In order to counteract the effect of the global warming, the United Nations has established the Sustainable Development Goals (SDGs). Specifically, a systemic effort has been addressed to the greenhouse gas (GHG) emissions, whose main goals consists in reducing the emission level to a zero net by the 2050.

The transport sector is one of the heavy demands for energy consumption; it requires a large amount of natural resources that traditionally come from non-renewable sources. For such reason, traditional vehicles (e.g., cars, and commercial vehicles) largely contribute to greenhouse emissions and air pollution, thus affecting both the environment and the health. In an attempt to reduce these impacts, several scholars and public administration encouraged new polices and solutions to favor the modal shift for both commuters (Comi and Polimeni, 2024; Nigro et al., 2024) and freight (Comi et al., 2024; Comi and Russo, 2022; Russo and Comi, 2023).

The introduction for digitalization and information communication technology (ICT) enabled the possibility to integrate the classical delivery problems thorough a new dynamic paradigm (Comi and Russo, 2022) and to introduce novel solutions towards a sustainable urban logistic (Knapskog and Browne, 2022). Of course the problem requires adequate models and methods (Cattaruzza et al., 2017; Crainic et al., 2023). Among the wide range of opportunity offered by innovation technology, the electric autonomous delivery robots represent a promising alternative in last-mile delivery (Arntz et al., 2023; Khoufi et al., 2019; Thibbotuwawa et al., 2020). Also, cooperative solutions, such as the integration between traditional vehicles and drones, have been evaluated in order to perform last-mile delivery to customers (Kyriakakis et al., 2023; Marinelli et al., 2018; Ren et al., 2023; Sacramento et al., 2019; Xiao et al., 2024). Although there are notable advantages in terms of efficiency and environmental sustainability compared to the performance offered by fossil fuel fleets, coordination between the two vectors still represents an open problem (Macrina et al., 2020).

The studies that dealt with the transition from fossil fuel-powered vehicles to electric vehicles (EVs) highlighted, first, the possibility of significantly reducing the current dependence on non-renewable energy sources (Jones et al., 2021; Napoli et al., 2021). As a matter of fact, in general, the EVs are responsible for the emission of fewer greenhouse gases (compared to traditional vehicles), especially in those cases where charging energy comes from renewable sources (such as solar or wind power).

In this paper, according to environmental sustainability issues, a freight distribution model based on the use of EVs, with charging system located at the depot, is proposed. The delivery plan represents the most critical phase in delivery operations; thus, customer visit scheduling is at the core of planning activities to minimize travel times, travel costs, and environmental impacts of freight delivery. This problem is well-known in literature as vehicle routing problem (VRP). Since its first formulation (truck dispatching problem, Dantzig and Ramser, 1959) this topic has been enriched with various formulations and solution procedures. In terms of formulation, a big efforts has been underpinned by focusing on the objective functions and problem constraints (e.g., Toth and Vigo, 2002). Concerning the objective function, the mathematical expression contains the components to be optimized (the travel time, the cost, and so on) and the decision variable(s). Regarding the constraints, alongside the classical ones (e.g., on variables, on size/capacity of the vehicle, on the route length), it is also relevant to mention other more specific formulations such as the VRP with time window (VRPTW, e.g., Bräysy et al., 2004) or the vehicle routing problem with delivery and backhaul options (VRPDB, e.g., Koç and Laporte, 2018). Concerning the solution procedures, as first, it is possible to share between the exacts and heuristics ones. An exact procedure provides an optimal solution of the problem, but often with a high computational effort and an unacceptable computation time, therefore the possibility to recur to an exact procedure is deeply connected the problem dimensions. However, over the years, thanks to the new powerful computer computing capacity, the analysts have extended the exact procedures to larger case studies. Likewise, the computational efforts pushed the development for the heuristic procedures, ranging from constructive algorithms (e.g., Clark and Wright algorithm) to more sophisticated ones (e.g., tabu search, simulated annealing, swarm optimization, and genetic algorithm). In this sense, formalizing a problem requires a careful analysis between solution accuracy and computational times (processing time) needed to obtain it. Within the set of heuristic procedures, a particular class is constituted by the metaheuristics, approximated procedures designed both for discrete and continuous variables (Dreo et al., 2006). Among these metaheuristics:

• Tabu search (Glover and Laguna, 1997) is an adaptive procedure; it is based on local search procedures that implement principles to avoid falling into local optimal solutions;

