Optimisation of Fuzzy Reverse Logistics Networks for Express Packaging Considering Recycling Rates

Abstract

The recycling and reuse of discarded express delivery cartons can yield environmental, economic, and social benefits. A key factor influencing the volume of express packaging collected is the uncertainty in the total amount of such packaging within the service range of each collection point. Additional uncertainties include the costs associated with the construction of recycling stations, operational expenses, transportation costs, additional recycling fees, and government subsidies. To address the issue of express packaging recycling, a fuzzy integer programming model for the reverse logistics network of express packaging is constructed. The model aims to minimise the total network cost and maximise the total recycling rate while enabling decisions regarding the location of recycling facilities and the flow between facilities. Then, a memetic algorithm based on dynamic local search is designed. Several alternative solution approaches were considered to evaluate the proposed algorithm, including the precision optimization method (CPLEX) and a hybrid priority-based genetic algorithm. The results confirm the feasibility of the memetic algorithm. Finally, the applicability of this fuzzy programming model is analysed and validated by changing the confidence level. The case study results reveal quantifiable trade-offs: as the confidence level (α) increases from 0.75 to 0.90 under a fixed recycling rate threshold (ε = 80%), the total network cost rises approximately linearly, while the required number of recycling stations increases, with their average facility level upgrading accordingly. Variations in confidence levels and the degree of total recycling rate achievement can significantly influence the increase in target values. Moreover, the magnitude of this influence exhibits irregularity, indicating that changes in confidence levels entail a certain degree of risk.

1. Introduction

With the development of networks, consumption patterns have changed considerably. China’s rapid development of e-commerce has boosted the growth of the courier industry, with the volume of express deliveries increasing 5.5 times in recent years, leading to significant changes in business revenues. Specifically, revenue from express delivery operations increased 3.8 times, total delivery volume remained the highest in the world for three consecutive years, and delivery volume increased significantly year-on-year. According to data from the State Post Bureau, the business volume of express delivery companies totalled 140.08 billion parcels in 2024, an increase of 22.3% compared with the previous year. According to Greenpeace, given the current development trend, the consumption of express delivery packaging will reach 412.705 million tons by 2025. This volume represents a considerable waste of resources, poses a threat to our environment, and is not conducive to the development of a green economy. Therefore, recycling packaging waste has attracted considerable attention from all sectors of society. Specifically, recycling 1 ton of waste paper saves 17 trees and approximately 3 cubic metres of landfill space, thereby reducing sewage discharge by about 100 cubic metres. In reality, the recycling rate of express packaging is less than 20%, while that of plastic packaging is 25% in developed countries that emphasise waste recycling, and the utilisation rate of paperboard packaging can even reach 45% [1]. The current volume of express delivery business in China is huge, and the problem of packaging waste is becoming increasingly severe. However, the overall recycling rate is still low, and there is a lack of systematic recycling models. In recent years, the government has successively introduced policies such as the “Management Measures for Mail and Express Packaging”, and the industry is also exploring green circular packaging. However, a large-scale and normalized recycling system has not yet been formed. This study aims to provide a reference for building an efficient and unified express packaging recycling path by analyzing the current situation and challenges.

Therefore, it is urgent to implement scientific and reasonable measures for the recycling and reuse of express delivery packaging, to reconcile economic development and environmental protection and build a sustainable, high-quality, and green express delivery industry. In recent years, national and international scholars have focused on waste disposal, including the treatment of used batteries, medical waste, and industrial waste. Currently, research on express delivery packaging recycling remains limited, and a comprehensive recycling and reuse system has not yet been established. To solve the problems of resource waste and environmental pollution, some domestic enterprises have implemented measures for recycling express delivery packaging [2]. However, there is currently no unified model or standardised recycling plan in China for express delivery packaging, resulting in many recycling-related problems, such as the low diversity of recycled packaging, the high cost of recycling and reuse, limited geographical coverage, and low public enthusiasm for recycling. Therefore, it is important to optimise China’s express delivery packaging recycling model. For the government, enterprises, and the general public, key related issues include establishing a scientific and reasonable express delivery packaging recycling model, reducing costs and improving the efficiency of the recycling and reuse process, and increasing the recycling rate. These issues are also essential for protecting the natural environment, saving material resources, and achieving sustainable development. Therefore, developing an efficient, low-cost, and high-recycling rate reverse logistics network for express delivery packaging is the key to solving this problem.

However, in practice, uncertainties affect decisions regarding logistics transportation. The recyclability of courier packaging plays a decisive role in optimising reverse logistics. The recycling volume of courier packaging is mainly influenced by residents’ willingness to recycle. As the recycling of courier packaging is still in its infancy, regulations and incentive mechanisms are incomplete, resulting in low recycling awareness among residents and a low recycling rate, ultimately affecting the location and service optimisation of courier packaging recycling stations [3]. Therefore, before designing a collaborative optimisation model, it is necessary to quantify residents’ willingness to recycle and then accurately describe the recycling volume of courier packaging. In addition, the land use conditions of different locations lead to uncertainty regarding the construction and operating costs of recycling stations, transportation costs per unit distance due to road complexity, enterprises’ additional return fees, government subsidies, the amount of discarded courier packaging at disposal points, and the capacity of recycling stations under construction. All of these factors cannot be ignored.

According to previous research results, common problems encountered in reverse logistics network design, such as facility location and path optimisation, are generally NP-hard problems [4]. The resolution time for such problems typically increases exponentially with the increase in scale and is often longer in an uncertain environment. Therefore, finding an optimal solution to these problems is also essential to ensure the optimality and feasibility of the related research methods and solutions. Given the complexity of optimisation problems, finding efficient and accurate solutions for the associated models is another key issue. Therefore, it is necessary to study the algorithm for solving the models for the above problems.

In conclusion, this paper makes the following contributions. (1) Based on the existing general reverse logistics model, this paper investigates the regional coverage planning problem of the express packaging recycling network, with a focus on the service correlation and system layout between disposal points and recycling stations. A regional coverage planning model is established based on various factors, such as the location of recycling facilities, the refund amount for recycling given to consumers, and government subsidies allocated to recycling enterprises. The principal innovation of this study lies in endogenizing consumer refunds and government subsidies—factors typically treated as exogenous parameters or post-considerations—and incorporating them as synchronized decision variables together with facility location and coverage, thereby jointly determining reverse flow allocation and network feasibility. (2) A fuzzy credibility-based multi-objective decision-making framework is developed for sustainable logistics network design under multi-source heterogeneous uncertainties. The novelty of the proposed framework is threefold: at the parameter level, it provides, for the first time, a uniform representation of both strategic economic parameters (e.g., subsidies, refunds) and operational physical parameters (e.g., costs, quantities) using fuzzy set theory, which better captures the imprecise knowledge inherent in the strategic planning phase; at the modelling level, it reformulates the objective of “maximizing the total recycling rate” in a fuzzy environment as “achieving an optimistic recovery level under credibility constraints,” thereby organically integrating environmental goals with uncertainty management; at the solution level, instead of a single “optimal” solution, the model generates a set of Pareto-optimal solutions that adapt to the decision-maker’s risk preference (i.e., credibility level), offering a spectrum of strategies for practical implementation. (3) A new memetic algorithm (MA) based on dynamic local search is proposed to solve the reverse logistics network, which is easier to implement and understand than other heuristic algorithms.

This article is structured as follows: Section 2 presents a literature review on reverse logistics network problems. Section 3 provides the mathematical model of the reverse logistics network problem studied. Section 4 presents the MA based on dynamic local search. Section 5 reports the numerical experiments, using different scales of data to test the efficiency of the proposed algorithm and conducting sensitivity analyses on changes in the confidence level and $\epsilon$-value. Finally, management insights are presented. Section 6 concludes the paper and provides recommendations for future research.

2. Literature Review

With the steady growth of Internet users and online shopping in China, the volume of express deliveries has grown exponentially in recent years, leading to a major packaging waste problem. Currently, there is no comprehensive recycling plan for express packaging, resulting in a low recycling rate and a large amount of packaging waste. Economically, this leads to a waste of social resources; socially, this leads to environmental pollution. Therefore, the packaging waste recycling problem must be solved quickly.

2.1. Express Packaging Recycling

In terms of empirical analysis, Lin et al. [5] conducted a comprehensive life cycle assessment enhanced by parameterization and global sensitivity analysis on China’s express packaging waste system managing cardboard, paper and envelopes, foam plastics, soft plastics, and woven bag fragments, to quantify the impact and carbon footprint reduction potential. Their global sensitivity analysis identified that the incineration proportion, the heat and power recovery rate in incinerators, and the recycling rate of cardboard are the primary factors contributing to outcome uncertainty. Yang et al. [6] developed a comprehensive conceptual framework to explore citizens’ recycling behaviors, identifying three prevalent patterns: collection station recycling (most common), followed by marketplace equipment recycling and door-to-door recycling. Their results indicate that social sanction, responsibility attribution, and centrality positively predict recycling behavior. Yang et al. [7] investigated the intention to adopt reusable express packaging, finding that perceived government support indirectly influences adoption intention through product engagement, which in turn directly and indirectly affects the recycling of express packaging. Wang et al. [8] examined key factors influencing consumers’ adoption of recycled express packaging, highlighting the interaction between altruistic values and rational considerations. Their results show that rational factors exert a greater influence than altruistic factors in shaping consumer acceptance of recyclable express packaging. Yang et al. [9] proposed an integrated model for assessing the sustainability of express packaging products. Their empirical study demonstrates that reusable express packaging products offer significant sustainability advantages, providing empirical evidence for governmental regulatory frameworks. Yang et al. [10] emphasized the habitual behaviors formed through consumers’ practical experience with disposable express packaging. They developed a theoretical model encapsulating the “social structure-behavior transformation habit construction” to identify intervention strategies for enhancing consumers’ transition responses away from disposable packaging. The aforementioned literature collectively elucidates the diversified empirical research pathways currently targeting China’s express packaging system, ranging from “environmental impact assessment” to “behavioral driving mechanisms.” Key findings include the identification of the recycling rate for critical materials (e.g., cardboard) as a pivotal factor affecting the uncertainty of outcomes like carbon emissions. Furthermore, citizen recycling behavior predominantly manifests as collection station recycling, supplemented by marketplace equipment and door-to-door recycling. Government support policies can indirectly boost adoption intention for reusable packaging by enhancing consumers’ “product engagement,” thereby providing a solid scientific basis for designing more efficient recycling systems.

