Research on Multi-Objective Green Vehicle Routing Problem with Time Windows Based on the Improved Non-Dominated Sorting Genetic Algorithm III

Abstract

To advance energy conservation and emissions reduction in urban logistics systems, this study focuses on the green vehicle routing problems with time windows (GVRPTWs), which remains underexplored in balancing environmental and service quality objectives. We propose a comprehensive multi-objective optimization framework that addresses this gap by simultaneously minimizing total distribution costs and carbon emissions while maximizing customer satisfaction, quantified based on the vehicle’s arrival time at the customer location. The rationale for adopting this tri-objective formulation lies in its ability to reflect real-world trade-offs between economic efficiency, environmental performance, and service level, which are often considered in isolation in previous studies. To tackle this complex problem, we develop an improved Non-Dominated Sorting Genetic Algorithm III (NSGA-III) that incorporates three key enhancements: (1) an integer-encoded initialization method to enhance solution feasibility, (2) a refined selection strategy utilizing crowding distance to maintain population diversity, and (3) an embedded 2-opt local search operator to prevent premature convergence and avoid local optima. Comprehensive validation experiments using Solomon’s benchmark instances and a real-world case demonstrate that the presented algorithm consistently outperforms several state-of-the-art multi-objective optimization methods across key performance metrics. These results highlight the effectiveness and practical relevance of our approach in advancing energy-efficient, low-emission, and customer-centric urban logistics systems.

1. Introduction

First formally conceptualized by Dantzig and Ramser in their seminal 1959 study on gasoline delivery optimization, the vehicle routing problem (VRP) constitutes a cornerstone of combinatorial optimization challenge in transportation logistics [1]. This NP-hard problem seeks to determine optimal vehicle routes that minimize operational costs while adhering to specific constraints. Amid the exponential growth of the global logistics market, the VRP has emerged as a critical research domain, particularly for urban distribution network optimization. Subsequent advancements have yielded specialized variants tailored to diverse operational needs, thereby fostering innovation in sustainable transportation systems and supply chain automation technologies.

The relentless surge in logistics demand has intensified the energy consumption of logistics operations, consequently increasing carbon dioxide emissions. In this context, the green vehicle routing problem (GVRP) has emerged as a critical research area aimed at mitigating climate change by optimizing vehicle routes to minimize environmental impact [2]. The significance of GVRP is underscored by the unsustainable nature of current production and distribution logistics strategies, which often prioritize economic efficiency over ecological sustainability. As a result, logistics policymakers are increasingly incorporating environmental, ecological, and social considerations into decision-making processes alongside traditional economic factors. Recent research on the GVRP employs a variety of methodologies to address its complex challenges. For instance, Soleimani et al. [3] developed a nonlinear multi-objective programming model to optimize the distribution of original and remanufactured products, effectively balancing economic and environmental objectives. Similarly, Macrina et al. [4] investigated energy consumption in mixed fleets of electric and conventional vehicles, proposing a heuristic algorithm based on a large neighborhood search that incorporates variables such as speed and acceleration to minimize energy usage. Additionally, Sadati and Çatay [5] tackled the complexities of the multi-depot GVRP by designing a hybrid algorithm that combines variable neighborhood search and tabu search, thereby enhancing the management of multi-depot operations and en-route refueling strategies. These studies highlight innovative approaches to the GVRP, emphasizing the need for continued research to advance sustainable logistics systems.

Although research on single-objective vehicle routing problems has reached a mature stage, the multi-objective GVRP (MOGVRP) continues to present significant challenges. The MOGVRP integrates various factors, such as vehicle specifications, environmental conditions, traffic dynamics, and driver behavior, to simultaneously address environmental, economic, and operational objectives. This complexity underscores its importance in advancing sustainable logistics practices that align with ecological and societal goals. Recent studies have proposed innovative methodologies to address these challenges. Toro et al. [6] developed a multi-objective mixed-integer linear programming model for the green capacitated location-routing problem, aiming to reduce fuel consumption, operational costs, and environmental impact. Poonthalir and Nadarajan [7] formulated a model to minimize fuel costs and travel distance by incorporating the effects of speed and load, which was solved using an enhanced particle swarm optimization algorithm. Long et al. [8] proposed a multi-objective model to minimize fuel usage and vehicle deployment under demand uncertainty, contrasting it with traditional objectives such as shortest distance and minimum time. Anderluh et al. [9] employed two distinct vehicle types for deliveries, aiming to optimize both delivery costs and carbon emissions. Ziaei and Jabbarzadeh [10] presented a multi-objective robust optimization method for green location routing in uncertain intermodal systems, focusing on minimizing carbon emissions, risks, and transportation costs. Addressing stochastic demand, Niu et al. [11] designed a multi-objective model to minimize delivery costs while maximizing customer satisfaction. In drone-assisted logistics, Zhang et al. [12] optimized collaborative routes for trucks and drones to reduce fuel and energy consumption. Yin et al. [13] further examined the interplay of load, speed, and emissions to minimize both emissions and costs. Additionally, recent efforts have extended the MOGVRP to electric vehicle routing, reflecting the increasing priority of zero-emission logistics. These advancements underscore the evolving complexity of MOGVRP and its pivotal role in sustainable logistics research.

As the increasing demand for enhanced logistics service quality, research interest in the GVRP with time windows (GVRPTWs) has grown significantly. This optimization challenge seeks to design efficient vehicle routing plans that simultaneously address three critical dimensions: (1) minimization of fuel consumption or carbon emissions, (2) adherence to customer-specified time windows, and (3) operational efficiency in logistics distribution systems. To better understand this problem, consider a logistics company that employs new energy vehicles to serve multiple urban customers. Each customer has a specific time window for service, and the vehicles must take into account energy consumption and carbon emissions. For example, a delivery vehicle may need to visit five customers in sequence, with moderate distances between them, widely varying time windows, and a limited driving range due to battery constraints. Planning a route that meets all time window requirements while minimizing total carbon emissions exemplifies the primary challenge addressed by the GVRPTW. Recent methodological advances demonstrate these principal directions of development. Rezaei et al. [14] proposed a mixed-integer linear programming model that incorporates heterogeneous vehicle fleets and refueling station constraints, aiming to minimize transportation costs and carbon emissions. Xu et al. [15] investigated the impact of speed variations during traffic congestion on customer satisfaction, and developed a vehicle routing model that accounts for non-linear, time-varying speeds. For the heterogeneous fleet GVRPTW, Yu et al. [16] introduced an improved branch and pricing algorithm that reduces computation time by utilizing multi-vehicle approximate dynamic programming and integer branching. Ren et al. [17] proposed an enhanced variable neighborhood search algorithm for the bi-objective mixed-fleet GVRPTW, improving solution quality and diversity through a selection mechanism. Liu et al. [18] developed a GVRPTW model focused on minimizing carbon emissions by considering the effects of time-dependent variations in vehicle travel distance and speed. These studies underscore the diverse methodologies being employed to optimize logistics operations while addressing environmental concerns and customer service requirements.

While the GVRPTW incorporates factors relevant to real-world logistics, its direct application in engineering practice remains limited. A critical gap lies in the oversimplification of single-objective formulations, which fail to capture the multi-criteria decision-making requirements inherent to practical vehicle routing problems. Therefore, research on the multi-objective GVRPTW (MOGVRPTW) has gained increasing significance. Ghannadpour and Zarrabi [19] modeled variations in fuel consumption across a heterogeneous fleet by accounting for road gradients, thereby enhancing the realism of their approach. Song et al. [20] developed an improved artificial fish swarm algorithm to solve a multi-objective model that balances fuel consumption and delivery costs for multiple vehicles. In the context of fuel distribution, Xu et al. [21] incorporated the impact of fuel unloading speed and arrival time on gas station satisfaction into their multi-objective model, addressing both operational efficiency and customer service. Zarouk et al. [22] provided a novel perspective by analyzing how driver fatigue influences carbon emissions, highlighting an often-overlooked factor in emissions reduction. Elgharably et al. [23] quantified the effects of customer satisfaction by integrating time windows and demand uncertainty into their model, offering a more comprehensive view of service quality. Furthermore, Kuo et al. [24] integrated supply chain management with GVRP by proposing a model that minimizes both supply chain costs and carbon emissions, which was solved using an enhanced multi-objective particle swarm optimization algorithm. Collectively, these studies underscore the growing importance of multi-objective approaches in addressing the complexities of sustainable logistics.

Although existing research provides valuable insights, few studies on the MOGVRPTW simultaneously consider delivery costs, carbon emissions, and customer satisfaction. Balancing economic and environmental objectives with customer satisfaction is essential for sustainable logistics. To address these gaps, this study formulates a mathematical model for MOGVRPTW that incorporates factors such as travel distance, load, and customer satisfaction. The study utilizes an improved Non-Dominated Sorting Genetic Algorithm III (INSGA-III) to solve the model and identify optimal delivery routes that balance costs, emissions, and customer satisfaction. The subsequent sections are organized as follows: Section 2 establishes a mathematical model for the MOGVRPTW, Section 3 introduces the INSGA-III, Section 4 carries out computational experiments, Section 5 provides a real-world case test, and finally Section 6 concludes this paper.

