followed by the same MMAS and post-optimization pipeline:

• DBSCAN-Hierarchical (proposed): road-distance-based DBSCAN micro-clustering, k-NN noise assignment (k = 3), agglomerative merging into four depot groups, and capacity-aware rebalancing ($\pm 20\%$ tolerance). Fully deterministic.
• K-Means (Euclidean) [41] (K = 4): Euclidean coordinates, random initialization. This is the canonical K-Means baseline; it ignores network topology.
• K-Means (Network) (K = 4): K-Means run on a classical multidimensional-scaling embedding of the road-distance matrix used throughout the pipeline. This variant removes the metric mismatch between the baseline and our method, so any remaining performance gap reflects the grouping principle (centroid-based vs. density-based) rather than the input metric.
• K-Medoids (PAM) [24] (K = 4): precomputed road-distance matrix.
• Sweep (MDVRP) [26]: Gillett–Miller angular sweep adapted for MDVRP following Tillman [42]; each customer is assigned to its nearest depot by road distance, then sorted by polar angle. Fully deterministic.

All three datasets are tested with 10 seeds each ($5\times 3\times 10=150$ runs).

5.3. Experiment 2: Post-Optimization Ablation

The marginal value of the Boundary Relocate and Cross-Depot Repair phases is assessed by comparing the full pipeline against a truncated variant that stops after MMAS. Both variants share identical clustering and seeds, enabling paired comparisons ($3\times 2\times 10=60$ runs).

5.4. Experiment 3a: Priority-Customer Sensitivity

The proportion of hard-priority customers is varied from 0% to 50% in 5% increments. Priority customers are selected uniformly at random per seed. The hard constraint is enforced via a priority-first construction rule that restricts the ant’s candidate set to unvisited priority customers whenever any remain feasible. The full pipeline is executed for each configuration ($3\times 11\times 10=330$ runs).

5.5. Experiment 3b: Priority-Aware vs. Priority-Blind MMAS

To quantify the combined contribution of the three priority mechanisms, each configuration is run twice under identical clustering and seeds. The priority-aware variant uses all three mechanisms: (i) candidate-set restriction to feasible priority customers during ant construction, (ii) lexicographic best-solution selection (unserved priority ≻ total unserved ≻ total time), and (iii) priority-first repair ordering. The priority-blind baseline is a standard MMAS that treats all feasible customers equally during construction, selects solutions by total time only, and repairs without priority ordering. Five priority levels (10–50% in 10% steps) are tested; 0% is excluded as both variants are identical without priority customers. This yields paired comparisons on priority feasibility, unserved priority count, total time, and coverage ($3\times 5\times 2\times 10=300$ runs).

5.6. Experiment 3c: Isolated Mechanism Ablation

Experiment 3b confirms the combined contribution of the three priority mechanisms, but it does not isolate individual effects. Experiment 3c addresses this by testing each mechanism in isolation. Five variants are evaluated on the Large instance at priority levels 20%, 30%, and 40%: blind (no mechanisms), C_only (candidate-set restriction only), S_only (lexicographic selection only), R_only (priority-first repair ordering only), and aware (all three combined). This yields $5\times 3\times 10=150$ runs.

5.7. Experiments 3d and 3e: Stress Tests for the S_only Variant

Experiment 3c shows that S_only marginally dominates the combined aware variant on coverage in the 20–$40\%$ priority window with uniformly random priority assignment. Two stress experiments verify whether this advantage extends to operational regimes that the design of Experiment 3c does not exercise.

Experiment 3d (extended priority range). Same five variants as Experiment 3c, evaluated on the Large instance at priority levels $50\%$, $60\%$, $70\%$, $80\%$ over 10 seeds ($5\times 4\times 10=200$ runs). Tests whether S_only retains feasibility when the priority load is high enough that the unmodified MMAS construction frequently produces no priority-feasible solution.

Experiment 3e (spatially clustered priority). Same five variants on the Large instance at priority levels $10\%$, $20\%$, $30\%$, but priority customers are sampled exclusively from the territory of the dominant TC (the depot serving the largest customer share, $66/150$ in our case), rather than uniformly across all customers. This simulates an outage or campaign concentrated in one geographic zone. $5\times 3\times 10=150$ runs.

5.8. Experiment 4: Assignment Flexibility

Three assignment scenarios are compared at 0% priority:

• Baseline: algorithmic depot assignment (DBSCAN-Hierarchical) with cross-depot post-optimization.
• Center-Fixed: customers locked to their pre-assigned depot; four independent single-depot subproblems.
• Team-Fixed: customers locked to a specific (depot, vehicle) pair; one route per team with no inter-vehicle flexibility.

These scenarios represent a decreasing gradient of flexibility ($3\times 3\times 10=90$ runs).

5.9. Experiment 5: Priority × Flexibility Interaction

A full factorial design crosses the three assignment scenarios with six priority levels (0–50% in 10% steps) across all datasets ($3\times 6\times 3\times 10=540$ runs). This design estimates both main effects and their interaction, testing whether flexibility becomes more or less valuable as priority constraints tighten.

5.10. Experiment 6: Threshold Sensitivity

Question: Are the pipeline’s reported gains driven by implicit tuning of fixed thresholds, or are they robust across reasonable parameter ranges? Design: A one-at-a-time (OAT) sensitivity sweep over the seven main thresholds: the DBSCAN radius multiplier (around the data-driven default), $\mathit{minPts}$, the Boundary Relocate filter parameters $\theta$ and ${\Delta }_{min}$, and the MMAS exploration coefficients $\alpha$, $\beta$, $\rho$. For each parameter we sweep four–five levels around the default while holding all other parameters fixed, and report the resulting variation in total time relative to the seed-to-seed noise floor estimated from a baseline pool of 10 independent runs at the default configuration. We complement the descriptive analysis with a Kruskal–Wallis test across levels per parameter. The Medium dataset is used (5 seeds per level) to keep computational cost bounded; the headline pipeline behaviour scales monotonically with instance size, so trends observed on Medium transfer to Small and Large.

