Green Vehicle Routing Problem Optimization for LPG Distribution: Genetic Algorithms for Complex Constraints and Emission Reduction

Abstract

This study develops a Green Vehicle Routing Problem (GVRP) model to address key logistics challenges, including time windows, simultaneous pickup and delivery, heterogeneous vehicle fleets, and multiple trip allocations. The model incorporates emissions-related costs, such as carbon taxes, to encourage sustainable supply chain operations. Emissions are calculated based on the total shipment weight and the travel distance of each vehicle. The objective is to minimize operational costs while balancing economic efficiency and environmental sustainability. A Genetic Algorithm (GA) is applied to optimize vehicle routing and allocation, enhancing efficiency and reducing costs. A Liquid Petroleum Gas (LPG) distribution case study in Yogyakarta, Indonesia, validates the model’s effectiveness. The results show significant cost savings compared to current route planning methods, alongside a slight increase in carbon. A sensitivity analysis was conducted by testing the model with varying numbers of stations, revealing its robustness and the impact of the station density on the solution quality. By integrating carbon taxes and detailed emission calculations into its objective function, the GVRP model offers a practical solution for real-world logistics challenges. This study provides valuable insights for achieving cost-effective operations while advancing green supply chain practices.

1. Introduction

Logistics is a significant and growing contributor to global Greenhouse Gas (GHG) emissions. Transport emissions alone increased at an average annual rate of 1.7% from 1990 to 2022, a growth rate surpassed only by the industrial sector [1]. This rise in emissions presents a critical obstacle to achieving the United Nations Sustainable Development Goals (SDGs), particularly Goal 13: Climate Action, which calls for urgent measures to combat climate change and its impacts. The Net Zero Emissions (NZE) by 2050 Scenario requires a reduction in CO₂ emissions from the transport sector by over 3% annually by 2030 [1]. While freight transport alone contributes significantly to this challenge, accounting for between 20% and 40% of the transportation sector’s carbon emissions [2,3], the broader transport sector, which includes passenger transport, aviation, and maritime activities, also drives overall emissions growth. Addressing this challenge requires targeted strategies to enhance sustainability across all transport segments, particularly in urban freight systems, where environmental impacts are most pronounced. Aligning with SDG targets, efforts to reduce emissions in freight transport also contribute to sustainable cities (Goal 11) and responsible consumption and production (Goal 12).

One of the significant challenges in logistics is the inefficiency in vehicle routing and allocation, which increases emissions, operational costs, and fuel consumption. When vehicles are misallocated, or routes are poorly planned, delivery times become prolonged, fuel usage rises, and expenses escalate, making it difficult for companies to balance profitability with environmental goals [4]. External factors, such as government regulations or supply shortages, further complicate logistics operations.

The Liquid Petroleum Gas (LPG) 3 kg shortage in Indonesia exemplifies how inefficiencies in the distribution system, such as poor governance and inadequate vehicle allocation, can exacerbate supply chain challenges [5]. As a subsidized energy source, LPG 3 kg plays a vital role in Indonesia’s energy security by ensuring inclusive and affordable access for low-income households. This program supports energy efficiency, reduces dependency on less efficient fuels such as kerosene, and strengthens energy equity across diverse social strata. The LPG 3 kg is part of a broader energy conversion initiative aimed at transitioning from kerosene to cleaner, more efficient energy sources, aligning with Indonesia’s commitment to sustainable development.

The Indonesian government has introduced regulations to ensure LPG subsidies are targeted accurately, preventing misuse and maintaining price stability and supply consistency. Optimizing the distribution routes of LPG 3 kg is critical for bolstering energy security, as it guarantees availability and accessibility across regions, reduces distribution costs to sustain subsidies, minimizes environmental impacts to promote energy sustainability, and enhances the overall energy efficiency and equity. These efforts contribute directly to achieving SDG-7, which advocates for affordable, reliable, sustainable, and modern energy for all.

In the context of LPG distribution, effective route planning is particularly crucial in urban areas, where time window constraints are frequently imposed [6,7]. Adherence to specific delivery timeframes is essential to avoid penalties, minimize customer dissatisfaction, and prevent unnecessary resource wastage [8]. By optimizing routes to meet these constraints, logistics operations can achieve significant cost savings, reduce travel distances, and improve sustainability, ensuring more efficient and environmentally friendly supply chain operations [9,10].

The complexity increases when managing heterogeneous vehicle fleets. Vehicles vary in size, speed, fuel efficiency, and emissions, making it essential to allocate the right vehicle for each delivery. Vehicle mismanagement—for instance, using large vehicles for small loads or long routes—can raise costs and emissions unnecessarily [7,11,12].

Another layer of complexity arises with simultaneous pickup and delivery operations. For instance, in LPG distribution, vehicles often need to deliver filled cylinders while collecting empty ones. Without careful planning, this dual operation can lead to underutilized vehicles, backtracking, and longer travel distances, increasing both costs and emissions [13,14]. To mitigate these inefficiencies, planning routes that balance delivery and pickup demands is crucial, ensuring the full utilization of vehicles in both directions.

Multi-trip logistics adds further complexity to planning. Vehicles may need to perform multiple trips during a delivery cycle, requiring precise scheduling to minimize downtime and ensure efficient resource utilization. Each trip introduces additional challenges, as factors such as cargo weight and route distance affect the vehicle capacity and operational efficiency, necessitating dynamic adjustments to routing and vehicle assignments [11,15].

In logistics, vehicle allocation and route planning are closely interconnected and critical for operational efficiency. Inefficient allocation, such as assigning vehicles with mismatched capacities to delivery routes, leads to delays, higher costs, and increased emissions. Effective vehicle allocation ensures that vehicles are appropriately matched to delivery demands, minimizing unnecessary trips, and maximizing the utilization of vehicle capacity. By optimizing vehicle assignments, logistics operations can reduce costs and emissions while improving the overall efficiency and sustainability [16,17,18].

This study aims to address these issues by developing a Green Vehicle Routing Problem (GVRP) model that integrates multiple logistics challenges to minimize transportation costs and emissions. The GVRP model incorporates key complexities such as time window constraints, heterogeneous fleets, multi-trip scheduling, and simultaneous pickup and delivery. These elements are optimized to enhance operational efficiency and reduce environmental impact. However, achieving these objectives involves a trade-off between minimizing costs and emissions, as improving one often leads to an increase in the other. Such a trade-off necessitates a balanced approach to simultaneously meet economic and environmental goals.

In order to solve the GVRP, this study employs a Genetic Algorithm (GA), a robust metaheuristic well-suited for tackling complex optimization problems [19]. The GA optimizes vehicle allocation to routes by encoding potential solutions as chromosomes, with each gene representing a specific vehicle-route assignment. Through processes such as selection, crossover, and mutation, the algorithm iteratively refines vehicle-route allocations, balancing factors like vehicle capacity, delivery time windows, multi-trip scheduling, and simultaneous pickup and delivery operations. Over successive iterations, the algorithm identifies optimal or near-optimal solutions that align vehicle assignments with route demands, effectively minimizing both operational costs and emissions.

This paper contributes to the existing literature by providing a practical solution to real-world logistics challenges, particularly optimizing vehicle allocation and routing, focusing on reducing operational costs and emissions. Its contributions include the following:

• The study integrates multiple logistics complexities, such as time windows, heterogeneous fleets, multi-trip scheduling, and simultaneous pickup and delivery, into a single GVRP model. In addition to these complexities, the model includes an emissions function that accounts for the actual load during pickups and deliveries, as well as the distance traveled. These calculated emissions are then translated into monetary values by applying a carbon tax, providing a comprehensive approach to balancing economic efficiency with environmental sustainability.
• This study applies a GA to optimize vehicle route allocation, achieving significant cost reductions and emission control. Validation through a case study on the LPG 3 kg shortage in Yogyakarta, Indonesia, underscores the model’s effectiveness in addressing real-world logistical challenges.
• Unlike prior studies relying on simulated datasets, this research utilizes actual LPG distribution data, enhancing the model’s practical relevance and applicability to real-world operations.

This paper is organized as follows: Section 2 reviews relevant literature on Vehicle Routing Problems (VRPs) and green logistics. Section 3 outlines the development of the GVRP model, including the integration of emissions-related costs such as carbon tax. Section 4 details the application of the GA as the optimization technique, focusing on its role in solving the GVRP model by optimizing vehicle allocation to routes. Section 5 presents the experimental results and analyzes the model’s performance, while Section 6 concludes with key findings, practical implications, and recommendations for future research.