2. Mathmatical Model of MOGVRPTW

2.1. Problem Description

This study addresses the MOGVRPTW within a framework that includes a single depot, multiple homogeneous vehicles, and multiple demand points. The objective is to optimize vehicle delivery routes to minimize total delivery costs and carbon emissions while maximizing customer satisfaction. All vehicles are assumed to depart from a single depot and return immediately upon completing their delivery tasks. Customer positions, time windows, service times, and demand quantities are known. Vehicles travel at a constant speed, and road conditions—such as traffic congestion, traffic lights, or gradients—are not considered. Deliveries must be completed to all customers, with each customer served by exactly one vehicle. Deliveries must occur within the customer’s specified time window; deviations will result in penalty costs and reduced customer satisfaction. All vehicles are identical, with a tare weight of 3.5 tons. This research aims to develop an optimization framework that balances minimizing operational costs, reducing carbon emissions, and maximizing customer satisfaction. Efficient route planning in urban logistics is crucial for improving cost-effectiveness, environmental sustainability, and customer service quality. Based on this problem description, we develop a multi-objective green vehicle routing optimization model with time windows. Relevant variables, parameters, and symbols are defined in

Table 1. Notions.

Symbol	Definition	Symbol	Definition
$\mathbb{V}$	Set of vehicles, $\mathbb{V}=\{1,2,\dots ,m\}$	$t_{ik}$;	Arrival time of the k-th vehicle at customer i;
$\mathbb{N}$	Set of customers, $\mathbb{N}=\{1,2,\dots ,n\}$. In particular, the depot is considered as customer 0, denoted as {0};	$t_{ijk}$	Driving time of the k-th vehicle on arc $(i,j)$;
${\sigma }_1$	Unit fuel cost	$s_i$;	Service time at customer i;
${\sigma }_2$	Unit vehicle activation cost;	$[E{T}_i,L{T}_{ij}]$	Specified service time window at customer i;
${\sigma }_3$	Unit distance transportation cost;	$[E{T}_i^{\alpha },L{T}_i^{\beta }]$	Expected service time window at customer i;
$c_{ij}$	Distance travel between customer i and j;	$\alpha$	Penalty cost per unit time for waiting before the customer’s expected service time $E{T}_i^{\alpha }$;
R	Vehicle capacity;	$\beta$	Penalty cost per unit time for waiting after the customer’s expected service time $L{T}_i^{\beta }$;
$d_i$	Demand of customer i;	$W_i\left(t_{ik}\right)$	Customer satisfaction function;
$d_{ijk}$	Actual load of the k-th vehicle on arc $(i,j)$;	$x_{ijk}$	If the k-th vehicle is traveling from customer i to j, the value is set to 1; otherwise, it is set to 0.
$v_{ij}$	Average speed of the vehicle on arc $(i,j)$	$y_{ik}$	If the k-th vehicle is in serve at customer i, the value is set to 1; otherwise, it is set to 0.

.

2.2. Objective Function

This study focuses on the MOGVRPTW, aiming to optimize three objective functions: minimizing delivery costs, reducing carbon emissions, and maximizing average customer satisfaction. For simplicity, let $f_1$, $f_2$, and $f_3$ denote the vehicle delivery cost, carbon emission, and average customer satisfaction, respectively.

2.2.1. Vehicle Delivery Cost

The delivery costs include fuel consumption, vehicle operation, travel expenses, and penalty fees. Therefore, in urban logistics, the vehicle delivery costs are as follows:

$$f_1=F_1+F_2+F_3+F_4,$$

(1)

where $F_1$ is the vehicle fuel consumption cost, $F_2$ is the vehicle operation cost, $F_3$ is the vehicle travel cost, and $F_4$ is the penalty cost.

Considering that fuel consumption is influenced by both travel distance and vehicle load, the cost of fuel consumption ($F_1$) is calculated using the Load-Dependent Fuel Consumption Model (LFCM) [25]. The mathematical formulation is as follows:

$$F_1={\sigma }_1\sum _{i=0}^n\sum _{j=0}^n\sum _{k=1}^m{c}_{ij}{x}_{ijk}\left({\rho }_0+{\displaystyle \frac{{\rho }^{*}-{\rho }_0}{R}}{d}_{ijk}\right),$$

(2)

where ${\rho }_0$ represents the fuel consumption rate of the vehicle in an unloaded state, while ${\rho }^{*}$ denotes the fuel consumption rate when the vehicle is fully loaded.

The operational costs ($F_2$) of a vehicle include the purchase cost, maintenance cost, and repair cost, all of which are directly proportional to the number of vehicles in use. The formula for calculating these costs is as follows:

$$F_2={\sigma }_2\sum _{j=1}^n\sum _{k=1}^m{x}_{0jk}$$

(3)

The cost of vehicle travel ($F_3$) is directly proportional to the distance traveled. The formula for calculating this cost is as follows:

$$F_3={\sigma }_3\sum _{i=0}^n\sum _{j=0}^n\sum _{k=1}^m{c}_{ij}{x}_{ijk}$$

(4)

The penalty cost is associated with the vehicle’s arrival time. As illustrated in Figure 1, customer satisfaction is maximized, and the penalty cost is zero when the vehicle arrives within the expected time window [$E{T}_i^{\alpha }$,$L{T}_i^{\beta }$]. If the vehicle arrives earlier than the expected time window [$E{T}_i$,$L{T}_i^{\alpha }$], waiting costs are incurred. The longer the vehicle arrives before the expected time, the higher the waiting costs. Conversely, if the vehicle arrives later than the expected time window [$E{T}_i^{\beta }$,$L{T}_i$], delay costs are incurred, which increase over time. The corresponding penalty cost ($F_4$) can be expressed by the following function:

$$\left\{\begin{array}{cc}\alpha \left(E{T}_i^{\alpha }-t_{ik}\right),\hfill & t_{ik}\in \left[E{T}_i,E{T}_i^{\alpha }\right]\hfill \\ 0,\hfill & t_{ik}\in \left[E{T}_i^{\alpha },L{T}_i^{\beta }\right]\hfill \\ \beta \left(t_{ik}-L{T}_i^{\beta }\right),\hfill & t_{ik}\in \left[L{T}_i^{\beta },L{T}_i\right]\hfill \end{array}\right.$$

(5)

In summary, the cost of the vehicle penalty can be calculated by using the following formula:

$$F_4=\alpha \sum _{i=1}^{n}max\left\{E{T}_i^{\alpha }-t_{ik},0\right\}+\beta \sum _{i=1}^{n}max\left\{t_{ik}-L{T}_i^{\beta },0\right\}$$

(6)

2.2.2. Vehicle Carbon Emission

In this study, we employ the MEET (Methodologies for Estimating Air Pollutant Emissions from Transport) model [26], a three-tiered emission estimation system, to formulate the carbon emission function for vehicles. The MEET model comprises three components: the speed-based carbon emission rate function $\varphi \left(v\right)$, the load correction function $\phi \left(v\right)$, and the road slope correction function $r\left(v\right)$. These functions, which depend on the vehicle’s average speed, are defined as follows:

$$\begin{array}{cc}\hfill \varphi \left(v\right)& ={\varphi }_0+{\varphi }_{1}v+{\varphi }_2{v}^2+{\varphi }_3{v}^3+{\displaystyle \frac{{\varphi }_4}{v}}+{\displaystyle \frac{{\varphi }_5}{v^2}}+{\displaystyle \frac{{\varphi }_6}{v^3}},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \phi \left(v\right)& ={\phi }_0+{\phi }_{1}v+{\phi }_2{v}^2+{\phi }_3{v}^3+{\displaystyle \frac{{\phi }_4}{v}}+{\displaystyle \frac{{\phi }_5}{v^2}}+{\displaystyle \frac{{\phi }_6}{v^3}}+{\displaystyle \frac{{\phi }_7}{v}},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill r\left(v\right)& =r_0+r_{1}v+r_2{v}^2+r_3{v}^3+r_4{v}^4+r_5{v}^5+r_6{v}^6,\hfill \end{array}$$

(9)

where ${\varphi }_0$ to ${\varphi }_6$ are the carbon emission correction coefficients, ${\phi }_0$ to ${\phi }_7$ are the load correction coefficients, $r_0$ to $r_7$ are the road slope correction coefficients, and v represents the vehicle’s average speed.

