1. Introduction

With growing global awareness of sustainability and the urgent need to reduce carbon emissions, the logistics industry is under increasing pressure to achieve a balance between operational efficiency and environmental responsibility. Cold-chain logistics plays a crucial role in ensuring the quality and safety of temperature-sensitive products such as food and pharmaceuticals ¹. However, cold-chain systems are characterized by high energy consumption, mainly owing to continuous refrigeration and complex routing processes, which lead to considerable operational costs and significant carbon emissions. Consequently, improving the efficiency and environmental performance of cold-chain logistics has become an important topic in sustainable logistics management.

The traditional vehicle routing problem (VRP) primarily focuses on minimizing logistics costs, including fuel consumption, vehicle operation, and driver expenses ². In contrast, additional factors such as refrigeration costs, cargo damage due to temperature fluctuations, and holding costs at distribution centers must be considered in cold-chain logistics ³. Furthermore, carbon emission costs arise not only from fuel consumption during transportation but also from the energy used for refrigeration during transportation and storage. Therefore, a comprehensive optimization approach is required to minimize both the economic and environmental impacts of cold-chain logistics.

This study addresses a dual-objective optimization problem in low carbon cold-chain logistics that incorporates logistics cost minimization and carbon emission cost reduction into vehicle routing planning. The proposed model integrates multiple cost components, including fixed, refrigeration, cargo damage, and holding costs, as well as the environmental impacts of transportation and refrigerated storage emission costs. Balancing cost efficiency with environmental sustainability is a major challenge. Thus, it is essential to develop an effective optimization framework to capture this solution.

To solve this complex problem, two representative multi-objective optimization algorithms were employed: non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective particle swarm optimization (MOPSO). NSGA-II, which is based on evolutionary principles, is known for its ability to maintain population diversity and achieve fast convergence through elitist selection and crowding distance mechanisms ⁴. The MOPSO, inspired by swarm intelligence, has a strong global search capability and convergence speed through cooperative information sharing among particles ⁵. Both algorithms have been successfully applied to logistics and supply chain optimization. In this study, they are applied to the same mathematical model and experimental setting to examine their performance in terms of solution quality, diversity, and convergence.

The main objectives of this study are as follows: (1) develop a mathematical model to integrate economic and environmental considerations in cold-chain logistics; (2) formulate the total cost and emission functions to represent fixed, refrigeration, cargo damage, and holding costs as well as emissions from transportation and storage; (3) implement the NSGA-II and MOPSO algorithms to solve the proposed model; (4) evaluate and compare their performance using established multi-objective indicators; and (5) test the proposed framework on benchmark datasets to provide insights into the relationship between cost control and emission reduction.

This study makes several contributions to existing literature. First, it proposes an integrated dual-objective model for cold-chain logistics that considers both economic and environmental concerns. Second, a detailed cost and emission function that incorporates refrigeration, cargo deterioration, and storage emissions is formulated. Third, two widely used multi-objective optimization algorithms are applied to obtain Pareto-optimal solutions, offering practical decision-making insights. Finally, the study provides results to illustrate the balance between cost savings and carbon reduction in cold-chain logistics, which can guide logistics operators and policymakers in designing sustainable and cost-effective transportation strategies.

The remainder of this paper is organized as follows. Section 2 presents a comprehensive literature review of low carbon cold-chain logistics and multi-objective optimization methods. Section 3 presents the mathematical formulation of the problem, including the problem description, modelling assumptions, and objective functions. Section 4 details the optimization algorithm and explains the evolutionary operations used to generate the Pareto-optimal solutions. Section 5 presents numerical experiments to validate the proposed model and analyzes the balance between logistics and carbon emission costs. Finally, Section 6 concludes the paper with the key findings and future research directions.

