Network Optimization of Fresh Products Cold Chain Considering Supply Disruption and Demand Fluctuation Under the Dual-Carbon Policy

Abstract

Against global industrial upgrading and China’s “dual-carbon” policy, the cold chain for fresh products faces numerous challenges such as supply disruptions, demand fluctuations, and low-carbon transformation. This study focuses on introducing the key optimization goals of the cold chain network for fresh products: maximizing the service level while minimizing operating costs and carbon emissions. To this end, this study proposes a high-dimensional multi-objective optimization model for the cold chain network of fresh products and designs four resilience strategies to address supply disruption and demand fluctuation scenarios. To solve this model, this study innovatively designs a hybrid algorithm combining neighborhood search and swarm intelligence, integrating the advantages of local exploration and global optimization to balance the relationships among multiple objectives efficiently. In addition, this study conducts a real-world case analysis to verify the effectiveness of the proposed model and the algorithm. Furthermore, by deeply exploring the comprehensive impacts of supply disruptions and demand fluctuations on the cold chain network for fresh products, the mechanism of action of resilience strategies in dealing with supply chain risks is highlighted. The research results provide valuable decision-making support for fresh cold chain enterprises to develop resilient and low-carbon network optimization strategies for cost reduction, efficiency improvement, and sustainable development.

1. Introduction

Against the backdrop of global industrial upgrading, cold chain logistics, as a core component of the modern supply chain system, is significant for ensuring the quality of fresh products and driving consumption upgrading. However, the industry’s current development is faced with a dual dilemma of environmental pressure and supply–demand risks. In 2023, the energy consumption of cold chain logistics accounted for more than 30% of the total energy consumption of the logistics industry, and the annual growth rate of carbon emissions in the circulation of fresh products reached 6.3% [1]. At the same time, frequent extreme weather events, geopolitical conflicts, and public health incidents have led to a 57% year-on-year increase in international cold chain supply disruptions from 2020 to 2022 [2]. The sustainability and stability of traditional cold chain logistics systems have been severely impacted. In this context, China is actively promoting the “dual-carbon” strategy and has incorporated the green and high-quality transformation of the cold chain logistics industry into the national sustainable development planning system. This study is grounded in the green-resilient supply chain (GRSC) theory, which integrates environmental sustainability with supply chain resilience to address complex operational risks [3]. There is significant room for optimization in all links of the cold chain network for fresh products, such as processing and preservation, distribution, and transportation. Therefore, under the “dual-carbon” policy framework, achieving the low-carbon transformation of cold chain logistics and enhancing supply chain resilience has become a key issue that the industry urgently needs to address.

Based on the above-mentioned development dilemmas in the industry, the core issue focused on in this study is how to construct an optimization model for the cold chain network of fresh products that considers the risks of supply disruptions and demand fluctuations while integrating low-carbon requirements. By implementing and adjusting targeted resilience strategies, optimizing facility location and transportation routes, this study aims to achieve the multi-objective coordination of minimizing operating costs, maximizing service levels, and controlling carbon emissions. This research can provide scientific and practical decision-making bases for enterprises in a complex and changeable operating environment, helping enterprises reduce costs and increase efficiency, and offer theoretical support and practical guidance for the sustainable development of the fresh cold chain industry.

Current research on cold chain network optimization exhibits three notable limitations. Firstly, existing studies predominantly operate within static parameter frameworks, thereby lacking the capacity to develop dynamic response mechanisms for supply disruptions triggered by extreme events such as pandemics and natural disasters. Secondly, demand forecasting models rely on deterministic assumptions, rendering them ineffective in capturing the sudden and nonlinear demand fluctuations associated with emerging consumption models, including e-commerce, live streaming, and community group buying. This limitation results in a deviation of over 19% between location-routing decisions and actual demand [4]. Thirdly, most of the literature addresses dynamic carbon emissions from transportation processes in isolation, overlooking the significant contribution of static carbon emissions from site facilities. This fragmented research paradigm often leads to trade-offs between carbon reduction objectives and operational efficiency in practical applications.

The resilience and low-carbon optimization of the cold chain network for fresh products has become a crucial issue that the industry urgently needs to address. The practical dilemmas of Walmart’s fresh food operations have typical research value [5]. As a leading global retail enterprise, Walmart’s fresh food supply chain is huge, vast, and complex. In recent years, its fresh food cold chain operation has faced severe challenges: firstly, at the supply level, extreme weather affects the stability of the supply chain. For example, the floods in the Midwestern United States in 2023 led to reduced production from suppliers, causing shortages in Walmart’s regional stores. This increased logistical costs and decreased customer satisfaction. Secondly, the demand side is highly volatile and uncertain. Emerging consumption models have exacerbated this situation, posing challenges to inventory management and distribution scheduling. Thirdly, the transportation and warehousing in the cold chain generate significant carbon emissions. Under the “dual-carbon” policy, low-carbon equipment upgrades require substantial upfront investment. Failure to transform may subject the company to policy-related and consumer-driven pressures. Therefore, constructing an optimized solution for the cold chain network of fresh products that considers both resilience and low-carbon requirements is of great practical significance for Walmart to overcome operational dilemmas, achieve cost control, and improve service quality.

This study proposes a high-dimensional multi-objective optimization model, RL-FPCN (resilient and low-carbon fresh products cold chain network), for the cold chain network of fresh products. The model takes maximizing the service level and minimizing the operating cost as dual objectives and considers carbon emissions as part of the operating cost to balance service, cost, and carbon emissions. This model entirely takes into account the scenarios of supply disruptions and demand fluctuations. It selects eight of the most representative scenarios and uses scenario probabilities to simulate the original uncertainty distribution. Furthermore, four resilience mechanisms are specifically designed to optimize supply chain risks dynamically. To solve this model, this study innovatively proposes a hybrid algorithm of neighborhood search and swarm intelligence, OTS-FICSMA (improved OPTICS algorithm considering temporal–spatial distance and improved slime mold algorithm with full iteration cycle). This algorithm integrates the advantages of the two in local exploration and global optimization, efficiently balancing the relationships among multiple objectives. This research is dedicated to providing strong methodological support for building a low-carbon and resilient cold chain network for fresh products, assisting enterprise decision-makers in achieving the optimization goals of cost reduction, efficiency improvement, and sustainability.

This research is structured as follows: Section 2 reviews the literature related to this topic. Section 3 illustrates the problems and their notations, and based on the L-FPCN model, four resilient strategies are integrated to construct the RL-FPCN model. Section 4 proposes a neighborhood search-swarm intelligence algorithm, OTS-FICSMA, to solve the mixed-integer programming model. In Section 5, a real-case simulation is conducted. The supply interruption and demand fluctuation scenarios are analyzed through data collection and scenario setting, and the impacts of the four resilient strategies on the network robustness are discussed. Furthermore, the calculation results are compared with other methods. Finally, Section 6 summarizes the research presented in this work and outlines some managerial implications and future research directions.

2. Literature Review

According to the research problem, this section reviews the related literature concerning the main issues, including the fresh product cold chain network and robust optimization of the supply chain network. Meanwhile, current research gaps are analyzed, and the contributions of this paper are presented.

2.1. Fresh Products Cold Chain Network (FPCN)

The fresh products cold chain network (FPCN) is a delivery network for fresh products that starts from agricultural production bases. These products undergo various stages, including processing, packaging, transportation, distribution, and retail, under specific temperature conditions, until they reach consumers’ tables [6]. The design of FPCN involves constructing and optimizing the network structure to improve overall efficiency while reducing operating costs.

Unlike other products, the characteristics of fresh products have multiple impacts on optimization for FPCN. Firstly, the cumulative effect of the freshness of fresh products on cold chain efficiency is significant over time. Scholars, represented by Yadav et al. [7] and Esteso et al. [8], have developed mathematical programming models with freshness-related constraints. They found that considering freshness in investment decisions can enhance the economic benefits of FPCN. Secondly, the dynamic changes in the freshness of fresh products bring complexity to inventory planning. Hiassat et al. [9] were the first to propose a location-inventory-routing model, using shelf-life limitations to determine warehouse-related parameters, inventory, and transportation routes; Paam et al. [10] constructed an inventory control optimization model for multiple periods, multiple products, and multiple warehouses to solve the problem of spoilage of fresh products in temperature-controlled warehouses and reduce inventory costs. Finally, the high requirement for the timeliness of delivery of fresh products directly affects the “last mile” transportation and distribution plans. Osvald and Stirn [11] first described the fresh vegetable distribution problem as a specific vehicle routing problem, considering the impact of perishability on distribution costs to reduce cargo losses; Pratap et al. [12] focused on the carbon footprint, production capacity, and time window constraints of fresh products and created a logistics route planning model to minimize greenhouse gas emissions.

With the continuous advancement of the “dual-carbon” initiative, green sustainability has become an essential consideration in designing fresh product cold chain networks (FPCNs). Peng et al. [13] proposed four optimization objectives for the multimodal cold chain network: minimizing total transportation time, cost, carbon emissions, and food waste. However, too many objectives may conflict, and the uncertainties in transportation were not fully considered. On this basis, Fathollahzadeh et al. [14] further refined the optimization objectives of the meat supply chain network. By integrating economic and environmental effects, they focused on optimizing waste and by-product management, maximizing profits, and minimizing carbon emissions. However, the model insufficiently considered other stakeholders in the supply chain. Different from the former two, in addition to paying attention to the economic and environmental goals of the coconut closed-loop supply chain network, Gholian-Jouybari et al. [15] also proposed the social sustainability dimensions of facility utilization and job opportunities. However, it is not easy to quantify and implement social goals. It is worth noting that in recent years, only Moghaddasi et al. [16] innovatively incorporated customer satisfaction as an optimization objective into designing the cold chain logistics network for perishable products. However, judging satisfaction based on time windows is too one-sided, and the model does not sufficiently explore other influencing factors.

In summary, the existing research mainly focuses on optimizing cold chain logistics networks under economic and environmental goals. However, the research on customer service levels in the supply chain system is still in its infancy. Currently, the service level is one of the critical indicators for measuring the cold chain network of fresh products, and it directly affects a company’s profitability and core competitiveness. In this regard, this study will focus on the goal of customer satisfaction and more realistically evaluate the service level through the maximum service distance of upstream nodes and the actual transportation volume. Therefore, the relevant model maximizes the service level and minimizes the operating cost as the dual-objective function and regards carbon emissions as part of the operating cost so as to balance the service level, operating cost, and carbon emissions.

2.2. Robust Optimization of Supply Chain Network

Supply disruptions and demand fluctuations can bring many negative impacts to the supply system, which highlights the great significance of integrating resilience strategies into the FPCN to enhance the stability of the supply chain. Christopher and Peck [17] first proposed that a resilient supply chain system could recover to regular operation more quickly after dealing with disruptions, and they developed a procedure for identifying and controlling supply chain risks for the first time, accelerating the response to demand fluctuations in the supply chain. On this basis, the resilience strategies of the supply chain network have been widely and deeply expanded. It mainly covers many important aspects, including facility strengthening [18], facility dispersion, semi-finished product preparation, alternative suppliers, storage backups, product redistribution, multi-route transportation [19], cross-level procurement [20], direct shipping, emergency inventory [21], inventory sharing, and multi-set covering strategies. However, in the current research on FPCNs, studies are scarce regarding the implementation and flexible adjustment of these strategies in the face of supply disruptions and demand fluctuations. Additionally, little research has been performed on how these strategies impact the optimization goals of the supply chain network.

When dealing with supply chain risks, the many dynamic factors that need to be evaluated and selected undoubtedly pose severe challenges to uncertainty management. Benhamida et al. [22] proposed a framework using the Fuzzy Multi-Criteria Decision-Making (MCDM) method, combining the Best-Worst Method (BWM) and the Fuzzy Analytic Hierarchy Process (TOPSIS) model to assess how different strategies perform in the face of uncertainties. This approach offers new ideas and methods for solving decision-making problems in uncertain situations. However, this study is mainly based on static analysis and fails to capture dynamic changes fully. To overcome this limitation, when handling the dynamic food delivery vehicle routing problem with demand fluctuations and time-varying speeds, Xie et al. [23] adopted the concept of Rolling Horizon Control (RHC) to convert dynamic challenges into static problems within a specific time horizon. Nevertheless, the model mainly considers the uncertainty of transportation time and gives less attention to other important dynamic factors in cold chain logistics. Nonetheless, related research continues to progress. In recent years, in the field of supply chain network optimization, studies that consider both supply disruptions and demand fluctuations have remained in their initial stages. Scholars, represented by Foroozesh et al. [24] and Mostaghim et al. [25], have modeled this problem using mixed-integer programming models and attempted to generate Pareto solutions with the ε-constraint method, thereby transforming the multi-objective optimization problem into a single-objective optimization problem with constraints. However, due to the excessive reliance of this method on parameter settings, it is difficult to comprehensively obtain the Pareto frontier. On the other hand, scholars, represented by Shekarabi et al. [26], have employed a robust optimization model. Although it addresses supply chain risks from different perspectives, it also has its drawbacks. It has high requirements for data quality and requires a large amount of high-quality data for support.