Let c represent the vehicle’s travel distance. The vehicle’s carbon emissions, denoted as C, can be expressed as follows:

$$C=\varphi \left(v\right)\phi \left(v\right)r\left(v\right)c$$

(10)

The case study focuses on a 3.5-ton payload box truck, a representative vehicle configuration for urban last-mile delivery operations. Given the predominantly flat terrain characteristics of urban logistics networks, road gradient impacts were excluded from this analysis. According to the data provided in reference [26], the carbon emission rate function and the load correction function of the vehicle on unit arc $(i,j)$ are given as follows:

$$\varphi \left(v_{ij}\right)=110+0.000375v_{ij}^3+{\displaystyle \frac{8702}{v_{ij}}}$$

(11)

$$\phi \left(v_{ij}\right)=1.27+0.0614v_{ij}-0.0011v_{ij}^3-{\displaystyle \frac{1.33235}{v_{ij}}}$$

(12)

In summary, the vehicle carbon emission function is defined as follows:

$$f_2=\sum _{i=0}^n\sum _{j=0}^n\sum _{k=1}^m\varphi \left(v_{ij}\right)\phi \left(v_{ij}\right){\displaystyle \frac{c_{ij}}{1000}}{x}_{ijk}$$

(13)

2.2.3. Average Customer Satisfaction

Customer satisfaction in the context of time-sensitive delivery systems depends on the timing of the delivery relative to two key time windows: the specified time window [$E{T}_i$, $L{T}_i$] and the expected time window [$E{T}_i^{\alpha }$,$L{T}_i^{\beta }$], where [$E{T}_i^{\alpha }$,$L{T}_i^{\beta }$] ⊆ [$E{T}_i$, $L{T}_i$]. The specified time window [$E{T}_i$, $L{T}_i$] defines the maximum allowable period within which the delivery must occur; deliveries outside this range result in zero customer satisfaction. Nested within [$E{T}_i$, $L{T}_i$], the expected time window [$E{T}_i^{\alpha }$,$L{T}_i^{\beta }$] represents the customer’s preferred delivery period. When the service is completed within [$E{T}_i^{\alpha }$,$L{T}_i^{\beta }$], customer satisfaction reaches its maximum of 100%. However, for deliveries occurring within [$E{T}_i$, $L{T}_i$] but outside [$E{T}_i^{\alpha }$,$L{T}_i^{\beta }$], satisfaction declines as the delivery time deviates further from the expected window. This relationship between delivery timing and customer satisfaction is illustrated in Figure 1.

It is important to note that the satisfaction function utilized in this article is symmetric. While this type of function may not accurately capture the true feelings of customers, it effectively represents the fluctuations in their actual emotions. Furthermore, from a temporal perspective, changes in people’s emotions inherently display symmetrical characteristics. This is one of the key reasons why this type of satisfaction function is widely employed in contemporary research.

Let $W_i\left(t_{ik}\right)$ represent the customer satisfaction function. Based on Figure 1, the function is defined as follows:

$$\left\{\begin{array}{cc}{\displaystyle \frac{t_{ik}-E{T}_i}{E{T}_i^{\alpha }-E{T}_i}}\times 100,\hfill & t_{ik}\in \left[E{T}_i,E{T}_i^{\alpha }\right]\hfill \\ 100,\hfill & t_{ik}\in \left[E{T}_i,E{T}_i^{\alpha }\right]\hfill \\ {\displaystyle \frac{L{T}_i-t_{ik}}{L{T}_i-L{T}_i^{\beta }}}\times 100,\hfill & t_{ik}\in \left[L{T}_i^{\beta },L{T}_i\right]\hfill \\ 0,\hfill & t_{ik}\in \left[0,E{T}_i\right]\cup \left[L{T}_i,+\infty \right]\hfill \end{array}\right.$$

(14)

Therefore, the average customer satisfaction can be expressed as:

$$f_3={\displaystyle \frac{1}{n}}\sum _{i=1}^n\sum _{k=1}^m{W}_i\left(t_{ik}\right)$$

(15)

2.3. Optimization Model

The MOGVRPTW addressed in this study is formally defined as a combinatorial optimization task where a homogeneous fleet of m vehicles must service n geographically dispersed customers under hard time window constraints with each location visited exactly once by a single vehicle. Different routing solutions yield varying total costs, carbon emissions, delivery times, and average customer satisfaction levels. Based on these considerations, we formulate the following optimization model:

$$f=\{min{f}_1,min{f}_2,max{f}_3\}$$

(16)

s.t.

$$\begin{array}{cc}\hfill \sum _{i=0}^n\sum _{k=1}^m{x}_{ijk}& =1,\forall j\in \mathbb{N}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \sum _{j=0}^n\sum _{k=1}^m{x}_{ijk}& =1,\forall i\in \mathbb{N}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \sum _{k=1}^m{y}_{ik}& =1,\forall i\in \mathbb{N}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \sum _{i=1}^n{d}_i{y}_{ik}& \le R,\forall k\in \mathbb{V}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \sum _{i=0}^n\sum _{k=1}^m{d}_{ijk}-\sum _{i=0}^n\sum _{k=1}^m{d}_{jik}& =d_j,\forall j\in \mathbb{N}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \sum _{j=1}^n\sum _{k=1}^m{x}_{0jk}& =\sum _{j=1}^n\sum _{k=1}^m{x}_{j0k}\le 1\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \sum _{j=1}^n\sum _{k=1}^m{x}_{0jk}& \le m\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \sum _{i=0}^n\sum _{k=1}^m{x}_{ijk}\left(t_{ik}+s_i+t_{ijk}\right)& =t_{jk},\forall j\in \mathbb{N}\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill t_{ijk}={\displaystyle \frac{c_{ij}}{v_{ij}}},\forall i,j\in \mathbb{N}\cup \left\{0\right\},& i\ne j,\forall k\in \mathbb{V}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill x_{ijk}& \in \{0,1\}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill y_{ik}& \in \{0,1\}\hfill \end{array}$$

(27)

In the above model, Equations (17) and (18) ensure that each customer is served by exactly one vehicle, while Equation (19) guarantees that every customer is visited for delivery. Equation (20) enforces the vehicle capacity constraint, ensuring that the total demand assigned to a vehicle does not exceed its capacity. Equation (21) confirms that the demand of each customer is fully satisfied. Additionally, Equation (22) mandates that all vehicles start and return to the depot, and Equation (23) restricts the number of vehicles used to the available fleet size. For time-related constraints, Equation (24) specifies the temporal relationship for vehicle k when departing from customer i and arriving at j, while Equation (25) quantifies the travel time along an arc (i, j). Finally, Equations (26) and (27) define the decision variables that are essential for optimizing the routing solution.

3. INSGA-III for MOGVRPTW

In this paper, we adopt the NSGA-III as the solution method to address the multi-objective optimization problem. NSGA-III is particularly effective for problems involving more than two conflicting objectives, as it employs a reference-point-based approach to maintain a well-distributed set of Pareto-optimal solutions. Compared to traditional multi-objective algorithms, such as SPEA2 or MOEA/D, NSGA-III exhibits superior capabilities in balancing convergence and diversity, especially in high-dimensional objective spaces.

Given that our model simultaneously optimizes three conflicting objectives, it is crucial to achieve a well-distributed set of solutions across the Pareto front. NSGA-III inherently supports this requirement without necessitating additional problem-specific adjustments or hybridization with other heuristics. This capability simplifies the algorithm’s design while ensuring strong solution quality and robustness. Furthermore, NSGA-III has been widely validated in solving complex combinatorial optimization problems, providing both high computational efficiency and scalability. These advantages make it particularly suitable for the problem settings addressed in this paper.

When solving multi-objective problems, NSGA-III is prone to getting trapped in local optima. To address this issue and improve the algorithm’s efficiency, this paper proposes an improved NSGA-III (INSGA-III). The INSGA-III combines the NSGA-III with a local search strategy to enhance population diversity and convergence speed.

• Population Initialization: The population is initialized using integer encoding, with individuals randomly generated, and the fitness value of each individual is calculated.
• Selection Process: The hybrid NSGA-II uses a combination of crowding distance-based selection and tournament selection to choose individuals from the current population, ensuring that high-quality genes are involved in the subsequent crossover and mutation processes.
• Crossover and Mutation: Leveraging the global search capability of NSGA-III, the population undergoes crossover and mutation, and reference points are generated.
• Local Search: The 2-opt local search strategy is applied to the selected high-quality individuals from the previous step to avoid losing superior individuals.
• Termination Check: Determine whether the algorithm has terminated by checking if the specified number of iterations has been reached. If so, the algorithm terminates; otherwise, it continues iterating.

3.1. Encoding and Decoding

Common chromosome encoding methods include binary encoding, decimal encoding, and integer encoding. In this paper, integer encoding is used to represent chromosomes, as illustrated in Figure 2. The depot is represented by 0. Suppose there are n customers and s vehicles, and after the delivery is completed, the vehicles return to the delivery center, resulting in a chromosome of length $s+n+1$. For example, consider the chromosome “0 2 10 0 7 1 5 0 9 3 6 0 4 8 0”, which represents the following delivery routes. Route 1: depot → customer 2 → customer 10 → depot. Route 2: depot → customer 7 → customer 1 → customer 5 → depot. Route 3: depot → customer 9 → customer 3 → customer 6 → depot. Route 4: depot → customer 4 → customer 8 → depot.   

