Introduction

Driven by global carbon neutrality goals and sustainable development strategies, enterprise supply chains are under increasing pressure to shift from efficiency-oriented models to greener, low-carbon approaches. With climate change intensifying and resource constraints mounting, Green Supply Chain Management (GSCM) has become central to advancing corporate sustainability1,2. Unlike conventional supply chains, GSCM pursues a dual objective: maximizing resource utilization while minimizing environmental impact across procurement, production, logistics, and waste management3. Consequently, achieving high supply chain performance while simultaneously reducing costs and emissions has emerged as a pressing research challenge.

Intelligent optimization algorithms—such as Particle Swarm Optimization (PSO)4, Genetic Algorithm (GA)5, and Ant Colony Optimization (ACO)6—have been widely applied in GSCM. However, they still face significant limitations in complex, multi-objective settings. Common problems include premature convergence, reduced population diversity, and declining search efficiency. These issues become even more pronounced in real-world environments characterized by nonlinearity, multiple constraints, and high-dimensional objectives. In addition, randomness in population initialization and static parameter tuning often hinder solution quality and diversity, making it difficult to balance competing goals in green decision-making. The Sparrow Search Algorithm (SSA) offers clear advantages in this context. It demonstrates strong global search capability and fast convergence, making it well-suited to nonlinear, high-dimensional, multi-objective supply chain problems7. Its dynamic role-based strategy—driven by discoverers, joiners, and watchers—maintains a balance between exploration and exploitation, thereby enhancing accuracy and stability. Furthermore, its simple structure and flexible parameters allow easy integration into existing multi-objective frameworks, extending its applicability in GSCM.

Building on these strengths, this study introduces a multi-objective collaborative optimization model. The core innovations and contributions of this study can be summarized in three areas. First, at the framework level, the Improved Sparrow Search Algorithm (ISSA) is integrated with the Non-dominated Sorting Genetic Algorithm III (NSGA-III) to form a hybrid intelligent model designed for multi-objective trade-offs in GSCM. Second, at the algorithmic level, three key enhancements are introduced to the standard SSA: (1) Tent chaotic mapping replaces random initialization to improve diversity and coverage of the initial population; (2) an adaptive periodic convergence factor enables a dynamic balance between global exploration and local exploitation; and (3) the combination of Lévy flight and elite opposition-based learning (EOBL) enhances the ability to avoid local optima. Third, at the empirical level, the model is tested on a real-world supply chain dataset. Comparative experiments and rigorous statistical analyses demonstrate that the proposed approach achieves significant improvements over multiple benchmark algorithms in reducing costs and carbon emissions, increasing resource utilization, and improving solution set quality. These results provide efficient and reliable technical support for complex green supply chain decision-making.

Recent related work

Current research on GSCM optimization

GSCM has emerged as a vital research area within supply chain studies. Its primary objective is to balance economic performance, environmental protection, and resource conservation. Current academic efforts typically address GSCM challenges through mathematical programming, heuristic algorithms, and simulation-based approaches. Similarly, Zhao and Zhou (2025)8 developed a learning-driven memetic algorithm for integrated scheduling of distributed production and transportation, highlighting the growing trend of applying hybrid intelligent algorithms to complex logistics problems.

For instance, Shi et al.9 proposed a green optimization model for cold chain logistics based on intelligent scheduling. Their model reduced energy consumption in transporting perishable goods and demonstrated strong practical applicability. Boskabadi et al.10 designed a green distribution network model for multiple products and periods under demand uncertainty, showing the adaptability of mathematical programming to complex logistics systems. Banihashemi et al.11 applied the fuzzy Best-Worst Method (BWM) to identify barriers to GSCM in the construction industry. Their findings revealed that policy constraints, limited awareness, and cost were major obstacles, providing valuable guidance for both policymakers and enterprises pursuing green transitions.

Similar multi-objective trade-off challenges also arise in other domains, particularly in engineering project management, where advanced optimization techniques have been applied to balance conflicting objectives. For example, Eirgash and Toğan12 introduced a multi-objective Aquila optimizer with dual EOBL to balance time, cost, quality, and carbon emissions in construction projects. In related work, Eirgash and Toğan and Eirgash et al.13,14 developed dynamic and novel golden ratio-based opposition learning strategies, integrating them with teaching–learning optimization to effectively address the “time–cost–environmental impact” trade-off. These studies demonstrate the effectiveness of hybrid optimization methods in achieving sustainable project management goals and provide valuable insights for applying similar strategies in GSCM.