Building upon the empirical findings concerning citizen recycling behavior and government support policies for express packaging, scholars have increasingly turned to game theory for further investigation. Cheng et al. [11] developed a computational evolutionary game model to analyze stakeholder behaviors in managing urban express packaging waste, focusing on the government, express companies, and consumers. Their results indicate that subsidies accelerate the convergence of both enterprises and consumers towards stable strategies. Li [12] established a tripartite game model involving merchants, logistics enterprises, and consumers to analyze the dynamic evolution of behavioral choices in response to overpackaging issues. Guo et al. [13] constructed an optimization model to evaluate two specific recycling policies: deposit-refund and reward-for-recovery. Their findings highlight the crucial role of consumers’ green awareness in the decision-making process. Lyu et al. [14] investigated four potential scenarios for platform retail collaborating with green packaging manufacturers (resale without recycling, resale with recycling, marketplace without recycling, and marketplace with recycling). Their study concludes that the optimal strategy for recycling and sales models is closely tied to green investment efficiency, recycling efficiency, and commission rates. Sui et al. [15] proposed a tripartite evolutionary game model to examine the interactions among the government, producers, and recyclers within an express packaging recycling network. The results show a negative correlation between recyclers’ disposal fees and subsidy caps, while an increase in the recycling rate corresponds directly to subsidies. Sun et al. [16] analyzed the evolutionary stability strategies of local governments and express enterprises by constructing an evolutionary game model, incorporating the bounded rationality and group behavior of decision-makers, as well as economies and diseconomies of scale in the operational costs of reusable express packaging. Yang et al. [17] examined the potential for resource reuse of express packaging by integrating echelon utilization into their model, testing closed-loop supply chain leadership models and coordination contracts aligned with corporate social responsibility (CSR) policies. Chen et al. [18] developed a game-theoretic model to assess three governance structures: direct third-party recycling, manufacturer-entrusted recycling, and logistics-entrusted recycling. Their findings demonstrate that the cost-sharing structure plays a decisive role in shaping environmental outcomes and stakeholder incentives. Yang et al. [19] highlight that understanding the evolutionary trends of express packaging waste recycling behaviors is crucial for effective waste management. By incorporating information policies and the factor of reference dependence, they constructed an evolutionary game model of express packaging waste recycling behavior to explore group decision-making under varying initial adoption rates. Collectively, this body of literature demonstrates the application of game theory—particularly evolutionary and tripartite games—to analyze the interactions among multiple stakeholders and the effectiveness of policies within express packaging recycling systems. These studies quantitatively elucidate how government policies, market mechanisms (e.g., cost-sharing, commissions), agent characteristics (e.g., green awareness), and business models jointly influence the long-term stable behavioral strategies of governments, enterprises, and consumers. This provides a rigorous theoretical foundation for designing incentive-compatible, self-enforcing recycling policies and market mechanisms.

The above studies simply added courier packaging recycling to the traditional logistics network model, assuming that all packaging would be recycled, but without considering the recycling rate. However, the ‘Report on the Current Situation and Trend of Green Packaging in China’s Courier Industry’ highlights that courier packaging waste accounts for 90% of the increase in household garbage, while the overall recycling rate is less than 10%. Specifically, the recycling rate of cardboard boxes does not exceed 50%, while that of packaging bags and internal buffers is basically zero [20]. Therefore, the design of reverse logistics networks for courier packaging should focus on improving the recycling rate of waste.

2.2. Reverse Logistics Network and Recycling Rates

Currently, few studies examine the reverse logistics network for courier packaging recycling. Wang et al. [21] designed an optimal collaboration strategy by addressing the synchronized pickup and delivery problem for ecological packaging. This strategy aims to minimize the total operational cost by establishing collaborative coalitions and efficiently allocating trucking resources based on time-space network properties. Shi et al. [22] studied the joint optimisation of courier packaging delivery and recyclable packaging collection to fairly distribute vehicle working time and operational costs. They developed a dual-objective mixed integer linear programming model, allowing a more feasible allocation for facility location and route planning. Zhou et al. [23] designed a multi-level reverse logistics network to collect environmentally friendly packages from customers, to solve the capacity–location–path problem in such networks using customer incentive mechanisms. Shi et al. [24] designed a courier packaging waste recycling network based on urban symbiosis strategies, including reuse, recycling, and replacement, for the treatment of courier packaging waste. By combining logistics analysis and optimisation methods, they established a multi-warehouse collaborative optimisation model. Shi et al. [25] provided a realistic simulation optimisation system to establish an efficient, green, and economical recycling logistics network through integrated optimisation methods and agent-based techniques. The planned solutions included decisions regarding facility location, service planning, and vehicle routes.

Few studies have investigated the design of reverse logistics networks based on recycling rates. Wang et al. [26] considered the issue of electric bicycle battery recycling and developed a model based on ‘Internet +’, showing that through the government’s reward and punishment mechanism, the waste battery recycling rate could increase by 2.59%. Trochu et al. [27] examined the uncertainty of the quality, quantity, and recycling rate of collected materials, to maximise expected profits and minimise landfill volume. They proposed a two-stage stochastic model to design an eco-efficient reverse logistics network to promote the recycling of more recyclable materials. Pan et al. [28] considered the objective conflicts between different stakeholders and the dynamic decision-making environment of the construction industry and developed a multi-period multi-objective mixed integer linear programming model to minimise construction and demolition waste disposal costs for contractors and maximise the profits of recycling companies and the recycling rate of construction and demolition waste. The above studies also considered the waste recycling rate in the reverse logistics network, calculated from two recycling rates or a variable representation and using simple random quantities to represent its uncertainty. However, in reality, the recycling rate is affected by multiple factors, such as the location of recovery facilities, the refund amount for recycling given to consumers, and government subsidies allocated to recycling enterprises. The impact of these factors on the recycling rate should be considered.

2.3. Uncertainty

Reverse logistics carries a high degree of uncertainty due to its intrinsic characteristics, making designing its network difficult. Therefore, uncertainty management constitutes an important research direction in the design of reverse logistics networks. For instance, Trochu et al. [29] pointed out that the quantity and recycling rate of materials were the main uncertainties and proposed a sample average approximation method to solve the proposed stochastic model, conducting sensitivity analyses on random parameters such as the quantity of recovered materials and the sample size. Kuşakcı et al. [30] considered the high uncertainty of return quantities and established a fuzzy mixed integer model for the reverse logistics network of scrapped vehicles in line with current Turkish regulations. Azizi et al. [31] addressed the multi-period reverse logistics network design problem with return and demand uncertainty and proposed a two-stage stochastic planning model for reverse logistics networks. Although the above studies established stochastic models for reverse logistics networks under conditions of uncertainty, their models integrate processes such as recovery, disassembly, and remanufacturing, but do not establish specific solutions for the most difficult and uncertain recovery process in reverse logistics.

2.4. Heuristic Algorithm

The logistics network problem involving site selection is an NP-hard problem [32,33], which is difficult to solve. The genetic algorithm (GA) is a heuristic algorithm commonly used in fields such as combinatorial optimisation, machine learning, signal processing, and adaptive control [34,35]. However, a simple GA usually lacks the ability to effectively search for solutions. Moscato and Norman [36] were the first to propose the concept of MA, a bionic intelligent algorithm, by integrating neighbourhood search into the GA. The MA combines the evolution of the biological and social layers and uses genes and memes as coding units of evolutionary information. This algorithm is widely used in various optimisation problems, especially to find approximate solutions to NP-hard problems [37,38,39,40,41]. Therefore, we propose an MA with dynamic local search (LSMA).

This study integrates the recovery rate of express packaging into the design of reverse logistics networks, thereby transforming environmental performance objectives from traditional qualitative constraints into quantifiable optimization targets. In response to the characteristics of the strategic planning phase—where historical data for key parameters (e.g., costs, subsidies, recovery volumes) is scarce and reliant on subjective judgment—a fuzzy mixed-integer programming model is constructed with the dual objectives of minimizing total cost and maximizing the total recovery rate. Cognitive uncertainty is addressed using credibility theory, and the ε-constraint method is employed to convert the bi-objective problem into a solvable single-objective formulation. Furthermore, a memetic algorithm with dynamic local search (LSMA) is designed for efficient solution. As summarized in

Table 1. Summary of the literature review.

Paper	Fuzzy Variable	Random Variable	Robust Optimization	Recycling Rate	Additional Recycling Fee	Government Subsidies	Reverse Logistics	Forward Logistics	Metaheuristic Method
Wang et al. [21]	No	No	No	No	No	No	Yes	Yes	Yes
Shi et al. [22]	No	No	No	No	No	No	Yes	Yes	Yes
Zhou et al. [23]	No	No	No	Yes	No	No	Yes	No	Yes
Shi et al. [24]	No	No	No	No	No	No	Yes	Yes	Yes
Shi et al. [25]	No	No	No	No	No	No	Yes	No	Yes
Wang et al. [26]	No	No	No	No	No	No	Yes	No	Yes
Trochu et al. [27]	No	No	No	Yes	No	No	Yes	Yes	Yes
Pan et al. [28]	No	No	No	No	No	No	Yes	Yes	Yes
Trochu et al. [29]	No	Yes	No	No	No	No	Yes	No	Yes
Kuşakcı et al. [30]	Yes	No	No	No	No	No	Yes	Yes	Yes
Azizi et al. [31]	No	Yes	No	No	No	No	Yes	Yes	Yes
This research	Yes	No	No	Yes	Yes	Yes	Yes	No	Yes

, existing research predominantly focuses on deterministic or stochastic optimization, often treating consumer refunds and government subsidies as exogenous parameters. While some literature addresses network design, it mainly employs mixed-integer linear programming models and employs heuristic methods like genetic algorithms for solving. However, most studies fail to systematically incorporate the “behavioral-policy” type of cognitive uncertainty and the endogenous, coordinated decision-making of economic incentive parameters into the reverse logistics network design framework. This omission may cause a disconnect between the model and the actual information structure of strategic decision-making, making it difficult to provide robust and resilient planning schemes under conditions of data insufficiency.

In conclusion, based on the traditional logistics network planning model, this paper takes into account the recycling process of courier packaging between disposal points and recycling stations. Considering factors such as the location of recycling facilities, the refund amount for recycling given to consumers, and government subsidies allocated to recycling enterprises, under the influence of uncertain parameters, this paper uses the fuzzy programming method to establish a fuzzy reverse logistics network planning model for courier packaging to minimise the total cost and maximise the total recycling rate.

3. Problem Description and Uncertainty Modelling

3.1. Problem Definition

This paper addresses the problem of planning the construction of a reverse logistics network for express packaging by a third-party logistics company (Figure 1). This network involves disposal points and candidate recycling stations. The disposal points are fixed locations for collecting packaging after its removal by recipients and can be located in convenient locations such as intelligent express cabinets in residential areas or schools. The candidate recycling stations can have different scale levels, each with a certain coverage range. Disposal points can only be served by recycling stations within their coverage range. Additionally, recycling stations have a fixed recycling rate, but thanks to additional return fees given by enterprises (used as a reward for the integrity of express packaging) and government subsidies (to increase the enthusiasm of recycling stations), their recycling rate will increase. Therefore, the recycling process of the reverse logistics network studied in this paper is as follows: after the collection of discarded express packaging by a disposal point, a recycling station within its coverage range collects this packaging.

The model has two objectives: (1) minimising the total cost of the recovery process, which includes the construction costs of recycling stations, the operating costs of discarded express packaging, transportation costs, recovery costs, and government subsidies allocated to recycling stations (contributing to cost reduction); and (2) maximising the total recycling rate. Due to limited resources, their use must be optimised, so recycling enterprises must increase their total recycling rate, calculated as total quantity recovered/total quantity discarded.