The model presented in this work is part of a more general framework, conceived as a two-level service: the first level is the urban freight delivery as usual; the second level consists of a service procedure to provide installation services or collect packaging. Solving this joint problem brings benefits both to the company (which can thus separate the delivery of the installation, thus being able to better employ the operators) and to the consumers (who, for example, no longer have the problem of getting rid of bulky packaging). In both cases, the problem can be formalized as a vehicle routing problem. In particular, in this work, a VRP with soft time windows (VRPSTW, Balakrishnan, 1993; Taillard et al., 1997) is formulated. Thus, each customer is associated with a time window, and the formulation allows a penalty into the vehicle cost function when the time window is not violated. In such a way, taking into account the delivery window, the formulation is not just a one-size-fits-all solution, and it makes the solving solution adaptable to different delivery scenarios, making it applicable in the field of urban freight delivery. Particularly, in the calculation of the objective function, the travel time, the service time (e.g., the time to carry out the parking, to make the delivery and/or provide the installation), and the penalty time (which is greater than zero if the freight vehicle arrives at the user before/after the time window opens/closes) are considered. This study also considers a constraint linked to battery capacity. In fact, the vehicles that make deliveries are fully electric, and it is assumed that each route can be operated without recharging the vehicle and that each vehicle starts from the depot with a full battery charge.

Therefore, this work offers some insights into the use of GA and a particular type/class of population-based SA (named PBSA) to solve the problem. A version of GA and PBSA, where a certain percentage of the initial population is generated by the Clarke and Wright (CW) algorithm, is proposed, and different combinations of algorithms parameters are tested. So, this paper is focused on testing these procedures, and provides a case study/toy model by considering only the first level of the framework. The twofold objective to test a procedure that allows finding a good solution (even if it may not be the optimal one) and that allows for minimizing energy consumption, thus improving vehicle performance and contributing to sustainable development.

The paper is structured as follows: Section 2 reports a concise literature review; Section 3 contains the formulation of the problem, while Section 4 relies on algorithms; Section 5 reports the results of a case study with the associated discussion; and, finally, Section 6 draws conclusions.

In the last 60 years, many variants of the VRP have been formulated and different procedures have been designed to solve them (Laporte, 2009). The success of this research topic also depends on the fact that it deals with the real and ever-present problem of urban goods distribution. Over the years, variants have been created not only to consider specific constraints (e.g., time windows) of the problem, but also to integrate new technologies (electric vehicles, drones).

In the context of the use of electric vehicles, in literature are provided formulations ad approaches to consider the peculiarity of the problem (Froger et al., 2019). Schneider et al. (2014) introduced the electric vehicle routing problem (EVRP) with time windows developing a heuristic approach to solve it. This problem can be extended by considering a fleet of heterogeneous electric vehicles, varying the capacity and the range (Hiermann et al., 2016). In more detailed models, energy consumption is a function of vehicle load (Lin et al., 2016; Goeke and Schneider, 2015). An aspect of the EVRP is the limited range of the vehicles respect to the traditional fueled vehicles. This implies that the vehicle may be charged during service, and the recharge can be full or partial (Desaulniers et al., 2016; Erdelić et al., 2019; Felipe et al., 2014; Keskin and Çatay, 2016). Other solutions explored in literature, alternative to recharge, are the battery swapping (Qian et al., 2024; Ren et al., 2023; Verma, 2018) and the use of the range as a constraint (Napoli et al., 2021). The current literature on EVRP and its variants is explored in depth by Kucukoglu et al. (2021).

In the context of the city logistics, the problem variants are formulated to consider specific aspects of the urban freight distribution. As an example, more cities impose restriction in accessing some areas (e.g., historical centers), in this case a suitable formulation of the problem is the VRP with Access Time Windows (Grosso et al., 2018; Zhou et al., 2024). Another problem formulation arise from the use of one (or more) urban distribution center(s) (Browne et al., 2005). Some authors formulated this problem considering two aspects: the location of the distribution center (Muñoz Villamizar et al., 2014) and the optimization of the routes (Cepolina and Farina, 2016; Musolino et al., 2019). In general, in this case, the vehicles are often eco-friendly vehicles with restrictive constraints on capacity and range (Díaz-Ramírez et al., 2023). A further aspect to consider in urban areas is that the travel time is not constant during the day, in this case the formulation of the problem takes into consideration this aspect (Ando and Taniguchi, 2006; Musolino et al., 2018). A recent challenge in VRP formulation is the use of unmanned vehicles in performing the service. Such vehicles can be ground robot (Chen et al., 2021; Chirala et al., 2023) or aerial drones (Huang et al., 2022; Kyriakakis et al., 2022). The use of unmanned vehicles imposes further constraints respect to the classical VRP. As an example, aerial drones can deliver the parcels up to a certain weight, there are limits due to the technology (e.g., the weight of the batteries), their (often limited) range depends on the load (this also applies to ground robots). Besides, aerial drones can be affected by weather, ground robots by congestion (Khoufi et al., 2019).

