Closed-Loop Supply Chain Network Design with Flexible Capacity under Uncertain Environment

Abstract

This paper incorporates flexible facility capacity and government subsidy factors into the consideration of the design of a closed-loop supply chain(CLSC) network based on an uncertain environment. Considering the minimization of economic cost and carbon emission, a multi-objective multi-period multi-product mixed integer linear programming model with fixed and flexible facility capacity is constructed respectively. The robust optimization method is applied to deal with the uncertain environment of demand, recycled product quality, and recycling rate faced by the CLSC, and the robust models under six uncertain sets are constructed respectively. For model solving, the designed algorithm uses the augmented $ϵ$-constraint method to handle multi-objective problems and introduces a three-stage method on top of the Benders decomposition algorithm to accelerate the efficiency of solving the main problem. Finally, through numerical cases, a CSLC with a flexible supply strategy can manage economic and environmental costs to cope with the negative impacts of an uncertain environment, while this paper verifies the effectiveness of the government subsidy strategy under different conditions and analyzes the potential limitations.

1. Introduction

Currently, supply chains encounter new challenges including increased globalization of market competition, market demand’s diversification and uncertainty, product life’s shortening, and intensified price competition. These developments require supply chains to seek new and efficient production methods [1]. To succeed and flourish in an intensely competitive environment, supply chains must explore cost-effective and fast production methods. A flexible facility capacity strategy can help achieve this goal. As society has developed and people’s understanding of sustainability has increased, constructing a supply chain is now more focused on balancing economic costs with environmental benefits and reducing carbon emissions. The implementation of CLSCs can decrease the processing and manufacturing steps of products, elongate the materials’ and products’ life periods, improve carbon sequestration capacity, and ultimately decrease energy consumption and carbon dioxide emissions incurred during raw material extraction, primary processing, product disposal, and reproduction [2]. The CLSC includes the use of scrap steel for energy recovery, for instance. Utilizing steel scrap as raw material saves 1.6 tons of carbon emissions, 350 kg of standard coal, 1.7 tons of new water, and 1.6 tons of concentrate powder while reducing emissions by 86% of exhaust gases, 76% of wastewater, and 92% of solid waste when compared to conventional steelmaking. Using scrap steel instead of iron ore as the raw material for steelmaking reduces the production of 1 ton of steel by approximately 1.6 tons of carbon dioxide emissions. According to reports, China’s 2020 scrap steel utilization reached about 260 million tons, which alone can decrease carbon dioxide emissions by around 416 million tons [3]. Developing a CLSC and improving remanufacturing, reprocessing, and energy recovery processes are essential to achieving ultra-low emissions and green development of the CLSC.

The paper aims to construct a CLSC network with flexible factory capacity. This means that factories in the supply chain can adopt flexible capacity strategies and regulate their capacity levels at various stages. The paper initially identifies the CLSC framework along with the related assumptions and parameter variables. Using this information, the paper constructs a mixed-integer linear programming model, which contains a fixed facility capacity and a flexible facility capacity model and is designed for a multi-objective, multi-period, multi-product environment. The study applies the augmented $ϵ$-constraint approach to address the multi-objective nature of the model. Furthermore, the study introduces the Benders decomposition algorithm(BDA), which decomposes the master problem to solve the model. The study evaluates the effects of six uncertain environments, including box, ellipsoidal, and polyhedral sets, on the CLSC. Additionally, it examines the impact of three sources of uncertainty, namely demand, recycling rate, and recycled product quality. Moreover, the study investigates the influence of flexible facility capacity strategies and government subsidy variations on the CLSC.

The contributions of this research are as follows:

1.

The flexible facility capacity strategy is applied to designing the CLSC network model;

2.

The CLSC network is considered to face three kinds of uncertainties, which are demand, recycling rate, and recycled product quality, and this paper applies the robust optimization method to construct the robust model under six different sets of uncertainties;

3.

This study investigates the effects of government subsidies on CLSCs in different situations.

The following sections detail the structure of this paper. Section 2 summarises existing research. Section 3 presents models for CLSC with fixed and flexible facility capacity, considering uncertainty. Section 4 applies robust optimization methods to improve the models. Section 5 presents the Benders decomposition algorithm to solve the model. In Section 6, management insights specific to businesses and governments are obtained through numerical case studies. Finally, Section 7 presents the conclusion of the paper.

2. Literature Review

2.1. Uncertainty in CLSCs

The aim of designing a CLSC network is to optimize the economic, environmental, and social benefits of the supply chain while meeting customer demand [4]. However, designing a CLSC network is affected by various uncertainties, such as information asymmetry, uncertain recycling quality, and unexpected events. These uncertainties increase the complexity and risk of the supply chain, which makes it challenging to develop an optimization model and solution methods. Therefore, performing CLSC network design under an uncertain environment is an important research topic. Fazli-Khalaf et al. [5] addressed demand uncertainty in CLSC network design by using a scenario-based, fuzzy stochastic programming approach. They extended this approach to control parameter uncertainty and level of risk aversion in output decisions effectively. However, it does not consider other types of uncertainties such as supply uncertainty or price volatility. According to Shi et al. [6], scrap recycling quantities are stochastic and non-linearly dependent on prices. They propose an optimization model that incorporates Lagrangian relaxation and a solution method to address demand and revenue uncertainty. In a subsequent study, Shi et al. [7] use demand and revenue uncertainty to construct a stochastic model of the CLSC network. Their proposed model integrates the costs of understocking and overstocking. However, they do not consider uncertainties in the reverse logistics process, such as return rates or quality of returned products. Alimoradi et al. [8] designed a CLSC model that accounts for uncertain product recovery rates in the reverse cycle. Moreover, it does not address other uncertainties like demand or cost. Selim and Ozkarahan [9] propose a solution for designing supply chain distribution networks based on interactive fuzzy objective programming, which considers uncertainties in demand and decision-maker objectives. Meanwhile, Talaei et al. [10] develop a bi-objective fuzzy mixed-integer linear programming model through the $ϵ$-constraint method. This model aims to minimize costs and emissions in supply chains while considering uncertainties in demand and costs. The researchers apply the model specifically to the copier industry. Francas and Minner [11] propose a two-stage stochastic model for configuring CLSC networks. They compare two manufacturing network options while factoring in uncertainties in demand and returns. Their two-stage stochastic model for configuring CLSC networks is innovative but it does not consider multiple types of uncertainties simultaneously.

The sustainable CLSC network designed by Zhen et al. [12] applied a scenario-based approach to cope with demand uncertainty and proposed a Lagrangian relaxation method to solve the model. An integrated MOMILP model for sustainable CLSC network design with cross-docking and uncertainty is proposed by Tavana et al. [13], and tested with simulated data using fuzzy goal programming. Resilience and quality uncertainty in the mining supply chain are considered by Arabi and Gholamian [14], using a reliable model and two-stage stochastic programming. An economic-environmental bi-objective CLSC model, which takes into account three types of uncertainty in the quality of the recycled product, the quantity, and the cost of recycling, is constructed by Boujelben et al. [15]. While these papers have significantly contributed to the field of CLSC network design under uncertainty, further improvement can be made by considering multiple types of uncertainties simultaneously. Hence, this paper’s constructed CLSC network operates under an uncertain environment, which includes uncertainties in product demand, recycled product quality, and recycling rates.

2.2. Influences of CLSC Network Design

2.2.1. Internal Factor

The CLSC network is a complex issue with various dimensions and perspectives. The factors that affect the design of a CLSC network can be categorized as internal and external factors. Internal factors refer to the supply chain’s characteristics and elements, including the members, nodes, links, processes, and strategies. These factors determine the operational efficiency, effectiveness, sustainability, and competitiveness of the supply chain [16]. Sadeghi Ahangar et al. [17] consider facility location and heterogeneous vehicles as intrinsic influencers in designing a MILP model for a green CLSC network. The performance of the proposed model was evaluated using data from an automotive parts company. The possibility of vehicle scheduling is considered by Nasr et al. [18] who propose a novel two-stage fuzzy model for sustainable CLSC network design and supplier selection in the apparel industry. Different strategies such as carbon cap as a constraint, carbon tax as a cost, and carbon cap and trade as both cost and constraint are considered by Xu et al. [19] in their multi-period model for hybrid CLSC network design. The spatial distribution of collection, recycling, reuse, and production is considered by Scheller et al. [20] in their construction of a novel CLSC production planning model. The case of lithium electronic batteries is used to validate the effectiveness of the CLSC transformation. In this study, Rahimi and Ghezavati [21] propose a model for the design of multi-period, multi-objective reverse logistics networks for construction and demolition waste, which incorporates facility capacity expansion. In their research, Melo et al. [22] study capacity expansion and contraction scenarios in the design of supply chain networks to manage demand fluctuations. Solvang and Yu [23] investigate the effect of system flexibility on reverse logistics and develop a model for constructing a multi-product, flexible, and sustainable reverse logistics network. While existing research has proposed models that consider various internal factors such as the spatial distribution of collection, recycling, reuse, and production; facility capacity expansion; and system flexibility, these factors have not been examined simultaneously in their models. This represents a gap in the literature.

Moreover, despite the growing attention given to the flexible facility capacity factor in reverse logistics network design research, only a few studies have explored the CLSC network design problem using the flexible facility capacity strategy.

2.2.2. External Factors

External factors pertain to the environment and context in which the supply chain operates, such as market demand, customer preferences, policies, and regulations, technological innovation, and competitive pressures. These factors determine the direction of development, the trend of change, and the adaptability and flexibility of the supply chain. Salehi-Amiri et al. [24] investigate the external social employment effects of a CLSC network in the avocado industry and propose a CLSC network that optimizes operational and social factors. Their aim is to minimize total costs and maximize job creation, and they have developed a MILP model to this end. Shahparvari et al. [25] develop models of reverse logistics networks with uncertain product flows and emphasize the effect of carbon tax changes on factory decisions. Marti et al. [26] explore the impact of environmental policies on supply chain networks by incorporating carbon footprint caps and carbon taxes as variables in the model. However, it does not consider other types of policies or regulations. Sazvar et al. [27] design a resilient supply chain network that can withstand variations in demand through an optimization framework that incorporates capacity planning and redundancy. Mishra and Singh [28] devise a nonlinear mixed-integer linear programming model that focuses on multi-country reverse logistics networks and explores the effectiveness of scaling up facility capacity as a strategy for improving the network’s efficiency. The research conducted by Eslamipirharati et al. [29] aims at developing a sustainable reverse supply chain network with two objectives, taking into account the effects of carbon tax policies and government subsidies. Most of the previous research studies on CLSC networks have not factored in government subsidies as an external variable. In contrast, the model presented in this paper incorporates government subsidies for remanufacturing, reprocessing, and energy recovery.

The literature above is summarized in

Table 1. Literature related to CLSC network.

Literature	CLSC	Multi-Obj	Multi-Period		Uncertainty		Flexible Facility Capacity	Government Subsidy	Method
				Demand	Recycling	Recycled Product
					Rate	Quality
Fazli-Khalaf et al. [5]	√	√							RFSP, CPLEX
Shi et al. [6]	√		√	√					NP, LRBA
Alimoradi et al. [8]	√	√			√				FMILP, GA
Talaei et al. [10]	√	√		√					RFP
Francas [11]	√			√	√				TSSP, MPA
Arabi [14]	√		√	√		√	√		TSSP, CPLEX
Boujelben et al. [15]	√	√			√	√			TSSP, NSGAII, LPR
Sadeghi et al. [17]	√	√	√		√				FP, CPLEX
Nasr et al. [18]	√	√							FGP, COUEENE
Xu et al. [19]	√		√	√					IMILP, CPLEX
Melo et al. [22]							√		MIP, CPLEX
Solvang [23]		√	√				√		TSSP, LINGO
Salehi-Amiri et al. [24]	√	√						√	MILP, LMM
Eslamipirharati et al. [29]	√	√				√		√	TSSP, NSGAII
Our paper	√	√	√	√	√	√	√	√	MIP, BDA, GUROBI

Note: RFSP, CPLEX, NP, LRBA, FMILP, GA, RFP, TSSP, MPA, NSGAII, LPR, FP, FGP, COUEENE, IMILP, MIP, LINGO, MILP, LMM, and GUROBI stand for robust fuzzy stochastic programming, CPLEX solver, nonlinear programming, lagrangian relaxation based approach, fuzzy mixed integer linear programming, genetics algorithm, robust fuzzy programming, two-stage stochastic programming, mathematical programming approach, non-dominated sorting genetic algorithm, linear programming relaxation, fuzzy programming, fuzzy goal programming, COUEENE solver, integrated mixed integer linear programming, mixed integer programming, LINGO solver, mixed integer linear programming, linear programming-metric method and GUROBI solver.