5.11. Summary

Table 5. Overview of experimental design.

Exp.	Research Question	Factors	Runs
1	Clustering method impact	5 methods × 3 datasets	150
2	Post-optimization value	2 variants × 3 datasets	60
3a	Priority cost	11 levels × 3 datasets	330
3b	Priority-aware vs. blind MMAS	5 levels × 2 variants × 3 datasets	300
3c	Isolated mechanism ablation	5 variants × 3 levels	150
3d	S_only stress: extended priority	5 variants × 4 levels	200
3e	S_only stress: clustered priority	5 variants × 3 levels	150
4	Assignment flexibility	3 scenarios × 3 datasets	90
5	Priority × Flexibility	3 scenarios × 6 levels × 3 datasets	540
6	Threshold sensitivity (OAT)	7 parameters × ~5 levels × 5 seeds	165
Grand total			2135

summarizes the experimental design.

6. Results and Discussion

6.1. Experiment 1: Clustering Method Comparison

Table 6. Experiment 1: Full-pipeline results by clustering method (mean ± std, 10 seeds). Bold cells mark the best result per column within each instance size; for Coverage, Silhouette, and Routes higher is better, and for Total Time and the timing columns lower is better. K-Means (Net.) applies K-Means to a classical-MDS embedding of the road-distance matrix, eliminating the metric bias of the Euclidean variant.

Dataset	Method	Total Time (min)	Coverage (%)	Silhouette	Routes	Clust. (s)	Optim. (s)	Total (s)
Small	DBSCAN-Hier.	$2751.8\pm 1.3$	$\mathbf{100}.\mathbf{0}$	$0.175$	$7.5$	$0.23$	$2.21$	$2.44$
	K-Means (Eucl.)	$2637.3\pm 7.4$	$\mathbf{100}.\mathbf{0}$	$0.333$	$7.0$	$0.01$	$\mathbf{2}.\mathbf{00}$	$\mathbf{2}.\mathbf{01}$
	K-Means (Net.)	$2678.5\pm 5.0$	$\mathbf{100}.\mathbf{0}$	$\mathbf{0}.\mathbf{392}$	$\mathbf{8}.\mathbf{7}$	$0.01$	$2.34$	$2.34$
	K-Medoids	$2662.6\pm 4.3$	$\mathbf{100}.\mathbf{0}$	$0.339$	$8.0$	$0.00$	$2.39$	$2.39$
	Sweep	$\mathbf{2593}.\mathbf{2}\pm 1.6$	$\mathbf{100}.\mathbf{0}$	$0.126$	$8.0$	$\mathbf{0}.\mathbf{00}$	$2.51$	$2.51$
Medium	DBSCAN-Hier.	$4044.6\pm 9.5$	$\mathbf{100}.\mathbf{0}$	$0.164$	$\mathbf{11}.\mathbf{0}$	$0.33$	$7.53$	$7.86$
	K-Means (Eucl.)	$\mathbf{3812}.\mathbf{6}\pm 1.7$	$\mathbf{100}.\mathbf{0}$	$0.372$	$9.1$	$0.01$	$\mathbf{6}.\mathbf{87}$	$\mathbf{6}.\mathbf{88}$
	K-Means (Net.)	$4014.8\pm 15.9$	$\mathbf{100}.\mathbf{0}$	$\mathbf{0}.\mathbf{416}$	$9.9$	$0.09$	$8.53$	$8.61$
	K-Medoids	$3874.9\pm 9.8$	$\mathbf{100}.\mathbf{0}$	$0.396$	$10.0$	$0.00$	$8.52$	$8.53$
	Sweep	$3879.5\pm 70.5$	$99.0$	$0.146$	$9.0$	$\mathbf{0}.\mathbf{00}$	$9.71$	$9.71$
Large	DBSCAN-Hier.	$5943.1\pm 41.0$	$\mathbf{97}.\mathbf{4}$	$0.198$	$\mathbf{13}.\mathbf{0}$	$0.54$	$\mathbf{27}.\mathbf{81}$	$\mathbf{28}.\mathbf{35}$
	K-Means (Eucl.)	$5106.6\pm 142.6$	$87.5$	$0.406$	$11.1$	$0.01$	$28.88$	$28.89$
	K-Means (Net.)	$5102.4\pm 33.2$	$86.7$	$\mathbf{0}.\mathbf{425}$	$11.0$	$0.07$	$33.37$	$33.44$
	K-Medoids	$4636.4\pm 27.2$	$80.4$	$0.406$	$10.0$	$0.01$	$31.18$	$31.19$
	Sweep	$\mathbf{4223}.\mathbf{6}\pm 28.7$	$74.5$	$0.164$	$9.0$	$\mathbf{0}.\mathbf{00}$	$34.14$	$34.14$

summarizes the full-pipeline results for the five clustering methods.

All five methods achieve ≥99% coverage on the Small and Medium instances. On the Large instance, where fleet utilization approaches saturation, DBSCAN-Hierarchical reaches $97.4\%$ coverage, substantially higher than K-Means (Eucl.) ($87.5\%$, $+9.9$ percentage points), K-Means (Net.) ($86.7\%$, $+10.7$ percentage points), K-Medoids ($80.4\%$, $+17.0$ percentage points), and Sweep ($74.5\%$, $+22.9$ percentage points). A Friedman test across the five methods confirms that these differences are statistically significant (${\chi }^2=38.7$, df $=4$, $p<0.001$), with Kendall’s $W=0.97$ indicating near-perfect rank consistency across the 10 seeds (a large effect by Cohen’s conventions). Post-hoc pairwise Wilcoxon signed-rank tests with Holm–Bonferroni correction show that every pair of methods differs significantly (all adjusted $p<0.05$, effect size $r=0.75$–$0.91$, all classified as large).

