Optimal Agent-Based Pickup and Delivery with Time Windows and Electric Vehicles

Abstract

The traditional methods of transporting goods and people in urban areas using vehicles powered by internal combustion engines are major contributors to pollution. As a result, an increasing number of logistics companies are transitioning to electric vehicles (EVs) for daily operations, replacing traditional engines. This shift opens research avenues regarding the integration of EVs into delivery workflows and how this can contribute to greener cities. This study tackles the EV routing problem, focusing on balancing battery constraints and optimizing routes. We formulated the problem as a pickup and delivery with time windows, incorporating electric energy consumption constraints, and utilized consensus mechanisms in an agent-based simulation context. Our evaluation used 15 scenarios, capturing variations in vehicle configurations, order generation rates, and battery and freight capacities. We compared two order allocation strategies: “Closest Allocation” and “Negotiation” consensus-based allocation. The results confirmed that the consensus-based strategy outperformed the “Closest Allocation” in metrics such as remaining orders, orders not handled in time, total distance traveled, total recharging cost, and total number of recharges. These findings have significant implications for urban planners, logistic companies, and policymakers, demonstrating that an agent-based simulation context for electric vehicles using consensus-based strategies can enhance delivery efficiency and promote sustainability.

1. Introduction

With the increasing complexity of demand for real-world transportation of goods and heightened environmental concerns, interest in improving transportation planning systems is now higher than ever. Given the vast and dynamic number of transportation requests in the real world, the proposed approach must provide feasible solutions. We introduce an agent-based approach that leverages a consensus mechanism as a decision-support tool to assist in human decision-making.

A vehicle routing problem (VRP) can be static if we consider route planning when the information is provided a priori. In other words, the static VRP deals with the planning of batches of orders, while the requests that occur while the plan is in progress have to wait and accumulate in a new batch until the system decides to solve a new route planning problem.

In the dynamic VRP, the requests are handled in the order in which they occur, with all available resources at that moment. Each new request is considered to define a new route planning problem to satisfy that order.

A recent report by the International Energy Agency revealed that 16% of global carbon emissions come from conventional road transport. However, the same study indicated that the number of electrical vehicles (EVs) surpassed 16.5 million in 2021 [1].

To achieve net zero emissions by 2050 [2], more logistics companies are transitioning to EVs for daily operations, replacing traditional internal combustion engines. In their research, the authors of [3] provide a study estimating the potential of powertrain electrification to reduce road freight CO₂ emissions and fossil fuel consumption. This shift has led to novel research opportunities in the field of VRP. Researchers are now exploring variations of traditional VRP that factor in battery charging times and the availability of charging stations. For example, the authors of [4] provide a solution method to the vehicle routing problem with time windows for EVs using a hybrid heuristic based on neighborhood search and tabu search algorithms. A similar approach using variable neighborhood search algorithms, but this time for the EV problem with simultaneous pickup and delivery, was introduced in [5].

Various factors, such as geographical locations, regulations, economics, and technological limitations, influence the utilization of EVs for logistics and car ride-sharing. Reference [6] underscores the importance of digital technology in optimizing routes for waste collection by employing geographical information systems and a combination of heuristics derived from the constructive genetic algorithm and tabu search. To optimize the routes of EVs while considering their limited range and energy consumption, the pickup and delivery problem with time windows and electric vehicles (PDPTW-EV) has been developed as a variation of the pickup and delivery problem with time windows (PDPTW). As the popularity of EVs continues to increase, PDPTW-EV has become an essential tool for efficient route planning. This model ensures that vehicles have sufficient charge to reach their intended destinations while considering uncertainties in travel times [7,8,9,10].

The growing interest in sustainable transportation solutions through the integration of EVs into logistics, delivery networks, or ride-sharing underscores the need for advanced decision-support tools capable of handling the complexity and dynamics of modern transportation systems. According to our literature review, most works propose various solutions based on heuristic, constraint-based, or bio-inspired algorithms [11,12], we address the challenge of dynamic environments by leveraging consensus mechanisms in an agent-based simulation context. Therefore, this research aims to evaluate the potential of this novel approach to better address the evolving demands of transportation planning in dynamic environments by proposing a more appropriate, sustainable, and efficient approach. The findings of this research are primarily relevant for industries looking to transition to more sustainable practices, providing a model for optimizing EV fleets in real-world scenarios.