. While previous studies have made significant contributions to the field of CLSC network design under uncertainty, most of them have solely investigated one or two uncertainty parameters, often ignoring the impact of three or more. Furthermore, limited attention has been given by scholars to the influence of flexible facility capacity strategy and government subsidies on CLSC network design.

This research aims to address gaps in understanding of CLSC network design by constructing a CLSC network that accounts for multiple uncertainties, including demand, recycling rate, and recycling quality. In addition, it incorporates the impact of flexible facility capacity strategy and government subsidies. This comprehensive approach yields a more nuanced understanding of CLSC network design, considering multiple internal factors simultaneously and exploring the impact of flexible facility capacity strategy.

3. Problem Formulation

3.1. Problem Description

To begin, raw materials are purchased by manufacturers from the market to create new products, which are subsequently transported to multiple distributors for dissemination to satisfy consumer demands in the market. When customers have completed using the product, a certain quantity of recycled product is generated, initiating the reverse supply chain. The collection center collects a portion of recycled products and subsequently transfers them to the inspection center for evaluation. The inspection center categorizes end-of-life products into four disposal types based on their quality and availability. The highest and second-highest quality recycled products are respectively sent to the Remanufacturing Center and the Recycling Center for remanufacturing and reprocessing. The resulting products can be used to various degrees in the production of new items. Lower-quality recycled products are transported to the energy center to replace the original smelting fuel and produce raw materials for production purposes. Unusable recycled products are transported to different landfill sites for disposal. This is illustrated in Figure 1.

This paper presents a model for a CLSC network that operates in a multi-objective, multi-product, multi-period uncertain environment. The model takes into consideration the government subsidies provided at different levels for the remanufacturing, reprocessing, and energy recovery segments of the reverse supply chain. The facility costs in a CLSC at a given capacity level include both fixed and variable costs. Fixed costs are incurred for building or leasing the facility, while variable costs are incurred for the daily operation of the facility on a period basis. Furthermore, this section examines various levels of flexible capacity for different facilities based on their handling capacity. The construction of a CLSC network, as presented in this paper, is affected by an uncertain environment, including uncertainty in market demand as well as the quantity and quality of recycled products.

3.2. Assumptions and Indices

This paper abstracts and simplifies certain mathematical problems, among which the specific assumptions are as follows:

Assumption 1.

Cross-level operation is not allowed in the CLSC network presented in this paper. In addition, the potential locations of facilities in the CLSC are predetermined for the CLSC to function properly;

Assumption 2.

Recycled products that go through the remanufacturing and reprocessing process can be used to meet the requirements of new products at varying levels. The quality of the new products made through this process is similar to that of the products made from raw materials;

Assumption 3.

This paper solely considers CO₂ emissions as the environmental assessment criterion. The unit CO₂ emissions from manufacturing new products along with remanufactured and reprocessed recycled products are equivalent in the flexible capacity model and the fixed capacity model;

Assumption 4.

To simplify modeling and calculations, this paper assumes that no products or materials are lost during processing and handling at any stage of the CLSC;

Assumption 5.

To enhance the model’s applicability, this paper assumes that recycled products found unusable after testing at the inspection center, are non-recoverable at the three stages of remanufacturing, reprocessing, and energy recovery, and can only be disposed of at landfills;

The sets, parameters, and decision variables are defined as follows.

(1) Set and indicator set:

- T: the set of CLSC network operation time period t;

- P: the set of product categories p;

- M: the set of potential new product manufacturers m;

- D: the set of distributors d;

- C: the set of existing customer locations c;

- O: the set of existing recycled product collection centers o;

- I: the set of potential recycled product inspection centers i;

- R: the set of potential recycled product remanufacturing facilities r;

- U: the set of potential recycled product recycling facilities u;

- E: the set of potential recycled product energy recovery facilities e;

- F: the set of existing non-recyclable product landfill sites f;

- L: the set of flexible capacity levels for manufacturers m, inspection centers i, remanufacturing facilities r, recycling facilities u, and energy recovery facilities e.

(2) Parameters and related variables:

- $\pi$: quality level of product p, with 1 being low quality, 2 being medium quality, and 3 being high quality;

- $G{S}_p^t$: Government subsidies for remanufacturing r, reprocessing u, and energy recovery e segments of the product p over the period t;

- ${\alpha }_p^t$: the proportion of recycled products reperiodd by collection centers for p products in period t as a percentage of products sold by distributors to customers;

- ${\gamma }_{pr},{\gamma }_{pu},{\gamma }_{pe}$: the proportion of product p that applies to remanufacturing r, reprocessing u, and energy recovery e;

- $R{P}_p$: the proportion of product p that can be used for remanufacturing, reprocessing, and energy recovery if the product is of good quality;

- $NR{P}_p$: the proportion of product p that is not available for remanufacturing, reprocessing, and energy recovery if the product is of good quality;

- $MRP$: the minimum percentage of products that can be used for recycling as required by the government for CLSC construction;

- ${\beta }_p^t$: the proportion of product p that is available for manufacturing new products after the remanufacturing step in period t;

- ${\lambda }_p^t$: the proportion of product p in period t that can be used to make new products after a reprocessing step;

- $D_c^t$: the demand for product p at customer point c in period t;

- $f{c}_m,f{c}_i,f{c}_r,f{c}_u,f{c}_e$: the fixed costs of building manufacturers m, inspection centers i, remanufacturing facilities r, recycling facilities u, and energy recovery facilities e;

- $f{c}_{lm},f{c}_{li},f{c}_{lr},f{c}_{lu},f{c}_{le}$: capacity level l fixed cost of construction of manufacturers m, inspection centers i, remanufacturing facility r, recycling facility u, energy recovery facility e;

- $v{c}_m^t,v{c}_i^t,v{c}_r^t,v{c}_u^t,v{c}_e^t$: variable costs of operating manufacturers m, inspection centers i, the remanufacturing facility r, the recycling facility u, and the energy recovery facility e over the period t;

- $v{c}_{lm}^t,v{c}_{li}^t,v{c}_{lr}^t,v{c}_{lu}^t,v{c}_{le}^t$: variable costs of operating manufacturers m, inspection centers i, remanufacturing facility r, recycling facility u, and energy recovery facility e at capacity level l over period t;

- $p{c}_p^t$: manufacturing cost per unit of product p in period t;

- $m{p}_p$: raw material cost per unit of product p;

- $r{c}_{pr}^t,r{c}_{pu}^t,r{c}_{pe}^t$: the cost of remanufacturing r, reprocessing u, and energy recovery e per unit of product p in period t;

- $l{c}_p^t$: the landfill cost per unit of product p in period t;

- $t{c}_p^t$: transportation cost per unit of product p transported per unit of distance in period t;

- $C{C}_m,C{C}_i,C{C}_r,C{C}_u,C{C}_e$: cost of capacity level change for manufacturer m, inspection center i, remanufacturing facility r, recycling facility u, energy recovery facility e;

- $C{Q}_{pm},C{Q}_{pi},C{Q}_{pr},C{Q}_{pu},C{Q}_{pe}$: capacity caps for manufacturers m, inspection center i, remanufacturing facility r, recycling facility u, energy recovery facility e;

- $C{Q}_{plm},C{Q}_{pli},C{Q}_{plr},C{Q}_{plu},C{Q}_{ple}$: capacity caps for manufacturers m, inspection center i, remanufacturing facility r, recycling facility u, and energy recovery facility e for l capacity level;

- $co{t}_{pm}^t,co{t}_{pi}^t,co{t}_{pr}^t,co{t}_{pu}^t,co{t}_{pe}^t,co{t}_{pf}^t$: carbon emissions generated in the manufacturing m, inspecting i, remanufacturing r, reprocessing u, energy recovery e, and landfill f segments for each unit of product p in period t;

- $co{t}_p^t$: carbon emissions generated per unit of product p transported per unit of distance in period t;

- $S_{md},S_{dc},S_{co},S_{oi},S_{ir},S_{iu},S_{ie},S_{if},S_{rm},S_{um}$: unit distance between facilities at all levels in the CLSC.

(3) Decision variables:

- $X_m,X_d,X_o,X_i,X_r,X_u,X_e$: a 0–1 variable that takes the value of 1 when the decision maker decides to build and use a manufacturer m, an inspection center i, a remanufacturing facility r, a recycling facility u, and an energy recovery facility e; otherwise, it takes the value of 0;

- $X_{lm},X_{li},X_{lr},X_{lu},X_{le}$: a 0–1 variable that takes the value of 1 when the decision maker decides to build a manufacturer m, an inspection center i, a remanufacturing facility r, a recycling facility u, and an energy recovery facility e with a capacity level of l; otherwise, it takes the value 0;

- $Y_m^t,Y_d^t,Y_o^t,Y_i^t,Y_r^t,Y_u^t,Y_e^t$: a 0–1 variable that takes the value of 1 when the decision maker decides to operate a manufacturer m, an inspection center i, a remanufacturing facility r, a recycling facility u, and an energy recovery facility e in period t; otherwise, it takes the value 0;

- $Y_{lm}^t,Y_{li}^t,Y_{lr}^t,Y_{lu}^t,Y_{le}^t$: a 0–1 variable that takes the value of 1 when the decision maker decides to operate the manufacturers m, the inspection center i, the remanufacturing facility r, the recycling facility u, and the energy recovery facility e with a capacity level of l in period t; otherwise, it takes the value 0;

- $C{L}_m^t,C{L}_i^t,C{L}_r^t,C{L}_u^t,C{L}_e^t$: a 0–1 variable that takes the value of 1 when the manufacturers m, the inspection facility i, the remanufacturing facility r, the recycling facility u, and the energy recovery facility e change their capacity level during period t; otherwise, it takes the value 0;

- $q_m,q_i,q_r,q_u,q_e,q_f$: continuous variable, the number of CLSC facilities manufacturing m, inspecting i, remanufacturing r, reprocessing u, energy recovery e, and landfill f products p in period t;

- $m{q}_m^t$: continuous variable, the quantity of raw materials purchased by manufacturers m for use in the production of product p in period t;

- $t{q}_{pmd}^t,t{q}_{pdc}^t,t{q}_{pco}^t,t{q}_{poi}^t,t{q}_{pir}^t,t{q}_{piu}^t,t{q}_{pie}^t,t{q}_{pif}^t,t{q}_{prm}^t,t{q}_{pum}^t$: a continuous variable that follows a CLSC process in period t from the manufacturers to the distributor, the distributor to the point of customer, the point of the customer to the collection center, the collection center to the inspection center, the inspection center to the (remanufacturing facility, recycling facility, energy recovery facility, landfill), (remanufacturing Facility, recycling facility) to manufacturers in terms of the quantity of product p transported.