This article focuses on the strategic layer design of reverse logistics network for express packaging. As shown in Figure 1, the recycling process involves the link from the disposal point to the recycling station. However, a complete model needs to balance complexity and practicality. In order to further explore the fundamental relationship between facility site selection, incentive policies, and network coverage efficiency, this article defines the scope of the model as follows: the core decision variables of the model are the site selection of recycling stations, the disposal points served by each recycling station (i.e., coverage relationship), and the corresponding unified collection and government subsidy amounts. This is an extension of a typical maximum-coverage site selection problem; The abstract treatment of the transportation process in this article recognizes the importance of transportation between disposal points and recycling stations but does not optimize it as an operational problem with variables such as path and frequency. On the contrary, the transportation process is reasonably abstracted as the unit transportation cost between two points, which is included in the total cost function. At the same time, this problem reflects the limitation of transportation feasibility by setting a maximum service distance constraint (i.e., only when the distance between the disposal point and the candidate recycling station is less than or equal to the maximum service distance is a coverage relationship allowed to be established). This abstraction is a common method in strategic planning, which allows us to evaluate the overall economic and environmental benefits of different network layouts and policy combinations without getting too caught up in operational details. Based on the above definition, the model in this article aims to answer a more strategic question: “Where should recycling stations be built, which areas should be covered, and what incentive policies should be implemented to maximize the recycling benefits of express packaging under limited budget?” rather than “How should vehicles be arranged for collection” at the operational level.

In reverse logistics, parameters such as recycling rates, processing costs, and redistribution demand are often difficult to characterize with precise historical data or probability distributions due to fluctuations in consumer behavior, variability in packaging conditions, and market dynamics. The associated uncertainty tends to reflect cognitive fuzziness (e.g., “the recycling rate is approximately 30–50%”) rather than objective randomness. Fuzzy programming addresses this type of subjective and imprecise semantic knowledge directly through membership functions, eliminating the need for extensive statistical data or assumed probability distributions, thereby circumventing the stringent data quality and modeling requirements of stochastic programming. Moreover, by incorporating flexible constraints that accommodate soft information, fuzzy programming offers a more balanced approach to economic feasibility and practicality compared to the overly conservative worst-case assumptions of robust optimization. Consequently, it achieves satisficing solutions rather than strictly optimal ones through adjustable membership degrees, making it particularly suitable for handling the multi-source, unstructured, and fuzzy uncertainties inherent in reverse logistics systems. Therefore, this paper assumes that some parameters are uncertain and uses gradient fuzzy numbers to represent changes in these parameters.

In summary, this paper aims to minimise the total cost and maximise the total recycling rate, taking into account uncertain parameters such as construction costs, operating costs, transportation costs, return fees, government subsidies, facility capacity, and the quantity of express packaging discarded at disposal points. As a result, the location of recycling stations in the reverse logistics network and the selection of logistics routes between disposal points and recycling stations are determined based on credibility theory in fuzzy mathematics. A fuzzy planning model for the reverse logistics network is then constructed.

3.2. Model Assumptions and Symbols

The assumptions made in this article are as follows. (1) There are several types of packaging materials used in a recycling cycle, such as cardboard boxes, plastic bags, woven bags, and internal buffers; (2) the number of facilities and their processing capacity are limited; (3) recycling stations have a base recycling rate, which may increase; and (4) each recycling station has a specific scale level, and additional collection fees and government subsidies are provided. The symbols required for the model in this article can be defined as follows (

Table 2. Definition of symbol.

Symbols	Description
Sets
$I$	Set of disposal points, $i\in I=\left\{1,2,\cdots ,NI\right\};$
$J$	Set of candidate recycling bins, $j\in J=\left\{1,2,\cdots ,NJ\right\};$
$L$	Set of levels for the candidate recycling bins, $l\in L=\left\{1,2,\cdots ,NL\right\};$
$H$	Set of courier packaging categories, $h\in H=\left\{1,2,3,4\right\},$ with categories 1 to 4 representing cardboard boxes, plastic bags, woven bags, and internal buffers, respectively.
Parameters
${\tilde{f}}_{lj}$	Fuzzy construction cost of recycling station $j$ of scale level $l,$ in RMB 10,000;
${\tilde{O}}_{hj}$	Fuzzy operational cost of express packaging h in recycling station j, in tons per CNY;
$\widetilde{ct}$	Fuzzy transportation cost per unit distance, in kilometres per RMB;
$\widetilde{cb}$	Fuzzy additional recycling fees provided by enterprises to recycling stations, in tons per RMB;
$\widetilde{cs}$	Fuzzy subsidy granted by the government to recycling stations, in tons per RMB;
${\tilde{q}}_{hi}$	Fuzzy discarded quantity of express packaging $h$ at disposal point $i,$ in tons;
$R_l$	Maximum coverage radius of recycling stations with capacity level $l,$ in kilometers;
$r_0^l$	baseline recycling rate within the central coverage radius of disposal points for recycling stations with capacity level $l,$ in percentage;
$r_1$	Incremental recycling rate improvement attributed to the additional recycling fees provided by enterprises, in percentage;
$r_2$	Incremental recycling rate improvement resulting from government subsidies, in percentage;
${\tilde{C}}_{lj}$	Fuzzy capacity of recycling station $j$ of scale level $l,$ in tons; $t_{ij}$ represents the transportation distance between disposal point $i$ and recycling station $j,$ in kilometres;
$g_{lij}$	Binary variable indicating whether disposal point $i$ falls within the coverage area of recycling station $j$ with capacity level $l,$ defined by radius $R_l,$ which takes a value of 1 if $t_{ij}\le R_l,\forall i\in I,j\in J,l\in L,$ and 0 otherwise.
Decision variables
$X_{lj}$	Binary variable set to 1 if recycling station $j$ of scale level $l$ is selected, and otherwise 0.
$Y_{lij}$	Binary variable set to 1 if disposal point $i$ is assigned to recycling station $j$ of scale level $l,$ and otherwise 0.
$Z_{ij}^1$	Binary variable set to 1 if the enterprise provides an additional return fee for courier packaging collected at disposal point $i$ and recycled by recycling station $j,$ and otherwise 0.
$Z_{ij}^2$	Binary variable set to 1 if the government grants a subsidy to recycling station $j,$ which recycles courier packaging from disposal point $i,$ and otherwise 0.
Process variable
${\eta }_{ij}$	Recycling rate of recycling station $j$ for disposal point $i.$ Specifically, ${\eta }_{ij}=\sum _{l\in L}{r}_0^l{g}_{lij}{Y}_{lij}+r_1{Z}_{ij}^1+r_2{Z}_{ij}^2,\forall i\in I,j\in J.$

).

3.3. Objective Function and Constraints of the Fuzzy Optimisation Model

In the planning of reverse logistics networks at the strategic level, key parameters such as government subsidies and consumer subsidies have significant forward-looking uncertainty in improving recycling rates. This type of uncertainty arises from policies that have not yet been finalized and markets that have not yet formed stable behavior patterns, so the information that can be relied upon in decision-making is often incomplete, semantic, and based on expert experience. Traditionally, probability theory has been a powerful tool for dealing with uncertainty, but its effectiveness relies on sufficient historical data to construct accurate probability distributions. For this research question, such data is often difficult to obtain. In contrast, fuzzy set theory provides a more natural framework for dealing with uncertainty based on subjective judgments, linguistic information, and cognitive imprecision. It characterizes the credibility of a parameter belonging to a “possible value” through membership functions, rather than its frequency of occurrence.

Considering the fuzziness of some parameters in the reverse logistics network design problem proposed in this paper, a fuzzy planning model based on credibility [42] is established. By combining the expected value method (used in the objective function) and the chance-constrained programming method (used in the constraints), the following fuzzy chance-constrained programming model (M1) based on credibility is established.

According to the problem description, the objective functions of the model are as follows:

(1)

Minimisation of the total cost

$$\begin{array}{cc}\hfill \min E\left[w_1\right]={\displaystyle \sum _{l\in L}}& {\displaystyle \sum _{j\in J}}E\left[{\tilde{f}}_{lj}\right]X_{lj}+{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}E\left[{\tilde{O}}_{hj}\right]\times E\left[{\tilde{q}}_{hi}\right]{\eta }_{ij}\hfill \\ & +E\left[\widetilde{ct}\right]{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}{t}_{ij}E\left[{\tilde{q}}_{hi}\right]{\eta }_{ij}+E\left[\widetilde{cb}\right]{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}E\left[{\tilde{q}}_{hi}\right]r_1{Z}_{ij}^1\hfill \\ & -E\left[\widetilde{cs}\right]{\displaystyle \sum _h}{\displaystyle \sum _i}{\displaystyle \sum _j}E\left[{\tilde{q}}_{hi}\right]r_2{Z}_{ij}^2\hfill \end{array}$$

(1)

(2)

Maximisation of the total recycling rate

$$\max E\left[w_2\right]={\displaystyle \frac{\sum _h\sum _i\sum _{j}E\left[{\tilde{q}}_{hi}\right]{\eta }_{ij}}{\sum _h\sum _i\sum _{j}E\left[{\tilde{q}}_{hi}\right]}}$$

(2)

According to the model’s assumptions, the decision variables of the objective functions must satisfy the following constraints:

$$\sum _l\sum _j{g}_{lij}{Y}_{lij}=1, \hspace{1em}\forall i\in I$$

(3)

$$Y_{lij}\le g_{lij}, \hspace{1em}\forall i\in I,j\in J,l\in L$$

(4)

$$\mathrm{C}\mathrm{r}\left\{\sum _{h\in H}\sum _{i\in I}{\tilde{q}}_{hi}{\eta }_{ij}\le \sum _{l\in L}{\tilde{C}}_{lj}{X}_{lj}\right\}\ge {\lambda }_j, \hspace{1em}\forall j\in J$$

(5)

$$\sum _{l\in L}{X}_{lj}\le 1, \hspace{1em}\forall j\in J$$

(6)

$$Y_{lij}\le X_{lj}, \hspace{1em}\forall i\in I,j\in J,l\in L$$

(7)

$$Z_{ij}^1\le \sum _{l\in L}{Y}_{lij}, \hspace{1em}\forall i\in I,j\in J$$

(8)

$$Z_{ij}^2\le \sum _{l\in L}{Y}_{lij}, \hspace{1em}\forall i\in I,j\in J$$

(9)

$$Z_{ij}^2\le Z_{ij}^1, \hspace{1em}\forall i\in I,j\in J$$

(10)

$$X_{lj},Y_{lij},Z_{ij}^1,Z_{ij}^2, \hspace{1em}\forall i\in I,j\in J,l\in L$$

(11)

Equations (3) and (4) indicate that each disposal point can only choose one level of recycling station; Equation (5) states that the amount of packaging waste collected by the recycling station cannot exceed its capacity and the credibility of this cannot be less than ${\lambda }_j\in \left[0, 1\right]$; Equation (6) indicates that each recycling station can only choose a certain level; Equation (7) states that only established recycling stations can carry out the recycling of waste collected at disposal points; Equations (8) and (9) indicate that recycling stations can only accept waste from disposal points if they recycle it; Equation (10) indicates that enterprises can only accept government subsidies after paying additional return fees; and Equation (11) restricts the values of the variables to 0 or 1.