2. Literature Review

VRPs have traditionally focused on minimizing operational costs, such as distance, time, and fuel consumption [20]. However, the increasing emphasis on sustainability has led to the emergence of the GVRP, which incorporates environmental objectives, such as reducing carbon emissions, fuel consumption, and energy usage, into routing decisions [21,22]. Carbon emissions are reduced to mitigate the environmental impact of routing solutions [23,24]. On the other hand, fuel consumption targets minimizing fossil fuel usage through cost-oriented optimization, often considered alongside other expenses such as vehicle operation and driver costs [12,25]. Energy consumption optimization in some studies is modeled as the product of the payload and distance traveled [26], while others focus specifically on electric energy requirements [27,28]. This shift reflects a broader global commitment to green logistics, where the challenge lies in balancing economic performance with environmental responsibility. Against this backdrop, this paper develops a framework that calculates carbon emissions as a function of the vehicle load and traveled distance, offering a practical approach to align economic efficiency with environmental sustainability. Recent studies have explored various dimensions of the GVRP, including time windows, heterogeneous vehicles, simultaneous pickup and delivery, and multi-trips, to address logistical complexities while reducing environmental impacts. This section categorizes the literature based on these key characteristics, showcasing advancements in green vehicle routing.

Time windows are essential for ensuring timely deliveries while minimizing environmental impact. Xu et al. [7] introduced a capacitated GVRP model integrating time windows, fuel consumption, and customer satisfaction, solved using Non-dominated Sorting Genetic Algorithm (NSGA-II). Additionally, Liu et al. [29] incorporated time windows into a Joint Distribution-Green Vehicle Routing Problem (JD-GVRP) model to optimize cold chain logistics under carbon trading policies. Their findings highlight the importance of collaboration among companies in reducing costs and emissions while indirectly reinforcing the significance of time-sensitive coordination to enhance delivery efficiency and sustainability. Wen et al. [30] proposed a Multi-Depot Green Vehicle Routing Problem with Time Windows (MDGVRPTW), employing an Adaptive Large Neighborhood Search (ALNS) algorithm to minimize carbon emissions, fuel consumption, and driver costs. Further advancing MDGVRPTW, Su et al. [31] incorporated customer satisfaction into the model and proposed a lightweight GA with Variable Neighborhood Search (VNS). Their approach demonstrated a superior scalability and optimization performance, making it suitable for environmentally conscious large-scale routing scenarios. Shifting the focus to time-dependent dynamics, Qi et al. [32] introduced the Time-Dependent Green Vehicle Routing Problem with Time Windows (TDGVRPTW), emphasizing the need to address varying travel times influenced by traffic congestion and other factors. Using a Q-learning-based Multi-Objective Evolutionary Algorithm (QMOEA), their study optimized energy consumption and vehicle duration while maintaining high-quality solutions across complex, time-sensitive logistics systems.

Urban logistics also benefits from time window integration. Chen et al. [33] proposed a cold chain logistics model with mixed fleets of electric and conventional vehicles, minimizing energy consumption and costs through an improved VNS algorithm. Zhou et al. [34] addressed time-dependent electric vehicle routing and scheduling, achieving significant energy savings with a mixed-integer programming model and VNS with Partial Model (VNS-PM). Liu et al. [35] further improved TDGVRPTW solutions with an ALNS algorithm enhanced by time discretization, efficiently reducing carbon emissions in large-scale scenarios. Lou et al. [36] introduced a low-carbon VRP incorporating high-granularity time-dependent speeds, speed fluctuations, and road conditions to address environmental concerns in logistics. Using a graph convolutional network for traffic prediction and a hybrid GA with adaptive VNS, their approach effectively minimizes carbon emissions, demonstrating strong performance in real-world scenarios.

Researchers have increasingly focused on heterogeneous vehicle fleets to address diverse logistical and environmental requirements. Kim et al. [24] expanded the scope of heterogeneous fleet routing by considering a multi-period approach. Their model incorporated carbon trading policies, where companies could buy or sell emission rights depending on their compliance in each period. Using a tabu search algorithm, they demonstrated that multi-period planning significantly reduced carbon emissions without increasing total costs, highlighting the importance of temporal optimization in heterogeneous fleet management. Wang and Wen [12] incorporated heterogeneous fleets into a low-carbon vehicle routing model for cold chain logistics, integrating carbon trading policies and optimizing emissions and costs using an Adaptive Genetic Algorithm (AGA). Islam et al. [37] introduced the mixed fleet green clustered logistics problem, combining hydrogen and conventional vehicles under CO₂ emission caps. Their hybrid Particle Swarm Optimization (PSO) and VNS approach achieved significant emission reductions, demonstrating the value of fleet diversity. In a follow-up study, Islam et al. [38] optimized fleet composition for logistics distribution, focusing on green and conventional vehicles under time-constrained deliveries. Using hybrid Ant Colony Optimization (ACO) and VNS, they achieved substantial CO₂ reductions and produced 21 new best-known solutions, showcasing the robustness of their approach. Gao et al. [26] developed a hybrid GA–Large Neighborhood Search (LNS) algorithm to optimize energy consumption in manufacturing systems by improving the management of heterogeneous automated guided vehicle fleets. Extending this application to urban logistics, Pak and Mun [25] applied heterogeneous fleets in small and medium cities, integrating time windows and multi-trip routing to optimize fuel consumption.

Simultaneous Pickup and Delivery (SPD) streamlines logistics by combining delivery and return operations, reducing the overall fuel consumption and emissions. Olgun et al. [39] developed a GVRP model with SPD, leveraging a Hyper-Heuristic algorithm (HH-ILS) to minimize fuel consumption while addressing greenhouse gas emissions. Santos et al. [40] extended SPD into reverse logistics, focusing on divisible deliveries and pickups. Their combination of ALNS and exact methods significantly reduced the emissions in load-constrained scenarios. Zhao et al. [41] advanced this approach by introducing a green split multiple-commodity VRP model, balancing transport costs, carbon emissions, and fuel consumption with a Two-Stage Search Quantum Particle Swarm Optimization (TSQPSO) algorithm. Similarly, Zhang et al. [21] addressed green truckload pickup and delivery with outsourcing, optimizing fuel consumption and speed using branch-and-price and column-generation algorithms.

Moreover, multi-trip routing, where vehicles perform multiple delivery rounds, is essential for improving fleet utilization and reducing emissions. Pak and Mun [25] specifically addressed multi-trips in their time-dependent VRP model for small and medium cities. Integrating heterogeneous fleets and time windows minimized the fuel consumption using a heuristic VNS-based approach, demonstrating substantial operational efficiency and sustainability.

As discussed above, the gap summary of previous research on the GVRP is presented in

Table 1. Gap of summary.

Authors	GVRP Characteristic				Decision Variables		Environmental Function	Model Objective	Model Solution	Case Study
	TW	HV	MT	SPD	VA	RP
Kim et al. [24]		√			√	√	CO2 emissions by distance; carbon trading	Min costs (travel and carbon trading)	Tabu search	Numerical example
Xu et al. [7]	√					√	Fuel-based vehicle load, capacity, and traffic conditions	Min fuel consumption; max customer satisfaction	Improved NSGA-II	Dataset from Xu et al. [42]
Wang and Wen [12]	√	√			√	√	CO2 as a function of fuel and electricity consumption; carbon trading	Min costs, carbon emissions; max customer satisfaction	Adaptive Genetic Algorithm	Cold chain logistics
Liu et al. [29]	√					√	CO2 as a function of fuel, distance, vehicle load; carbon trading	Min costs (fixed, travel, damage, refrigeration, and emission)	SA	Cold chain logistics
Chen et al. [43]	√					√	CO2 as a function of fuel and electricity consumption; carbon cost/tax	Min costs (fixed, travel, refrigeration, and emission)	VNS	Dataset from Solomon [44]
Islam et al. [37]	√	√		√	√	√	CO2 emissions by load, speed, and distance; emission cap	Min travel distance under emission constraint	Hybrid PSO and Neighborhood Search	Dataset from Gelinas et al. [45]
Islam et al. [38]	√	√			√	√	CO2 emissions by fuel use, total weight, and distance	Min travel distance under emission constraint	Hybrid ACO and VNS	Dataset from Solomon [44]
Olgun et al. [39]				√		√	Fuel-based vehicle load and distance	Min fuel costs	Hyper-Heuristic algorithm	Dataset from [46]; Dethloff [47]
Wen et al. [30]	√					√	CO2 emissions by fuel use	Min costs (carbon, fuel, vehicle rental, and driver salaries)	ALNS	Dataset from Solomon [44]
Qi et al. [32]	√					√	CO2 emissions by fuel use	Min time duration vehicles and energy consumption; max customer satisfaction	QMOEA	Dataset from Solomon [44]
Gao et al. [26]	√	√			√	√	Energy consumption-based distance	Min distance and energy	Hybrid GA- LNS	Dataset from Solomon [44]
Zhao et al. [41]	√			√		√	CO2 emissions by distance	Min costs (transport, carbon, penalty, loading-unloading, and fuel consumption)	Two-stage search quantum PSO	Dataset from Solomon [44]
Liu et al. [35]	√					√	CO2 emissions by load, speed, and distance	Min carbon emissions	ALNS	Dataset from Solomon [44]
Santos et al. [40]				√		√	CO2 emissions by fuel use	Min route costs; min emission	ALNS	Dataset from Hoff et al. [48]
Chen et al. [33]	√	√			√	√	CO2 emissions by fuel; carbon prices	Min costs (fixed, energy consumption, damage, and carbon)	VNS	Cold supply chain
Su et al. [31]	√					√	CO2 emissions by fuel use	Min costs (distance; penalties, fuel; carbon emissions; and vehicle rental)	Hybrid GA and VNS	Dataset from Cordeau et al. [49]; Homberger and Gehring [50]
Lou et al. [36]	√					√	CO2 emissions by fuel use and speed	Min carbon emissions	Hybrid GA and VNS	Logistic distribution, China
Zhang et al. [21]	√			√		√	CO2 emissions by fuel use	Min costs (fuel and outsource)	Heuristic algorithm	Numerical example of small–medium cities
Zhou et al. [34]	√					√	Energy consumption	Min costs (fixed, travel distance, and energy)	VNS with partial model (VNS-PM)	Lu et al. [51]
Pak and Mun [25]	√	√	√		√	√	Fuel consumption by speed	Min fuel consumption	VNS	Numerical example of small–medium cities
This paper	√	√	√	√	√	√	CO2 emissions by vehicle load and distance; carbon tax	Min costs (travel, and carbon emission)	GA	LPG distribution