2. Literature Review

With deepening global concern for sustainable development and carbon emission control, the logistics industry has been encouraged to adopt greener and more energy-efficient operations. Among the various branches of logistics, cold-chain logistics has attracted particular attention because of its high energy demand and complex temperature control requirements ⁶. It is estimated that refrigeration and transportation processes account for a large proportion of the total energy consumption in cold-chain systems, making this sector one of the key areas for carbon reduction efforts.

The VRP has long served as a fundamental framework for studying logistics distribution planning. Classical VRP models mainly focus on minimizing transportation costs or the total travel distance. However, in cold-chain logistics, the products are often perishable and sensitive to temperature, which introduces additional operational challenges ⁷. Huai et al. constructed a model to minimize the distribution and cargo damage costs for the multi-vehicle path problem in cold-chain logistics ⁸, whereas Deng et al. introduced the effect of temperature change on transported products to construct a distribution cost model to rationally plan the transport path to reduce the total distribution cost ⁹.

As environmental awareness grows, attention has gradually shifted from cost minimization to the integration of environmental objectives, particularly the reduction of greenhouse gas emissions ¹⁰. To address the dual pressures of total distribution and carbon emission costs, Liu et al. constructed a green vehicle path model combining joint distribution and carbon trading mechanism to achieve the dual goals of economic and environmental benefits ¹¹. Meanwhile, the drastic increase in cold-chain orders further aggravates the complexity of route optimization and the carbon emission burden. To address this challenge, Tao et al. constructed a cold-chain logistics route optimization model to minimize the comprehensive cost of improving distribution efficiency ¹².

Because the cold-chain logistics routing problem usually involves conflicting objectives such as minimizing costs and reducing carbon emissions, traditional single-objective methods cannot effectively capture the balance among these goals. Consequently, multi-objective optimization has become an essential approach for solving such problems ¹³. Ransikarbum et al. developed a multi-objective routing model for the biofuel supply chain by integrating spatial clustering and route optimization to improve overall efficiency ¹⁴. Recently, a multi-objective differential evolution algorithm integrating a directional generation mechanism has shown enhanced convergence and diversity in solving complex optimization problems ¹⁵. Among the existing algorithms, NSGA-II is widely adopted due to its strong ability to explore diverse Pareto-optimal solutions. Bai et al. applied an improved NSGA-II to balance carbon and operational costs in cold-chain logistics, offering insights into both enterprise decisions and policy design ¹⁶.

In parallel, MOPSO has gained attention owing to its rapid convergence and efficient search through swarm collaboration. Li et al. demonstrated the effectiveness of MOPSO in optimizing total costs and greenhouse gas emissions in logistics systems ¹⁷. Because NSGA-II is based on genetic evolution whereas MOPSO relies on swarm intelligence and dynamic learning, understanding their comparative strengths is essential for selecting suitable algorithms for practical logistics applications.

To summarize, research on cold-chain logistics routing has evolved from traditional single-objective models emphasizing logistics cost to multi-objective frameworks that also consider product deterioration, temperature control, energy consumption, and carbon emissions. With the growing emphasis on green logistics and low carbon policies, strategies such as joint distribution and carbon trading have been incorporated to enhance economic and environmental performance while maintaining product quality. Therefore, multi-objective optimization algorithms have been widely applied owing to their strong capability to explore solution spaces and construct Pareto fronts.

Despite these advancements, most existing studies have concentrated on cost and emission optimization during the transportation stage, with limited attention paid to the energy consumption and emissions arising from the holding process. To address these limitations, this study developed a comprehensive dual-objective cold-chain routing model and applied both NSGA-II and MOPSO for comparative analysis. This study evaluated the effectiveness of each algorithm in terms of convergence, distribution uniformity, and overall solution quality, thereby providing practical insights for selecting suitable optimization strategies for sustainable cold-chain logistics planning.