In the studies mentioned above, fuzzy multi-criteria decision-making, probability analysis, robust optimization, and stochastic programming are commonly used methods for designing under uncertain risks. However, these methods have some drawbacks when optimizing the actual cold chain network of fresh products, specifically in terms of parameter settings and data quality, which makes some computational experiments difficult to handle. Meanwhile, the joint design for supply disruptions and demand fluctuations is still in its infancy. Tools used to solve the problems include Swarm Intelligence Algorithms (SIA), Business Software (BS), including Gurobi 11.0 and CPLEX 20.1.0, Neighborhood Search (NS), etc. Using these tools alone is highly susceptible to the dimensionality of the problem, easily falling into local optimal solutions, thus leading to a decrease in the overall optimization level. Therefore, this study will comprehensively consider the impacts of supply disruptions and demand fluctuations on FPCNs, select the most representative scenarios, and use scenario probabilities to simulate the original uncertainty distribution, thereby reducing the dependence on parameter settings and data quality. At the same time, a new Neighborhood Search-Swarm Intelligence Algorithm will be designed to solve uncertain problems effectively.

2.3. Research Gaps and Their Contributions

To summarize, some research gaps remain in previous studies.

(a)

In the field of multi-objective optimization of the supply chain, studies, represented by Pratap et al. [12] and Gholian-Jouybari et al. [15], mainly focus on operational cost control and greenhouse gas emission reduction. Although these studies encompass multi-dimensional low-carbon optimization measures, such as static facility location and dynamic route planning, the absence of customer service level in the research system renders it impossible to construct a comprehensive optimization loop for cost reduction, efficiency improvement, and the sustainable development of the FPCN.

(b)

In the field of supply chain risk research, operational risks such as uncertain customer demand and strategic-level supply disruptions, especially those in the fresh cold chain industry, have received less attention in previous studies. Foroozesh et al. [24] and Mostaghim et al. [25] focused on the chain reactions of supply disruptions on society and the economy. In contrast, Shekarabi et al. [26] proposed various strategies to mitigate the disruption risks of FPCN. Therefore, a significant gap remains in the research on resilient strategies applicable to FPCN in scenarios of supply disruptions and demand fluctuations.

(c)

Regarding the solution algorithms for optimization models, the optimization algorithms developed by current scholars are not yet mature. Using commercial business software or neighborhood search to solve robust optimization models may have many drawbacks, as shown in the relevant studies of Paam et al. [10], Foroozesh et al. [24], and Mostaghim et al. [25], which makes it difficult to effectively adapt to relatively complex situations of supply chain robust optimization problems.

In response to these issues, this study aims to bridge the gaps and contribute innovative ideas and discussions to the development of fresh products cold chain optimization. The new contributions of this study are summarized as follows.

(A)

To design a practical RL-FPCN model, the bi-objective function constructed in this work combines three indicators, namely the environmental indicator (reducing dynamic and static carbon emissions), the economic indicator (minimizing the total operating cost), and the demand satisfaction indicator (maximizing the service level).

(B)

This study innovatively designs and optimizes the RL-FPCN, considering the supply chain risks of supply disruptions and demand fluctuations simultaneously. The supply disruptions include the capacity loss of facility nodes and the unavailability of paths between facility layers. Meanwhile, this research adopts four resilience strategies for optimization, namely site strengthening, emergency inventory, cross-level procurement, and multi-route transportation, to effectively address the mentioned supply and demand risks.

(C)

This study provides a new idea for solving models. It innovatively integrates the improved OPTICS algorithm, considering temporal–spatial distance and the improved slime mold algorithm with a full iteration cycle, thus constructing the neighborhood search–swarm intelligence algorithm OTS-FICSMA [27]. The nodes with similar time and space dimensions are first clustered through this improvement to generate the initial solution. Then, the intervention optimization is carried out on the full cycle of the initial solution iteration to improve the quality of the solution continuously. Finally, the solution performance of the algorithm is greatly improved.

(D)

At the practical application level, taking a real case in Chengdu, China as an example, the scenario analysis method is used to evaluate the impacts of the four resilience strategies and their collaborative performance in mitigating scenarios of supply disruptions and demand fluctuations. This work further compares the solution performance of the proposed OTS-FICSMA with those of algorithms with similar structures. The comparison verifies that the OTS-FICSMA effectively solves problems, providing strong support for the practical application of logistics distribution network optimization.

Table 1. Comparisons between related literature and this study.

Related Literature	Carbon Emission		Resilience Analysis			Customer Service Level	Solution Approach	Real-Case Analysis
	Static	Dynamic	Disruption	Fluctuation	Strategies
Leng et al. (2024) [6]		√		√		√	SIA
Paam et al. (2022) [10]	√			√	√		BS	√
Pratap et al. (2022) [12]	√	√		√			SIA
Peng et al. (2024) [13]		√		√	√		SIA	√
Fathollahzadeh et al. (2024) [14]		√		√	√		SIA	√
Gholian-Jouybari et al. (2024) [15]	√	√			√		SIA	√
Moghadda-si et al. (2023) [16]		√		√		√	BS
Hasani et al. (2021) [18]			√	√	√	√	NS	√
Sabouhi et al. (2021) [20]			√		√		BS	√
Alikhani et al. (2021) [21]			√	√	√		BS	√
Benhamida et al. (2025) [22]			√	√	√		BWM+ TOPSIS	√
Xie et al. (2024) [23]				√	√		SIA	√
Foroozesh et al. (2022) [24]		√	√		√		BS	√
Mostaghim et al. (2024) [25]		√	√		√		BS	√
Shekarabi et al. (2024) [26]			√		√		BS
This Study	√	√	√	√	√	√	NS-SIA	√

summarizes the main characteristics of these studies, including carbon emissions, resilience analysis, customer service level, solution methods, and real-case analysis. Among them, static carbon emissions mainly consider the operation and production processes of station facilities, while dynamic carbon emissions emphasize the vehicle transportation process during the distribution of goods. Distinguishing between these two aspects helps decision-makers in cold chain logistics formulate targeted emission reduction measures comprehensively and achieve the goal of reducing carbon emissions more effectively. In terms of resilience, the indicators include disruption factors, fluctuation factors, and whether resilience strategies are proposed. The customer service level mainly discusses whether the model takes it as a key optimization objective. Regarding the solution methods, this study uses the criterion of which tool is used to solve the model. The real-case analysis focuses on whether the study has carried out a series of simulation analyses on real cases.

3. Problem Description and Model Construction

To describe the proposed RL-FPCN problem, Figure 1 shows a classic four-layer FPCN. It comprises direct procurement suppliers, processing centers, regional distribution centers (RDCs), and store warehouses. As the starting point of FPCN, the direct procurement suppliers provide primary fresh products by relying on the direct procurement base of fresh products. Cold chain processing centers need to refrigerate, process, and package the raw materials to ensure the consistency of their quality and hygiene standards, thus providing consumers with more distinctive fresh products. The locations of supply and demand for FPCN fresh products are usually scattered and far apart. To respond quickly and effectively to the personalized needs of terminals, RDCs need to undertake the essential functions of cargo collection and distribution, as well as last-mile distribution. The storage warehouses serve as the endpoints of the FPCN and receive processed fresh products from the RDC. In summary, the proposed RL-FPCN aims to achieve the minimization of the total cost of the supply network through location decision and route planning, and maximize the service level of store warehouses, enhancing the resilience and low-carbon sustainability of the entire fresh cold chain in the face of complex situations such as supply disruptions and demand fluctuations.

Hence, this study considers four supply disruption scenarios (earthquake O1, flood O2, fire O3, and public health incident O4) and four demand fluctuation scenarios (gradual demand increase N1, sudden demand surge N2, gradual demand decrease N3, and sharp demand decline N4). At the same time, the situation of no disruption O0 and normal demand N0 are set as reference benchmarks for comparison.

Through the mixed-integer programming model of L-FPCN and the scenario settings, we have developed an optimization model to reflect the resilient and low-carbon strategies for the network optimization of fresh products cold chain (RL-FPCN).

Among them, the four resilience strategies proposed in this research include:

I.

Site strengthening strategy: this is reflected by investing more funds in processing centers and RDCs to reduce the proportion of capacity loss in disruption scenarios. In general, the level of strengthening is proportional to the operating costs and inversely proportional to capacity losses.

II.

Emergency inventory strategy: this means keeping additional emergency stock in processing centers and RDCs. In this case, when a node loses capacity, the emergency inventory will meet customer demand. Maintaining emergency inventory incurs emergency inventory costs.

III.

Cross-level procurement strategy: this allows fresh products to be directly transported from processing centers to store warehouses, saving the operation cost of RDCs and enhancing flexibility. However, the unit transportation cost of direct delivery to store warehouses is generally higher than through the RDCs.

IV.

Multi-route transportation strategy: this allows nodes at a certain level to receive raw materials or products from multiple sites at the previous level. Different route selections at various levels will incur the corresponding costs.

3.1. Model Assumptions

This work makes the following assumptions to model the proposed RL-FPCN problem.

• When supply disruptions occur, a site’s inventory capacity or production capacity is lost, or certain routes between sites become unavailable.
• When demand fluctuations occur, the demand for a certain fresh product in the store warehouse will vary within a reasonable range.
• Supply disruptions and demand fluctuations are considered independent events.
• Carbon emissions are incorporated into the model as part of the operating costs.
• The carbon emission costs generated by on-site operations, production activities, and route transportation in FPCN are estimated by considering dynamic and static carbon emission tax rates.
• The emissions are related to the attributes of site operations and the quantity of production or transportation.

3.2. Symbols and Variables

To mathematically describe the RL-FPCN problem, this study defines the symbols and variables, as shown in

Table 2. Symbols and variables.

Sets
		Set of direct procurement suppliers, $i\in I$
$J$		Set of processing centers, $j\in J$
$K$		Set of RDCs, $k\in K$
$L$		Set of store warehouses, E
$R$		Set of raw materials produced by suppliers, $r\in R$
$P$		Set of fresh products, $p\in P$
$O$		Set of supply disruption scenarios, $o\in O$
$N$		Set of demand fluctuation scenarios, $n\in N$
$V$		Strengthening levels of sites under the site strengthening strategy, $v\in V$
$W$		Set of optional routes under the multi-route transportation strategy, $w\in W$
Parameters related to the low-carbon network
Route planning	$di{s}_{ij}$	Actual distance from direct procurement supplier i to processing center j
	$di{s}_{jk}$	Actual distance from processing center j to RDC k
	$di{s}_{kl}$	Actual distance from RDC k to store warehouse l
	$t{s}_{ij}^r$	Unit transportation cost of transporting raw material r from direct procurement supplier i to processing center j
	$t{p}_{jk}^p$	Unit transportation cost of transporting product p from processing center j to RDC k
	$t{r}_{kl}^p$	Unit transportation cost of transporting product p from RDC k to store warehouse l
	$et{s}_{ij}$	Transportation carbon emission factor from direct procurement supplier i to processing center j
	$et{p}_{jk}$	Transportation carbon emission factor from processing center j to RDC k
	$et{r}_{kl}$	Transportation carbon emission factor from RDC k to store warehouse l
Location decision	$f{p}_j$	Fixed operating cost of processing center j
	$f{r}_k$	Fixed operating cost of RDC k
	$o{s}_p$	Ordering cost of product p
	$o{q}_k^p$	Quantity of product p ordered by RDC k each time
	$s{t}_k$	Inventory holding cost rate of RDC k
	$es{p}_j$	Operational carbon emission factor of processing center j
	$es{r}_k$	Operational carbon emission factor of RDC k
	$ew{f}_j^p$	Production carbon emission factor of processing center j for producing product p
	$p{c}_j^p$	Unit cost of processing center j for producing product p
Others	$hsdi{s}_{kl}$	Maximum service distance from RDC k to store warehouse l
	$pe{n}_{lm}^p$	Penalty cost for the unmet demand of customer m for product p shipped from RDC l
	$c_p$	Unit price of product p
	$ce$	Fixed carbon tax
Parameters related to theresilience network
Route planning	$di{s}_{jl}$	Actual distance from store warehouse l to processing center j under the cross-level procurement strategy
	$r{s}_{ij}$	Route selection cost from direct procurement supplier i to processing center j under the multi-route transportation strategy
	$r{p}_{jk}$	Route selection cost from processing center j to RDC k under the multi-route transportation strategy
	$r{k}_{jl}$	Route selection cost from processing center j to store warehouse l under the cross-level procurement and multi-route transportation strategies
	$r{r}_{kl}$	Route selection cost from RDC k to store warehouse l under the multi-route transportation strategy
	$t{k}_{jl}^p$	Unit transportation cost of transporting product p from processing center j to store warehouse l under the cross-level procurement strategy
	$et{k}_{jl}$	Transportation carbon emission factor from processing center j to store warehouse l under the cross-level procurement strategy
Location decision	$f{p}_j^v$	Fixed operating cost of processing center j after being strengthened with the strengthening level v
	$f{r}_k^v$	Fixed operating cost of RDC k after being strengthened with the strengthening level v
	$rc{p}_j^p$	Unit procurement cost of emergency inventory of product p purchased from processing center j
	$rc{r}_k^p$	Unit procurement cost of emergency inventory of product p purchased from CDC k
Others	${\phi }^o$	Probability of the occurrence of the supply disruption scenario o
	${\phi }^n$	Probability of the occurrence of the demand fluctuation scenario n
	$d_l^{pon}$	Demand quantity of product p at store warehouse l under the supply disruption scenario o and demand fluctuation scenario n
Decision variables
Route planning	$se{r}_{kl}$	Binary variable = 1, if the service distance from RDC k to store warehouse l is less than or equal to [the maximum service distance]; and 0 otherwise
	$z{s}_{ij}^w$	Binary variable = 1, if the route w from direct procurement supplier i to processing center j is selected to operate; and 0 otherwise
	$z{p}_{jk}^w$	Binary variable = 1, if the route w from processing center j to RDC k is selected to operate; and 0 otherwise
	$z{r}_{kl}^w$	Binary variable = 1, if the route w from RDC k to store warehouse l is selected to operate; and 0 otherwise
	$z{k}_{jl}^w$	Binary variable = 1, if the route w from processing center j to store warehouse l is selected; and 0 otherwise
	$x{s}_{ij}^{ron}$	Continuous variable: quantity of raw material r from direct procurement supplier i to processing center j under the supply disruption scenario o and demand fluctuation scenario n
	$x{p}_{jk}^{pon}$	Continuous variable: quantity of product p from processing center j to RDC k under the supply disruption scenario o and demand fluctuation scenario n
	$x{r}_{kl}^{pon}$	Continuous variable: quantity of product p from RDC k to store warehouse l under the supply disruption scenario o and demand fluctuation scenario n
	$x{k}_{jl}^{pon}$	Continuous variable: quantity of product p from processing center j to store warehouse l under the supply disruption scenario o and demand fluctuation scenario n
Location decision	$y{p}_j^v$	Binary variable = 1, if processing center j strengthened with the strengthening level v is selected to operate; and zero otherwise
	$y{r}_k^v$	Binary variable = 1, if RDC k strengthened with the strengthening level v is selected to operate; and zero otherwise
	$z_l^{pon}$	Continuous variable: quantity of product p in short supply at store warehouse l under the supply disruption scenario o and demand fluctuation scenario n
	$v{p}_j^{pno}$	Continuous variable: quantity of emergency inventory of product p purchased from processing center j under the supply disruption scenario o and demand fluctuation scenario n
	$v{r}_k^{pno}$	Continuous variable: quantity of emergency inventory of product p purchased from RDC k under the supply disruption scenario o and demand fluctuation scenario n