3.4. Defuzzified Model Formulation

3.4.1. Transformation of Clear Equivalent Models

$\mu \left(x\right)$ is the membership function of fuzzy variable $\tilde{\xi }$, and $\tau$ is a real number. Then, the credibility measure [42] is

$$Cr\left\{\tilde{\xi }\le \tau \right\}={\displaystyle \frac{1}{2}}\left(\underset{x\le \tau }{\mathit{sup}} \mu \left(x\right)+1-\underset{x>\tau }{\mathit{sup}} \mu \left(x\right)\right)$$

(12)

The expected value of fuzzy variable $\tilde{\xi }$ based on the credibility measure is

$$E\left[\tilde{\xi }\right]={\int }_0^{+\infty }Cr\left\{\tilde{\xi }\ge \tau \right\}d\tau -{\int }_{-\infty }^{0}Cr\left\{\tilde{\xi }\le \tau \right\}d\tau$$

(13)

Suppose that $\tilde{\xi }$ is a trapezoidal fuzzy number, then the quadruple formed by its crisp numbers can be expressed as $\tilde{\xi }=\left({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4\right),{\tau }_1\le {\tau }_2\le {\tau }_3\le {\tau }_4$. According to Equation (12), the expected value of $\tilde{\xi }$ is $E\left[\tilde{\xi }\right]=\left({\tau }_1+{\tau }_2+{\tau }_3+{\tau }_4\right)/4$, and the corresponding credibility measure is

$$Cr\left\{\tilde{\xi }\le \tau \right\}=\left\{\begin{array}{c}0, \tau \in \left(\left.-\infty ,{\tau }_1\right]\right.\\ {\displaystyle \frac{\tau -{\tau }_1}{2\left({\tau }_2-{\tau }_1\right)}}, \tau \in \left(\left.{\tau }_1,{\tau }_2\right]\right.\\ {\displaystyle \frac{1}{2}}, \tau \in \left(\left.{\tau }_2,{\tau }_3\right]\right.\\ {\displaystyle \frac{\tau -2{\tau }_3+{\tau }_4}{2\left({\tau }_4-{\tau }_3\right)}}, \tau \in \left(\left.{\tau }_3,{\tau }_4\right]\right.\\ 1, \tau \in \left({\tau }_4,+\infty \right)\end{array}\right.$$

(14)

$$Cr\left\{\tilde{\xi }\ge \tau \right\}=\left\{\begin{array}{c}1, \tau \in \left(\left.-\infty ,{\tau }_1\right]\right.\\ {\displaystyle \frac{2{\tau }_2-{\tau }_1-\tau }{2\left({\tau }_2-{\tau }_1\right)}}, \tau \in \left(\left.{\tau }_1,{\tau }_2\right]\right.\\ {\displaystyle \frac{1}{2}}, \tau \in \left(\left.{\tau }_2,{\tau }_3\right]\right.\\ {\displaystyle \frac{{\tau }_4-\tau }{2\left({\tau }_4-{\tau }_3\right)}}, \tau \in \left(\left.{\tau }_3,{\tau }_4\right]\right.\\ 0, \tau \in \left({\tau }_4,+\infty \right)\end{array}\right.$$

(15)

According to Equations (14) and (15), the following theorem holds:

If $\tilde{\xi }$ is a trapezoidal fuzzy number, with $\tilde{\xi }=\left({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4\right)$ and ${\tau }_1\le {\tau }_2\le {\tau }_3\le {\tau }_4$, for a given confidence level $\alpha$ and $\in \left(\left.0.5, 1\right]\right.$, then

$$Cr\left\{\tilde{\xi }\le \tau \right\}\ge \alpha ⟺\tau \ge 2\left(1-\alpha \right){\tau }_3+\left(2\alpha -1\right){\tau }_4$$

(16)

$$Cr\left\{\tilde{\xi }\ge \tau \right\}\ge \alpha ⟺\tau \le \left(2\alpha -1\right){\tau }_3+2\left(1-\alpha \right){\tau }_2$$

(17)

From Equations (16) and (17), along with the expected value of the trapezoidal fuzzy number, model M2, a clear equivalent model to model M1 can be obtained:

$$\begin{array}{cc}\hfill \min E\left[w_1\right]={\displaystyle \sum _{l\in L}}& {\displaystyle \sum _{j\in J}}{\displaystyle \frac{f_{lj}^1+f_{lj}^2+f_{lj}^3+f_{lj}^4}{4}}{X}_{lj}\hfill \\ & +{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}{\displaystyle \frac{Q_{hj}^1+Q_{hj}^2+Q_{hj}^3+Q_{hj}^4}{4}}\times {\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{\eta }_{ij}\hfill \\ & +{\displaystyle \frac{{ct}_1+{ct}_2+{ct}_3+{ct}_4}{4}}{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}{t}_{ij}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{\eta }_{ij}\hfill \\ & +{\displaystyle \frac{{cb}_1+{cb}_2+{cb}_3+{cb}_4}{4}}{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{r}_1{Z}_{ij}^1\hfill \\ & -{\displaystyle \frac{{cs}_1+{cs}_2+{cs}_3+{cs}_4}{4}}{\displaystyle \sum _h}{\displaystyle \sum _i}{\displaystyle \sum _j}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{r}_2{Z}_{ij}^2\hfill \end{array}$$

(18)

$$\min E\left[w_2\right]={\displaystyle \frac{\sum _{h\in H}\sum _{i\in I}\sum _{j\in J}\frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}{\eta }_{ij}}{\sum _{i\in I}\sum _{j\in J}\frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}}$$

(19)

s.t.

$$\begin{array}{c}{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}\left[2\left(1-{\lambda }_j\right)q_{hi}^3+\left(2{\lambda }_j-1\right)q_{hi}^4\right]{\eta }_{ij}\le {\displaystyle \sum _{l\in L}}\left[\left(2{\lambda }_j-1\right)C_{li}^1+2\left(1-{\lambda }_j\right)C_{li}^2\right]X_{lj},\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \forall j\in J\hfill \end{array}$$

(20)

and Equations (3), (4), and (6)–(11).

For model M2, the confidence level of the chance constraint satisfies $0.5\le {\lambda }_j\le 1,\forall j\in J$.

3.4.2. Transformation of the Single-Objective Model

Although M2 is already an integer linear programming model, the presence of multiple objectives makes it difficult to solve. This paper linearises the model using the $\epsilon$-value constraint method, which allows us to obtain the solution and its Pareto solution. According to the method for solving multi-objective problems, the most important objective is taken as the main objective, while all other objectives are considered as constraints [43]. In this model, the total cost is considered the main objective, while objective function (2) (maximising the total recycling rate) becomes a constraint condition of the new model. Then, M2 becomes M3, a single-objective integer linear programming model that can be solved directly:

$$\begin{array}{cc}\hfill \min E\left[w_1\right]={\displaystyle \sum _{l\in L}}& {\displaystyle \sum _{j\in J}}{\displaystyle \frac{f_{lj}^1+f_{lj}^2+f_{lj}^3+f_{lj}^4}{4}}{X}_{lj}\hfill \\ & +{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}{\displaystyle \frac{Q_{hj}^1+Q_{hj}^2+Q_{hj}^3+Q_{hj}^4}{4}}\times {\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{\eta }_{ij}\hfill \\ & +{\displaystyle \frac{{ct}_1+{ct}_2+{ct}_3+{ct}_4}{4}}{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}{t}_{ij}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{\eta }_{ij}\hfill \\ & +{\displaystyle \frac{{cb}_1+{cb}_2+{cb}_3+{cb}_4}{4}}{\displaystyle \sum _{h\in H}}{\displaystyle \sum _{i\in I}}{\displaystyle \sum _{j\in J}}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{r}_1{Z}_{ij}^1\hfill \\ & -{\displaystyle \frac{{cs}_1+{cs}_2+{cs}_3+{cs}_4}{4}}{\displaystyle \sum _h}{\displaystyle \sum _i}{\displaystyle \sum _j}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{r}_2{Z}_{ij}^2\hfill \end{array}$$

(21)

s.t.

$$\sum _{h\in H}\sum _{i\in I}\sum _{j\in J}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}{\eta }_{ij}\ge \epsilon \sum _{i\in I}\sum _{j\in J}{\displaystyle \frac{q_{hi}^1+q_{hi}^2+q_{hi}^3+q_{hi}^4}{4}}$$

(22)

and Equations (3), (4), (6)–(11), and (20).

By modifying the constraint on the minimum value on the right side of objective function (2), an effective solution for this model can be obtained. Decision-makers can then make decisions based on all solution results [43].

4. Solution Method

As M3 is difficult to solve directly using solvers such as LINGO, CPLEX, and GAMS in large-scale problems, this paper proposes an MA based on dynamic local search to solve M3.

4.1. Encoding and Decoding Operation

This paper adopts a priority encoding method. The advantage of this method is that it does not require any repair operations for encoding during the subsequent search process [28]. This paper modifies the original priority encoding method [44] to make it applicable to the focal problem. The specific encoding operations are as follows (Algorithm 1). The symbols here are used to illustrate the process of the program and are independent of the symbols used in the model of this article:

Algorithm 1 Encoding operation

Input:

$M$: the total number of disposal points $i$ belonging to the set, $i\in I=\left\{1,2,\cdots ,M\right\}$;

$N$: the total number of items in recycling bin $j$, $j\in J=\left\{1,2,\cdots ,N\right\}$;

$P$: the total number of levels in the recycling bin.

Output:

${level}_j$: the level of the recycling bin;

$B_{i,2}$: represents a 0–1 matrix with $i$ rows and 2 columns. The first column indicates whether an additional handling fee is provided, and the second column indicates whether a government subsidy is given.

$v_{ij}$: the priority number of the two nodes.

For $j=1,2,\cdots ,N$

Step 1: ${level}_j=$ the integer between $\left[1,P\right]$;

End

Step 2: Each element in matrix $B_{i,2}$ is randomly assigned a value of 0 or 1 with a 50% probability.

Step 3: Randomly arrange the numbers from 1 to $M\times N$ on the empty matrix $R_{ij}$ to obtain an $M$-row $N$-column matrix.

Step 4: let $v_{ij}=R_{ij}$.

Based on the above encoding operation, if $M$ is 4, $N$ is 5, and $P$ is 3, then the individual code can be expressed as follows:

In

Table 3. A Feasible Individual Code.

		$level$
$B_{i,2}$	$v_{ij}$	2	1	2	3	1
1	1	9	13	7	5	12
1	0	17	1	16	18	3
0	0	6	19	2	10	20
1	0	11	8	14	4	15

, the numbers in row 2 and columns 3 to 7 represent ${level}_j$, the numbers in rows 3 to 6 and columns 1 and 2 represent $B_{i,2}$, and the numbers in rows 3 to 6 and columns 3 to 7 represent $v_{ij}$.