The solution procedures used to solve the VRP (and its variants) range from exact to heuristic algorithms. The following review skips the topic of exact algorithms and focuses only on heuristics and metaheuristics (without the presumption of being exhaustive).

Concerning constructive procedures, a first heuristic that can be cited is the Clarke and Wright (CW) algorithm, a procedure designed to produce a solution that maximizes a variable called saving: The aim is to put customers in the solution to maximize the ‘saved’ cost due to the aggregation of customers. Proposed by Clarke and Wright (1964), this algorithm was improved by introducing a parametric approach (Gaskell, 1967; Yellow, 1970) and is often used in combination with other procedures. As an example, Caccetta et al. (2013) hybridized the algorithm with a domain reduction procedure, demonstrating that the hybrid procedure significantly improves the results obtained. Robbins and Turner (1979) combined the CW algorithm with a 2-opt procedure: the objective is to update the solution provided by CW (the CW procedure individuates the solution rigidly, without further updates). Other examples of constructive heuristics are the sweep algorithm (Gillett and Miller, 1974) and the petal algorithm (Foster and Ryan, 1976; Renaud et al., 1996).

Regarding the improvement proposed in the domain of heuristic procedures, local search (intended as an approach that can explore the search space starting from an initial solution and trying to improve it) and its variations are often used to solve the VRP (e.g., Erdoğan, 2017). Mladenović and Hansen (1997) introduced the concept of variable local search, in this case the idea is to change the neighborhood during the search operations. Toth and Tramontani (2008) proposed a local search algorithm in which the neighborhood of the solution is explored using an integer linear programming procedure. Brandão (2020) defined an iterated local search procedure capable of remembering previous moves, memory use allows for more efficient exploration of the solution space and can prevent falling into local optima. Ropke and Pisinger (2006) and Pisinger and Ropke (2019) proposed an adaptive large neighborhood search heuristic that uses some methods for removing/inserting customers in the solution until a stopping criterion is met. This procedure can be used to solve different variants of the VRP (Pisinger and Ropke, 2007).

The Tabu Search (TS) (Glover, 1989) is an iterative algorithm with memory that allows us to improve an initial solution by applying a certain number of moves that cause local changes in the current solution (Cordeau and Laporte, 2005; Brandão, 2009). Cordeau et al. (1997) and Cordeau et al. (2001) proposed a TS algorithm capable of solving the VRP and some variants, the aim being to have a simple procedure that reduces the number of parameters to consider. Jia et al. (2013) proposed an improved form of TS by adding some local search strategies and a mutation operator. A further possible improvement of the TS consisted in the introduction of a parallel approach, which allows one to simultaneously consider more than one neighborhood of the solution (Badeau et al., 1997; Caricato et al., 2003; Cordeau and Maischberger, 2012; Garcia et al., 1994).

Ant Colony Optimization (ACO) simulates the movements of a set of artificial ants; each ant is independent from the others and exchanges information using a trail of pheromones. The generic ant chooses which node to reach next, depending on the amount of pheromones. The solution is thus built incrementally, each time adding an element to a partial solution. Donati et al. (2008) proposed a parallel approach in which two colonies of ants operate in parallel, optimizing two different aspects of the problem. Yu et al. (2009) introduced a mutation operator in construction to move users from one solution to another.

Simulated Annealing (SA) was originally proposed by van Laarhoven and Aarts (1987), the aim is to minimize a function that makes changes to an initial solution (the results of the algorithm are based on a set of operators used to modify the solutions). A new solution is accepted as the current solution with a certain probability (derived from the Boltzmann distribution): this implies that it is possible to choose a worse solution than the best one found (this mechanism allows us to better explore the space of solutions). Some authors (e.g., Bräysy et al., 2008) used a deterministic approach, with the aim of speeding up the procedure by eliminating randomity due to the probabilistic choice. Osman (1993) proposed a SA procedure in which the generation of new solutions is based on the swap/shift of users from one route to another. Yu et al. (2009) proposed an SA with a random choice of operators used in the search for solutions, a better solution than the current is automatically chosen as the current solution, and a worse one is chosen with a probability obtained from a Boltzmann distribution (that is, the solution is chosen if the probability is higher than a threshold value). However, simulated annealing (SA) is a solo-search algorithm, and the results found by SA depend on the selection of the starting point and the decisions to move to the new solution or not SA makes. To overcome the drawbacks of being trapped in local minima and taking a long computational time to find a reasonable solution, Askarzadeh et al. (2016) proposed a population-based simulated annealing algorithm (PBSA), in which each solution memorizes its best experience and stores it in the population memory. Additionally, Shaabani & Kamalabadi (2016) used a PBSA algorithm compared to genetic algorithms and simulated annealing, which shows the superiority of the PBSA algorithm.