3.2. Selection Operator

Despite the prevalence of tournament selection in evolutionary algorithms, its effectiveness diminishes significantly when applied to many-objective optimization problems. The fundamental limitation stems from insufficient selection pressure in high-dimensional objective spaces, where conventional tournament selection fails to adequately balance convergence and diversity. A single selection method often proves inadequate; it may not be sufficient to maintain both convergence and diversity within the solution set. This necessitates the implementation of additional criteria to ensure effective and diverse selection outcomes. To address this challenge, we propose a dual-criterion tournament selection framework that synergistically integrates fast non-dominated sorting with crowding distance metrics from the NSGA-III paradigm. The method proceeds as follows:

• Calculate the Pareto dominance ranks and normalized crowding distances simultaneously for all population members to establish dual fitness criteria, the non-dominated rank, and crowding distance for each individual in the population.
• For each selection event, randomly select a group of individuals (tournament size), and select the individual with the lowest non-dominated rank from this group.
• If there are multiple individuals within the tournament group that have the same lowest non-dominated rank, select the one with the largest crowding distance to enhance diversity in the solution set.

3.3. Crossover Operator

A range of solutions for vehicles’ tasks is created by using the order-based crossover (OBX) operator. This operator facilitates the generation of unique combinations of parental genes, expanding the scope of exploration within the solution space. First, we identify the positions in paternal chromosome 2 that correspond to the genes selected in paternal chromosome 1. Then, offspring 1 is constructed using the unselected genes in paternal chromosome 2 while keeping their positions aligned. Subsequently, the selected genes from parent 1 are methodically inserted into the vacant positions of offspring 1.

For example, if the 7-5-4 genes are randomly selected from parent chromosome $p_1$, these genes are temporarily omitted from parent chromosome $p_2$. The remaining sequences, e.g., 3-2-6-1-8-9, form the base structure of child $c_1$. The selected genes (e.g., 7-5-4) are then sequentially integrated into the corresponding vacancies in offspring $c_1$, while retaining their original sequences in parent $p_1$. Figure 3 illustrates these specific processes.

3.4. Mutation Operator

In this study, we introduced a hybrid mutation strategy combining three-point inversion mutation and two-point swap mutation operators [27]. During the mutation phase, each chromosome undergoes a probabilistic selection mechanism based on its predetermined mutation threshold $P_m$. The selection mechanism triggers based on real-time probability evaluation. When a randomly selected chromosome exhibits a mutation probability below threshold $P_m$, the three-point inversion operator is activated. As illustrated in Figure 4a, this process randomly determines three distinct mutation loci within the chromosome’s genetic sequence and subsequently reverses the intervening allele order, thereby generating a novel genotypic configuration. For chromosomes with mutation probability equal to or exceeding $P_m$, the two-point swap mutation is implemented. This protocol randomly selects two non-adjacent positions, and then exchanges the genes at these locations to generate a new chromosome, as demonstrated in Figure 4b. The dual-operator mechanism establishes a balanced exploration-exploitation dynamic: the inversion operator facilitates macro-level structural diversity through segment reversal, while the swap operator enables micro-level local refinement via targeted gene exchange.

3.5. Local Search Strategy

NSGA-III has strong global search capabilities; however, its local search ability is relatively weak, and during the iteration process, it is prone to getting stuck in local optima. To find potentially better solutions, this paper adopts a hybrid approach combining the 2-opt local search strategy [28] with the NSGA-III algorithm. The 2-opt strategy is a widely used local search strategy for solving the traveling salesman problem, which improves the quality of the solution by swapping two nodes in the path. It is simple, easy to implement, has relatively low computational complexity, and can accelerate the convergence of the algorithm. Although the MOGVRPTW studied in this article is not exactly the same as the traveling salesman problem, 2-opt can still be used as a useful local search strategy. In this study, 2-opt can help the genetic algorithm jump out of local optima and explore other potential solutions in the solution space, thereby improving the quality of the solution.

When addressing the MOGVRPTW, nodes can be considered customer points, and vehicle access routes can be generated by swapping the positions of these customer points. When implementing the 2-opt strategy, select a segment of the route between two nodes for the exchange and subsequently update the individual’s fitness value. If the new individual is superior to the original individual, it is replaced with the current individual to achieve local optimization. This process can be summarized as pseudocode as follows Algorithm 1:

Algorithm 1: 2-Opt. Swap Function

Input: $Node(i,j)$; Output: Best $Node(i^{*},j^{*})$; function    2-Opt.Swap($Node$, i, j)    do                    $New.Node$←$Node$;                    $Route\_New.Node$←$Route\_Node[1:i-1]$                                                    +$Reverse(Route\_Node[i:j\left]\right)$+$Route\_Node[j:1]$;                    Calculate $Fitness\_New.Node$;                    Return $New.Node$; end for    $i=1$  to  $Size\_Node-1$    do         for    $j=i+1$  to  $Size\_Node$    do                  $New.Node=2-Opt.Swap(Node,i,j)$;                  if    $Fitness\_New.Node<Fitness\_Node$    then                         $Node=New.Node$;                  end          end end Return the best $Node$ ($i^{*},j^{*}$).

4. Computational Experiments

In this study, we evaluate the performance of the INSGA-III algorithm by comparing it with five widely used multi-objective optimization algorithms: NSGA-III, NSGA-II, SPEA2, MOPSO, and MOEA/D. All algorithms were implemented in MATLAB R2020a and executed on a computer equipped with an Intel(R) Core(TM) i7-12700H processor (2.30 GHz) and 16.0 GB of RAM in the Windows 10 platform.

4.1. Experimental Parameter Settings

In this study, the model necessitates consideration of time windows and cargo demand for each customer. To this end, the standard Solomon Benchmark [29] dataset for the VRPTW is employed for testing. This dataset is categorized into three types—C, R, and RC—which exhibit concentrated, dispersed, and mixed distributions of spatial and time window characteristics, respectively. For the purposes of this research, nine representative instances are selected, encompassing C-type, R-type, and RC-type instances with 25, 50, and 100 customer locations. Given that the benchmark was originally designed for single-objective VRP models, modifications are implemented to adapt it for the multi-objective scenario. Following the methodology outlined in [27], the original time windows in the dataset are interpreted as service time windows for customer satisfaction, with an additional 30 min extended at both ends to define the acceptable time window for customers. The fundamental experimental parameters are detailed in

Table 2. Experimental parameters.

Parameter	Value	Parameter	Value
${\sigma }_1$	10 CNY/L	$\alpha$	20 CNY/h
${\sigma }_2$	100 CNY/vehicle	$\beta$	60 CNY/h
${\sigma }_3$	2 CNY/km	${\rho }_0$	0.122 L/km
R	400 kg	${\rho }^{*}$	0.388 L/km
$v_{ij}$	40 km/h

.

4.2. Algorithm Performance Metrics

In this study, we evaluate the performance of the proposed algorithm using several key metrics tailored for multi-objective optimization problems. These include Hypervolume (HV) [30], Spacing (SP) [31], Inverted Generational Distance (IGD) [32], Set Coverage (SC) [33], and Diversification Metric (DM) [34]. Each metric is chosen to comprehensively assess the algorithm’s effectiveness across complex optimization scenarios.

In multi-objective optimization, the HV metric is widely employed to quantify the coverage extent of an approximate Pareto solution set in the objective space. It measures the volume of the objective space that is dominated by the solution set and bounded by a reference point, thereby providing a comprehensive assessment of both the convergence towards the true Pareto front and the diversity of the solution set. The HV value correlates positively with the algorithm’s multiobjective optimization capability—higher values signify superior convergence–diversity balance. The calculation of this metric is typically formulated as follows:

$$HV=\delta \left(\bigcup _{i=1}^n{v}_i\right)$$

(28)

where $\delta (·)$ represents the Lebesgue measure, quantifying the dominated hypervolume space, n denotes the cardinality of Pareto-optimal solutions in the non-dominated set, and $v_i$ represents the hypercube formed between the reference point and the i-th solution.

The SP metric is employed to assess the uniformity of the solution set. It measures the standard deviation of the distances between consecutive solutions in the approximate Pareto front, after sorting the solutions along one objective. A smaller SP value indicates a more uniform distribution, which is essential for offering decision-makers a diverse range of trade-off solutions. The formula for calculating the SP metric is as follows:

$$SP=\sqrt{{\displaystyle \frac{1}{\left|n\right|-1}}\sum _{i=1}^{\left|n\right|}{\left(\overline{d}-d_i\right)}^2}$$

(29)

where n denotes the total number of solutions in the set, $d_i$ represents the minimum distance from the i-th solution to any other solution, and $\overline{d}$ is defined as the mean of all $d_i$.