Current applications of the SSA and its variants

As an emerging swarm intelligence optimization algorithm, the SSA offers strong global search capabilities and adaptive characteristics. It has been applied in diverse fields, including function optimization, image recognition, mechanical parameter tuning, and energy system scheduling. Gao et al. 15 developed a multi-strategy enhanced version of SSA by introducing adaptive step sizes and hybrid update mechanisms, which significantly improved convergence speed and robustness and extended its applicability to complex multi-objective problems. Wu et al.16 combined an improved SSA with a stochastic configuration network to design a flame recognition system, enhancing both image feature extraction accuracy and computational efficiency, thereby demonstrating SSA’s engineering value in image recognition. Awadallah et al.17 conducted a systematic review of SSA variants across engineering optimization, image processing, and intelligent control. They emphasized SSA’s scalability for high-dimensional problems, while also noting its susceptibility to local optima. Babalik and Babadag18 proposed a binary SSA for feature selection in X-ray security imaging, which improved classification accuracy and reduced dimensionality, illustrating SSA’s adaptability in discrete feature spaces. Liu et al.19 applied SSA to scheduling in a renewable energy–hydrogen–ammonia synthesis system, achieving efficient operations under multiple constraints and highlighting SSA’s potential in coupled energy systems. Du et al.20 introduced a two-stage SSA framework for large-scale multi-objective problems, enhancing both exploration and exploitation to improve solution efficiency and convergence accuracy. The applicability of intelligent optimization algorithms has also extended into highly specialized engineering domains. For example, Karasahin21 applied SSA to the characterization and design optimization of mechanical systems in miniature swashplate-free flying robots. This underscores the versatility and strong potential of such computational tools for addressing complex challenges across different fields.

Overview of NSGA-III applications in Multi-Objective optimization

NSGA-III, the third generation of the NSGA family, was designed to address high-dimensional multi-objective optimization problems and has been widely applied. Tanhadoust et al.22 developed a two-stage optimization framework based on NSGA-III for reinforced concrete structures, balancing structural cost with seismic performance. Their study demonstrated the algorithm’s effectiveness in handling trade-offs in civil engineering. Gao et al.23 built a hybrid ANN–NSGA-III model to optimize the geometry of high-altitude diesel engine combustion chambers, achieving balanced improvements in fuel efficiency and emissions. Jia et al.24 coupled a response surface model with NSGA-III to optimize emissions performance in diesel engines under oxygen-enriched fuel and pilot injection strategies. Their results showed effective control of NOx and particulate emissions, reinforcing NSGA-III’s role in greener energy systems. Mao et al.25 proposed a U-NSGA-III-based calibration framework for the SWAT hydrological model, improving both parameter identification accuracy and simulation performance. Collectively, these studies highlight NSGA-III’s adaptability across engineering and environmental domains, where conflicting objectives must be balanced.

Research gaps and innovations of this study

Despite these advances, GSCM requires even higher performance from multi-objective optimization algorithms, and several gaps remain. First, single algorithms often become trapped in local optima when solving constrained multi-objective problems and lack effective mechanisms to escape. Second, population initialization and parameter tuning are usually static, limiting adaptability in dynamic environments. Third, few integrated optimization frameworks are designed specifically for the unique requirements of green supply chains. To address these gaps, this study proposes a collaborative optimization model that combines the ISSA with NSGA-III. The model is designed to capture trade-offs among cost, carbon emissions, and resource utilization in GSCM. It also demonstrates a novel application of multi-objective intelligent optimization methods to complex decision-making systems.

GSCM and coordinated optimization methods

Modeling and analysis of multi-objective GSCM

Traditional supply chain optimization has largely focused on minimizing costs while overlooking environmental impacts and resource efficiency. This narrow focus is insufficient for the goals of sustainable development. To overcome these limitations, the current study develops a multi-objective green supply chain optimization model (Fig. 1). The model integrates economic performance, environmental protection, and resource utilization, thereby promoting coordinated development within green supply chains.

Fig. 1
figure 1

Schematic diagram of the multi-objective green supply chain optimization model.