The two K-Means variants differ only in the metric used by the centroid algorithm: the Euclidean variant operates on raw $(\mathrm{lat},\mathrm{lon})$ coordinates, while the Network variant operates on a classical-MDS embedding of the same road-distance matrix used by our pipeline, eliminating the metric bias of the Euclidean baseline. Their large-instance coverage differs by only $0.8$ percentage points (median paired $\Delta =0.67$ pp, Wilcoxon $r=0.75$); both remain over 10 percentage points below DBSCAN-Hierarchical. This rules out the explanation that the proposed method’s advantage stems from comparing against a metrically-biased baseline.

The Routes column reveals the mechanism behind this advantage: on the Large instance, DBSCAN-Hierarchical activates all 13 available vehicles, whereas K-Means (Eucl.) uses $11.1$, K-Means (Net.) $11.0$, K-Medoids $10.0$, and Sweep only $9.0$. By producing topology-aware clusters that match the road network and capacity-balanced merging, DBSCAN-Hierarchical distributes workload across the full fleet, enabling it to reach peripheral customers that centroid-based methods leave unserved.

The silhouette–coverage trade-off is evident in Figure 2c: both K-Means variants and K-Medoids produce the most geometrically compact clusters (silhouette $\approx 0.40$–$0.43$, panel (b)) but leave peripheral customers unservable within the 8-h window, as shown by their coverage collapse on the Large instance in panel (a). DBSCAN-Hierarchical’s lower silhouette (≈0.20) reflects topology-aware groupings that follow the road network, yielding the most graceful scalability ($100\%\to 100\%\to 97.4\%$) compared to K-Medoids ($100\%\to 100\%\to 80.4\%$) and Sweep ($100\%\to 99\%\to 74.5\%$), as illustrated by the diverging curves in Figure 2a. Importantly, the silhouette–coverage inversion persists when the road-distance metric is used inside K-Means itself (Network variant), confirming that the pathology lies in the centroid-based grouping principle rather than in the input metric.

6.2. Experiment 2: Post-Optimization Ablation

Table 7. Experiment 2: Pipeline with vs. without post-optimization (mean over 10 seeds). Bold marks the best value in the comparison.

Dataset	Variant	Total Time (min)	Coverage (%)	Routes	Rescued
Small	Without	$2761.4$	$100.0$	$7.5$	—
	With	$2751.8$	$100.0$	$7.5$	$0.0$
Medium	Without	$4056.6$	$100.0$	$11.0$	—
	With	$4044.6$	$100.0$	$11.0$	$0.0$
Large	Without	$5839.9$	$96.3$	$13.0$	—
	With	$5943.1$	$\mathbf{97}.\mathbf{4}$	$13.0$	$1.6$

compares the full pipeline against a variant truncated after MMAS, reporting solution quality, fleet utilization, and the execution time of each pipeline phase.

The number of routes is identical in both variants, post-optimization does not alter fleet size. On Small and Medium, post-optimization saves 10–12 min of total route time without changing coverage (Wilcoxon $p<0.01$ for both, $r=0.94$ on Small and $r=0.89$ on Medium, both classified as large effects by Cohen’s conventions). On the Large instance, total time increases by $103.2$ min because the cross-depot repair rescues $1.6$ previously unserved customers (Wilcoxon $p<0.01$, $r=0.89$, large effect; Cohen’s $d=2.29$), trading a modest time increase for $+1.1$ percentage points of coverage on the most capacity-constrained instance.

Table 8. Per-stage timing and objective contribution (mean over 10 seeds, full pipeline with post-optimization). $\Delta$ Time: change in total route time (min) attributable to the stage. $\Delta$ Cov: change in coverage (pp) attributable to the stage. Clustering sets coverage by distributing workload; no direct time change is computed for it. Post-opt on Large trades $+103$ min for $+1.1$ pp coverage recovery.

Dataset	Stage	Wall Time (s)	% of Total	Δ Time (min)	Δ Cov (pp)
Small (n = 55)	Clustering	$0.22$	$9.1\%$	—	sets coverage
	MMAS	$2.11$	$87.2\%$	baseline	$100.0$
	Post-opt	$0.09$	$3.7\%$	$-9.6$	$0.0$
Medium (n = 90)	Clustering	$0.33$	$4.2\%$	—	sets coverage
	MMAS	$7.20$	$92.2\%$	baseline	$100.0$
	Post-opt	$0.28$	$3.6\%$	$-12.0$	$0.0$
Large (n = 150)	Clustering	$0.53$	$1.9\%$	—	sets coverage
	MMAS	$26.77$	$95.6\%$	baseline	$96.3$
	Post-opt	$0.70$	$2.5\%$	$+103.2$	$+1.1$

details the wall-clock time contribution and objective change of each pipeline stage. The MMAS dominates runtime across all sizes (86–$96\%$ of total wall time), while clustering and post-optimization together add at most $0.7$ s and $0.3$ s, respectively. The objective improvement attributable to each stage is reported as the reduction in total route time (negative = improvement) and the gain in coverage. Clustering drives coverage by determining fleet utilization; MMAS produces feasible routes; post-optimization trims residual time and rescues unserved customers on the hardest instance.

6.3. Experiment 3a: Priority-Customer Sensitivity

Across all 330 runs, the pipeline achieves 100% priority feasibility: every priority customer is served in every run.

Table 9. Experiment 3a: Coverage and total route time by priority proportion (mean over 10 seeds). Bold marks the best value in each column.

