Green Supply Chain Optimization Based on Two-Stage Heuristic Algorithm

Abstract

Green supply chain management is critical for driving sustainable development and addressing escalating environmental challenges faced by companies. However, due to the multidimensionality of cost–benefit analysis and the intricacies of supply chain operations, strategic decision-making regarding green supply chains is inherently complex. This paper proposes a green supply chain optimization framework based on a two-stage heuristic algorithm. First, anchored in the interests of intermediary core enterprises, this work integrates upstream procurement and transportation of products with downstream logistics and distribution. In this aspect, a three-tier green complex supply chain model incorporating economic and environmental factors is developed to consider carbon emissions, product non-conformance rates, delay rates, and transportation costs. The overarching goal is to comprehensively optimize the trade-off between supply chain costs and carbon emissions. Subsequently, a two-stage heuristic algorithm is devised to solve the model by combining the cuckoo search algorithm with the brainstorming optimization algorithm. Specifically, an adaptive crossover–mutation operator is introduced to enhance the search performance of the brainstorming optimization algorithm, which caters to both global and local search perspectives. Experimental results and comparison studies demonstrate that the proposed method performs well within the modeling and optimization of the green supply chain. The proposed method facilitates the efficient determination of ordering strategies and transportation plans within tight deadlines, thereby offering valuable support to decision-makers in central enterprises for supply chain management, ultimately maximizing their benefits.

1. Introduction

Greenhouse gas emissions have been increasing at an alarming rate, significantly contributing to global warming and giving rise to unprecedented climate system changes worldwide, which manifests in various regions as frequent extreme weather events and rising sea levels, among other consequences. Supply chain operations are one of the major contributors to greenhouse gas emissions [1]. For retail enterprises, carbon emissions from activities such as manufacturing, packaging, transportation, and other procurement processes account for approximately 80–90% of the total carbon footprint [1]. Furthermore, according to statistics from the International Energy Agency (IEA), global carbon emissions increased by 65% from 1990 to 2021 [2]. In the transportation sector in China, for instance, carbon emissions surged from approximately 94 million tons to around 960 million tons, marking a nine-fold increase. Hence, the optimization of carbon emissions in supply chain operations and transportation is critical for environmental preservation.

The concept of a green supply chain (GSC) has been proposed to adapt to sustainable development and to meet the growing environmental challenges by integrating environmentally sustainable practices into traditional supply chains [3]. In this respect, the collaboration between upstream and downstream can meet the growing environmental awareness in the supply chain, reduce production costs, stimulate economic growth, and alleviate ecological concerns. Furthermore, a growing number of stakeholders are demanding that companies take responsibility for the environmental damage caused by their supply chains. It has become necessary for companies to expand their environmental commitments and incorporate environmental requirements throughout supplier selection activities, product manufacturing, and final customer delivery processes [4]. Effective green supply chain management (GSCM) can enhance the competitiveness of a company [5].

Numerous efforts have been made to incorporate carbon emissions into various stages of supply chain operations. Haeri et al. [6], considering environmental factors, proposed a comprehensive gray-based green supplier selection model. Luo et al. [7], with the objective of reducing carbon emissions, studied time-varying green vehicle transportation problems accounting for traffic congestion. However, most of the existing studies predominantly focus on individual aspects of green procurement or green logistics, with limited efforts to holistically investigate the entire green supply chain. The integrated carbon emissions associated with procurement, transportation, inventory, and overall logistics have not been taken into account. The efficiency of the supply chain operations and transportation logistics for intermediary enterprises can be significantly enhanced by integrating the procurement of the upstream suppliers with the downstream logistics and distribution. Such integration has the potential to substantially reduce the costs and carbon emissions of intermediary enterprises. Furthermore, the joint optimization and coordinated scheduling of the entire green supply chain, integrating procurement, transportation, and distribution processes, can significantly reduce unnecessary resource wastage, enhance system efficiency, and boost production yields. Simultaneously, this approach helps in mitigating carbon emissions, reducing transportation costs, and minimizing inventory holding rates.

Optimization methods for addressing a GSC can be primarily categorized into mathematical programming techniques and heuristic algorithms. Zhen et al. [8] introduced a dual-objective green supply chain network considering the trade-off between carbon emissions and operational costs. They employed a Lagrangian relaxation method for network optimization. Kaur et al. [9] proposed a fuzzy-MCDM-based approach for minimizing total costs and environmental impacts in the context of elastic and sustainable integrated production and procurement models. Mitrai et al. [10] proposed a two-stage stochastic programming method to determine the optimal investment for supply chain networks and conducted a study based on actual cases. Due to the complexity of supply chain structures and relationships, conventional mathematical optimization approaches often contend with intricate model structures and large sets of decision variables, resulting in exponential growth in solution time as problem size increases. Therefore, adaptive and robust heuristic algorithms with strong search capabilities have emerged as potent tools for GSC problem-solving. Homayouni et al. [11] proposed a novel robust heuristic optimization method for a dual-objective green supply chain model. The method can effectively handle uncertainties in demand and economics, thereby reducing pollution in the supply chain. Chalmardi et al. [12] investigated sustainable supply chain network design, constructing a novel two-tiered optimization model and employing a simulated annealing algorithm to significantly reduce the environmental impact of the supply chain network. However, existing methods do not take into account the inadequate solving of models and operator capabilities at different stages, leading to a lack of searching ability and solution accuracy.

To this end, this paper proposes a multi-objective GSC optimization framework that integrates economic and environmental factors. In this framework, a three-tier supply chain network model that encompasses procurement, transportation, inventory, and distribution is established. In the procurement phase, the model is used to identify the optimal suppliers and order quantities from all potential suppliers by taking into account factors such as supplier costs, distances, product reliability, and carbon emissions. In the distribution phase, the model is utilized to optimize routing decisions to reduce costs and carbon emissions during the distribution process. Compared with existing approaches, the contributions of the proposed research are highlighted as follows:

(1)

A novel multi-objective GSC model is constructed by incorporating economic and environmental factors. Both procurement and distribution are considered within this comprehensive framework, ensuring the integration of environmental factors throughout the entire supply chain.

(2)

A two-stage heuristic algorithm is devised to solve the GSC model by combining the cuckoo search algorithm with the brainstorming algorithm, which can generate efficient transportation, inventory, and vehicle routing decisions. Additionally, an adaptive crossover–mutation operator is introduced to enhance the search performance of the brainstorming optimization algorithm.

(3)

Sensitivity analysis of the algorithm parameters was conducted to determine the optimal iteration settings. The algorithm model was then validated using a supply chain case study, resulting in supply chain procurement–distribution decisions, which can provide decision support for intermediate enterprises in procurement, transportation, and logistics design.

The rest of the paper is organized as follows: In Section 2, some significant literature is reviewed related to the study. Section 3 expounds on the proposed mathematical model for green supply chains. Section 4 introduces the theoretical foundation of the two-stage heuristic algorithm employed. Furthermore, in Section 5, a practical case study is presented to analyze and demonstrate the proposed model and algorithm, with the goal of minimizing the total cost and carbon emissions of intermediate enterprises while maximizing their benefits. Finally, Section 6 summarizes the research work and offers prospects for future studies.