This paper is structured as follows. In Section 2, a comprehensive literature review covers existing research on agent-based modeling for vehicle routing problems and various consensus approaches used in agent-based systems. In Section 3, we detail the mathematical model of PDPTW-EV formulation as an extension of the classical PDPTW in which we introduce some constraints related to battery consumption. Section 4 describes the agent-based simulation mode (ABSM), including an in-depth explanation of the simulation environment and the consensus-based algorithm. The experimental setup and the obtained results are depicted in Section 5. Finally, this paper summarizes the key findings, implications, and possible future research directions in Section 6.

2. Derivation of Hypotheses and Related Works

2.1. Agent-Based Models for Vehicle Routing Problems

Agent-based approaches over VRPs and their variations have become popular solutions for complex optimization problems, particularly in dynamic environments with uncertainty and many distributed actors. Continuous information exchange and the need to reach a consensus over the negotiation of local solutions are essential to solving these problems and obtaining feasible global solutions.

The authors of [13] presented an agent-based approach for optimization problems using a decomposition model. Their results included estimates of the potential performance gain using a distributed implementation.

In an ABSM for VRP, multiple autonomous agents represent the vehicles that interact and coordinate to solve the routing problem collaboratively. Each agent is aware of its current location, its assigned customers, and its constraints. Agents can usually exchange this information to solve the route optimization problem.

Two approaches are distinguished by the authors in [14]: centralized and decentralized. The first refers to the centralized approach in which a central entity (“controller”, “coordinator”) makes a decision based on the information obtained from all other agents. Usually, a coordinator agent has a global overview of the system by gathering information and constraints from every agent to determine the best solution. Since this approach is a single decision-making entity, the decisions tend to be consistent with the global objective. The drawbacks of the centralized approach are the higher computational complexity, where the agents need to process the information from all other agents to make a decision, and the system may also be less adaptive to changes in dynamic contexts. The second approach is decentralization, where agents make decisions based on their local information or locally available data, such as from their neighbors or local environments. In this approach, the agents usually interact in a peer-to-peer fashion. The computation is distributed across agents, which may reduce the computational complexity and allow parallel processing. The authors proposed the usage of the contract net protocol (CNP) in a decentralized approach to order handling.

The idea of using agent-based approaches to solve dynamic optimization problems has been researched for a long time in the literature. For example, an agent-based approach to solving optimization problems using the Dantzig–Wolfe column generation scheme is proposed in [13]. In [15], the authors used CNP as a negotiation mechanism between vehicle agents to minimize the number of used vehicles.

Agent-based approaches to freight transportation problems were identified in many studies. For example, in [16], the authors proposed a knowledge-based freight broker based on agents and constraint programming, showing relevant results. The authors in [17] considered a multi-agent approach in combination with a heuristic local search to solve vehicle routing problems with time windows (VRPTW). Their evaluations were conducted on Solomon benchmark instances, which showed results comparable to those in the literature.

2.2. Consensus Mechanisms

A consensus mechanism is a process that involves a group of individuals or entities coming to a joint decision or agreement through iterative interactions. The goal of the consensus mechanism is to reach a solution that satisfies all parties involved, even if it is not the ideal solution.

The most commonly used consensus mechanisms in the literature include voting [18], averaging, and negotiation. Each approach is defined by various algorithms and techniques, each with strengths and weaknesses. In the voting consensus mechanism, the individuals vote on a proposal and select the one with the highest number of votes. The averaging method allows individuals to update their proposals based on the other proposals received from other individuals. Through negotiation techniques, individuals search for an agreement representing the shared solution.