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This study focuses on a typical three-tier supply chain network comprising suppliers, manufacturers, and distribution centers. The model defines decision variables to represent the logistics relationships and resource allocation strategies between each node in the supply chain. Specifically, i, j, and k denote the indices for suppliers, manufacturers, and distribution centers, respectively. \({x_{ij}}\) represents the quantity of products supplied from supplier i to manufacturer j. \({y_{jk}}\) denotes the quantity of products delivered from manufacturer j to distribution center k. \({z_i}\) is a binary variable, where \({z_i}=1\) indicates that supplier i is selected; otherwise, Zi=0. \(C_{{ij}}^{{proc}}\) refers to the unit purchasing cost from supplier i. \(C_{j}^{{prod}}\) represents the unit production cost of manufacturer j. \(C_{{jk}}^{{trans}}\) denotes the unit transportation cost of products from manufacturer j to distribution center k. \({I_j}\) represents the inventory level of manufacturer j, while \(C_{j}^{{inv}}\) is the unit inventory holding cost for manufacturer j. \(E_{j}^{{prod}}\) indicates the carbon emission factor per unit of product produced by manufacturer j. \(E_{{jk}}^{{trans}}\) represents the unit carbon emission factor for transporting products from manufacturer j to distribution center k. \(C_{j}^{{\hbox{max} }}\) denotes the maximum production capacity of manufacturer j, and \(T_{{jk}}^{{\hbox{max} }}\) specifies the maximum transportation capacity from manufacturer j to distribution center k.

The model incorporates three interrelated objectives—economic, environmental, and efficiency optimization:

The total cost in this study includes procurement, production, transportation, and inventory costs. The total cost function \({f_1}(x)\) is defined as in Eq. (1):

$${f_1}(x)=\sum\limits_{{i=1}}^{{{n_s}}} {\sum\limits_{{j=1}}^{{{n_m}}} {C_{{ij}}^{{proc}} \cdot {x_{ij}}} } +\sum\limits_{{j=1}}^{{{n_m}}} {C_{j}^{{prod}} \cdot \left( {\sum\limits_{{i=1}}^{{{n_s}}} {{x_{ij}}} } \right)} +\sum\limits_{{j=1}}^{{{n_m}}} {\sum\limits_{{k=1}}^{{{n_d}}} {C_{{jk}}^{{trans}} \cdot {y_{jk}}} } +\sum\limits_{{j=1}}^{{{n_m}}} {C_{j}^{{inv}} \cdot {I_j}}$$
(1)

Carbon emissions mainly originate from transportation and manufacturing processes. The total emission function \({f_2}(x)\) is defined as in Eq. (2):

$${f_2}(x)=\sum\limits_{{i=1}}^{{{n_s}}} {\sum\limits_{{j=1}}^{{{n_m}}} {E_{{ij}}^{{trans}} \cdot {x_{ij}}} } +\sum\limits_{{j=1}}^{{{n_m}}} {E_{j}^{{prod}} \cdot \left( {\sum\limits_{{i=1}}^{{{n_s}}} {{x_{ij}}} } \right)} +\sum\limits_{{j=1}}^{{{n_m}}} {\sum\limits_{{k=1}}^{{{n_d}}} {E_{{jk}}^{{trans}} \cdot {y_{jk}}} }$$
(2)

Resource utilization measures the efficiency of production capacity and transportation. The objective is to maximize the efficient use of resources. The resource utilization function \({f_3}(x)\) is defined as in Eq. (3):

$${f_3}(x)=\sum\limits_{{j=1}}^{{{n_m}}} {\frac{{\sum\nolimits_{{i=1}}^{{{n_s}}} {{x_{ij}}} }}{{C_{j}^{{\hbox{max} }}}}} +\sum\limits_{{j=1}}^{{{n_m}}} {\sum\limits_{{k=1}}^{{{n_d}}} {\frac{{{y_{jk}}}}{{T_{{jk}}^{{\hbox{max} }}}}} }$$
(3)

To ensure the rationality and feasibility of the model, the following constraints are designed:

  1. (1)

    Demand Satisfaction Constraint: Ensures that the demand at each distribution center is fully met, as shown in Eq. (4):

    $$\sum\limits_{{j=1}}^{{{n_m}}} {{y_{jk}}} \leqslant {D_k},\quad \forall k=1, \ldots ,{n_d}$$
    (4)

    where \({D_k}\) is the demand of distribution center k (in units).

  2. (2)

    Supplier Supply Capacity Constraint: The supply volume from each supplier must not exceed its maximum capacity and is only allowed if the supplier is selected, as shown in Eq. (5):

    $$\sum\limits_{{j=1}}^{{{n_m}}} {{x_{ij}}} \leqslant S_{i}^{{\hbox{max} }} \cdot {z_i},\quad {z_i} \in \{ 0,1\} ,\forall i=1, \ldots ,{n_s}$$
    (5)

    where \(S_{i}^{{\hbox{max} }}\) is the maximum supply capacity of supplier i.