2. Literature Review

A supply chain is a complex system that connects the supply and demand sides. It encompasses the entire process from production to consumption of raw materials, components, intermediate products, and final products. It consists of various activities such as planning, procurement, production, logistics, sales, and after-sales. Throughout the movement of materials and products in these activities, pollution control and environmental protection often face regulatory and market pressures. Therefore, GSCM has gradually emerged as an ecological modernization tool for core companies [13], balancing productivity and environmental resources. Its aim is to protect resources, reduce pollution, and achieve sustainable environmental development [14]. In addition, both developed and developing countries are actively promoting carbon neutrality policies and establishing relevant laws and regulations. This has accelerated the cooperation and development between the environment and the supply chain, further driving the application of GSCM in different industries [15]. GSCM has become an important means to enhance environmental image, meet public expectations for environmental protection, and improve supply chain operations.

GSCM incorporates environmental factors into supply chain management [16], which mainly include the following: (1) international environmental management; (2) green purchasing; (3) collaboration between customers and the environment; (4) green logistics; (5) ecological design [17]. In the supply chain network, both suppliers in the upstream and demand customers in the downstream play crucial roles. Tseng et al. [18] summarized GSCM and indicated that the GSCM network includes suppliers, manufacturers, customers, and logistics service providers, forming a closed-loop system. They also highlighted the development trends and future challenges of GSCM. Kumar et al. [19] developed a green supply chain model that involved manufacturers and retailers. They focused on the manufacturing and remanufacturing processes to deliver consumer goods rapidly, considering carbon emission costs as part of manufacturers’ and suppliers’ holding and degradation costs. Martí et al. [20] considered demand uncertainty and supply chain response decisions under different carbon policies. They constructed a green supply chain network and proved the effectiveness of GSCM. Diabat et al. [21] proposed a facility location model for supply chains, taking into account carbon emissions in manufacturing, warehousing, and transportation processes. This model aimed to minimize fixed and distribution costs to the greatest extent possible.

Meanwhile, selecting appropriate green suppliers is crucial for node enterprises to combat pollution and environmental pressures. Therefore, Yang et al. [22] analyzed the green innovation criteria of GSCM and established an integrated multi-criteria decision model with fuzzy hierarchical analysis to select the best cooperative suppliers. Govindan et al. [23] applied GSCM in the paper industry, focusing on paper recycling and green purchasing. They proposed models for optimal green supplier selection and order allocation to reduce greenhouse gas emissions and costs in the production process. The effectiveness of the models was demonstrated by reducing carbon emissions by 26.2% in a paper manufacturing company. Islam et al. [24] proposed a three-stage solution framework combining deep learning, principal component analysis, and optimization techniques to address supplier selection and order allocation problems. They conducted experiments using a real dataset from the Canadian meat industry to validate their approach. Aiming at the supplier selection problem of chemical enterprises, Wang et al. [25] proposed a kind of spherical fuzzy analytic hierarchy process based on compromise decisions, which can help enterprises realize the sustainability of the supply chain. Tirkolaee et al. [26] considered three levels of the supply chain: suppliers, central warehouses, and retailers. They explored sustainable supplier selection methods for a two-tier supply chain based on fuzzy decision-making and multi-objective programming, aligning with GSCM standards. Most of these studies focused on the sustainability index of suppliers and how to allocate orders. However, it is equally important to consider carbon emissions during the transportation process after the order allocation between suppliers and focal firms.

Green logistics is also an important component of GSCM, often manifested as the green vehicle routing problem (GVRP) [27]. Tseng et al. [18] emphasized green logistics in GSCM, aiming to improve environmental performance by minimizing greenhouse gas emissions through logistics operations. Liu et al. [28] studied green logistics problems with time windows, considering the impact of time-varying factors such as vehicle speed, passenger load, driving speed, and driving time on carbon emissions. They proposed an improved ant colony algorithm based on congestion avoidance for problem-solving. Rauniyar et al. [29] considered the minimization of carbon dioxide emissions and total distance in green logistics. They utilized a new variant of NSGA-II, which is a multi-objective approach, to obtain the optimal solutions. Li et al. [30] incorporated energy costs and greenhouse gas emission costs into green logistics for cold chain management. They applied an improved particle swarm optimization algorithm to solve the problem and discussed the impact of changing the vehicle’s maximum load on the total supply chain cost and greenhouse gas emissions. Zhang et al. [31] aimed to reduce carbon emissions in logistics distribution by establishing a logistics distribution model with optimization objectives including carbon emissions and product damage. They utilized an improved heuristic algorithm to solve the model. Yu et al. [32] proposed an improved branch-and-price (BAP) algorithm to precisely solve the time-dependent heterogeneous fleet green routing problem. This algorithm significantly reduces the number of branches and computational time. It can be observed that many existing research studies on green logistics mainly employ traditional heuristic methods and their variations, such as genetic algorithms, ant colony algorithms [33], grasshopper optimization algorithms [34], and particle swarm algorithms [30], while few utilize new heuristic algorithms.

3. Green Supply Chain Model

In the distribution stage of the supply chain, from the perspective of the focal company in the supply chain network, there are two main actions that it can undertake: purchasing goods from upstream suppliers and selling goods to downstream demanders, thereby completing the procurement and sales process and earning intermediary profits while minimizing carbon emissions. Therefore, determining how to procure goods from upstream suppliers and how to supply goods to downstream demanders becomes critical in improving both profit and operational environmental performance for the focal company. A supply chain with the focal company as the central node and upstream and downstream entities forms a typical three-tier supply chain system, as shown in Figure 1.

3.1. Problem Description

In a typical three-tier supply chain system, there are $n$ suppliers marked as $S_i$, where $i=1,2,\dots ,n$ and $m$ demand firms marked as $D_j$, where $j=1,2,\dots ,m$ The node enterprises are denoted by $C_1$.

Therefore, the optimization problem of the green supply chain can be described as follows: Upon receiving product demand from downstream demand firms $D_j$, node enterprises $C_1$ must procure products from various suppliers, taking into account the varying procurement costs, product quality, and environmental benefits offered by these suppliers $S_i$. Following procurement, they must select the optimal method and route for product delivery to downstream demand firms $D_j$, aiming to minimize the economic cost and environmental impact of the entire supply chain.

We make the following assumptions:

(1)

The products are purchased from suppliers without any quantity discounts.

(2)

The demand remains constant, and no supplier is allowed to have a shortage in product supply.

(3)

The spatial locations of the suppliers, intermediate companies, and demand sides are known.

(4)

The demand and time windows of the intermediate companies and demand sides are known.

(5)

The products are delivered from the intermediate companies to multiple demand sides, and the delivery vehicles must return to the intermediate companies after completing the delivery task.

(6)

The delivery vehicles travel at a constant speed, disregarding traffic congestion, and their transportation volume must not exceed their maximum payload.

(7)

The demands of the intermediate companies and demand sides must be met, and each demand customer is served by only one delivery vehicle. The remaining symbol explanations are shown in

Table 1. Nomenclature of constants and variables.