The literature provides various consensus mechanisms applied in agent-based systems to improve decision-making support in domains such as production planning, control, or supply chain management. Ref. [19] provided a comprehensive study of existing methods and theories for achieving consensus in multi-agent systems (MASs). In [20], the authors described a consensus mechanism that allowed agents to coordinate the decision-making process, focusing on locally available information. In Ref. [21], the authors described the usage of the nearest neighbor rule in coordinating groups of mobile autonomous agents.

3. PDPTW-EV Formulation

The PDPTW-EV can be stated as the problem of finding optimal routes by minimizing the distance and the number of vehicles used in a given set of customer locations using mixed-integer linear programming (MILP). The model can be depicted as follows:

• Variables

• k denotes the number of vehicles;
• n denotes the number of pickup locations;
• m denotes the number of delivery locations;
• s denotes the number of charging stations.
• $q_i$ denotes the goods to pick up or deliver at location i; positive value for pickup location and negative value for delivery locations; $q_0=q_{n+m+1}=0$;
• $e_i$ denotes the lower time limit to start a service at location i;
• $l_i\ge e_i$ denotes the upper time limit to start a service at location i;
• $d_i$ denotes the service duration at location i;
• $c_{ij}^k$ denotes the travel cost of vehicle k from location i to location j;
• $Q^k$ denotes the capacity of vehicle k;
• $t_{ij}^k$ denotes the time that vehicle k spends traveling from location i to location j;
• $C^k$ denotes the maximum battery capacity of vehicle k;
• $L_i^k$ denotes the remaining battery level of vehicle k at location i.

• Sets

• P denotes the set of pickup locations $P=\{1,\dots ,n\}$;
• D denotes the set of delivery locations $D=\{1,\dots ,m\}$;
• K denotes the set of vehicles $K=\{1,\dots ,k\}$;
• S denotes the set of charging stations $S=\{1,\dots ,s\}$.

• Decision variables

• $x_{ij}^k$ is 1 if vehicle k travels from location i to location j, else it is 0;
• $Q_i^k$ is the current load of the vehicle at location i;
• $B_i^k$ - start of the service of vehicle k at location i;
• $L_{ij}^k$ is the remaining battery level of vehicle k after traveling from location i to location j.

Let $G=(V,E)$ be an undirected graph with $V=P\cup D\cup \{0,n+m+1\}\cup S$ as the set of nodes and $E=\left\{\right(i,j\left)\right|i,j\in V\}$ as the set of edges that denote the shortest paths between locations. The goal is to optimize vehicle fleet usage by minimizing the total travel cost to deliver all customer orders within their required time windows.

The MILP formulation of the problem is given by the following:

$$\begin{array}{cc}\hfill \mathrm{minimize}& \sum _{k\in K}\sum _{i\in V}\sum _{j\in V}{c}_{ij}{x}_{ij}^k\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathrm{subject}\mathrm{to}& \sum _{k\in K}\sum _{j\in V}{x}_{ij}^k=1,\forall i\in P\cup D\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \sum _{j\in V}{x}_{0j}^k=1,\forall k\in K\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \sum _{i\in V}{x}_{i,n+m+1}^k=1,\forall k\in K\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \sum _{i\in V}{x}_{ij}^k-\sum _{i\in V}{x}_{ji}^k=0,\forall j\in P\cup D,k\in K\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & B_j^k\ge x_{ij}^k(B_i^k+d_i),\forall i,j\in V,k\in K\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & B_i^k\le B_{n+i}^k,\forall i\in P,k\in K\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & Q_j^k\ge x_{ij}^k(Q_i^k+q_i),\forall i,j\in V,k\in K\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \max \{0,q_i\}\le Q_i^k\le \min \{Q^k,Q^k+q_i\},\forall i\in P,k\in K\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \sum _{j\in V}{x}_{ij}^k-\sum _{j\in V}{x}_{n+i,j}^k=0,\forall i\in P,k\in K\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & 0\le L_{ij}^k\le R^k,\forall i,j\in V,k\in K\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & L_{ij}^k=L_i^k-{\displaystyle \frac{t_{ij}^k}{R^k}},\forall i,j\in V,k\in K\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \sum _{i\in V}{x}_{ij}^k-\sum _{i\in V}{x}_{ji}^k=0,\forall j\in S,k\in K\hfill \end{array}$$