Based on the encoded information obtained and combined with the characteristics of M2, the specific steps of the decoding operation are as follows (Algorithm 2):

Algorithm 2 Encoding operation

Input:

$i$: Set of disposal points;

$j$: Set of recycling bins;

$b_i$: Demand quantity for point $i$, $\forall i\in I$;

$a_j$: The maximum capacity of recycling bin $j$, $\forall j\in J$;

$g_{ij}$: A 0–1 binary matrix that determines whether disposal point $i$ is within the coverage area of recycling station $j$;

$v_{ij}$: The priority number of the two nodes.

Output:

$x_{ij}$: The transportation volume between disposal point $i$ and recycling station $j$.

Step 1: let $x_{ij}=0,\forall i\in I,j\in J$ While $\sum _{i\in I}{b}_i>\mathbf{0}$

For $i\in I,j\in J$

Step 2: If $g_{ij}=0$ then $v_{ij}=0$

End

Step 3: $\left(i^{*},j^{*}\right)=arg max\left\{v_{ij},i\in I,j\in J\right\},\forall i^{*}\in I,j^{*}\in J$

Step 4: If $a_{j^{*}}<b_{i^{*}}$ then $v_{i^{*}{j}^{*}}=0$, and return to Step 3

Step 4: $x_{i^{*}{j}^{*}}=min\left\{a_{j^{*}},b_{i^{*}}\right\}$ Update the demand quantity and maximum capacity: $a_{j^{*}}=a_{j^{*}}-x_{i^{*}{j}^{*}}$, $b_{i^{*}}=b_{i^{*}}-x_{i^{*}{j}^{*}}$

End

4.2. Fitness and Local Search Selection Operations

In the MA population, as a fitness function is needed to represent the superiority or inferiority of the evolutionary direction of each cultural gene, it is necessary to evaluate the fitness of each cultural gene by calculation. The fitness calculation method is the reciprocal of the target function value proposed by the model (Equation (20)).

When choosing the local search operation, the roulette wheel method is applied because it is the simplest and most commonly used. Its basic idea is that the probability of each individual being selected is proportional to its fitness value. That is, the greater the fitness of an individual, the greater the probability that it will be selected. After each roulette selection, the population size doubles. Therefore, the cultural gene with the highest fitness in the offspring should be selected for the subsequent crossover operation to ensure that the population size remains unchanged.

4.3. Crossover Operation

Using the single-point crossover method:

(1)

Generate a random integer $d$ in the interval [1, 3] to determine which matrix (i.e., ${level}_j$, $B_{i,2}$, or $v_{ij}$) will be used for individual exchange. For example, if $d=3$

Parent individual 1:

		$level$
$B_{i,2}$	$v_{ij}$	2	1	2	3	1
1	1	9	13	7	5	12
1	0	17	1	16	18	3
0	0	6	19	2	10	20
1	0	11	8	14	4	15

Parent individual 2:

		$level$
$B_{i,2}$	$v_{ij}$	3	2	1	2	3
0	0	20	2	11	5	4
1	1	8	6	14	19	3
1	1	1	18	13	15	9
0	0	7	17	16	12	10

(2)

After crossing, we obtain

Offspring individual 1:

		$level$
$B_{i,2}$	$v_{ij}$	2	1	2	3	1
1	1	20	2	11	5	4
1	0	8	6	14	19	3
0	0	1	18	13	15	9
1	0	7	17	16	12	10

Offspring individual 2:

		$level$
$B_{i,2}$	$v_{ij}$	3	2	1	2	3
0	0	9	13	7	5	12
1	1	17	1	16	18	3
1	1	6	19	2	10	20
0	0	11	8	14	4	15

This operation does not have any crossover probability. Therefore, any two parent cultural genes can undergo this crossover operation. Furthermore, based on the fitness values of the two resulting offspring individuals, a neighbourhood search will be performed to determine whether to search for the best solution.

4.4. Dynamic Neighbourhood Search

Select the sub-cultural gene with the best fitness value for neighbourhood search to obtain a more optimised result. This operation can be performed on a specific cultural gene based on a probability calculation (set to 70% in this paper). There are three neighbourhood search methods: 2-opt, 3-opt, and capacity adjustment. The neighbourhood search process concerns the $v_{ij}$ matrix. The 2-opt process goes as follows: First, generate a random integer in the range [1, 8] and call it $r$. Next, randomly select two numbers from the second layer matrix of the $r$-th section and swap their positions. Similarly, 3-opt randomly selects three numbers from the second layer matrix of the $r$-th section and swaps their positions, so that none of these numbers are in their original position.

Capacity adjustment is applied to $level$ and ${\mathrm{B}}_{\mathrm{i},2}$. For $level$, with a 50% probability, randomly select two elements and swap them, or randomly select one element from $level$ and replace it with another number; for $B_{i,2}$, with a 50% probability, randomly select two elements and swap them, or randomly select one element from $B_{i,2}$ and replace it with another number (e.g., replacing 0 with 1 or 1 with 0).

The dynamic neighbourhood search designed in this paper is mainly used to obtain a large number of individuals during a single iteration and accelerate the improvement direction of these individuals. In the next iteration, it will explore more carefully and obtain better individuals. Here is the dynamic neighbourhood search process (Algorithm 3):

Algorithm 3 Dynamic neighbourhood search

With a 75% probability, the following steps will be executed:

For $i=1 to (10+k/8)$

Step 1: If $i<4$ apply 3-opt neighbourhood search Step 2: Else apply 2-opt neighbourhood search

Step 3:   For $j=1 to (4\ast (N/2)+k/8)$

use capacity adjustment neighbourhood search

Step 4:    If the fitness value of the new neighbourhood cultural gene is superior to that of the current cultural gene, then the new neighbourhood cultural gene will be used to replace the current one. Step 5:    Else with a probability of (0.05*$pw$), use the newly obtained neighborhood cultural genes to replace the current cultural genes.

End

End

Here, $k$ represents the current iteration number, $pw = (tun - k)/(tun \ast 3)$, $tun$ is the total number of iterations, and $N$ is the maximum number of levels.

In summary, the LSMA steps are as follows (Algorithm 4):

Algorithm 4 LSMA Algorithm

Step 1: Initialisation Randomly generate 2*(pop_size) cultural genes

Step 2: Calculate the fitness of each individual

Step 3: Selection operation

Step 4: Crossover operation

Step 5: Dynamic neighbourhood search

Step 6: End Determine whether the current number of iterations is greater than the maximum number of iterations. If so, output this value; otherwise, return to Step 2.

5. Numerical Experiments

5.1. Input Data Generation

To evaluate the performance of the LSMA proposed in Section 4 in solving the reverse logistics network design problem for express packaging, this algorithm can be compared with the HGA used in [45]. For consistency, the HGA is applied to the encoding and decoding method used in this paper. Additionally, the LSMA and the CPLEX12.8 solver are used to solve M3, and their results are then compared. The codes for the LSMA and HGA are written using MATLAB7.0 software. The operating environments for MATLAB7.0 and CPLEX12.8 are a desktop computer with an Intel^(R) Core^(TM) i5-6500 processor (Dell Inc. Shanghai, China) at 3.20 GHz, 4 GB of memory, and the Windows 7 (64 bit) operating system. As shown in

Table 4. Size of Test Questions (unit: number).

Test Question	Disposal Point	Recycling Bin	Recycling Bin Level	Express Packaging Category
1	30	10	4	4
2	60	20	4	4
3	120	40	4	4
4	240	80	4	4
5	480	160	4	4
6	720	240	4	4
7	960	320	4	4

and

Table 5. Range of Parameter Values in Test Questions.

Parameter	Range	Parameter	Range
${\tilde{f}}_{lj}$	Uniform (20,000, 50,000)	$R_l$	Uniform (30, 50)
${\tilde{O}}_{hj}$	Uniform (2, 6)	$r_0^l$	Uniform (0.4, 0.55)
$\widetilde{ct}$	Uniform (2, 5)	$r_1$	Uniform (0.25, 0.3)
$\widetilde{cb}$	Uniform (100, 120)	$r_2$	Uniform (0.15, 0.2)
$\widetilde{cs}$	Uniform (40, 50)	${\tilde{C}}_{lj}$	Uniform (2000, 5000)
${\tilde{q}}_{hi}$	Uniform (100, 400)	$t_{ij}$	Uniform (8, 60)

, seven test questions of different sizes are randomly generated.

5.2. Evaluation of Candidate Solution Approaches

Set the confidence level to 1 and the $\epsilon$ value of objective function (2) to 0.5. Use two algorithms to perform 10 calculations on the seven randomly generated test questions, and obtain the results in

Table 6. Calculation Results.

	CPLEX		LSMA			HGA
Test questions (population size, number of iterations)	Objective value	CPU time (seconds)	Objective value	Average value	CPU time (seconds)	Objective value	Average value	CPU time (seconds)
1 (50, 100)	570,772	45.63	570,772	546,326.47	68.28	570,772	563,479.43	60.54
2 (50, 100)	1,082,391	102.63	1,082,391	1,085,472.26	182.16	1,086,283	1,095,459.05	162.47
3 (50, 150)	2,274,623	462.72	2,281,210	2,286,564.08	286.53	2,290,309	2,323,217.61	237.76
4 (100, 150)	4,287,517	5208.36	4,295,238	4,296,149.03	373.13	4,386,327	4,520,146.79	321.43
5 (100, 200)	8,834,158 *	>10,458.42	8,803,265	8,805,384.53	481.24	8,942,475	9,014,238.42	409.15
6 (100, 200)	heuristic method	--	12,186,415	12,190,326.45	635.83	13,277,326	14,127,249.74	571.52
7 (100, 300)	--	--	17,538,692	17,551,501,63	926.58	20,274,049	21,146,156.09	637.27

Note: Bold numbers represent optimal solutions and * indicates feasible solutions.

(the algorithm parameters (population size, number of iterations) are adjusted as the problem size increases to ensure the quality of the solution). Plot the iterative convergence curve for test question 3 in Table 6 (Figure 2).

Table 6 shows that for test questions 1 and 2, the objective function values obtained by CPLEX and LSMA are the same, but CPLEX takes longer than LSMA, indicating that the LSMA is suitable for solving small-scale problems, but its efficiency is not high. For test questions 3 and 4, compared with CPLEX, the LSMA achieves a higher objective function value while reducing the running time, indicating that it is less accurate but more efficient in solving medium-sized test problems. For test question 5, the objective function value obtained with the LSMA is lower than that obtained with CPLEX, and its running time is much shorter than that of the HGA. Regarding test questions 6 and 7, CPLEX does not provide effective feasible solutions, while the LSMA solves them relatively quickly with good results. These results demonstrate the efficiency of the LSMA.

