Truck-multidrone same-day delivery strategies: On-road resupply vs depot return

Expert Systems With Applications 272 (2025) 126757 Available online 6 February 2025 0957-4174/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Truck-multidrone same-day delivery strategies: On-road resupply vs depot return David Sanchez-Wells a,* , Jos´ e L. Andrade-Pineda b , Pedro L. Gonzalez-R a a Department of Industrial Engineering and Management Science, School of Engineering, University of Seville, Camino de los Descubrimientos, s/n., 41092 Seville, Spain b Robotics, Vision &Control Group, School of Engineering, University of Seville, Camino de los Descubrimientos, s/n., 41092 Seville, Spain ARTICLE INFO Keywords: Truck-Multidrone Logistics Genetic Algorithm Makespan Truck Mileage Last-mile Delivery Resupply ABSTRACT This paper explores an enhanced two-waved same-day delivery (SDD) system that leverages a mothership truck equipped with multiple drones supported by an auxiliary “resupply”truck. Under standard SDD operations, this mothership truck, also capable of performing deliveries, must return to the depot to reload, incurring extra travel time and mileage. In contrast, the proposed resupply strategy enables the second delivery wave by dispatching a secondary vehicle to meet the mothership truck on-road, reloading parcels without interrupting ongoing deliveries by the drones. A single unified routing framework, the Genetic Algorithm with Iterated Estimations for Resupply (GAIER), is presented to optimise both strategies under two selectable criteria: minimising total service time or total truck mileage. In tests with benchmark networks of different sizes (20, 50, and 75 nodes), incorporating a resupply truck reduced every selected criterion when compared to the strategy where the mothership vehicle returns to the depot. Subsequent comparative analysis points an average reduction of 17 % in service time and 21 % in truck mileage while statistical analyses support the strategy choice significancy, confirming resupply strategy’s potential for cost savings and reduced environmental impact. These findings bolster our proposition that incorporating a resupply truck into hybrid truck-multidrone systems enhances flexibility in drone delivery scheduling and improves the system’s ability to meet urban demand. 1. Introduction Last-mile delivery is currently under growing pressure to commit to the shorter delivery time requirements of increasingly demanding customers. In particular, there is a rising demand for same-day delivery (SDD) services, often referring to goods that arrive to a central depot over time—e.g., in make-to-order operations, such as pharmaceuticals, perishable goods, or online shopping orders—and to be delivered in urban areas. While asking customers to fetch the parcel is an alternative, whether leaving the goods at predetermined collection centres (Hong et al., 2019) or at mobile parcel lockers (Peppel et al., 2024), customers are more willing to wait for a suitable home delivery, even in cases where no specific delivery time slot has been explicitly agreed upon. However, along with the implicit claim for a short delivery time, customers have a limited willingness to pay for instant deliveries, what has led the courier industry to reorganise their SDD service in dispatch waves (Klapp et al., 2018). When considering the standard system concerning deliveries that are made with only trucks or vans, the key issue is hence deciding when to return to the central depot to reload for subsequent deliveries, by seeking for the trade-off between waiting for additional products and dispatching the vehicle as soon as possible (Klapp et al., 2018). However, while maximising load consolidation leads to lowering the number of trips, it can result not only in greater mileage, but also in increased idle time and delayed deliveries. For that reason, more disruptive attempts are appearing every day, which redesign existing delivery processes to better adapt to the new scenarios. 1.1. Emergence of drone-based solutions on SDD One of the most promising avenues for innovation on same-day delivery lies in the incorporation of drones (or unmanned aerial autonomous vehicles, UAVs) that despite coming with their own set of limitations—e.g., payload and battery constraints (Tadi´ c et al., 2024; Bi et al., 2024)—can fly back and forth in middle-sized cities within 15 km (Shankland, 2022). Lamb et al. (2022) compared the traditional truckonly delivery with a UAV delivery system used in combination with * Corresponding author. E-mail addresses: [email protected] (D. Sanchez-Wells), [email protected] (J.L. Andrade-Pineda), [email protected] (P.L. Gonzalez-R). Contents lists available at ScienceDirect Expert Systems With Applications journal homepage: www.elsevier.com/locate/eswa https://doi.org/10.1016/j.eswa.2025.126757 Received 22 March 2024; Received in revised form 30 January 2025; Accepted 2 February 2025 Expert Systems With Applications 272 (2025) 126757 2 short-to-medium term capacitated inventory storage facilities (referred to as micro-fulfilment centres) to mitigate the UAV range limitations. Recently, some works (Ghiasvand et al., 2024; Li et al., 2024) have considered that van/truck role consists of moving orders from the depot/s to a set of satellite, no-capacitated, fixed locations, from which UAV flights are scheduled to complete the delivery. Symmetric roles have been applied in recent literature (Dayarian et al., 2020; Pina-Pardo et al., 2021; McCunney &Cauwenberghe, 2019; Dienstknecht et al., 2022; Pina-Pardo et al., 2024a; Pina-Pardo et al., 2024b). These authors innovate the SDD service by assuming that parcels are truck-deliveredonly, whereas drones are reserved the role-play of auxiliary vehicles for “resupply”whose flights are scheduled for injecting newly arrived orders into the ongoing truck distribution route. Dynamic collaborative routing strategies, such as en-route synchronisation between trucks and drones—which we name “on-road”to highlight that our mothership truck is reloaded while delivering—have shown significant potential in optimising delivery times and resource utilisation under random requests (Cui et al., 2024). Thus, the extent to which drones interact with trucks differs, whereas resupplying the truck directly when meeting it en-route at meeting points (Pina-Pardo et al., 2021; Pina-Pardo et al., 2024b) or leaving its load at transshipment points from which the truck picks up the parcels (McCunney &Cauwenberghe, 2019). According to Moshref-Javadi et al. (2023), this drone resupply concept can result in a greater number of packages delivered within a specific time window, since avoiding truck returns to the depot to pick up newly available orders reduces the delivery truck downtime, thereby benefiting the overall operational efficiency (Moshref-Javadi et al., 2020a; Murray &Chu, 2015). On their recent work, Alkaabneh &Sutharson (2024) consider that the assessment of the resupply operational mode should take into consideration both the total mileage of ground vehicle/s as well as the total time that the latest ground vehicle requires for returning to the depot after serving all the clients. In this paper, we consider that the SDD service could gain in operational efficiency twofold: (i) by fully exploiting the flexibility of drone package deliveries, and (ii) by using a ground vehicle for executing resupply tasks. The resupply role could therefore be given to both a van or a truck, ground vehicles suited to carry a greater quantity of orders—aligning with the courier industry’s primary goals. Additionally, we propose innovating the SDD service by leveraging the drones’ability to bypass ground traffic and provide rapid, flexible delivery options in particular, multi-drop flight missions. The latter is perfectly possible in current drone technology, as presented in the last-mile hybrid delivery system by Masmoudi et al. (2022), referred to as the vehicle routing problems with drones equipped with multi-package payload compartments, which involves multiple tandems of truck-drone pairs and multivisit drone trips. In fact, in contexts other than the SDD, further topologies of hybrid truck-drone delivery systems have considered the use of drones to deliver packages directly to customers; e.g., taking off from trucks acting as mobile depots (Murray &Chu, 2015; Murray &Raj, 2020) or from the depot itself (Ham, 2018). 1.2. Research gap and proposed approach Around the SDD problem, and on the assumption that the actual demand for the next day is unknown, Ghiasvand et al. (2024) tackle uncertainty on customer demands through a kernel-based machine learning approach to minimise the total customer waiting time, while Li et al. (2024) configures a fast-delivery strategy for online grocery retailing, fitting to the uncertainty of demands through a two-step stochastic procedure aimed at minimising the total delivery costs. This is achieved firstly by consolidating a subset of truckloads to designate the satellite locations from which drones will take the goods to deliver to customers, and secondly by marking the remaining groceries to be routed along with future arrivals. Ulmer &Thomas (2018) have considered an SDD system combining truck and drone deliveries, distributing the served areas so that drones are used for delivery customers in the outermost locations, whereas trucks are still preferred to service the downtown areas close to the depot. However, deploying drones to serve customers only based on their location is not fully exploiting the flexibility that could be gained from considering a truckdrone tandem system where drones can be dispatched from the carrying truck to visit customers. Dayarian et al. (2020) and Pina-Pardo et al. (2021; 2024a) proposed models for drone-supporting delivery for the SDD, where all customers are served by trucks while drones solely provide support for trucks with parcel resupply. This mode suits dynamic ordering by preventing trucks from repeatedly returning to the depot to pick up newly arrived orders. In our research, we propose addressing one of such scenarios, but using combined truck and drone delivery of parcels with parcel resupply by means of a ground vehicle. Specifically, we propose a novel strategy for the SDD that explores the synergy between a simple resupply truck (RT), and a more sophisticated drone-commanding truck (DCT). The DCT manages a variety of drones that take-off from and land from it along with its route—i.e., customer locations taken as the rendezvous points to synchronise with it—to perform multi-drop drone missions, while the RT is able to perform deliveries to customers while returning to the depot after reloading the DCT. This feature has been formerly addressed, although not always along the use of combined truck and drone delivery (Jeong &Lee, 2023; Poikonen &Golden, 2020). Poikonen &Golden (2020) stated the k-multi-visit drone routing problem (kMVDRP), where kdrones are assigned to serve a set of customers and the truck acts as a mobile platform for the drone to both replenish its energy or to pick the packages up before flying to deliver to customers. Jeong & Lee (2023) add the flexible launching at locations separate from customer sites, although again under the assumption of trucks not visiting customers but moving between launch positions. Different to the above-mentioned works, our strategic approach holds on: (i) the active delivery role of trucks and drones, both with capacity constraints and (ii) the drone service range limit. In the hybrid truck-drone research area, our truck-based resupply strategy is a novelty, which we propose for a more extensive use of a DCT liberated from returning to depot. This allows us to design a SDD service that combines the strengths of both trucks while ensuring parcels are readily accessible to drones without relocating them from the surroundings of delivery zones. We have also outlined a unified VRP-based mathematical formulation in order to decide between both strategies—depot return and on-road resupply—by rigorously capturing the interplay between different constraints and variables, such as the capacities of each type of vehicle, as well as drone autonomy, among others. 