(13)

where the objective function is Equation (1) and it minimizes total travel costs across all vehicles and locations. Equation (2) ensures that each pickup or delivery location is visited by a vehicle exactly once. Equation (3) states that each vehicle starts its route from the depot, while Equation (4) ensures that the route ends at the depot. Equation (5) guarantees that each location is either the start or the end of a route segment for each vehicle. The next two equations are related to time constraints. Equation (6) ensures that each vehicle’s service start at each location is correctly determined by considering the service duration and travel time, while Equation (7) ensures that the start of service at pickup locations is not after the start of service at the corresponding delivery locations. By Equation (8), we ensure that the load on each vehicle at each location is correctly updated based on pickup and delivery actions, and by Equation (9), we ensure that the load on each vehicle at each pickup location stays within the vehicle’s capacity. Next, Equation (10) ensures that the load on each vehicle at each delivery location is consistent with the load at the corresponding pickup location. The last three equations describe the electrical vehicle constraints as follows: Equation (11) ensures that the remaining battery level of each vehicle after traveling between locations is within its maximum range. Equation (12) calculates the remaining battery level of each vehicle after traveling between locations based on its initial level and travel distance. Finally, Equation (13) ensures that the flow of vehicles at charging stations is balanced, meaning that the number of vehicles entering and leaving each station is the same.

Figure 1 depicts the solution example of two vehicle paths. Vehicle 1’s path, colored in blue, starts from the depot (the red square) and reaches the pickup location represented by the diamond shape. Next, it reaches the delivery location and finally returns to the depot. The path of vehicle two, colored in green, depicts the path including a charging station hop (green triangle). It starts from the depot, reaches the pickup point, then reaches the delivery location. Given that the remaining battery will not guarantee it will reach the depot, it takes a stop to charge at the charging station, as represented by the green triangle.

4. Agent-Based Simulation System

In this section, we present the details of the agent-based system for simulation. We first focus on the general agent-based simulation model and then we delve into the agent-based consensus allocation strategy.

4.1. Agent-Based Simulation Model

The representation of each vehicle as an agent within the simulation offers several advantages. Each agent operates autonomously yet communicates and collaborates with other agents to achieve their goals. The agent-based approach allows for flexibility and scalability in the simulation. As the number of vehicles and orders increases, the model can easily accommodate without significant loss in performance. Each agent in the simulation processes information and makes decisions independently, reducing the computational overhead as in centralized approaches. The agents’ autonomy and collaborative capabilities can lead to a more adaptive and resilient solution in optimizing plans for EVs in dynamic environments.

The ABSM described below is required to handle the dynamic PDPTW-EV. We consider a set of initial orders assigned to each vehicle using a simple assignment strategy. Then, new incoming orders are inserted into the vehicle’s initial route plan after establishing a consensus of each vehicle on how to deal with the order. In the context of solving the PDPTW-EV, we represent each vehicle, order, and charging station as agents in a grid environment. The agents and their interactions can be observed in Figure 2.

Table 1. Description of the order agent.

Order:
unique_id	A unique identifier for the order.
pos	Position of the order.
demand	The demand associated with the order.
ready_time	The time when the order is ready.
due_time	The due date for the order.
service_time	The time taken to service the order.
partner_id	Identifier for the partner associated with the order.
pickup	Indicates whether the order is a pickup or a delivery.

depicts the properties defining the order agent, including time-based properties, location properties, and links to partner order agents within the pickup and delivery context.

The charging station agent is the agent that represents a charging station and it is characterized by the properties defined in

Table 2. Description of the charging station agent.

Charging Station
unique_id	A unique identifier for the order.
pos	Position of the order in the grid.
recharging_rate	The recharging rate at the charging station recharges a vehicle.