The IGD metric evaluates the proximity of an algorithm’s solution set to the true Pareto front. This metric is essential as it quantitatively measures the algorithm’s performance in approximating the true Pareto front, which is critical for assessing and comparing optimization methods. A smaller IGD value indicates that the solution set is closer to the true Pareto front, reflecting better convergence of the algorithm. It can be calculated by using the following formula:

$$IGD={\displaystyle \frac{1}{\left|n^{*}\right|}}\sum _{x\in n^{*}}d(x,n)$$

(30)

where $n^{*}$ represents the true Pareto front, n is the solution set obtained by the algorithm, and $d(x,n)$ denotes the minimum Euclidean distance form a solution x∈$n^{*}$ to any solution in the set n.

The SC metric quantifies the extent to which one non-dominated solution set dominates another, facilitating comparisons between algorithms. For two solution sets (A and B) obtained from distinct algorithms, $SC(A,B)$ is defined as the proportion of solutions in B dominated by at least one solution in A, expressed mathematically as follow equation:

$$SC(A,B)={\displaystyle \frac{1}{\left|B\right|}}\left|\{y\in B\mid \exists x\in A:x>y\}\right|$$

(31)

In Equation (31), $x>y$ means that the solution x in A dominates the solution y in B. The term $SC(A,B)$ represents the percentage of solutions in B that are dominated by at least one solution in A. If $SC(A,B)$ is equal to 1, it means that all solutions in B are dominated by those in A. Conversely, if $SC(A,B)$ is equal to 0, it indicates that no solutions in B are dominated by solutions in A. To comprehensively evaluate algorithmic performance, both $SC(A,B)$ and $SC(B,A)$ should be assessed. If $SC(A,B)$ is greater than $SC(B,A)$, it suggests that A exhibits greater coverage than B and vice versa.

The DM can be used to assess the diversity and distribution of a non-dominated solution set obtained by an algorithm within the objective space. A higher $DM$ value signifies greater diversity, indicating superior algorithmic performance in capturing a broad range of trade-offs. It is calculated as follows:

$$DM=\sqrt{\sum _{i=1}^m{\left({\displaystyle \frac{max{f}_i-min{f}_i}{max{f}_{i,\mathrm{all}}-min{f}_{i,\mathrm{all}}}}\right)}^2}$$

(32)

where $max{f}_i$ and $min{f}_i$ denote the maximum and minimum values of the i-th objective function within the solution set obtained by the selected algorithm, respectively, while $max{f}_{i,\mathrm{all}}$ and $min{f}_{i,\mathrm{all}}$ indicate the corresponding extremes of the i-th objective function across all evaluated algorithms, including the comparison algorithms.

4.3. Algorithm Parameter Settings

The performance of the INSGA-III algorithm is influenced by its parameter settings. To address this, we employ the orthogonal experimental method to determine the optimal settings for the main parameters of INSGA-III, including the population size $nPop$, crossover probability $Pc$, and mutation probability $P_m$. Each parameter is evaluated at three distinct levels, as detailed in

Table 3. Algorithm parameter level settings.

Parameters	Parameter Levels
	1	2	3
$nPop$	40	60	80
$P_c$	0.6	0.7	0.8
$P_m$	0.1	0.2	0.3

.

The experimental design is based on the $L_9$($3^3$) orthogonal table. We use the medium-scale instance C102, which has 50 customer points, as the test case. The INSGA-III algorithm is executed for 200 generations in each experiment. To ensure statistical robustness, each configuration is replicated 10 times. Performance is assessed using the average HV value ($H{V}_{avg}$), where a higher $H{V}_{avg}$ indicates better convergence and diversity of the solution set. The outcomes of these orthogonal experiments are presented in

Table 4. Orthogonal experimental results.

Experiments	Parameters			${\mathit{H}\mathit{V}}_{\mathit{a}\mathit{v}\mathit{g}}$
	$\mathit{n}\mathit{P}\mathit{o}\mathit{p}$	${\mathit{P}}_{\mathit{c}}$	${\mathit{P}}_{\mathit{m}}$
1	40	0.6	0.1	0.8011
2	40	0.7	0.3	0.7820
3	40	0.8	0.2	0.6713
4	60	0.6	0.4	0.7245
5	60	0.7	0.2	0.8103
6	60	0.8	0.3	0.7403
7	80	0.6	0.0	0.7962
8	80	0.7	0.1	0.8911
9	80	0.8	0.3	0.8033

.

As demonstrated in Figure 5, the crossover probability $P_c$ exerts a substantial influence on algorithmic performance, whereas an optimal $P_c$ value effectively maintains population diversity. The population size $nPop$ possesses a dual effect: adequate populations maintain solution space coverage, whereas insufficient sizes increase risks of premature convergence and local optimum entrapment. Parameter combinations yielding elevated $H{V}_{avg}$ values signify superior algorithm performance, as this metric comprehensively reflects both convergence accuracy and distribution uniformity. Based on these analyses, the parameter configuration for the INSGA-III was rigorously determined as: $nPop$ = 80, $P_c$ = 0.7, and $P_m$ = 0.1. To ensure comparative validity, identical parameter values were systematically applied across all five benchmark algorithms where protocol consistency permitted. Specifically for MOPSO implementation, the inertia weight and acceleration coefficients were, respectively, set to 0.7 and 2, following established practices in swarm intelligence optimization.

4.4. Algorithm Comparison Experiment

The experimental framework systematically evaluates INSGA-III against five established multi-objective optimization algorithms across different problem scales, yielding nine distinct experimental configurations. To ensure statistical reliability, each configuration was independently executed 10 times with subsequent calculation of metric averages, where optimal values across comparative algorithms are distinctly marked in bold. As quantitatively documented in

Table 5. Comparison of algorithm performance.

Algorithm	Metric	25 Customer Points			50 Customer Points			100 Customer Points
		C102	R201	RC202	C102	R201	RC202	C102	R201	RC201
INSGA-III	$AvgHV$	0.8612	0.8487	0.8176	0.8581	0.8299	0.8113	0.8285	0.7526	0.8035
	$AvgSP$	0.0646	0.0534	0.0322	0.0609	0.0326	0.0374	0.0209	0.0279	0.0319
	$AvgIGD$	0.0754	0.0463	0.0518	0.1228	0.1238	0.1127	0.1733	0.2062	0.2569
	$AvgDM$	1.2332	1.5519	1.5007	1.5277	1.5343	1.4416	1.2087	1.3454	1.1194
	$AvgS{C}_1$	0.7344	0.5198	0.6604	0.6581	0.6300	0.5113	0.6285	0.7526	0.6035
	$AvgS{C}_2$	0.5248	0.5987	0.5889	0.6609	0.6326	0.6374	0.6209	0.7879	0.6519
	$AvgS{C}_3$	0.5552	0.6669	0.6001	0.5228	0.6238	0.5900	0.6733	0.6662	0.5969
	$AvgS{C}_4$	0.7115	0.6224	0.7009	0.8581	0.8229	0.8113	0.8285	0.8269	0.8069
	$AvgS{C}_5$	0.7905	0.8021	0.8432	0.9800	0.7736	0.9493	0.9227	0.8001	0.8889
NSGA-III	$AvgHV$	0.8307	0.7400	0.7154	0.7741	0.7020	0.7948	0.8132	0.6034	0.6853
	$AvgSP$	0.0799	0.0718	0.0473	0.0634	0.0441	0.0391	0.0428	0.0482	0.0357
	$AvgIGD$	0.0998	0.1975	0.1537	0.1795	0.2007	0.1275	0.1861	0.2419	0.2649
	$AvgDM$	1.0079	1.3763	1.4035	1.2873	1.1271	1.0954	1.2207	1.0768	0.7915
	$AvgS{C}_1$	0.4556	0.3224	0.4445	0.2977	0.3218	0.3511	0.3369	0.2237	0.5214
NSGA-II	$AvgHV$	0.8449	0.7197	0.7309	0.7373	0.7596	0.7553	0.7789	0.5302	0.6921
	$AvgSP$	0.1724	0.0631	0.0549	0.0980	0.0386	0.0502	0.0448	0.0366	0.0368
	$AvgIGD$	0.0858	0.1544	0.1718	0.1633	0.1514	0.1703	0.1801	0.2771	0.2815
	$AvgDM$	1.5378	1.1901	1.4302	0.9474	1.4216	1.4320	1.2862	0.9700	0.8674
	$AvgS{C}_2$	0.4008	0.3664	0.3997	0.3337	0.3689	0.3766	0.3264	0.3998	0.2916
SPEA2	$AvgHV$	0.8282	0.8059	0.6707	0.7620	0.7707	0.7891	0.6234	0.7095	0.6639
	$AvgSP$	0.1100	0.0913	0.1082	0.1188	0.0903	0.0492	0.0402	0.1032	0.0849
	$AvgIGD$	0.0856	0.0867	0.1908	0.2200	0.1409	0.1373	0.3271	0.2889	0.2642
	$AvgDM$	0.7682	1.3905	1.0673	1.5508	1.1836	1.3122	0.7633	1.1061	0.9816
	$AvgS{C}_3$	0.4112	0.2334	0.3445	0.3658	0.3564	0.3236	0.3967	0.3226	0.3015
MOPSO	$AvgHV$	0.6228	0.6530	0.5543	0.5067	0.5737	0.4935	0.4915	0.3750	0.3873
	$AvgSP$	0.0706	0.0623	0.0614	0.1186	0.0682	0.0673	0.0529	0.0972	0.0811
	$AvgIGD$	0.2507	0.2296	0.3067	0.5744	0.3977	0.5101	0.5715	0.8011	0.6921
	$AvgDM$	1.1987	1.1027	0.8892	0.7300	1.1156	0.9496	0.6604	1.1061	0.9571
	$AvgS{C}_4$	0.1446	0.2664	0.3001	0.2097	0.2698	0.2359	0.2016	0.1854	0.1993
MOEA/D	$AvgHV$	0.5805	0.7579	0.7484	0.5476	0.7861	0.7570	0.5267	0.5978	0.5622
	$AvgSP$	0.1572	0.0973	0.0393	0.0782	0.1020	0.0799	0.0650	0.1143	0.0880
	$AvgIGD$	0.2969	0.1516	0.2106	0.1572	0.1342	0.1499	0.5352	0.3007	0.3690
	$AvgDM$	1.2156	0.8455	1.1315	0.7462	1.0015	0.9947	0.6734	1.1236	0.9678
	$AvgS{C}_5$	0.6235	0.4677	0.4610	0.3632	0.5563	0.5246	0.3231	0.4977	0.5220