1.3. Contribution statement and organisation of this paper This study assesses the efficiency gains achievable through the integration of a resupply truck supporting a drone-commanding truck equipped with a multidrone fleet in a SDD scenario. While the mathematical modelling of the explained assumptions—i.e., autonomy limitation of drones while all vehicles are capacity-limited and perform the same type of delivery—is considered intrinsic, in order to evaluate such efficiency gains we need to design and make use of a problem-solving framework capable of minimising one objective at a time: the duration of the service—the time elapsed until final gathering at the depot—or the total mileage of the trucks, recognising the latter as a key environmental impact factor. This mileage criterion is pertinent both to electric trucks, where it serves as a driver for electrical battery life cycle consumption, and to combustion engine trucks, where it gauges the level of emissions. These factors will be integral to the design criteria of the novel joint delivery strategy proposed herein. The research questions are as follows: D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 3 •How will the adoption of an auxiliary “resupply”truck affect the performance of a SDD service based on a truck deploying a fleet of drones? •Do the instances influence in the comparative performance of the standard and the resupply strategy? As previously stated, in order to support the comparison of both strategies, and having into account the computational burden from the stated NP-hard mathematical model, we have devised a unified way of generating high-quality solutions within a multidrone and capacitylimited context, the so-called GAIER (Genetic Algorithm with Iterated Estimations for Resupply). Importantly, the GAIER framework adheres to the established formulations to the purpose of facilitating the comparative assessment of the strategies rather than introducing novel mathematical constructs. GAIER is an optimisation framework that utilises a unique blend of heuristic solution evolution methods, combining a Genetic Algorithm (GA) and an Iterated Greedy (IG) heuristic (Ruiz &Stützle, 2007), all under the global guidance of Simulated Annealing (SA). GAIER also leverages an efficient binary-customised encoding to minimise the combinatorial solution space and introduces a novel vector-based coding evolved from previous works (Gonzalez-R et al., 2020; Gonzalez-R et al., 2024). The GAIER framework stands out for its innovative use of a hybrid optimisation technique that combines the strengths of multiple heuristics, not only accelerating the solution process but also improving the quality of the solutions found. In short, GAIER is particularly effective in handling the complex dynamics of a multi-modal delivery system involving both trucks and drones. In providing an original and comprehensive approach to optimising last-mile delivery operations, this paper seeks to make a significant contribution to both scholarly discourse and practical implementation in the rapidly evolving landscape of same-day delivery logistics. In particular, from the application of our GAIER tool we can claim that the SDD service can be redesigned with a supporting RT for attaining a more dynamic route adjustment on the DCT by means of its replenishment of parcels, which ultimate leads to reducing the DCT idle time and increasing the number of deliveries that can be made within a given time frame. Furthermore, the latter is attained while getting a reduction in DCT truck mileage—let’s recall here, costs of a DCT mile is typically high, from its more sophisticated features—which directly translates to lower emissions and fuel consumption, aligning with sustainability goals in logistic industries that require rapid delivery of perishable goods or urgent medical supplies. The remainder of the paper is structured as follows. Section 2 provides a comprehensive review of the existing literature in the domains of last-mile drone logistics, resupply in last-mile delivery systems, and genetic algorithms, pinpointing the gaps that this research aims to fill. Section 3 delves into the mechanics of the proposed resupply strategy, and Section 4 elucidates the GAIER framework. Section 5 presents a rigorous case study and a sensitivity analysis to empirically validate the effectiveness of the resupply strategy. Finally, Section 6 provides a synthesis of the findings, outlines the contributions to academia and industry, and suggests directions for future research. 2. Literature review This literature review provides a comprehensive overview of recent research focusing on the relevant studies regarding the use of drones on SDD and last-mile delivery, as well as how genetic algorithms are applied today to the last-mile truck-drone delivery problem. Table 1 reports on the more directly connected works arisen from our literature review. 2.1. Same-day delivery and drones Nowadays, SDD service companies are struggling to decide how to dispatch orders that become ready during the day of work to better meet the increased customer expectations. In exploring the trade-off between delivery cost and time, various new drone-based operational models have emerged. Ulmer &Thomas (2018) presented a dynamic vehicle routing problem that combines conventional vehicles and drones for SDD, deciding based on customer location whether the vehicle or the drone will be the servicing vehicle. Specifically, in this work, drones are employed to make autonomous deliveries from the depot to the customers located in areas far from the depot, since they can reach them in a shorter time (with the higher travel speed resulting from the avoidance of traffic). However, in this work drone range limitation is ignored, which is indeed a very crucial assumption in drone routing. There are other operational models that are based on exploiting a two-echelon delivery network: the drone tranships its load at transshipment locations which are in turn collected by trucks for last-mile delivery. McCunney &Cauwenberghe (2019) applied this focus in a simulation study, concluding that the drone resupply policy outperforms the traditional one, where trucks have to return to the depot to pick up parcels, both on delivery time and distance for serving all customers. Moshref-Javadi et al. (2023) analysed two case studies for an urban and suburban area upon the simulation approach by McCunney &Cauwenberghe (2019) but extended to the consideration of multi-trip flight missions for resupply drones that leave their load at the transhipment locations. Their proposal consists of a hybrid truck and drone twoechelon location routing problem whose objective is the minimisation of the total travel time for trucks and drones, decomposed into two stages: truck routing decisions and drone resupply events to occur at any of the potential transhipment locations. As highlighted in recent systematic reviews of drone delivery systems, integrating drone resupply mechanisms into urban logistics can revolutionise last-mile delivery solutions (Jazairy et al., 2025). The seminal work by Dayarian et al. (2020) introduces the VRP with resupply to study how a fleet of delivery trucks is dispatched from a fulfilment centre to perform home deliveries of online orders whereas a sort of drones working in tandem with them are regularly resupplying the delivery truck by meeting with it at any location at which the truck is stationary to transfer the newly arrived packages. Specifically, the authors focused on how best to schedule truck-drone meetings at any customer location along the truck’s route to maximise the percentage of orders served and minimise total operational costs (including the transportation costs due to both types of vehicles). The complexity of the problem that arises led researchers to consider only the single delivery truck and a single drone case, concluding that the drone resupply reduces both truck travel and delivery times compared to the truck-only operational model. Arguing that there is a limited capacity of the truck, Dienstknecht et al. (2022) considered that only a subset of orders can be loaded on the truck when it departs from the depot and that the reminders are subsequently resupplied by the drone to the truck. Noticeably, they assume that drones can carry only one package on each trip. They state the Travelling Salesman Problem (TSP) with drone resupply, whose resolution is first focused on the drone resupply subproblem (dynamic programming to find the optimal drone schedule for a given truck route) and then aimed at the exploration of different algorithms to determine efficient truck routes (genetic algorithms and various construction heuristics). However, the reported numerical experiments indicate that truck returns are still needed. The impossibility of carrying the entire set of packages in the delivery truck can also be due to a delivery time for the order at the fulfilment centre or depot. Pina-Pardo et al. (2021) studied an SDD scenario where they assume that the release times and due dates of orders are known in advance. They provide a MILP formulation that is solved throughout a two-stage decomposition heuristic, firstly determining the truck route and secondly defining the occurrence of resupply events at customer locations. The same SDD scenario is further studied by Pina-Pardo et al. (2024a), but extended to multiple trucks and D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 4 Table 1 Literature classification (own elaboration). Related literature Vehicles Interaction Capacity limits Drone operation features Modelling &Solving Features Number of Trucks Number of Drones DroneTruck direct interaction Truck carrying drones to service area Truck can serve any customer Drone serving any customer Limited truck capacity Limited drone capacity Multidrop flights Launch retrieval Drone flight range/ time varies Flexible docking Modelling approach Minimised objective Solving methods &algorithms Instances size Gonzalez-R et al. (2020) 1 1 ✓ ✓ ✓ ✓✓ ✓   MILP Time IG +SA 250 Mara et al. (2022) 1 1 ✓ ✓ ✓ ✓ ✓ ✓ ✓   MILP Time COTS +ALNS 200 Murray &Raj (2020) 1 m ✓ ✓ ✓ ✓  ✓ ✓ MILP Time 3-PH-HEUR 8 to 100 Leon-Blanco et al. (2022) 1 m ✓ ✓ ✓ ✓✓ ✓   Agents Time AGENTS 500 Gu et al. (2022) m m ✓ ✓ ✓ ✓✓ ✓ ✓ MILP Time, Cost ILS +VNS 200 Jiang et al. (2024) m m ✓ ✓ ✓ ✓  ✓✓MILP Cost COTS +ALNS 100 Masmoudi et al. (2022) m m ✓ ✓ ✓✓ ✓   ✓Profit (Max) TS +SA 200 Kitjacharoenchai et al. (2020) m m ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓   MILP