Note: TW: time window; HV: heterogeneous vehicle; MT: multi-trips; SPD: simultaneous pickup and delivery; PSO: Particle Swarm Optimization; GA: Genetic Algorithm; SA: simulated annealing; ALNS: adaptive large-scale Neighborhood Search; VNS: Variable Neighborhood Search; ACO: Ant Colony Optimization; QMOEA: Q-learning-based Multi-Objective Evolutionary Algorithm.

. As shown in Table 1, most GVRP studies incorporate time windows as a core characteristic. However, only six studies—Wang and Wen [12], Islam et al. [37,38], Gao et al. [26], Chen et al. [33], and Pak and Mun [25]—integrate time windows with heterogeneous vehicles in their models. Among these, only Kim et al. [24] explored heterogeneous vehicles in a multi-period context, providing unique insights but in a distinct application. Additionally, few studies combine time windows with simultaneous pickup and delivery (SPD), such as Islam et al. [37] and Zhao et al. [41]. In contrast, others, such as Olgun et al. [39] and Santos et al. [40], only focus on SPD without time windows. The most comprehensive work, Pak and Mun [25], integrates time windows, heterogeneous vehicles, and multi-trips into a single model. Despite its robustness, it does not address SPD. This study bridges this gap by incorporating all four characteristics—time windows, heterogeneous vehicles, multi-trips, and SPD—into a single GVRP model, addressing the logistical complexities inherent in LPG distribution systems.

Previous studies have primarily focused on optimizing route planning under specific constraints. However, when heterogeneous vehicles are considered, the decision variables expand beyond route planning to include vehicle allocation, as demonstrated by Kim et al. [24], Wang and Wen [12], Islam et al. [37,38], Gao et al. [26], Chen et al. [33], and Pak and Mun [25]. The inclusion of vehicle allocation introduces an additional layer of complexity, requiring the model to allocate the appropriate vehicle types to specific routes effectively. This study incorporates vehicle allocation as a critical decision variable, reflecting the complexities of managing LPG delivery fleets of heterogeneous vehicles serving varied demands in both urban and rural settings.

Most studies address carbon emissions as functions of factors such as fuel usage, distance, load, speed, and energy. Several mechanisms have been adopted to translate carbon emissions into monetary terms, including carbon trading (Kim et al. [24], Wang and Wen [12], and Liu et al. [29]), emission caps (Islam et al. [37,38]), and emission taxes (Chen et al. [33,43]). Among these, the carbon tax method stands out for its simplicity and practicality. Our study adopts the carbon tax approach, calculating emission costs based on the vehicle load and distance traveled. This approach provides a more precise framework for balancing economic and environmental objectives, supporting the goals of sustainable logistics.

GVRP models are classified as NP-Hard, and most prior studies rely on meta-heuristic methods such as ALNS, PSO, SA, and ACO to solve these problems effectively. This study employs a GA known for its versatility in solving complex optimization problems. GA excels in exploring large solution spaces, maintaining solution diversity, and avoiding premature convergence—critical advantages for solving GVRP models.

Additionally, while many studies validate their models using benchmark datasets like Solomon or random numerical examples, few utilize real case datasets. Notable exceptions include Wang and Wen [12], Liu et al. [29], Chen et al. [33], and Lou et al. [36], which leverage cold supply chain data. Following this approach, this study validates its model using real case data from an LPG distribution network in Yogyakarta, Indonesia, demonstrating its applicability in addressing real-world logistical challenges.

This study addresses key gaps in GVRP research by integrating four critical characteristics—time windows, heterogeneous vehicles, multi-trips, and SPD into a single, comprehensive GVRP model. These features allow the model to reflect better the complexities of real-world logistics operations, particularly those involving the allocation and utilization of heterogeneous fleets. The inclusion of vehicle allocation as a decision variable further strengthens the model’s ability to handle diverse logistical challenges.

The case study focuses on the distribution of LPG 3 kg in Yogyakarta, Indonesia, a critical supply chain system with complex operational requirements. This case was selected due to its relevance in Indonesia, where LPG is a primary energy source, and its logistical network poses challenges such as tight time windows and the need for multi-trip deliveries. The choice of Yogyakarta provides a practical context for validating the model while demonstrating its applicability to essential commodity distribution systems.

Adopting a carbon tax approach aligns with Indonesia’s Presidential Regulation No. 98 of 2021, which mandates the implementation of Carbon Economic Value mechanisms, including carbon taxes, to reduce greenhouse gas emissions and achieve Nationally Determined Contribution targets. By internalizing emissions-related costs, the model incorporates real-world regulatory considerations, encouraging balanced decision-making that addresses both economic and environmental objectives.

In addressing the GVRP, this study employs a GA, which offers a robust and scalable method for optimizing complex logistical systems. The model is validated using real case data, demonstrating its ability to reduce operational costs, manage emissions, and comply with emerging carbon pricing regulations in Indonesia and similar economies transitioning toward sustainability.

Overall, this research significantly contributes to theoretical advancements in the GVRP and practical applications in green logistics. By combining sustainability considerations with operational efficiency, this study offers valuable insights into essential commodity distribution networks, emphasizing the importance of balancing economic goals with environmental responsibility.

3. Problem Definitions and Model Formulations

This section presents problem definitions and mathematical models of the GVRP that consider carbon emission, time windows, heterogeneous vehicles, multi-trips, and simultaneous pickup and delivery.

3.1. Problem Descriptions

As illustrated in Figure 1, real-world scenarios reveal inefficiencies within the current distribution system that lead to supply disruptions. Several issues must be addressed to enhance the management of LPG 3 kg deliveries. Firstly, distribution from depots to customers involves a range of vehicle types with varying capacities, requiring an optimized allocation of vehicles to routes that maximize load utilization and minimize operational costs. Secondly, delivery schedules often fail to align with each customer’s time windows, creating delays; this issue can be mitigated by implementing route planning that prioritizes timely deliveries. Additionally, vehicle underutilization is a frequent problem, with vehicles often returning to the depot after serving only one customer, resulting in increased operational expenses and emissions. Vehicle underutilization can be addressed by designing multi-customer delivery routes that reduce empty travel. Moreover, in practical cases, deliveries involve transporting full LPG cylinders and collecting empty ones, where emissions need to be carefully managed by calculating emissions based on the travel distance and vehicle load. Lastly, in some instances, deliveries can follow multi-trip schedules. If time permits, vehicles may make additional trips; however, emissions should still be minimized to maintain environmental efficiency. Given these challenges, this paper aims to develop a GVRP model that addresses the complexities of LPG distribution by integrating emission constraints, time windows, heterogeneous vehicles, multi-trip capabilities, and simultaneous pickup and delivery. The model is solved using GA to determine optimal vehicle allocation and route planning, minimizing both transportation costs and carbon emissions through a tax-based approach.