Notations		Meaning
Decision variables	$s_i$	$\mathrm{Binary} \mathrm{variable}, \mathrm{whether} \mathrm{supplier} S_i$ is selected
	$x_{i1}$	$\mathrm{Quantity} \mathrm{of} \mathrm{supply} \mathrm{from} \mathrm{suppliers} S_i$ $\mathrm{to} \mathrm{core} \mathrm{enterprise} C_1$
	$z_j$	$\mathrm{Binary} \mathrm{variable}, \mathrm{whether} \mathrm{demand} \mathrm{firm} D_j$ is selected
	$z_{jk}$	Binary variable, whether the vehicle reaches node $k$ from node $j$
Constants	$I$	Total number of suppliers
	$E$	Set of nodes for both the supplier enterprises and demand firm nodes
	$I_1$	Unit holding cost for core enterprise
	$p_f$	Price per liter of fuel
	$p_c$	Carbon price per kilogram of carbon dioxide
	$p_d$	Driver’s wage per unit distance traveled
	$p_t$	Vehicle rental cost
	$q_1^{\prime }$	Demand quantity for node enterprises (including safety stock and downstream demand)
	$p_i$	Unit procurement cost for suppliers  $i$
	$p_i^{\prime }$	Ordering cost for supplier $i$ (including handling fees, insurance fees, driver wages, etc.)
	$f_i$	Product failure rate of supplier  $i$
	$l_i$	Product delay rate of supplier  $i$
	$c_i$	The carbon emission index of the production unit product of the supplier  $i$
	$t_j$	Time to reach the demand firm  $j$
	$q_j$	Demand quantity of the demand firm  $j$
	$(e{t}_j,l{t}_j)$	Time window of the demand firm  $j$

.

3.2. Calculation Method for Vehicle Carbon Emissions

Demir et al. [35] investigated existing models for carbon emissions and fuel consumption and proposed a comprehensive modal emissions model (CMEM), which calculates the instantaneous fuel usage FR in grams during second t as follows:

$$P_t=0.5C_a{\rho }_{a}A{\nu }_t^3+(w+f){\nu }_t(g\sin \phi +g{C}_{r}cos\phi +a_t)$$

(1)

where $C_a$ and $C_r$ are the air drag coefficient and rolling resistance coefficient, ${\rho }_a$ is the air density constant, $A$ is the frontal area of the delivery vehicle, $v_t$ represents the velocity of the vehicle, $\omega$ is the curb weight of the vehicle, $f$ is the payload of the vehicle, $g$ is the gravitational constant, $\phi$ is the road gradient, and $a_t$ is the acceleration. Therefore, the instantaneous fuel usage (unit: grams) is as follows:

$$F{R}_t=\frac{\zeta }{\kappa }\left(C_f{C}_e{C}_d+\frac{P_t}{1000\epsilon \overline{\omega }}\right)$$

(2)

where $\zeta$ represents the mass ratio of fuel to air, $\kappa$ is the heating value of typical fuel, $C_f$ is the fictitious coefficient of the engine, $C_e$ is the engine speed, $C_d$ is the engine displacement, $\overline{\omega }$ is the engine efficiency parameter, and $\epsilon$ represents the vehicle transmission efficiency. Considering that the vehicle mass $M_a=\omega +f$ plays an important role, the instantaneous fuel consumption at moment $t$ can be divided into a term linearly correlated with $M_a$ and a term $F{R}_t(0)$ that is linearly independent. Thus, the formula for $F{R}_t(M_a)$ is obtained as follows:

$$F{R}_t(M_a)=\frac{\zeta }{\kappa }\times \frac{{\nu }_t(\mathrm{g} \sin \phi +g{C}_r\cos \phi +a_t)}{1000\epsilon \overline{\omega }}{M}_a+F{R}_t(0)$$

(3)

In the formula, $F{R}_t\left(0\right)$ represents the emissions unrelated to vehicle mass, as shown in the following equation:

$$F{R}_t(0)=\frac{\zeta }{\kappa }\times \left(C_f{C}_e{C}_d+\frac{0.5C_a{\rho }_{a}A{\nu }_t^3}{1000\epsilon \overline{\omega }}\right)$$

(4)

Therefore, for nodes $i$ and $j$, given the distance between the two points $d_{ij}$ and the average driving speed of the vehicle $v$, the total fuel consumption $F_{ij}$ for crossing $arc(i,j)$ is equal to the fuel consumption rate $F{R}_t(M_a)$ multiplied by the travel time $d_{ij}/v_t$.

$$F_{ij}={\alpha }_1\times d_{ij}/v_t+(\omega +f_{ij})\times {\alpha }_2\times d_{ij}/v_t$$

(5)

where ${\alpha }_1=\frac{\zeta }{\kappa }\times \left(C_f{C}_e{C}_d+\frac{0.5C_a{\rho }_{a}A{\nu }_t^3}{1000\epsilon \overline{\omega }}\right)$ and ${\alpha }_2=\frac{\zeta }{\kappa }\times \frac{{\nu }_t(\mathrm{g} \sin \phi +g{C}_r\cos \phi +a_t)}{1000\epsilon \overline{\omega }}$.

After the total fuel consumption $F_{ij}$ is obtained, the carbon emission index $R$ can be used to convert the total fuel consumption $F_{ij}$ of $arc(i,j)$ into carbon emissions. Therefore, the total carbon emissions $E_{ij}$ on $arc(i,j)$ can be described as follows:

$$E_{ij}=R\times F_{ij}=R\left[{\alpha }_1\times d_{ij}/v+(\omega +f_{ij})\times {\alpha }_2\times d_{ij}/v\right]$$

(6)

3.3. Green Supply Chain Mathematical Model

To optimize the green supply chain, the overall objective is to minimize the cost, minimize carbon emissions, and maximize customer satisfaction, while ensuring maximum profits for the node enterprises. The specific objectives include the following: (1) minimizing the total cost of procurement, holding, and transportation; (2) minimizing the non-conformance rate of products; (3) minimizing product delivery delays; (4) minimizing carbon emissions during procurement and transportation processes.

Based on the characteristics of the supply chain, it can be divided into two stages: supplier selection and procurement stage (upper-level model) and product transportation and distribution stage (lower-level model). By minimizing the costs in both stages, the overall profit can be maximized.

$$\min Z=\min \left(Z_1+Z_2\right)$$

(7)

where $Z_1$ represents the optimal value of the upper-level model and $Z_2$ represents the optimal value of the lower-level model.

3.3.1. Upper-Level Model

The upper-level model primarily focuses on supplier selection and the procurement stage. The mathematical model for this stage is as follows:

$$\min Z_{11}=\min {\displaystyle \sum _{i=1}^I\left(p_i{x}_{i1}+I_1{x}_{i1}+p_i^{\prime }{s}_i+F_{\mathrm{i}1}{p}_f{s}_i\right)}$$

(8)

$$\min Z_{12}=\min {\displaystyle \sum _{i=1}^I{E}_{i1}{p}_c{s}_i}$$

(9)

$${\min \mathrm{Z}}_{13}=\min {\displaystyle \sum _{i=1}^I{f}_i{x}_{i1}}$$

(10)

$$\min {\mathrm{Z}}_{14}=\min {\displaystyle \sum _{i=1}^I{l}_i{x}_{i1}}$$

(11)

$$\min {\mathrm{Z}}_{15}=\min {\displaystyle \sum _{i=1}^I{c}_i{x}_{il}}$$

(12)

$$s.t.{\displaystyle \sum _{i=1}^I{x}_{i1}\ge }{q}_1^{\prime }$$

(13)

$${\displaystyle \sum _{i=1}^I{f}_i{s}_i\le f_{\max }}$$

(14)

$${\displaystyle \sum _{i=1}^I{l}_i{s}_i\le l_{\max }}$$

(15)

$$x_{i1}\le C_{1\max }$$

(16)

$$x_{i1}\ge 0\hspace{0.17em}and\hspace{0.17em}integer$$

(17)

Equations (8)–(12) represent the minimization of various objectives, including the minimum cost (including procurement cost, holding cost, ordering cost, and transportation cost), minimum carbon emissions during transportation, minimum non-conformance rate, minimum product delay rate, and minimum supplier carbon emission index. Equation (13) states that the quantity purchased from all suppliers must satisfy the demand quantity of the node enterprise. Equation (14) sets a maximum limit on the non-conformance rate of purchased products, denoted as $f_{\max }$, which must be lower than the allowable maximum non-conformance rate of the node enterprise. Equation (15) sets a maximum limit on the product delay rate, denoted as $l_{\max }$, which must be lower than the allowable maximum delay rate of the node enterprise. Equation (16) represents the constraint for the maximum payload of vehicles during the procurement stage, where $C_{1\max }$ represents the maximum payload of vehicles in the procurement stage. Equation (17) represents the non-zero constraint for the quantity of purchased products.