  3. (3)

    Manufacturer Production Capacity Constraint: Total production volume of each manufacturer must not exceed its capacity, as shown in Eq. (6).

    $$\sum\limits_{{i=1}}^{{{n_s}}} {{x_{ij}}} \leqslant C_{j}^{{\hbox{max} }},\forall j=1, \ldots ,{n_m}$$
    (6)
  4. (4)

    Transportation Capacity Constraint: The transportation volume must not exceed the maximum carrying capacity of the transportation channel, as shown in Eq. (7).

$${y_{jk}} \leqslant T_{{jk}}^{{\hbox{max} }},\;\forall j=1, \ldots ,{n_m},k=1, \ldots ,{n_d}$$
(7)

Design of the ISSA

The SSA is a recently developed swarm intelligence optimization method inspired by the cooperative foraging and vigilance behaviors of sparrows26. While promising, the traditional SSA has notable limitations, including premature convergence to local optima, insufficient diversity in population initialization, and inflexible parameter tuning. To address these challenges, this study introduces an ISSA that enhances adaptability and robustness in multi-objective green supply chain optimization. The core workflow of ISSA is illustrated in Fig. 2.

Fig. 2
figure 2

Main process of the ISSA algorithm.

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In this study, the core structure of the ISSA includes the following components:

Initial Population Generation: Traditional SSA initializes the population using a uniform random distribution, which can lead to uneven sample distribution and reduced search quality. To enhance population diversity and solution space coverage, this study introduces Tent chaotic mapping to generate the initial population27. The Tent map is defined as follows (Eq. 8):

$${x_{k+1}}=\left\{ {\begin{array}{*{20}{c}} {\frac{{{x_k}}}{\mu },}&{0 \leqslant {x_k}<\mu } \\ {\frac{{1 - {x_k}}}{{1 - \mu }},}&{\mu \leqslant {x_k} \leqslant 1} \end{array}} \right.$$
(8)

where µ ∈ (0,1) is the control parameter, commonly set to 0.7, and \({x_k}\) denotes the k-th value in the chaotic sequence. Through iterative mapping, the sequence achieves good traversal and uniform distribution within the [0,1] interval. Based on this, the initial position of an individual in the population is defined as (Eq. 9):

$$X_{i}^{{(0)}}={X_{{\text{min}}}}+{x_i} \cdot ({X_{{\text{max}}}} - {X_{{\text{min}}}}),i=1,2, \ldots ,N$$
(9)

where \({X_{{\text{min}}}}\) and \({X_{{\text{max}}}}\) are the lower and upper bounds of the decision variables, \({x_i}\) is the Tent map output, and N is the population size.

Adaptive Periodic Convergence Factor: In SSA, the convergence factor plays a crucial role in balancing exploration (searching new areas) and exploitation (refining known solutions). To improve adaptability, this study designs an adaptive periodic convergence factor \(r(t)\), which dynamically adjusts the focus throughout iterations. It is defined as (Eq. 10):

$$r(t)={r_{{\text{min}}}}+({r_{{\text{max}}}} - {r_{{\text{min}}}}) \cdot {\text{sin}}(\frac{{\pi t}}{T})$$
(10)

where t is the current iteration, T is the maximum number of iterations, and \({r_{{\text{min}}}}\) and \({r_{{\text{max}}}}\) are the minimum and maximum values of the convergence factor, respectively.

Lévy Flight and EOBL: To enhance the algorithm’s ability to escape local optima, ISSA incorporates two mechanisms: Lévy flight28 and EOBL29.

Lévy Flight: This technique introduces random step sizes based on heavy-tailed probability distributions, enabling jump-based exploration and enhancing global search ability. It is formulated as (Eq. 11):

$$X_{i}^{{t+1}}=X_{i}^{t}+\alpha \cdot Levy(\lambda )$$
(11)

where α is the step size control parameter, and \(Levy(\lambda )\) represents the Lévy-distributed random step, defined as (Eq. 12):

$$Levy\sim \frac{u}{{{{\left| v \right|}^{1/\lambda }}}},u\sim N(0,{\sigma ^2}),v\sim N(0,1)$$
(12)

The calculation of \(\sigma\) is shown in Eq. (13):

$$\sigma ={\left[ {\frac{{\Gamma (1+\lambda ) \cdot {\text{sin}}(\pi \lambda /2)}}{{\Gamma ((1+\lambda )/2) \cdot \lambda \cdot {2^{(\lambda - 1)/2}}}}} \right]^{1/\lambda }}$$
(13)

where λ ∈ (1,3], typically set to 1.5.

EOBL: To increase diversity and avoid premature convergence, EOBL generates opposite solutions for elite individuals.