3. Mathematical Model

This section introduces a mathematical model for the low carbon cold-chain vehicle routing problem. The model aims to optimize two objectives: minimizing the total logistics cost and reducing the carbon emission cost while ensuring delivery via refrigerated transport. This section is divided into three parts: the problem description, model assumptions and parameter definitions, and mathematical formulation.

3.1. Problem Description

Cold-chain logistics plays an essential role in the transportation of perishable products, ensuring that products remain within the required temperature ranges to maintain their quality and reduce spoilage. However, the high operational costs associated with refrigeration and transportation, along with increasing concerns over carbon emissions, present significant challenges for optimizing cold-chain logistics ¹⁸. In this study, a dual-objective optimization model was developed to address the VRP in low carbon cold-chain logistics, with the aim of minimizing both the total logistics and carbon emission costs.

The problem considered in this study focuses on the optimization of distribution routes from a single warehouse to multiple distribution centers, ensuring the timely delivery of perishable products while minimizing economic cost and environmental impact. As shown in Fig. 1, each distribution center has a specific demand, and the available fleet consists of refrigerated vehicles with limited capacity. The total logistics cost includes multiple components, such as fixed costs, refrigeration expenses, cargo damage costs due to temperature deviations, and holding costs at the distribution centers. Simultaneously, the environmental impact was evaluated based on fuel consumption and emissions generated during transportation, as well as the energy consumption of refrigeration units at storage facilities.

Given the complexity of this dual-objective optimization problem, an effective and robust solution approach is required to achieve a balance between cost efficiency and sustainability in low carbon cold-chain logistics. The optimization model ensures that decisions regarding vehicle routing consider financial efficiency and the long-term goal of reducing the carbon emissions in logistics operations. By integrating these objectives, the proposed approach aims to support decision-makers in designing low carbon, cost-effective cold-chain logistics networks.

4. Solution Methodology

To solve the proposed dual-objective cold-chain logistics routing problem, two evolutionary multi-objective optimization algorithms were employed: NSGA-II and MOPSO. These algorithms are among the most widely applied approaches for handling complex optimization problems involving conflicting objectives. Both are capable of approximating Pareto-optimal fronts with high accuracy and preserving the population diversity during the search process. NSGA-II emphasizes elitism and crowding distance to maintain convergence and diversity, whereas MOPSO utilizes cooperative information sharing among particles to accelerate convergence toward an optimal balance between the cost and emission objectives ²⁴.

4.1. NSGA-II for Dual-Objection Problem

NSGA-II is an evolutionary algorithm based on the genetic operations of selection, crossover, and mutation. It uses a fast non-dominated sorting mechanism to classify solutions according to Pareto dominance and introduces elitism to retain high-quality solutions across generations ²⁵. A crowding distance metric was used to maintain diversity within the population and prevent premature convergence.

In this study, the NSGA-II encodes each solution as a feasible vehicle route and evaluates individuals using two objectives: total logistics cost and carbon emission cost ²⁶. Tournament selection is used to select parent solutions, followed by crossover and mutation to generate offspring. After each generation, the parent and offspring populations were combined and ranked by dominance, and the best individuals were selected to form the next generation. This process was repeated until the maximum number of generations was reached. The following presents the detailed pseudocode of the NSGA-II implementation for the proposed problem, which helps to clarify the algorithmic process (see Algorithm 1).

4.2. MOPSO for Dual-Objection Problem

MOPSO extends the classical particle swarm optimization to handle multiple objectives. Each particle represents a potential solution, and its position and velocity are iteratively updated based on personal experience and information from the best global solutions. The algorithm uses an external archive to store non-dominated solutions and applies a crowding distance to maintain diversity ²⁷.

In this study, MOPSO was adapted to the same problem representation as NSGA-II. The position of each particle corresponds to a routing plan, and its fitness is evaluated using the same dual objectives. During the iterations, the particles move toward the global and personal best solutions, gradually improving the balance between the cost and emission objectives. Mutations are occasionally introduced to enhance exploration and to avoid stagnation. The following pseudocode outlines the MOPSO algorithm used to solve the cold-chain vehicle routing problem, providing a clear description of its procedural steps (see Algorithm 2).