.

3.3. Construction of the Basic Model

The L-FPCN model maximizes the service level and minimizes the operating costs as its two objectives. It makes decisions on multiple aspects like site location and quantity, transportation routes, and inventory levels, and incorporates carbon emissions from various processes. The relevant objective function and constraints of L-FPCN are as follows.

(1)

Objective Function

Based on the study of Leng et al. [6] and Zeng et al. [28], the maximization of the service level in the L-FPCN model is defined as follows:

$$L{Z}_1=Max({\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}x{r}_{kl}^{pon}}}}z{r}_{kl}}}se{r}_{kl}/{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{d}_l^{pon}}}}})$$

(1)

The minimization of operating costs in the L-FPCN model is defined as follows.

$$L{Z}_2=Min\{C{F}_1+C{F}_2+C{F}_3\}$$

(2)

$$\begin{array}{ll}C{F}_1=& {\displaystyle \sum _{j\in J}f{p}_{j}y{p}_j+}{\displaystyle \sum _{k\in K}f{r}_{k}y{r}_k}+{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{r\in R}{\phi }^o{\phi }^{n}p{c}_j^{p}x{s}_{ij}^{ron}}}}}}}\\ & +{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{d}_l^{pon}o{s}_p/o{q}_k^p}}}}}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}x{c}_{kl}^{pon}{c}_{p}s{t}_l/2}}}}}\\ & +{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{d}_l^{pon}{c}_p}}}}\end{array}$$

(3)

$$\begin{array}{ll}C{F}_2=& {\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\phi }^o{\phi }^n}({\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{r\in R}di{s}_{ij}t{s}_{ij}^{r}x{s}_{ij}^{ron}+{\displaystyle \sum _{j\in J}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{p\in P}di{s}_{jk}t{p}_{jk}^{p}x{p}_{jk}^{pon}}}}}}}}\\ & +{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}(di{s}_{kl}t{c}_{kl}^{p}x{c}_{kl}^{pon}+pe{n}_{kl}^p{z}_l^{pon}))}}}\end{array}$$

(4)

$$\begin{array}{ll}C{F}_3=& ce({\displaystyle \sum _{j\in J}es{p}_{j}y{p}_j+{\displaystyle \sum _{k\in K}es{r}_{k}y{r}_k}})+ce{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\phi }^o{\phi }^n}} \\ & ({\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{r\in R}ew{f}_j^{p}x{s}_{ij}^{\mathrm{ro}n}}}}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{r\in R}et{s}_{ij}x{s}_{ij}^{ron}}}}+\\ & {\displaystyle \sum _{j\in J}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{p\in P}et{p}_{jk}x{p}_{jk}^{pon}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}et{r}_{kl}x{r}_{kl}^{p\mathrm{o}n}}}}}}})\end{array}$$

(5)

In the objective Equation (1), LZ1 represents the maximization of the customer service level of the L-FPCN within a specific service scope; in the objective Equation (2), LZ2 represents the minimization of the comprehensive operating costs of the L-FPCN, including site selection costs (CF1), transportation costs (CF2), and comprehensive carbon emission costs (CF3). The objective of Equation (3) is to determine the site selection costs. Among them, the first to second terms are the fixed operating costs of the processing centers and RDCs, the third term is the production cost of the processing centers, and the fourth to sixth terms constitute the inventory costs of the RDCs, which are the ordering cost, the product inventory holding cost of the RDCs, and the material cost, respectively. The objective of Equation (4) is to determine the transportation costs. The first to third terms are the sums of the transportation costs between points at various levels under different scenarios, and the fourth term is the sum of the penalty costs for unmet customer demands under different scenarios. Equation (5) aims to calculate the comprehensive carbon emission costs. The first to second terms are the carbon emissions during site operation processes, the third term is the carbon emissions during the production process, and the fourth to sixth terms are the carbon emissions during the transportation process.

(2)

Constraints

In addition to the typical constraints that ensure the model’s integrity, this paper innovatively proposes the customer demand constraints (f), which are used to meet customer demands and the minimum service level.

a.

Route selection constraints: Equations (6)–(11) ensure that the transportation routes can only be selected among the open sites, that is, all the nodes covered by both ends of the transportation routes at all levels are the chosen sites.

$$z{s}_{ij}\le y{s}_i \forall i\in I,j\in J$$

(6)

$$z{s}_{ij}\le y{p}_j \forall i\in I,j\in J$$

(7)

$$z{p}_{jk}\le y{p}_j \forall j\in J,k\in K$$

(8)

$$z{p}_{jk}\le y{c}_k \forall j\in J,k\in K$$

(9)

$$z{c}_{kl}\le y{c}_k \forall k\in K,l\in L$$

(10)

$$z{c}_{kl}\le y{r}_l \forall k\in K,l\in L$$

(11)

b.

Site capacity constraints: Equations (12) and (13) state that the product flowing out of the accessible site cannot exceed its remaining capacity in the disruption and fluctuation scenarios. $rca{p}_j^{po}$ and $rca{r}_k^{po}$ represent the capacities of processing center j and RDC k for storing product p under the supply disruption scenario o, respectively.

$${\displaystyle \sum _{k\in K}x{p}_{jk}^{pon}}\le y{p}_{i}rca{p}_j^{po} \forall j\in J,p\in P,o\in O,n\in N$$

(12)

$${\displaystyle \sum _{l\in L}x{r}_{kl}^{pon}}\le y{r}_{k}rca{r}_k^{po} \forall k\in K,p\in P,o\in O,n\in N$$

(13)

c.

Traffic flow balance constraints: Equations (14)–(16) indicate that a product flow is generated between two nodes only if the path between them is selected and available. There are no capacity constraints on these routes. $dr{s}_{ij}^o$, $dr{p}_{jk}^o$, and $dr{r}_{kl}^o$ are all binary variables, representing the road conditions from supplier i to factory j, from factory j to RDC k, and from RDC k to store warehouse l, respectively, under the supply disruption scenario o. If the road between the corresponding nodes is disrupted, the value is 1; otherwise, it is 0.

$$x{s}_{ij}^{ron}\le z{s}_{ij}\left(1-dr{s}_{ij}^{\mathrm{o}}\right)\beta \forall i\in I,j\in J,r\in R,\mathrm{o}\in O,n\in N$$

(14)

$$x{p}_{jk}^{pon}\le z{p}_{jk}\left(1-dr{p}_{jk}^o\right)\beta \forall j\in J,k\in K,p\in P,o\in O,n\in N$$

(15)

$$x{r}_{kl}^{pon}\le z{r}_{kl}\left(1-dr{r}_{kl}^o\right)\beta \forall k\in K,l\in L,p\in P,o\in O,n\in N$$

(16)

d.

Single route constraints: Equations (17)–(19) ensure a single transportation route between nodes at various levels.

$${\displaystyle \sum _{i\in I}z{s}_{ij}}\le 1 \forall j$$

(17)

$${\displaystyle \sum _{j\in J}z{p}_{jk}}\le 1 \forall k$$

(18)

$${\displaystyle \sum _{k\in K}z{r}_{kl}}\le 1 \forall l$$

(19)

e.

Goods flow balance constraints: Equations (20) and (21), respectively, indicate the flow balance constraints of the processing centers and RDCs. $u_{rp}$ represents the conversion rate from raw material r to product p.

$${\displaystyle \sum _{i\in I}x{s}_{ij}^{r\mathrm{o}n}}={\displaystyle \sum _{p\in P}{\displaystyle \sum _{k\in K}{u}_{rp}x{p}_{jk}^{pon}}} \forall j\in J,r\in R,o\in O,n\in N$$

(20)

$${\displaystyle \sum _{j\in J}x{p}_{jk}^{pon}}={\displaystyle \sum _{l\in L}x{d}_{kl}^{pon}} \forall k\in K,p\in P,o\in O,n\in N$$

(21)

f.

Customer demand constraints: Equation (22) shows that the actual demand for the storehouse is composed of the quantity of received and unsatisfied goods. Equation (23) reflects the minimum service level constraint of the storehouse, which stipulates that the ratio of the amount of goods received to the actual demand shall not be lower than the minimum service standard.

$${\displaystyle \sum _{k\in K}x{r}_{kl}^{pon}}+z_l^{pon}=d_l^{pon} \forall l\in L,p\in P,o\in O,n\in N$$

(22)

$${\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}x{r}_{kl}^{pon}}}}/{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{d}_l^{pon}}}\ge \alpha \forall o\in O,n\in N$$

(23)

g.

Equations (24) and (25) describe the binary variables, and Equation (26) ensures the positivity of the output results.

$$y{p}_j,y{c}_k\in \left\{0,1\right\}$$

(24)

$$z{s}_{ij},z{p}_{jk},z{r}_{kl}\in \left\{0,1\right\}$$

(25)

$$x{s}_{ij}^{ron},x{p}_{jk}^{pon},x{r}_{kl}^{pon},z_l^{pon}\ge 0$$

(26)

3.4. Robust Model Construction

Integrating these four resilient strategies into the L-FPCN model, a mixed integer programming model of RL-FPCN is formed to enhance its robustness and reliability. The objective of the RL-FPCN model is to maximize the service level while minimizing the operating costs. The model decisions include the selection of sites with different locations and strengthening levels; the choice of transportation routes between nodes at various levels; the transportation decisions of raw materials and products; the decision on the amount of emergency inventory; and the decision on the shortage quantity.

The model of RL-FPCN maximizes the service level and minimizes the operating costs, which are two objectives. It makes decisions on multiple aspects like site selection and strengthening levels, transportation routes, emergency inventory, and shortage quantities. Integrating four resilient strategies into the L-FPCN model enhances its robustness and reliability.