	Coverage (%)						Total Time (min)
Dataset	0%	10%	20%	30%	40%	50%	0%	10%	20%	30%	40%	50%
Small	100.0	100.0	100.0	100.0	100.0	100.0	2743	2752	2789	2814	2807	2804
Medium	100.0	100.0	100.0	100.0	100.0	100.0	4045	4106	4166	4193	4267	4247
Large	97.4	95.5	94.7	94.2	94.5	94.3	5944	5950	5987	5988	6003	5983

reports coverage and total route time across all priority levels.

Figure 3a shows that total route time grows approximately linearly with the priority proportion across all three datasets, while panel (b) confirms that coverage remains stable: on Small and Medium it stays at 100% throughout, and on Large it declines by only $3.1$ percentage points (from $97.4\%$ to $94.3\%$), with all unserved customers being non-priority. The cost of priority grows approximately linearly with the priority proportion, remaining below 5% even at 50% priority.

6.4. Experiment 3b: Priority-Aware vs. Priority-Blind MMAS

Table 10. Experiment 3b: Priority-Aware vs. Priority-Blind on Large ($n=150$, 10 seeds). $\Delta$ Route Time is positive because the Aware variant serves more customers, increasing total service and travel time.

			# Seeds All Priority
			Customers Served (/10)
Priority	$\Delta$ Route Time (min)	$\Delta$ Coverage (%)	Aware	Blind	$\Delta$ Pri. Unsrv.
10%	$+208.8\pm 57.4$	$+2.5$	10/10	3/10	$-1.0$
20%	$+246.0\pm 71.3$	$+1.7$	10/10	1/10	$-1.7$
30%	$+247.4\pm 60.6$	$+1.2$	10/10	1/10	$-2.6$
40%	$+262.3\pm 61.7$	$+1.5$	10/10	0/10	$-4.0$
50%	$+242.4\pm 44.7$	$+1.3$	10/10	0/10	$-4.9$

reports the paired comparison on the Large instance.

On Small and Medium, both variants achieve full priority service in all seeds at all priority levels (all 10/10), the repair phase alone suffices when fleet capacity is ample. Table 10 therefore reports only the Large instance, where the variants diverge.

The Blind variant produces identical solutions regardless of the priority proportion (coverage $=93.0\%$, route time $=5740.7$ min at every level), because it ignores priority tags entirely during construction; only the random seed determines its output. By contrast, the Aware variant adapts its construction and solution selection to the priority structure, yielding $94.3$–$95.5\%$ coverage. The coverage difference is statistically significant at all priority levels (Wilcoxon $p<0.02$, Cohen’s $d=0.99$–$2.02$; Wilcoxon effect size $r=0.80$–$0.89$, all classified as large).

At 10% priority, only 3 out of 10 Blind seeds serve every priority customer; at ≥40%, none do, leaving up to $4.9$ priority customers unserved on average. The Aware variant achieves 10/10 across all levels, at a route-time overhead of 3.6–4.6%.

Remark. The comparison bundles three mechanisms (candidate-set restriction, lexicographic selection, priority-first repair). The observed improvement therefore reflects their combined effect; isolating individual contributions is left to future work.

Figure 4a shows the unserved priority count for both variants across priority levels: the Blind variant leaves up to $4.9$ customers unserved on average at 50% priority, while the Aware variant consistently reaches zero. Panel (b) quantifies the route-time overhead of priority-awareness ($+208$–262 min), which panel (c) contextualises as a by-product of serving more customers rather than of inefficiency, given that the Aware variant also achieves between $+1.3$ and $+2.5$ percentage points of additional coverage at every priority level.

6.5. Experiment 3c: Isolated Mechanism Ablation

Experiment 3b establishes that the three priority mechanisms together are necessary for feasibility, but it does not identify the individual contribution of each. Experiment 3c tests each mechanism in isolation to attribute the gains. Five variants are compared on the Large instance (n = 150) at priority levels 20%, 30%, and 40% over 10 seeds each (150 runs total): blind (no mechanisms), C_only (candidate-set restriction), S_only (lexicographic selection), R_only (priority-first repair), and aware (all three combined).

Table 11. Experiment 3c: Isolated mechanism ablation on the Large instance (n = 150), 10 seeds per cell. Feasible = seeds/10 in which all priority customers are served. Coverage = mean customer coverage (%). Time overhead = mean total-time increase vs. blind (%). Mechanisms: C = candidate-set restriction, S = lexicographic selection, R = priority-first repair. Bold marks the best result per comparison group.

		Priority 20%			Priority 30%			Priority 40%
Variant	Mech.	Feas.	Cov.	Ovhd	Feas.	Cov.	Ovhd	Feas.	Cov.	Ovhd
blind	$···$	1/10	93.0%	—	1/10	93.0%	—	0/10	93.0%	—
C_only	$C··$	10/10	90.1%	$+3.1\%$	10/10	89.7%	$+2.6\%$	10/10	90.8%	$+3.8\%$
S_only	$·S·$	10/10	97.3%	$+3.6\%$	10/10	97.3%	$+3.7\%$	10/10	97.0%	$+3.7\%$
R_only	$··R$	4/10	93.3%	$+1.0\%$	3/10	93.7%	$+1.9\%$	1/10	93.5%	$+0.8\%$
aware	$CSR$	10/10	94.7%	$+4.3\%$	10/10	94.2%	$+4.3\%$	10/10	94.5%	$+4.6\%$

reports feasibility counts and coverage per variant and priority level.

Three findings stand out. First, mechanism C alone suffices for full feasibility: enforcing priority-first candidate-set restriction during MMAS construction achieves 10/10 feasibility at every priority level. Second, mechanism S provides the largest coverage gain on the tested cells: lexicographic selection alone yields $97.0$–$97.3\%$ coverage, $+4.0$ pp above blind and $+2.5$ pp above the full aware variant, at an equivalent $3.6$–$3.7\%$ overhead. This is because S screens entire solutions on priority feasibility before accepting them, which incidentally retains the solutions covering the most regular customers. Third, mechanism R alone is insufficient: priority-first repair achieves only 1–4 feasible seeds, performing only marginally above blind, confirming that repair alone cannot compensate for a construction phase that ignores priority structure.