, the systematic comparisons employ five SC metrics: $S{C}_1$ quantifies performance differentials between INSGA-III and conventional NSGA-III, $S{C}_2$ compares against NSGA-II, $S{C}_3$ evaluates performance relative to SPEA2, $S{C}_4$ contrasts with MOPSO, and $S{C}_5$ assesses the performance difference with MOEA/D.

The comprehensive experimental findings demonstrate that, across problems of varying scales, INSGA-III statistically outperforms the other five comparison algorithms in terms of HV, SP, IGD, and SC. However, for C-type instances, INSGA-III exhibits slightly inferior performance on the Diversification Metric (DM) compared to the other algorithms. This discrepancy is likely attributable to the concentration of solutions within the solution space for these instances. Despite this, INSGA-III achieves acceptable performance on the DM for the remaining instance types. This phenomenon suggests that the algorithm’s emphasis on convergence-intensity balance may slightly compromise extreme solution spread in particular constraint-dense scenarios.

To evaluate algorithm performance variations across problem scales, the C102 benchmark instance was systematically analyzed under small, medium, and large-scale configurations. A comparative analysis of HV and SP distributions, illustrated through boxplot visualizations in Figure 6 and Figure 7, reveals that INSGA-III does not consistently attain the highest values across all instances and metrics. However, its boxplots exhibit narrower interquartile ranges and median values closer to the optima, indicating that the non-dominated solution set from 10 comparative runs shows reduced variation in HV and SP. This suggests greater stability in INSGA-III’s solutions. Additionally, INSGA-III excels in other metrics, such as IGD and SC, underscoring its superior convergence, uniformity, and diversity compared to the other five comparison algorithms.

Additionally, we compared five algorithms on C102 instances of varying sizes by plotting their convergence curves for three objectives: total delivery cost $f_1$, carbon emissions $f_2$, and average customer dissatisfaction (minimized) $f_3$. The results are presented in Figure 8, Figure 9 and Figure 10. In a specific case with 50 customer points, INSGA-III started with the highest initial total delivery cost but converged the fastest, overcoming local optima within approximately 19 s to reach the optimal solution. For carbon emissions and average customer dissatisfaction, although its initial values were comparable to those of other algorithms, INSGA-III quickly converged to better solutions in the early stages of the search. Overall, across all instance sizes, INSGA-III achieved the best values for all three objectives, with SPEA2 and MOPSO ranking second and last, respectively. This suggests that INSGA-III not only has a faster convergence speed but also a superior ability to escape local optima, leading to optimal values for each objective.

We conducted experiments to explore the mean and best performance characteristics of five algorithms on diverse instances of varying sizes. Each algorithm was run on each instance ten times to capture variability in performance. For each algorithm–instance pair, we calculated the mean and the best value achieved across these ten runs for three objectives: total delivery cost, carbon emissions, and customer satisfaction. The best value is defined as the minimum for total delivery cost and carbon emissions (minimization objectives) and the maximum for customer satisfaction (maximization objective). The results are presented in

Table 6. Extreme experimental results.

Algorithm	Objective	25 Customer Points			50 Customer Points			100 Customer Points
		C102	R201	RC202	C102	R201	RC202	C102	R201	RC201
INSGA-III	$min{f}_1$	2205	8072	6768	12,025	15,491	15,882	34,458	33,544	47,715
	$min{f}_2$	257.46	660.59	457.98	175.81	486.88	503.62	552.44	991.36	1070.13
	$max{f}_3$	0.9972	0.8436	0.9465	0.9265	0.8651	0.9095	0.9124	0.8651	0.8549
NSGA-III	$min{f}_1$	3812	14,681	10,597	30,150	23,108	27,938	182,880	49,890	61,729
	$min{f}_2$	318.41	757.17	620.31	223.57	553.93	513.61	652.32	1097.36	1223.73
	$max{f}_3$	0.9335	0.8036	0.9154	0.7984	0.7965	0.7989	0.6608	0.7300	0.7470
NSGA-II	$min{f}_1$	5042	14,965	9422	35,704	21,448	27,903	153,007	47,380	60,521
	$min{f}_2$	280.53	738.42	594.12	212.82	532.56	544.80	661.33	1115.94	1150.09
	$max{f}_3$	0.9022	0.7819	0.8865	0.8241	0.8027	0.8348	0.6928	0.7735	0.7891
SPEA2	$min{f}_1$	9621	9187	14,089	26,840	18,002	21,207	94,899	41,947	59,150
	$min{f}_2$	287.52	674.51	545.42	233.65	505.13	526.54	1101.46	1008.47	1293.55
	$max{f}_3$	0.8647	0.8346	0.7815	0.8574	0.7998	0.8654	0.7009	0.8199	0.7882
MOPSO	$min{f}_1$	7646	17,346	14,094	57,373	25,734	34,568	159,934	68,789	84,336
	$min{f}_2$	383.56	811.06	926.69	405.46	631.18	840.06	1471.97	1200.49	1633.65
	$max{f}_3$	0.9229	0.6965	0.7603	0.7249	0.7007	0.6934	0.6094	0.6340	0.6574
MOEA/D	$min{f}_1$	3729	9927	7593	14,047	15,866	16,261	69,715	43,167	52,717
	$min{f}_2$	327.00	739.95	502.98	385.00	507.88	691.06	1278.43	1102.83	1444.01
	$max{f}_3$	0.9511	0.7613	0.8639	0.8511	0.8575	0.8846	0.7701	0.8192	0.7929

and

Table 7. Mean experimental results.

Algorithm	Objective	25 Customer Points			50 Customer Points			100 Customer Points
		C102	R201	RC202	C102	R201	RC202	C102	R201	RC201
INSGA-III	$Avg{f}_1$	2456	8451	7265	13,495	16,389	16,004	54,962	41,019	50,715
	$Avg{f}_2$	305.96	683.46	490.63	204.14	512.44	529.65	580.95	1084.68	1296.83
	$Avg{f}_3$	0.9152	0.8196	0.9066	0.8998	0.8394	0.8849	0.9064	0.8316	0.8269
NSGA-III	$Avg{f}_1$	4051	15,396	11,884	32,004	25,170	28,654	194,561	51,498	62,875
	$Avg{f}_2$	350.45	780.53	660.05	250.68	580.66	546.33	686.94	1235.64	1345.85
	$Avg{f}_3$	0.8842	0.7844	0.8863	0.7694	0.7698	0.7767	0.6433	0.7198	0.7297
NSGA-II	$Avg{f}_1$	5236	16,143	10,123	36,874	22,587	29,006	158,598	49,423	61,362
	$Avg{f}_2$	315.47	750.88	613.55	229.78	559.33	583.77	702.69	1256.82	1365.69
	$Avg{f}_3$	0.8633	0.7615	0.8440	0.7859	0.7852	0.8160	0.6789	0.7598	0.7766
SPEA2	$Avg{f}_1$	9938	10,112	16,051	28,758	19,964	24,159	102,357	43,156	61,005
	$Avg{f}_2$	326.38	710.60	578.33	264.45	526.48	550.65	1192.54	1156.56	1387.53
	$Avg{f}_3$	0.8344	0.8019	0.7516	0.7583	0.7698	0.8423	0.6894	0.8056	0.7648
MOPSO	$Avg{f}_1$	9469	18,574	15,163	58,346	27,459	34,905	167,854	70,547	88,489
	$Avg{f}_2$	413.89	830.59	1005.83	424.38	656.85	887.44	1569.33	1301.58	1735.69
	$Avg{f}_3$	0.8455	0.6547	0.7342	0.7015	0.6867	0.6698	0.5865	0.6080	0.6136
MOEA/D	$Avg{f}_1$	4089	12,784	8042	15,244	17,769	17,444	73,004	46,987	55,021
	$Avg{f}_2$	384.05	799.98	550.16	404.50	580.64	720.04	1456.37	1301.45	1566.88
	$Avg{f}_3$	0.9002	0.7213	0.7926	0.7969	0.8013	0.7998	0.7043	0.7344	0.7642