5. Experiments and Results

This section presents the numerical experiments conducted to evaluate the effectiveness of the proposed dual-objective optimization model and compare the performances of the NSGA-II and MOPSO algorithms. The experiments were performed using the well-known Solomon benchmark dataset ²⁸. The experimental scenario consisted of a single warehouse and multiple distribution centers, each with specific demand and service time constraints. All transported products require temperature-controlled conditions, and the objective is to determine the optimal delivery routes that minimize both logistics and carbon emission costs.

The model parameters used in the experiments are listed in Table 1. These parameters include the vehicle capacity, fuel consumption rate, refrigeration energy consumption, and other technical coefficients essential to the formulation of the cost and emission functions. All values were selected based on realistic assumptions made by industrial cold-chain logistics standards and validated through relevant literature to ensure the applicability and accuracy of the model.

For both algorithms, the population (or swarm) size was set to 500 and the maximum number of iterations was fixed at 100. The crossover and mutation rates for NSGA-II were set to 0.9 and 0.1, respectively. For MOPSO, the learning coefficients were \(c_1=c_2=2\), and the inertia weight \(w\) linearly decreased from 0.9 to 0.4. To evaluate and compare the two algorithms, three standard performance indicators were adopted: hypervolume (HV), spread (\(\Delta\)), and convergence (C).

Table 2 presents the comparative performances of the NSGA-II and MOPSO across the six benchmark instances for the dual-objective cold-chain vehicle routing problem. Both algorithms successfully generated high-quality Pareto fronts; however, their performance characteristics differed across the datasets. NSGA-II achieved lower total logistics costs (LC) and superior convergence values in most cases, indicating its stronger ability to approach the true Pareto-optimal region. In contrast, MOPSO produced slightly lower carbon emission (CE) values for several clustered and hybrid instances, suggesting better adaptability to problems with mixed or structured spatial patterns.

Regarding the multi-objective performance indicators, NSGA-II consistently achieved higher HV values across all datasets, demonstrating a broader Pareto front coverage and better overall optimization quality. The spread values of the MOPSO were smaller in certain cases, implying a more compact but less diverse solution distribution. In terms of convergence, NSGA-II outperforms MOPSO, exhibiting significantly lower values across all instances, confirming its strong convergence stability, robustness, and higher computational accuracy.

NSGA-II demonstrated a superior balance between convergence precision and solution diversity compared with MOPSO. Its consistent performance across all benchmark instances reflects a stronger ability to approximate the true Pareto-optimal set while maintaining population diversity through its elitism and crowding-distance mechanisms. The lower convergence values achieved by NSGA-II indicate that its search process is more stable and less prone to premature stagnation, thereby producing solutions that are both accurate and evenly distributed along the Pareto front. In contrast, MOPSO exhibits rapid convergence in the early stages of the optimization and a greater capacity for global exploration, enabling it to identify a broader range of extreme solutions. However, its solution distribution tends to be less uniform, suggesting weaker diversity preservation. The comparative results confirmed that NSGA-II is particularly effective when high precision and solution stability are required, whereas MOPSO is better suited for applications that demand faster exploration and wider coverage of the objective space in cold-chain routing optimization.

The RC1 instance was selected as a representative example to further illustrate the optimization behavior of both algorithms. The RC-type instances in the Solomon benchmark set feature a hybrid customer distribution that combines randomly scattered and clustered customer locations. This mixed spatial structure increases routing complexity and enhances the interaction between logistics costs and carbon emission objectives. Therefore, RC1 provides a comprehensive and challenging scenario that effectively reflects the balancing characteristics of cold-chain vehicle routing problems, making it suitable for analyzing the evolutionary behavior of both NSGA-II and MOPSO.