The selection of EOBL over traditional EOBL rests on several key considerations. Conventional OBL generates opposite solutions for either the entire population or randomly selected individuals. Although this approach enhances diversity, it introduces two drawbacks. First, applying it to all individuals leads to high computational overhead. Second, producing opposite solutions for poorly performing individuals often results in exploration of inferior regions, lowering efficiency. EOBL addresses these issues by focusing only on elite individuals, i.e., the best-performing solutions in the population. It assumes that the opposite positions of these elite solutions may also contain promising regions. Concentrating the search around these candidates reduces computational cost, improves efficiency, and helps the algorithm escape local optima, thereby accelerating convergence toward the global Pareto front.

For a given elite solution \({X^ * }\), its opposite solution \(X_{{opp}}^{ * }\) is computed as (Eq. 14):

$$X_{{opp}}^{ * }={X_{{\text{min}}}}+{X_{{\text{max}}}} - {X^ * }$$
(14)

At each iteration, the algorithm compares the fitness of the elite individual and its opposite, retaining the better one. This mechanism improves both the diversity and global exploration capacity of the solution set.

To ensure the robustness of the improved strategies, this study analyzed key parameters:

  1. 1.

    Adaptive Periodic Convergence Factor r(t): This factor introduces periodic oscillations via a sine function, and its dynamic adjustment is critical for algorithm stability. In the early iterations, r(t) is relatively large, promoting extensive global exploration and effectively avoiding premature convergence to local optima. As iterations proceed, r(t) decreases periodically, guiding the algorithm to perform fine-grained local exploitation around the current best solutions. Compared with linear decay or fixed parameters, this periodic exploration–exploitation balance better adapts to complex optimization environments, enhancing the algorithm’s ability to escape local optima at different stages and improving both convergence stability and solution quality.

  2. 2.

    Lévy Flight Parameters (α and λ): The performance of Lévy flight mainly depends on the step size control parameter α\alphaα and the distribution exponent λ (typically set to 1.5). The parameter α\alphaα governs the magnitude of jumps, allowing the algorithm to perform both fine local searches and large-scale global transitions. The exponent λ determines the “tail thickness” of the Lévy distribution, affecting the probability of long-distance jumps. By introducing this heavy-tailed random walk, ISSA can more effectively escape local optima, significantly enhancing its global search capability. Although these parameters are somewhat sensitive, preliminary experiments and fine-tuning ensure they provide stable contributions throughout the optimization process.

Design and analysis of the ISSA-NSGA-III collaborative optimization algorithm

To enhance solution quality and convergence in multi-objective green supply chain optimization, this study integrates the improved ISSA into the NSGA-III framework30. The ISSA-NSGA-III model maintains Pareto optimality while significantly improving convergence speed and the ability to escape local optima. The collaborative optimization process consists of five main stages, as illustrated in Fig. 3.

Fig. 3
figure 3

Schematic of the collaborative optimization framework based on ISSA-NSGA-III.

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In this model, the first stage is population initialization, a process critical to the performance of multi-objective optimization. This study employs Tent chaotic mapping to generate the initial solutions, enhancing both the diversity and coverage of the solution space. This approach ensures a more uniform distribution of individuals across the search space, thereby improving the quality of the initial population and providing a strong baseline for subsequent non-dominated sorting.

The second stage involves fitness evaluation and non-dominated sorting. In each generation, the multi-objective fitness of all individuals is evaluated based on the three objective functions defined in Eq. (1) through (3): total cost \({f_1}(x)\), carbon emissions \({f_2}(x)\), and resource utilization \({f_3}(x)\).

NSGA-III then ranks the individuals using an extended non-dominated sorting mechanism, dividing them into multiple non-dominated fronts. The first front \({F_1}\) consists of all solutions not dominated by any others. The second front \({F_2}\) contains solutions dominated only by those in \({F_1}\), and so forth.

The dominance relationship between two solutions is determined as follows: solution a is said to dominate solution b if, for all objectives \({f_i}\), \({f_i}(a) \leqslant {f_i}(b)\), and there exists at least one objective \({f_j}(a)<{f_j}(b)\).

The third stage implements a reference-point-based diversity preservation mechanism. To address NSGA-II’s limitations—such as poor distribution and clustering effects in high-dimensional (more than three objectives) scenarios—NSGA-III introduces a reference point mechanism based on hyperplane projection. This enables a more uniform spread of solutions across the objective space.