. When a vehicle agent that requires recharging arrives at the charging station, it fully recharges at the defined recharging_rate.

Finally,

Table 3. Description of the vehicle agent.

Vehicle
unique_id	A unique identifier for the vehicle.
speed	The vehicle’s speed (cells per step).
capacity	The freight capacity of the vehicle.
battery_capacity	The maximum charge level of the battery.
battery_consumption_rate	The battery consumption rate per simulation step.
orders	The list of orders for the vehicle.
charge	The vehicle’s current battery charge level.
recharge_rate	The vehicle’s charge rate per simulation step.
pos	The position of the vehicle.
path	The path taken by the vehicle.
plan	The plan of the vehicle.
load	The current load of the vehicle.
time	The current time of the vehicle.
current_order	The order currently served by the vehicle.
current_order_index	The index of the current order to be handled in the order list.
needs_charging_threshold	This is used to determine when a vehicle needs recharge
needs_charging	Indicates if the vehicle needs to charge its battery based on the $needs\_charging\_threshold\times battery\_capacity$.
total_orders	The total number of orders for the vehicle.
distance_travelled	The total distance traveled by the vehicle.
number_of_charges	The total number of times the vehicle has been charged.

describes the properties of a vehicle agent. The vehicle agent represents an EV by properties such as speed, battery capacity, list of orders to be handled, transport capacity, and the plan.

Figure 3 describes the simulation process as an activity diagram. The simulation starts by generating the orders. The first step generates a pair of pickup and delivery orders. Following, the order is assigned to a vehicle agent based on the chosen simulation strategy and is added to its order list. Next, the vehicles handle delivering considering constraints depicted by Equations (2)–(10), ensuring enough battery charging or reaching the charging station for a complete recharge to fulfill their assigned list of orders as stated by Equations (11)–(13). After finishing all their orders, the vehicles are returned to the depot. The simulation can be configured to handle two types of order assignation: “Closest Allocation”, in which the order is allocated to the closest vehicle agent, and the “Negotiation” consensus mechanism, where the vehicle agents negotiate for handling the pickup and delivery orders pair.

4.2. Agent-Driven Consensus-Based Allocation

The consensus mechanism is built as a negotiation-based process as depicted in Figure 4. The negotiation occurs between all Vehicle Agents when a new pickup and delivery orders pair is generated. Next, the pickup and delivery orders are submitted to all the vehicle agents. Subsequently, each vehicle agent evaluates its cost based on the constraints defined by the model in Section 3 and generates a bid value as the total distance traveled by the vehicle (Equation (1)), considering the new orders. The best bid value is then used to allocate the pickup and delivery orders to a vehicle agent. The whole process is described by Algorithm 1.

Algorithm 1 Negotiation-based consensus mechanism for assessing new orders.

1: $V\leftarrow$ List of vehicles

2: $O\leftarrow$ List of newly generated orders

3: for o in O do

4:       $candidates\leftarrow \varnothing$

5:       for v in V do

6:            if v can feasibly handle o then

7:                  $candidates\leftarrow candidates\cup \left\{v\right\}$

8:            end if

9:       end for

10:       if $candidates$ is not empty then

11:            $selected\_vehicle\leftarrow$ Negotiate($candidates$)

12:            Assign o to $selected\_vehicle$

13:       else

14:            Reject o

15:       end if

16: end for

17: function Negotiate($candidates$)

18:       $best\_vehicle\leftarrow$ Randomly select a candidate from $candidates$

19:       for v in $candidates$ do

20:            if v’s proposal is better than $best\_vehicle$’s then

21:                  $best\_vehicle\leftarrow v$

22:            end if

23:       end for

24:       return $best\_vehicle$

25: end function

5. Experiments and Results

The experiments were conducted in a Mesa agent-based modeling framework in Python [22]. The ABSM was implemented to deal with the dynamic PDPTW-EV, where vehicle agents handle pickup and delivery orders in a grid-like environment to enhance the total distance traveled by vehicles.