Table 6 also compares the LSMA and the HGA. For test problem 1, the target values obtained with both algorithms are the same, while for test problems 2 and 7, the target values obtained with the LSMA are lower than those obtained with the HGA. When comparing the running time, for all test problems, the LSMA is much slower than the HGA. Furthermore, as the problem size increases, the running time of the LSMA also increases, because it incorporates dynamic neighbourhood search to enhance its local search capability. Although this process increases the running time, it allows the algorithm to obtain better quality solutions. As shown in Figure 2, with the same encoding method, the LSMA provides better solution results than the HGA and is more likely to jump out of local optimisation solutions. This proves that the LSMA has some advantages. The quality of its results is higher than that of the HGA, demonstrating the high accuracy of the LSMA.

The comparison with the Universal Precision Solver (CPLEX) and the Classic Heuristic Algorithm (HGA) shows that LSMA exhibits unique advantages in solving the fuzzy mixed integer programming model proposed in this study. It should be emphasized that for traditional deterministic site selection coverage problems, there are a large number of specialized and efficient algorithms that utilize the special structure of the problem. However, the M3 model in this article has a more complex structure due to the transformation of fuzzy constraints and endogenous incentive decisions. The value of LSMA as a general heuristic framework that does not rely on specific problem structures lies in its ability to effectively address the solving challenges brought about by innovative modeling. As shown in Table 6, LSMA has validated its ability to find the optimal solution for small-scale problems that CPLEX can still efficiently solve; LSMA provides a better ‘time quality’ trade-off for solving medium to large-scale problems with significantly reduced efficiency in CPLEX; In the case of large-scale problems where CPLEX cannot provide effective solutions within a reasonable time, LSMA has become a feasible and superior solution approach compared to traditional heuristic methods (HGA). Therefore, the core contribution of LSMA is to provide a reliable solution tool for the large-scale application of this new type of fuzzy programming model, rather than aiming to replace all specialized algorithms in classical scenarios.

To further scrutinize the superiority of the algorithms, the facility location solutions obtained by CPLEX and the proposed LSMA within a 600 s time limit for Test Problems 4, 5, 6, and 7 are presented in

Table 7. Selection of Recycling Stations for CPLEX and LSMA.

Test Questions	CPLEX	LSMA
4	[4 2 0 0 2 2 4 2 4 2 0 4 0 4 0 3 4 2 4 1 0 4 1 0 4 0 0 4 4 4 3 4 4 4 3 2 0 4 2 0 0 2 3 2 4 1 1 1 4 3 0 4 3 0 2 0 2 0 3 3 0 0 0 3 4 1 2 2 4 4 1 0 2 0 0 1 0 3 2 4]	[0 4 4 3 4 2 0 0 0 4 0 4 2 4 2 2 2 0 0 0 4 2 3 0 4 2 3 4 0 0 2 0 4 2 4 4 4 0 2 4 3 0 0 0 4 3 3 2 0 0 2 4 2 0 2 4 4 0 0 4 0 0 4 0 0 0 4 0 2 0 2 4 3 0 0 4 3 4 0 0]
5	[1 4 3 4 0 0 2 2 0 2 1 4 1 0 0 4 1 0 0 4 3 1 1 1 1 1 0 4 2 2 4 4 2 1 2 4 4 2 0 1 0 0 2 4 2 0 0 4 2 4 4 2 0 0 4 0 0 3 3 2 0 2 2 4 4 3 2 3 0 2 0 0 2 4 2 2 0 4 3 1 3 3 0 0 0 4 0 3 4 4 0 0 4 4 3 0 1 4 4 0 2 2 3 4 3 0 0 2 2 1 0 4 4 0 1 4 0 4 4 2 3 0 0 4 2 2 1 4 3 4 4 0 2 4 4 0 0 1 0 2 0 4 0 4 4 3 2 2 4 2 4 4 4 0 4 4 4 0 2 2]	[3 4 0 4 4 0 4 3 0 4 2 0 0 0 2 0 4 4 2 0 4 3 0 2 2 4 3 0 4 0 2 4 4 2 0 4 0 4 0 0 0 4 0 2 2 0 0 0 4 0 2 0 0 0 2 0 2 0 0 2 0 0 4 0 0 4 4 0 4 3 2 4 4 2 0 0 0 4 0 2 4 4 0 0 0 4 3 4 4 4 0 3 4 0 2 3 4 4 0 4 4 0 0 0 4 4 2 0 2 0 0 4 0 3 0 0 4 0 4 0 2 0 2 0 2 4 2 0 2 2 4 2 3 0 2 2 4 4 3 0 2 4 3 0 3 2 0 4 4 4 3 0 0 2 0 4 3 2 3 0]
6	[0 1 0 4 4 4 1 4 0 0 4 4 4 4 2 2 0 0 2 4 4 4 4 0 0 2 3 0 1 4 3 3 4 2 0 1 0 4 2 0 4 1 4 4 1 0 0 0 3 0 2 4 2 0 4 2 4 0 2 0 3 0 4 2 0 0 4 0 1 4 4 1 3 0 2 0 2 1 0 0 3 0 2 4 1 2 3 4 2 4 0 0 4 2 2 1 2 4 2 4 4 4 0 4 4 4 2 3 3 2 1 4 4 4 2 0 4 0 0 4 0 1 0 4 0 0 3 4 2 2 1 4 2 2 0 2 4 0 4 4 1 2 4 2 2 0 4 3 3 3 4 4 4 4 2 0 1 0 4 2 4 3 3 0 0 4 2 0 0 4 0 2 0 4 0 2 1 0 0 2 4 4 1 0 3 2 1 4 0 2 0 0 1 2 0 2 0 0 0 2 2 4 2 0 3 2 4 4 2 3 4 0 0 4 2 0 0 1 3 1 1 2 0 4 4 4 1 2 3 3 4 0 3 3 0 2 0 4 4 1]	[2 0 0 0 0 3 4 3 0 2 2 3 3 4 4 0 0 0 0 4 4 4 4 2 0 4 4 0 3 0 2 4 0 0 0 0 2 4 2 0 4 0 0 2 2 4 2 2 0 4 2 0 0 0 4 0 4 4 4 0 2 2 0 4 4 2 0 0 0 0 4 0 4 2 4 3 3 2 4 0 0 4 2 4 0 4 2 0 2 4 2 2 0 0 0 3 4 0 0 4 3 0 4 0 0 0 4 2 0 0 0 0 2 4 0 2 0 2 0 4 3 0 4 0 0 4 2 2 0 4 0 4 0 0 4 4 3 2 0 2 2 0 2 0 4 0 4 4 4 2 0 3 3 0 0 0 3 0 4 0 4 4 2 4 4 0 2 3 0 0 3 2 0 0 3 2 2 4 0 0 4 2 3 4 4 3 2 4 2 4 2 2 0 0 0 0 0 4 4 4 2 4 0 4 3 0 4 0 4 2 0 4 4 0 0 0 2 3 0 0 4 4 0 4 2 0 0 0 3 0 2 4 0 4 4 3 4 3 0 4]
7	[0 0 4 0 1 0 3 0 2 1 2 1 2 0 4 1 1 4 2 3 4 0 4 4 4 4 4 3 0 0 4 0 0 4 2 0 0 1 3 4 4 1 3 4 4 4 3 0 2 0 0 4 4 3 2 2 1 2 2 3 2 4 2 3 4 0 0 0 4 4 0 2 0 3 1 0 4 2 2 2 4 2 0 3 1 2 4 0 0 1 4 4 1 2 4 0 4 2 0 0 2 2 4 0 4 4 4 3 2 0 2 0 4 0 2 4 4 2 4 4 2 4 2 0 0 2 0 0 0 3 4 2 1 0 0 4 4 2 0 4 0 0 4 3 2 0 4 2 3 4 0 0 4 1 2 4 4 0 4 4 4 0 4 0 2 1 4 0 4 4 0 3 3 3 1 2 2 2 0 1 4 4 2 4 0 0 0 0 4 1 4 0 2 3 0 2 0 0 4 2 0 3 0 0 4 4 1 3 4 2 0 4 0 4 2 2 4 0 2 4 0 2 4 0 4 2 4 4 2 0 2 4 4 4 0 1 0 2 4 2 2 4 2 1 1 1 0 0 1 4 3 4 2 3 4 2 3 4 1 1 3 4 0 2 0 3 0 0 0 1 0 0 4 4 4 0 3 4 4 0 4 0 0 1 0 0 3 4 0 4 0 4 1 4 3 1 3 2 0 4 4 0 0 1 2 2 0 2 0 2 3 2 4 1 1 2 0 3 2 2]	[2 4 0 4 0 2 0 0 4 0 3 0 3 4 0 0 0 4 0 0 2 4 0 0 4 0 0 4 3 4 0 2 0 0 0 2 2 2 0 2 0 0 3 2 4 3 0 0 0 4 3 4 0 2 0 2 4 4 3 4 4 0 4 4 3 0 2 4 2 0 0 3 4 4 4 0 2 0 0 0 4 2 4 4 2 4 0 2 2 0 2 0 0 2 2 0 2 3 0 2 0 2 2 2 0 4 2 4 0 4 0 0 4 0 0 0 3 4 0 3 0 4 4 4 4 2 4 2 4 3 2 4 3 4 2 4 4 2 4 4 4 4 0 0 0 4 0 4 2 4 3 0 2 0 2 4 3 2 3 4 2 0 3 0 4 3 0 0 2 4 0 0 4 0 4 2 4 2 4 0 0 3 0 0 0 4 0 4 2 2 0 2 3 4 4 4 0 0 0 0 2 0 4 0 0 2 0 2 0 2 0 0 4 4 0 4 0 0 2 0 3 4 0 0 0 4 0 4 4 0 0 2 4 0 4 4 4 3 4 0 0 4 2 2 4 4 4 4 0 4 0 0 0 2 0 0 0 2 3 3 3 0 3 3 0 0 0 4 0 0 3 2 4 0 4 0 4 4 4 0 4 0 2 4 0 4 4 0 4 4 0 2 4 0 3 0 0 0 4 2 3 2 0 0 2 4 0 2 0 0 0 2 0 2 2 0 2 0 4 3]

Note: The number 0 in square brackets indicates that the recycling bin is not selected, while 1–4 indicate that recycling bins of levels 1–4 are selected.

. As shown in Table 7, the solutions generated by CPLEX consistently involve a larger number of selected facilities, with some also assigned higher capacity grades compared to those from LSMA. This observation, however, underscores the superiority in optimization efficacy of the LSMA. The LSMA successfully identifies network configurations with significantly lower total costs while satisfying all constraints, including coverage requirements, capacity credibility, and the recovery rate target. This indicates that the LSMA, through its more refined global search, can pinpoint efficient network structures that operate effectively without necessitating over-construction (i.e., fewer facilities with moderate grades). Consequently, it avoids the resource redundancy and capital overinvestment potentially present in the CPLEX solutions. This capability directly demonstrates the effectiveness of the dynamic local search mechanism embedded within the LSMA, proving that it provides a more economically competitive and preferable solution for strategic investment planning.