3.3. Fixed-Capacity Model

This section presents a model of a CLSC network with fixed facility capacity. The model has two objective functions, the first of which is to minimize the total economic cost of operating the CLSC. The goal of this objective is to determine the optimal location and configuration of each facility, the movement of products between facilities, and as a result to reduce the total cost. The total costs include the expenses of building or leasing facilities, operating them on a cyclical basis, disposing of product manufacturing, remanufacturing, reprocessing, energy recovery, and landfilling, transporting products between facilities, and procurement of raw materials for new products. The second objective is to minimize the total carbon emissions from the CLSC’s operations, which include the carbon emissions from the manufacture, remanufacture, reprocessing, energy recovery, and landfilling of products, and product transportation between facilities at all levels.

$$\begin{array}{cc}\hfill Min{Z}_1& =\sum _{m\in M}f{c}_m{X}_m+\sum _{d\in D}f{c}_d{X}_d+\sum _{o\in O}f{c}_o{X}_o+\sum _{i\in I}f{c}_i{X}_i+\sum _{r\in R}f{c}_r{X}_r+\sum _{u\in U}f{c}_u{X}_u\hfill \\ \hfill & +\sum _{e\in E}f{c}_e{X}_e+\sum _{t\in T}\left[\left(\sum _{m\in M}v{c}_m{Y}_m^t+\sum _{d\in D}v{c}_d{Y}_d^t+\sum _{o\in O}v{c}_o{Y}_o^t+\sum _{i\in I}v{c}_i{Y}_i^t+\sum _{r\in R}v{c}_r{Y}_r^t\right.\right.\hfill \\ \hfill & \left.+\sum _{u\in U}v{c}_u{Y}_u^t+\sum _{e\in E}v{c}_e{Y}_e^t\right)+\sum _{p\in P}\left(\sum _{m\in M}p{c}_p^t{q}_{pm}^t+\sum _{i\in I}r{c}_{pi}^t{q}_{pi}^t+\sum _{r\in R}\left(r{c}_{pr}^t-G{S}_{pr}^t\right)q_{pr}^t\right.\hfill \\ \hfill & +\sum _{u\in U}\left(r{c}_{pu}^t-G{S}_{pu}^t\right)q_{pu}^t+\sum _{e\in E}\left(r{c}_{pe}^t-G{S}_{pe}^t\right)q_{pe}^t+\sum _{f\in F}l{c}_p^t{q}_{pf}^t+t{c}_p^t(\sum _{m\in M}\sum _{d\in D}{S}_{md}t{q}_{pmd}^t\hfill \\ \hfill & +\sum _{d\in D}\sum _{c\in C}{S}_{dc}t{q}_{pdc}^t+\sum _{c\in C}\sum _{d\in D}{S}_{co}t{q}_{pco}^t+\sum _{o\in O}\sum _{i\in I}{S}_{oi}t{q}_{poi}^t+\sum _{i\in I}\sum _{r\in R}{S}_{ir}t{q}_{pir}^t\hfill \\ \hfill & +\sum _{i\in I}\sum _{u\in U}{S}_{iu}t{q}_{piu}^t+\sum _{i\in I}\sum _{e\in E}{S}_{ie}t{q}_{pie}^t+\sum _{i\in I}\sum _{f\in F}{S}_{if}t{q}_{pif}^t+\sum _{r\in R}\sum _{m\in M}{S}_{rm}t{q}_{prm}^t\hfill \\ \hfill & \left.\left.+\sum _{u\in U}\sum _{m\in M}{S}_{um}t{q}_{pum}^t)+\sum _{m\in M}m{c}_{p}m{q}_{pm}^t\right)\right].\hfill \end{array}$$

The primary objective is to minimize the total economic cost of the CLSC. Terms one through seven represent the total fixed costs of establishing the manufacturer, distributor, collection center, inspection center, remanufacturing center, recycling center, and energy center. Terms eight through fourteen represent the total variable costs of operating the above facilities on a cyclical basis. The fifteenth item represents the total cost of producing new products, while the sixteenth item represents the total cost of inspecting recycled products. The seventeenth to nineteenth items represent the total cost of remanufacturing, reprocessing, and energy recovery of recycled products, after accounting for government subsidies. The twentieth item signifies the overall cost associated with landfilling the end-of-life product. The next set of costs, items twenty-one through thirty, demonstrate the overall cost of transporting the product between different facility levels as depicted in Figure 1 showcasing the CLSC sequence. The thirty-first term represents the total cost of raw material purchases made by the manufacturer for the product. The energy recovery is subtracted from carbon emissions, resulting in carbon savings.

$$\begin{array}{cc}\hfill Min{Z}_2& =\sum _{t\in T}\sum _{p\in P}\left[\left(\sum _{m\in M}{cor}_{pm}^t{q}_{pm}^t+\sum _{i\in I}{cor}_{pi}^t{q}_{pi}^t+\sum _{r\in R}{cor}_{pr}^t{q}_{pr}^t+\sum _{u\in U}{cor}_{pu}^t{q}_{pu}^t\right.\right.\hfill \\ \hfill & \left.+\sum _{e\in E}{cor}_{pe}^t{q}_{pe}^t+\sum _{f\in F}co{r}_{pf}^t{q}_{pf}^t\right)+\left(\sum _{m\in M}\sum _{d\in D}{cot}_p^t{S}_{md}t{q}_{pmd}^t\right.\hfill \\ \hfill & +\sum _{d\in D}\sum _{c\in C}{cot}_p^t{s}_{dc}t{q}_{pdc}^t+\sum _{c\in C}\sum _{o\in O}{cot}_p^t{S}_{co}t{q}_{pco}^t+\sum _{o\in O}\sum _{i\in I}{cot}_p^t{S}_{oi}t{q}_{poi}^t\hfill \\ \hfill & +\sum _{i\in I}\sum _{r\in R}{cot}_p^t{S}_{ir}t{q}_{pir}^t+\sum _{i\in I}\sum _{u\in v}{cot}_p^t{S}_{iu}t{q}_{pim}^t-\sum _{i\in I}\sum _{e\in E}{cot}_p^t{S}_{ie}t{q}_{pie}^t\hfill \\ \hfill & \left.\left.+\sum _{i\in I}\sum _{f\in F}{cot}_p^t{S}_{if}t{q}_{pif}^t+\sum _{r\in R}\sum _{m\in M}{cot}_p^t{S}_{rm}t{q}_{prm}^t+\sum _{u\in U}\sum _{m\in M}{cot}_p^t{S}_{um}t{q}_{pum}^t\right)\right].\hfill \end{array}$$

The second objective function aims to reduce the total carbon emissions of the CLSC. The first term represents the total carbon emissions resulting from the production of new products, while the remaining terms (second to sixth) represent the total carbon emissions arising from the inspecting, remanufacturing, reprocessing, energy recovery, and landfilling of recycled products. Lastly, the seventh to sixteenth terms (illustrated in Figure 1 for the CLSC) reflect the total carbon emissions resulting from the transport of products across different levels of the facility.

$$s.t.\sum _{r\in R}{\beta }_p^{t}t{q}_{prm}^{t-1}+\sum _{u\in U}{\lambda }_p^{t}t{q}_{pum}^{t-1}+m{q}_{pm}^t=q_{pm}^t\forall m\in M\forall t\in T\forall p\in P,$$

(1)

$$\sum _{d\in D}t{q}_{pmd}^t⩽q_{pm}^t\forall m\in M\forall t\in T\forall p\in P,$$

(2)

$$\sum _{c\in C}t{q}_{pdc}^t\le \sum _{m\in M}t{q}_{pmd}^t\forall d\in D\forall t\in T\forall p\in P,$$

(3)

$$\sum _{d\in D}t{q}_{pdc}^t⩾{\tilde{D}}_{pc}^t\forall t\in T,$$

(4)

$$\sum _{o\in O}t{q}_{pco}^t⩽\sum _{d\in D}\tilde{\alpha }t{q}_{pdc}^t\forall c\in C\forall t\in T\forall p\in p,$$

(5)

$$\sum _{c\in C}t{q}_{pco}^t=\sum _{i\in I}t{q}_{poi}^t\forall o\in O\forall t\in T\forall p\in P,$$

(6)

$$\sum _{o\in O}t{q}_{poi}^t=q_{pi}^t\forall t\in T\forall p\in P\forall i\in I,$$

(7)

$$\sum _{i\in I}t{q}_{pir}^t=q_{pr}^t\forall t\in T\forall p\in P\forall r\in R,$$

(8)

$$\sum _{i\in I}t{q}_{piu}^t=q_{pu}^t\forall t\in T\forall p\in P\forall u\in U,$$

(9)

$$\sum _{i\in I}t{q}_{pie}^t=q_{pe}^t\forall t\in T\forall p\in P\forall e\in E,$$

(10)

$$\sum _{i\in I}t{q}_{pif}^t=q_{pf}^t\forall t\in T\forall p\in P\forall f\in F,$$

(11)

$$\sum _{m\in M}t{q}_{prm}^t=q_{pr}^t\forall t\in T\forall p\in P\forall r\in R,$$

(12)

$$\sum _{m\in N}t{q}_{pum}^t=q_{pu}^t\forall t\in T\forall p\in P\forall u\in U,$$

(13)

$$q_{pi}^t=\sum _{r\in R}t{q}_{pir}^t+\sum _{u\in U}t{q}_{piu}^t+\sum _{e\in E}t{q}_{pie}^t+\sum _{f\in f}t{q}_{pif}^t\forall t\in T\forall p\in P\forall i\in I,$$

(14)

$$q_{pm}^t⩽Y_m^{t}C{Q}_{pm}\forall t\in T\forall p\in P\forall m\in M,$$

(15)

$$\sum _{m\in M}t{q}_{pmd}^t⩽Y_d^{t}C{Q}_{pd}\forall t\in T\forall p\in P\forall d\in D,$$

(16)

$$\sum _{c\in C}t{q}_{pco}^t⩽C{Q}_{po}\forall t\in T\forall p\in P\forall o\in O,$$

(17)

$$q_{pi}^t⩽Y_i^{t}C{Q}_{pi}\forall t\in T\forall p\in P\forall i\in I,$$

(18)

$$q_{pr}^t⩽Y_r^{t}C{Q}_{pr}\forall t\in T\forall p\in P\forall r\in R,$$

(19)

$$q_{pu}^t⩽Y_u^{t}C{Q}_{pu}\forall t\in T\forall p\in P\forall u\in U,$$

(20)

$$q_{pe}^t⩽Y_e^{t}C{Q}_{pe}\forall t\in T\forall p\in P\forall e\in E,$$

(21)

$$X_m⩾Y_m^t\forall t\in T\forall m\in M,$$

(22)

$$Y_m^{t+1}⩾Y_m^t\forall t\in T\forall m\in M,$$

(23)

$$X_d⩾Y_d^t\forall t\in T\forall d\in D,$$

(24)

$$Y_d^{t+1}⩾Y_d^t\forall t\in T\forall d\in D,$$

(25)

$$X_o⩾Y_o^t\forall t\in T\forall o\in O,$$

(26)

$$Y_o^{t+1}⩾Y_o^t\forall t\in T\forall o\in O,$$

(27)

$$X_i⩾Y_i^t\forall t\in T\forall i\in I,$$

(28)

$$Y_i^{t+1}⩾Y_i^t\forall t\in T\forall i\in I,$$

(29)

$$X_r⩾Y_r^t\forall t\in T\forall r\in R,$$

(30)

$$Y_r^{t+1}⩾Y_r^t\forall t\in T\forall r\in R,$$

(31)

$$X_u⩾Y_u^t\forall t\in T\forall u\in U,$$

(32)

$$Y_u^{t+1}⩾Y_u^t\forall t\in T\forall u\in U,$$

(33)

$$X_e⩾Y_e^t\forall t\in T\forall e\in E,$$

(34)

$$Y_e^{t+1}⩾Y_e^t\forall t\in T\forall e\in E,$$

(35)

$$q_{pm}^t⩾U{R}_{pm}C{Q}_{pm}{Y}_m^t\forall t\in T\forall p\in P\forall m\in M,$$

(36)

$$q_{pi}^t⩾U{R}_{pi}C{Q}_{pi}{Y}_i^t\forall t\in T\forall p\in P\forall i\in I,$$

(37)

$$q_{pr}^t⩾U{R}_{pr}C{Q}_{pr}{Y}_r^t\forall t\in T\forall p\in P\forall r\in R,$$

(38)

$$q_{pu}^t⩾U{R}_{pu}C{Q}_{pu}{Y}_u^t\forall t\in T\forall p\in P\forall u\in U,$$

(39)

$$q_{pe}^t⩾U{R}_{pe}C{Q}_{pe}{Y}_e^t\forall t\in T\forall p\in P\forall e\in E,$$

(40)

$${\pi }_p{\tilde{\gamma }}_{pr}{q}_{pi}^t⩾\sum _{r\in R}t{q}_{pir}^t\forall p\in P\forall t\in T\forall i\in I,$$

(41)

$${\pi }_p{\tilde{\gamma }}_{pu}{q}_{pi}^t⩾\sum _{u\in U}t{q}_{piu}^t\forall p\in P\forall t\in T\forall i\in I,$$

(42)

$${\pi }_p{\tilde{\gamma }}_{pe}{q}_{pi}^t⩾\sum _{e\in E}t{q}_{pie}^t\forall p\in P\forall t\in T\forall i\in I,$$

(43)

$$q_{pi}^t\left(NR{P}_p+\left(1-{\pi }_p\right)R{P}_p\right)⩽\sum _{f\in F}t{q}_{pif}^t\forall t\in T\forall p\in P\forall i\in I,$$

(44)

$$\sum _{r\in R}t{q}_{pir}^t+\sum _{u\in U}t{q}_{piu}^t+\sum _{e\in E}t{q}_{pie}^t⩾{\pi }_{p}R{P}_{p}MR{P}_p{q}_{pi}^t\forall t\in T\forall p\in P\forall i\in I,$$

(45)

$$\begin{array}{cc}\hfill & X_m,X_d,X_o,X_i,X_r,X_u,X_e,Y_m^t,Y_d^t,Y_o^t,Y_i^t,Y_r^t,Y_u^t,Y_e^t\in \left\{0,1\right\},\hfill \\ \hfill & q_m,q_i,q_r,q_u,q_e,q_f,t{q}_{pmd}^t,t{q}_{pdc}^t,t{q}_{pco}^t,t{q}_{poi}^t,t{q}_{pir}^t,t{q}_{piu}^t,t{q}_{pie}^t,t{q}_{pif}^t,t{q}_{prm}^t,t{q}_{pum}^t,m{q}_{pm}^t⩾0.\hfill \end{array}$$

(46)

Equation (1) restricts the production of new products. Equation (2) limits the quantity of new products sent from the manufacturer to the distributor. Equation (3) governs the flow of new products between the manufacturer, distributor, and customer. Equation (4) mandates that distributors meet product demand at each customer point. Equation (5) states that collection centers will collect a proportion of recycled products. Equation (6) refers to the flow constraint of recycled products between customer points, collection centers, and inspections. Equations (7)–(13) represent the flow constraints of the reverse supply chain in the CLSC. Equation (14) ensures that inspected recycled products are transported to remanufacturing centers, recycling centers, energy centers, and landfill sites. Equations (15)–(21) establish capacity and production limits for various facilities. Equations (22)–(35) restrict the location and operation of facilities. Equations (36)–(40) mandate that facilities maintain a minimum utilization level. Equations (41)–(43) establish the amount of recycled products to be transported based on availability. Equation (44) governs the disposal of unsuitable recycled products at landfills. Equation (45) establishes the maximum amount that can be transported to a landfill. Finally, Equation (46) defines the domain of the decision variable. These constraints guide the optimization process in an uncertain environment, balancing economic costs with environmental benefits and reducing carbon emissions.