Because all objectives cannot be achieved at the same time, in order to solve this multi-objective model, a normalization weighted method is used to convert it into a single-objective model, as shown in the following equation:

$$\min \hspace{0.33em}{Z}_1=\min \left(\begin{array}{l}{\beta }_1\frac{Z_{11}-Z_{11\min }}{Z_{11\max }-Z_{11\min }}+{\beta }_2\frac{Z_{12}-Z_{12\min }}{Z_{12\max }-Z_{12\min }}\\ +{\beta }_3\frac{Z_{13}-Z_{13\min }}{Z_{13\max }-Z_{13\min }}+{\beta }_4\frac{Z_{14}-Z_{14\min }}{Z_{14\max }-Z_{14\min }}\\ +{\beta }_5\frac{Z_{15}-Z_{15\min }}{Z_{15\max }-Z_{15\min }}\end{array}\right)$$

(18)

Here, ${\beta }_1~{\beta }_5$ are the corresponding weights for each objective.

3.3.2. Lower-Level Model

The lower-level model primarily involves the transportation and distribution stage of the product. Its mathematical model is as follows:

$$\min Z_2=\min \left(Z_{21}+Z_{22}+Z_{23}\right)$$

(19)

$$Z_{21}={\displaystyle \sum _{(j,k)\in E}{E}_{jk}{p}_c{z}_{jk}}$$

(20)

$$\hspace{0.17em}{Z}_{22}=\hspace{0.17em}{\displaystyle \sum _{(j,k)\in E}{p}_f{F}_{jk}{z}_{jk}}+{\displaystyle \sum _{(j,k)\in E}{p}_d{d}_{jk}{z}_{jk}}+p_{t}h$$

(21)

$$Z_{23}=a{\displaystyle \sum _{i\in E}\max (}{t}_i-e{t}_i,0)+b{\displaystyle \sum _{i\in E}\max (}l{t}_i-t_i,0)$$

(22)

$$s.t.{\displaystyle \sum _{(j,k)\in E}{z}_{jk}=1}$$

(23)

$$0<{\displaystyle \sum _{j\in E}{z}_j{q}_j}<C_{2\max }$$

(24)

$$h\le C_{3\max }$$

(25)

Equation (19) is the objective function for the transportation and distribution stage of the product. Equation (20) is the objective function for minimizing the carbon emission index. Equation (21) represents the minimization of fuel consumption cost, driver wages, and vehicle rental. Equation (22) is the penalty function for time window violations. Equation (23) states that all demand points can only be serviced by one vehicle. Equation (24) is the constraint on the maximum vehicle load, where $C_{2\max }$ represents the maximum vehicle load for the transportation and distribution stage. Equation (25) is the constraint on the number of vehicles, ensuring that the number of participating vehicles does not exceed the total number of vehicles owned by the nodes, where $C_{3\max }$ represents the total number of vehicles.

4. Two-Stage Heuristic Optimization Algorithm

Considering the characteristics of the green supply chain, we designed a two-stage heuristic optimization algorithm to solve the problem. For the upper-level model, we employed the cuckoo search algorithm, inspired by the parasitic behavior of cuckoos. This algorithm is renowned for its simplicity and efficiency in exploring the search space, effectively balancing global exploration and local exploitation. Its robustness and efficiency make it suitable for identifying the best suppliers, as it avoids local minima and converges to the global optimum. For the lower-level model, we adopted an improved brainstorm optimization algorithm. This algorithm uses clustering to search for local optima and then compares them to find the global optimum, ensuring solution diversity through the concept of mutation. Its flexibility allows it to adapt to changing problem dynamics and constraints, accurately representing the mathematical model of the lower-level problem and enabling rapid convergence to high-quality solutions.

4.1. The Cuckoo Search Algorithm

The cuckoo search (CS) algorithm is a metaheuristic algorithm proposed by Yang and Deb [36] in 2009, inspired by the brood parasitism behavior of cuckoo birds. The population renewal formula adopts the Levy flight strategy as shown in the following equation:

$$Y_i^{(t+1)}=Y_i^{(t)}+\alpha \otimes Levy(\lambda )$$

(26)

where $Y_i^{(t)}$ represents the position of the $i\left(i=1,2,\dots ,d\right)$ nest in the $t$ generation. $\otimes$ denotes element-wise multiplication. $\alpha$ represents the step size control quantity used to control the size of the step, typically taking the value of $\alpha =1$. $Levy(\lambda )$ denotes the random search path using the Levy flight mechanism, where the step lengths follow a heavy-tailed stable distribution.

$$Levy~\mu =s^{-\lambda },1<\lambda \le 3$$

(27)

In the equation, $s$ represents the random step length in the Levy flight search, which satisfies the following distribution:

$$s=\frac{U}{{\left|V\right|}^{1/\lambda }}U~N(0,{\sigma }^2),V~N(0,1)$$

(28)

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

(29)

After generating the next generation of nests, calculate the fitness value. If it is superior to the previous generation, update the nest positions; otherwise, keep them unchanged. After updating the positions, compare a randomly generated number $\epsilon$, following a uniform distribution from 0 to 1, with the detection probability $p_a$. If $\epsilon$ is greater than $p_a$, randomly change the nest positions in $Y_i^{(t)}$:

$$Y_i^{(t+1)}=Y_i^{(t)}+\alpha s\otimes H(p_a-\epsilon )\otimes (Y_j^{(t)}-Y_k^{(t)})$$

(30)

where $H\left(\cdot \right)$ represents the unit step function. $\epsilon$ represents a randomly selected number from a uniform distribution. $Y_j^{(t)}$ and $Y_k^{(t)}$ are two points randomly selected at generation $t$. This is a local search. Relative to the randomness of Levy flight, the local random walk has a certain directionality, enabling the maximum utilization of the positional information of existing points.

Check if the algorithm meets the maximum iteration limit. If it does, output the optimal nest position $Y_{best}^t$ and the optimal solution $f_{\min }$. Otherwise, continue the optimization process.

The CS algorithm flow is shown in

Table 2. Cuckoo search algorithm pseudocode.