3.2. Assumptions

The assumptions of this proposed model are given as follows:

• The total number of vehicles in the fleet is fixed and predetermined before optimization begins.
• Each vehicle has a fixed capacity that does not change over time, ensuring consistent load handling.
• Vehicles travel at a constant average speed.
• The demand at each station (or node) is constant, known in advance, and free from uncertainty or variability.
• The distances between each pair of stations are predetermined and remain unchanged.
• Travel times between stations are calculated based on the fixed distance and the constant average speed of the vehicle without accounting for external delays such as traffic congestion.
• Each station has a predefined delivery time window, which is fixed and must be respected.
• Vehicles may begin service at a station before the time window opens if pre-arranged with the station owner, but this incurs a penalty cost.
• All stations are visited precisely once, and all delivery and pickup demands are satisfied within the planned routes.
• Each vehicle performs simultaneous pickup and delivery operations, assuming that the items picked up (e.g., empty cylinders) are the same type as those delivered (e.g., full cylinders).

3.3. Notations

The following notations are introduced in

Table 2. Notations of the model.

Notations	Descriptions
Sets and Indices
$M$	Set of stations and depot, where 0 denotes the depot and 1, 2, 3, ..., $m$ are the stations (nodes).
$V$	Set of vehicles, indexed as $v = 1, 2, 3, \dots , V.$
$i, j$	Indices for stations (nodes), where $i$, $j$ ∈ M.
$R$	Set of trips (or rounds) each vehicle can make, where $r = 1, 2, 3, \dots , R$ represents individual trips by each vehicle.
Parameters
$T_i$	Number of full LPG cylinders delivered to station $i$ $\in$ $M.$
$R_i$	Number of empty LPG cylinders picked up from station $i \in M$ (assumed $R_i=T_i$ for balance).
$C^v$	Capacity of vehicle $v \in V$, in terms of the maximum number of LPG cylinders (full + empty) it can carry.
$W_{full}$	Weight of a single full LPG cylinder.
$W_{empty}$	Weight of a single empty LPG cylinder.
$W_{truc{k}_{empty}}$	Tare weight (empty weight) of the vehicle at the depot.
$W_i^{v,r}$	Total weight on vehicle v at station $i$ on trip $r$, including both full and empty LPG cylinders.
$d_{ij}$	Distance between station $i \in M$ and station $j \in M$, in kilometers.
$S^v$	Average speed of single vehicle $v$.
$f_i,h_i$	Time window for station $i$, with $f_i$ as the earliest and $h_i$ as the latest delivery times.
${LD}_{full}$	Loading time per 10 units gallon of full LPG cylinders.
${UD}_{full}$	Unloading time per 10 units gallon of full LPG cylinders.
${LD}_{empty}$	Loading time per 10 units gallon of empty LPG cylinders.
${UD}_{empty}$	Unloading time per 10 units gallon of empty LPG cylinders.
$B_{early}$	Penalty cost per unit of time due to early arrivals.
$B_{late}$	Penalty cost per unit of time due to late arrivals.
$B_{late}^{\left(0\right)}$	Penalty cost per unit time for late returns to the depot.
$c_{km}$	Cost per kilometer traveled.
$X_i^{v,r}$	Arrival time of vehicle $v$ at station $i$ on trip $r$.
$X_0^{v,r}$	Vehicle $v$ returns to the depot after completing trip $r.$
$T_{final}^v$	Total time for vehicle $v$ to complete all trips.
$E{F}_{{CO}_2}$	Emission factor for CO2, in kgCO2/short ton-mile.
$E{F}_{{CH}_4}$	Emission factor for CH₄, in gCH₄/short ton-mile.
$E{F}_{N_{2}O}$	Emission factor for N2O, in gN2O/short ton-mile.
$GW{P}_{{CO}_2}$	Global Warming Potential for CO2.
$GW{P}_{{CH}_4}$	Global Warming Potential for CH₄.
$GW{P}_{N_{2}O}$	Global Warming Potential for N2O.
${{CO}_{2e}}_{ij}^{v,r}$	Total CO2 equivalent emissions for vehicle $v$ traveling from station $i$ to station $j$ on trip $r$, in ton CO2e.
$CT$	Carbon tax rate, representing the cost per ton of CO2e emitted.
Variables
$y_{ij}^{v,r}$	Binary decision variable, where $y_{ij}^{v,r}$ = 1 if vehicle $\mathrm{v}$ is allocated to travel from station $i$ to station $j$ on trip $r$, and $y_{ij}^{v,r}$ = 0 otherwise.
Conversion factors
$k{m}_{t{o}_{miles}}$	0.621371: Conversion factor from kilometers (km) to miles.
$k{g}_{t{o}_{short-ton}}$	0.00110231: Conversion factor from kilograms (kg) to short tons.
$g_{t{o}_{kg}}$	0.001: Conversion factor from grams to kilograms.

to develop the mathematical model:

3.4. Mathematical Formulation

Since the model considers some complexities of LPG distribution, as discussed in the problem descriptions, the GVRP model can be formulated as shown in Equation (1). The objective function in Equation (1) minimizes the total travel cost, carbon emission cost, and time-based penalties.

$$Z=\sum _{v \in V}\sum _{r \in R}\sum _{i, j \in M^{v,r}}{c}_{km}·d_{ij}·y_{ij}^{v, r}+CT\sum _{v \in V}\sum _{r \in R}\sum _{i, j \in N^{v, r}}{{CO}_{2e}}_{ij}^{v, r}+P_{early}+P_{late}+P_{late}^{\left(0\right)}$$

(1)

Subject to

$$P_{\mathrm{early}}=\sum _{i \in M, \hspace{0.17em}i \ne 0}\sum _{v \in V}\sum _{r \in R}\left\{\begin{array}{c}{B}_{\mathrm{early}}⌈\max \left(0,f_i-X_i^{v, r}\right)⌉\\ 0\end{array}\right.\begin{array}{c}If \left(f_i-X_i^{v, r}\right)\ge 1 \mathrm{h}\\ If \left(f_i-X_i^{v, r}\right)<1 \mathrm{h}\end{array}$$

(2)

$$P_{\mathrm{late}}=\sum _{i \in M, \hspace{0.17em}i \ne 0}\sum _{v \in V}\sum _{r \in R}\left\{\begin{array}{c}{B}_{\mathrm{late}}⌈\max \left(0,X_i^{v, r}-h_i\right)⌉\\ 0\end{array}\right.\begin{array}{c}If \left(X_i^{v, r}-h_i\right)\ge 1 \mathrm{h}\\ If \left(X_i^{v, r}-h_i\right)<1 \mathrm{h}\end{array}$$

(3)

$$P_{\mathrm{late}}^{\left(0\right)}=\sum _{i \in M, \hspace{0.17em}i = 0}\sum _{v \in V}\sum _{r \in R}\left\{\begin{array}{c}{B}_{\mathrm{late}}^{\left(0\right)}⌈\max \left(0,X_0^{v, r}-h_0\right)⌉\\ 0\end{array}\right.\begin{array}{c}If \left(X_0^{v, r}-h_0\right)\ge 1 \mathrm{h}\\ If \left(X_0^{v, r}-h_0\right)<1 \mathrm{h}\end{array}$$

(4)

$$\sum _{i \in N}\left(T_i+R_i\right)\cdot y_{ij}^{v, r}\le C^v, \forall v\in V, \forall r\in R$$

(5)

$$\sum _{v \in V}\sum _{r \in R}\sum _{j \in M}{y}_{ij}^{v, r}=1, \forall i \in M, i \ne 0$$

(6)

$$f_i\le X_i^{v, r}\le h_i, \forall i\in M^{v, r}, \forall v\in V, \forall r\in R$$

(7)

$$X_j^{v, r}=\left\{\begin{array}{c}{f}_0+{\displaystyle \frac{d_{0j}}{S_v}}+{\displaystyle \frac{{\sum }_{ i \in M, i = 0 }{T}_i{y}_{ij}^{v, r}}{10}}L{D}_{full}+{\displaystyle \frac{T_j}{10}}U{D}_{full}+{\displaystyle \frac{R_j}{10}}{LD}_{empty}, if i=0, j\ne 0 \\ X_i^{v, r}+{\displaystyle \frac{d_{ij}}{S_v}}+{\displaystyle \frac{T_j}{10}}U{D}_{full}+{\displaystyle \frac{R_j}{10}}{LD}_{empty}, if i\ne 0, j=0 \\ X_i^{v, r}+{\displaystyle \frac{d_{i0}}{S_v}}+{\displaystyle \frac{{\sum }_{ i \in M, i \ne 0 }{R}_i{y}_{ij}^{v, r}}{10}}{UD}_{empty}, if j=0 \left(\mathrm{i}.\mathrm{e}., X_0^{v, r}\right) \end{array}\right.$$

(8)

$$T_{final}^v={max}_{r \in R}\left(X_0^{v, r}\right)$$

(9)

$$W_0^{v, r}=W_{truc{k}_{empty}}+\sum _{i \in N^{v, r}}{T}_i\cdot W_{full}$$

(10)

$$W_j^{v, r}=W_i^{v, r}-\left(T_i\cdot W_{full}\right)+\left(R_i\cdot W_{empty}\right)$$