A set of reference points \(R=\{ {r_1},{r_2},...,{r_K}\}\) is generated on a unit hyperplane, and each solution’s objective vector is projected onto its nearest reference point. If too many solutions are associated with one reference point, those closest to it are retained; if no solution maps to a reference point, the nearest one is selected to fill the gap.

In the fourth stage, the ISSA search mechanism is embedded to enhance local escape capabilities. To compensate for NSGA-III’s relatively weak local search ability and susceptibility to converging to suboptimal Pareto fronts, this model incorporates ISSA’s local enhancement strategies. During the position update of the discoverer individuals, Lévy-distributed long-tailed random perturbations are introduced to encourage larger jumps and improve the ability to escape local optima. In addition, an elite opposition-based solution \({X_{elite}}\) is generated for some high-performing individuals, defined as follows (Eq. 15):

$${X_{opp}}={X_{{\text{max}}}}+{X_{{\text{min}}}} - {X_{elite}}$$
(15)

The better solution between the original and its opposite is retained, which helps the algorithm escape local optima and expand the search boundaries.

Furthermore, a cyclic convergence control function \(c(t)\) is introduced, as shown in Eq. (16):

$$c(t)=co{s^2}(\frac{{\pi t}}{{2T}})$$
(16)

where t is the current iteration number, and T is the maximum number of iterations. In early stages, a high \(c(t)\) promotes global exploration; in later stages, a reduced \(c(t)\) shifts focus toward local exploitation.

The fifth and final stage involves updating the population and checking the termination criteria. At the end of each generation, NSGA-III’s elitism strategy selects the highest-ranked and well-distributed individuals to form the parent population for the next generation. The algorithm terminates when either the maximum number of iterations T is reached or the Pareto front shows no significant improvement over several consecutive generations.

Upon termination, the algorithm outputs a set of non-dominated solutions that approximate the Pareto front. This solution set provides multiple coordinated optimization strategies for cost, carbon emissions, and resource efficiency, enabling decision-makers to select the most suitable scheduling scheme according to their specific priorities.

The pseudocode of the ISSA-NSGA-III-based coordinated green supply chain optimization model is presented in Fig. 4.

Fig. 4
figure 4

Pseudocode workflow of the ISSA-NSGA-III-based green supply chain coordination optimization algorithm.

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To evaluate the computational efficiency of the proposed algorithm, this section analyzes the time complexity of the ISSA-NSGA-III model. Let N denote the population size, M the number of objective functions, T the maximum number of iterations, and D the dimensionality of the decision variables.

  1. 1.

    Time Complexity of the ISSA: During initialization, Tent chaotic mapping generates an initial population of size N and dimension D, which has a time complexity of O(N·D). In each iteration, evaluating the multi-objective fitness for all individuals requires O(N·M) time. Updating the positions of discoverers, followers, and watchers depends primarily on the population size and decision variable dimension, with complexity O(N·D). Additionally, the Lévy flight and EOBL strategies, introduced to enhance global search, operate only on a subset of individuals and thus have a lower complexity than O(N·D). Overall, the ISSA search mechanism contributes approximately O(N·D) per iteration.

  2. 2.

    Time Complexity of NSGA-III: The computational cost of NSGA-III arises mainly from two steps. First, the non-dominated sorting procedure, which ranks the population layer by layer to assign Pareto levels, has a complexity of O(M·N2) and represents the most computationally intensive part. Second, the reference-point-based selection mechanism, used to maintain diversity within the same rank, involves associating individuals with predefined reference points and performing niche preservation, with a complexity of O(M·N).

  3. 3.

    Overall Complexity of ISSA-NSGA-III: Since the ISSA search mechanism is embedded in each iteration of NSGA-III, the total time complexity per iteration is the sum of the constituent costs: O(M·N2) + O(N·D). In typical multi-objective optimization scenarios, the population size N is much larger than the number of objectives M and decision variable dimension D, making the non-dominated sorting O(M·N2) the dominant term. Considering T iterations, the overall computational complexity is approximately O(T·M·N2). This analysis shows that the model’s computational cost is mainly determined by the maximum number of iterations, the number of objectives, and the population size. These insights provide valuable guidance for evaluating its feasibility in practical applications of varying scales.