5.3. Sensitivity Analysis and Managerial Insights

The reverse logistics network model constructed in this article contains fuzzy parameters. To explore their impact, test problem 1 is used for analysis. Suppose that there are 30 disposal points, 10 candidate recycling stations, and 4 recycling station levels in the network, with $r_1$ and $r_1$ being 21% and 22%, respectively. $\widetilde{ct}$ is (2, 3, 4, 5), $\widetilde{cb}$ is (90, 100, 110, 120), and $\widetilde{cs}$ is (40, 50, 60, 70). The fuzzy operating costs of recycling stations for express packaging are cardboard boxes (2, 3, 4, 5), plastic bags (0.9, 1.5, 2, 2.6), woven bags (1, 2, 3, 4), and internal buffers (3, 4, 5, 6), respectively. The values of ${\tilde{q}}_{hi}$ follow the range shown in Table 3, and the remaining data are shown in

Table 8. Relevant Data of Recycling Bins.

Level	Fuzzy Construction Cost (10,000 Yuan)	Fuzzy Ability (1000 Tons)	Maximum Coverage Radius (km)	Basic Recycling Rate
1	(2, 2.5, 3, 3.5)	(2, 2.5, 3, 3.5)	8	43%
2	(2.5, 3, 3.5, 4)	(2.5, 3, 3.5, 4)	11	46%
3	(3, 3.5, 4, 4.5)	(3, 3.5, 4, 4.5)	13	49%
4	(3.5, 4, 4.5, 5)	(3.5, 4, 4.5, 5)	15	52%

,

Table 9. Coordinates of Disposal Points.

Number	Horizontal Axis X-Axis	Vertical Axis Y-Axis	Number	Horizontal Axis X-Axis	Vertical Axis Y-Axis
1	18.04	6.15	16	18.47	31.85
2	21.02	16.63	17	12.66	41.57
3	14.95	11.48	18	6.99	37.84
4	9.78	13.62	19	16.64	51.67
5	7.43	7.78	20	5.55	30.10
6	13.49	13.20	21	36.12	19.33
7	30.97	23.35	22	42.99	11.68
8	27.21	31.74	23	47.90	10.48
9	35.77	38.20	24	42.12	16.51
10	29.58	24.25	25	45.43	48.82
11	37.18	27.09	26	38.92	42.54
12	46.23	36.22	27	40.32	48.02
13	14.79	32.46	28	26.94	36.41
14	17.93	43.24	29	30.23	38.09
15	9.63	40.36	30	26.07	53.97

and

Table 10. Location of Recycling Bins.

Number	Horizontal Axis X	Vertical Axis Y
1	14.09	38.76
2	31.27	32.20
3	14.16	11.36
4	46.15	21.92
5	22.75	55.82
6	27.58	40.50
7	40.18	43.26
8	24.84	19.86
9	41.65	16.58
10	15.69	21.32

.

Take a confidence level between 50% and 100%, set $\epsilon$ between 50% and 90%, and use the LSMA to solve the model. The calculation results are shown in

Table 11. Total Cost with Changes in Confidence Level and $\epsilon$ Value.

	$\mathit{\epsilon }$ Value
Confidence Level	50%	55%	60%	65%	70%	75%	80%	85%	90%
50%	531,174	582,593	639,816	704,149	776,520	858,285	939,043	1,042,567	1,145,317
55%	531,174	583,317	641,967	711,496	785,541	864,884	944,362	1,045,818	1,151,222
60%	533,431	587,240	649,371	719,930	793,997	869,175	953,213	1,052,513	1,158,217
65%	542,275	589,511	651,950	723,762	797,694	874,229	971,255	1,068,401	1,174,871
70%	544,217	589,511	654,062	723,762	800,380	886,525	975,636	1,074,010	1,188,861
75%	555,284	606,037	668,933	735,657	808,019	886,525	980,444	1,083,679	1,192,079
80%	556,925	607,678	669,084	736,531	808,019	886,525	986,531	1,094,956	1,195,258
85%	557,321	612,350	678,134	750,795	828,904	916,057	1,008,417	1,104,286	1,199,238
90%	565,617	616,193	680,317	753,181	836,158	919,126	1,015,542	1,113,870	1,223,375
95%	565,617	619,942	684,962	759,387	845,633	930,280	1,030,913	1,130,614	1,268,174
100%	570,772	625,023	684,962	761,267	845,633	930,280	1,034,522	1,130,614	1,272,711

and

Table 12. Selection of Recycling Stations with Changes in Confidence Level and $\epsilon$ Value.

	$\mathit{\epsilon }$ Value
Confidence Level	50%	55%	60%	65%	70%	75%	80%	85%	90%
50%	[3 2 2 0 1 0 3 0 2 0]	[3 2 4 0 1 0 2 0 2 0]	[4 3 4 0 1 0 2 0 2 0]	[4 3 4 0 1 0 2 0 3 0]	[4 4 4 0 1 0 4 0 4 0]	[4 2 4 0 1 0 3 2 4 0]	[4 4 4 0 1 0 4 4 4 0]	[4 2 4 0 1 3 4 2 4 0]	[4 3 4 0 3 4 4 4 4 0]
55%	[3 2 2 0 1 0 3 0 2 0]	[3 2 4 0 1 0 2 0 2 0]	[3 3 4 0 1 0 3 0 2 0]	[4 4 4 0 1 0 3 0 4 0]	[4 4 4 0 1 0 4 0 4 0]	[4 4 4 0 1 0 4 1 3 0]	[4 4 4 0 1 0 4 4 4 0]	[4 2 4 0 2 3 3 2 4 0]	[4 4 4 0 1 2 3 0 4 4]
60%	[4 2 2 0 1 0 2 0 2 0]	[4 2 3 0 1 0 2 0 2 0]	[4 3 4 0 1 0 2 0 3 0]	[4 3 4 0 1 0 4 0 3 0]	[4 2 4 0 1 0 4 1 3 0]	[4 4 4 0 1 0 4 2 3 0]	[4 4 4 0 1 0 4 4 4 0]	[4 2 4 0 2 3 4 2 4 0]	[4 4 4 0 1 0 4 2 4 4]
65%	[4 4 2 0 1 0 2 0 2 0]	[4 3 4 0 1 0 2 0 2 0]	[4 4 4 0 1 0 3 0 2 0]	[4 4 4 0 1 0 4 0 4 0]	[4 2 4 0 1 0 4 2 4 0]	[4 4 4 0 1 0 4 2 4 0]	[4 1 4 0 1 4 4 2 4 0]	[4 4 4 0 1 0 4 2 4 4]	[4 4 4 0 3 0 4 4 4 4]
70%	[4 3 4 0 1 0 2 0 2 0]	[4 3 4 0 1 0 2 0 2 0]	[4 4 4 0 1 0 3 0 2 0]	[4 4 4 0 1 0 4 0 4 0]	[4 3 4 0 1 0 4 2 4 0]	[4 4 4 0 1 0 4 4 4 0]	[4 2 4 0 1 2 4 3 4 0]	[4 4 4 0 2 2 4 4 4 0]	[4 2 4 0 1 2 4 3 4 4]
75%	[3 3 3 0 1 0 3 0 3 0]	[4 3 4 0 1 0 2 0 3 0]	[4 3 4 0 1 0 4 0 3 0]	[4 4 4 0 1 0 2 1 4 0]	[4 4 4 0 1 0 4 1 4 0]	[4 4 4 0 1 0 4 4 4 0]	[4 3 4 0 1 3 4 3 4 0]	[4 4 4 0 1 4 4 0 4 4]	[4 3 4 0 3 4 4 4 4 4]
80%	[3 3 3 0 1 0 3 0 3 0]	[4 3 4 0 1 0 2 0 3 0]	[4 3 4 0 1 0 3 1 2 0]	[4 4 4 0 1 0 4 1 3 0]	[4 4 4 0 1 0 4 1 4 0]	[4 4 4 0 1 0 4 4 4 0]	[4 4 4 0 1 4 4 4 4 0]	[4 4 4 0 1 4 4 0 4 4]	[4 3 4 0 3 3 4 3 4 4]
85%	[3 4 3 0 1 0 2 0 3 0]	[3 3 3 0 1 0 3 1 2 0]	[3 3 3 0 1 0 4 1 4 0]	[3 4 4 0 1 0 4 2 4 0]	[4 4 4 0 2 0 4 3 4 0]	[4 4 4 0 4 0 4 4 4 0]	[4 4 4 0 3 4 4 4 4 0]	[4 3 4 0 2 1 4 3 4 4]	[4 3 4 0 2 3 4 4 4 4]
90%	[4 3 2 0 1 0 2 1 2 0]	[4 4 4 0 1 0 3 0 3 0]	[4 3 3 0 1 0 4 1 3 0]	[4 3 4 0 1 0 4 3 4 0]	[4 4 4 0 4 0 4 3 3 0]	[4 3 4 0 2 3 4 2 4 0]	[4 4 4 0 1 0 4 4 4 4]	[4 3 4 0 2 3 4 4 4 4]	[4 4 4 0 3 3 4 4 3 4]
95%	[4 3 2 0 1 0 2 1 2 0]	[4 4 4 0 1 0 2 1 2 0]	[4 4 4 0 1 0 3 1 4 0]	[4 3 4 0 1 0 4 4 4 0]	[4 1 4 0 2 2 4 4 4 0]	[4 4 4 0 2 4 4 4 4 0]	[4 4 4 0 2 1 4 4 4 4]	[4 4 4 0 2 4 4 4 4 4]	[4 4 4 3 2 4 4 4 4 4]
100%	[4 4 4 0 2 0 2 0 3 0]	[4 3 4 0 1 0 2 1 4 0]	[4 4 4 0 1 0 3 1 4 0]	[4 4 4 0 1 0 4 4 4 0]	[4 1 4 0 2 2 4 4 4 0]	[4 4 4 0 2 4 4 4 4 0]	[4 2 4 0 2 1 4 4 4 4]	[4 4 4 0 2 4 4 4 4 4]	[4 4 4 3 3 4 4 4 4 4]

Note: The number 0 in square brackets indicates that the recycling bin is not selected, while 1–4 indicate that recycling bins of levels 1–4 are selected.

, and Figure 3 is based on Table 10.

As shown in Table 11 and Table 12, the overall trend of network changes is as follows: at constant $\epsilon$ and increasing confidence level, or at a constant confidence level and increasing $\epsilon$, the location of recycling stations in the network differs, the total number of selected recycling stations increases, and their levels improve. The number of times enterprises pay additional recycling fees for discarded express packaging and the government allocates subsidies to recycling stations increases (due to the lack of specific data in the article), and the target value (total cost) increases. Indeed, increasing the confidence level increases the amount of express packaging reliably recycled and ensures that express packaging whose recycling is uncertain (i.e., not systematic) is recycled. Alternatively, increasing the $\epsilon$ value may require recycling a sufficient amount of express packaging to meet the required recycling rate, increasing the network’s logistics volume and transportation costs. Furthermore, improving the level of recycling stations or increasing their number can meet this part of the logistics volume, increasing their fixed and operating costs and ultimately increasing the target value.