(1)

Objective Function

The maximization of the service level in the RL-FPCN model is defined as shown in Equation (27).

$$RL{Z}_1=Max({\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{w\in W}{\displaystyle \sum _{p\in P}x{r}_{kl}^{pon}}}}}z{r}_{kl}^w}}se{r}_{kl}/{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{d}_l^{pon}}}}})$$

(27)

The minimization of operating costs in the RL-FPCN model is defined as shown in Equations (28)–(33).

$$RL{Z}_2=Min\{C{F}_6+C{F}_7+C{F}_8+C{F}_9+C{F}_{10}\}$$

(28)

$$C{F}_6={\displaystyle \sum _{j\in J}{\displaystyle \sum _{v\in V}f{p}_j^{v}y{p}_j^v}}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{v\in V}f{r}_k^{v}y{r}_k^v}}+{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{r\in R}{\phi }^o{\phi }^{n}p{c}_j^{p}x{s}_{ij}^{ron}}}}}}}$$

(29)

$$\begin{array}{ll}C{F}_7=& {\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{d}_l^{pon}o{s}_p/o{q}_k^p}}}}}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}x{r}_{kl}^{pon}{c}_{p}s{t}_l/2}}}}}\\ & +{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{d}_l^{pon}{c}_p}}}} +{\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\displaystyle \sum _{p\in P}({\displaystyle \sum _{j\in J}{\phi }^o{\phi }^{n}rc{p}_j^{p}v{p}_j^{pno}}}}}\\ & +{\displaystyle \sum _{k\in K}{\phi }^o{\phi }^{n}rc{r}_k^{p}v{r}_k^{pno}})\end{array}$$

(30)

$$\begin{array}{ll}C{F}_8=& {\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{w\in W}r{s}_{ij}z{s}_{ij}^w}}}+{\displaystyle \sum _{j\in J}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{w\in W}r{p}_{jk}z{p}_{jk}^w}}}+{\displaystyle \sum _{j\in J}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{w\in W}r{k}_{jl}z{k}_{jl}^w}}}\\ & +{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{w\in W}r{r}_{kl}z{r}_{kl}^w}}}\end{array}$$

(31)

$$\begin{array}{ll}C{F}_9=& {\displaystyle \sum _{o\in O}{\displaystyle \sum _{n\in N}{\phi }^o{\phi }^n}({\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{r\in R}di{s}_{ij}t{s}_{ij}^{r}x{s}_{ij}^{ron}+{\displaystyle \sum _{j\in J}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{p\in P}di{s}_{jk}t{p}_{jk}^{p}x{p}_{jk}^{pon}}}}}}}}\\ & +{\displaystyle \sum _{j\in J}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}di{s}_{jl}t{k}_{jl}^{p}x{k}_{jl}^{pon}}}}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}(di{s}_{kl}t{r}_{kl}^{p}x{r}_{kl}^{pon}}}}+pe{n}_{kl}^p{z}_l^{pon}))\end{array}$$

(32)

$$\begin{array}{ll}C{F}_{10}=& ce({\displaystyle \sum _{j\in J}{\displaystyle \sum _{v\in V}es{p}_j^{v}y{p}_j^v}}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{v\in V}es{r}_k^{v}y{r}_k^v}})\\ & +ce{\displaystyle \sum _{\mathrm{o}\in O}{\displaystyle \sum _{n\in N}{\phi }^o{\phi }^n(}}{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{p\in P}{\displaystyle \sum _{r\in R}ew{f}_j^{p}x{s}_{ij}^{r\mathrm{o}n}}}}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{r\in R}et{s}_{ij}x{s}_{ij}^{ron}}}}\\ & +{\displaystyle \sum _{j\in J}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{p\in P}et{p}_{jk}x{p}_{jk}^{pon}}}}+{\displaystyle \sum _{j\in J}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}et{k}_{jl}x{k}_{jl}^{pon}}}}+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{l\in L}{\displaystyle \sum _{p\in P}et{r}_{kl}x{r}_{kl}^{pon}}}})\end{array}$$

(33)

In Equation (27), RLZ1 represents the maximization of the customer service level of the RL-FPCN with the multi-route transportation strategy within a specific service scope. In Equation (28), RLZ2 represents the minimization of the comprehensive operating costs of the RL-FPCN. Equation (29) is the site selection cost under the site strengthening strategy, where the first two terms are the fixed operating costs of processing centers and RDCs, and the third term is the production cost of processing centers. Equation (30) represents the inventory cost under the emergency inventory strategy, where the first three terms correspond to the inventory costs of RDCs, and the last two terms describe the procurement costs associated with emergency inventory. Equation (31) is the route selection cost under the multi-route transportation strategy, and Equation (32) is the transportation cost under the cross-level procurement strategy. Equation (33) is the comprehensive carbon emission cost, and since its modeling idea is the same as that in the L-FPCN, no repeated description is provided.

(2)

Constraints

The RL-FPCN model is developed from the L-FPCN model by integrating four resilient strategies. Hence, this section only presents the constraints related to these strategies. The design of other constraints is consistent with that of the L-FPCN model and will not be repeated.

a.

Site strengthening constraints: Equations (34) and (35) indicate that at most one degree of strengthening can be chosen for the sites selected for strengthening.

$${\displaystyle \sum _{v\in V}y}{p}_j^v\le 1 \forall j\in J$$

(34)

$${\displaystyle \sum _{v\in V}y}{r}_k^v\le 1 \forall k\in K$$

(35)

b.

Emergency inventory constraints: Equations (36)–(39) indicate that emergency stock configuration is triggered only when a processing center or RDC is selected. In addition, the ratio between the emergency inventory capacity and the remaining capacity of the site is constrained by the maximum original capacity. $ca{p}_j^p$ and $ca{r}_k^p$ represent the capacities of processing center j and RDC k for storing product p, respectively.

$$v{p}_j^{pno}\le {\displaystyle \sum _{v\in V}y}{p}_j^{v}ca{p}_j^p \forall j\in J,p\in P,o\in O,n\in N$$

(36)

$$v{r}_k^{pno}\le {\displaystyle \sum _{v\in V}y}{r}_k^{v}ca{r}_k^p \forall k\in K,p\in P,o\in O,n\in N$$

(37)

$$v{p}_j^{pno}+rca{p}_j^{pvo}\le ca{p}_j^p \forall j\in J,p\in P,v\in V,o\in O,n\in N$$

(38)

$$v{r}_k^{pno}+rca{r}_k^{pvo}\le ca{r}_k^p \forall k\in K,p\in P,v\in V,o\in O,n\in N$$

(39)

c.

Cross-level procurement constraints: Equations (40)–(42) supplement the route selection from factories to RDCs and the freight flow information.

$$z{k}_{jl}^w\le {\displaystyle \sum _{v\in V}y}{p}_j^v \forall j\in J,l\in L,w\in W$$

(40)

$$z{k}_{jl}^w\le {\displaystyle \sum _{v\in V}y}{r}_l^v \forall j\in J,l\in L,w\in W$$

(41)

$${\displaystyle \sum _{i\in I}x{s}_{ij}^{ron}}={\displaystyle \sum _{p\in P}{u}_{rp}\left({\displaystyle \sum _{k\in K}x{p}_{jk}^{pon}+{\displaystyle \sum _{l\in L}x{p}_{jl}^{pon}}}\right) }\forall j\in J,r\in R,o\in O,n\in N$$

(42)

d.

Multi-route transportation constraints: Equation (43) supplements the traffic flow information from factories to RDCs. $dr{k}_{jl}^o$ is a binary variable, representing the road conditions from factory j to store warehouse l under the supply disruption scenario o. If the road between the corresponding nodes is disrupted, the value is 1; otherwise, it is 0. In addition, $\beta$, as a model adjustment factor, is a constant value of infinity.

$$x{k}_{jl}^{pon}\le z{k}_{ij}^w\left(1-dr{k}_{jl}^o\right)\beta \forall j\in J,l\in L,p\in P,o\in O,n\in N,w\in W$$

(43)

4. Proposed OTS-FICSMA

To solve the NP-hard problem of RL-FPCN, referring to the development idea of the two-stage heuristic algorithm with swarm intelligence algorithm as the core proposed by Sun et al. [29] and Wang et al. [27], this paper designs a new Neighborhood Search–Swarm Intelligence Algorithm, namely OTS-FICSMA.

On the one hand, the nodes in the fresh cold chain network exhibit significant spatiotemporal characteristics. To address this, we propose an OPTICS algorithm with spatiotemporal distance. This algorithm clusters nodes with similar spatiotemporal distances, generating an initial solution and reducing the dimensionality of the problem. Compared to other clustering methods like the K-Means and DBSCAN algorithms, our proposed algorithm has distinct advantages. It can automatically identify clusters of varying densities using the neighborhood radius and density. This effectively overcomes the problem of parameter sensitivity and enables better adaptation to the complexity of the fresh cold chain network.

On the other hand, we introduce and improve the SMA to conduct a global search for the optimal initial solution. The SMA has a strong global search ability in the solution space, and its unique search and encirclement mechanism helps to explore a broader solution space. However, when dealing with complex high-dimensional models, the original algorithm has problems, such as a single optimization mechanism in the early stage and being easily disturbed in the later optimization stage. Therefore, this study optimizes the entire iteration cycle of the SMA.

The following part provides a detailed introduction to the design idea and details of the calculation of the OTS-FICSMA.

4.1. Improved OPTICS Algorithm Considering Temporal–Spatial Distance (OTS)

This section improves the OPTICS algorithm [30] in accordance with the characteristics of the RL-FSCSN model. Specifically, we introduce the “temporal–spatial distance” to modify the calculation of the core distance and reachability distance in OPTICS. By clustering downstream nodes with similar spatiotemporal dimensions, we can partition the customer allocation areas of different spatiotemporal clusters and generate an initial solution. The detailed improvement steps are as follows:

First, evaluate the spatial distance and temporal distance between the two downstream nodes $C_i$ and $C_j$, respectively.

For the spatial distance, we use the Euclidean distance between $C_i$ and $C_j$ to measure it. For the temporal distance, assuming that the time windows of $C_i$ and $C_j$ are $[E{T}_i,L{T}_i]$ and $[E{T}_j,L{T}_j]$, respectively. Suppose that the sum of the earliest departure time and transportation time is $Q$, and the latest departure time and transportation time is $E$. Then, there are four situations for the interval $[Q,E]$, as shown in Figure 2 and Equation (44).

$${\mathrm{d}}_{\mathrm{ij}}^T=\left\{\begin{array}{l}{I}_1(E{T}_j-E),E<E{T}_j\\ I_2(Q-L{T}_j),Q>\mathrm{L}{T}_j\\ I_3(t_i+t_{sp}),Q\le E{T}_j<E\cup Q<L{T}_j<E\\ t_i+t_{sp},E{T}_j<Q<E<L{T}_j\end{array}\right.$$

(44)

In Figure 2, there are four scenarios as follows. For the first case, the vehicle arrives earlier than the specified time window as a whole, and $I_1$ represents the early arrival penalty function. For the second case, the vehicle arrives later than the specified time window as a whole, and is subject to the late arrival penalty function. For the third case, the vehicle’s arrival time partially overlaps with the specified time window, and $I_3$ stands for the deviation coefficient. For the fourth case, the vehicle’s arrival time exactly matches the time window. In this case, the difference between the vehicle’s arrival times can be regarded as the time distance between the two nodes.

Secondly, the temporal distance and spatial distance are normalized to obtain the integrated spatiotemporal distance. The formula is shown in Equation (45). Among them, ${\omega }_1$ and ${\omega }_2$ represent the weight ratio of space–time distance, respectively.

$${\mathrm{d}}_{ij}^{TS}={\omega }_1{d}_{ij}^T+{\omega }_2{d}_{ij},{\omega }_1+{\omega }_2=1$$

(45)

Finally, based on the spatiotemporal distance, analyze the core distance and reachable distance of each node. The formulas are shown in Equations (46) and (47).

$$cd(x)=\left\{\begin{array}{l}Undefined,\left|N_{Epsilon}(x)\right|<MinPts\\ d\left(x,N_{Epsilon}^i(x)\right),\left|N_{Epsilon}(x)\right|\ge MinPts\end{array}\right.$$

(46)

$$rd(y,x)=\left\{\begin{array}{l}Undefined,\left|N_{epsilon}(x)\right|<MinPts\\ \max \left\{cd(x),d(x,y)\right\},\left|N_{epsilon}(x)\right|\ge MinPts\end{array}\right.$$

(47)

The calculation is carried out with $Epsilon$ as the neighborhood radius and $MinPts$ as the neighborhood density threshold. Let $N_{epsilon}(x)$ be the neighborhood of $x$ within a radius of $Epsilon$. The variable $d$ is used to measure the distance between two nodes. Let $N_{Epsilon}^i(x)$ denote the i-th nearest solution to $x$ in the set, and the parameter $i$ ranges within $[1,MinPts]$.