Experimental evaluation

The dataset used in this study was obtained from the publicly available Supply Chain Management dataset on Kaggle (2020) (https://www.kaggle.com/datasets/lastman0800/supply-chain-management). It contains comprehensive records from various companies, detailing their supply chain management (SCM) practices, performance metrics, and green logistics initiatives. The dataset includes key information on agile SCM, lean manufacturing, cross-docking, and quantitative indicators such as supplier count, inventory turnover, lead time (days), order fulfillment rate (%), and customer satisfaction rate (%). To tailor the dataset for green multi-objective optimization modeling, 20 critical decision variables related to procurement, production, transportation, inventory, and carbon emissions were selected. Fields with over 30% missing data were excluded. To enhance model performance, all numerical variables were normalized to a [0,1] range. Missing values were imputed using the median or mean, depending on the variable’s distribution. Categorical variables such as “logistics level” and “transportation mode” were transformed into numerical format via one-hot encoding. Records with logical inconsistencies (e.g., supply quantities exceeding production capacity or negative carbon emissions) were removed, and class imbalance was addressed through resampling. The final preprocessed dataset was split into training (80%) and testing (20%) subsets.

All experiments were conducted on a high-performance workstation running Ubuntu 22.04, equipped with an Intel Core i9-13900 H CPU, 32 GB RAM, and an NVIDIA RTX 4080 GPU. The algorithms were implemented in Python 3.10, using libraries such as NumPy, Pandas, Scikit-learn, Matplotlib, and the Platypus framework. Detailed experimental hyperparameter settings are shown in Table 1. All parameters were fine-tuned through preliminary experiments and kept consistent across all comparative tests.

Table 1 Hyperparameter settings.
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To comprehensively evaluate the performance of the proposed ISSA-NSGA-III model in green supply chain optimization, this study conducts a comparative analysis. The model is tested against several benchmark algorithms: NSGA-III31, NSGA-II32, MOPSO33, the standard SSA, and the two-stage improved Sparrow Search Algorithm (TS-SSA) proposed by Du et al. (2025). Five key evaluation metrics are used for analysis: Total Cost evaluates the model’s economic efficiency across procurement, production, transportation, and inventory stages. Total Carbon Emissions measure the environmental impact generated mainly from logistics and manufacturing processes. Resource Utilization Rate assesses how efficiently production and transportation capacities are used, reflecting the sustainability of the optimization results. Spacing Metric indicates the uniformity of the Pareto front distribution. Lower values suggest more balanced and diverse solutions, reducing the risk of clustering or sparsity that could impair solution representativeness and generalizability. Hypervolume (HV) represents the volume of the objective space covered by the Pareto front, relative to a reference point. A higher HV value signifies better overall solution quality and broader coverage of the objective space. Together, these metrics offer a well-rounded evaluation framework, capturing the economic, environmental, and algorithmic performance dimensions of green supply chain optimization models.

Results and discussion

Analysis of green supply chain optimization results across algorithms

The optimization results obtained using the proposed ISSA-NSGA-III model are compared against those from the other algorithms. Key outcomes for total cost, total carbon emissions, and resource utilization rate are illustrated in Figs. 5, 6 and 7.

Fig. 5
figure 5

Total cost results across different algorithms.

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Fig. 6
figure 6

Total carbon emissions results across different algorithms.

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Fig. 7
figure 7

Resource utilization results across different algorithms.

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Figures 5, 6 and 7 show the comparative results for total cost, carbon emissions, and resource utilization across different algorithms. The proposed ISSA-NSGA-III consistently outperforms all benchmarks in these three key metrics. For total cost (Fig. 5), all algorithms start at similar values at iteration 0. As iterations proceed, ISSA-NSGA-III exhibits a rapid cost decline, reaching 765,300 CNY by iteration 300—significantly lower than the final costs of other models. This demonstrates its clear advantage in economic optimization. The improved strategies contribute substantially to this performance: Tent chaotic mapping generates a high-quality, well-distributed initial population, enabling early exploration of promising solutions. Lévy flight enhances global search, allowing rapid traversal of unproductive regions and quick identification of low-cost solutions. In contrast, standard SSA and MOPSO curves flatten after roughly 150 iterations, suggesting entrapment in local optima. ISSA-NSGA-III, with its adaptive convergence factor and EOBL, continues fine-grained local exploitation in later stages, escaping local traps and converging toward a more globally optimal solution.