In Table 11 and Table 12, the bold numbers indicate that for the same $\epsilon$ value, the target values for adjacent confidence levels and the selection of recycling bins are identical. When the $\epsilon$ value is the same, and the target values for adjacent confidence levels are different, the selection of recycling bins is the same, such as a value of 55% and confidence levels of 75% and 80%. In the case of the same $\epsilon$ value and different target values of adjacent confidence levels, although the number of recycling stations is the same, their location differs, and their levels are reduced. For example, when the $\epsilon$ value is 80%, a confidence level of 85% selects seven level 4 recycling stations and one level 3 recycling station, while a confidence level of 90% selects seven level 4 recycling stations and one level 1 recycling station. Although the overall trend indicates that for the same $\epsilon$ value, the total number and level of recycling stations decrease with decreasing confidence level, there may be situations where stations are not affected by the confidence level. Therefore, the above results indicate that in this paper, the confidence level has no systematic impact on the structure of the reverse logistics network for express packaging.

Furthermore, in Table 11 and Table 12, as the value of confidence level increases, the target value (total cost) strictly increases. The location of recycling stations in the network differs, the total number of selected recycling stations strictly increases, and their levels increase. The number of times that enterprises pay additional recycling fees for discarded express packaging and the government allocates subsidies to recycling stations also strictly increases. These results indicate that the level of achievement of the limited total recycling rate has a significant impact on the reverse logistics network for express packaging.

In Figure 3, for the same $\epsilon$ value, as the confidence level increases, the total cost also increases. Moreover, the confidence level within a certain range is relatively close to the magnitude of the increase in total cost, showing an approximately linear proportional relationship. For the same $\epsilon$ value, the total cost increases with the value of $\epsilon$, and the increase rate varies. The above explanation shows that changes in the confidence level and overall recycling rate can significantly affect the target values, and that the magnitude of this increase is irregular, indicating that there is a certain degree of risk associated with changes in the confidence level.

Table 11 shows that under a fixed recycling rate threshold, increasing the confidence level leads to a systematic rise in total cost. A note in Table 11 clarifies that this cost increase is partly attributable to the “increased frequency of government subsidy allocations to recycling stations.” This reveals a core relationship: when decision-makers pursue higher system robustness, they not only need to construct more and more expensive facilities but also require both the government and enterprises to provide greater incentives (subsidies and refunds) to “activate” and secure the expected recovery volume. Table 12 demonstrates how the number, level, and location of recycling stations change under different combinations of the recycling rate threshold and confidence level. Since the disbursement of subsidies is directly linked to whether a facility is selected and processes recyclables (Constraints (8) and (9)), the implied intensity of the subsidy policy (embedded within the model optimization) directly influences which levels and locations of facilities are incorporated into the optimal network. A system anticipating subsidy support is more likely to justify the construction of higher-level, more widely covering, yet costlier facilities. Figure 3 and the associated analysis indicate that the total cost increases in an “approximately linear” yet “irregular” manner as the confidence level (α) rises. This irregularity stems partly from the discrete decision-making nature of the incentive policies (subsidies and refunds). The introduction of subsidies means that, when responding to uncertainty, cost increases are not smooth and continuous. Instead, upon reaching certain thresholds, costs exhibit a piecewise jump triggered by the activation of subsidy payments.

Based on the aforementioned analysis, integrated with practical logistics management, policy formulation, and corporate decision-making, the following managerial insights are provided:

(1) Confidence Level Serves as a Core Management Lever for Quantifying the “Risk-Cost” Trade-off.

Sensitivity analysis results indicate that, under a fixed recycling rate target, the total network cost increases approximately linearly with the confidence level (see Figure 3 and Table 11 in the document). For instance, when the recycling rate threshold (ε) is 80%, the total cost increases significantly as the confidence level rises from 75% to 90%. This provides managers with a clear decision-making tool: quantifying “the additional marginal cost required to improve system operational robustness (i.e., reliability) by one percentage point.” Decision-makers (e.g., logistics network planners or policymakers) can adjust this intuitive parameter—the confidence level—based on their organization’s risk tolerance (e.g., how much extra budget they are willing to allocate for “90% assurance” versus “75% assurance”) to generate corresponding optimized network investment plans. This transforms strategic decision-making from the vague notion of “improving robustness” into a precise “cost–risk” exchange.

(2) Network Facility Layout Should Possess “Resilience” Aligned with Strategic Objectives.

Results from Table 12 show that different combinations of confidence levels and recycling rate targets lead to entirely different facility locations, quantities, and level configurations. This implies that a network pursuing low cost with acceptable risk (low confidence level) is fundamentally different in physical form from one pursuing high assurance and a high recycling rate (high confidence level and/or high ε value). The managerial implication is that enterprises should not seek a single “optimal” static network but should plan for a portfolio of solutions with “resilience.” In the early investment stage, managers can use the model to evaluate optimal layouts under different development scenarios, thereby making more forward-looking infrastructure decisions. It also clarifies the specific pathways and costs for network upgrades or adjustments when future objectives change.

(3) Algorithmic Tools Enable a Shift in Decision-Making Paradigm from “Cost Center” to “Value Investment.”

Algorithm comparison results (Table 6) demonstrate that the proposed LSMA algorithm can generate high-quality solutions for large-scale, complex problems within a reasonable timeframe. Its performance is superior to traditional heuristic algorithms and remains effective even when exact solvers fail. This provides significant managerial tool value: it enables enterprises to conduct rapid, systematic “simulation-optimization” analysis of reverse logistics networks. Managers can use this tool to evaluate the effects of different incentive policies (refunds, subsidies), calculate the minimum cost to comply with new environmental regulations, or compare the benefits of building their own recovery network versus third-party cooperation. Consequently, packaging recycling operations can be transformed from a passive, hard-to-quantify “cost center” into a “strategic investment component” that can be proactively managed and its value assessed through optimization models.

(4) Subsidy Policies Require Coordinated Design with Network Planning.

The findings indicate that “where to locate facilities” and “how much subsidy to provide” are inseparable decisions requiring joint optimization. The optimal subsidy strategy is highly dependent on the specific network layout. Consequently, an effective policy should not rely on setting a uniform subsidy rate. Instead, it should formulate differentiated subsidy schemes based on optimization simulations of the integrated network (encompassing facility location and consumer incentives) to maximize funding efficiency.

Overall, the impact of changes in recycling rate thresholds (i.e., values) and confidence levels on the location of express packaging logistics networks is significant. Managers should objectively evaluate their enterprise’s recycling level and other relevant information regarding express packaging to determine whether the enterprise should invest in recycling and whether it can benefit from government subsidies to improve the recycling rate of express packaging. They should also consider the level of risk (i.e., choose a reasonable confidence level) and develop a feasible express packaging recycling plan. The model and algorithm proposed in this study provide managers with a data-driven decision support system. Its core managerial value lies in: quantifying “risk” and “environmental targets” and incorporating them into cost accounting; revealing how the network structure evolves with strategic objectives, thereby supporting resilient planning; and offering an efficient computational tool that makes the optimization of complex sustainable logistics networks an actionable decision-making process.

6. Conclusions

This article examines the reverse logistics network for recycling express packaging, considering uncertain network parameters such as construction costs, operating costs, transportation costs, collection fees, and government subsidies. By combining the expected value method and the opportunity-constrained programming method, a fuzzy mixed linear programming model is constructed, with the aim of minimising the total cost of the recycling network and maximising the recycling rate. To solve this model, an LSMA is proposed, and its effectiveness and applicability are demonstrated through numerical examples. The results indicate that the model and algorithm can effectively handle the uncertainty in the recycling process, significantly improve the system recycling rate while controlling costs, and provide a reliable tool for optimizing reverse logistics networks.

The results of this article have positive social and environmental impacts. By systematically improving the efficiency of express packaging recycling, it can directly reduce waste pollution and resource waste and promote the development of the circular economy. This not only reduces the overall environmental governance cost of society but also cultivates the awareness of green consumption and production among the public and enterprises, making a substantial contribution to promoting the construction of “waste-free cities” and achieving the “dual carbon” goals.

The current research has limitations. Firstly, the model focuses on the reverse recycling network from the consumer end to the recycling station, without including the logistics links of reprocessing, remanufacturing, or reusing packaging after recycling. Although this makes the research question more focused, it fails to reveal the overall cost and facility location changes under the fully closed-loop logistics system. Practical calibration of parameters and assumptions: The estimation methods for various uncertain parameters in the model, such as various costs and recovery amounts, are mainly based on theoretical settings or historical data deduction in the article and have not been fully combined with real-time operational data or expert judgment in specific regions for calibration. This may affect the absolute accuracy of the model in specific application scenarios. In addition, the LSMA algorithm adopted by the research institute performed well in the test cases, but its computational efficiency and stability in larger-scale networks, such as city-level or national-level nodes, have not been fully validated. Finally, numerical analysis is based on specific parameter settings, and the generalizability of the conclusions needs to be further tested under different market environments and policy conditions.

Therefore, it is recommended that policymakers employ the proposed fuzzy optimization model and algorithm as a core decision-support system for regional recycling network planning. Its value lies in providing a quantified framework that integrates decision-makers’ risk preferences (confidence levels) directly into cost–benefit analysis. This enables authorities to assess the explicit marginal cost of increasing system reliability (e.g., the additional investment required to raise the confidence level from 75% to 90%), thereby transforming strategic trade-offs from qualitative judgments into precise “cost–risk” balances; generate a portfolio of resilient network layouts adapted to different development scenarios (e.g., low-cost/high-risk vs. high-investment/high-reliability strategies), supporting forward-looking and flexible infrastructure planning; and co-design subsidy policies and facility locations through optimized simulations, thereby maximizing the efficiency of fiscal incentives and guiding enterprises to invest in an intelligent, network-based recovery infrastructure that is both economically viable and environmentally effective. For logistics and e-commerce enterprises, it is recommended to gradually integrate reverse logistics networks, use algorithm tools to optimize the layout of recycling nodes and transportation paths, and transform packaging recycling from a cost centre to a strategic link that combines environmental benefits and long-term economic value. Although this article examines the recycling process of express packaging, it does not consider the logistics transportation process for reusing express packaging after recycling. Therefore, future research should fully integrate the logistics network of express delivery packaging reuse based on the recycling process, while considering social and environmental factors, to promote the efficient and healthy development of the express delivery packaging recycling network. In addition, the research scope will be expanded from the recycling network to a complete closed-loop logistics system that includes sorting, remanufacturing, and redistribution, exploring the impact of packaging reuse on facility location, transportation routes, and overall economic benefits. It will also explicitly introduce social and environmental goals, such as carbon emissions and employment impact, in addition to cost and recycling rates, and construct multi-party game or collaborative decision-making models for government, enterprises, and consumers. Next, it will develop efficient decomposition algorithms, reinforcement learning frameworks, or hybrid metaheuristic algorithms for large-scale networks at the city or national level to obtain high-quality solutions within an acceptable time. It will also internalize policy variables such as government subsidy standards, carbon taxes, and extended producer responsibility systems, quantitatively analyze the impact of different policy tools on network structure and behavior of all parties, and provide a direct basis for policy design. Finally, it will explore the feasibility and economic and environmental benefits of new business models based on optimization results, such as deposit systems, shared packaging pools, and targeted recycling incentives.