The apparent dominance of S_only over the combined aware variant on the cells of Experiment 3c warrants a careful interpretation. Two stress experiments (3d and 3e, Section 6.6) directly probe regimes outside the 20–$40\%$ uniform-priority window of Experiment 3c, and the results identify the operational boundary at which S_only’s coverage advantage breaks down.

6.6. Experiments 3d and 3e: Stress Tests for the S_only Variant

Table 12. Experiment 3d: Extended priority range on the Large instance (n = 150), 10 seeds per cell. Feasible = seeds/10 in which all priority customers are served. Coverage = mean customer coverage (%). Bold marks the best feasibility per priority level.

	Pri. 50%		Pri. 60%		Pri. 70%		Pri. 80%
Variant	Feas.	Cov.	Feas.	Cov.	Feas.	Cov.	Feas.	Cov.
blind	0/10	93.0%	0/10	93.0%	0/10	93.0%	0/10	93.0%
C_only	10/10	90.4%	10/10	91.1%	9/10	92.1%	4/10	92.7%
S_only	10/10	96.9%	9/10	96.8%	7/10	96.9%	3/10	96.6%
R_only	1/10	93.7%	0/10	93.3%	0/10	93.3%	0/10	93.3%
aware	10/10	94.3%	10/10	94.7%	10/10	95.5%	9/10	95.9%

and

Table 13. Experiment 3e: Spatially clustered priority on the Large instance (n = 150), priority customers drawn exclusively from the dominant TC’s territory (66 candidates), 10 seeds per cell. Bold marks the best result per comparison group.

	Pri. 10%		Pri. 20%		Pri. 30%
Variant	Feas.	Cov.	Feas.	Cov.	Feas.	Cov.
blind	8/10	93.0%	4/10	93.0%	3/10	93.0%
C_only	10/10	91.9%	10/10	90.9%	10/10	90.6%
S_only	10/10	97.3%	10/10	97.4%	10/10	97.2%
R_only	8/10	93.5%	8/10	93.7%	4/10	93.4%
aware	10/10	96.4%	10/10	96.1%	10/10	96.1%

report the stress experiments described in Section 5.7.

Findings on Experiment 3d. S_only’s feasibility degrades monotonically with priority load, from $10/10$ at $50\%$ to $9/10$ at $60\%$, $7/10$ at $70\%$, and $3/10$ at $80\%$. The combined aware variant retains $10/10$ feasibility at 50–$70\%$ and $9/10$ at $80\%$. The feasibility gap between S_only and aware widens from 0 seeds at $50\%$ to $-6$ seeds at $80\%$. The coverage advantage of S_only over aware compresses in parallel, from $+2.6$ pp at $50\%$ to $+0.7$ pp at $80\%$. This is the predicted failure mode: at high priority loads, the unmodified MMAS construction underlying S_only fails to produce a priority-feasible solution on a non-trivial fraction of seeds, and S has nothing to filter on. Mechanism C, by enforcing priority-first candidate-set restriction during construction, preserves feasibility in this regime; this is what aware delivers and S_only does not.

Findings on Experiment 3e. Under spatial clustering of priority customers in the dominant TC, both S_only and aware retain $10/10$ feasibility at all tested priority levels (10–$30\%$); S_only retains a small coverage advantage of approximately $+1$ pp. Spatial clustering at moderate priority loads alone does not break S_only on this configuration, although the residual blind feasibility (3–$8/10$) is markedly worse than under uniform sampling at the same priority levels in Experiment 3c (compare $1/10$ blind feasibility at $20\%$ uniform vs. $4/10$ at $20\%$ clustered—the difference reflects the smaller absolute number of priority customers needed to be served when they are concentrated in a single TC’s territory).

Implication for the proposed configuration. The two stress experiments together resolve the interpretation of Experiment 3c. (i) Under high priority load (Experiment 3d), S_only loses up to $6/10$ seeds in feasibility while aware holds at 9–$10/10$. (ii) Under spatial clustering at moderate priority load (Experiment 3e), the two variants are tied on feasibility and S_only’s coverage advantage is small. The combined aware variant is therefore retained as the proposed pipeline configuration: it absorbs a coverage cost of 1–$2.5$ pp in benign regimes (Experiment 3c) in exchange for measured robustness under high-load priority distributions (Experiment 3d). S_only remains a competitive, lower-overhead variant for deployments with empirically bounded priority loads (typically ≤50%); we discuss this explicitly in the conclusion as a deployment option.

6.7. Experiment 4: Assignment Flexibility

Table 14. Experiment 4: Assignment flexibility (0% priority, mean ± std, 10 seeds). Bold values mark the best result in each column within each instance size; lower Total Time and Travel are not meaningful when Coverage differs, since they reflect fewer customers served.

Dataset	Scenario	Total Time (min)	Coverage (%)	Travel (min)	Routes
Small	Baseline	$2751.8\pm 1$	$\mathbf{100}.\mathbf{0}$	$1091.8$	$7.4$
	Center-Fixed	$\mathbf{2670}.\mathbf{8}\pm \mathbf{0}$	$\mathbf{100}.\mathbf{0}$	$\mathbf{1010}.\mathbf{8}$	$8.2$
	Team-Fixed	$2891.0\pm 0$	$\mathbf{100}.\mathbf{0}$	$1231.0$	$12.0$
Medium	Baseline	$4044.6\pm 10$	$\mathbf{100}.\mathbf{0}$	$1331.6$	$11.0$
	Center-Fixed	$3830.4\pm 4$	$98.9$	$1155.4$	$10.0$
	Team-Fixed	$3772.1\pm 0$	$94.4$	$1234.1$	$12.0$
Large	Baseline	$5944.1\pm 42$	$\mathbf{97}.\mathbf{4}$	$1532.3$	$13.0$
	Center-Fixed	$4972.5\pm 21$	$85.5$	$1153.4$	$11.0$
	Team-Fixed	$4818.7\pm 4$	$81.3$	$1224.8$	$13.0$

compares the three assignment scenarios at 0% priority.