Cuckoo Search Algorithm
$\mathbf{Input}: \mathrm{objective} \mathrm{function} f(x)$$, x={(x_1,\dots ,x_d)}^T$$, \mathrm{discovery} \mathrm{probability} p_a$,
$\mathrm{maximum} \mathrm{number} \mathrm{of} \mathrm{iterations} N_{\max }$$, \mathrm{optimal} \mathrm{nest} \mathrm{position} Y_{best}^0$$, \mathrm{optimal} \mathrm{solution} f_{\min }$
$\mathbf{Output}: \mathrm{optimal} \mathrm{nest} \mathrm{position} Y_{best}^0$$, \mathrm{optimal} \mathrm{solution} f_{\min }$
$1: \mathrm{generating} \mathrm{the} \mathrm{initial} \mathrm{positions} X_i\left(i=1,2,\dots ,n\right)$ of $n$ nests.
$2: \mathrm{while} (\mathrm{number} \mathrm{of} \mathrm{iterations} t<N_{\max }$) do
$3: \mathrm{randomly} \mathrm{select} \mathrm{a} \mathrm{nest} Y_i^{(t-1)}$$, \mathrm{whose} \mathrm{fitness} \mathrm{is} f_i^{(t-1)}$.
$4: \mathrm{update} Y_i^{(t-1)}$$\mathrm{used} \mathrm{by} \mathrm{Equations} \left(26\right) \mathrm{and} \left(27\right) \mathrm{to} \mathrm{obtain}{Y}_i^{(t)}$$, \mathrm{whose} \mathrm{fitness} \mathrm{is} f_i^{(t)}$.
$5: \mathrm{if} f_i^{(t-1)}$$>f_i^{(t)}$, then
$6: \mathrm{replace} Y_i^{(t-1)}$$\mathrm{with} Y_i^{(t)}$.
7:          end
$8: \mathrm{if} \mathrm{rand}(0,1) > p_a$, then
$9: \mathrm{update} Y_i^{(t)}$ used by Equation (30)
10:        end
$11: \mathrm{update} Y_{best}^{\left(t\right)}$$\mathrm{and} f_{\min }$
$12: t=t+1$
13: end
$14: \mathrm{return} Y_{best}^{\left(t\right)}$$\mathrm{and} f_{\min }$

.

4.2. Improved Brainstorming Optimization Algorithm

The brainstorm optimization algorithm (BSO) [37], proposed by Professor Yu Hui Shi in 2011, is an optimization algorithm based on the brainstorming strategy of human minds. This algorithm primarily draws inspiration from the core ideas of brainstorming processes, including deferred judgment, bold assumptions, cross-fertilization, and the idea of quantity winning. BSO treats individuals in the search process as ideas generated by the human brain and clusters them into several clusters. The best individual in each cluster is set as the cluster center. During the solution generation process, individuals with better fitness values are retained. BSO mainly consists of three strategies: clustering strategy, new individual generation strategy, and individual selection strategy.

4.2.1. Clustering Strategy

The purpose of clustering is to converge all solutions to smaller regions. BSO can utilize various clustering algorithms, with the commonly used one being the $K-means$ algorithm. The $K-means$ algorithm is a common unsupervised method for dividing a set of data points into $K$ different clusters. The basic idea is to minimize the sum of squared distances between each data point and its corresponding cluster center, thereby placing the data points into different clusters. The objective function for the $K-means$ algorithm is as follows:

$$J={\displaystyle \sum _{i=1}^k{\displaystyle \sum _{x\in C_i}(x-u_i{)}^2}}$$

(31)

where $k$ represents the number of clusters, $C_i$ represents the $i$ cluster, and $u_i$ represents the average value of the $i$ cluster. The specific operational steps are as follows:

(1)

Randomly select $K$ initial cluster center points.

(2)

For each data point, calculate its distance to all cluster centers and assign it to the nearest cluster.

(3)

For each cluster, recalculate its centroid by taking the average value of the coordinates of all points in the cluster as the new centroid.

$$u_i=\frac{1}{\left|C_i\right|}{\displaystyle \sum _{x\in C_i}x}$$

(32)

(4)

Repeat steps (2) and (3) until the cluster center points no longer change or the maximum number of iterations is reached.

4.2.2. Improved New Entity Generation Strategy

To generate new individuals in BSO, a predetermined probability value $p_2$ is used to determine whether a new individual is generated from one or two old individuals. Generating new individuals from a single cluster helps refine the search area and improve search efficiency. On the other hand, generating new individuals from two or more clusters enhances global search capability, increasing population diversity.

Using the probabilities $p_3$ and $p_4$, the new individual generated from the cluster center or from outside the cluster center can be determined when generating new individuals from a single cluster or multiple clusters. The general strategy for generating new individuals is to apply Gaussian mutation, as represented in the following equation:

$$x_{new}=x_{selected}+\xi n(\mu ,\sigma )$$

(33)

In this context, $x_{new}$ and $x_{selected}$ represent the newly generated individual and the selected individual, respectively. $n(\mu ,\sigma )$ denotes the Gaussian random function, which is used to generate random values. $\xi$ represents the step size, which is calculated as follows:

$$\xi =\log sig\frac{(\frac{E}{2}-e)}{K}\theta$$

(34)

where $\log sig()$ represents the logarithmic transfer function on the interval $\left(0,1\right)$, $E$ denotes the maximum number of iterations, and $e$ represents the current iteration number. $K$ is used to change the slope of the $\log sig()$ function, and $\theta$ is a random value within $\left(0,1\right)$.

The Gaussian mutation technique has the advantage of having good local search capability. However, its disadvantage is that it lacks the ability to escape local optima, leading to poor global convergence. Therefore, the crossover and mutation process of genetic algorithms is introduced here to balance the global and local optimization capabilities, resulting in a good iterative performance. When two clusters are randomly selected, the adaptive crossover operation is performed on two individuals (individuals encoded with information about the order of visiting demand points) in the two clusters based on the crossover probability. The steps are as follows:

(1)

Select a demand point as the starting point for sub-individual 1.

(2)

Calculate the fitness between the right neighboring point and the demand point for both individuals. Choose the closer point as the second encoded point for sub-individual 1.

(3)

Repeat the process to obtain all the encoded points for sub-individual 1.

(4)

Go back to step 1 and find the left neighboring point. Repeat the above steps to obtain sub-individual 2. The crossover is complete.

When a single cluster is randomly selected, perform the mutation operation on one individual in the cluster based on the mutation probability. The mutation operation is conducted using a multi-point mutation method. To maintain the usability of solutions, select four encoded points and perform pairwise swapping to complete the mutation.

4.2.3. Selection Strategy

After the generation of new individuals in each generation, a selection strategy is employed to preserve better solutions between the new and original individuals. This ensures the convergence capability of the algorithm. On the other hand, the clustering strategy and generation strategy are used to add new solutions to the population, thereby maintaining the diversity of the entire population.

The workflow of the BSO algorithm is as follows:

(1) Define the objective function $f(x)$, $x={(x_1,\dots ,x_d)}^T$ and initialize the population by randomly generating $n$ individuals $X_i\left(i=1,2,\dots ,n\right)$. Set the number of clusters $K$, and define the selection probabilities $p_1$, $p_2$, $p_3$, and $p_4$.

(2) Employ the $K-means$ clustering method to cluster the individuals, and designate the best individuals of each cluster as the cluster centers.

(3) Use the probability $p_1$ to determine whether to replace the best individuals of each cluster with ordinary individuals, thereby increasing the randomness of the algorithm.