(11)

$$W_{i, short-ton}^{v, r}=W_i^{v, r}\cdot k{g}_{t{o}_{short-ton}}$$

(12)

$$d_{ij}^{miles}=d_{ij}\cdot k{m}_{t{o}_{miles}}$$

(13)

$${{CO}_2}_{ij}^{v, r}=d_{ij}^{miles}\cdot W_{i, short-ton}^{v, r}\cdot E{F}_{{CO}_2}\cdot GW{P}_{{CO}_2}$$

(14)

$${{CH}_4}_{ij}^{v, r}=d_{ij}^{miles}\cdot W_{i, short-ton}^{v, r}\cdot E{F}_{{CH}_4}\cdot GW{P}_{{CH}_4}\cdot g_{t{o}_{kg}}$$

(15)

$$N_2{O}_{ij}^{v, r}=d_{ij}^{miles}\cdot W_{i, short-ton}^{v, r}\cdot E{F}_{N_{2}O}\cdot GW{P}_{N_{2}O}\cdot g_{t{o}_{kg}}$$

(16)

$${{CO}_{2e}}_{ij}^{v, r}={\displaystyle \frac{{{CO}_2}_{ij}^{v, r}+{{CH}_4}_{ij}^{v, r}+N_2{O}_{ij}^{v, r}}{1000}}$$

(17)

$$y_{ij}^{v, r}=\left\{\mathrm{0,1}\right\}, \forall i, j \in M^{v, r}, \forall v\in V, \forall r\in R$$

(18)

The constraint of the model is described in

Table 3. Constraint of the model.

Constraint	Descriptions
Objective model	Objective model (1) aims to minimize total operational costs, which include travel costs, carbon emission costs, and penalty costs incurred due to early or late arrivals.
Time-based penalties	Constraint Equation (2): A penalty is applied if a vehicle arrives at a station before its designated time window opens.
	Constraint Equation (3): A penalty is applied for late arrivals after the time window closes.
	Constraint (4): A penalty is incurred if a vehicle returns to the depot later than the allowed time
Vehicle capacity	Constraint (5) ensures that vehicles respect their capacity limits, accounting for the simultaneous transport of full cylinders for delivery and empty cylinders for pickup.
Station visits	Constraint (6) ensures that each station (excluding the depot) is visited exactly once, satisfying all its delivery and pickup demands.
Time window compliance	Constraint (7) enforces that station vehicle arrival times are within the specified time windows.
Arrival time calculations	Constraint (8a): Calculates arrival time from the depot to the first station, accounting for travel time, unloading full cylinders, and loading empty cylinders.
	Constraint (8b): Calculates travel between two consecutive stations ($i$ and $j$), considering the completion time at station $i$, travel time, and operational times at station $j$.
	Constraint (8c): Calculates travel time from the final station of the trip back to the depot, including unloading collected empty cylinders.
Cumulative completion time	Constraint (9) ensures that the cumulative completion time of all trips is considered, including the final trip for each vehicle.
Vehicle weight management	Constraint (10): Defines the initial vehicle weight as the empty vehicle weight plus the load of full cylinders for delivery.
	Constraint (11): Updates the vehicle’s weight dynamically after each delivery or pickup at a station
Carbon Emissions Calculation based on GHG Protocol [52]	Constraints (12): Convert vehicle weight from kg to short tons.
	Constraints (13): Convert distances from km to miles.
	Constraints (14–16): Convert CO2, CH₄, and N2O emissions into CO2 equivalents using Global Warming Potential (GWP).
	Constraint (17): Aggregate emissions for all trips into total CO2e (tons) to assess the environmental impact of the model.

.

4. Genetic Algorithm

The proposed GVRP model, while incorporating additional variables and constraints, remains a derivative of the classical VRP model, sharing similar calculation principles and complexities. Since the VRP is an NP-hard problem [53], the GVRP is also NP-hard, necessitating heuristic or metaheuristic algorithms for efficient resolution. This study employs GA for its ability to handle complex optimization problems. The GA is particularly effective in exploring large solution spaces through genetic operations like selection, crossover, and mutation, maintaining population diversity and reducing the risk of premature convergence to suboptimal solutions. Unlike simulated annealing, which focuses on deep local searches, GA balances exploration and exploitation, enabling it to discover global optima more reliably.

In this approach, each individual GA represents a feasible vehicle route, with the key decision variable $y_{ij}^{v, r}$ indicating whether vehicle $v$ travels from station $i$ to $j$ during trip $r$. The GA minimizes the objective function through iterative refinement while ensuring all constraints are satisfied, as illustrated in Figure 2.

4.1. Population Initialization

In this step, an initial population of chromosomes ($P$, population size) is generated, where each chromosome represents a feasible vehicle allocation to routes ($y_{ij}^{v, r}$). The process begins by randomly generating a set of unique node sequences $(i, j)$ corresponding to the problem’s constraints, ensuring no duplicate nodes within each route. These nodes are then allocated to vehicles $\left(v\right)$ using optimization rules within the GA framework. Each vehicle’s route is further divided into multiple trips $\left(r\right)$ to comply with its capacity limits, as the vehicles in this problem are heterogeneous. The chromosome representation captures the allocation of vehicles to specific routes and trips, as illustrated in Figure 2. This representation allows the GA to search for optimal configurations effectively.

4.2. Evaluate Fitness

The fitness of each chromosome is evaluated using the objective function represented in Equation (1), which combines transportation costs, emissions-related costs, and penalties associated with constraint violations. For each chromosome, the algorithm calculates the total travel costs, estimates the emissions-related costs, and incorporates penalties for unfulfilled constraints or inefficient routing. The resulting fitness value reflects the effectiveness of the chromosome in meeting the objectives of the problem, guiding the GA toward more optimized solutions.

4.3. Selection

The next step involves selection, where parent chromosomes are chosen using a tournament selection method. This process randomly selects a subset of chromosomes (the tournament pool) and identifies the one with the highest fitness as a parent. This method ensures a balance between exploration and exploitation by giving fitter chromosomes a higher likelihood of contributing to the next generation while maintaining diversity within the population.

4.4. Crossover

The crossover operation aims to combine the genetic material of parent chromosomes to produce offspring. The representation of this crossover process is depicted in Figure 3.

In this study, a two-cut point crossover is used with a probability $P_c$. For each pair of parent chromosomes, two cut points are randomly selected within the allocation sequence $y_{ij}^{v, r}$. The segments between these cut points are swapped between the parents to create offspring. After crossover, offspring are checked for feasibility to ensure they meet the constraints of the problem, such as vehicle capacity and route continuity. The crossover allows the GA to explore new solutions by combining the strengths of the parent chromosome.

4.5. Mutation

Mutation introduces variability into the population by altering the genetic material of offspring. In this study, two mutation operators are used: creep and jump applied with a probability $P_m.$

• Creep mutation: slightly adjusts the vehicle allocation or sequence in the chromosome, representing small, localized changes.
• Jump mutation: makes more considerable changes, such as reallocating a vehicle to a completely different route or swapping nodes between trips.

Both types of mutation ensure the offspring remain feasible and respect all constraints. Mutation is critical for preventing premature convergence by introducing new genetic material. The process of mutation is illustrated in Figure 4.

4.6. Evaluate Fitness Across Generations

After crossover and mutation, the fitness of the offspring is evaluated using the same objective function $Z$ from Equation (1). The offspring are then combined with the parent population to form a new generation. The best individual chromosomes are selected based on fitness to form the next generation. This process is repeated for a predefined number of generations $\left(G_n\right)$, ensuring that the population evolves toward better solutions over time. At the end of $G_n$ generations, the chromosome with the highest fitness is selected as the optimal solution, representing the best allocation of vehicles to routes.

5. Computational Experiment

In this section, we evaluate the performance of the proposed GVRP model using a GA approach. The model was implemented in Microsoft Excel version 2021 and solved utilizing the GA add-in available through XL Optimizer version 1.2.4.0 Pro (https://xloptimizer.com/, accessed on 25 August 2024). To validate the effectiveness of this method, several numerical experiments were conducted on a PC equipped with an Intel^® Core™ i5-10500H CPU @ 2.5 GHz processor and 8 GB of RAM, running Windows 10 Professional. The experiments were based on a real case study of LPG distribution in Yogyakarta, and the results obtained from the GA approach were compared against the actual distribution routes to assess its practical applicability and performance.