4.2. Improved Slime Mold Algorithm with Full Iteration Cycle (FICSMA)

4.2.1. Basic Solving Principle of the Slime Mold Algorithm

As a swarm intelligence algorithm, SMA is divided into two stages: search and encirclement [27]. It can transmit reflected waves according to the mass of the food source to search for the target solution. When the surrounding food has a high-quality coefficient, the slime mold forms a thicker vein network and converges quickly towards the target. Otherwise, it converges slowly over a larger area. The slime mold’s position update formula at this stage is shown in Equation (48).

$$X_{t+1}=\left\{\begin{array}{l}{X}_t^{best}+vb(W{X}_t^A-X_t^B),r<p\hfill \\ vc{X}_t,r\ge p\hfill \end{array}\right.$$

(48)

Among them, $t$ represents the iterations; $X_t^{best}$ denotes the position of the individual with the highest fitness in the population; $vb$ represents the iteration parameter within the range of $[-a,a]$; $W$ represents the weight coefficient in the searching process of the slime mold; $X_t^A$, $X_t^B$ denote the positions of two random individuals A and B in the population; $vc$ represents the control parameter within the range of [0, 1]; $r$ represents a random number within the range of [0, 1]; $p$ represents the position parameter for the slime mold’s update iteration, and its formula is shown in Equation (49).

$$p=\tanh |S_i-DF|$$

(49)

Among them, $S_i$ represents the fitness value of the i-th slime mold individual in the slime mold population, and $DF$ represents the best fitness value of the slime mold individuals under the current algorithm iteration.

The calculations of the range threshold $a$ for $vb$ and the weight coefficient $W$ are shown in Equations (50) and (51), where $t_{\max }$ represents the maximum iterations.

$$a=\mathrm{arc}\tanh \left(-\left({\displaystyle \frac{t}{t_{\max }}}\right)+1\right)$$

(50)

$$W(SmellIndex(i))=\left\{\begin{array}{l}1+r\log \left({\displaystyle \frac{F^{best}-S_i}{F^{best}-F^{worst}}}+1\right),\mathrm{if} i=Condition\hfill \\ 1-r\log \left({\displaystyle \frac{F^{best}-S_i}{F^{best}-F^{worst}}}+1\right),\mathrm{if} i=Others\hfill \end{array}\right.$$

(51)

Among them, $F^{best}$ and $F^{\mathit{worst}}$ represent the maximum and minimum fitness values in the current iterative process. When $i=Condition$, it represents the individuals whose fitness ranks in the top 50% of the slime mold population, and when $i=Others$, it represents the remaining individuals. $SmellIndex(i)$ represents the position index of the fitness value sequence of the slime mold individuals.

When slime molds search out food, they will continue to search randomly to seek food sources that offer higher quality. The position update formula for some slime mold individuals at this stage is shown in Equation (52).

$$X_{t+1}^{rand}=\left\{\begin{array}{l}rand(UB-LB)+LB,rand<z\hfill \\ X_t^{best}+vb(W{X}_t^A-X_t^B),rand\ge z\cap r<p\hfill \\ vc{X}_t,rand\ge z\cap r\ge p\hfill \end{array}\right.$$

(52)

Among them, $UB$ and $LB$ represent the upper and lower limits of the search range of the slime molds, rand represents a random number within the range of [0, 1], and $z$ represents the proportional parameter of the slime mold individuals performing random searches in the slime mold population, usually within the range of [0.02, 0.06].

4.2.2. Improvement Path of the Slime Mold Optimization Algorithm with Full Iteration Cycle

(1)

Multi-objective problem balancing mechanism

The RL-FPCN proposed in this study involves optimizing service level, operating cost, and carbon emissions in the cold chain network of fresh products. In response, we have constructed a dual-objective function that maximizes the service level and minimizes the operating cost. Moreover, we have included the carbon emission cost as part of the operating cost, simplifying the model while considering all three factors. However, the trade-off of the weights of these objectives is closely related to decision-makers’ preferences. Therefore, we have introduced an index-based balance mechanism for multi-objective problems. Specifically, we have constructed a ranking matrix to rank the optimization results of each solution, aiming to evaluate and integrate the comprehensive fitness values of multiple objectives. The steps are as follows: firstly, calculate the initial values of all objective functions for each solution. Secondly, the ranking matrix is used to rank the performance of each solution in each objective function. Finally, the results are converted into a ranking of the comprehensive fitness values of each solution.

Suppose the optimization solution for RL-FSCN is $OS(h)$, the initial value of the dual-objective function is $f_n$, and the comparison result arranged in ascending order is $R{C}_n(h)$, where n represents the n-th objective function, and h represents the h-th optimization solution. Then, the performance ranking process is shown in

Table 3. The performance rankings of each solution in various objective functions.

Objects	Optimization Scheme
	$OS(1)$	$OS(2)$	$OS(3)$	$OS(4)$	···	$OS(h)$
$RL{Z}_1$	$R{C}_1(2)$	$R{C}_1(h)$	$R{C}_1(4)$	$R{C}_1(3)$	···	$R{C}_1(1)$
$RL{Z}_2$	$R{C}_2(h)$	$R{C}_2(4)$	$R{C}_2(1)$	$R{C}_2(3)$	···	$R{C}_2(2)$

, and the calculations related to the ranking are detailed in Formulas (53) and (54).

$$f_n(h)=\left\{\begin{array}{l}{(H-O{S}_n(h))}^2,O{S}_n(h)>1\hfill \\ A{H}^2,O{S}_n(h)=1\hfill \end{array}\right.$$

(53)

$$f(h)=f_1(h)+f_2(h)$$

(54)

Formula (53) is the fitness ranking result of the n-th objective function, where $H$ is the total number of optimization schemes. Formula (54) is the comprehensive fitness ranking result of the h-th optimization scheme, where $A$ is a variable with a value of 1 to 2, used to increase the fitness of the optimal individual.

The ranking matrix constructed in this study transforms multi-objective numerical values with different dimensions into relative rankings, enabling objective and intuitive comparisons within a unified framework. Additionally, OTS-FICSMA can utilize the ranking results to simplify the process of searching for optimal solutions. By selecting better solutions based on the comprehensive fitness values of the solutions, it increases the probability of obtaining the Pareto front solution set. This approach ensures that the algorithm can find better multi-objective balance solutions in the complex solution space, enhancing the quality and reliability of the optimization results.

(2)

Pre-Stage Optimization Mechanism of Standard Brownian Motion

During the early search stage of the SMA, individual positions were updated based on $r<p$, determined by the fittest individual and two random ones, causing slime molds to gravitate towards the fittest. However, this simplistic mechanism fails as objective functions and constraints become more complex, resulting in a significant slowdown in convergence and a high likelihood of being trapped in local optima.

Brownian motion is characterized by continuous time and state space, and the motion of its particles is highly random. Figure 3 illustrates the trajectories and distributions of a large number of Brownian particles, initially positioned at the origin, in a MATLAB 2021A software test case [27]. It can be seen that the overall spatial position of these particles at time t follows a normal distribution of $(0,{\sigma }^{2}It)$. Therefore, incorporating the Brownian motion mechanism in the early stage of algorithm iteration will disrupt the fixed search pattern where individual slime molds are drawn to limited objectives. Instead, they will randomly move in all directions with a probability following a normal distribution. This approach remarkably enhances the positional diversity of individual slime molds, allowing them to explore more areas of the solution space that would otherwise be easily missed under the conventional search mode. Consequently, it effectively averts the algorithm from becoming trapped in a local optimum prematurely and boosts the likelihood of attaining the global optimum solution.

Consequently, the probability density function of the normal distribution can be used to ascertain the displacement $R_B$ of the particles during the standard Brownian motion, with the function’s mean $\mu =0$ and variance ${\sigma }^2=1$, as shown in Equation (55).

$$f_B(x;\mu ,\sigma )={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}\exp \left(-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}\right)\infty$$

(55)

Equation (56) details the position update mechanism for SMA once the standard Brownian motion has been incorporated. Among them, $X_t^C$ is a randomly selected position from the first N/2 individuals after the SMA ranks the fitness of all current slime mold individuals; ${\alpha }_B$ is the step size control factor.

$$X_{t+1}=X_t^{best}+vb(W{X}_t^{best}-X_t^C)+{\alpha }_B\otimes R_B$$

(56)

(3)

Post-stage Optimization Mechanism of the Ant Colony Optimization Algorithm

During the late search stage of the SMA, slime mold positions update based on $r\ge p$, guided by the convergence factor $vc$. As iterations increase, $vc$ linearly drops from 1 to 0. This linear change cannot accurately reflect the real quality-concentration feedback. When the optimal solution is not at the origin, SMA’s performance is disrupted, reducing solution accuracy.

The Ant Colony Optimization (ACO) algorithm, which conducts information interaction based on the concentration of pheromones, provides an effective solution to the feedback dilemma in the later stage of the SMA. This can be explained in two aspects.

Firstly, in the FICSMA that integrates the ACO mechanism, when individual slime molds update their positions, they simulate the path selection method of the ant colony based on transition probabilities, replacing the convergence factor vc in the SMA with the ant colony transition probability of the ACO. According to the solution logic of the ACO, when an ant colony selects a path, it comprehensively considers the pheromone concentration and the heuristic function. Paths with a higher pheromone concentration have a greater probability of being selected. This characteristic enables the slime mold population to focus on the areas where high-quality solutions may exist during the later stage of the search in the algorithm. In the RL-FPCN problem, individual slime molds release pheromones by simulating the path selection behavior of the ant colony for solution paths with high fitness. As the search progresses, the pheromones on the solution paths gradually accumulate, increasing the probability of the paths being selected, attracting more individuals, and driving the algorithm to converge towards high-quality solutions.

Secondly, the ant colony records the visited nodes through a taboo list to avoid repeated searches and ensure the efficiency of the process. After a round of search is completed, the path pheromones are dynamically updated: the old pheromones evaporate proportionally, and the new pheromones are strengthened according to the path selection situation. This mechanism not only retains the information of high-quality paths but also prevents the algorithm from relying too much on certain paths, effectively balancing the exploration and exploitation capabilities.

Therefore, this paper integrates ACO’s search strategy to enhance SMA’s performance in the late stages. The basic ACO search strategy is formulated as shown in Equation (57).

$$p_{ij}^k(t)=\left\{\begin{array}{l}{\displaystyle \frac{{\tau }_{ij}^{\alpha }(t){\eta }_{ij}^{\beta }(t)}{{\displaystyle \sum _{s\in D_k}{\tau }_{is}^{\alpha }(t){\eta }_{is}^{\beta }(t)}}},\mathrm{if} j\in D_k\hfill \\ 0,\mathrm{if} j\in others\hfill \end{array}\right.$$

(57)

In this context, $p_{ij}^k\left(t\right)$ denotes the transition probability for ant $k$ to select path $(i,j)$ at time $t$. ${\tau }_{ij}(t)$ represents the pheromone concentration between the two endpoints of path $(i,j)$ at time $t$. The heuristic function ${\eta }_{ij}$ is the reciprocal of the distance of path $(i,j)$. The heuristic factors $\alpha$ and $\beta$ signify the relative importance of pheromone concentration and path distance in the ants’ path-selection process, respectively. Moreover, the nodes traversed by an ant are recorded in the taboo list $Tab{e}_k$, and $D_k$ is the set of nodes that are not in the taboo list. These nodes in $D_k$ are the available choices for the ant.

The SMA, integrated with the ACO, replaces the convergence factor $vc$ with the transition probability $p_{ij}^k\left(t\right)$, and its position update is shown in Equation (58). The position iteration situations of the SMA in two-dimensional and three-dimensional spaces are shown in Figure 4 [27].

$$X_{t+1}=p_{ij}^k(t)X_t=\left\{\begin{array}{l}{\displaystyle \frac{{\tau }_{ij}^{\alpha }(t){\eta }_{ij}^{\beta }(t)X_t}{{\displaystyle \sum _{s\in D_k}{\tau }_{is}^{\alpha }(t){\eta }_{is}^{\beta }(t)}}},\mathrm{if} j\in D_k\hfill \\ 0,\mathrm{if} j\in others\hfill \end{array}\right.$$

(58)

Once all ants have traversed all nodes according to the transition rules, the nodes each ant visits are recorded in the taboo list, generating a feasible path solution. Subsequently, the pheromones on the path undergo two operations: previously introduced pheromones are weakened, while newly introduced ones are strengthened. The rules are shown in Equations (59) and (60).

$${\tau }_{ij}(t+n)=(1-{\rho }_v){\tau }_{ij}(t)+\Delta {\tau }_{ij}(t,t+n)$$

(59)

$$\Delta {\tau }_{ij}(t,t+n)={\displaystyle \sum _{k=1}^{k_{\max }}\Delta {\tau }_{ij}^k(t,t+n)}$$

(60)

Among them, ${\tau }_{ij}(t+n)$ is the pheromone concentration on the path $(i,j)$ after the update; $n$ is the number of nodes; ${\rho }_v$ is the pheromone evaporation factor; $\Delta {\tau }_{ij}(t,t+n)$ is the amount of newly added pheromones on the path $(i,j)$ after the iteration. $\Delta {\tau }_{ij}^k(t,t+n)$ is the amount of pheromone left by the ant on the path $(i,j)$.

From Figure 4, we can clearly observe the iterative process of FICSMA throughout the entire cycle. “Evaluate W” is used to assess the quality of food sources. X1, X2, X3, and Xbest represent the magnitudes of the food source quality. That is, the darker the color of the circle, the higher the quality of the target solution. For the solution X3 with the lowest quality, individual slime molds are in the vein initialization stage, which is the population initialization stage. For the solution X1 with the highest quality, individual slime molds will form main veins for comprehensive search, which is the early search stage. For the solution X2 of medium quality, individual slime molds will form branch veins for slow convergence over a larger range, which is the later encirclement stage.