Time LNS 76 Kitjacharoenchai et al. (2019) m m ✓ ✓ ✓ ✓    ✓✓MILP Time ADAP. HEUR 100 Luo et al. (2021) 1 m ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ MILP Time TS +NEIGH 100 Poikonen & Golden (2020) 1 m ✓ ✓ ✓  ✓ ✓     RouteTransform Shortest 50 Poikonen & Golden (2020) 1 m ✓ ✓ ✓ ✓ ✓ ✓     B&B+GREEDY HEUR  Moshref-Javadi et al. (2020a) 1 m ✓ ✓ ✓ ✓    ✓   Time TS +SA 100 Moshref-Javadi et al. (2020b) 1 m ✓ ✓ ✓ ✓ ✓ ✓ ✓     Time 100 Sacramento et al. (2019) m m ✓ ✓ ✓ ✓ ✓ ✓   Cost 200 Dayarian et al. (2020) 1 1 ✓✓  ✓    Dyn. Opt. Customers HEUR 60 Dienstknecht et al. (2022) 1 1 ✓✓✓ ✓     Dyn. Prog. +Sim. Cost DYN OPT ++CONSTR. HEUR 350 Pina-Pardo et al. (2021) 1 m ✓✓  ✓    MILP Time 2-STEP-HEUR 50 Li et al. (2024) m m   ✓   ✓   Stoch. Cost PSO +Reinfor. Learning 25 Ghiasvand et al. (2024) m m   ✓✓     MILP + MDP Time Data-Driven-Opt <100 Pina-Pardo et al. (2024b) m m ✓✓✓ ✓     MILP + MDP Time 100 Pina-Pardo et al. (2024a) m m ✓✓✓ ✓     MILP Time MATHEUR 100 Gao et al. (2023) m m ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓   MILP Cost COL GEN+BENDERS <35 Kuo et al. (2022) m m ✓   ✓  ✓  MILP VNS 200 D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 5 multiple drones, where newly released orders can be collected by trucks at the depot or resupplied en-route via drones. The assumption in recent works (Pina-Pardo et al., 2021; Pina-Pardo et al., 2024a) is that all customer information—i.e., delivery locations and order arrival times—is known before the operation starts, and hence they concern a static routing problem. Going further in the same SDD context, Pina-Pardo et al. (2024b) claim that in view that the dispatcher must immediately decide whether to accept or reject a customer request for same-day delivery of an order, dynamic decision concerning to order acceptance under uncertain order release can be attained using a route-based Markov Decision Process and an efficient online policy to dynamically route a truck that can receive newly arrived orders along its route via drones dispatched from a depot. In these three referred works, both types of vehicles are allowed to perform several routes—i.e., return to the depot—during the delivery horizon, synchronising these vehicles spatially and temporally for en-route resupply operations. However, as regards as the multi-trip feature that we are applying in the present paper, we must point out that the multi-trip feature that the authors endorse to their operational model is unclear as regards the drones: the common meaning of multi-trip regards as multi-drop capability, and this is not assumed in (Pina-Pardo et al., 2021; Pina-Pardo et al., 2024a; Pina-Pardo et al., 2024b). In every synchronised meeting, the drone has taken off from the depot and left all its load on the corresponding truck and then proceeded to fly back to the depot. 2.2. Drones for last-mile delivery There is a research stream that for years has been considering that drone technology allows for transforming the last-mile delivery by dispatching drones from trucks to deliver packages to customers. Within the scope of last-mile delivery involving hybrid truck-drone logistics, the scholarly literature shows two predominant viewpoints on the role of ground vehicles or trucks. One school of thought posits trucks mainly as mobile depots to aid drones in operations such as battery replacement (Karak &Abdelghany, 2019; Ferrandez et al., 2016). Another perspective attributes a more active role to trucks where, in addition to serving as launch and recovery platforms for drones, they also participate in the delivery of packages to customers (de Freitas &Penna, 2020; Dell’Amico et al., 2022; Ha et al., 2018; Murray &Chu, 2015). There has been a proliferation of research that explores the interaction between a single truck and multiple drones. For instance, Chang &Lee (2018) and Moshref-Javadi et al. (2020b) focus on scenarios where drones are deployed and then collected by the same truck but potentially at a different location from that it was launched, which appoints to the synchronisation of truck-drone meetings as crucial decisions (Moshref-Javadi et al., 2020a; Yoon, 2018). Other works such as Murray &Raj (2020) and Raj &Murray (2020) extend the seminal FSTSP to the case of an active delivery truck equipped with multiple drones (mFSTSP) and model the energy consumption of drones as a nonlinear function of parcel weight, speed and operation time. More recently, drone launch/retrieval in route has been addressed in Marinelli et al. (2018), Schermer et al. (2019) and Li et al. (2022), although they all are limited to the case of one drone route covering exactly one customer. Conversely, the current paper follows Gonzalez-R et al. (2020) and Gonzalez-R et al. (2024) assumption of multi-drop missions, where drones serve multiple customers per flight and synchronisation events occur always at customer locations in a so-called Truck-multiple Drones Tandem Logistics (TmDTL) system. It is noteworthy that other researchers such as Poikonen &Golden (2020; 2020), Luo et al. (2021), Leon-Blanco et al. (2022), and Gu et al. (2022) have also explored the multicustomer per fly case. Poikonen & Golden (2020) consider multi-trip drone missions in tandem with a single truck that can also deliver to customers, while Poikonen &Golden (2020) presented an interesting TSP-D with multiple drones considering adjustable speeds and battery consumption rate as a function of the payload, drones that can be launched and retrieved by a truck at any point in its way between two locations, although they assumed the truck serves solely as a mobile depot (drone primary) which does not deliver packages to customers. Multi-drop drone deliveries are considered by Jeong &Lee (2023), adding the flexible launching at locations separate from customer sites, although again under the assumption of trucks not visiting customers but moving between launch positions. Luo et al. (2021), which have addressed a system similar to TmDTL, although their computational experience is severely limited by the number of drones in the fleet (solely two drones per truck). The connection is direct with Leon-Blanco et al. (2022) and Gonzalez-R et al. (2024). The former used an agent-based heuristic solved large TmDTL instances with the makespan minimisation as the single objective criterion, while the latter adopted a multiobjective approach for the TmDTL based on a bivector coding scheme and a simulated annealing algorithm to generate approximate Pareto fronts for a biobjective problem, with truck mileage and service completion time as key performance indicators. Differently to the above-mentioned works, we are concerned by the active delivery role of trucks and drones, both with capacity constraints. In view of the conducted review, we found that there is a promising new operational model for transforming the SDD, which, to the best of our knowledge, has never been studied. Why not using a ground resupply vehicle that coordinates with the delivery truck? Similar to Pina-Pardo et al. (2021) and Pina-Pardo et al. (2024a), we aim at static routing for defining the resupply meetings, that in our case involve an auxiliary truck and not resupply drones. Since we know it could outperform the traditional truck-only system, why not fully exploiting a last-mile delivery based on a multidrone fleet dispatched from the delivery truck? Our paper deepens into this issue by addressing a unified CTmDTL formulation to identify which is the best strategy. Like in our former work (Gonzalez-R et al., 2024), in this research we also approach the operational model, aiming at an efficient resolution method for attaining the holistic optimisation of our novel logistical system fitting to instances according to single depot in a service area with from 25 to 75 locations to serve (Li et al., 2024; Pina-Pardo et al., 2021; Poikonen & Golden, 2020; Dayarian et al., 2020; Kitjacharoenchai et al., 2020). 2.3. Genetic algorithms use in the last-mile truck-drone delivery problem The resupply concept in last-mile logistics is promising. However, the computational complexity of solving these problems cannot be overlooked, nor the need to develop efficient algorithms to solve drone resupply problems ignored. For example, Ponza (2016) used a simulated annealing algorithm to solve the Flying Sidekick Travelling Salesman Problem (FSTSP), a problem related to drone and truck logistics that serves up to 200 customers. Gu et al. (2022) developed Iterative Local Search and Variable Neighbourhood Descent algorithm in their VRPD with multi-visits model in which a fleet of truck-drone tandems to serve up to 200 customers is solved in less than 250 s. Mara et al. (2022) have designed an adaptive large neighbourhood search (Ropke &Pisinger, 2006), abbreviated as ALNS, heuristic for addressing the same problem in Gonzalez-R et al. (2020), which they call the multi-drop FSTSP. Algorithms based on ALNS have also been developed to solve multiple variants of resupply problems, demonstrating scalability and efficiency (Moshref-Javadi et al., 2020b; Sacramento et al., 2019). Moshref-Javadi et al. (2020b) proposed an integrated Tabu Search and Simulated Annealing (TS-SA) showing that their hybrid solution (single truck deploying multiple drones returning to the truck at a location different from where they are originally launched) could achieve significant time savings in waiting, as compared to the truck-only model. Sacramento et al. (2019) proposed an ALNS with several destroy and repair operators for solving a VRPD generalisation from the FSTSP that, like us used capacitated truck and drone, but they consider that some locations are to be served by the truck owing to the weight of the load, limits the synchronisation at truck stops among a preselected subset of candidate locations and further, that drone can only visit one customer per trip. That was the case also in Kitjacharoenchai et al. (2019) which have used an D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 6 Adaptive Insertion algorithm along with several mTSP construction heuristics (one of them based on GAs) for solving a benchmark of Solomon-like instances sized up to 100 customers to be served by a fleet of truck-drone tandems, each drone with unitary payload and with the chance to be retrieved at a different truck to that from it took off. This feature, referred to as flexible docking has been reported along with multi-drop capability in drone delivery in recent multidrone literature (Masmoudi et al., 2022; Jiang et al., 2024). Masmoudi et al. (2022) have applied a multi-start TS with SA heuristic for solving instances with up to 200 customers for maximising the profit from respecting their due date with a multi-truck distribution system using a fleet of supportive (not delivery) truck equipped with a drone to serve customers and then allowed to return to a different truck. Jiang et al. (2024) have developed an ALNS heuristic to solve instances of up to 100 locations to served, arisen from the rich VRPD the authors formulated to model how a truck and drone fleet are deployed taking advantage of the drone flexible docking feature to fulfil pickup and delivery in rural areas at minimal total routing costs. In this context, it is worth noting that GAs are increasingly used in the context of truck-drone delivery, such as Lu et al. (2022), in contrast with other optimisation