3.4. Flexible-Capacity Model

Efficient facility utilization at all levels of the CLSC can effectively reduce overall costs during network design. Facilities with larger capacities in the fixed capacity model tend to make it difficult to achieve higher facility utilization, thus leading to increased total supply chain costs. The flexible capacity model has facilities at all levels with varying fixed and variable costs based on their size, which can be large, medium, or small. Flexible shifting of facilities at different capacity levels is a way to improve CLSC operations’ performance. Capacity levels in the model are denoted by the variable l, and the set L can be considered to represent any capacity level. Modifying l for a specific facility alters the capacity of that facility and its overall capacity. Additionally, it affects the fixed costs of constructing the lease and the variable costs of operating the facility. Although most aspects of the flexible facility capacity model are the same as those in the fixed capacity model, the cost objective function, capacity constraints, and facility utilization constraints will differ when manufacturers, inspection centers, remanufacturing centers, recycling centers, and energy centers adopt a flexible facility capacity strategy. The flexible capacity model introduces a new set of constraints to ensure that facility capacity operates at only a single level over a period. Furthermore, it takes into account the expenses associated with modifying facility capacity levels.

$$\begin{array}{cc}\hfill Min{Z}_1& =\sum _{t\in T}\sum _{l\in L}\left(\sum _{m\in M}v{c}_{lm}^t{Y}_{lm}^t+\sum _{i\in I}v{c}_{li}^t{Y}_{li}^t+\sum _{r\in R}v{c}_{lr}^t{Y}_{lr}^t\right.\hfill \\ \hfill & \left.+\sum _{u\in U}v{c}_{lu}^t{Y}_{lu}^t+\sum _{e\in E}v{c}_{le}^t{Y}_{le}^t\right)+\sum _{t\in T}\left(\sum _{m\in M}C{C}_{m}C{L}_m^t\right.\hfill \\ \hfill & \left.+\sum _{i\in I}C{C}_{i}C{L}_i^t+\sum _{r\in R}C{C}_{r}C{L}_r^t+\sum _{u\in U}C{C}_{u}C{L}_u^t+\sum _{e\in E}C{C}_{e}C{L}_e^t\right).\hfill \end{array}$$

The objective function of the flexible facility capacity model modifies the fixed and variable cost functions of the objective function of the fixed facility capacity model for facilities that can vary in capacity, such as manufacturers, inspection centers, remanufacturing centers, recycling centers, and energy centers. Additionally, it includes the cost function for the alteration of the facility capacity level.

$$s.t.q_{pm}^t⩽\sum _{l\in L}C{Q}_{plm}{Y}_{lm}^t\forall t\in T\forall p\in P\forall m\in M,$$

(47)

$$q_{pi}^t⩽\sum _{l\in L}C{Q}_{pli}{Y}_{li}^t\forall t\in T\forall p\in P\forall i\in I,$$

(48)

$$q_{pr}^t⩽\sum _{l\in L}C{Q}_{plr}{Y}_{lr}^t\forall t\in T\forall p\in P\forall r\in R,$$

(49)

$$q_{pu}^t⩽\sum _{l\in L}C{Q}_{plu}{Y}_{lu}^t\forall t\in T\forall p\in P\forall u\in U,$$

(50)

$$q_{pe}^t⩽\sum _{l\in L}C{Q}_{ple}{Y}_{le}^t\forall t\in T\forall p\in P\forall e\in E,$$

(51)

$$q_{pm}^t⩾U{R}_{pm}\sum _{l\in L}C{Q}_{plm}{Y}_{lm}^t\forall t\in T\forall p\in P\forall m\in M,$$

(52)

$$q_{pi}^t⩾U{R}_{pi}\sum _{l\in L}C{Q}_{pli}{Y}_{li}^t\forall t\in T\forall p\in P\forall i\in I,$$

(53)

$$q_{pr}^t⩾U{R}_{pr}\sum _{l\in L}C{Q}_{plr}{Y}_{lr}^t\forall t\in T\forall p\in P\forall r\in R,$$

(54)

$$q_{pu}^t⩾U{R}_{pu}\sum _{l\in L}C{Q}_{plu}{Y}_{lu}^t\forall t\in T\forall p\in P\forall u\in U,$$

(55)

$$q_{pe}^t⩾U{R}_{pe}\sum _{l\in L}C{Q}_{ple}{Y}_{le}^t\forall t\in T\forall p\in P\forall e\in E,$$

(56)

$$\sum _{l\in L}{Y}_{lm}^t⩽1\forall t\in T\forall m\in M,$$

(57)

$$C{L}_m^t⩾Y_{l^{*}m}^t+\sum _{l\in L\setminus \left\{l^{*}\right\}}{Y}_{lm}^{t-1}-1\forall t\in T\forall m\in M\forall l^{*}\in L,$$

(58)

$$\sum _{l\in L}{Y}_{li}^t⩽1\forall t\in T\forall i\in I,$$

(59)

$$C{L}_i^t⩾Y_{l^{*}i}^t+\sum _{l\in L\setminus \left\{l^{*}\right\}}{Y}_{li}^{t-1}-1\forall t\in T\forall i\in I\forall l^{*}\in L,$$

(60)

$$\sum _{l\in L}{Y}_{lr}^t⩽1\forall t\in T\forall r\in R,$$

(61)

$$C{L}_r^t⩾Y_{l^{*}r}^t+\sum _{l\in L\setminus \left\{l^{*}\right\}}{Y}_{lr}^{t-1}-1\forall t\in T\forall r\in R\forall l^{*}\in L,$$

(62)

$$\sum _{l\in L}{Y}_{lu}^t⩽1\forall t\in T\forall u\in U,$$

(63)

$$C{L}_u^t⩾Y_{l^{*}u}^t+\sum _{l\in L\setminus \left\{l^{*}\right\}}{Y}_{lu}^{t-1}-1\forall t\in T\forall u\in U\forall l^{*}\in L,$$

(64)

$$\sum _{l\in L}{Y}_{le}^t⩽1\forall t\in T\forall e\in E,$$

(65)

$$C{L}_e^t⩾Y_{l^{*}e}^t+\sum _{l\in L\setminus \left\{l^{*}\right\}}{Y}_{le}^{t-1}-1\forall t\in T\forall e\in E\forall l^{*}\in L,$$

(66)

$$Y_{lm}^t,Y_{li}^t,Y_{lr}^t,Y_{lu}^t,Y_{le}^t,C{L}_m^t,C{L}_i^t,C{L}_r^t,C{L}_u^t,C{L}_e^t\in \left\{0,1\right\}.$$

(67)

Equations (47)–(56) have been adapted from the fixed capacity model for use in the flexible facility capacity model. According to Equations (57), (59), (61), (63) and (65), facilities at the manufacturer, inspection center, remanufacturing center, recycling center, and energy center levels must operate at a single capacity level during the current period. Equations (58), (60), (62), (64) and (66) record changes in capacity levels at each facility level over different periods, allowing for accurate assessment of costs arising from changes in capacity levels. These objective functions and constraints ensure that the model accurately analyzes changes in facility capacity levels over different periods. It’s important to note that if there’s no cost for changing a facility’s capacity levels, the decision variables linked to changes in capacity levels will be assumed to be identical to 1 ($C{L}_{m,i,r,u,e}=1$), regardless of any actual changes in the facility’s capacity levels. This implies that under zero cost for changes to facility capacity levels, the CLSC decision is not affected by the flexible capacity strategy.

4. Robust Model

This paper utilizes a robust optimization approach to handle uncertain parameters in the proposed CLSC problem. This paper discusses the robust optimization method proposed by Li et al. [30] and presents a conversion of the MILP model, proposed in the previous section, into a mixed-integer robust model. When there is uncertainty about product demand, recycled product quality, and recycling rate, the objective of the robust optimization set-based approach is to select the optimal solution that does not violate the uncertainty situation from the uncertainty of possible solutions. The following definitions are provided for the uncertain parameters in the MIP problem.

Definition 1.

To simplify the representation, it is common to consider the following MILP model (see Equation (68)):

$$\begin{array}{cc}\hfill Max& \sum _m^{}{c}_m{x}_m\hfill \\ \hfill s.t.& \sum _m{\tilde{a}}_{im}{x}_m\le {\tilde{d}}_i\forall i.\hfill \end{array}$$

(68)

Since the product demand, recycled product quality, and recycling rate uncertainties in this paper involve continuous variables and constant constraint terms, this section performs robust equivalent transformations of these two terms of the MIP model. First, consider the uncertainty of the parameters in the constraints of the MIP model, where x and d denote the continuous variable and constant constraint terms, respectively, and ${\tilde{a}}_{im}$, ${\tilde{d}}_i$ denote the true values of the parameters that may be affected by the uncertainty. Considering the ith constraint of the above model, assume that the uncertain parameter in the ith limit is defined as in (69):

$$\begin{array}{cc}\hfill & {\tilde{a}}_{im}=a_{im}+{\xi }_{im}{\widehat{a}}_{im},\forall m\in M_i,\hfill \\ \hfill & {\tilde{d}}_i=d_i+{\xi }_{i0}{\widehat{d}}_i,\hfill \end{array}$$

(69)

where $M_i$ denotes the subset containing indices of continuous variables whose corresponding coefficients are subject to uncertainty; $a_{im}$, $p_i$ denote the nominal values of the parameters; ${\widehat{a}}_{im}$, ${\widehat{p}}_i$ denote the positive constant perturbations; and ${\xi }_{im}$, ${\xi }_{i0}$ are the random variables affected by uncertainty. using Definition 1 above, the ith constraint of the original model (68) can be rewritten as Equation (70):

$$\sum _{m\notin M_i}^{}{a}_{im}{x}_m+\sum _{m\in M_i}^{}{\tilde{a}}_{im}{x}_m\le {\tilde{d}}_i,$$

(70)

The uncertainty term in the Equations (69) and (70) is collectively referred to as (71):

$$\sum _m^{}{a}_{im}{x}_m+\left\{-{\xi }_{i0}{\widehat{d}}_i+\sum _{m\in M_i}^{}{\xi }_{im}{\widehat{a}}_{im}{x}_m\right\}\le d_i.$$

(71)

Set-induced robust optimization methods involve a predefined set of uncertainties denoted by U. The aim is to find a feasible solution for any x in the predefined uncertainty set to avoid infeasibility, as demonstrated in Equation (72):

$$\sum _j{a}_{ij}{x}_j+\left[\underset{\xi \in U}{max}\left\{-{\xi }_{i0}{\widehat{d}}_i+\sum _{j\in J_i}{\xi }_{ij}{\widehat{a}}_{ij}{x}_j\right\}\right]\le d_i.$$

(72)

The original model’s constraints (68) are replaced with the corresponding robust counterpart constraints. As a result, the robust dual counterpart Equation (73) for the original MIP model is obtained:

$$\begin{array}{cc}\hfill Max& \sum _m{c}_m{x}_m\hfill \\ \hfill s.t.& \sum _m{a}_{im}{x}_m+{max}_{\xi \in U}\left\{-{\xi }_{i0}{\widehat{d}}_i+\sum _{m\in M_i}{\xi }_{im}{\widehat{a}}_{im}{x}_m\right\}\le d_i{\forall }_i.\hfill \end{array}$$

(73)

The formulation of robust optimization models depends on the selection of uncertainty sets that represent distinct environments of uncertainty for the model’s parameters. This section introduces three uncertainty sets-box, ellipsoidal, and polyhedral. To simplify the complexity, we eliminate the constraint index i from the random vector ξ. We consider the data to be uncertain within a specific type of uncertainty set. Our objective in this paper is to identify optimal solutions from among the feasible candidates that are realizable for all data within the uncertainty set, i.e., the ones that are not subjected to the effects of uncertainty. We present below three definitions of uncertainty sets.