(4)

Instant Detection Mechanism of Iterative Variation

In the SMA’s encirclement phase, the parameter z determines the proportion of random searches. However, in the original parameter setting, the value of this parameter is small, which means that the SMA has a weak ability for random search. When dealing with complex multi-objective function problems, it may become stuck in a local optimum and fail to consider the overall situation. In this regard, this paper proposes an iterative change detection mechanism based on the proportion parameter z, and the scheme is as follows:

First, calculate the fitness difference between the t-th iteration and the 0.95t-th iteration of the algorithm, and denote it as $g_f$. The formula is shown in Equation (61).

$$g_f(t)=\left|f[x(t)]-f[x(0.95t)]\right|$$

(61)

Secondly, set the minimum fitness threshold as the detection factor, denoted as $g_{\min }$. The formula is shown in Equation (62).

$$g_{\min }(t)={\displaystyle \frac{(UB+LB)}{2}}\exp \left[{\displaystyle \frac{-10(0.95t)}{t_{\max }}}\right]$$

(62)

Finally, compare the magnitude relationship between $g_f$ and $g_{\min }$. If $g_f<g_{\min }$, it is regarded as an iterative anomaly in the SMA. At this time, the algorithm will select one-fifth of the individuals from the slime mold population to conduct another round of random searches. The formula is shown in Equation (63).

$$X_{t+1}^{rand}=\left\{\begin{array}{l}{\displaystyle \frac{1}{5}}rand(UB-LB)+LB,g_f<g_{\min }\hfill \\ X_t,g_f\ge g_{\min }\hfill \end{array}\right.$$

(63)

Upon detecting iterative anomalies, the proposed detection mechanism triggers partial swarm re-initialization. This action raises the percentage of randomly behaving individuals. The enhanced flexibility and diversity in the distribution of individuals enable the SMA to effectively escape local optima and facilitate further exploration of global solutions.

4.3. Analysis of the Algorithm Time Complexity

The pseudocode of the OTS-FICSMA designed in this study is shown in Appendix A. Before analyzing the time complexity of the algorithm, we will conceptually compare the OTS-FICSMA with the benchmark SMA. As a hybrid algorithm that integrates neighborhood search and swarm intelligence, the computational content of OTS-FICSMA can mainly be divided into seven processes: (1) calculate the spatiotemporal distances and perform clustering; (2) initialize the positions of the slime mold population; (3) evaluate the comprehensive fitness of individual slime molds and rank them; (4) calculate the weight coefficients of all individual slime molds; (5) update the positions of individual slime molds; (6) generate new positions through the pheromone guidance mechanism of ACO; (7) conduct iterative anomaly detection.

The calculation of the standard SMA mainly depends on four components: (1) population initialization; (2) fitness value evaluation and ranking; (3) weight update; (4) individual position update.

In summary, the computational process of the SMA is relatively basic and simple, while the OTS-FICSMA, through the integration of multiple strategies, has significantly improved in terms of search richness and algorithm comprehensiveness. Based on the conceptual differences, the following further analyzes the performance of the two algorithms in terms of time complexity.

Assuming the size of store warehouses is $C$, the population size of SMA is $S$, the dimension of the problem is $D$, and the maximum number of iterations is $t\max$. Based on the process of the OTS-FICSMA, when facing iterative anomalies, its time complexity is calculated as: $D+C+t_{\max }(11/5S+D+S\log S+2DS+CS+1)$. In normal cases, its time complexity is calculated as: $D+C+t_{\max }(2S+D+S\log S+2DS+CS+1)$. Meanwhile, referring to the solution process of the standard SMA [27], its time complexity can be analyzed as: $D+t_{\max }(S+S\log S+2DS)$. Therefore, compared with the standard algorithm, the time complexity of the proposed OTS-FICSMA increases by at most $C+t_{\max }(6/5S+D+CS+1)$. However, in examples of a general scale, the value of $2t_{\max }DS$ is much larger than $C+t_{\max }(6/5S+D+CS+1)$. This indicates that the time complexity of OTS-FICSMA is roughly the same as that of the standard algorithm, and the algorithm optimized through the mentioned path will not significantly increase the degree of time redundancy.

5. Case Simulation

5.1. Case Data Collection and Scenario Setting

5.1.1. Data Collection

In this study, HM, a large fresh cold chain enterprise in Chengdu, Sichuan Province, is selected as the object of case analysis to conduct an in-depth study of the proposed model and algorithm. As a key player in the fresh cold chain industry, the scale of HM’s fresh cold chain business has continued to expand over the past five years. From 2023 to 2024, it has accounted for more than 70% of the fresh product cold chain business in Southwest China and has established a four-level cold chain network covering multiple direct procurement suppliers, processing centers, RDCs, and store warehouses. In addition, the enterprise actively explores the business model combining online-to-offline (O2O) with the in-store warehouse experience, making useful attempts in expanding the market and enhancing the customer experience.

However, the enterprise’s operations are confronted with numerous thorny problems. Firstly, the market demand for fresh products fluctuates violently, and the frequent occurrence of supply chain risks such as supply disruptions and sudden demand changes severely impacts the enterprise’s cold chain network scheduling system. Secondly, although the HM enterprise has been equipped with relatively complete cold chain logistics facilities, which provide basic support for its business operations, these facility resources are still limited in the context of business expansion and supply chain risks. Finally, influenced by the “dual-carbon” policy, the enterprise is facing increasing pressure of carbon emissions in its cold chain logistics operations and is in urgent need of exploring a green and efficient network layout. Evidently, the difficulties faced by the HM enterprise are highly representative within the industry, reflecting the bottlenecks that the entire fresh cold chain industry urgently needs to break through during its development process.

Accurate and comprehensive data serve as the foundation for case analysis and model application. In this study, multi-channel and multi-method exploration has been carried out in data collection. The case data in this section mainly consists of two parts. The first part of the data, related to the cold chain network nodes, is sourced from the 2024 annual financial report [31] of the group to which the HM enterprise belongs. This report details the network scale, node attributes, and supply–demand situation in the region. The second part of the data, related to the main parameters of the model, is primarily obtained from the 2024 local statistical yearbook [32] and the annual report data of the largest competitor in the same industry [33]. For the remaining parameters not disclosed in official materials, estimations are made based on the data from relevant references [6,27,28] to ensure that the data can effectively support the case analysis and model application.

The situation of the network nodes is shown in Figure 5. The actual distance between two nodes is calculated by calling the longitude and latitude through the Google Maps plugin. The original network operation situation of this enterprise in this region is shown in Figure 6. For other relevant data and model parameters related to this case, please refer to Appendix B

Table A1. Data related to the facility nodes of HM enterprise.

Facility Nodes		Fixed Operation Cost (CNY)	Capacity (kg)	Operation Carbon Emission Factor	Production Carbon Emission Factor	Unit Production Cost (CNY)
Processing centers	P1	12,500	6125	0.13	0.36	50
	P2	17,000	6425	0.21	0.45	40
	P3	22,500	6875	0.37	0.55	35
	P4	28,000	7250	0.42	0.58	35
	P5	25,500	7100	0.18	0.64	35
	P6	14,000	6375	0.48	0.38	50
RDC	R1	17,500	2250	0.09	——	——
	R2	33,000	3050	0.16	——	——
	R3	24,000	2500	0.23	——	——
	R4	25,500	2650	0.28	——	——
	R5	37,000	3200	0.12	——	——
	R6	29,500	2850	0.26	——	——

,

Table A2. Data related to the store warehouse of HM enterprise.

Store Warehouse	Demand (kg)	Store Warehouse	Demand (kg)
C1	421	C13	389
C2	537	C14	602
C3	265	C15	788
C4	456	C16	367
C5	578	C17	634
C6	289	C18	812
C7	489	C19	398
C8	590	C20	667
C9	301	C21	834
C10	511	C22	403
C11	620	C23	705
C12	237	C24	856

and

Table A3. Other variables and parameters related to the model.

Parameters	Values	Parameters	Values
$o{s}_p$	¥50	$et{p}_{jk}$	0.5 kgCO2e/(t·km)
$c_p$	¥180	$et{r}_{kl}$	0.3 kgCO2e/(t·km)
$t{s}_{ij}^r$	¥15	$ce$	45 ¥/(tCO2e)
$t{p}_{jk}^p$	¥20	$u_{rp}$	0.65
$t{r}_{kl}^p$	¥25	$pe{n}_{lm}^p$	¥500
$et{s}_{ij}$	0.6 kgCO2e/(t·km)	$hsdi{s}_{kl}$	150 km

.

It is worth noting that the HM enterprise is fully aware of the potential risks faced by the supply chain, and has given special consideration to the possible threats that the four major supply disruption factors (earthquakes, floods, fires, and public health incidents) and the four major demand fluctuation factors (competitors entering the market, economic slowdown, economic recovery, and entering the peak demand season) may pose to the supply network. Based on this, this paper applies the proposed RL-FPCN model to the HM enterprise, comprehensively examining the specific impacts of supply disruptions and demand fluctuations on the enterprise’s supply chain and conducting an in-depth evaluation of the effectiveness of resilient strategies in addressing these challenges. In actual operation, the HM enterprise is committed to optimizing the entire process, from raw material procurement to the processing, production, and transportation of fresh products, by making informed decisions on facility location selection and transportation routes.

5.1.2. Scenario Setting

This paper adopts the scenario planning method to establish supply disruption and demand fluctuation scenarios in the FPCN. By analyzing the dynamics and results that affect future development, the resilience of the system in the face of future uncertainties is enhanced. Referring to the research results of Zeng et al. [28], this paper sets out the relevant parameters of four resilient strategies. This paper assumes that there are three different reinforcement levels available for the processing centers and RDCs. When the reinforcement levels are v1 and v2, the fixed operation costs of the processing centers and RDCs are 1.5 times and 1.8 times, respectively, higher than those without reinforcement (i.e., v0). The carbon emissions generated by the facilities at v1 and v2 are 1.2 times and 1.4 times, respectively, higher than those without reinforcement. The processing centers and RDCs may lose a certain proportion of their capacities after destructive events, and facilities with higher reinforcement levels will lose a smaller proportion of their capacities. For example, a facility with a reinforcement level of v0 will lose 75% of its capacity after an earthquake, while facilities with reinforcement levels of v1 and v2 will lose 50% and 25% of their capacities, respectively. At the same time, this paper collected the risk data [1], statistical yearbook [32], and enterprise annual report [31,33] of the fresh product industry in the past two years, and set the occurrence probability for the scenario, as shown in

Table 4. Influence of disruption and fluctuation scenarios on the SSCN and their occurrence probabilities.

Scenario Classification			Influence	Probability
Supply Disruption Scenario Set O	o0	Normal	The sites and routes are in a normal state.	60%
	o1	Earthquake	The sites within the earthquake-affected area are disrupted; that is, they will lose 75% of their capacity.	5%
	o2	Flood	The routes between the affected sites are disrupted; that is, some paths will be unavailable for selection.	15%
	o3	Fire	The production capacity of the processing factory decreases by 75%.	15%
	o4	Public Health Incident	Some sites and the routes between sites are disrupted.	5%
Demand Fluctuation Scenario Set N	n0	Normal	Customers meet the normal market demand.	60%
	n1	Competitors Entering the Market	The demand decreases by 75%.	10%
	n2	Economic Growth Slowdown	The demand decreases by 25%.	10%
	n3	Economic Recovery	The demand increases by 25%.	10%
	n4	Entering the Peak Demand Season	The demand increases by 75%.	10%

. Based on the above settings of supply disruption and demand fluctuation scenarios, the network operation status of the basic model (L-FPCN) solved by the OTS-FICSMA is shown in Figure 7.

5.2. Analysis of Supply Disruption Scenarios

(1)

Analysis of Service Level and Risk Resistance Ability

The comparison of service levels between the L-FPCN and RL-FPCN models under various supply disruption scenarios is shown in

Table 5. Comparison of service levels between L-FPCN and RL-FPCN under supply disruption scenarios.

Supply Disruption Scenario Set O		o0	o1	o2	o3	o4
Comparison of Service Levels	L-FPCN	73.45%	69.12%	57.23%	46.51%	55.48%
	RL-FPCN	94.01%	86.23%	87.12%	90.89%	86.34%
	Gap	27.99%	24.75%	52.23%	95.42%	55.62%

and Figure 8. Under different disruption scenarios, the service level of RL-FPCN is higher than that of L-FPCN. It is worth noting that the optimization degree of RL-FPCN is the most remarkable in the o3 fire scenario, and the gap in service level between it and L-FPCN reaches 95.42%. This indicates that when the supply is disrupted due to a fire, the supply capacity of L-FPCN will be significantly affected, making it challenging to maintain a high service level. However, RL-FPCN maintains the service level at a relatively high state through practical strategies, which significantly reduces the impact of the fire on the supply. At the same time, in the o2 flood scenario and the o4 public health incident scenario where the supply is severely affected, the robustness of the RL-FPCN model is also effectively verified. In these two scenarios, the gaps in service levels between RL-FPCN and L-FPCN reach 52.23% and 55.62%, respectively.