methods such as Gravitational Search Algorithms (GSA) by Rashedi et al. (2009), Particle Swarm Optimisation (PSO) as explained in Rasouli (2024), or the Inclined Planes System (IPS) method proposed by Mozaffari et al. (2016). The main reason for this is the combinatorial nature of the problem to be solved. Unlike GSA and PSO, which excel in continuous optimisation but face significant challenges when adapted for discrete problems, GAs offer a flexible framework that naturally handles the discrete nature of combinatorial tasks (Nhu et al., 2023; Raut et al., 2024). While IPS optimisation introduces innovative heuristic strategies, it also requires extensive customisation to maintain efficiency in combinatorial contexts (Yang et al., 2024). Studies have demonstrated that GAs consistently outperform GSA, IPS, and PSO in terms of solution quality and computational efficiency in combinatorial settings (Raut et al., 2024; Le et al., 2024). Additionally, in the case of Kataoka et al. (2022), the authors refer to the GA methodology described by Hazama et al. (2022). Furthermore, while none of the reviewed documents directly addresses a problem identical to the single-criterion truck-drone delivery problem we are concerned with, two of them provide valuable insight into the application of GAs in solving drone routing problems. Consecutively, we focus on highlighting the most relevant and impactful studies to build a solid case for the application of GAs, providing a comprehensive overview of how GAs can effectively address the unique challenges posed by this problem. Zhang et al. (2022) proposed an evolution of the Non-dominated Sorting Genetic Algorithm II or NGSA-II (Deb et al., 2002) known as ENSGA-II. This approach is designed to handle multiobjective problems related to truck-drone delivery. In ENSGA-II, solutions are represented by a single vector (referred to as the “giant route”), which is subsequently processed by two distinct algorithms: a truck route split algorithm and a drone route construction algorithm. The crossover operation in ENSGA-II incorporates both partially matching crossover (PMX) and order crossover (OX) operators in a random manner to promote diversity within the population. Furthermore, a multidimensional local search is applied to the Pareto front generated at each GA iteration to enhance solution quality and diversity. Furthermore, Cai &Qian (2022) address the No-wait Drone Scheduling Travelling Salesman Problem (NW-DSTSP) using a hybrid approach that combines Adaptive Particle Swarm Optimisation (APSO) with Genetic Algorithms (GAs). Their solution methodology involves a three-step process, beginning with the formulation of the problem as a Mixed Integer Linear Problem (MILP). A Greedy strategy is employed to generate an initial solution to a simplified version of the problem. Subsequently, a GA crossover operator is applied to refine the solution. An adaptive adjustment of the learning factor and inertia factor weights is also performed to improve the convergence behaviour. Unfortunately, specific implementation details are not available. Adding to existing studies, Karak¨ ose (2024) introduces an innovative approach to the hybrid truck-multidrone problem. This research expands on the use of more generic GAs by developing a method that optimises delivery time in scenarios involving single and multiple drones in conjunction with a truck, creating various delivery scenarios where drones can only deliver one single client. The results of this study show a notable inverse relationship between the number of drones and delivery time, a finding that complements the approaches of Zhang et al. (2022) and Cai &Qian (2022). As a conclusion, we propose to put GAs in the centre of our resolution framework, noting that the application of GAs in addressing the truckdrone delivery problem is an active yet still underdeveloped area of research. Zhang et al. (2022) ENSGA-II, Cai &Qian (2022) hybrid APSOGA, and Karak¨ ose (2024) generic GA approaches demonstrate the potential of GAs to optimise complex truck-drone delivery operations both for single and multiple criteria optimisations. While further research is needed to refine these methodologies and adapt them to specific realworld scenarios, ultimately contributing to the advancement of lastmile logistics using drones, GAs were deemed the most suitable for our optimisation framework, providing a balanced approach to exploration and exploitation, crucial for solving the complex combinatorial optimisation problem presented in this research (Wen et al., 2024). In summary, the decision to use GAs in our optimisation framework is based on their proven ability to balance exploration and exploitation, which is crucial for solving complex combinatorial optimisation problems. This approach aligns with the methodologies successfully applied in other logistics and delivery optimisation studies. While it is true that some studies are not directly comparable to our specific problem, the general success and adaptability of GAs in related scenarios strongly support their use in our research. 2.4. Addressing main research gaps Our model integrates the resupply truck (RT), the dronecommanding truck (DCT), and drones into a cohesive system that dynamically adapts to different demands. This integration allows for more efficient coordination and utilisation of resources, addressing the limitations of previous hybrid models that treated these components more independently. Firstly, rather than just serving as resupply vehicles (Dayarian et al., 2020; Pina-Pardo et al., 2021; McCunney &Cauwenberghe, 2019; Dienstknecht et al., 2022; Pina-Pardo et al., 2024a; Pina-Pardo et al., 2024b), our goal is fully exploiting the capabilities of drones by allowing them to perform direct deliveries to customers. This approach maximises the flexibility and speed advantages of drones, especially in urban areas with high traffic congestion. In this manner, the multi-modal SDD model proposed aims to control costs by optimising the fleet size and usage, ultimately creating economic value by enhancing delivery responsiveness and efficiency while keeping operational expenses in check. We take the standard DCT return-to-depot strategy as a baseline to compare with, while we explicitly decide not accounting for other models that we judge would draw a scenario with complexities beyond what is being of widespread adoption in current last-mile hybrid delivery operation. For instance, we are aware of models such as that by Gao et al. (2023), where drones address both picking and delivery and are deployed from several trucks, or that in Stodola &Kutˇ ej (2024), where trucks are operated from differentiated (decentralised) depots, each truck in tandem with one drone. Secondly, we found that the impact of the truck capacity is somehow overlooked in most of the studies on drone-truck combined operations, except for Sacramento et al. (2019) and Kitjacharoenchai et al. (2020). Sacramento et al. (2019) studied a VRPD generalisation from the FSTSP that, like us used capacitated truck and drone, but they consider that some locations are to be served by the truck owing to the weight of the load, limits the synchronisation at truck stops among a preselected subset of candidate locations and further, that drone can only visit one D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 7 customer per trip. Kuo et al. (2022) proposed a VRPTWD model by considering the presence of customer time windows within the VRPD model of Sacramento et al. (2019), thereby adding capacity constraints to truck but not considering the multi-drop drone missions that are within the scope of our investigation. Kitjacharoenchai et al. (2020) considered both trucks and drones’capacities when stating a variation of the classic CVRP which, denoted as two-echelon vehicle routing problem with drones (2EVRPD), deploys multiple trucks and drones with the capability of multiple visits. They aimed their resolution at finding optimal routes of both trucks and drones to minimise the arrival at depot of all the vehicles after completing the deliveries. Hence, in our study we explicitly incorporate the capacity limitations of both trucks and drones thereby providing a more realistic and practical framework for evaluating the efficiency of the delivery operations. Finally, we point out that the research in truck-drone delivery operations often prioritises optimisation of an only objective, such as minimising either delivery time or operational cost. There is a need for studies where algorithms show flexibility to optimise aiming to different objectives in a flexible manner by choice of the user. Existing works like those by Leon-Blanco et al. (2022) and Poikonen &Golden (2020) tend to focus on singular optimisation goals, which does not provide a holistic view of operational efficiency. While our algorithm optimises either makespan or truck mileage in separate runs, it can solve both standard and resupply strategies without modifications. This flexibility allows decision makers to choose the optimisation criterion that best aligns with their operational goals. The capability to switch between objectives without changing the algorithm underscores its robustness and adaptability, being highly beneficial for the case study and analysis conducted in Section 4 to demonstrate the resupply strategy’s superior performance in reducing delivery times and truck mileage compared to the standard strategy: a DCT returning to the depot for reloading and then moving again to the delivery area. Overall, the proposed resupply strategy not only enhances operational efficiency and reduces costs but also contributes to environmental sustainability. These improvements provide a competitive edge for logistics companies in the rapidly evolving landscape of same-day delivery services. 3. Capacitated truck-multiple drones tandem logistics Capacitated Truck-multiple Drones Tandem Logistics (CTmDTL) is a delivery system based on the TmDTL system detailed in Gonzalez-R et al. (2024) with the addition of capacity constraints. As in the cited work, the CTmDTL system aims to serve parcels to a set of customers by means of a DCT and a fleet of autonomy-limited drones that work collaboratively, but on this occasion, all ground and aerial vehicles are capacityconstrained. Therefore, drones do not only refill their autonomy but also the necessary load for their next multi-drop mission when synchronising with the truck. This is what we call the “standard”DCT return-to-depot strategy, as represented in Fig. 1. Hence, in the context of TmDTL, trucks and drones, despite both serving customers, differ significantly in terms of autonomy, travel speed, and, according to the recent restriction, delivery capacity. Constant travel speeds of drones and trucks have been assumed, similar to other studies (Gonzalez-R et al., 2020). We assume drone speed is always higher than truck speed at the operational level, as usual in this type of approach at a planning decision level (Luo et al., 2021; Moshref-Javadi et al., 2020b). The impact of acceleration and its implications on take-off and landing, as well as the effect of wind on drone energy consumption, is beyond the scope of this study. 