Definition 2.

The box uncertainty set can be denoted in terms of an infinite number of paradigms of uncertain data vectors. The variable parameter controlling the size of the uncertainty set can be represented by Ψ, as denoted in Equation (74):

$$U_{\infty }=\left\{\xi \mid {\parallel \xi \parallel }_{\infty }\le \Psi \right\}=\left\{\xi \parallel {\xi }_j\mid \le \Psi ,{\forall }_j\in J_i\right\}.$$

(74)

If the uncertainty parameter is bounded within a specific interval ${\tilde{a}}_{ij}\in \left[a_{ij}-{\tilde{a}}_{ij}{a}_{ij}+{\tilde{a}}_{ij}\right]\forall j\in J_i$, the uncertainty can be represented by ${\tilde{a}}_{ij}=a_{ij}+{\xi }_{ja}ij$, when $\psi =1$ ($U_{\infty }=\left\{\xi \right||{\widehat{\xi }}_j\mid \le 1,$$\forall j\in J_i\}$), for the special case of the box uncertainty set. In this paper, we use interval uncertainty sets to denote box sets for $Psi=1$, and box uncertainty sets to denote adjustable bounded sets in general.

Definition 3.

The ellipsoidal uncertainty set is expressed in terms of the 2-paradigm of the uncertain data vector, and Ω is the variable parameter controlling the size of the uncertainty set, expressed as in Equation (75):

$$U_2=\left\{\xi \mid {\parallel \xi \parallel }_2\le \Omega \right\}=\left\{\xi \mid \sqrt{\sum _{j\in J_i}{\xi }_j^2}\le \Omega \right\}.$$

(75)

Definition 4.

The polyhedral uncertainty set is represented by the 1-paradigm of the uncertain data vector and Γ is the variable parameter controlling the size of the uncertainty set, expressed as in Equation (76):

$$U_1=\left\{\xi \mid {\parallel \xi \parallel }_1\le \Gamma \right\}=\left\{\xi \mid \sum _{j\in J_i}\left|{\xi }_j\right|\le \Gamma \right\}.$$

(76)

The three uncertainty sets, box, ellipsoidal, and polyhedral, can be further combined to generate new uncertainty sets. Bounded uncertainty is an important feature of uncertainty that has received considerable attention in practice. Subsequently, this paper introduces three sets of uncertainty, comprising box, ellipsoidal, and polyhedral which are referred to as the ‘box + ellipsoidal’ set, the ‘box + polyhedral’ set, and the ‘box + ellipsoidal + polyhedral’ set respectively, as uncertainty sets.

Definition 5.

“Box + ellipsoidal” uncertainty set This type of uncertainty set is the intersection between an ellipsoidal and a box, as defined in (77):

$$U_{2\cap \infty }=\left\{\xi |\sum _{j\in J_i}{\xi }_j^2\le {\Omega }^2,|{\xi }_j\mid \le \Psi ,\forall j\in J_i\right\}.$$

(77)

According to geometry, the adjustable box defined in Section 3.2 and the adjustable ellipsoidal defined in Section 3.3 must fulfill the relation (78) to guarantee that their intersection does not reduce to a single shape:

$$\Psi \le \Omega \le \Psi \sqrt{\left|J_i\right|},$$

(78)

When $\Psi =1$, the above set (78) defines the intersection between the interval and the ellipsoidal, which is referred to in this paper as the “interval + ellipsoidal” uncertainty set.

Definition 6.

“Box + Polyhedral” Uncertainty Set The described uncertainty set is formed at the intersection of a polyhedral set and an interval set. The defining values of the interval set are the 1-paradigm and infinity-paradigm, as illustrated in Equation (79):

$$U_{1\cap \infty }=\left\{\xi |\sum _{j\in J_i}|{\xi }_j|\le \Gamma ,|{\xi }_j\mid \le \Psi ,\forall j\in J_i\right\}.$$

(79)

If the parameters satisfy the relation (80), the intersection between the set of adjustable boxes, described in Definition 2, and the set of adjustable polyhedral, explained in Definition 4, cannot be reduced to either of the two adjustable sets, similar to the previously mentioned case.

$$\Psi \le \Gamma \le \Psi \sqrt{\left|J_i\right|},$$

(80)

When $\Psi =1$, the above set (80) defines the intersection between intervals and polyhedral, which is referred to in this paper as the “interval + polyhedral” uncertainty set.

Definition 7.

“Box + ellipsoidal + polyhedral” Uncertainty Set This type of uncertainty set is the intersection between a polyhedral and an interval set defined by the 1-paradigm and the infinity-paradigm, as shown in (81):

$$U_{1\cap 2\cap \infty }=\left\{\xi |\sum _{j\in J_i}\left|{\xi }_j\right|\le \Gamma ,\sum _{j\in J_i}{\xi }_j^2\le {\Omega }^2,\right.\left.|{\xi }_j|\le \Psi ,\forall j\in J_i\right\}.$$

(81)

If the adjustable parameters satisfy relation (82), the intersection between the polyhedral and the ellipsoidal for this type of uncertainty set is not reduced to either of them.

$$\begin{array}{cc}\hfill & \Psi \le \Omega \le \Psi \sqrt{\left|J_i\right|},\hfill \\ \hfill & \Omega \le \Gamma \le \Omega \sqrt{\left|J_i\right|},\hfill \end{array}$$

(82)

The initial equation confirms the intersection of the ellipsoidal and the box, while the second one confirms the intersection of the ellipsoidal with the polyhedral. This paper presents general considerations for deriving the robust counterpart equation under various uncertainty sets. For the CLSC network model constructed in this paper as market demand and recycling rate and quality uncertainty, the uncertain parameters of the robust counterpart equation appear on the left and right sides of the ith constraint equation, which are assumed to be subjected to bounded uncertainty and there are 10% of demand coefficients, 8% of recycling rate coefficients, and 15% of recycling quality coefficients that deviate from their maximum deviation from their nominal values. The robust counterpart of the three uncertain parameters under the six uncertainty sets is given below as box, ellipsoidal, polyhedral, ‘interval + ellipsoidal’, ‘interval + polyhedral’, and ‘interval + ellipsoidal + polyhedral’.

Box set: Transform Equations (4), (5), (41), (42) and (43) into robust equivalents for the set of box uncertainty, as shown in Equations (83)–(87):

$$\sum _{d\in D}t{q}_{pdc}^t⩾D_{pc}^t+\Psi {\widehat{D}}_{pc}^t\forall p\in P\forall t\in T\forall c\in C,$$

(83)

$$\left\{\begin{array}{c}\sum _{o\in O}t{q}_{pco}^t-\sum _{d\in D}{\alpha }_p^{t}t{q}_{pdc}^t+\Psi \left(0.08\sum _{0\in 0}t{q}_{pco}^t+\sum _{d\in D}{\widehat{\alpha }}_p^{t}t{q}_{pdc}^t\right)⩽0\hfill \\ \forall t\in T\forall p\in P\forall c\in C,\hfill \end{array}\right.$$

(84)

$$\left\{\begin{array}{c}\sum _{u\in U}t{q}_{piu}^t-{\pi }_p{\gamma }_{pu}{q}_{pi}^t+\Psi \left(0.15\sum _{u\in U}t{q}_{piu}^t+{\widehat{\gamma }}_{pu}{\pi }_p{q}_{pi}^t\right)⩽0\hfill \\ \forall p\in P\forall t\in T\forall i\in I,\hfill \end{array}\right.$$

(85)

$$\left\{\begin{array}{c}\sum _{u\in U}t{q}_{piu}^t-{\pi }_p{\gamma }_{pu}{q}_{pi}^t+\Psi \left(0.15\sum _{u\in U}t{q}_{piu}^t+{\widehat{\gamma }}_{pu}{\pi }_p{q}_{pi}^t\right)⩽0\hfill \\ \forall p\in P\forall t\in T\forall i\in I,\hfill \end{array}\right.$$

(86)

$$\left\{\begin{array}{c}\sum _{p\in E}t{q}_{pie}^t-{\pi }_p{\gamma }_{pe}{q}_{pi}^t+\Psi \left(0.15\sum _{p\in E}t{q}_{pie}^t+{\widehat{\gamma }}_{pe}{\pi }_p{q}_{pi}^t\right)⩽0\hfill \\ \forall p\in P\forall t\in T\forall i\in I.\hfill \end{array}\right.$$

(87)

Ellipsoidal set: Transform Equations (4), (5), (41), (42) and (43) into robust equivalents for the set of ellipsoidal uncertainty, as shown in Equations (88)–(92):

$$\sum _{d\in D}t{q}_{pdc}^t⩾D_{pc}^t+\Omega \sqrt{{\left(D_{pc}^t\right)}^2}\forall c\in C\forall t\in T\forall p\in P,$$

(88)

$$\left\{\begin{array}{c}\sum _{o\in O}t{q}_{pco}^t-\sum _{d\in D}{\alpha }_p^{t}t{q}_{pdc}^t\hfill \\ +\Omega \sqrt{{\sum }_{o\in O}{\left(0.08\right)}^2{\left(t{q}_{pdc}^t\right)}^2+{\sum }_{d\in E}{\left({\widehat{\alpha }}_p^t\right)}^2{\left(t{q}_{pdc}^t\right)}^2}⩽0\hfill \\ \forall c\in C\forall t\in T\forall p\in P,\hfill \end{array}\right.$$

(89)

$$\left\{\begin{array}{c}\sum _{r\in R}t{q}_{pir}^t-{\pi }_p{\gamma }_{pr}{q}_{pi}^t+\Omega \sqrt{{\sum }_{r\in R}{\left(0.15\right)}^2{\left({tq}_{pir}^t\right)}^2+{\left({\widehat{\gamma }}_{pr}\right)}^2{\left({\pi }_p{q}_{pi}^t\right)}^2}⩽0\hfill \\ \forall p\in P\forall t\in T\forall i\in I,\hfill \end{array}\right.$$

(90)

$$\left\{\begin{array}{c}\sum _{u\in U}t{q}_{piu}^t-{\pi }_p{\gamma }_{pu}{q}_{pi}^t+\Omega \sqrt{{\sum }_{u\in v}{\left(0.15\right)}^2{\left({tg}_{piu}^t\right)}^2+{\left({\widehat{\gamma }}_{pu}\right)}^2{\left({\pi }_p{q}_{pi}^t\right)}^2}⩽0\hfill \\ \forall p\in P\forall t\in T\forall i\in I,\hfill \end{array}\right.$$

(91)

$$\left\{\begin{array}{c}\sum _{e\in E}t{q}_{pie}^t-{\pi }_p{\gamma }_{pe}{q}_{pi}^t+\Omega \sqrt{{\sum }_{e\in E}{\left(0.15\right)}^2{\left({tq}_{pie}^t\right)}^2+{\left({\widehat{\gamma }}_{pe}\right)}^2{\left({\pi }_p{q}_{pi}^t\right)}^2}⩽0\hfill \\ \forall p\in P\forall t\in T\forall i\in I.\hfill \end{array}\right.$$

(92)

Polyhedral set: Transform Equations (4), (5), (41), (42) and (43) into robust equivalents for the set of ellipsoidal uncertainty, as shown in Equations (93)–(97):

$$\left\{\begin{array}{c}\sum _{d\in D}t{q}_{pdc}^t⩾D_{pc}^t+h_{pc}^t\Gamma \forall t\in T\forall p\in P\forall c\in C,\hfill \\ h_{pc}^t⩾{\widehat{D}}_{pc}^t\forall t\in T\forall p\in P\forall c\in C,\hfill \end{array}\right.$$