From a measurable perspective, the improvement of the service level directly reflects the stronger resilience of RL-FPCN. Taking the o3 fire scenario as an example, due to the limited supply capacity in the fire scenario, the service level of L-FPCN is only 57.23%. In contrast, RL-FPCN significantly improves the service level through a series of strategies, among which the emergency inventory strategy plays a crucial role. When the source supply is blocked due to the occurrence of a fire, the emergency inventory is promptly put into use, effectively filling the supply gap caused by the reduction in production capacity. At the same time, the site strengthening strategy enhances the disaster resistance of processing centers and RDCs, reducing the degree of damage to facilities in the fire. The multi-route transportation and cross-level procurement strategies ensure the smooth transportation of goods and the flexibility of supply. With the coordination of multiple strategies, the service level of RL-FPCN is increased to 87.12%, effectively enhancing its resilience in coping with supply disruption scenarios.

(2)

Analysis of Operating Costs and Carbon Emissions

The comparison of cost structures and changing trends in total costs and weights, as well as carbon emission costs between the L-FPCN and RL-FPCN models under supply disruption scenarios, is shown in

Table 6. Comparison of cost structures between L-FPCN and RL-FPCN under supply disruption scenarios.

Supply Disruption Scenario O		Total Cost	Location Selection Cost	Inventory Cost	Path Selection Cost	Transportation Cost	Comprehensive Carbon Emission Cost
o0	L-FPCN	396,244	182,946	144,788	0	46,955	21,555
	RL-FPCN	325,172	196,339	45,524	9300	54,564	19,445
	Gap	−17.94%	7.32%	−68.56%	0	16.20%	−9.79%
o1	L-FPCN	458,430	208,906	154,858	0	80,363	14,303
	RL-FPCN	351,966	201,290	52,689	9538	66,240	22,209
	Gap	−23.22%	−3.65%	−65.98%	0	−17.57%	55.28%
o2	L-FPCN	507,644	268,798	175,442	0	41,982	21,422
	RL-FPCN	405,267	272,705	67,193	10,699	33,880	20,790
	Gap	−20.17%	1.45%	−61.70%	0	−19.30%	−2.95%
o3	L-FPCN	740,314	462,549	228,165	0	30,278	19,322
	RL-FPCN	418,523	202,691	100,990	8872	79,980	25,990
	Gap	−43.47%	−56.18%	−55.74%	0	164.15%	34.51%
o4	L-FPCN	535,000	263,220	201,374	0	41,409	28,997
	RL-FPCN	439,675	252,066	80,153	12,003	68,941	26,512
	Gap	−17.82%	−4.24%	−60.20%	0	66.49%	−8.57%

and Figure 9 and Figure 10. Under different disruption scenarios, the total cost of RL-FPCN is always lower than that of L-FPCN. The cost savings of RL-FPCN are primarily reflected in the inventory cost, with a significant reduction in inventory cost in each scenario. Similarly to the comparison results of service level optimization, the cost optimization of RL-FPCN is the most remarkable in the o3 fire scenario. The total cost optimization reaches 43.47%, among which the location selection cost optimization reaches 56.18% and the inventory cost optimization reaches 55.74%. This indicates that the relevant strategies provided by RL-FPCN can effectively adjust the cost structure and play a significant role in cost control in a severe supply disruption scenario, such as a fire. As for the transportation cost, different scenarios exhibit varying trends. In the o3 fire and o4 public health incident scenarios, the transportation costs of RL-FPCN increase significantly, reaching 164.15% and 66.49%, respectively. This may be due to the adoption of more flexible transportation strategies to cope with supply disruptions. In the o1 earthquake and o2 flood scenarios, the transportation costs decrease.

We divided the experimental results of carbon emission costs into two categories for analysis.

In the first category, under the scenarios of o0 normal, o2 flood, and o4 public health incident, the comprehensive carbon emission costs of RL-FPCN are reduced by 9.79%, 2.95%, and 8.57%, respectively, compared with those of L-FPCN. This is attributed to the cross-level procurement and emergency inventory strategies. The former reduces the additional transportation trips caused by road blockages, and the latter avoids the increase in warehousing energy consumption due to inventory backlogs, effectively reducing carbon emissions.

In the second category, under the scenarios of o1 earthquake and o3 fire, the comprehensive carbon emission costs of RL-FPCN are higher than those of L-FPCN, by 55.28% and 34.51%, respectively. This is mainly due to the implementation of the site strengthening and multi-route transportation strategies. To reduce the production capacity loss of the sites, relevant nodes adopt the site strengthening strategy to increase the lower limit of production capacity. However, a higher level of strengthening means more carbon emissions. At the same time, although the multi-route transportation strategy ensures the supply of goods, it involves the transportation of goods from multiple superior nodes to a certain inferior node, further increasing the carbon emissions in the transportation link. The changes in carbon emissions brought about by these strategic adjustments are reflected through the cost data, which reflects the trade-off between emission reduction and resilience of RL-FPCN in extreme scenarios.

5.3. Analysis of Demand Fluctuation Scenarios

(1)

Analysis of Service Level and Risk Resistance Ability

The comparison of service levels between the L-FPCN and RL-FPCN models under various demand fluctuation scenarios is shown in

Table 7. Comparison of service levels between L-FPCN and RL-FPCN under demand fluctuation scenarios.

Demand Fluctuation Scenario N		n0	n1	n2	n3	n4
Service Level Comparison	L-FPCN	73.45%	56.12%	46.61%	47.18%	54.76%
	RL-FPCN	94.01%	80.30%	83.67%	77.24%	78.61%
	Gap	27.99%	43.09%	79.51%	63.71%	43.55%

and Figure 11. The optimization degree of RL-FPCN is the most remarkable in the n2 economic growth slowdown scenario, and the gap in service level between it and L-FPCN reaches 79.51%, compared with the optimization of 43.09% in the n1 scenario. This suggests that the impact resulting from a targeted decrease in demand is more severe than that in the demand shock scenario of competitors entering the market. This may be because when demand decreases appropriately, the enterprise fails to adjust the product supply chain structure in a timely manner, resulting in a supply chain that still maintains a relatively high level of resource allocation and production capacity. However, due to the decrease in demand, the production capacity utilization rate decreases, which in turn affects inventory and transportation efficiency, ultimately leading to an extension of delivery times and a reduction in customer satisfaction rates.

In the n3 economic recovery and n4 entering the peak demand season scenarios, RL-FPCN also shows sound optimization effects. The gaps in service levels between it and L-FPCN reach 63.71% and 43.55%, respectively, indicating that RL-FPCN can also cope with demand fluctuations and ensure a high service level in scenarios with rising demand.

(2)

Analysis of Operating Costs and Carbon Emissions

The comparison of cost structures and changing trends in total costs and quantities, as well as carbon emission costs, between the L-FPCN and RL-FPCN models under demand fluctuation scenarios is shown in

Table 8. Comparison of cost structures between L-FPCN and RL-FPCN under demand fluctuation scenarios.

Demand Fluctuation Scenario N		Total Cost	Location Selection Cost	Inventory Cost	Path Selection Cost	Transportation Cost	Comprehensive Carbon Emission Cost
n0	L-FPCN	396,244	182,946	144,788	0	46,955	21,555
	RL-FPCN	325,172	196,339	45,524	9300	54,564	19,445
	Gap	−17.94%	7.32%	−68.56%	0	16.20%	−9.79%
n1	L-FPCN	324,919	143,582	112,162	0	59,038	10,137
	RL-FPCN	275,568	162,200	41,335	8928	50,291	12,814
	Gap	−15.19%	12.97%	−63.15%	0	−14.82%	26.41%
n2	L-FPCN	364,544	142,027	141,261	0	70,211	11,045
	RL-FPCN	297,159	188,192	31,023	8885	62,641	6418
	Gap	−18.48%	32.50%	−78.04%	0	−10.78%	−41.89%
n3	L-FPCN	435,869	196,359	139,522	0	87,915	12,073
	RL-FPCN	337,959	153,670	68,403	13,586	77,967	24,333
	Gap	−22.46%	−21.74%	−50.97%	0	−11.32%	101.55%
n4	L-FPCN	471,530	240,716	116,327	0	104,444	10,043
	RL-FPCN	352,095	175,414	87,602	11,548	55,666	21,865
	Gap	−25.33%	−27.13%	−24.69%	0	−46.70%	117.71%

, Figure 12 and Figure 13. The total costs of both L-FPCN and RL-FPCN generally show an increasing trend with the growth of demand, and the gap in total costs between the two also roughly follows this rule. Under demand fluctuation scenarios, the cost savings of RL-FPCN primarily stem from reduced inventory costs. In each demand fluctuation scenario, the reduction range of inventory costs is quite apparent. For example, in the n2 economic growth slowdown scenario, the inventory cost optimization rate reaches −78.04%. This may be because the transportation and procurement strategies adopted by RL-FPCN enhance the flexibility of transportation, reduce inventory backlogs, and consequently lower inventory costs.

Regarding the carbon emission situation, in most demand fluctuation scenarios, RL-FPCN demonstrates optimized performance in terms of comprehensive carbon emission costs. For example, in the n0 normal and n2 economic growth slowdown scenarios, the comprehensive carbon emission costs are reduced in both cases. However, in the n3 economic recovery and n4 entering the peak demand season scenarios, the carbon emission costs of RL-FPCN increase significantly, with the increase rates reaching 101.55% and 117.71%, respectively. By comparing the cost structures, it is found that in these two scenarios, the transportation cost optimization rates of the resilient strategies are −11.32% and −46.70%, respectively. This suggests that the initiated cross-level transportation and related strategies effectively respond to the rising demand, but may lead to an increase in transportation frequency and an imbalance in the production plans of sites, resulting in higher carbon emissions.

5.4. Comparison of Algorithm Performance

To further verify the comprehensive performance of the proposed algorithm, taking the supply chain network of the fresh cold chain enterprise HM as an example, this paper selects relevant algorithms with similar component structures and widely used in logistics network optimization, ISMA-EOQI [34] and SMA [35], and compares them with the proposed OTS-FICSMA to jointly solve the RL-FPCN model under supply disruptions and demand fluctuations. The relevant parameters of the three algorithms are set according to the original studies [27,34,35], as shown in

Table 9. Internal parameter settings of the three algorithms.

Algorithm	Parameter Settings
OTS-FICSMA	$\mathrm{z}$ = 0.003, ${\mathsf{\alpha }}_{\mathrm{B}}$ = 0.002, ${\mathrm{m}}_{\mathrm{a}\mathrm{n}\mathrm{t}}$ = 45, ${\mathsf{\tau }}_0$ = 0.35, ${\mathsf{\rho }}_{\mathrm{v}}$ = 0.85, $\mathsf{\alpha }$ = 4, $\mathsf{\beta }$ = 3, $\mathsf{\gamma }$ = 1.5
ISMA-EOQI	$\mathrm{z}$ = 0.003, ${\mathrm{C}\mathrm{R}}_1$∈ [0.15, 0.30], ${\mathrm{C}\mathrm{R}}_2$∈ [0.5, 0.7]
SMA	$\mathrm{z}$ = 0.003

in detail.

The simulation experiment environment for this case is as follows: Ultra 9 185H CPU, with 32 GB of memory, a Windows 11 64-bit operating system, and the running software is MATLAB R2024. To ensure the comparability of the simulation results, the population size $S$ is set to 40, the problem dimension $D$ to 30, and the maximum number of iterations $t\max$ to 400 for all three algorithms. The final optimization results for the three algorithms in RL-FPCN are presented in

Table 10. Comparison of supply chain network optimization schemes of the three algorithms.

Indicator	OTS-FICSMA	ISMA-EOQI	SMA
RLZ1	84.86%	71.23%	66.51%
RLZ2	359,776	424,857	511,469
Site Strengthening Strategy	P4	RDC: R6	RDC: R1, R6
Emergency Inventory Strategy	P5 RDC: R2	P2, P3, P6	P4, P5 RDC: R2
Cross-Level Procurement Strategy	P3 → C5, C6, C10 P5 → C22, C23, C24 P6 → C19, C20	P2 → C2, C3 P3 → C5, C6 P6 → C19, C20	P4 → C13, C17, C18
Multi-Route Transportation Strategy	P4 + P6 → R5	P3 + P5 → R4	S1 + S3 → P3 P3 + P5 → R4
Effective Number of Iterations	94	149	233

. The optimization schemes of the supply chain network, as solved by the three algorithms, along with the iterative curves of the comprehensive fitness values, are shown in Figure 14, Figure 15, Figure 16 and Figure 17.