, which show both mean and best values for each algorithm across different instance scales. In these tables, the optimal value for each objective is highlighted in bold. From these tables, it is evident that INSGA-III consistently outperforms the other algorithms across all problem scales (small, medium, and large) in terms of both mean and best values for all three objectives. For instance, from an average perspective, in the C102 instance with 25 customer points: (1) The mean total delivery cost for INSGA-III is 2456 CNY, representing savings of 39.37%, 53.09%, 75.29%, 74.06%, and 39.94% compared to NSGA-III, NSGA-II, SPEA2, MOPSO, and MOEA/D, respectively. (2) The mean carbon emissions for INSGA-III are 305.96 kg, resulting in reductions of 12.7%, 3.01%, 6.26%, 26.08%, and 20.33% compared to NSGA-III, NSGA-II, SPEA2, MOPSO, and MOEA/D, respectively. (3) For customer satisfaction, INSGA-III achieves a mean value of 0.9972, which is higher than that of other algorithms. Thus, INSGA-III demonstrates superior performance in both average and best-case scenarios for all evaluated objectives.

To facilitate a more intuitive comparison of the distribution of non-dominated solutions obtained by the algorithms, we present scatter plots of the non-dominated solution sets for three C102 instances of varying scales in Figure 11, Figure 12 and Figure 13. Non-dominated solutions, also known as Pareto-optimal solutions, are those where no objective can be improved without worsening at least one other objective. As shown in these figures, although INSGA-III may yield fewer non-dominated solutions in certain instances compared to other algorithms, its solutions are more concentrated in the upper-left region of the plots. This concentration suggests that, in most cases, the quality of INSGA-III’s non-dominated solutions is superior to that of other algorithms. Moreover, the distribution of these solutions indicates that INSGA-III exhibits enhanced performance in terms of solution distribution.

Figure 14, Figure 15 and Figure 16 show the optimal vehicle routes under three different scenarios for the C102 benchmark instance with varying scales. From left to right, the figures represent the optimal routes for the scenarios of minimum total delivery cost, minimum carbon emissions, and maximum average customer satisfaction, respectively.

5. Real-World Case Study

To comprehensively evaluate the performance of the algorithm proposed in this paper, we designed a dual validation approach that includes comparative experiments on benchmark datasets and implementation in a real-world industrial case study. For the practical implementation, we selected a private enterprise in City B, China, referred to as Company A. This modern company integrates multi-modal operations, including cargo warehousing, long-distance logistics transportation, urban distribution within the same city, and value-added supply chain services. Its distribution network spans multiple cities and towns across China. Specifically, within City B, Company A operates more than 1500 urban distribution outlets, enabling it to provide fully integrated logistics services to its customers. Furthermore, the company’s nationwide service coverage and complex operational ecosystem provide an ideal testbed for evaluating the algorithm’s scalability and adaptability under large-scale real-world conditions.

5.1. Experimental Data Processing

We selected the logistics network of Company A, located in a district of City B, as our case study. The selected service area serves 39 customer points, whose spatial distribution patterns are illustrated in Figure 17. Notably, clusters of customers exhibit significant spatial proximity, presenting opportunities for route optimization through node consolidation. To enhance delivery efficiency, we processed the original customer point data using a hierarchical clustering method, merging the nearest customer points into single points. Specifically, the following sets of customer points were each consolidated into a single new point: {1, 3, 4, 5}, {9, 10}, {15, 29}, {16, 17, 18}, {21, 22}, {24, 34}, and {26, 27}. Consequently, the original 39 customer points were reorganized into 29 ones. This simplified the network structure while retaining basic spatial service characteristics.

To facilitate spatial analysis and data processing, it is necessary to convert the geographic coordinates of Company A’s customer points into a planar coordinate system. The planar Cartesian coordinates (x, y) of the customer nodes were obtained through projection transformation. After data preprocessing, it was determined that the case study in this research involves delivery service demands for 29 customer points.

Table 8. Distribution center and customer point information.

Customer Point	X Coordinate	Y Coordinate	Customer Demand (ton)	Service Time (min)	Time Window (h)
0	30.956	8.842	0	0	$[6,12]$
1	18.692	1.528	1.08	22	$[7,10]$
2	22.197	4.345	1.68	25	$[6,10]$
3	26.275	5.505	1.5	23	$[7.5,12]$
4	12.078	5.638	0.7	14	$[6,7.5]$
5	9.72	8.348	0.58	12	$[7,10]$
6	8.279	8.691	0.31	6	$[6,8]$
7	24.772	8.793	0.8	16	$[6,9]$
8	15.096	8.924	0.83	12	$[6.5,9.5]$
9	8.991	10.484	0.5	10	$[8,11]$
10	18.086	11.073	0.98	20	$[8,10]$
11	9.182	14.126	0.5	10	$[7,11]$
12	17.378	23.541	0.87	17	$[7,10]$
13	20.219	14.498	1.45	22	$[8,11]$
14	11.988	12.037	0.65	13	$[7,11]$
15	20.771	9.748	1.6	24	$[6,8.5]$
16	23.793	8.079	2.21	33	$[6,9]$
17	22.388	11.148	1.75	26	$[8,12]$
18	24.273	6.217	2.3	35	$[6,9]$
19	20.904	6.195	1.63	24	$[7,11]$
20	17.467	18.673	0.9	14	$[7,11]$
21	8.738	18.799	0.43	9	$[8,11]$
22	18.621	8.057	1.05	21	$[6,9]$
23	3.311	9.515	1.63	24	$[7,11]$
24	7.661	23.979	0.25	5	$[7.5,11]$
25	14.63	12.356	0.76	15	$[7,11]$
26	15.677	22.054	0.83	12	$[7,10]$
27	19.378	9.56	1.1	22	$[6,9.5]$
28	19.745	7.877	1.2	24	$[6.5,9.5]$
29	20.008	1.984	1.25	19	$[8,11]$

presents the spatial location information and delivery task parameters.

Based on the urban development level and residents’ consumption data of City B, we set the vehicle-related parameters as follows: ${\sigma }_1$ = 10 CNY/L, ${\sigma }_2$ = 100 CNY/vehicle, ${\sigma }_3$ = 2 CNY/km, R= 4000 kg, ${\nu }_{ij}$ = 40 km/h, $\alpha$ = 20 CNY/h, $\beta$ = 60 CNY/h, ${\rho }_0$ = 0.122 L/km, ${\rho }^{*}$ = 0.388 L/km. Through a series of orthogonal experiments, we identified the optimal INSGA-III configuration for the current problem domain: $nPop$ = 80, $Pc$ = 0.7, and $P_m$ = 0.2. To ensure a fair comparison across algorithms, parameters common to multiple methods were assigned identical values where applicable. Specifically, for the MOPSO algorithm, we set the inertia coefficient to 0.7 and the acceleration coefficient to 2. All computational experiments were conducted on the hardware platform detailed in Section 4, maintaining computational environment consistency.

5.2. Algorithm Comparison Experiment

As outlined in Section 4, each algorithm was executed ten times in the case study to ensure the stability of the results. The experimental outcomes are presented in

Table 9. Experimental results of algorithm performance.

Algorithm	$\mathit{A}\mathit{v}\mathit{g}\mathit{H}\mathit{V}$	$\mathit{A}\mathit{v}\mathit{g}\mathit{S}\mathit{P}$	$\mathit{A}\mathit{v}\mathit{g}\mathit{I}\mathit{G}\mathit{D}$	$\mathit{A}\mathit{v}\mathit{g}\mathit{D}\mathit{M}$	${\mathit{A}\mathit{v}\mathit{g}\mathit{S}\mathit{C}}_1$	${\mathit{A}\mathit{v}\mathit{g}\mathit{S}\mathit{C}}_2$	${\mathit{A}\mathit{v}\mathit{g}\mathit{S}\mathit{C}}_3$	${\mathit{A}\mathit{v}\mathit{g}\mathit{S}\mathit{C}}_4$	${\mathit{A}\mathit{v}\mathit{g}\mathit{S}\mathit{C}}_5$
INSGA-III	0.8053	0.1146	0.6540	1.3320	0.8145	0.6368	0.5263	0.7745	0.8045
NSGA-III	0.7156	0.2551	1.8678	1.0609	0.1256	N/A	N/A	N/A	N/A
NSGA-II	0.6042	0.5369	0.6755	0.9780	N/A	0.1511	N/A	N/A	N/A
SPEA2	0.7745	0.2736	0.7552	0.6364	N/A	N/A	0.4091	N/A	N/A
MOPSO	0.4415	0.3244	3.0230	1.0145	N/A	N/A	N/A	0.1646	N/A
MOEA/D	0.4595	0.3091	0.6945	0.9987	N/A	N/A	N/A	N/A	0.5000

, which demonstrates that the INSGA-III algorithm outperforms the other algorithms across all five metrics. These findings indicate that INSGA-III achieves greater solution diversity and dominates the coverage of competing algorithms’ solution spaces.