(93)

$$\left\{\begin{array}{c}\sum _{o\in O}t{q}_{pco}^t-\sum _{d\in D}{\alpha }_p^{t}t{q}_{pdc}^t+h_{pc}^t\Gamma \le 0\forall t\in T\forall p\in P\forall c\in C,\hfill \\ h_{pc}^t⩾0.08t{q}_{pco}^t\forall o\in O\forall t\in T\forall p\in P\forall c\in C,\hfill \\ h_{pc}^t⩾{\widehat{\alpha }}_p^{t}t{q}_{pdc}^t\forall d\in D\forall t\in T\forall p\in P\forall c\in C,\hfill \end{array}\right.$$

(94)

$$\left\{\begin{array}{c}\sum _{r\in R}t{q}_{pir}^t-{\pi }_p{\gamma }_{pr}{q}_{pi}^t+h_{pi}^t\Gamma ⩽0\forall t\in T\forall p\in P\forall i\in I,\hfill \\ h_{pi}^t⩾0.15q_{pir}^t\forall t\in T\forall p\in P\forall i\in I\forall r\in R,\hfill \\ h_{pi}^t⩾{\widehat{\gamma }}_{pr}{\pi }_p{q}_{pi}^t\forall t\in T\forall p\in P\forall i\in I\forall r\in R,\hfill \end{array}\right.$$

(95)

$$\left\{\begin{array}{c}\sum _{u\in U}t{q}_{piu}^t-{\pi }_p{\gamma }_{pu}{q}_{pi}^t+h_{pi}^t\Gamma ⩽0\forall t\in T\forall p\in P\forall i\in I,\hfill \\ h_{pi}^t⩾0.15q_{piu}^t\forall t\in T\forall p\in P\forall i\in I\forall u\in U,\hfill \\ h_{pi}^t⩾{\widehat{\gamma }}_{pu}{\pi }_p{q}_{pi}^t\forall t\in T\forall p\in P\forall i\in I\forall u\in U,\hfill \end{array}\right.$$

(96)

$$\left\{\begin{array}{c}\sum _{e\in E}t{q}_{pie}^t-{\pi }_p{\gamma }_{pe}{q}_{pi}^t+h_{pi}^t\Gamma ⩽0\forall t\in T\forall p\in P\forall i\in I,\hfill \\ h_{pi}^t⩾0.15q_{pie}^t\forall t\in T\forall p\in P\forall i\in I\forall e\in E,\hfill \\ h_{pi}^t⩾{\widehat{\gamma }}_{pe}{\pi }_p{q}_{pi}^t\forall t\in T\forall p\in P\forall i\in I\forall e\in E.\hfill \end{array}\right.$$

(97)

Robust counterparts for the three uncertainty sets ‘interval + ellipsoidal’, ‘interval + polyhedral’, and ‘interval + ellipsoidal + polyhedral’ are shown in Appendix A.

5. Solution Method

5.1. Augmented $ϵ$-Constraint Method

Multi-objective programming involves multiple objective functions, and it is usually not possible to find a single optimal solution that optimizes all of them simultaneously. The decision-maker, when confronted with such a situation, attempts to choose the most optimal solution from the available options. In multi-objective programming, the concept of optimality is replaced by Pareto optimality, which also goes by the names ‘efficient solution’ and ‘non-dominated solution.’ It pertains to a scenario where improving one objective function leads to a deterioration in at least one other objective function. As per Beckmann et al. [31], the a posteriori approach is a more effective way to solve multi-objective programming. In this method, valid solutions to the problem are generated first, and then the decision-maker selects the one that aligns with their most preferred choice. This paper’s bi-objective programming model utilizes the $ϵ$-constraint method (also known as the a posteriori method) to solve the problem, but the said traditional method has some drawbacks, including (1) difficulty in computing the objective function over a range of valid sets, (2) lack of assurance regarding the solution’s efficiency, and (3) significant time increase for problems with multiple objective functions. As a solution to the issues with the traditional $ϵ$-constraint method, this paper proposes utilizing the incremental $ϵ$-constraint method, which was suggested by Mavrotas [32]. This method is shown in the model (98).

$$\begin{array}{cc}\hfill Max& \left(f_1\left(\mathbf{x}\right)+eps\times \left(s_2/r_2+s_3/r_3+\cdots +s_p/r_p\right)\right)\hfill \\ \hfill s.t.& f_2\left(\mathbf{x}\right)-s_2=e_2,\hfill \\ \hfill & f_3\left(\mathbf{x}\right)-s_3=e_3,\hfill \\ \hfill & \dots \hfill \\ \hfill & f_{\mathrm{k}}\left(\mathbf{x}\right)-s_k=e_k,\hfill \\ \hfill & \mathbf{x}\in \mathrm{S},s_i\in {\mathrm{R}}^{+},e_k={\mathrm{lb}}_{\mathrm{k}}+\left({\mathrm{i}}_{\mathrm{k}}\times {\mathrm{r}}_{\mathrm{k}}\right)/{\mathrm{g}}_{\mathrm{k}}.\hfill \end{array}$$

(98)

In this section, we consider $eps$ to be a sufficiently small number taking a value of ${10}^{-3}$, where $eps$ is usually selected from the range of values $[{10}^{-3}$, ${10}^{-6}]$. Here, $l{b}_k$ represents the lower bound on objective function k, $r_k$ represents the interval of the objective function k, S represents the feasible domain of the original problem, and $s_k$ is a non-negative relaxation (residual) variable for $k=2$ to p. The augmented $ϵ$-constraint method is applied as follows: firstly, obtain $p-1$ domains from the lexicographic optimization payoff matrix to be utilized as constraints for objective functions. Then, divide the domain of the ith objective function into $q_i$ domains using $q_i-1$ equidistant grid points. Hence, a total of $q_i+1$ grid points are used to change the right term $e_i$ of the ith objective function parametrically. The total count of multi-objective programming solution computations is then $(q_2+1)\times (q_3+1)\times ...$ The expression is given by multiplying $q_p+1$. One of the benefits of using the augmented $ϵ$-constraint method is that we can regulate the density of the effective set representation by assigning suitable values to $q_i$. Increasing the number of grid points leads to a more dense representation of the active set, but also increases the computation time. Therefore, we can achieve a balanced approach between the density of the active set and computation time.

5.2. Benders Decomposition

This paper utilizes the Benders decomposition algorithm to solve the proposed model. The algorithm splits the problem into a master problem with 0–1 integer variables and a subproblem with continuous variables. However, the complexity of the master problem can lead to inefficient solutions and affect the algorithm’s convergence efficiency. To address this, we introduce a three-stage method to enhance the traditional Benders decomposition algorithm. This method accelerates the solution of the master problem, thereby improving the overall convergence efficiency of the algorithm. The original problem, represented by fixed and flexible facility capacity MILP models, is divided into a master problem and a sub-problem. For illustrative purposes, we use the flexible facility capacity model as an example.

First is the sub-problem:

$$\begin{array}{cc}\hfill Min{Z}_{PSP}& =\sum _{t\in T}\left[\left(\sum _{p\in P}\sum _{m\in M}p{c}_p^t{q}_m^t+\sum _{i\in I}r{c}_{pi}^t{q}_{pi}^t+\sum _{r\in R}\left(r{c}_{pr}^t-GS\right)q_{pr}^t\right.\right.\hfill \\ \hfill & \left.\left.+\sum _{p\in P}\sum _{u\in U}r{c}_{pu}^t-GS\right)q_{pu}^t+\sum _{e\in E}\left(r{c}_{pe}^t-GS\right)q_{pe}^t+\sum _{f\in F}l{c}_p^t{q}_{pf}^t\right)\hfill \\ \hfill & +\sum _{p\in P}\left(\sum _{m\in M}\sum _{d\in D}t{c}_p^t{S}_{md}t{q}_{pmd}^t+\sum _{d\in D}\sum _{c\in C}t{c}_p^t{S}_{dc}t{q}_{pdc}^t+\sum _{c\in C}\sum _{o\in O}t{c}_p^t{S}_{co}t{q}_{pco}^t\right.\hfill \\ \hfill & +\sum _{o\in O}\sum _{i\in I}t{c}_p^t{S}_{oi}t{q}_{poi}^t+\sum _{i\in I}\sum _{r\in R}t{c}_p^t{S}_{ir}t{q}_{pir}^t+\sum _{i\in I}\sum _{u\in U}t{c}_p^t{S}_{iu}t{q}_{piu}^t\hfill \\ \hfill & +\sum _{i\in I}\sum _{e\in E}t{c}_p^t{S}_{ie}t{q}_{pie}^t+\sum _{i\in I}\sum _{f\in F}t{c}_p^t{S}_{if}t{q}_{pif}^t+\sum _{r\in R}\sum _{m\in M}t{c}_p^t{S}_{rm}t{q}_{prm}^t\hfill \\ \hfill & \left.\left.+\sum _{u\in U}\sum _{m\in M}t{c}_p^t{S}_{um}t{q}_{pum}^t+\sum _{m\in M}m{c}_{p}m{q}_{pm}^t\right)\right]-eps(s_2/r_2).\hfill \end{array}$$

$s.t.$ fixed facility capacity model constraints Equations (1)–(21), constraints Equations (36)–(45), flexible facility capacity model constraints Equations (47)–(56).

$$f_2\left(x\right)+s_2=e_2.$$

(99)

Define V as a dual variable and the dual sub-problem is expressed as follows:

$$\begin{array}{cc}\hfill Max{Z}_{DSP}& =e_2{V}^0+\sum _{t\in T}\sum _{p\in P}\left[\sum _{l\in L}\left(\sum _{m\in M}C{Q}_{plm}{\overline{Y}}_{lm}^t{V}_{tpm}^1+\sum _{i\in I}C{Q}_{pli}{\overline{Y}}_{li}^t{V}_{tpi}^2\right.\right.\hfill \\ \hfill & +\sum _{r\in R}C{Q}_{plr}{\overline{Y}}_{lr}^t{V}_{tpr}^3+\sum _{u\in U}C{Q}_{plu}{\overline{Y}}_{lu}^t{V}_{tpu}^6+\sum _{e\in E}C{Q}_{ple}{\overline{Y}}_{le}^t{V}_{tpe}^7\hfill \\ \hfill & +\sum _{m\in M}U{R}_{pm}C{Q}_{plm}{\overline{Y}}_{lm}^t{V}_{tpm}^8+\sum _{i\in I}U{R}_{pi}C{Q}_{pli}{\overline{Y}}_{li}^t{V}_{tpi}^9+\sum _{r\in R}U{R}_{pr}C{Q}_{plr}{\overline{Y}}_{lr}^t{V}_{tpr}^{10}\hfill \\ \hfill & \left.+\sum _{u\in U}U{R}_{pu}C{Q}_{plu}{\overline{Y}}_{lu}^t{V}_{tpu}^{11}+\sum _{e\in E}U{R}_{pe}C{Q}_{ple}{\overline{Y}}_{le}^t{V}_{tpe}^{12}\right)+\sum _{d\in D}C{Q}_{pd}{\overline{Y}}_d^t{V}_{tpd}^4\hfill \\ \hfill & \left.+\sum _{o\in O}C{Q}_{po}{\overline{Y}}_o^t{V}_{tpo}^5+\sum _{c\in C}{\tilde{D}}_{pc}^t{V}_{tpc}^{16}\right].\hfill \end{array}$$

The master problem is presented below:

$$\begin{array}{cc}\hfill Min{Z}_{MP}& =\eta +\sum _{t\in T}\sum _{l\in L}\left(\sum _{m\in M}v{c}_{lm}^t{Y}_{lm}^t+\sum _{i\in I}v{c}_{li}^t{Y}_{li}^t+\sum _{r\in R}v{c}_{lr}^t{Y}_{lr}^t+\sum _{u\in U}v{c}_{lu}^t{Y}_{lu}^t\right.\hfill \\ \hfill & \left.+\sum _{e\in E}v{c}_{le}^t{Y}_{le}^t\right)+\sum _{t\in T}\sum _{d\in D}v{c}_d^t{Y}_d^t+\sum _{t\in T}\sum _{o\in O}v{c}_o^t{Y}_o^t+\sum _{t\in T}\left(\sum _{m\in M}C{C}_{m}C{L}_m^t\right.\hfill \\ \hfill & \left.+\sum _{i\in I}C{C}_{i}C{L}_i^t+\sum _{r\in R}C{C}_{r}C{L}_r^t+\sum _{u\in U}C{C}_{u}C{L}_u^t+\sum _{e\in E}C{C}_{e}C{L}_e^t\right)+\sum _{m\in M}f{c}_m{X}_m\hfill \\ \hfill & +\sum _{d\in D}f{c}_d{X}_d+\sum _{o\in O}f{c}_o{X}_o+\sum _{i\in I}f{c}_i{X}_i+\sum _{r\in R}f{c}_r{X}_r+\sum _{u\in U}f{c}_u{X}_u+\sum _{e\in E}f{c}_e{X}_e.\hfill \end{array}$$