The Genetic Algorithm (GA) simulates, in a simplified manner, the evolution of life forms employing three main operators: selection, crossover, and mutation (Goldberg, 1989). Thus, the basis of the GA is the idea of evolving a population of solutions until a stopping test (for example, the number of iterations) has been satisfied. This algorithm, given its flexibility, is suitable for solving the VRP and different implementations (in terms of operators) have been proposed so far. As an example, Baker and Ayechew (2003) proposed a basic genetic approach demonstrating that GA is competitive (in terms of computation time and solution quality) with other solution procedures. Alba and Dorronsoro (2006) developed a cellular GA (a form of genetic algorithm in which the solutions are placed on a grid and each of them can interact only with those in his neighborhood) to solve the VRP. Yusuf et al. (2014) tested the use of a rank-based operator for selection and a different crossover operator to solve VRP. Nazif and Lee (2012) implemented an improved crossover operator capable of generating two children at the same time. Berger and Barkaoui (2003) proposed a hybrid form of GA to solve the VRP, the basic idea is to develop two populations simultaneously (swap of solutions from one population to another is possible) to minimize the total distance traveled. Ho et al. (Ho et al., 2008) also proposed a hybrid form of the algorithm, combining GA with some heuristics to improve convergence. Vidal et al. (2012) hybridized GA with the introduction of the education operator: it is a form of mutation that uses local search to improve the solution. Ochelska-Mierzejewska et al. (2021) performed experiments to determine the best combination of genetic operators to solve the VRP. The recent lines of research tend to improve the solutions and find better individuals combining the genetic algorithms with deterministic or heuristic methods or combining the work of the genetic algorithm with other metaheuristics, such as Deterministic Annealing or Tabu Search. For example, Xu et al. (2011) proposed an improved GA to solve the classical VRP, incorporating SA into GA. Zhu et al. (2021) proposed an improved neighbor routing initialization method for the adaptive elitist genetic algorithm. Mrad et al. (2021), instead, proposed a two-step procedure that uses GA to find the assignment of companies to depots and CW algorithm to determine the routes from each depot to customers.

Let G (N, A) be a directed graph, where N = 0, 1,…n, is the set of vertex and A = {(i, j): i ≠ j; i, j ∈ N} the set of edges (an edge represents the path connecting two vertex). At each edge (i, j), there is an associated travel time t _ij . The set N contains the customers and the depot d (labeled vertex 0). At each vertex there are associated (Figure 1):

• a service time (s _i ),

A fleet V of vehicles, with homogeneous capacity Q ^v is involved in the process. The problem consists of designing a set of vehicle routes where each costumer is reached only once by a single vehicle. A soft constrained time windows are assumed, and a vehicle pays a penalty p _i for late/early arrival at vertex i.

The formulation is provided in Equation 1, where it is defined as a minimization problem, the objective being to minimize the sum of the time components introduced above: F = ∑ i ∈ N ∑ j ∈ N ∑ v ∈ V t i j v + s i v + p i v ⋅ x i j v (1)

subject to: ∑ v ∈ V ∑ j ∈ N x i j v = 1   ∀ i ∈ N c ; i , j ≠ d (2) ∑ v ∈ V ∑ j ∈ N x d j v = V   j ≠ d (3) ∑ v ∈ V ∑ j ∈ N x j d v = V   j ≠ d (4) ∑ i ∈ N ∑ j ∈ N q j ⋅ x i j v ≤ Q v   ∀ v ∈ V ; i ≠ j (5) ∑ i ∈ N ∑ j ∈ N h i j ⋅ x i j v ≤ R v   ∀ v ∈ V ; i ≠ j (6) ∑ i ∈ N ∑ j ∈ N t i j v + s i v + p i v ⋅ x i j v ≤ D v   ∀ v ∈ V ; i ≠ j (7) τ v ≤ τ 0   ∀ v ∈ V (8) p i v = a i − t i v   i f   t i v ≤ a i t i v − b i   i f   t i v > b i ∀ i ∈ N , i ≠ d (9) x i j v ∈ 0 , 1 (10) where:

• N is a set including the customers and the depot;