3.1. Our proposal: A resupply strategy On its side, the “resupply”strategy proposed in this paper works under the assumption that an auxiliary ground vehicle (with the same capacity limitation as the DCT truck) is included in the mentioned system to help convey to the delivery area those parcels beyond the capacity of the DCT (or that arrived later at the depot) in two different ways: (i) synchronisation with the DCT at any customer location to reload its capacity and (ii) direct delivery to customers on its way back to the depot. The reason for this last point is twofold: firstly, aiming to optimise efficiency by ensuring that the DCT and the drone fleet operate at or near full capacity for most of their routes, reducing the number of trips required, and optimising fuel usage; and secondly, by refilling the DCT’s capacity, the operation can minimise the time spent returning to the depot for additional cargo. This can lead to faster delivery times, improving customer satisfaction, and allowing for more deliveries within a given period. This strategy is visually represented in Fig. 2. As a result, while the DCT and its accompanying drones synchronise to overpass drone autonomy and capacity limitations, there is a parallel need of the DCT to synchronise with the resupply vehicle to address its own capacity constraint. This assumes that the vehicle that arrives first will need to wait for the other, ensuring a cohesive operational flow. Despite the added complexity introduced by this new synchronisation, the incorporation of the auxiliary cargo vehicle is identified as having the potential to decrease the total service time (makespan) and/or minimise the total mileage covered by the trucks in the logistics system. A relevant detail is that, as in TmDTL, drones have the option to remain parked on the DCT. Upon reaching the next node, they can takeoff for a new service route or stay parked on the DCT for multiple trips. Consequently, a noteworthy consideration is that a portion of the drone fleet in the DCT may remain unused, emphasising the importance of accurately sizing the fleet. Our study focuses on assessing the impact of resupply in two-waved SDD scenarios, where the standard method requires only one return to the depot for capacity replenishment. On the contrary, in the resupply strategy, a single synchronisation between the DCT and the resupply truck is sufficient to adhere to the capacity constraints of the DCT. Fig. 1. CTmDTL standard strategy routing diagram. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 8 3.2. Visit assumptions To summarise the previously mentioned conditions in a straightforward and interpretable manner, we establish two sets of rules, each outlining the behaviour for the standard and resupply strategies. Based on the terminology introduced by Gonzalez-R et al. (2024), the rules contained in this list will be called “visit assumptions”. The visit assumptions for the standard strategy are stated as follows: 1) All vehicles meet at the depot, as it is assumed that it is the necessary starting and end node for a complete service. 2) Any vehicle can serve any node, and all nodes must be served. 3) If a node is not visited by the DCT, only a single drone can visit and serve it. 4) If a node is visited by the DCT, any number of drones can meet it at that node, fully recovering autonomy and reloading capacity. From this point on, there are two possibilities for any drone: a. Restart again to continue to serve the nodes. b. Land in the DCT without consuming any autonomy until the DCT reaches its next node. 5) The synchronisation time is considered negligible when compared to the mission time, and, as such, it has been disregarded. 6) Meeting at a node necessarily involves waiting time, as all vehicles in need of synchronising will have to wait until the rest reach that node. These waiting times do not consume any autonomy for the drones. Therefore, hovering is not considered. 7) The truck technology allows it to navigate any potential route among the set of locations, irrespective of its length. Therefore, the autonomy of the truck is assumed to be unlimited. 8) The capacity of drones is limited and homogeneous, with the only method of replenishing their capacity through synchronisation with the DCT. 9) The capacity of the DCT is limited, and the only method of replenishing it is visiting the depot. In this study, the number of visits to the depot is restricted to one. As expected, when stating the visit assumptions for the resupply strategy, many of the previous remain. On the other hand, we must make some adjustments. Specifically, all rules of the standard strategy can be kept except visit assumption 9, which refers to truck capacity, and an additional one (visit assumption 10) must be added. Therefore, for the resupply strategy, the last two rules would be stated as follows. 10) The capacity of the DCT is limited, and the only method of replenishing it is by synchronising with the resupply truck. Number of synchronisations is limited to one in this study. 11) The resupply truck can only serve other nodes than the synchronisation one after it has synchronised with the DCT, therefore, on its way back to the depot. 3.3. Mathematical outline: A unified modelling approach In this section, we present a comprehensive mathematical formulation of the CTmDTL problem, integrating both the standard depot return and on-road resupply strategies under a single unified framework. The objective is to create a model flexible enough to handle a range of operational constraints—battery autonomy, vehicle capacities, and route synchronisations—while allowing us to optimise for either makespan or truck mileage. This formalisation is essential for understanding how we later design and implement an effective, heuristicbased resolution method. Since the model generalises the Vehicle Routing Problem—already known to be NP-hard—the CTmDTL formulation inherits this complexity. Therefore, we do not aim to obtain exact solutions for large instances via classical branch-and-bound techniques. Instead, we provide a robust mathematical representation that can be tackled by specialised algorithms or heuristics. The model captures the timing, capacity usage, drone autonomy, and synchronisation required for multi-vehicle, multi-drone routes. It also accommodates two alternative objectives (minimising the makespan or minimising total truck travel) to be chosen by the decision makers, which allows to align the optimisation goal with their specific operational priorities. By embedding both strategies—depot return and on-road resupply- —into a single formulation, we enable the strategy to be chosen by the solver itself with the same methodological tool. In practice, the problem can be approached with heuristic or matheuristic methods that incorporate the relevant load-transfer and synchronisation constraints. These constraints extend earlier capacity restrictions inspired by Kara et al. (2007) and drone battery models adapted from Gonzalez-R et al. (2020). Our emphasis is on demonstrating how each strategy performs under a consistent set of assumptions, rather than competing on solver implementations or exhaustive algorithmic benchmarks. In what follows we present the modelling choices and notation used (e.g., binary routing variables, load transfer variables, and time tracking variables) before addressing our design of the GAIER solution framework in Section 4. 3.3.1. Sets and parameters Sets •N: Set of nodes N= {0,1,2,⋯,n}, where 0 is the depot and {1,⋯,n} customers. •Λ: Set of arcs Λ= {(i,j)|i,j∈N,i∕= j}. •V: Set of ground vehicles V= {v1,v2},where v1denotes the single Drone-Commanding Truck (DCT) and v2(only active in the resupply strategy) denotes the resupply truck (RT). •D: Set of drones D= {d1,⋯,dm}, where each drone h∈Dexhibits homogeneous autonomy and load capacity features. Parameters •qi: Load demand at customer i. •Q(g) v1: Maximum load capacity of DCT. Fig. 2. CTmDTL resupply strategy routing diagram. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 9 •Q(g) v2: Maximum load capacity of RT (if used). •Q(a) h: Maximum capacity of any drone. • α : Maximum uninterrupted flight time (or distance budget) before needing to resynchronise with the DCT, also defined as autonomy of every drone in the homogenous fleet. • τ (a) i,j: Time that any aerial vehicle (drone) needs for moving from node ito node j. • τ (g) i,j: Time that any ground vehicle (DCT or RT) needs to travel from node ito node j. •disti,j: Distance for a vehicle traveling from ito j. •M: Sufficiently large positive constant used in constraints to enable or disable specific conditions based on the values of associated binary variables. 3.3.2. Decision variables Binary variables •β: Strategy selection flag, where β=0 indicates that the standard depot return strategy is chosen, and β=1 indicates that the on-road resupply strategy is chosen. •Ri: RT to DCT resupply node flag, where Ri=1 indicates that the DCT v1meets and receives load from the RT v2at node i, zero otherwise. •Zi,h: DCT and Drone meeting node flag, where Zi,h=1 indicates that the drone hmeets the DCT v1at node i, zero otherwise. This meeting needs to happen for drone landing, drone take-off, drone autonomy replenishment and/or drone capacity reload. •Xi,j,v: Truck arc flag, where Xi,j,v=1 indicates if truck vtravels directly from node ito node j, zero otherwise. •Yi,j,h: Drone arc flag, where Yi,j,h=1 indicates if drone hflies directly from node ito node j, zero otherwise. •S(g)i,v: Truck service flag, where S(g)i,v=1 indicates if truck vserves node i, zero otherwise. •S(a)i,h: Drone service flag, where S(a)i,h=1 indicates if drone hserves node i, zero otherwise. Continuous variables •L(g)i,j,v: weight loaded in vehicle vif it goes from ito j, zero otherwise. L(g)i,j,v≥0,L(g)i,j,v≤Q(g) v,∀(i,j) ∈ Λ,v∈V. •L(a)i,j,h: weight loaded in drone hif it goes from ito j, zero otherwise. L(a)i,j,h≥0,L(a)i,j,h≤Q(a) h,∀(i,j) ∈ Λ,h∈D. •Δi: amount of load transferred from v2(the RT) to v1(the DCT) at node i. Δ0=0,Δi≥0,∀i∈N, •θi,h: amount of load transferred from v1(the DCT) to drone hat node i. θ0,h=0,θi,h≥0,∀i∈N,h∈D, •T: total service time (makespan) by which all deliveries are completed, and all vehicles have completed their routes and returned to the depot. •Ai,v: arrival time of vehicle vat node i. •Di,v: departure time of vehicle vfrom node i. •Ai,h: arrival time of drone hat node i. •Di,h: departure time of drone hfrom node i. •b+i,h: remaining autonomy of drone hon its arrival to node i. b+i,h≥0,b+i,h≤ α ,∀i∈N,h∈D, •b−i,h: remaining autonomy of drone hon its departure from node i. b−i,h≥0,b−i,h≤ α ,∀i∈N,h∈D. 3.3.3. Objective functions Minimise Total Service Time (Makespan) min T.(1) This objective focuses on completing all deliveries as early as possible, measured by the time Tat which the last vehicle (truck or drone) finally returns to the depot. Minimise Total Truck Distance min(∑(i,j)∈Λdisti,j⋅Xi,j,v1+∑(i,j)∈Λdisti,j⋅Xi,j,v2).(2) This alternative objective focuses on reducing ground vehicle mileage by penalising every distance unit travelled by any of the trucks. In settings where fuel costs, vehicle wear, or carbon emissions are the primary concern, minimising total truck distance helps shift deliveries to drones or shorter truck trips, even if it does not necessarily minimise the overall completion time. 3.3.4. Key constraints A Single Servicing Vehicle per Customer ∑v∈VS(g)i,v+∑h∈DS(a)i,h=1,∀i∈ {1,⋯,n}.(3) Servicing Vehicle Must Visit the Customer S(g)i,v≤∑j:(j,i)∈ΛXj,i,v,∀i∈ {1,⋯,n},v∈V,(4) S(a)i,h≤∑j:(j,i)∈ΛYj,i,h,∀i∈ {1,⋯,n},h∈D.