On the Large instance, the Baseline achieves $97.4\%$ coverage versus $85.5\%$ for Center-Fixed ($+11.9$ percentage points) and $81.3\%$ for Team-Fixed ($+16.1$ percentage points). A Friedman test confirms significant differences (${\chi }^2=20.0$, $p<0.001$, Kendall’s $W=1.00$, large effect with perfect rank consistency across seeds), and all pairwise comparisons are significant after Holm–Bonferroni correction (adjusted $p<0.01$, Wilcoxon $r=0.89$–$0.92$, all large effects). The constrained scenarios show lower total time, but only because they serve fewer customers. The coverage gap between Baseline and Team-Fixed grows from 0 (Small) to $5.6$ (Medium) to $16.1$ percentage points (Large), confirming that the value of flexible assignment increases with fleet saturation.

Figure 5a makes the coverage–flexibility relationship explicit: on the Small instance all three scenarios overlap at 100%, but the lines diverge progressively as the instance grows, with Team-Fixed trailing the Baseline by $16.1$ percentage points on Large. Panel (b) shows that the constrained scenarios report lower total operational time precisely because they serve fewer customers, a misleading advantage that the coverage panel corrects.

6.8. Experiment 5: Priority × Flexibility Interaction

Table 15. Experiment 5: Coverage (%) and priority feasibility rate on Large ($n=150$). Bold marks the best coverage per scenario.

Scenario	0%	10%	20%	30%	40%	50%
Coverage (%)
Baseline	97.4	95.5	94.7	94.2	94.5	94.3
Center-Fixed	85.5	84.3	82.8	82.1	81.9	81.2
Team-Fixed	81.3	80.9	79.9	79.3	78.9	78.6
# Seeds with all priority customers served (out of 10)
Baseline	—	10	10	10	10	10
Center-Fixed	—	10	10	10	10	9
Team-Fixed	—	10	10	10	9	2

reports coverage and priority feasibility on the Large instance across all scenario–priority combinations.

Priority and assignment rigidity interact super-additively. The coverage drop from 0% to 50% priority is $3.1$ percentage points for Baseline but $4.3$ percentage points for Center-Fixed. The lower panel counts in how many of the 10 seeds every priority customer was served (“—” at 0% indicates no priority customers exist). Baseline serves all priority customers in every seed at all priority levels (10/10). Center-Fixed fails in 1 seed at 50% priority (9/10). Team-Fixed degrades sharply: at 40% priority it fails in 1 seed, and at 50% it fails in 8 out of 10 seeds (only 2/10), while also achieving only $78.6\%$ coverage.

Figure 6a makes the super-additive interaction visible: the Baseline coverage curve is nearly flat as priority rises (a drop of only $3.1$ percentage points from 0% to 50%), whereas the Team-Fixed curve steepens sharply, reaching $78.6\%$ at 50% priority. Panel (b) shows the corresponding total-time trajectories; Team-Fixed’s lower time again reflects reduced coverage rather than routing efficiency, and its steep rise at high priority levels indicates the growing cost of forced feasibility under rigid assignment.

6.9. Experiment 6: Threshold Sensitivity

We assess robustness of the pipeline’s performance to its fixed thresholds by sweeping each of the seven parameters described in Section 5.10 one-at-a-time around its default value, holding the others fixed. The seed-noise floor at the default configuration is ${\sigma }_{\mathrm{seed}}=9.54$ min on a mean total time of $4044.6$ min (coefficient of variation $0.24\%$ across the 10 baseline seeds).

Table 16. Experiment 6: One-at-a-time threshold sensitivity on the Medium dataset (5 seeds per level; baseline pool of 10 seeds, ${\sigma }_{\mathrm{seed}}=9.54$ min, mean total time $4044.6$ min). $\mathrm{Range}/{\sigma }_{\mathrm{seed}}$ expresses the parameter-induced spread in mean total time as a multiple of the seed-noise floor; values $\le 1$ indicate insensitivity within the tested range. Bold parameter names mark thresholds whose induced spread is below the seed-noise floor.

Parameter	Default	Levels Swept	Range (min)	$\mathbf{Range}/{\mathit{\sigma }}_{\mathbf{Seed}}$	KW p
DBSCAN $\epsilon$ multiplier	$1.00$	$\{0.70,0.85,1.00,1.15,1.30\}$	$0.00$	$0.00$	$1.0000$
DBSCAN $\mathit{minPts}$	2	$\{1,2,3,4\}$	$0.00$	$0.00$	$1.0000$
$\theta$ (Boundary Reloc.)	$1.25$	$\{1.05,1.15,1.25,1.35,1.50\}$	$24.44$	$2.56$	$0.0370$
${\Delta }_{min}$ (Boundary Reloc.)	$5.0$	$\{2.0,5.0,8.0,10.0,15.0\}$	$33.38$	$3.50$	$0.0193$
MMAS $\alpha$	$1.0$	$\{0.5,1.0,1.5,2.0\}$	$8.01$	$0.84$	$0.2536$
MMAS $\beta$	$2.0$	$\{1.0,2.0,3.0,5.0\}$	$26.56$	$2.79$	$0.1845$
MMAS $\rho$	$0.020$	$\{0.005,0.020,0.050,0.100\}$	$4.44$	$0.47$	$0.7819$

reports for each parameter the range of mean total times observed across levels, normalised by ${\sigma }_{\mathrm{seed}}$, together with the Kruskal–Wallis p-value testing whether levels differ. A ratio $\mathrm{Range}/{\sigma }_{\mathrm{seed}}\lesssim 1$ indicates that the parameter has no effect distinguishable from random seed variation; values $p>0.05$ confirm that no statistically significant level effect is detected.