5.1. Case Study

The case study focuses on the LPG distribution network in Yogyakarta, addressing complex logistical challenges such as heterogeneous vehicle availability, underutilized capacity, strict time windows, multi-trip scheduling, and simultaneous pickup and delivery requirements. The proposed GVRP model aims to minimize both transportation costs and environmental emissions by optimizing the allocation of vehicles to specific routes. To test the model, operational data were collected from the actual LPG distribution system, as provided at the following link: https://bit.ly/4cN3WfV, accessed on 10 December 2024. This dataset includes demand information $T_i$ from 89 locations (comprising 1 depot and 88 distribution stations), a distance matrix $d_{ij}$, real-world routes, and time window ($f_i, h_i)$ constraints.

Additional operational data used in the study are presented in

Table 4. Data on vehicles.

Vehicle Type, $\mathit{V}$	${\mathit{C}}^{\mathit{v}}$ (Units)	${\mathit{S}}^{\mathit{v}}$ (km/hr)	${\mathit{c}}_{\mathit{k}\mathit{m}}$ (IDR/km)
1	560	50	1410
2	560	50
3	360	55
4	250	65

and

Table 5. Information on distribution operations and emission-related data.

Parameter	Value	Parameter	Value
$B_{early}$	IDR 5000/hour	$E{F}_{{CO}_2}$	0.2970 kg CO2/short ton-mile
$B_{late}$	IDR 5000/hour	$E{F}_{{CH}_4}$	0.0035 g CH₄/short ton-mile
$B_{late}^0$	IDR 10000/hour	$E{F}_{N_{2}O}$	0.0027 g N2O/short ton-mile
${LD}_{full}$	39.29 s/10 gallon	$GW{P}_{{CO}_2}$	1
${UD}_{full}$	32.56 s/10 gallon	$GW{P}_{{CH}_4}$	28
${LD}_{empty}$	28.34 s/10 gallon	$GW{P}_{N_{2}O}$	265
${UD}_{empty}$	24.45 s/10 gallon	$k{m}_{t{o}_{miles}}$	0.6213
$W_{full}$	8 kg/gallon	$k{g}_{t{o}_{short-ton}}$	0.0011
$W_{empty}$	5 kg/gallon	$g_{t{o}_{kg}}$	0.001
		$CT$	IDR 30,000/ton CO2e

, detailing vehicle capacities, transportation operations, and emission-related parameters. Specifically, emission parameters were obtained from the GHG Protocol [52]. By incorporating these practical constraints and data, the model offers a comprehensive solution for improving route efficiency, reducing costs, and mitigating environmental impacts, representing a significant advancement over current distribution practices.

5.2. Model Evaluation

The GVRP model was evaluated through a series of scenarios designed to assess the robustness of the GA in solving problems of varying complexity. Each scenario was tested using carefully selected GA parameter combinations to ensure robustness and identify the optimal configuration for minimizing transportation, emission, and penalty costs. No direct comparison could be made with other studies, as no prior research currently integrates all four characteristics—time windows, heterogeneous vehicle fleets, multi-trips, and simultaneous pickup and delivery—into a single GVRP model. Additionally, while many studies focus on solving the VRP or GVRP using standard datasets like Solomon’s, this study utilized real-world data from LPG distribution to validate the practical applicability of the proposed model. This approach ensures that the findings not only contribute to theoretical advancements but also demonstrate their relevance to addressing real-world logistics challenges.

The first scenario involved testing the model on a large-scale problem with 88 stations and 1 depot, aligning with the real-world LPG distribution routes in Yogyakarta. This scenario was prioritized to ensure direct comparability with actual operational data. In the second scenario, the model was tested on medium-scale problems consisting of 40 and 60 stations with one depot, allowing for an evaluation of the GA’s efficiency under moderately complex conditions. Finally, the third scenario examined the model’s performance on small-scale problems with 20 stations and 1 depot, providing insights into its scalability and adaptability across problem sizes.

5.2.1. Parameter GA Tuning

For large-scale testing with 88 stations, three GA parameter combinations were selected to evaluate the algorithm’s performance by balancing exploration and exploitation while addressing the complexities of the GVRP model. The parameters (population size ($P$), crossover probability ($P_c.$), and mutation probability ($P_m.$)) were strategically varied to assess their impact on solution quality and efficiency. Each combination was tested over five independent runs to ensure robustness and provide a comprehensive evaluation of the GA’s effectiveness. Additionally, the number of generations ($G_n$ = 200) was kept constant across all runs, while the key GA parameters were varied as follows:

• Combination 1: $P$ = 100, $P_c.$ = 0.80, and $P_m.$ = 0.025.
• Combination 2: $P$ = 125, $P_c.$ = 0.65, and $P_m.$ = 0.005.
• Combination 3: $P$ = 150, $P_c.$ = 0.95, and $P_m.$ = 0.001.

The results of this initial large-scale evaluation, summarized in

Table 6. Performance evaluation of GA parameter combinations for large-scale GVRP testing.

Fitness Function in IDR (Transportation Costs + Emission Costs + Penalty Costs)
Experiment	I	II	III	IV	V	Average	SD
Combination 1	1,641,025.40	1,702,377.38	1,766,129.97	1,572,360.49	1,700,177.97	1,676,414.24	73,087.73
Combination 2	1,695,196.24	1,561,700.50	1,695,456.43	1,728,208.59	1,604,355.88	1,656,983.53	70,466.14
Combination 3	1,553,785.04	1,613,426.07	1,541,177.54	1,665,997.19	1,601,173.77	1,595,111.92	50,034.25

, provide valuable insights into the impact of parameter settings on the algorithm’s performance in solving real-world routing problems. The repeated trials under each parameter set helped identify consistent trends and ensure the reliability of the GA. These findings set the stage for further analysis of medium- and small-scale scenarios, highlighting the robustness of the proposed GVRP model and GA approach.

The performance of each GA parameter combination across five experiments is evaluated based on the average fitness value and standard deviation (SD), as shown in Table 6. Combination 3 achieves the lowest average fitness value (IDR 1,595,111.92), indicating superior performance in minimizing transportation, emission, and penalty costs. The SD for Combination 3 is significantly lower (50,034.25), demonstrating more excellent stability and consistency compared to Combination 1 (73,087.73) and Combination 2 (70,466.14). An ANOVA test was conducted to determine whether the differences between the combinations are statistically significant, as shown in

Table 7. Tukey’s simultaneous tests for differences of means.

Difference of Levels	Difference of Means	SE of Difference	95% CI	T-Value	Adjusted p-Value
Combination 2—Combination 1	−19,431	41,329	(−129,606; 90,744)	−0.47	0.886
Combination 3—Combination 1	−81,302	41,329	(−191,477; 28,873)	−1.97	0.163
Combination 3—Combination 2	−61,872	41,329	(−172,046; 48,303)	−1.50	0.327

Individual confidence level = 97.94%.

.

Based on Table 7, the statistical analysis of the GA parameter combinations reveals key insights into the algorithm’s performance and parameter effectiveness. Although none of the pairwise comparisons show statistical significance (all adjusted p-values > 0.05), the mean differences highlight notable trends. Combination 3 consistently outperformed Combination 1 and 2, with a mean difference of −81,302 and −61,872, respectively, in the fitness function. These trends indicate that Combination 3 tends to produce better results in minimizing costs, even though the differences are not statistically significant. Furthermore, while encompassing zero, the 95% confidence intervals suggest a practical advantage for Combination 3, as its average results remain consistently lower than those of the other combinations.

In addition to achieving the lowest average fitness value, Combination 3 exhibited a robust performance with minimal variation across multiple trials. Its lower SD compared to that of Combination 1 and 2 supports its stability and reliability in solving the GVRP model effectively. While statistical significance was not observed, the consistent trends and reduced variability clearly establish the practical superiority of Combination 3. This parameter configuration effectively balances exploration and exploitation, addressing the complexities of large-scale GVRP instances.

Based on comparative analysis and statistical evaluation, Combination 3 ($P$ = 150, $P_c.$ = 0.95, and $P_m.$ = 0.001) is identified as the most effective and reliable parameter set for solving the proposed GVRP model. Its consistent performance underscores its suitability for optimizing the model while minimizing costs and ensuring robustness across diverse scenarios.

The graph in Figure 5 shows the best result from the third replication of GA’s optimal parameter settings in Combination 3. By running for 200 generations over 27 min and 49 s, the algorithm found the optimal solution at generation 59, costing approximately IDR 1,541,177.54. These findings underscore the efficiency of GA in solving large-scale global optimization problems.

Running for 200 generations in 27 min and 49 s, the algorithm found the optimal solution by generation 59, costing approximately IDR 1,541,177.54. This result highlights GA’s efficiency in solving large-scale global optimization problems, quickly converging to high-quality solutions. The steep initial cost reduction reflects vigorous exploration, while the plateau after generation 59 demonstrates stability and effective exploitation. This performance underscores GA’s robustness in handling complex, real-world routing challenges.