(5) Vehicles Flow Continuity ∑j:(i,j)∈AXi,j,v=∑k:(k,i)∈AXk,i,v,∀i∈ {1,⋯,n},v∈V,(6) ∑j:(i,j)∈AYi,j,h=∑k:(k,i)∈AYk,i,h,∀i∈ {1,⋯,n},h∈D.(7) Quantity of Vehicles Departures From the Depot Depending on the Strategy ∑(0,j)∈ΛX0,j,v1=2−β,∑(j,0)∈ΛXj,0,v1=2−β,(8) ∑(0,j)∈ΛX0,j,v2=β,∑(j,0)∈ΛXj,0,v2=β,(9) ∑(0,j)∈ΛY0,j,h=1,∑(j,0)∈ΛYj,0,h=1,∀h∈D.(10) Activation of Meeting Flags involving DCT Zi,h≤∑k:(k,i)∈ΛXk,i,v1,Zi,h≤∑k:(k,i)∈ΛYk,i,h,∀i∈N,h∈D,(11) Ri≤∑k:(k,i)∈ΛXk,i,v1,Ri≤∑k:(k,i)∈ΛXk,i,v2,∀i∈ {1,⋯,n}.(12) RT to DCT Load Transfer is Limited to Resupply Event Node Δi≤Q(g) v2⋅Ri,∀i∈ {1,⋯,n}.(13) RT to DCT Load Transfer Must be Performed at the First Node Visited by RT Δi+∑j∈{1,⋯,n}L(g)i,j,v2≤Q(g) v2⋅X0,i,v2,∀i∈ {1,⋯,n},(14) D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 16 different Two-Factor ANOVAs (one for each response variable) have been performed. This form of statistical analysis allows us to examine the effect of two different factors (Strategy and Instance Number) on each outcome simultaneously, ideally by rejecting the null hypothesis (there is no significant difference among the average response values for the strategies) and supporting the alternative hypothesis (there is significant difference among the strategies). Factors and factor interactions that have a significant impact are denoted with an asterisk (*) and are further emphasised through their pvalues. According to the methodology adopted and the predetermined significance level of 5 % for this study, factors or interactions that exhibit a p-value less than 0.05 are considered significant. Furthermore, the F-ratio values are indicative of the relative significance and impact of these factors or interactions on the overall performance of the system. In this case, the results of the ANOVAs reject the null hypothesis and provide support for the alternative hypothesis. Now, we can execute post-hoc checks to gain further insights and draw conclusions. Furthermore, Fig. 5 shows the marginal means plots, which facilitate the indication of an average effect for each level on every factor. It is important to note that, according to the evaluated response, smaller values mean better performance. A fast visual analysis of the figure allows to easily identify that in all cases the application of the resupply strategy to CTmDTL provides better results, both when looking at the makespan or truck mileage response variables. The rest of the factors’behaviour, while not statistically studied, is as expected: regarding the instance size factor, bigger size means also greater makespan and truck mileage values. The behaviour of the optimisation criterion factor is also logical: optimising for a specific response variable lowers the resulting values of the variable and provides worse results for the other response variable. At last, we refer the reader to Fig. 6 for an example of a GAIER execution as an example of GASGA convergence and the evolution of the algorithm’s performance. Noticeably, while the result is highly unstable in the initial iterations of the algorithm, it tends to stabilise after a certain number of iterations. No further improvement was identified after the shown value for this execution. 6. Discussion of the results When examining the conclusions drawn from the above ANOVA analysis, we must interpret the findings with careful attention to the operational implications while focused specifically on the strategy factor, as it is the main contribution of this paper. It is important to note that the selection of control parameters in our study was made according to preliminary pilot experimentation. This experimentation helped us identify suitable parameter values that optimise the performance of our proposed system within the given constraints. While we acknowledge that a comprehensive sensitivity analysis of these parameters could provide deeper insights for a specific case study, such an analysis falls outside the scope of the current work due to space limitations and the specific focus of our study. Future research could extend our findings by exploring the sensitivity of the system to a wider range of parameter variations. Table 8 DOE configuration. Factor Description Levels INInstance Number {61,62,⋯,90} OcOptimisation Criterion {Makespan,Mileage} SStrategy {Standard,Resupply} Response variables Description MMakespan TTruck Mileage DOE configuration Order of data collection Random Design 120-run Full Factorial Replications 5 Total trials 600 Alpha level 0.05 Table 9 Average metrics of strategies result for 5 replications of each instance. Strategy Nodes Optimising Average makespan Average truck mileage Standard 20 Makespan 401 1216 Truck Mileage 530 608 50 Makespan 659 3823 Truck Mileage 851 1785 75 Makespan 851 6775 Truck Mileage 1093 2868 Resupply 20 Makespan 245 1001 Truck Mileage 337 517 50 Makespan 516 2931 Truck Mileage 788 1266 75 Makespan 783 5369 Truck Mileage 1188 1924 Table 10 Makespan and Truck Mileage response variables Two-Factor ANOVA analyses. Response Variable M Source SS DF MS F p-value F crit S579,703 1 579,703 236.1 * <0.001 3.9 IN18,428,655 29 635,471 258.8 * <0.001 1.5 S×IN507,420 29 17,497 7.1 * <0.001 1.5 Within (error) 589,398 240 2456     TOTAL 20,105,175 299      Response Variable T Source SS DF MS F p-value F crit S34,456,138 1 34,456,138 295.4 * <0.001 3.9 IN583,097,627 29 20,106,815 172.4 * <0.001 1.5 S×IN17,675,021 29 609,483 5.2 * <0.001 1.5 Within (error) 27,992,719 240 116,636     TOTAL 663,221,505 299      D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 17 Concerning the makespan response variable (Table 10), strategy significance is pronounced with a p-value less than 0.001, indicating that changes in strategy affect completion time (namely, a feature of steadiness no matter that the impact is not the largest). The interaction of the strategy with the chosen instance is also statistically significant: significant F-ratios and low p-values indicate that the synergistic effects of these factors should not be ignored. In practical terms, this implies that the effectiveness of selecting a resupply strategy is contextdependent and can be enhanced or diminished by the levels of other factors in the operational setting. Turning to the truck mileage response variable (Table 10), the strategy factor remains statistically significant with a p-value less than 0.001. This reinforces the notion that selecting an optimal strategy has significant, influence on truck mileage. The instance factor interaction effects are also statistically significant in this case, suggesting that the impact of the strategy on truck mileage does vary with different instances as well. This can be quite insightful for operational decisions, as it suggests that the proposed strategy may not have a predictable standalone effect on truck mileage. In a broader last-mile logistics context, the consistent significance of strategy selection across different response variables denotes the potential of the resupply strategy as a lever for operational improvement. Furthermore, for operational research and logistics planning, the findings could be transformative, as they suggest that decision makers should also consider how the strategy interacts with other operational elements to maximise its full potential. At last, the uniformity of the effect of the strategy factor across different performance metrics implies that it could be a stable building block in the complex structure of lastmile logistics optimisation. 6.1. Direct comparison between strategies The metrics used for the quality assessment in this section, Makespan Improvement and Truck Mileage Improvement, both measured in percentage, are even comparison metrics for both strategies. The values are calculated by dividing each of the resupply strategy metric values by the standard strategy metric values for Makespan and Truck Mileage optimisation trials that correspond to the same demand pattern. Intuitively, both percentages are crucial in evaluating any possible advantages in operational efficiency and environmental impact when comparing strategies. The analysis of the same set of a total of 600 experiments used for generating Table 9 as well as in the DOE draws the results that can be Fig. 5. Makespan and Mileage response variables. Marginal means response averages plot. Fig. 6. Truck mileage evolution in 75 nodes size instance optimised using GASGA and resupply strategy. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 18 seen in Fig. 7. The high makespan improvement at lower node counts (38 % for 20 nodes) suggests that, for smaller delivery networks, the use of a resupply truck significantly reduces delivery times. This is likely due to the reduced need for the mothership truck to return to the depot, which can be time-consuming. As the number of nodes increases, the makespan improvement decreases until stabilising at approximately 20 % for both 50 and 75 nodes. This could be due to the increased complexity in coordinating delivery and resupplying as the size of the network increases. On the other hand, the improvement in mileage is steadier between different node sizes. In particular, for 75 nodes, mileage improvement is the highest (30 %), suggesting that, in denser networks, optimising for mileage can substantially reduce total truck travel distances, potentially leading to cost savings in fuel and vehicle maintenance. The negative improvement of makespan at 75 nodes when optimising for mileage (−10 %) implies that focusing solely on reducing mileage can lead to inefficiencies in time, possibly due to a more extensive use of the drones for performing deliveries, which directly implies more but also longer waiting times for the synchronisations. Finally, the interaction between the number of nodes and the applied optimisation strategy is complex. At lower node counts, makespan improvement is dominant, but as the number of customers grows, the advantage shifts towards mileage improvement. This change may be influenced by factors such as routing complexity and logistical challenges of coordinating multiple deliveries in the same area while respecting all constraints. 6.2. Managerial lessons Certainly, the previous comparison analysis offers valuable insights that can be translated into actionable managerial lessons that are presented henceforth. 6.2.1. Efficiency in small-scale operations For operations with fewer delivery nodes (e.g., 20 nodes), the significant increases in the makespan (38 %) indicate the opportunity to greatly reduce delivery times by incorporating a resupply truck. In timesensitive scenarios (e.g., perishable goods), using a resupply truck to avoid depot returns significantly increases efficiency. Additionally, the minimisation of mileage is also an improvement source when coping with low number of customers, as it also leads to savings in operational costs. 6.2.2. Strategies for larger networks As the number of customers increases (e.g., 75 nodes), the bigger opportunity for improvement gradually shifts towards optimising truck mileage, where we see the highest improvement (30 %). This approach can substantially reduce travel distances, leading to lower fuel costs and reduced vehicle wear and tear. However, the negative effect on makespan at higher node counts when focusing solely on mileage underscores the need for a balanced strategy. Managers are advised to approach optimisation with a holistic perspective, ensuring that emphasis on one operational aspect does not inadvertently compromise the overall balance and efficacy of organisational performance. It is important to proceed with caution, recognising the intricate interplay between different operational dimensions, to achieve a harmonious outcome. 