Based on the data comparison in Table 10, the OTS-FICSMA demonstrates significant advantages in both core indicators and strategic optimization. In terms of service level and cost control, OTS-FICSMA achieves the highest service rate of 84.86% and the lowest operating cost of 359,776, which is an improvement of 19.1%/15.3% compared with ISMA-EOQI (71.23%/424,857) and 27.6%/29.8% compared with SMA (66.51%/511,469). Its global optimization mechanism effectively balances the core contradiction between service efficiency and cost compression. At the same time, OTS-FICSMA requires only ninety-four iterations to converge, which is significantly better than the 233 iterations of SMA, highlighting the algorithm’s superiority in terms of computational resources and efficiency.

From a strategic perspective, OTS-FICSMA achieves precise efficiency improvements at the front end of the supply chain through data-driven site upgrades of the processing center P4. In contrast, ISMA-EOQI and SMA focus on the redundant layout of RDC nodes and have weak cost control capabilities. Its emergency inventory strategy only needs to cover two key nodes, P5 and R2 (compared to the three processing centers of ISMA-EOQI and the combination of R2, P4, and P5 in SMA), reducing redundant risk with a straightforward inventory layout. In addition, its cross-level procurement is directly associated with processing centers and end customers (such as P3 → C4–C6, P5 → C22–C24), and compresses intermediate levels through multi-route combined transportation (such as P4 + P6 → R5), achieving a dual improvement in the response speed and resilience of the supply chain. The comprehensive effect of these strategies enables OTS-FICSMA to quickly identify bottlenecks and dynamically adjust resource allocation in complex supply chain scenarios, thereby forming a competitive global advantage in terms of service, cost, and efficiency.

Through the comparison of the solution performance of the three algorithms in Figure 17, it can be seen that in the early stage of iteration, both ISMA-EOQI and SMA experience multiple search detours and stagnation states. At the same time, as the number of detours increases, the algorithms are more likely to become trapped in local optima. In contrast, the iterative performance of OTS-FICSMA is more continuous and smoother, and it has a more efficient solution ability when dealing with complex scenarios such as supply disruptions and demand fluctuations. In the later stages of iteration, both ISMA-EOQI and SMA enter bottleneck periods to varying degrees, exhibiting significant differences in integrity and continuity compared to OTS-FICSMA.

5.5. Sensitivity Analysis

To evaluate the impact of key model parameters, such as the carbon emission factor and the probability of supply chain risks, on network optimization decisions, we adopted the approach of adjusting parameter sets for sensitivity analysis to simplify the difficulty of parameter processing. Among them, there is parameter set $E=\left\{et{s}_{ij},et{p}_{jk},et{r}_{kl},et{k}_{jl},es{p}_j,es{r}_k,ew{f}_j^p\right\}$ for the carbon emission factor, which includes the transportation, operation, and production carbon emission factors involved in RL-FSCN. And there is parameter set $\Phi =\left\{({\phi }^o,{\phi }^n)|o,n\in \left\{1,2,3,4\right\}\right\}$ for the probability of supply chain risks, which includes the scenario probabilities of four types of supply disruptions and four types of demand fluctuations. Based on the parameter sets E and Φ, we set four different percentage coefficients, respectively, for sensitivity analysis. In addition, since ${\phi }^o$ and ${\phi }^n$ in the risk probability set are independent of each other, when converting the percentage coefficients, it is necessary to ensure that the sum of each series is 1, that is, ${\displaystyle {\sum }_{o=0}^4{\phi }^o=1}$ and ${\displaystyle {\sum }_{n=0}^4{\phi }^n=1}$. The solution results of OTS-FICSMA are shown in

Table 11. Sensitivity analysis of the parameter set of carbon emission factors.

Indicator	$\mathit{E}$	$50\mathbf{\%}\mathit{E}$		$75\mathbf{\%}\mathit{E}$		$125\mathbf{\%}\mathit{E}$		$150\mathbf{\%}\mathit{E}$
	BKS	Cmin	GAP	Cmin	GAP	Cmin	GAP	Cmin	GAP
RLZ1	84.86%	90.57%	6.73%	87.08%	2.62%	80.21%	−5.48%	78.21%	−7.84%
RLZ2	359,776	319,523	−11.19%	335,269	−6.81%	375,416	4.35%	389,692	8.32%
Site Strengthening Strategy	P4	P2, R5		P5		R1		R5
Emergency Inventory Strategy	P5, R2	P5, R1		R1, R5		P5, R2		P5
Cross-Level Procurement Strategy	P3 → C5, C6, C10 P5 → C22, C23, C24 P6 → C19, C20	P2 → C1, C2, C3 P5 → C22, C23, C24 P6 → C19, C20		P2 → C1, C2, C3 P3 → C5, C6, C10 P5 → C22, C23, C24		P3 → C5, C6, C10		P5 → C22, C23, C24
Multi-Route Transportation Strategy	P4 + P6 → R5	P3 + P5 → R4 P4 + P6 → R5		S1 + S2 → P3 P4 + P6 → R5		P3 + P5 → R4		S1 + S2 → P3

and

Table 12. Sensitivity analysis of the parameter set of supply chain risks.

Indicator	$({\mathit{\phi }}^{\mathit{o}},{\mathit{\phi }}^{\mathit{n}})$	$50\mathbf{\%}({\mathit{\phi }}^{\mathit{o}},{\mathit{\phi }}^{\mathit{n}})$		$75\mathbf{\%}({\mathit{\phi }}^{\mathit{o}},{\mathit{\phi }}^{\mathit{n}})$		$125\mathbf{\%}({\mathit{\phi }}^{\mathit{o}},{\mathit{\phi }}^{\mathit{n}})$		$150\mathbf{\%}({\mathit{\phi }}^{\mathit{o}},{\mathit{\phi }}^{\mathit{n}})$
	BKS	Cmin	GAP	Cmin	GAP	Cmin	GAP	Cmin	GAP
RLZ1	84.86%	94.23%	11.04%	90.96%	7.19%	77.35%	−8.85%	75.47%	−11.07%
RLZ2	359,776	300,194	−16.56%	323,742	−10.02%	382,572	6.34%	396,827	10.30%
Site Strengthening Strategy	P4	R5		R4		P2, R5		P4, R1, R6
Emergency Inventory Strategy	P5, R2	R2		P5		P5, R1		P2, R4
Cross-Level Procurement Strategy	P3 → C5, C6, C10 P5 → C22, C23, C24 P6 → C19, C20	P5 → C22, C23, C24		P3 → C5, C6, C10 P6 → C18, C19, C20		P2 → C1, C2, C3 P5 → C22, C23, C24		P2 → C1, C2, C3 P3 → C5, C6, C10 P4 → C13, C17, C18
Multi-Route Transportation Strategy	P4 + P6 → R5	P3 + P5 → R4		P4 + P6 → R5		P3 + P5 → R4 P4 + P6 → R5		S1 + S2 → P3 P4 + P6 → R5

. $BKS$ represents the optimal solution based on the conditions given in the case, $C_{\min }$ represents the optimal solution under the univariate condition, and $GAP$ represents the deviation between the two.

As the carbon emission factors in Table 11 increase, the four resilience strategies will make more conservative adjustments, leading to a decrease in the service level. Compared with the service level, the total cost is more sensitive to changes in the carbon emission factors. In Table 12, the optimal value of the objective function decreases as the probability of supply chain risks increases. When the risk probability coefficient increases by 50%, the model has to intensify the intervention of the four resilience strategies. Although this causes the operating cost to rise by up to 10.30%, it also stabilizes the service level within an acceptable range of over 75%. It should be noted that the impact of the probability of supply chain risks on RL-FSCN is more significant than that of the carbon emission factors, and the deviation between the upper and lower limits of the service level can reach 18.76%. In conclusion, the proposed model can effectively reflect the impact of key parameters on the optimization decisions of the cold chain network, providing a powerful basis for enterprises to formulate strategies.

6. Conclusions and Future Research

6.1. Conclusions

Considering the numerous challenges currently faced by the fresh cold chain, such as supply disruptions, demand fluctuations, and low-carbon transformation, and the many deficiencies in existing research, we propose a comprehensive optimization solution for the fresh product cold chain network.

Firstly, this solution conducts an in-depth analysis of various scenarios of supply disruptions and demand fluctuations and constructs a high-dimensional multi-objective optimization model, RL-FPCN. This model aims to maximize the service level and minimize the operating cost (including carbon emission cost). It integrates four resilience strategies: site strengthening, emergency inventory, cross-level procurement, and multi-route transportation, taking into account the improvement of resilience, cost control, and carbon emission reduction.

Secondly, to solve the model, we design the OTS-FICSMA hybrid algorithm. The initial solution is generated by clustering through the OPTICS algorithm considering spatiotemporal distances, and the SMA is optimized throughout the entire iteration cycle. The optimization measures include: establishing a ranking matrix to balance multiple optimization objectives, increasing the search diversity with standard Brownian motion in the early stage, enhancing the accuracy by integrating the ant colony optimization algorithm in the later stage, and using an iteration change detection mechanism to enhance stability, thus improving the solution quality.

Finally, we conduct an in-depth case study on HM enterprise in Chengdu. The results show that the RL-FPCN model can effectively address the optimization problems of the fresh product cold chain network, improve the supply chain resilience, stabilize the service level during supply and demand changes, optimize the cost structure, and reasonably control carbon emissions. The OTS-FICSMA has prominent advantages over similar algorithms. It has a faster solution speed, can better balance multiple objectives, and achieve the optimization of resilience, cost, and carbon emissions. In addition, the solution also conducts a sensitivity analysis of carbon emission factors and supply chain risk probabilities based on the optimization results. The performance of the proposed model and algorithm has been effectively verified under different scenarios and parameter combinations.

6.2. Management Insights

This study provides a multi-faceted innovative perspective for the optimization of the fresh product cold chain network at the theoretical level:

(a)

In terms of multi-objective optimization, the constructed RL-FPCN model aims to maximize the service level and minimize the operating cost, while also taking carbon emissions into account. This breaks through the limitations of traditional research that only focuses on economic and environmental objectives. Additionally, the study explores the balance relationship and collaborative optimization mechanism among multiple objectives, thus improving the application of multi-objective optimization theory in the cold chain field.

(b)

In terms of supply chain risk management, the four resilience strategies designed for supply disruptions and demand fluctuations clarify the response methods under different risk scenarios and their impact mechanisms on the supply chain, providing a more comprehensive perspective for the study of supply chain risk response strategies.

(c)

In terms of the design of hybrid heuristic algorithms, the OTS-FICSMA innovatively integrates neighborhood search and swarm intelligence algorithms, improving the solution efficiency and quality of the algorithm in complex scenarios, and providing a novel idea for algorithm design to solve complex optimization problems.

This study provides highly valuable guidance for decision-makers in fresh cold chain enterprises.

(a)

In the face of potential risks of supply disruptions and demand fluctuations, decision-makers can use the RL-FPCN model to make scientific location and routing decisions based on the risk probabilities. By flexibly applying the four resilience strategies, they can optimize the scheduling of facility resources, improve the service level, and reduce operating costs and carbon emissions.

(b)

When decision-makers are faced with conflicting objectives, they can use the ranking matrix proposed in this study to transform multi-objective numerical values with different dimensions into relative rankings, and conduct objective and intuitive comparisons within a unified framework, so as to achieve the balance of multiple objectives in the optimization scheme. In addition, when faced with the choice among different combinations of resilience strategies, they can rely on the solution results of the proposed algorithm, use it as a guide, and prioritize the selection of the best strategy under the current conditions.

(c)

In research modeling, relevant parameters and decision variables are categorized by facility location and routing, offering strong extensibility. SMEs’ decision-makers can adjust parameters as per businesses’ needs. This study provides clear references for key parameters to guide practical decision-making.

6.3. Limitations and Future Research

However, this study also has certain limitations. On the one hand, although the study is conducted against the backdrop of the “dual-carbon” policy, it fails to conduct a critical discussion on the policy mechanisms or restrictive factors. During the actual implementation of the policy, issues such as the effectiveness of regulatory incentives, data availability, and compliance will all have an impact on the realization of the goals of cost reduction and efficiency improvement for fresh cold chain enterprises. On the other hand, this study lacks discussion on the issue of supply chain fairness. In the fresh product supply chain, factors such as supply chain disruptions and carbon policies will have varying degrees of impact on different supply chain stakeholders. For example, suppliers, processing centers, retailers, and consumers will face different impacts and opportunities when confronted with these factors, and such differences should be understood and balanced.

Based on this study, future research on the optimization of the fresh product cold chain network can be deepened from two aspects. Firstly, data-based demand forecasting research should be conducted and combined with real-time tracking technology. Multi-source data should be integrated, using cloud adoption and artificial intelligence [36] to accurately grasp the dynamics of demand. Operational strategies should be adjusted according to real-time information to optimize the resilient and sustainable supply chain. Secondly, the sources of model uncertainty should be expanded by incorporating factors such as the quality of raw materials and weather disturbances, making the model more in line with reality. At the same time, in-depth research should be conducted on policy carbon accounting methods [37], especially on the impact of cap-and-trade programs on model decisions. This can help enterprises achieve a win–win situation with economic and environmental benefits, and promote the sustainable development of industry.