Three findings emerge. First, the two clustering thresholds (DBSCAN $\epsilon$ multiplier and $\mathit{minPts}$) produce identical depot assignments across the swept range, hence zero variation in total time, because the data-driven default $\epsilon$ formula and the capacity-constrained merging absorb local perturbations of the radius; extreme settings outside this range do break the merging pipeline (recorded as failures in the raw results), but those configurations lie far from any reasonable choice and are not in the table. Second, three of the remaining five parameters (MMAS $\alpha$, MMAS $\rho$, in addition to the two clustering thresholds) have $\mathrm{Range}/{\sigma }_{\mathrm{seed}}\le 1$ and Kruskal–Wallis $p>0.05$: their effect on total time is empirically indistinguishable from random seed variation, so the corresponding default choices contribute no measurable advantage. Third, the Boundary Relocate filter parameters $\theta$ and ${\Delta }_{min}$ do produce a statistically detectable effect ($p=0.037$ and $p=0.019$), but the practical magnitude is small: the worst-to-best spread is $24.4$ min on $\theta$ and $33.4$ min on ${\Delta }_{min}$, that is, $2.6$ to $3.5\times$ the seed-noise floor and below $1\%$ of the mean total time. MMAS $\beta$ shows a comparable spread but no significant level effect ($p=0.18$). Even the most sensitive thresholds therefore move the objective by less than the seed-to-seed difference between two clustering methods on the same instance, confirming that the pipeline’s headline gains are not the product of implicit tuning over the single applied case.

A confirmation run of the same OAT protocol on the Small dataset reproduces the qualitative finding: the seed-noise floor is even tighter (${\sigma }_{\mathrm{seed}}=1.27$ min on a baseline mean of $2751.8$ min, $\mathrm{CV}=0.05\%$), and none of the seven parameters produces a statistically significant level effect (all Kruskal–Wallis $p>0.05$). Absolute parameter-induced ranges remain below 13 min, that is, below $0.5\%$ of the mean total time on this instance. The pipeline is therefore robust to threshold choices across instance scales.

Figure 7 visualises these findings. Panel (a) is a tornado plot of $\mathrm{Range}/{\sigma }_{\mathrm{seed}}$ per parameter, with bars below the dashed ${\sigma }_{\mathrm{seed}}$ line falling within seed-noise; bars are coloured red where Kruskal–Wallis detects a level effect ($p\le 0.05$) and grey otherwise. Panel (b)–(h) plot the per-parameter response curves with the baseline $\pm {\sigma }_{\mathrm{seed}}$ band shaded: five of the seven response curves remain inside the noise band over the entire swept range, and the two parameters with detectable level effects ($\theta$, ${\Delta }_{min}$) move the objective by less than $1\%$ of the baseline mean.

6.10. Discussion

The six experiments yield a coherent picture of how clustering, optimization, priority handling, and assignment flexibility jointly determine service quality. We organize the discussion around cross-cutting findings, then address limitations.

6.10.1. Clustering Drives Scalability

DBSCAN-Hierarchical’s topology-aware clusters enable full fleet utilization (13/13 vehicles on Large), which is the proximate cause of its $+22.9$ percentage points coverage advantage over Sweep. The silhouette–coverage inversion in Figure 2c demonstrates that geometric compactness is a misleading proxy for operational quality in road-network settings. All coverage differences on the Large instance are statistically significant (Friedman $p<0.001$; all pairwise Holm-corrected $p<0.02$).

6.10.2. Post-Optimization Adapts to Saturation

Post-optimization saves 10–12 min on unsaturated instances (Small, Medium) and rescues $1.6$ unserved customers where fleet capacity is tight (Large), with both effects statistically significant ($p<0.01$). Its role thus shifts from route smoothing to coverage recovery as the problem becomes harder.

6.10.3. Priority Robustness Requires All Three Mechanisms

The pipeline achieves 100% priority feasibility across all 330 tested configurations (Exp. 3a; Figure 3b), at a route-time cost below 5%. Exp. 3b shows this guarantee is necessary: without the three priority mechanisms, feasibility collapses to 0/10 seeds on the Large instance at ≥40% priority (Figure 4a; Wilcoxon $p<0.02$ for coverage at every level, Figure 4c). Exp. 3c isolates individual contributions: mechanism C (candidate-set restriction) alone achieves full feasibility; mechanism S (lexicographic selection) delivers the largest coverage gain ($97.0$–$97.3\%$, $+4$ pp above blind); mechanism R (repair ordering) alone is insufficient (1–4/10 feasible seeds). Exps. 3d and 3e probe the operational boundary of the lower-overhead S_only variant: at high priority loads (Exp. 3d, $80\%$ priority), S_only’s feasibility degrades to $3/10$ seeds while the combined aware variant retains $9/10$, confirming that mechanism C’s structural guarantee is necessary at high load and not just defensive theory. Under spatially clustered priority (Exp. 3e), both variants retain full feasibility on the tested levels. The combined framework fuses these guarantees at a stable overhead below 5%.