Equation (1) defines the objective function, whose purpose is to minimize the cost of all routes. Constraint Equation 2 requires that a user must be reached by only one vehicle. Constraint Equation 3 imposes that all vehicles start from the depot, while constraint Equation 4 imposes that all vehicles return to the depot. Constraint Equation 5 is on the vehicle capacity and indicates the total quantity delivered is less than the vehicle capacity. Constraint Equation 6 indicates that the maximum length of a route is less than the vehicle range, while constraint Equation 7 indicated the maximum duration of a route. Constraint Equation 8 is on the departure time from the depot and indicates that all vehicles can start the service after a certain time. Constraint Equation 9 refers the penalty associated to a vehicle for an early or late by arrival to the user location. Finally, constraint Equation 10 defines the domain of decision variables.

The optimization procedure uses two different algorithms: a genetic algorithm (GA) and a population-based simulated annealing (PBSA), both methods based on the creation of an adaptive population. This characteristic is essential to maintain the best solutions of the previous population along with the subsequent generations. In general, a SA algorithm, if not based on population, is not useful to the scope of the procedure because of it does not update the memory with the best experiences and simply returns the best fitness value (of different solutions) based on the randomly selected initial solution.

The route first cluster second (Beasley, 1983) principle is adopted to determine the vehicle route: as starting solution the classical is adopted by relaxing all the constraints of the problem. This solution can be generated randomly or optimized with standard (heuristic or metaheuristic) procedures such as Clark and Wright savings or tabu search.

The various routes are then obtained by breaking the entire solution into “unique” routes to satisfy the constraint conditions for each of them. Furtherly, the solution must necessarily be decoded after the operators who create the new populations work because of the modifications on the positions of consumers to find better solutions; the values of capacity, distance, and time also change and, therefore, must be recreated groups of consumers that respect the constraints.

In addition, in order to check the quality of the obtained solutions, a tool from literature, based on ALNS algorithm, is used to solve the problem (Erdoğan, 2017). To apply this tool, it was necessary to relax one of the problem constraints. Two test applications will be presented in the following: first, comparing the results from the ALNS, GA and PBSA (this also allowed to calibrate the parameters of the algorithms) and a second assuming that constraints have been restored (in this case it is not possible to compare the results with ALNS).

Freight delivery in urban areas is at the core of the political agenda of many countries and municipalities, and new technologies offer more appealing solutions to develop novels and more sustainable strategies. Thus, urban freight distribution is undertaking the path to a radical transition, thus switching from traditional vehicles with internal combustion engines to electric vehicles (this also to accomplish the Sustainable Development Goals defined by the United Nations). These electric vehicles have limited range, and it is therefore essential to optimize their routes to minimize time and energy consumption.

A general algorithm considering the problem of vehicle routing with electric vehicles (EVRP) has been proposed to support the transition to zero emissions vehicles in urban freight distribution and explore the possibility enabled by these means of transport. This paper has tested two procedures (GA and PBSA) to solve, in an urban context, a VRPSTW with EVs with the aim to minimize the total time with a constraint on the tour distance, due to the restricted battery capacity of electric vehicles; the objective function is based on some time components (travel time, delivery time and a possible penalty). Both the procedures are population-based, which allowed one to update more than one solution over the iterations. The procedures are tested on a small problem, compared with each other and with a literature procedure. It is underlined that both procedures manage to find the solutions due to the ability to maintain and/or modify the best solutions of each iteration in the following iterations. For both GA and PBSA, the route-first cluster-second principle is used, and a decoding procedure (with respect the problem constraints) is implemented to obtain the routes for each vehicle With the aim of reducing the number of iterations, good-quality solutions are introduced in the population by using the CW algorithm.

To test the proposed procedures, two cases on the same test problem are considered. In the first case, a procedure from the literature is used to solve the problem. However, to do this, it was necessary to relax one of the constraints of the problem. This allowed us to obtain: an initial evaluation of the performance of the procedures, and a calibration of the GA and PBSA parameters to obtain good results in acceptable times. In the second case, the problem (without any relaxation) is solved with the parameters determined previously. From this test, the PBSA demonstrated the best performances, but further analyses are required (as an example, improving the GA operators).

The obtained output show that when solving a problem by relaxing the constrained starting time, the total time and the total trip length are reduced, even if the number of vehicles used is the same. Thus, also reflecting the possibility to reduce the environmental costs associated with deliveries. Nevertheless, more specific analyses and further developments that concern improving the proposed procedures (e.g., developing new operators) and the exploring the micro-hub cooperation services are necessary.