$s.t.$ Equations (22)–(35), Equations (57)–(66),

$$\begin{array}{cc}\hfill & e_2{\widehat{V}}^{0\left(k\right)}+\sum _{t\in T}\sum _{p\in P}\left[\sum _{l\in L}\left(\sum _{m\in M}C{Q}_{plm}{Y}_{lm}^t{\widehat{V}}_{tpm}^{1\left(k\right)}+\sum _{i\in I}C{Q}_{pli}{Y}_{li}^t{\widehat{V}}_{tpi}^{2\left(k\right)}\right.\right.\hfill \\ \hfill & +\sum _{r\in R}C{Q}_{plr}{Y}_{lr}^t{\widehat{V}}_{tpr}^{3\left(k\right)}+\sum _{u\in U}C{Q}_{plu}{Y}_{lu}^t{\widehat{V}}_{tpu}^{6\left(k\right)}+\sum _{e\in E}C{Q}_{ple}{Y}_{le}^t{\widehat{V}}_{tpe}^{7\left(k\right)}\hfill \\ \hfill & +\sum _{m\in M}U{R}_{pm}C{Q}_{plm}{Y}_{lm}^t{\widehat{V}}_{tpm}^{8\left(k\right)}+\sum _{i\in I}U{R}_{pi}C{Q}_{pli}{Y}_{li}^t{\widehat{V}}_{tpi}^{9\left(k\right)}+\sum _{r\in R}U{R}_{pr}C{Q}_{plr}{Y}_{lr}^t{\widehat{V}}_{tpr}^{10\left(k\right)}\hfill \\ \hfill & \left.+\sum _{u\in U}U{R}_{pu}C{Q}_{plu}{Y}_{lu}^t{\widehat{V}}_{tpu}^{11\left(k\right)}+\sum _{e\in E}U{R}_{pe}C{Q}_{ple}{Y}_{le}^t{\widehat{V}}_{tpe}^{12\left(k\right)}\right)+\sum _{d\in D}C{Q}_{pd}{Y}_d^t{\widehat{V}}_{tpd}^{4\left(k\right)}\hfill \\ \hfill & \left.+\sum _{o\in O}C{Q}_{po}{Y}_o^t{\widehat{V}}_{tpo}^{5\left(k\right)}+\sum _{c\in C}{\tilde{D}}_{pc}^t{\widehat{V}}_{tpc}^{16\left(k\right)}\right]⩽\eta ,\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill & e_2{\widehat{V}}^{0\left(k\right)}+\sum _{t\in T}\sum _{p\in P}\left[\sum _{l\in L}\left(\sum _{m\in M}C{Q}_{plm}{Y}_{lm}^t{\widehat{V}}_{tpm}^{1\left(k\right)}+\sum _{i\in I}C{Q}_{pli}{Y}_{li}^t{\widehat{V}}_{tpi}^{2\left(k\right)}\right.\right.\hfill \\ \hfill & +\sum _{r\in R}C{Q}_{plr}{Y}_{lr}^t{\widehat{V}}_{tpr}^{3\left(k\right)}+\sum _{u\in U}C{Q}_{plu}{Y}_{lu}^t{\widehat{V}}_{tpu}^{6\left(k\right)}+\sum _{e\in E}C{Q}_{ple}{Y}_{le}^t{\widehat{V}}_{tpe}^{7\left(k\right)}\hfill \\ \hfill & +\sum _{m\in M}U{R}_{pm}C{Q}_{plm}{Y}_{lm}^t{\widehat{V}}_{tpm}^{8\left(k\right)}+\sum _{i\in I}U{R}_{pi}C{Q}_{pli}{Y}_{li}^t{\widehat{V}}_{tpi}^{9\left(k\right)}+\sum _{r\in R}U{R}_{pr}C{Q}_{plr}{Y}_{lr}^t{\widehat{V}}_{tpr}^{10\left(k\right)}\hfill \\ \hfill & \left.+\sum _{u\in U}U{R}_{pu}C{Q}_{plu}{Y}_{lu}^t{\widehat{V}}_{tpu}^{11\left(k\right)}+\sum _{e\in E}U{R}_{pe}C{Q}_{ple}{Y}_{le}^t{\widehat{V}}_{tpe}^{12\left(k\right)}\right)+\sum _{d\in D}C{Q}_{pd}{Y}_d^t{\widehat{V}}_{tpd}^{4\left(k\right)}\hfill \\ \hfill & \left.+\sum _{o\in O}C{Q}_{po}{Y}_o^t{\widehat{V}}_{tpo}^{5\left(k\right)}+\sum _{c\in C}{\tilde{D}}_{pc}^t{\widehat{V}}_{tpc}^{16\left(k\right)}\right]⩽0.\hfill \end{array}$$

(101)

The constraints for the master problem comprise two types of facility capacity model constraints: fixed capacity model constraints Equation (22) through Equation (35) and flexible capacity model constraints Equations (57) through Equation (66). The optimal cuts Equation (100) and feasible cuts Equation (101) are added to the master problem after solving the dual sub-problem iterations.

When using BDA to solve large-scale MILP problems, the inefficiency of solving the master solution may affect the convergence of the bounds due to the computational complexity; therefore, this paper introduces a three-stage approach to solving the master problem in each iteration.

Stage 1: This stage involves the decision between the manufacturer and the customer point, which includes the selection of the location of the manufacturer’s and distributor’s facilities, the operation of the facilities, and the choice of the manufacturer’s flexible capacity level change. The mathematical model for this stage is as follows:

$$\begin{array}{cc}\hfill Min{Z}_{MP}^1& =\sum _{m\in M}f{c}_m{X}_m+\sum _{d\in D}f{c}_d{X}_d+\sum _{t\in T}\sum _{l\in L}\sum _{m\in M}V{C}_{lm}^t{Y}_{lm}^t\hfill \\ \hfill & +\sum _{t\in T}\sum _{d\in D}V{C}_d^t{Y}_d^t+\sum _{t\in T}\sum _{m\in M}C{C}_{m}C{L}_m^t+{\eta }_1.\hfill \end{array}$$

$s.t.$ Equations (22)–(25), Equations (57) and (58),

$$\begin{array}{cc}\hfill {\eta }_1& ⩾\sum _{t\in T}\sum _{p\in P}\sum _{l\in L}\sum _{m\in M}\left(C{Q}_{plm}{Y}_{lm}^t{\widehat{V}}_{tpm}^{1\left(k\right)}+U{R}_{pm}C{Q}_{plm}{Y}_{lm}^t{\widehat{V}}_{tpm}^{8\left(k\right)}\right)\hfill \\ \hfill & +\sum _{t\in T}\sum _{p\in P}\sum _{d\in D}C{Q}_{pd}{Y}_d^t{\widehat{V}}_{tpd}^{4\left(k\right)}.\hfill \end{array}$$

(102)

Stage 2: In this stage, decisions need to be made between the customer point and the inspection based on the decisions made in Stage 1. Stage 2 includes decisions on the location of the recycling and inspections, operations, and changes in the level of flexible capacity at the inspection. The mathematical model for this phase is as follows:

$$\begin{array}{cc}\hfill Min{Z}_{MP}^2& =\sum _{i\in I}f{c}_i{X}_i+\sum _{o\in O}f{c}_o{X}_o+\sum _{t\in T}\sum _{l\in L}\sum _{i\in I}V{C}_{li}^t{Y}_{li}^t\hfill \\ \hfill & +\sum _{t\in T}\sum _{o\in O}V{C}_o^t{Y}_o^t+\sum _{t\in T}\sum _{i\in I}C{C}_{i}C{L}_i^t+{\eta }_2.\hfill \end{array}$$

$s.t.$ Equations (26)–(29), Equations (59) and (60),

$$\begin{array}{cc}\hfill {\eta }_2⩾& e_2{\widehat{V}}^0+\sum _{t\in T}\sum _{p\in P}\sum _{l\in L}\sum _{i\in I}\left(C{Q}_{pli}{Y}_{li}^t{\widehat{V}}_{tpi}^{2\left(k\right)}+U{R}_{pi}C{Q}_{pli}{Y}_{li}^t{\widehat{V}}_{tpi}^{9\left(k\right)}\right)\hfill \\ \hfill & +\sum _{t\in T}\sum _{p\in P}\sum _{o\in O}C{Q}_{po}{Y}_o^t{\widehat{V}}_{tpo}^{5\left(k\right)}+\sum _{t\in T}\sum _{p\in P}\sum _{c\in C}{\tilde{D}}_{pc}^t{\widehat{V}}_{tpc}^{16\left(k\right)}.\hfill \end{array}$$

(103)

Stage 3: Based on the decisions made in Stages 1 and 2 to make decisions in Stage 3 and complete the CLSC operation. The third stage includes the location and operation of remanufacturing centers, recycling centers, and energy centers, as well as changes in the level of flexible capacity. The mathematical model for this stage is as follows:

$$\begin{array}{cc}\hfill Min{Z}_{MP}^3& =\sum _{r\in R}f{c}_r{X}_r+\sum _{u\in U}f{c}_u{X}_u+\sum _{e\in E}f{c}_e{X}_e+\sum _{t\in T}\sum _{l\in L}\left(\sum _{r\in R}V{C}_{lr}^t{Y}_{lr}^t\right.\hfill \\ \hfill & \left.+\sum _{u\in U}V{C}_{lu}^t{Y}_{lu}^t+\sum _{e\in E}V{C}_{le}^t{Y}_{le}^t\right)+\sum _{t\in T}\left(\sum _{r\in R}C{C}_{r}C{L}_r^t+\sum _{u\in U}C{C}_{u}C{L}_u^t\right.\hfill \\ \hfill & \left.+\sum _{e\in E}C{C}_{e}C{L}_e^t\right)+{\eta }_3.\hfill \end{array}$$

$s.t.$ Equations (30)–(35), Equations (61)–(66),

$$\begin{array}{cc}\hfill {\eta }_3⩾& \sum _{t\in T}\sum _{p\in P}\sum _{l\in L}\left(\sum _{r\in R}C{Q}_{plr}{Y}_{lr}^t{\widehat{V}}_{tpr}^{3\left(k\right)}+\sum _{u\in U}C{Q}_{plu}{Y}_{lu}^t{\widehat{V}}_{tpu}^{6\left(k\right)}+\sum _{e\in E}C{Q}_{ple}{Y}_{le}^t{\widehat{V}}_{tpe}^{7\left(k\right)}\right.\hfill \\ \hfill & \left.+\sum _{r\in R}U{R}_{pr}C{Q}_{plr}{Y}_{lr}^t{\widehat{V}}_{tpr}^{10\left(k\right)}+\sum _{u\in U}U{R}_{pu}C{Q}_{plu}{Y}_{lu}^t{\widehat{V}}_{tpu}^{11\left(k\right)}+\sum _{e\in E}U{R}_{pe}C{Q}_{ple}{Y}_{le}^t{\widehat{V}}_{tpe}^{12\left(k\right)}\right).\hfill \end{array}$$

(104)

In this section, we apply the augmented generalization $ϵ$-constraint method introduced above to deal with the multi-objective model and combine it with the designed enhanced Benders decomposition algorithm to improve the solving efficiency and design a comprehensive solution algorithm to solve the multi-objective robust model, the pseudo-code is shown in Algorithm 1.

Algorithm 1 Model Solving Algorithm

Creating a payoff matrix by solving a model using the enhanced Benders Decomposition algorithm ($f_k\left(x\right)$, for k = 1...p) For k = 2...p set the lower bound $l{b}_k$ and set the number of grid points $g_k$ (k = 2...p) Initialization parameters: $i_2$...$i_{p-1}$, $i_p$ = 0, $neff=0$ while $i_2<g_2$ do    $i_2=i_2+1$    while $i_{p-1}<g_{p-1}$ do      $i_{p-1}=i_{p-1}+1$      while $i_p<g_p$ do         $i_p=i_p+1$         Solving models using enhanced Benders decomposition algorithm         if Calculation produces feasible solutions then           neff = neff + 1           record neff         else           $i_p=g_p$         end if      end while      $i_p=0$      $i_{p-1}=0$    end while end while

6. Numerical Analysis

This paper analyzes the Solvang [23] dataset and related research in a numerical case study, and extends its data logic to a multi-period CLSC model with flexible facility capacity. The following computational analysis incorporates this model. The model examines two products operating over three periods, including five candidate manufacturers, six candidate distributors, four existing consumer sites, six candidate collection centers, five candidate inspection centers, four remanufacturing centers, four recycling centers, four energy centers, and one landfill site. The models with fixed facility capacities and flexible facility capacities display discrepant capabilities for manufacturers, inspection centers, remanufacturing centers, recycling centers, and energy centers. The flexible facility capacity model considers three facility capacity levels ($l_1$, $l_2$, and $l_3$): low, medium, and high. The fixed facility capacity model assumes that all facilities are set to a high capacity level to meet the maximum demand in the market. Building and operating facilities at high capacity levels incur higher fixed and variable costs than at low and medium capacity levels. Changing a facility’s capacity level incurs less fixed cost than building a new facility with a different capacity level. This enforces making capacity level changes within a feasible time horizon.