5.2.2. Performance Comparison of the Optimized Route Model with Actual Data Route

The comparison between the optimized and actual routes, as shown in Figure 6, provides valuable insights into the interplay between cost efficiency and environmental impact. The optimized route, derived from the third replication of Combination 3, achieved a total cost of IDR 1,541,177.54, representing a significant reduction of 16.85% compared to the actual route cost of IDR 1,853,665.17. This substantial cost saving demonstrates the effectiveness of the proposed GVRP model in optimizing logistical operations under real-world conditions. The actual route cost was calculated by inputting operational records (https://bit.ly/4cN3WfV, accessed on 10 December 2024) into the GVRP model, ensuring a valid and realistic comparison.

However, the analysis of carbon emissions reveals a nuanced trade-off. The actual route from the real data resulted in slightly lower emissions, producing 0.95-ton CO₂e compared to the optimized route’s 0.96-ton CO₂e—a marginal difference of 1.08%. While the cost savings achieved by the optimized route highlight its economic efficiency, this slight rise in emissions underscores the inherent challenge of balancing financial and environmental objectives in real-world logistics. In this case, carbon emissions translate directly into financial implications via carbon tax, with the actual route incurring a tax cost of IDR 28,513.67, compared to IDR 28,822.54 for the optimized route. Although the cost difference is relatively minor (IDR 308.87), it illustrates how even minimal changes in emissions can influence overall expenses, particularly in contexts with stringent carbon tax policies.

The slight increase in CO₂ emissions generated by the optimized route must also be evaluated in the context of government-defined CO₂ emission thresholds. If the total emissions remain below the regulatory threshold, the increase is acceptable as the environmental impact is still within permissible limits. In cases where emissions exceed the threshold, the savings from reduced transportation costs could be reallocated to pay for additional carbon tax penalties, ensuring compliance with environmental regulations. This approach balances the trade-offs between economic savings and environmental responsibility.

In scenarios where transportation cost savings are insufficient to cover the additional carbon tax due to exceeded thresholds, the economic objective of minimizing transportation costs may need to be relaxed. This adjustment allows for the optimization model to prioritize reducing CO₂ emissions to meet the regulatory limits. By adopting such a flexible approach, the proposed GVRP model ensures compliance with emissions regulations while maintaining a balance between economic efficiency and environmental sustainability. This flexibility allows businesses to align their operations with evolving regulatory frameworks and sustainability goals, ensuring a holistic approach to green logistics.

The performance comparison between the actual and optimized routes, as presented in Figure 6, is significantly influenced by the vehicle allocation to routes determined through the GA. This allocation process is detailed in

Table 8. The optimized route searched by GA.

$\mathit{V}$	Trip $\mathit{r}$	Route Planning	${\mathit{d}}_{\mathit{i}\mathit{j}}$ $\left(\mathbf{k}\mathbf{m}\right)$	Capacity Usage (%)	Visualization
1	Trip 1 Trip 2 Trip 3	0$\to$71$\to$87$\to$8$\to$20$\to$47$\to$67$\to$18$\to$0 0$\to$60$\to$39$\to$28$\to$2$\to$30$\to$48$\to$62$\to$0 0$\to$50$\to$81$\to$1$\to$83$\to$84$\to$41$\to$0	82.70 74.40 53.10	92.86 83.93 96,43
2	Trip 1 Trip 2 Trip 3 Trip 4 Trip 5 Trip 6 Trip 7	0$\to$56$\to$64$\to$9$\to$65$\to$53$\to$85$\to$0 0$\to$73$\to$38$\to$3$\to$82$\to$54$\to$22$\to$0 0$\to$77$\to$43$\to$32$\to$42$\to$86$\to$75$\to$0 0$\to$7$\to$37$\to$34$\to$19$\to$68$\to$14$\to$0 0$\to$49$\to$58$\to$33$\to$24$\to$80$\to$0 0$\to$69$\to$45$\to$78$\to$4 88$\to$25$\to$79$\to$0 0$\to$29$\to$10$\to$66$\to$17$\to$57$\to$6$\to$0	103.90 53.00 90.80 68.80 42.90 99.70 49.90	91.07 87.50 96,43 75,00 85,71 98,21 76,79
3	Trip 1 Trip 2 Trip 3 Trip 4 Trip 5 Trip 6	0$\to$26$\to$70$\to$76$\to$0 0$\to$21$\to$36$\to$27$\to$15$\to$0 0$\to$44$\to$12$\to$23$\to$55$\to$0 0$\to$72$\to$74$\to$61$\to$16$\to$0 0$\to$51$\to$52$\to$11$\to$63$\to$0 0$\to$5$\to$35$\to$0	34.40 30.20 66.80 51.20 48.20 44.30	86,11 86,11 88,89 86,11 75,00 63,89
4	Trip 1 Trip 2	0$\to$31$\to$59$\to$46$\to$0 0$\to$40$\to$13$\to$0	58.70 12.50	92,00 88,00

, which outlines the specific trips assigned to each vehicle, the distance covered in each trip, and a visual representation of the optimized routes. These optimized routes were carefully designed to maximize vehicle utilization and streamline operations. In contrast, the actual route data, available through the provided link, reflect less efficient allocation and trip planning. Table 8 further analyzes the impact, summarizing metrics such as trip time, time window penalties, distances, delivery quantities, costs, and emissions per trip. The optimized allocation improves efficiency by reducing trips and maximizing vehicle utilization with minimal underutilized capacity. For instance, the optimized model consolidates trips and reduces the total trips, as shown in Table 8. Additionally, it minimizes penalties by adhering more closely to delivery time windows, enhancing service reliability.

Moreover, the optimized routes in

Table 9. Logistics operations based on optimized route results.

Vehicle	Trip	Time	Status	Penalty Costs (IDR)	Total Distance	Delivery Unit	Total Costs (IDR)	Emissions (Ton CO2e)
Vehicle 1	Trip—1	11:01	Start	0.00	82.70	520.00	116,607.00	0.08
	Trip—2	14:05	Continue	0.00	74.40	470.00	104,904.00	0.06
	Trip—3	16:55	Continue	0.00	53.10	540.00	74,871.00	0.05
	Total			0.00	210.20		296,382.00	0.20
Vehicle 2	Trip—1	11:25	Start	0.00	103.90	510.00	146,499.00	0.11
	Trip—2	14:09	Continue	0.00	53.00	490.00	74,730.00	0.05
	Trip—3	17:44	Continue	0.00	90.80	540.00	128,028.00	0.10
	Trip—4	10:50	Start	0.00	68.80	420.00	97,008.00	0.06
	Trip—5	13:16	Continue	0.00	42.90	480.00	60,489.00	0.04
	Trip—6	17:03	Continue	0.00	99.70	550.00	140,577.00	0.10
	Trip—7	10:31	Start	0.00	49.90	430.00	70,359.00	0.05
	Total			0.00	509.00		717,690.00	0.51
Vehicle 3	Trip—1	09:27	Start	10,000.00	34.40	310.00	48,504.00	0.03
	Trip—2	11:03	Continue	0.00	30.20	310.00	42,582.00	0.03
	Trip—3	13:20	Continue	0.00	66.80	320.00	94,188.00	0.05
	Trip—4	15:19	Continue	0.00	51.20	310.00	72,192.00	0.04
	Trip—5	17:07	Continue	0.00	48.20	270.00	67,962.00	0.04
	Trip—6	09:36	Start	0.00	44.30	230.00	62,463.00	0.03
	Total			10,000.00	275.10		387,891.00	0.21
Vehicle 4	Trip—1	09:30	Start	0.00	58.70	230.00	82,767.00	0.03
	Trip—2	10:26	Continue	0.00	12.50	220.00	17,625.00	0.01
	Total	10:26		0.00	71.20		100,392.00	0.04

highlight several key operational improvements and constraints. One significant observation is that vehicles do not always complete deliveries within a single day, particularly in the case of Vehicle 2 and Vehicle 3. These vehicles required two days to finalize their delivery tasks due to the time constraints imposed by the depot and station operational hours, which typically end at 17:00. This scheduling constraint emphasizes the practical limitation of time windows in real-world logistics operations.

The allocation of vehicles to routes in the optimized model minimizes the overall trip distances, resulting in efficient route planning. For instance, Vehicle 4, despite its smaller capacity, completed its deliveries in just two trips, covering a total distance of 71.2 km. This result contrasts with the actual route’s vehicle capacity usage inefficiency, frequently resulting in underutilized trips or excessive multi-tripping. A notable advantage of the optimized routes lies in the effective utilization of vehicle capacity, ensuring no vehicle operates beyond its designated limits. The fleet in this study comprises four vehicles with underscore capacities of 560, 560, 360, and 250 cylinders, respectively. As shown in Table 8, the capacity usage rates for each vehicle across all trips remain within allowable thresholds, demonstrating efficient load management. Furthermore, Table 9 highlights that the delivery weight for each trip is carefully allocated to fully utilize the available capacity without exceeding the limits.