Figure 18 presents boxplots comparing the HV and SP metrics after running each of the six algorithms ten times. For the HV metric, the INSGA-III algorithm exhibits a smaller interquartile range and a more centralized distribution, indicating superior convergence and diversity compared to the other five algorithms. Similarly, for the SC metric, while all six algorithms show a small interquartile range, the INSGA-III algorithm achieves better values. This suggests that although the SP metric values are stable across all algorithms after ten runs on Company A’s case, the INSGA-III algorithm produces a more uniform and well-distributed solution set.

We executed all six algorithms using the case study of Company A and generated convergence curves for three objective functions: total distribution cost, carbon emissions, and customer satisfaction. To simplify plotting, we redefined average customer satisfaction as the minimization of average customer dissatisfaction, as depicted in Figure 19.

For total distribution cost, the INSGA-III algorithm begins with a higher initial value of approximately 6500 CNY compared to some other algorithms. Nevertheless, it demonstrates the fastest convergence rate within the first 10 s of the search process, followed closely by MOEA/D. After approximately 25 s, the INSGA-III algorithm surpasses local optima to attain the optimal solution. Regarding carbon emissions, the INSGA-III algorithm exhibits initial values and convergence speeds comparable to those of the other algorithms. However, it delivers the superior final outcome, achieving an optimal solution of approximately 140 kg after about 18 s. For customer satisfaction, the INSGA-III algorithm outperforms the other algorithms in convergence speed, reaching 100% satisfaction in less than one second. These findings confirm that the INSGA-III algorithm efficiently identifies optimal solutions for the 29 customer points in Company A’s case study, highlighting its exceptional performance in both convergence speed and solution quality across all evaluated objectives.

Table 10. Extreme and mean experimental results.

Algorithm	Objective	Results	Objective	Results	Algorithm	Objective	Results	Objective	Results
INSGA-III	$min{f}_1$	2262	$Avg{f}_1$	2448	NSGA-III	$min{f}_1$	3852	$Avg{f}_1$	4012
	$min{f}_2$	140.65	$Avg{f}_2$	160.39		$min{f}_2$	169.85	$Avg{f}_2$	180.64
	$max{f}_3$	1.0000	$Avg{f}_3$	0.9436		$max{f}_3$	0.9278	$Avg{f}_3$	0.9006
NSGA-II	$min{f}_1$	4179	$Avg{f}_1$	4342	SPEA2	$min{f}_1$	3472	$Avg{f}_1$	3801
	$min{f}_2$	171.97	$Avg{f}_2$	191.36		$min{f}_2$	150.37	$Avg{f}_2$	162.37
	$max{f}_3$	0.9775	$Avg{f}_3$	0.9463		$max{f}_3$	0.9203	$Avg{f}_3$	0.9067
MOPSO	$min{f}_1$	5413	$Avg{f}_1$	5991	MOEA/D	$min{f}_1$	2834	$Avg{f}_1$	3069
	$min{f}_2$	202.50	$Avg{f}_2$	245.34		$min{f}_2$	159.91	$Avg{f}_2$	176.89
	$max{f}_3$	0.9478	$Avg{f}_3$	0.9133		$max{f}_3$	0.9654	$Avg{f}_3$	0.9436

presents the extreme and mean values for six algorithms across three optimization objectives: total distribution cost, carbon emissions, and customer satisfaction. The mean values were derived from running each algorithm 10 times. For the case study of Company A, the INSGA-III algorithm demonstrates superior performance. Specifically, for total distribution cost, the minimum and mean costs of the optimal route identified by the INSGA-III algorithm are 2262 CNY and 2448 CNY, respectively, which are significantly lower than those of the other algorithms. Regarding carbon emissions, the INSGA-III algorithm achieves a minimum of 140.65 kg and a mean of 160.39 kg, both of which are less than those of the competing algorithms. Thus, the INSGA-III algorithm outperforms the other algorithms in both total distribution cost and carbon emissions. For customer satisfaction, while the INSGA-III algorithm reaches a maximum satisfaction of 100% in some runs, its mean satisfaction is 0.9436. This is higher than that of NSGA-III, SPEA2, and MOPSO but comparable to NSGA-II and MOEA/D. This similarity likely arises because all algorithms utilize approximately ten vehicles, thereby largely meeting the customers’ time requirements. Overall, the INSGA-III algorithm exhibits greater stability and consistency across multiple runs compared to the other algorithms.

Figure 20 presents the spatial scatter plot and its two-dimensional projections of the non-dominated solution sets obtained by various algorithms for Company A’s case study. The INSGA-III algorithm generates fewer solutions compared to some other algorithms, with the MOEA/D algorithm producing the fewest. However, the solutions from the INSGA-III algorithm are predominantly located in the upper left region of the plot, which corresponds to lower costs and higher satisfaction levels. This distribution suggests that the overall quality of the INSGA-III algorithm’s solution set surpasses the other ones.

Table 11. Optimal delivery sequence for minimum total cost.

Route	Customer Visit Sequence	Route Length (km)
Route 1	0-16-15-0	20.8810
Route 2	0-7-4-6-28-0	46.8854
Route 3	0-20-26-12-11-24-21-0	74.8665
Route 4	0-25-19-3-0	36.6570
Route 5	0-27-22-8-0	32.7749
Route 6	0-17-10-0	26.2375
Route 7	0-13-9-27-0	50.8931
Route 8	0-18-2-0	19.8214
Route 9	0-1-5-23-14-0	60.3349

,

Table 12. Optimal delivery sequence for minimum carbon emission.

Route	Customer Visit Sequence	Route Length (km)
Route 1	0-18-2-0	19.8214
Route 2	0-7-1-5-29-0	51.9434
Route 3	0-20-12-26-24-21-9-0	67.6993
Route 4	0-16-19-0	21.0472
Route 5	0-15-17-0	21.2370
Route 6	0-22-4-8-28-0	39.8151
Route 7	0-11-25-10-13-0	47.9913
Route 8	0-3-0	11.4974
Route 9	0-27-6-23-14-0	56.0404

and

Table 13. Optimal delivery sequence for maximum average customer satisfaction.

Route	Customer Visit Sequence	Route Length (km)
Route 1	0-20-25-23-11-0	65.1575
Route 2	0-3-0	11.4974
Route 3	0-7-27-6-28-0	45.5127
Route 4	0-19-5-14-1-0	52.8591
Route 5	0-17-29-0	31.2595
Route 6	0-13-10-21-9-0	58.6432
Route 7	0-18-2-0	19.8214
Route 8	0-15-4-8-0	40.1627
Route 9	0-16-22-0	24.7295
Route 10	0-26-12-24-0	59.9664

present the delivery sequences corresponding to three distinct objective scenarios extracted from the Pareto-optimal solution set generated by INSGA-III. Specifically, the selected solutions represent (1) the route with the minimum total distribution cost, (2) the route with the lowest carbon emissions, and (3) the route with the highest average customer satisfaction. These representative solutions enable a comparative analysis of the impact of varying optimization preferences on routing strategies.

In conclusion, our findings conclusively demonstrate that the INSGA-III algorithm proposed in this study exhibits superior performance compared to the other five multi-objective optimization algorithms across multiple key performance metrics, particularly in terms of convergence speed, solution diversity, and computational efficiency. Comprehensive computational experiments conducted on benchmark instances and a real-world case consistently validate the algorithm’s effectiveness in handling complex vehicle routing scenarios characterized by multiple constraints. These results highlight the effectiveness of INSGA-III in addressing the MOGVRPTW.

6. Conclusions

This paper addresses the environmental impact of fuel consumption and carbon emissions during vehicle delivery processes by constructing time penalty cost functions and customer satisfaction functions using time windows. The objective is to minimize total delivery cost, minimize carbon emissions, and maximize average customer satisfaction, leading to the development of a multi-objective green vehicle routing problem model with time windows. The problem is solved using MATLAB programming, and an improved NSGA-III algorithm is designed. This algorithm is compared with five other multi-objective optimization algorithms. The experimental results demonstrate that the improved NSGA-III algorithm has significant advantages in terms of computational efficiency and optimization performance. Furthermore, the optimized total delivery cost objective not only reduces the economic costs for businesses but also enhances environmental benefits through reduced carbon emissions. Maximizing average customer satisfaction further improves social benefits. This research provides a reference for logistics companies in optimizing vehicle routing paths while reducing delivery costs, cutting carbon emissions, and enhancing customer satisfaction. The study has strong practical application value and aligns with the real-world needs of enterprise development.