(4) Update the individuals. Based on the probability $p_2$, determine whether to select one or two clusters. Use $p_3$ and $p_4$ to decide whether to operate on the cluster centers or ordinary individuals. To enhance the global search ability of the algorithm while maintaining good convergence, the original approach of adding Gaussian random values to the selected individuals for updating is abandoned. Instead, the crossover and mutation processes of genetic algorithms are introduced. If a single cluster is randomly selected, the mutation operator is applied based on the mutation probability $P_m$. If two clusters are randomly selected, the crossover operator is used based on the crossover probability $P_c$.

(5) To prevent the destruction of outstanding individuals and further improve the optimization accuracy, dynamic crossover probability $P_c$ and dynamic mutation probability $P_c$ are employed to achieve adaptive changes. The expressions are as follows:

$$\begin{array}{l}{P}_c=\left\{\begin{array}{cc}{P}_{c1}-\frac{P_{c1}(f_{avg}-f_1)}{f_{\max }-f_{avg}}& f_1<f_{avg}\\ P_{c1}& f_1\ge f_{avg}\end{array}\right.\\ P_m=\left\{\begin{array}{cc}{P}_{m1}-\frac{P_{m1}(f_{avg}-f_2)}{f_{\max }-f_{avg}}& f_2<f_{avg}\\ P_{m1}& f_2\ge f_{avg}\end{array}\right.\end{array}$$

(35)

In the expressions, $f_{avg}$ represents the average fitness value, $f_{\max }$ represents the maximum fitness value, $f_1$ represents the higher fitness value between the two individuals in the crossover, and $f_2$ represents the fitness value of the mutated individual. $P_{c1}$ denotes the original crossover probability, and $P_{m1}$ denotes the original mutation probability. When an individual’s fitness value is lower than the average fitness value, it indicates that the individual is relatively superior. According to the formula, the mutation and crossover probabilities are adaptively reduced, thereby preserving the superior individuals to a greater extent.

(6) After the new individuals are obtained, an elite preservation mechanism is introduced. If the new individual is better than the original solution, it is retained. Otherwise, the original solution is preserved to improve the convergence performance of the algorithm.

(7) Update the best solution. If the maximum number of iterations is reached, output the best solution, which represents the optimal distribution plan for the product. Otherwise, continue iterating.

4.3. Two-Stage Heuristic Method

In this paper, the typical three-layer supply chain is divided into two stages, and different optimization methods are used to solve each stage. The optimization objective for the first stage is given by Equation (18), and the optimization objective for the second stage is given by Equation (19). The solution process is as follows:

(1)

Upper-level model solution process

(a)

Set $Z_1$ as the fitness function and use real number encoding to generate feasible solutions for the purchasing plans of suppliers $S_i$ and node enterprises $C_1$. Set population size, number of iterations, and other model hyperparameters.

(b)

Update the nest positions according to Equations (26) and (27).

(c)

Generate a random number $\epsilon$ and compare it with $p_a$. If $\epsilon >p_a$, update the nest positions using Equation (30).

(d)

Check the stopping condition and output the optimal solution $Z_{1-\min }$ and the optimal nest position, which represents the purchasing plan for the first stage.

(2)

Two-stage model solution process

(a)

Set $Z_2$ as the fitness function and use real number encoding to generate distribution plans for node enterprise $C_1$ and demand firm $D_j$. Set the number of clusters, selection probabilities, and other model hyperparameters.

(b)

Use $K-means$ for clustering and select the best fitness individuals as the cluster centers.

(c)

Perform selection based on probabilities and update the individuals using the crossover and mutation operators of genetic algorithms. Preserve the elite individuals and update the best solution.

(d)

Check the stopping condition and output the optimal solution $Z_{2-\min }$, as well as the best individual representing the optimal distribution plan.

By combining the results from both stages, the optimal goods procurement–sales route for node enterprise $C_1$ can be generated.

5. Case Analysis

In this section, we utilize a supply chain case study to validate the effectiveness of the model and assess the performance of the algorithms employed in the green supply chain model. We first describe the parameter selection and test cases, followed by presenting the research results. Finally, we conduct an analysis and summary. All tests were conducted on a single processor of an Intel(R) Core(TM) i5-12400 processor sourced from Chengdu, China with a frequency of 2.50 GHz. The RAM of the system running on Windows 10 was 32.0 GB.

5.1. Parameter Selection and Calculation

5.1.1. Model Parameters

By referring to the freight rates of a logistics company [38], we can obtain the vehicle rental cost $p_t$ and the driver wages per kilometer $p_d$. Additionally, based on the data from the gasoline price website, we have set the price per liter of fuel as $p_f$. The remaining parameters are shown in

Table 3. The model parameters.

Notation	Description	Typical Value
$C_a$	Air resistance coefficient	0.7
$C_r$	Rolling resistance coefficient	0.01
${\rho }_a$	Air density constant	1.2041 $\mathrm{kg}/{\mathrm{m}}^3$
$A$	Frontal surface area of delivery vehicles	4 ${\mathrm{m}}^2$
$v_t$	Average vehicle speed	72 $\mathrm{km}/\mathrm{h}$
$\omega$	Vehicle weight	9400 $\mathrm{kg}$
$g$	Acceleration due to gravity	9.81 $\mathrm{m}/{\mathrm{s}}^2$
$\phi$	Road gradient or road camber	0
$a_t$	Vehicle acceleration	0
$\zeta$	Mass ratio of fuel to air	1
$\kappa$	Typical fuel calorific value	44 $\mathrm{kJ}/\mathrm{g}$
$C_f$	Engine fictional coefficient	0.2
$C_e$	Engine speed	40 $\mathrm{rev}/\mathrm{s}$
$C_d$	Engine displacement	5 $\mathrm{L}$
$\overline{\omega }$	Engine efficiency parameters	0.9
$\epsilon$	Vehicle transmission efficiency	0.4
$R$	Carbon emission coefficient	3.164
$p_c$	Price per kilogram of carbon	CNY 0.22329
$p_f$	Price per liter of fuel	CNY 8.17
$p_d$	Driver’s wage per kilometer	CNY 3
$p_t$	Vehicle rental cost	CNY 100
$f_{\max }$	Maximum failure rate	0.05
$l_{\max }$	Maximum delay rate	0.05
$C_{1\max }$	Maximum payload capacity of vehicles of first stage	15,000 $\mathrm{kg}$
$C_{2\max }$	Maximum payload capacity of vehicles of second stage	15,000 $\mathrm{kg}$
$C_{3\max }$	Total number of vehicles	6

.

Because node coordinates are usually given in latitude and longitude, the distance between two points can be calculated using the great circle distance formula. First, the latitude and longitude coordinates need to be converted to radians, as follows:

$$\begin{array}{l}la{t}_i=x_i\cdot \pi /180\\ lo{n}_i=y_i\cdot \pi /180\end{array}$$

(36)

where $(x_i,y_i)$ represents the latitude and longitude coordinates of the nodes. Using the Earth radius $r=6371.393 \mathrm{km}$, the distance between the two nodes can be calculated as follows:

$$d_{ij}=r\cdot \arccos \left[\sin \left(la{t}_i\right)\cdot \sin \left(la{t}_j\right)\right.+\left.\cos \left(la{t}_i\right)\cdot \cos \left(la{t}_j\right)\cdot \cos \left(lo{n}_i-lo{n}_j\right)\right]$$

(37)

5.1.2. Algorithm Parameters

(1)

The parameter of CS

The population size m in the cuckoo search algorithm is a crucial parameter that directly affects the algorithm’s performance and efficiency. When the population size is small, the algorithm’s search space coverage is limited, and the few sample points may not adequately capture the problem’s complexity, leading to instability. Conversely, when the population size is too large, it significantly increases the consumption of computational resources and results in excessive redundant searches. Therefore, we tested values of 50, 100, 150, 200, and 250, and observed their convergence efficiency, as shown in Figure 2.