This precise load distribution ensures that no vehicle is underutilized or overloaded, a typical inefficiency observed in the actual routes. By avoiding excess trips and optimizing vehicle allocations, the model minimizes unnecessary trips and reduces the overall delivery time. This operational efficiency translates into substantial cost savings, as evidenced by the significant reduction in the total logistics costs. The integration of load optimization into the GVRP model not only enhances the practicality of the solution but also ensures compliance with vehicle capacity constraints, reinforcing the reliability and scalability of the proposed approach for real-world logistics systems.

While the optimized routes demonstrate significant cost savings through effective vehicle capacity utilization, an environmental trade-off is apparent. Despite reducing penalty and delivery costs, the optimized routes result in a slight increase in emissions compared to the actual routes. This increase can be attributed to the higher number of trips performed by certain vehicles, such as Vehicle 2, which completed seven trips in total. Although the model minimizes overall delivery costs, the additional trips slightly elevate emissions, illustrating the inherent challenge of achieving a perfect balance between financial and environmental objectives.

5.2.3. Sensitivity Analysis on Different Scenarios

In practice, logistics and distribution companies often operate at varying scales, serving different numbers of stations. To assess how the station count impacts the performance of the GVRP model, we tested scenarios with 20 stations for small-scale problems and 40 and 60 stations for medium-scale problems. These tests were conducted using the optimal GA parameter setting (Combination 3): $P$ = 150, $P_c.$ = 0.95, and $P_m.$ = 0.001. Each scenario was run five times, and the best result for each station count (20, 40, and 60) was selected, as shown in

Table 10. Vehicle allocation to route planning results for 20 stations.

Vehicle	Trip	Route Planning	Total Distance (km)	Delivery Unit	Total Costs (IDR)	Emissions (Ton CO2e)	Emission Costs (IDR)
1	1	0$\to$56$\to$53$\to$85$\to$70$\to$76$\to$0	71.10	460	375,550.28	0.26	7681.28
2	1 2	0$\to$71$\to$64$\to$9$\to$31$\to$8$\to$20$\to$0 0$\to$73$\to$38$\to$3$\to$82$\to$21$\to$0	143.10	530 450
3	1	0$\to$26$\to$65$\to$87$\to$59$\to$0	46.70	320

,

Table 11. Vehicle allocation to route planning results for 40 stations.

Vehicle	Trip	Route Planning	Total Distance (km)	Delivery unit	Total Costs (IDR)	Emissions (Ton CO2e)	Emission Costs (IDR)
1	1 2	0$\to$71$\to$31$\to$8$\to$20$\to$47$\to$67$\to$18$\to$0 0$\to$77$\to$44$\to$75$\to$7$\to$37$\to$0	112.90	530 410	647,195.22	0.42	12,554.22
2	1 2	0$\to$64$\to$87$\to$85$\to$70$\to$73$\to$38$\to$0 0$\to$3$\to$82$\to$22$\to$43$\to$32$\to$12$\to$23$\to$0	169.70	550 510
3	1 2 3	0$\to$56$\to$9$\to$26$\to$65$\to$0 0$\to$53$\to$59$\to$76$\to$27$\to$0 0$\to$60$\to$42$\to$86$\to$0	132.00	350 350 290
4	1 2	0$\to$21$\to$54$\to$0 0$\to$36$\to$15$\to$0	35.50	230 90

and

Table 12. Vehicle allocation to route planning results for 60 stations.

Vehicle	Trip	Route Planning	Total Distance (km)	Delivery Unit	Total Costs (IDR)	Emissions (Ton CO2e)	Emission Costs (IDR)
1	1	0$\to$64$\to$9$\to$87$\to$8$\to$59$\to$76$\to$0 0$\to$77$\to$37$\to$34$\to$19$\to$51$\to$52$\to$80$\to$0	138.80	510 560	1,024,945.40	0.65	19,490.90
2	1 2 3 4 5	0$\to$56$\to$26$\to$65$\to$53$\to$85$\to$70$\to$0 0$\to$73$\to$38$\to$3$\to$82$\to$21$\to$54$\to$0 0$\to$36$\to$27$\to$67$\to$18$\to$43$\to$32$\to$60$\to$0 0$\to$42$\to$55$\to$68$\to$39$\to$28$\to$2$\to$58$\to$0 0$\to$33$\to$24$\to$30$\to$48$\to$0	338.75	540 560 530 560 300
3	1 2 3	0$\to$71$\to$31$\to$20$\to$47$\to$15$\to$0 0$\to$44$\to$12$\to$75$\to$23$\to$7$\to$0 0$\to$14$\to$72$\to$74$\to$61$\to$16$\to$0	167.60	350 350 360
4	1 2	0$\to$22$\to$86$\to$0 0$\to$49$\to$0	57.30	170 150

. This approach highlights the GA’s efficiency in finding near-optimal solutions across different operational scales.

The results in Table 10, Table 11 and Table 12 reveal an important insight regarding the relationship between the number of stations and vehicle utilization. As the number of stations increases, the complexity of the route allocation grows, necessitating a higher number of vehicles and trips to service all delivery points effectively. For the 20-station scenario, only three vehicles were needed to complete all trips, indicating efficient vehicle utilization due to fewer delivery points and a relatively compact service area. This efficient vehicle utilization results in lower total distances (260.9 km) and minimal costs (IDR 375,550.28), including emission costs.

In contrast, the 40-station scenario demonstrates an increase in both the number of trips and vehicles. Here, four vehicles are allocated to manage the additional stations, leading to a total distance of 450.10 km and a significant rise in total costs to IDR 647,195.22. Similarly, the 60-station problem requires even more extensive vehicle deployment to handle the dispersed stations efficiently. The five vehicles in this scenario accumulate a total distance of 702.45 km, with total costs rising to IDR 1,024,945.40. This trend reflects a direct correlation between the number of stations and the scale of logistical operations.

The smaller the number of stations, the fewer vehicles and trips are required, demonstrating the scalability of the GA in optimizing vehicle allocation as the problem complexity varies. This relationship demonstrates the principle of reducing the operational redundancy and maximizing the resource efficiency for more minor problem scales, as widely discussed in the logistics optimization literature [54,55]. Conversely, as the station count grows, more vehicles are needed to maintain service levels, highlighting the challenge of balancing cost efficiency and service quality in more extensive logistics networks. By optimizing vehicle capacity and route planning, the GA minimizes underutilized trips in smaller station scenarios. This finding is consistent with the results from multi-depot VRPs, where effective vehicle allocation is crucial to minimizing operational costs and emissions. This observation supports the notion that optimization techniques can dynamically adjust to varying scales of logistical challenges, ensuring sustainability and efficiency across diverse operational scopes.

6. Conclusions

This study presents a comprehensive analysis of the GVRP by integrating real-world complexities, including time windows, heterogeneous vehicle capacities, simultaneous pickup and delivery, multiple trips, and emissions-related costs. The proposed model, implemented with a GA, optimizes vehicle allocation and route planning to address logistical challenges in LPG distribution. Validation against actual routes highlights its practical relevance, with optimized routes achieving significant reductions in the total transportation costs, penalty costs for time window violations, and emissions-related costs from carbon taxes. These findings illustrate the model’s ability to enhance cost efficiency while promoting environmental sustainability.

The robustness and scalability of the model are evident across different problem scales. For smaller station networks, optimization maximizes vehicle utilization, reducing unnecessary trips and minimizing operational emissions. For more extensive networks, the system dynamically allocates additional vehicles to maintain service levels, balancing logistical effectiveness and cost efficiency. These results, validated against real-world data, reinforce the practicality of the proposed GVRP model and the versatility of GA in solving complex, multi-constraint logistics problems.

Beyond LPG distribution, the model’s adaptability enables its application in other industries, such as beverage logistics, particularly for mineral water gallon and glass-bottled beverage deliveries. These applications face similar challenges, including heterogeneous vehicle capacities and varying station demands. The model’s flexibility also allows it to adapt to changing economic and environmental parameters, such as fluctuating fuel prices and stricter carbon tax regulations, ensuring continued relevance in dynamic contexts. Future studies could explore further enhancements to the model, particularly its applicability in more diverse industries or under increasingly stringent sustainability requirements.

Future research could expand the model’s applicability by incorporating social dimensions, such as driver fatigue, equitable resource distribution, and customer satisfaction, to address sustainability further. Integrating real-time data, such as traffic conditions or dynamic demand changes, could enhance the model’s responsiveness and adaptability. Moreover, combining GA with other metaheuristic algorithms, such as PSO or ACO, could strengthen its capability to balance exploration and exploitation. These advancements would not only enhance the model’s performance but also broaden its applicability across diverse logistics scenarios, contributing to the future of sustainable logistics.