Figures 2(a) and (b) show the optimization process of NSGA-II. Fig. 2(a) shows the evolution of logistics and carbon emission cost over 100 generations. Both objectives exhibit a rapid decrease during the early stages, followed by a steady convergence trend as the algorithm progresses. This demonstrates that NSGA-II efficiently explores the search space and gradually refines the solution set through elitism and crowding-distance preservation. Fig. 2(b) shows the generation-wise evolution of the Pareto front. As the iterations proceed, the non-dominated solutions move toward the lower-left region of the objective space, indicating simultaneous reductions in cost and emissions. The final Pareto front is well distributed and continuous, showing that NSGA-II maintains a balance between convergence accuracy and population diversity.

Figures 3(a) and (b) show the optimization behavior of the MOPSO algorithm under identical experimental settings. As shown in Fig. 3(a), both logistics cost and carbon emission cost exhibit a rapid and substantial decrease within the first 30 iterations, demonstrating the strong global exploration ability and rapid convergence of the swarm-based optimization mechanism. Following this initial improvement phase, the decline in both objectives becomes gradual, indicating that the MOPSO tends to converge early and may experience stagnation in later iterations. Fig. 3(b) shows the evolutionary process of the non-dominated solutions generated by MOPSO. The results show that the algorithm can identify a diverse range of solutions during early iterations; however, the final Pareto front appears relatively uneven and sparsely distributed compared with that of NSGA-II, suggesting that its capability to maintain population diversity weakens as convergence progresses.

The visualization results for the RC1 instance provided valuable insights into the distinct search dynamics of the two algorithms. NSGA-II demonstrates a smoother convergence process and generates a more evenly distributed Pareto front, whereas MOPSO achieves faster early-stage optimization and stronger global search capability. These observations are consistent with the quantitative metrics listed in Table 2, confirming that NSGA-II exhibits convergence stability and diversity preservation, whereas MOPSO is advantageous for the rapid exploration and discovery of extreme solutions. The complementary characteristics of these algorithms indicate their combined potential for solving complex low carbon cold-chain routing problems.

6. Conclusion

This study developed a dual-objective optimization model for low carbon cold-chain logistics routing that simultaneously minimizes total logistics cost and carbon emission cost. The model comprehensively integrates transportation, refrigeration, cargo damage, and holding costs together with emissions arising from fuel consumption and refrigeration energy use. Two representative multi-objective evolutionary algorithms, NSGA-II and MOPSO, were employed and compared under identical experimental conditions to efficiently solve the model. Benchmark tests based on the Solomon dataset further verify the validity, robustness, and practical applicability of the proposed framework.

The experimental analysis demonstrates that both algorithms can effectively generate Pareto-optimal solutions; however, they differ in their optimization characteristics. NSGA-II achieved a higher convergence accuracy and produced more uniformly distributed Pareto fronts, indicating strong stability and reliable diversity preservation. MOPSO exhibited faster convergence in the early stages and stronger global exploration ability, discovering a broader range of solutions across the objective space. Comparisons based on the hypervolume, spread, and convergence indicators confirmed that NSGA-II performed better in terms of convergence precision and uniformity, whereas MOPSO was more efficient in early-stage exploration. Visualization using the RC1 instance further confirmed these patterns: NSGA-II converged smoothly and generated balanced Pareto fronts, whereas MOPSO reached near-optimal regions quickly but with a less uniform distribution.

From a practical perspective, the proposed model provides a quantitative decision tool for balancing economic efficiency and environmental sustainability in cold-chain logistics. These results suggest that NSGA-II is preferable when stable and well-distributed solutions are required, whereas MOPSO is advantageous when rapid convergence and broad exploration are required. Future work may focus on combining the strengths of both algorithms or introducing adaptive learning mechanisms to further improve the convergence and diversity. Extending the framework to large-scale and real-time cold-chain networks will also enhance its value for sustainable logistics planning.