As observed, when the population size is 50 or 100, the algorithm may not sufficiently explore the entire solution space, leading to local optima. When m is 200 or 250, the convergence efficiency of the algorithm is comparable, but a larger population size results in increased computational resource consumption and longer computation times. Therefore, after comparison, we chose m = 200 as the population size for the cuckoo search algorithm.

(2)

The parameter of IBSO

In the context of the brainstorming algorithm, the parameter $p_2$ represents the probability of choosing between one cluster or two clusters. This parameter plays a pivotal role in striking a balance between global search performance and local search performance. A smaller value of $p_2$ diminishes the likelihood of selecting a single cluster, thereby affecting the interaction of information among individuals, which, in turn, determines the diversity of solutions produced by the algorithm. Thus, the parameter $p_2$ holds relative significance within the algorithm. In this study, we consider a range of values for $p_2$, specifically 0.1, 0.2, 0.4, 0.6, and 0.8. The convergence efficiency of the algorithm is observed, as depicted in Figure 3.

It can be observed that when $p_2$ is set to 0.1, the probability of selecting a single cluster is low, leading to strong global search performance. Consequently, the algorithm exhibits rapid convergence during the early iterations, but slower convergence in the later stages. Conversely, when $p_2$ is set to 0.8, the global search performance weakens, while its local search performance strengthens, resulting in premature convergence and trapping in local optima. Upon comparative analysis, it becomes evident that a value around $p_2=0.4$ strikes a better balance between global and local search performance, offering a more optimal convergence behavior.

5.2. Case Study

The supply chain in the case study consists of 15 suppliers, 1 distributor and 28 retail enterprises. The objective is to optimize the purchasing and distribution path using the multi-stage heuristic optimization algorithm to achieve cost control while minimizing carbon emissions in the entire supply chain. Some of the pertinent information implicated in the case is presented in

Table 4. Supply chain information.

Category	Number	Longitude	Latitude	Price (CNY)	Ordering Cost (CNY)	Quantity Demanded	Inventory Cost (CNY/Piece)	Delay Rate	Failure Rate	Carbon Emission Index	Time Window
suppliers	1	113.2069	22.9691	2400	11,000	/	/	0.0036	0.0042	1.71	/
	2	114.9941	22.9635	2300	12,000	/	/	0.0031	0.0036	1.58	/
	3	113.5448	22.7285	2100	13,000	/	/	0.0035	0.0037	1.67	/
	4	113.2365	22.9203	2200	12,000	/	/	0.0024	0.0044	1.70	/
	5	113.3231	23.1240	2300	11,000	/	/	0.0035	0.0039	1.87	/
	6	113.1397	23.4730	2300	12,000	/	/	0.0022	0.0045	1.41	/
distributor	1	114.0065	34.7238	3000	/	1200	140	/	/	/	/
retail enterprises	1	126.6165	45.7503	/	/	40	/	/	/	/	(0, 48)
	2	125.3499	43.8337	/	/	14	/	/	/	/	(0, 72)
	3	123.3488	41.7588	/	/	10	/	/	/	/	(24, 72)
	4	116.3361	39.7530	/	/	34	/	/	/	/	(0, 72)
	5	117.1927	39.2032	/	/	40	/	/	/	/	(24, 72)
	6	114.5694	38.0103	/	/	40	/	/	/	/	(48, 72)
	7	106.3077	38.4503	/	/	40	/	/	/	/	(24, 96)
	8	112.4820	37.8558	/	/	54	/	/	/	/	(48, 96)

(not all nodes are listed in the table due to the vast amount of data, as detailed in Appendix A). The location information of all nodes, including suppliers, distributors, and retail enterprises, is shown in Figure 4.

The average weight of each product is 50 kg. For the distributor’s upstream procurement stage, the weight coefficient $\beta$ is set to 0.2, and the CS algorithm is set with the population size of 200, 50 iterations, and the discovery probability $p_a=0.3$. For the distributor’s transportation and distribution stage to downstream retail companies, the IBSO is set with a population size of 200, 500 iterations, and selection probabilities $p_1=0.1$, $p_2=0.4$, $p_3=0.3$, and $p_4=0.2$.

The two-stage heuristic algorithm was executed 100 times, and the statistical parameters of the algorithm are shown in

Table 5. The statistical parameters of the algorithm.

Algorithm	Minimum of the Fitness Function	Maximum of the Fitness Function	Mean	Standard Deviation
CS	0.08	0.15	0.11	0.021
IBSO	41,536	42,088	41,706	167.7

.

The standard deviations of both algorithms are relatively small. Due to the different dimensions of the upper model’s objective functions, the fitness function value of the upper model is the normalized weighted objective function value. For the lower model, since the dimensions of the objective functions are consistent, the fitness function value is the total cost of the lower model.

Considering the integrity of the scheme, the optimal scheme in 100 experiments is presented here. The total cost is CNY 2,998,288 and the planned path diagram is shown in Figure 5.

The results obtained from the upper-level model show that suppliers 11 and 15 each ordered 600 units of goods. The total cost, which includes procurement costs, ordering costs, holding costs, and transportation costs, is CNY 2,956,465 (rounded to the nearest whole number). The carbon emission price during transportation is CNY 287. There were five units of non-conforming products and a total of three units of product delay. The total carbon emission index for the suppliers is 1446.

The results obtained for the lower-level model (with 0 representing the distributor itself) are as follows: The optimal cost is CNY 41,536 (all integers are retained). This includes the use of four vehicles, carbon emission price of CNY 374, driver’s wage expenditure of CNY 36,126 fuel cost of CNY 4328, vehicle rent of CNY 400, and time penalty of CNY 307. The specific distribution plan is shown in

Table 6. Distribution routing table.

Path Sequence Number	Distribution Route
1	0→17→26→27→28→24→23→22→0
2	0→9→10→3→2→1→5→4→6→12→0
3	0→16→15→25→13→14→7→8→0
4	0→11→21→20→19→18→0

.

5.3. Comparison Experiment

5.3.1. Algorithm Performance Comparison

To demonstrate the effectiveness of the improved brainstorm optimization algorithm (IBSO), we compared the improved algorithm with the genetic algorithm (GA), particle swarm optimization (PSO), and the original brainstorm optimization algorithm (BSO) using this case study. The population size for each algorithm was set to 200, with a maximum of 500 iterations. For the genetic algorithm, the crossover probability was set to 0.8, and the mutation probability was 0.05. For the particle swarm optimization, the individual learning factor was set to 0.1, and the social learning factor was set to 0.075. The parameters for the brainstorm optimization algorithm were set the same as those for IBSO. Each algorithm was executed 100 times. The iteration results of a certain experiment are shown in Figure 6. To maintain the integrity of the solution, the specific results of the optimal solution are provided in

Table 7. Algorithm comparison.

Algorithm Name	Number of Vehicles	Carbon Emission (CNY)	Fuel Price (CNY)	Driver Salary Expenditure (CNY)	Time Penalty (CNY)	Optimal Cost (CNY)
GA	5	383	4437	37,728	298	43,347
PSO	4	396	4588	38,024	302	43,711
BSO	4	399	4609	36,814	304	42,525
IBSO	4	374	4328	36,126	307	41,536

, and the statistical results are shown in

Table 8. The statistical parameters of the algorithms.