Based on the proposed model, we evaluate two consensus mechanisms. One involves exchanging some of the remaining orders between vehicles when a new order is generated, and the other entails assigning the order to the vehicle plan that is the most suitable to handle. The objective is to minimize the total traveled distance.

In the experiments conducted, the charging stations were placed at the beginning of the simulation, and we did not employ any placement strategies. The charging stations can handle charging for any number of vehicles and have infinite charging capacity.

The experiments aim to compare two different allocation strategies, “Closest Allocation” and “Negotiation”, across 15 randomly generated scenarios. Algorithm 2 describes the scenario generation process. The orders are generated in a given distance range. The GenerateValidPosition procedure guarantees that the generated order position is in the grid and the given range. To ensure that the generated order times are in future steps of the simulation and not in the past, the CreateOrder procedure must define the range in terms of steps for the order’s time-related properties and link the order to the corresponding one.

The metrics used to evaluate the performance of these strategies include Total Orders Generated, Remaining Orders, Orders Not Handled In Time, Total Distance Traveled, Number of Charges, and Total Recharging Cost.

Algorithm 2 Order generation.

1: $Min\_Distance\leftarrow$ Minimum distance

2: $Max\_Distance\leftarrow$ Maximum distance

3: $S\leftarrow$ Number of simulation steps

4: $O\leftarrow$ List of newly generated orders

5: for o in O do

6:       $Pickup\_Position\leftarrow$GenerateValidPosition(∅)

7:       $Delivery\_Position\leftarrow$GenerateValidPosition([$Pickup\_Position$])

8:       $Pickup\_Order\leftarrow$CreateOrder(o, $Pickup\_Position$, $true$)

9:       $Delivery\_Order\leftarrow$CreateOrder(o, $Delivery\_Position$, $false$)

10:       Place $Pickup\_Order$ on the grid

11:       Place $Delivery\_Order$ on the grid

12: end for

13: function GenerateValidPosition($ExistingPositions$)

14:       $Position\leftarrow$ Random position within the grid

15:       while any position in $ExistingPositions$ is within $Min\_Distance$ of $Position$ do

16:            $Position\leftarrow$ Random position within the grid

17:       end while

18:       return $Position$

19: end function

20: function CreateOrder(o, $Position$, $IsPickup$)

21:       $ID\leftarrow$ Increment current order ID

22:       $Demand\leftarrow$ Random demand within the given range for o

23:       Set the ready time within the next S steps for o

24:       Set the due date within S steps of the ready time for o

25:       Set the service time within a given range for o

26:       if $IsPickup$ then

27:            $PartnerID\leftarrow$ ID + 1

28:       else

29:            $PartnerID\leftarrow$ ID − 1

30:       end if

31:       return Order($ID$, $Position$, $Demand$, $ReadyTime$, $DueDate$, $ServiceTime$, $PartnerID$, $IsPickup$)

32: end function

5.1. Evaluation Scenarios

The simulation scenarios are depicted in

Table 4. Scenarios for evaluating vehicle allocation mechanisms.

Scenario	Vehicles	Order Rate	Battery Cap.	Freight Cap.
1	10	100	1000	200
2	5	50	500	150
3	15	200	1500	300
4	20	50	300	100
5	8	150	800	250
6	12	120	1200	220
7	6	80	600	180
8	25	250	2000	400
9	10	75	750	200
10	18	180	1800	350
11	7	70	700	170
12	20	100	1000	220
13	15	120	1300	300
14	10	40	900	180
15	22	220	1600	360

.

The scenarios involve a variety of vehicle configurations, order generation rates, battery capacities, and freight capacities. We considered the same charging station placements for each scenario. The number of properties ranges from low to high.

Most configurations with a large number of vehicles typically exhibit high order generation rates, along with moderate to high battery and freight capacities. In contrast, scenarios with fewer vehicles usually feature lower order generation rates and battery capacities but maintain moderate freight capacities.

5.2. Results

The simulation results based on the proposed scenarios are depicted in this subsection.