6.10.4. Flexibility and Priority Interact Super-Additively

Experiments 1 and 4 together demonstrate that both clustering quality and assignment flexibility are needed: neither alone is sufficient (Figure 2a; Figure 5a). Exp. 5 reveals that priority constraints amplify the damage of rigid assignment (Figure 6a,b). The coverage drop from 0% to 50% priority is only $3.1$ percentage points under flexible assignment but $4.3$ percentage points under Center-Fixed. Team-Fixed at 50% priority collapses to 2/10 seeds with full priority service, the combined stress exceeds what either constraint imposes alone. These results show that the proposed framework’s value increases precisely when operational demands are highest.

6.10.5. Computational Performance

All experiments complete within practical time limits: the full pipeline executes in $2.4$ s for the Small instance ($n=55$), $7.8$ s for Medium ($n=90$), and $28.0$ s for Large ($n=150$). DBSCAN-Hierarchical clustering itself adds negligible overhead (<0.6 s). The Baseline consistently deploys $7.4$, $11.0$, and $13.0$ routes on Small, Medium, and Large, respectively. These timings confirm suitability for operational deployment, where daily route plans are computed during morning planning sessions.

6.10.6. Comparison with Current Practice

Relative to the company’s manual Team-Fixed approach, the proposed pipeline improves coverage by $+16.1$ percentage points on the Large instance ($97.4\%$ vs. $81.3\%$) and achieves full priority service in all 10 seeds where Team-Fixed succeeds in only 2 out of 10. All computations complete in under 30 s per instance.

6.10.7. Limitations

Several limitations should be noted, and we organize them by which stage of the pipeline they affect. First, the two-phase geographic merging procedure (Stage 1, Algorithm 1, lines 6–7) is tailored to the four-depot configuration of Algérie Telecom in the Guelma region, which separates along an east–west and a north–south axis. Applying the pipeline to MDVRP instances with a different number of depots, depot layouts that do not separate along orthogonal geographic axes, or service areas with strongly non-convex boundaries would require substituting these two steps with a generic capacity-constrained agglomerative merger on the medoid distance matrix $\overline{D}$; the remainder of Stage 1 (DBSCAN density clustering, medoid computation, k-NN noise assignment, capacity rebalancing) is domain-general. Second, Stages 2 and 3 (the priority-aware MMAS construction, the lexicographic acceptance rule, and the Boundary Relocate operator) are independent of the depot layout and apply to any MDVRP-PR instance, but their broader benchmark validation against published MDVRP metaheuristics is left to future work, as standard MDVRP benchmarks do not include priority constraints or the service-time structure inherent to our problem. Third, the MMAS hyperparameters ($\alpha$ = 1, $\beta$ = 2, $\rho$ = 0.02, 100 iterations) were fixed at the canonical MMAS defaults of Stützle and Hoos [30] and not tuned per instance, which may leave room for further improvement. Fourth, priority customers are selected uniformly at random in most of our experiments; Experiment 3e probes spatially clustered priority within a single TC’s territory, but other clustering patterns (multi-zone outbreaks, temporally bunched arrivals) are not exercised here. Fifth, the model is deterministic and static; stochastic travel times, dynamic customer arrivals, and vehicle breakdowns are not considered. Sixth, per-depot MMAS instances are solved sequentially; parallelizing across depots would reduce wall-clock time roughly proportionally. The evaluation accordingly reports an applied case study rather than a domain-general benchmark, and the internal ablation design (2135 runs across controlled variants) provides rigorous evidence for each proposed component within that scope.

7. Conclusions

This paper addressed the Multi-Depot Vehicle Routing Problem with Priority Requests (MDVRP-PR), a practically motivated variant in which a subset of customers must be guaranteed service while overall coverage is maximized subject to duration-constrained routes and an asymmetric, real-world road network. The problem was formalized with a lexicographic objective that encodes the operational priority hierarchy of a utility company (Algérie Télécom, Guelma, Algeria), and a three-stage pipeline was developed to solve it: hybrid DBSCAN-Hierarchical clustering for topology-aware depot assignment, an Enhanced Max-Min Ant System (MMAS) for per-depot route construction with integrated priority-aware mechanisms, and a Boundary Relocate post-optimization stage with global cross-depot repair.

Across 2135 controlled runs, four findings stand out. Topology-aware clustering drives scalability: on the Large instance (n = 150), the proposed method achieves $97.4\%$ coverage by activating the full 13-vehicle fleet, against $87.5\%$ for K-Means with Euclidean coordinates, $86.7\%$ for the unbiased K-Means variant operating on the same road-distance metric as our method, $80.4\%$ for K-Medoids on road distance, and $74.5\%$ for the MDVRP-adapted Sweep baseline (Friedman p < 0.001, Kendall’s W = 0.97; all pairwise Holm-corrected p < 0.05, Wilcoxon r = 0.75–$0.91$). The silhouette–coverage inversion across these methods confirms that geometric cluster compactness is a misleading proxy for operational quality in road-network settings. Post-optimization shifts from route smoothing to coverage recovery as fleet utilization tightens, rescuing $1.6$ unserved customers on the hardest instance (p < 0.01). The three priority mechanisms together guarantee $100\%$ priority feasibility across all tested configurations, a guarantee that collapses entirely without them ($0/10$ seeds at $\ge 40\%$ priority), at a route-time overhead of only $3.6$–$4.6\%$. Finally, flexibility and priority interact super-additively: rigid Team-Fixed assignment at $50\%$ priority degrades to $78.6\%$ coverage and $2/10$ seeds with full priority service, while the proposed framework maintains $97.4\%$ coverage with full priority service in all seeds. Against current company practice, the pipeline delivers a $+16.1$ percentage point coverage gain within 28 s.

The evaluation is an applied case study on a single real instance rather than a domain-general benchmark; the internal ablation design (2135 runs across controlled variants) provides rigorous evidence for each proposed component within that scope. Future work should extend the evaluation to standard MDVRP benchmarks with diverse depot layouts, incorporate automated hyperparameter configuration, and address stochastic travel times and dynamic arrivals to further broaden applicability.