Algorithm Name	Minimum	Maximum	Mean	Standard Deviation
GA	43,347	56,818	48,640	3027.7
PSO	43,711	60,070	50,145	2926.5
BSO	42,525	54,833	47,871	2579.2
IBSO	41,536	42,088	41,706	167.7

.

It can be observed from Figure 6 that the IBSO outperforms GA, PSO, and BSO in terms of iteration speed and searching for optimal results. Due to the rapid reduction in the diversity of individuals within the population during the early stages of evolution in the GA, the algorithm tends to converge prematurely. This can lead to the loss of meaningful search points and optimal solutions, causing the algorithm to fall into local optima. The PSO algorithm has a slower iteration speed due to the limitations of retail enterprise scale and problem constraints. The IBSO algorithm introduces crossover and mutation operators in updating individual parts and adapts crossover and mutation probabilities, which enables it to balance global and local search performance, thereby demonstrating good effectiveness. Therefore, as shown in Table 7 and Table 8, IBSO has a smaller standard deviation and higher stability compared to other algorithms.

5.3.2. The Impact of Vehicle Types

To investigate the impact of vehicle types on carbon emissions and optimal costs, five vehicles with different load capacities were selected for comparative analysis, as shown in

Table 9. Vehicle types.

Vehicle Type	Vehicle Weight (kg)	Load Capacity (kg)
C1	3500	6000
C2	6800	9800
C3	9400	15,000
C4	13,500	18,500
C5	18,000	31,000

.

A total of five types of vehicles were selected, and five sets of experiments were conducted. Each algorithm was run 100 times in each set of experiments, and the IBSO experimental results were compared with those of the GA, PSO, and BSO algorithms. The specific results are shown in Figure 7 and Figure 8.

From the figures, it can be observed that as the vehicle types change, the fuel consumption of distribution vehicles increases due to the increase in vehicle weight and maximum load capacity, leading to a gradual rise in carbon emissions. Conversely, the increase in maximum load capacity results in a decrease in the number of distribution vehicles and the total distance traveled by all vehicles, leading to a gradual decrease in optimal costs. It is noteworthy that we found carbon emissions to increase slowly in the early stages of vehicle type variation, followed by a rapid increase, while optimal costs exhibit the opposite trend. This suggests that when the load capacity of distribution vehicles is low, the increase in carbon emissions is not significant, and optimal costs decrease substantially. However, when the load capacity of distribution vehicles is high, carbon emissions increase significantly, while cost reduction is less pronounced. Therefore, managers should make rational choices regarding the types of distribution vehicles to achieve a balance between carbon emissions and economic costs. Additionally, due to the relatively low carbon price of 0.22329 CNY/kg, which results in a low proportion of carbon emissions price in the total cost, the costs of vehicle types C4 and C5 in the figure show only a slow decrease despite the drastic increase in carbon emissions. Therefore, it is advisable to increase the carbon emissions price appropriately to increase the level of importance given to environmental protection by the enterprise.

Meanwhile, from the two box plots, it can be seen that the improved brainstorm optimization algorithm (IBSO) proposed in this paper, due to the improvement of the update operator, exhibits good local and global search performance. Therefore, under different vehicle types, the algorithm can calculate relatively optimal results. Additionally, from the figures, it can be observed that due to the adaptive operation of IBSO on crossover and mutation probabilities, compared to other algorithms, the standard deviation of different experimental results is smaller, indicating higher stability.

5.3.3. Model Performance Comparison

To assess the optimization performance in terms of carbon footprints, this case study involves the exclusion of carbon emissions and fuel consumption from the model’s objective function for comparison, while also calculating the actual carbon emissions during distribution transportation. The upper-level model yielded a carbon emissions value of 352 units, which represents an increase of approximately 22%. In comparison to GA, PSO, and BSO, the results are illustrated in Figure 9.

It is evident that, with the inclusion of carbon emissions in the model, core enterprises, when making distribution decisions, manage to reduce the carbon emissions during their transportation processes to a certain extent. Overall, by incorporating carbon emissions into the model’s objectives, we observed an approximate 10% reduction, demonstrating the model’s effectiveness in assisting company decision-makers in lowering carbon emissions in supply chain operations.

5.4. Analysis of Management and Environmental Sustainability

This paper primarily elucidates the optimization of a two-stage green supply chain, aiming to generate efficient transportation and distribution decisions by minimizing the model’s cost function and carbon emissions. Building upon this, the following analysis regarding management and the environment can be provided:

• The proposed GSC model comprehensively considers transportation carbon emissions, inventory costs, and transportation costs. It utilizes an integrated approach for solution, which not only provides effective procurement decisions in the first stage but also improves distribution decisions for vehicle routes in the second stage. Therefore, this integrated optimization can offer industries efficient procurement and transportation designs.
• Additionally, managers prefer to make decisions in a short period. The two-stage heuristic algorithm with improved update operators can effectively balance global and local search capabilities, thereby significantly improving iteration efficiency and convergence speed to solve practical problems. It can provide better solutions to managers in a short period. Moreover, due to the algorithm’s versatility, decision-makers can use it to solve problems of various scales.
• The GSC model, with its various cost components (carbon emissions, transportation, inventory, and time costs), serves as objectives that aid managers in making more precise decisions regarding the environment and economy. This flexibility allows decision-makers to alter the priorities of different costs according to their needs, enabling the development of GSC models with varying cost–benefit profiles.
• When calculating carbon emissions, the model primarily considers the carbon emissions from suppliers and the carbon emissions from transportation and distribution, ensuring the integration of environmental factors throughout the entire supply chain. In calculating transportation carbon emissions, both distance and load factors are considered to achieve environmental sustainability.
• During the experimental process, it was observed that carbon emissions during the distribution process are primarily influenced by factors such as vehicle types, quantity, route length, and the retail distribution order. Therefore, managers can reduce carbon emissions and achieve a balance between environmental and economic factors by properly arranging vehicle types and distribution plans.

6. Conclusions

This paper focuses primarily on the optimization challenges pertaining to green supply chains. Commencing from the standpoint of intermediary businesses, a three-tier green supply chain model is proposed, which takes into account economic and environmental factors, integrating upstream product procurement and transportation with downstream product distribution. Experiments have shown that the model can reduce carbon emissions by about 10%. To tackle this green supply chain model, a two-stage heuristic algorithm has been devised for solutions. In order to enhance the search performance of the lower-level model brainstorming optimization algorithm, genetic algorithm crossover and mutation phases have been introduced to balance global and local search characteristics. Furthermore, in order to minimize disruption to excellent individuals and expedite algorithm convergence, an elite preservation strategy has been implemented, and adaptive operations have been applied to crossover and mutation probabilities. The improved algorithm exhibits significantly enhanced convergence speed by about 60% and superior search performance. Experimental results demonstrate that the proposed approach performs well within this green supply chain model, enabling rapid acquisition of ordering strategies and transportation routes under limited time constraints. This, in turn, assists intermediary businesses in making informed decisions related to logistics design, inventory control, and risk management.

In the future, the green supply chain model can be further expanded by considering various factors such as road congestion and different carbon emissions for different vehicle types, in order to better fit practical scenarios. This expansion will guide decision-makers of central enterprises in optimizing and refining the supply chain.