Total orders generated: This represents the total number of orders generated in each scenario.

In Figure 5, we can see that the number of generated orders is the same for each scenario. The blue bars represent the “Closest Allocation” strategy, while the orange bars depict the “Negotiation” strategy. The x-axis represents the scenarios labeled “Scenario 1” to “Scenario 15” and the y-axis represents the number of orders.

Remaining orders: This metric depicts the number of orders not completed by the end of the simulation. As can be observed in Figure 6, the results show that “Closest Allocation” generally has higher remaining orders compared to “Negotiation”, indicating that “Negotiation” might be more efficient in completing orders.

The results in Figure 6, where the x-axis represents the scenarios labeled “Scenario 1” to “Scenario 15” and the y-axis represents the number of orders, shows that the “Closest Allocation” generally has more remaining orders compared to “Negotiation”, indicating that the last is more efficient in completing orders. For example, in Scenario 4, “Closest Allocation” has 20 remaining orders, while “Negotiation” has significantly fewer. The pattern is consistent across most scenarios.

Orders not handled in time: This represents the number of orders not completed within the specified time windows. Figure 7 depicts the results for each scenario where the x-axis represents the scenarios and the y-axis represents the number of orders not handled in time. Blue bars correspond to “Closes Allocation” and the orange bars to “Negotiation”. We can observe that, in several scenarios, the “Closest Allocation” strategy has more orders not handled in time than the “Negotiation” approach, suggesting that the latter leads to better performance in handling orders within their time windows.

Total distance traveled: This metric shows the total distance traveled by all vehicles. As depicted in Figure 8, where the x-axis represents the scenarios labeled “Scenario 1” to “Scenario 15” and the y-axis represents the distance traveled, in some scenarios, the results show longer distances for the “Closest Allocation”, while in others, they show longer distances for “Negotiation”. Although this metric may show that the vehicles traveled a longer distance, it needs to be looked at along with the above-mentioned metrics for order handling.

Number of charges: This indicates how often vehicles need to recharge. As depicted in Figure 9, “Closest Allocation” often requires more charges compared to “Negotiation”, suggesting that “Negotiation” may be more efficient in managing vehicle battery levels. The x-axis represents the scenarios labeled “Scenario 1” to “Scenario 15”, and the y-axis represents the number of charging station visits.

Total recharging cost: This represents the total cost incurred for recharging the vehicles. In most scenarios, as depicted in Figure 10, “Negotiation” has a lower recharging cost than “Closest Allocation”. The x-axis represents the scenarios labeled “Scenario 1” to “Scenario 15” and the y-axis represents the recharging cost.

The results of the depicted ABSM suggest that the “Negotiation” approach may be more efficient in handling orders within time windows and managing recharging efficiency. While a “Closest Allocation” strategy may lead to higher operational costs and less efficient order completion.

6. Conclusions and Future Work

This paper presents an ABSM for dealing with the dynamic PDPTW-EV using a consensus-based technique to achieve near-optimal solutions. The ABSM was built by extending the classical PDPTW formulation by introducing eclectic vehicle constraints related to battery consumption and recharging. By introducing those constraints, we tailored the model to reflect the challenges posed by EVs.

The initial results demonstrate that the “Negotiation” consensus-based strategies consistently outperformed the “Closest Allocation” strategy across all evaluated metrics, highlighting the effectiveness of collaborative decision-making in dynamic routing environments. These research findings have significant implications for industries relying on EV fleets, such as logistics and transportation, where efficient route planning directly impacts operational costs.

The insights gained in this study could be valuable for EV operations. By adopting the “Negotiation” strategy, an efficient and sustainable logistic operation can be achieved, leading to reduced energy consumption and lower operation costs while improving customer satisfaction.

In future work, we plan to extend our research by employing other negotiation mechanisms that could further enhance the performance of the agent-based simulation in industrial applications. The agent model can be integrated with real-world industrial systems, facilitating the reuse and adaption of our findings in various domains such as ride-sharing, delivery systems, and automated logistics networks.