For total carbon emissions (Fig. 6), ISSA-NSGA-III reduces emissions from 232.6 tons to 171.17 tons, consistently outperforming all other algorithms throughout the process. This highlights its effectiveness in promoting environmental sustainability. In terms of resource utilization (Fig. 7), ISSA-NSGA-III reaches 87.93% after 300 iterations, substantially higher than NSGA-II (78.9%) and MOPSO (78.1%). This demonstrates its superior efficiency in managing production and transportation resources. Overall, ISSA-NSGA-III achieves balanced optimization across economic and environmental objectives while improving resource allocation. Compared with the standard SSA, it reduces total cost by approximately 14.02%, lowers carbon emissions by 14.16%, and increases resource utilization by 15.4%, clearly demonstrating its superiority over existing algorithms. The carbon emissions and resource utilization trends reinforce the cost convergence results, highlighting the model’s robustness in multi-objective optimization. ISSA-NSGA-III consistently converges toward optimal solutions across all three conflicting objectives. Guided by the NSGA-III framework, it effectively balances economic performance, environmental impact, and resource efficiency, steadily progressing toward a comprehensive optimal equilibrium.

Analysis of solution set quality across algorithms

To further assess solution quality, this study evaluates each algorithm using the Spacing Metric and HV, as shown in Figs. 8 and 9, respectively.

Fig. 8
figure 8

Spacing Metric results for green supply chain optimization across algorithms.

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Fig. 9
figure 9

HV results for green supply chain optimization across algorithms.

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Figures 8 and 9 illustrate the solution set quality of the ISSA-NSGA-III algorithm. For the Spacing Metric (Fig. 8), ISSA-NSGA-III achieves a value of 0.081, significantly outperforming NSGA-II (0.167), NSGA-III (0.174), and MOPSO (0.190). The Spacing Metric measures the uniformity of solutions along the Pareto front. A lower value indicates that solutions are evenly distributed without clustering, providing decision-makers with a diverse set of trade-off options. ISSA-NSGA-III attains the lowest spacing value, reflecting the even distribution of its solutions across the objective space. This advantage stems from the NSGA-III framework’s reference-point-based diversity preservation and ISSA’s powerful search capabilities, which generate high-quality, diverse candidate solutions.

Regarding HV (Fig. 9), ISSA-NSGA-III reaches 0.857, substantially higher than TS-SSA (0.749) and NSGA-II (0.697). This indicates broader coverage of the objective space and confirms the high quality of its Pareto solutions. Compared with the baseline SSA, ISSA-NSGA-III improves the Spacing Metric by 62.15% and increases HV by 28.5%, demonstrating consistent gains in both diversity and convergence. The highest HV value indicates that the algorithm’s Pareto front is not only closer to the theoretical optimum but also spans the largest effective decision space. Overall, the combination of ISSA’s convergence capability and NSGA-III’s diversity management enables ISSA-NSGA-III to generate a superior Pareto-optimal solution set, making it the most effective algorithm in this multi-objective optimization task.

Statistical significance test

To statistically verify that ISSA-NSGA-III’s superior performance is not due to chance, a non-parametric Friedman test was conducted. This test does not assume normality and is suitable for comparing multiple algorithms on the same problem. Each algorithm was independently run 30 times, and the final values of five key performance indicators—total cost, total carbon emissions, resource utilization, Spacing Metric, and HV—after 300 iterations were used as sample data.

The null hypothesis (H₀) assumed no significant differences among algorithms for each metric, with a significance level of α = 0.05. The results are presented in Table 2.

Table 2 Friedman test average rankings (α = 0.05).
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All p-values are well below 0.001, strongly rejecting the null hypothesis and confirming that the observed performance differences among the algorithms are statistically significant. A closer look at the average rankings further highlights the comprehensive advantage of ISSA-NSGA-III. The model achieves the top rank for each individual metric and attains an overall average rank of 1.13, substantially outperforming the second-best TS-SSA (2.30) and all other benchmark algorithms. This clearly demonstrates its leading position. In summary, the statistical analysis confirms that ISSA-NSGA-III’s superior performance is both robust and significant. These results provide strong evidence of the model’s reliability and effectiveness in addressing complex green supply chain optimization problems.

Conclusion

This study proposes a novel multi-objective evolutionary model, ISSA-NSGA-III, which embeds an ISSA within the NSGA-III framework to address the complexities of green supply chain optimization. Empirical evaluation using a real-world supply chain dataset shows that the model significantly outperforms benchmark algorithms—including SSA, NSGA-II, and MOPSO—across key metrics such as total cost, carbon emissions, resource utilization, and Pareto front quality. ISSA-NSGA-III effectively balances economic efficiency with environmental sustainability, demonstrating robustness and strong capability in solving complex multi-objective optimization problems. However, the study has certain limitations. It does not fully account for dynamic parameter changes under uncertain conditions, and its validation is limited to a single supply network scale. Future research could explore the integration of real-time, data-driven mechanisms, extend the model to multi-tier supply chain structures, and incorporate reinforcement learning techniques to enhance adaptability and intelligent decision-making in complex green supply chain environments.