6.2.3. Adaptive planning and route optimisation The results suggest that the optimal strategy might vary depending on the size of the delivery network. Managers should employ adaptive route planning tools that can adjust strategies based on daily delivery volume and network size. Definitely, using advanced routing algorithms like the GAIER we have applied in our research will be helpful in dynamically discovering routing planning with feasible values for both, makespan and truck mileage. 6.2.4. Economic impact The application of the resupply strategy demonstrates substantial improvements in both makespan and truck mileage across various scenarios. This translates to economic gains by reducing the number of trips required, optimising trucks usage, and increasing the number of deliveries within a given period. These improvements are indicative of the resupply strategy’s potential to significantly reduce operational costs and enhance the overall value created in the delivery process, which aligns with findings by Xiao et al. (2024), who demonstrated the importance of dynamic drone resupply in enhancing the efficiency and responsiveness of same-day delivery systems. Fig. 7. Resupply Improvement vs. Number of Nodes and Optimisation Target. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 19 6.2.5. Broader implications for practical applications The findings from our study have significant implications for practical applications in truck routing optimisation. Our multi-modal delivery model, which includes a resupply truck, demonstrates clear benefits in reducing delivery times while controlling costs. This strategy is particularly advantageous in industries where same-day delivery is critical, such as pharmaceuticals and perishable goods, as these industries often face tight delivery windows and heavy traffic, where drones can bypass congestion and improve efficiency. By reducing the need for trucks to return to the depot, the strategy enables faster, more flexible deliveries, making it a viable solution in dense, high-demand areas. By limiting the fleet size of trucks equipped with drones and introducing a simple resupply van, our model enhances delivery efficiency with a modest increase in operational costs. This approach not only optimises resource utilisation but also provides a scalable solution adaptable to various delivery contexts, promising improved service levels and customer satisfaction. Additionally, the 17 % reduction in makespan and 21 % decrease in truck mileage demonstrate that the proposed strategy can potentially lead to real-world benefits, including faster deliveries, reduced fuel consumption, and lower operational costs. These improvements also support sustainability goals by cutting emissions and extending vehicle lifespan. Overall, the resupply strategy enhances both economic efficiency and environmental impact, providing a clear advantage for logistics companies. 7. Conclusions and future research In an age in which the shorter delivery time is more and more becoming a critical factor in fast-paced same-day delivery scenarios—e. g., delivery of meals and perishable goods—we propose exploiting a hybrid truck-drone system but adding an auxiliary resupply truck to prevent the primary truck’s detour back to the depot. Different to other recent approaches relegating the resupply role only to drones, we reserve it to a ground vehicle that serves the purpose of providing the main truck (mothership truck for launching and retrieving of the drones under its control) with the newly arrived orders. Building on the CTmDTL formulation, we have developed a unified framework to identify the optimal strategy while adhering to capacity constraints, drone autonomy, and route synchronisation under two distinct strategies for generating solutions that reduce makespan or truck mileage in realistic same-day delivery networks. We have compared the standard and the resupply strategies through detailed analyses, bringing to light the significant improvements in both efficiency and environmental sustainability that can be achieved by integrating a resupply truck into a truck-multidrone logistics system. The use of a resupply truck allows for a considerable decrease in the makespan within small-scale delivery networks, which also arises when addressing bigger delivery scenarios but with lower intensity. In the latter, the operational gaining shift towards the savings in truck travel total distance, fuel usage, and vehicle maintenance emerging from the lower truck mileage attained. Thus, our study aims to provide actionable insights for decision makers through an in-depth sensitivity analysis. The economic analysis of our approach indicates that the resupply strategy not only improves delivery efficiency but also results in notable cost savings. By minimising the number of depot returns and optimising the use of delivery resources, this strategy enhances the creation of value for logistics companies, offering a clear pathway to increased profitability and competitive advantage. In this sector where timely and reliable delivery is key, the enhancements brought forth by the resupply strategy can significantly improve customer experience, therefore offering businesses a competitive edge that can become a key differentiator. This study also contributes to the fields of logistics and supply chain management by validating the theoretical proposition that integrating resupply strategies enhances operational efficiency in small-scale delivery networks. The findings support dynamic routing and real-time adaptive logistics strategies, demonstrating significant improvements in makespan and truck mileage. Additionally, the research highlights the need for balanced optimisation in larger networks, contributing to theoretical models that address trade-offs between operational metrics such as makespan and mileage. The application of advanced routing algorithms, specifically the GAIER algorithm, reinforces the theoretical frameworks advocating for sophisticated computational techniques to optimise delivery routes dynamically. Moreover, our study offers a foundation for economic impact and cost-benefit analysis in logistics aiming to quantify cost savings and efficiency gains from resupply strategies in particular cases. The insights into multi-modal delivery models, particularly those incorporating resupply trucks and drones, contribute to emerging discussions on hybrid delivery systems and their scalability. By providing a foundation for analysing the scalability and adaptability of different logistical strategies, our research bridges a gap and opens new possibilities to the incorporation of resupply-based strategies, enhancing both academic knowledge and practical outcomes in logistics optimisation. In this direction, future research could build on our findings by exploring specific cases, long-term impacts and cross-industry applications. In the realm of future research, emerging approaches, such as Generative Artificial Intelligence (GAI), also present exciting avenues for exploration. While GAI has shown remarkable capabilities in data generation and predictive tasks, its application in solving complex combinatorial optimisation problems, such as the truck-drone delivery problem, remains underexplored. Future work could investigate how GAI could complement existing methods like GAIER by generating dynamic delivery scenarios or creating hybrid optimisation frameworks possibly based on enhanced mathematical models—for example, from adding multiple auxiliary trucks or partial, multi-node resupply rules. Although GAI is not yet widely adopted for these specific problems, it may offer novel insights for addressing larger, more complex networks or real-time adaptive logistics environments. Besides, our analysis further supports the need for adaptive planning and dynamic route optimisation, catering to the rapidly evolving last-mile logistics landscape. As the last-mile logistics business landscape is rapidly evolving, the ability to modify strategies based on fluctuating delivery volumes and network sizes is critical. While no direct empirical comparation between GAIER and other methods is provided in this study, it is crucial to note that the effectiveness of GAs in solving combinatorial optimisation problems similar to the truck-drone delivery problem has been extensively documented in the literature, where their use showed significant improvements in solution quality and computational efficiency. Future research could explore further economic implications, risk studies, comparative analyses or better initialisation values for algorithm parameters, as well as the integration of multiple resupply trucks to further enhance system efficiency and flexibility. This approach may offer increased scalability for larger or more dynamic delivery regions, potentially improving operational dynamics and environmental sustainability. Such developments could further refine our strategy, promising even greater improvements in delivery performance and customer satisfaction. In conclusion, our resupply strategy for truck-with-multidrone system offers a promising approach to optimising last-mile delivery operations. This model substantially improves delivery times and lowers costs, making it highly applicable to various delivery contexts while increasing logistics performance and customer satisfaction. This research makes a significant contribution to the field of last-mile logistics, proposing the resupply strategy as an effective method to improve operational efficiency and environmental sustainability which carries practical implications for the logistics industry. These findings set the stage for further research and development of strategies to optimise lastmile delivery in the increasingly complex logistics environment. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 20 Declaration of Generative AI and AI-assisted technologies in the writing process During the preparation of this work the authors used Generative Pretrained Transformer 4 (GPT-4) in order to improve language and readability. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication. CRediT authorship contribution statement Pedro L. Gonzalez-R: Conceptualization, Formal analysis, Visualization, Supervision. David Sanchez-Wells: Methodology, Software, Validation, Computational Evaluation, Writing –original draft. Jos´ e L. Andrade-Pineda: Conceptualization, Formal analysis, Investigation, Resources, Data curation, Writing –review &editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. . Design of GASGA genetic algorithm The Genetic Algorithm Sanchez-Gonzalez-Andrade (GASGA) is designed to address complex optimisation problems by leveraging a dual-vector individual representation, specialised genetic operators, and adaptive mechanisms. GASGA, which flowchart is represented in Fig. 8, is particularly effective for problems where solution components can be naturally divided into distinct vectors, such as sequence and modes vectors. This annex provides a comprehensive and code-extensive read of the GASGA algorithm, detailing its implementation and unique features. A.1 GASGA key insights GASGA adopts a novel approach to the traditional single-vector individual representation by utilising two distinct vectors: a sequence vector (nodes vector) and a modes vector. The nodes vector represents a permutation sequence where each element is unique, reflecting a sequence of locations. In contrast, the modes vector allows repetitions, representing a set of visit modes. A.1.1 Dual-Vector representation •Nodes Vector: The nodes vector is always a permutation vector where no repetition is allowed, ensuring a unique sequence of locations. This uniqueness is crucial for maintaining the integrity of the sequence, which is typically required in routing problems and other combinatorial optimisation challenges. •Modes Vector: The modes vector does not require uniqueness, allowing for repeated visit modes across the vector. This flexibility enables the algorithm to represent scenarios where multiple visits to the same mode or state are possible, which can be critical in scheduling and allocation problems. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 21 Fig. 8. GASGA Algorithm Flowchart. A.1.2 Specialised genetic operators •Operators for Nodes Vector: Crossover operators such as Uniform Partially Matched, Partially Matched, and Ordered are chosen to preserve the permutation property. These operators ensure that the offspring maintain the necessary constraints of the nodes vector, facilitating effective exploration of the solution space without violating the problem’s constraints. •Operators for Modes Vector: Crossover operators like Two Point and Uniform are selected for their ability to effectively combine vectors without avoiding repetitions. These operators allow for the flexible recombination of modes, which is essential for exploring different combinations and configurations in the solution space. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 22 A.2 Detailed GASGA implementation This section provides an extensive, detailed presentation of the GASGA algorithm. A.2.1 Population initialisation The initialisation of the population involves generating a set of random individuals that serve as the starting point for the evolutionary process. Each individual in the population is composed of a sequence vector and a modes vector, adhering to the problem’s specific constraints and requirements. This initial population forms the basis for the subsequent evolutionary operations. It is generated randomly, having into account the constraints of both the nodes and modes vectors. A.2.2 Evaluation function The evaluation function assesses the fitness of each individual in the population. This function is crucial as it determines how well an individual solves the problem at hand. The fitness evaluation considers both the nodes vector and the modes vector, ensuring that the solution respects the unique constraints of each vector. The process involves generating routes, resolving the routes to compute objectives and penalties, and summarising the overall fitness score. Higher fitness values indicate better solutions, guiding the selection process for future generations. Algorithm 3 GASGA Evaluation Function 1: Receive individual 2: Copy nodes and modes vectors from the individual 3: Insert starting and ending points into the vectors 4: Generate routes for the vehicles using the nodes and modes vectors 5: Resolve the routes to compute the objective value and penalties 6: Compute the overall fitness score based on objective and penalties 7: Assign the fitness score to the individual 8: end function A.2.3 Hall of fame initialisation The hall of fame is initialised to keep track of the best individuals encountered during the evolutionary process. This mechanism ensures that the highest quality solutions are preserved and available for comparison against current population members. The hall of fame provides a way to monitor the progress of the algorithm and guarantees that the best solution found can be returned at the end of the process. Its initialisation is as straightforward as creating an empty list. A.2.4 Main evolution loop The main evolution loop is the core of the GASGA algorithm, where the evolutionary process iterates through generations. In each iteration, parents are selected using a stochastic selection method, and offspring are generated through crossover and mutation operations. The new offspring are then evaluated, and the population is updated. The hall of fame is also updated with the best individuals from the current generation. This loop continues until a termination condition is met. Algorithm 4. GASGA Main Evolution Loop 1:while elapsed_time <time_limit do 2: Select parents from population using stochastic selection method 3: Initialise offspring as an empty list 4: for each pair of parents do 5: Determine vector type (nodes or modes) for crossover and mutation 6: Select appropriate crossover operator based on vector type 7: Apply crossover to produce children with probability cxpb 8: Select appropriate mutation operator based on vector type 9: Apply mutation to children with probability mutpb 10: Add modified children to offspring 11: end for 12: Evaluate offspring 13: Update population with offspring 14: Update hall of fame with best individuals from current generation 15: Adjust crossover_prob and mutation_prob based on performance 16:end while A.2.5 Adaptive mechanism The adaptive mechanism dynamically adjusts the probabilities of crossover and mutation based on the evolutionary context. If the algorithm detects stagnation or insufficient progress, it increases these probabilities to inject more diversity into the population and encourage broader exploration. Conversely, if improvements are consistently being made, it reduces these probabilities to allow for more intensive exploitation of promising areas of the solution space. This self-tuning capability ensures a balanced approach to exploration and exploitation throughout the evolutionary process. D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 23 Algorithm 5. GASGA Adaptive Mechanism 1:if stagnation_detected or insufficient_progress then 2: Increase crossover_prob and mutation_prob 3:else if consistent_improvement then 4: Decrease crossover_prob and mutation_prob 5:end if A.3 Randomised deep search integration An in-depth look at the Randomised Deep Search heuristic and its integration into GASGA. A.3.1 Randomised deep search The Randomised Deep Search heuristic is a sophisticated search strategy designed to enhance the GASGA algorithm’s efficiency. It introduces problem-specific knowledge and strategies into the evolutionary process, enabling more effective navigation through challenging regions of the search space. This heuristic is conditionally applied, depending on specific criteria within the evolutionary loop, ensuring that it complements the genetic operations without dominating them. Check Algorithm 2 in the paper for the Randomised Deep Search pseudocode. A.3.2 Conditional application The conditional application of the Randomised Deep Search heuristic ensures that it is used judiciously within the main GASGA loop. The heuristic is triggered based on certain conditions, such as stagnation in the evolutionary progress, specific performance thresholds or with a certain probability. This strategic application helps the algorithm escape local optima and explore more diverse solution spaces, improving the overall quality of the solutions found. A.4 Crossover and mutation operators Detail of the various tailored crossover and mutation operators used in GASGA. A.4.1 Tailored Ordered crossover (OX) Used for the nodes vector to ensure that the permutation property is maintained. This operator swaps sub sequences between two parent individuals, producing offspring that inherit the sequence structure while preserving uniqueness. The tailored OX maintains the core idea of selecting two crossover points and copying the segment between them from one parent to the offspring. Additionally, it uses a function to ensure the rest of the positions are filled with genes from the other parent while preserving the order and avoiding duplicates. This tailored approach emphasises maintaining the permutation properties and handles the complexity of filling the remaining positions explicitly. A.4.2 Tailored Partially Matched crossover (PMX) Another operator for the nodes vector that ensures permutations are preserved. It swaps segments between parents and maps remaining elements to maintain the sequence’s validity. The tailored PMX follows the same initial steps of selecting two crossover points and copying the segment. It includes an explicit function to handle the mapping of values between the parents, ensuring that conflicts are resolved, and no duplicates are present in the offspring. The tailored version places a clear emphasis on the process of swapping values to maintain permutation properties, providing a structured way to manage conflicts. A.4.3 Tailored uniform partially matched crossover (UPMX) This crossover operator randomly selects elements to swap between parents, preserving the permutation property and creating diverse offspring. The tailored UPMX applies a probability to decide whether to swap each gene. It also includes an explicit function to manage the swapping process, ensuring that the permutation properties are preserved, and no duplicates are introduced. This tailored approach emphasises maintaining the integrity of the sequence by using structured swapping of values, providing clarity and consistency. A.4.4 Non-tailored two point mutation Applied to the modes vector, allowing changes at two randomly selected positions. This operator introduces variability without violating the vector’s repetition constraints. A.4.5 Non-tailored uniform mutation Randomly alters elements in the modes vector, introducing new values while respecting the vector’s characteristics. This mutation helps explore different combinations of modes. A.5 Summary and conclusion The Genetic Algorithm Sanchez-Gonzalez-Andrade (GASGA) represents a significant advancement in the field of genetic algorithms, particularly for complex optimisation problems that benefit from a dual-vector representation. This annex has detailed the various components and innovative features of the GASGA algorithm, providing a comprehensive understanding of its implementation and functionality. Key points of the GASGA algorithm include: D. Sanchez-Wells et al. Expert Systems With Applications 272 (2025) 126757 24 •Dual-Vector Representation •Specialised Genetic Operators •Dynamic Crossover and Mutation Strategies •Adaptive Mechanism •Randomised Deep Search Integration In conclusion, GASGA’s innovative aspects—from its dual-vector individual representation and dynamic operator selection to its adaptive genetic operations and the integration of deep search heuristics—collectively contribute to a highly adaptable and efficient algorithm. GASGA is adept at tackling complex optimisation problems by balancing nuanced solution space exploration with targeted exploitation and problem-specific interventions. These features enhance the algorithm’s ability to deliver high-quality solutions, making it an invaluable tool in genetic algorithm research and optimisation. 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