In the numerical case, parameters are randomly generated following a uniform distribution. The reason for choosing the uniform distribution is that it ensures an equal probability of the defined range of values. Additionally, uniform distribution is commonly used in the numerical analysis of supply chains. As an example, Solvang and Yu [23], Rahimi and Ghezavati [21], and Zarbakhshnia et al. [33] have employed uniform demand distributions to define the relevant parameters in their network design problems. Below, the various parameter tables are presented.

Table 2. Parameters related to facility capacity levels and capacity ranges in CLSC.

Parameter (Unit Product)	Product 1	Product 2
$C{Q}_{pm}$(fixed)	U(50,000, 60,000)	U(50,000, 60,000)
$C{Q}_{plm}$(flexibility: low)	U(25,000, 30,000)	U(25,000, 30,000)
$C{Q}_{plm}$(flexibility: medium)	U(40,000, 45,000)	U(40,000, 45,000)
$C{Q}_{plm}$(flexibility: high)	U(50,000, 60,000)	U(50,000, 60,000)
$C{Q}_{pi}$(fixed)	U(35,000, 40,000)	U(35,000, 40,000)
$C{Q}_{pli}$(flexibility: low)	U(20,000, 24,000)	U(20,000, 24,000)
$C{Q}_{pli}$(flexibility: medium)	U(26,000, 30,000)	U(26,000, 30,000)
$C{Q}_{pli}$(flexibility: high)	U(35,000, 40,000)	U(35,000, 40,000)
$C{Q}_{pr}$(fixed)	U(25,000, 30,000)	U(25,000, 30,000)
$C{Q}_{plr}$(flexibility: low)	U(12,000, 15,000)	U(12,000, 15,000)
$C{Q}_{plr}$(flexibility: medium)	U(18,000, 21,000)	U(18,000, 21,000)
$C{Q}_{plr}$(flexibility: high)	U(25,000, 30,000)	U(25,000, 30,000)
$C{Q}_{pu}$(fixed)	U(25,000, 30,000)	U(25,000, 30,000)
$C{Q}_{plu}$(flexibility: low)	U(12,000, 15,000)	U(12,000, 15,000)
$C{Q}_{plu}$(flexibility: medium)	U(18,000, 21,000)	U(18,000, 21,000)
$C{Q}_{plu}$(flexibility: high)	U(25,000, 30,000)	U(25,000, 30,000)
$C{Q}_{pe}$(fixed)	U(25,000, 30,000)	U(25,000, 30,000)
$C{Q}_{ple}$(flexibility: low)	U(12,000, 15,000)	U(12,000, 15,000)
$C{Q}_{ple}$(flexibility: medium)	U(18,000, 21,000)	U(18,000, 21,000)
$C{Q}_{ple}$(flexibility: high)	U(25,000, 30,000)	U(25,000, 30,000)
$C{Q}_{pd}$(fixed)	U(45,000, 50,000)	U(45,000, 50,000)
$C{Q}_{po}$(fixed)	U(50,000, 55,000)	U(50,000, 55,000)

displays the facility capacity levels and their respective ranges.

Table 3. Parameters related to carbon emissions in CLSC.

Parameter	t = 1		t = 2		t = 3
(g)	Product 1	Product 2	Product 1	Product 2	Product 1	Product 2
$co{r}_{pm}^t$	150	150	135	135	120	120
$co{r}_{pi}^t$	20	20	17	17	15	15
$co{r}_{pr}^t$	35	35	30	30	27	27
$co{r}_{pu}^t$	30	30	28	28	25	25
$co{r}_{pe}^t$	70	70	65	65	60	60
$co{r}_{pf}^t$	45	45	45	45	45	45
$co{t}_p^t$	0.12	0.12	0.09	0.09	0.06	0.06

illustrates the period-related parameters associated with the supply chain operations. Furthermore,

Table 4. Period-related parameters in CLSC(1).

Parameter	t = 1		t = 2		t = 3
	Product 1	Product 2	Product 1	Product 2	Product 1	Product 2
$v{c}_m^t$	U(160, 170)	U(160, 170)	U(152, 162)	U(152, 162)	U(144, 154)	U(144, 154)
$v{c}_{lm}^t$	U(77, 84)	U(77, 84)	U(73, 80)	U(73, 80)	U(69, 76)	U(69, 76)
$v{c}_{lm}^t$	U(98, 105)	U(98, 105)	U(93, 100)	U(93, 100)	U(88, 95)	U(88, 95)
$v{c}_{lm}^t$	U(126, 133)	U(126, 133)	U(120, 126)	U(120, 126)	U(114, 119)	U(114, 119)
$v{c}_d^t$	U(60, 65)	U(60, 65)	U(57, 62)	U(57, 62)	U(54, 59)	U(54, 59)
$v{c}_o^t$	U(55, 60)	U(55, 60)	U(52, 57)	U(52, 57)	U(49, 54)	U(49, 54)
$v{c}_i^t$	U(90, 95)	U(90, 95)	U(86, 89)	U(86, 89)	U(82, 85)	U(82, 85)
$v{c}_{li}^t$	U(49, 53)	U(49, 53)	U(47, 50)	U(47, 50)	U(45, 48)	U(45, 48)
$v{c}_{li}^t$	U(63, 67)	U(63, 67)	U(60, 64)	U(60, 64)	U(57, 61)	U(57, 61)
$v{c}_{li}^t$	U(70, 74)	U(70, 74)	U(66, 70)	U(66, 70)	U(63, 67)	U(63, 67)
$v{c}_r^t$	U(90, 95)	U(90, 95)	U(86, 89)	U(86, 89)	U(82, 85)	U(82, 85)
$v{c}_{lr}^t$	U(49, 53)	U(49, 53)	U(47, 50)	U(47, 50)	U(45, 48)	U(45, 48)
$v{c}_{lr}^t$	U(63, 67)	U(63, 67)	U(60, 64)	U(60, 64)	U(57, 61)	U(57, 61)
$v{c}_{lr}^t$	U(70, 74)	U(70, 74)	U(66, 70)	U(66, 70)	U(63, 67)	U(63, 67)
$v{c}_u^t$	U(90, 95)	U(90, 95)	U(86, 89)	U(86, 89)	U(82, 85)	U(82, 85)
$v{c}_{lu}^t$	U(49, 53)	U(49, 53)	U(47, 50)	U(47, 50)	U(45, 48)	U(45, 48)
$v{c}_{lu}^t$	U(63, 67)	U(63, 67)	U(60, 64)	U(60, 64)	U(57, 61)	U(57, 61)
$v{c}_{lu}^t$	U(70, 74)	U(70, 74)	U(66, 70)	U(66, 70)	U(63, 67)	U(63, 67)
$v{c}_e^t$	U(60, 65)	U(60, 65)	U(57, 62)	U(57, 62)	U(54, 59)	U(54, 59)

and

Table 5. Period-related parameters in CLSC(2).

Parameter	t = 1		t = 2		t = 3
	Product 1	Product 2	Product 1	Product 2	Product 1	Product 2
$v{c}_{le}^t$	U(35, 38)	U(35, 38)	U(33, 36)	U(33, 36)	U(31, 34)	U(31, 34)
$v{c}_{le}^t$	U(42, 46)	U(42, 46)	U(40, 44)	U(40, 44)	U(38, 42)	U(38, 42)
$v{c}_{le}^t$	U(49, 53)	U(49, 53)	U(47, 50)	U(47, 50)	U(45, 48)	U(45, 48)
$p{c}_p^t$	U(50, 70)	U(50, 70)	U(48, 67)	U(48, 67)	U(46, 64)	U(46, 64)
$r{c}_{pi}^t$	U(20, 25)	U(20, 25)	U(19, 24)	U(19, 24)	U(18, 23)	U(18, 23)
$r{c}_{pr}^t$	U(35, 40)	U(35, 40)	U(33, 38)	U(33, 38)	U(31, 36)	U(31, 36)
$r{c}_{pu}^t$	U(40, 45)	U(40, 45)	U(38, 43)	U(38, 43)	U(36, 41)	U(36, 41)
$r{c}_{pe}^t$	U(20, 30)	U(20, 30)	U(19, 28)	U(19, 28)	U(18, 26)	U(18, 26)
$l{c}_p^t$	U(4, 7)	U(4, 7)	U(3.8, 6.6)	U(3.8, 6.6)	U(3.6, 6.3)	U(3.6, 6.3)
$t{c}_p^t$	U(0.8, 1.2)	U(0.8, 1.2)	U(0.72, 1.08)	U(0.72, 1.08)	U(0.65, 1.02)	U(0.65, 1.02)
$G{S}_{pr}^t$	U(100, 200)	U(100, 200)	U(90, 180)	U(90, 180)	U(76, 150)	U(76, 150)
$G{S}_{pu}^t$	U(150, 300)	U(150, 300)	U(135, 270)	U(135, 270)	U(115, 230)	U(115, 230)
$G{S}_{pe}^t$	U(150, 300)	U(150, 300)	U(135, 270)	U(135, 270)	U(115, 230)	U(115, 230)
${\beta }_p^t$	0.6	0.5	0.7	0.65	0.8	0.75
${\theta }_p^t$	0.3	0.25	0.4	0.35	0.5	0.45
$D_{pc}^t$(unit of product)	U(40,000, 60,000)	U(25,000, 30,000)	U(50,000, 80,000)	U(35,000, 40,000)	U(60,000, 90,000)	U(45,000, 50,000)
${\alpha }_p^t$	0.5	0.45	0.58	0.51	0.62	0.56
$v{c}_{lu}^t$	U(70, 74)	U(70, 74)	U(66, 70)	U(66, 70)	U(63, 67)	U(63, 67)
$v{c}_e^t$	U(60, 65)	U(60, 65)	U(57, 62)	U(57, 62)	U(54, 59)	U(54, 59)

illustrate the period-independent parameters, and the carbon emissions resulting from each link of the supply chain are shown in Table 5.

Table 6. Other pertinent parameters related to CLSC.

Parameter	Product 1	Product 2	Parameter	Product 1	Product 2
$f{c}_m$	U(1600, 1700)	U(1600, 1700)	$U{R}_{pm}$	U(0.8, 0.9)	U(0.8, 0.9)
$f{c}_{lm}$	U(1100, 1200)	U(1100, 1200)	$U{R}_{pi}$	U(0.75, 0.85)	U(0.75, 0.85)
$f{c}_{lm}$	U(1400, 1500)	U(1400, 1500)	$U{R}_{pr}$	U(0.65, 0.75)	U(0.65, 0.75)
$f{c}_{lm}$	U(1800, 1900)	U(1800, 1900)	$U{R}_{pu}$	U(0.65, 0.75)	U(0.65, 0.75)
$f{c}_d$	U(600, 650)	U(600, 650)	$U{R}_{pe}$	U(0.65, 0.75)	U(0.65, 0.75)
$f{c}_o$	U(550, 600)	U(550, 600)	${\gamma }_{pr}$	0.3	0.3
$f{c}_i$	U(900, 950)	U(900, 950)	${\gamma }_{pu}$	0.4	0.4
$f{c}_{li}$	U(700, 750)	U(700, 750)	${\gamma }_{pe}$	0.3	0.3
$f{c}_{li}$	U(900, 950)	U(900, 950)	$R{P}_p$	0.8	0.8
$f{c}_{li}$	U(1000, 1050)	U(1000, 1050)	$NR{P}_p$	0.2	0.2
$f{c}_r$	U(900, 950)	U(900, 950)	$MR{P}_p$	0.75	0.75
$f{c}_{lr}$	U(700, 750)	U(700, 750)	${\pi }_p^L$	U(0.5, 0.65)	U(0.5, 0.65)
$f{c}_{lr}$	U(900, 950)	U(900, 950)	${\pi }_p^M$	U(0.65, 0.75)	U(0.65, 0.75)
$f{c}_{lr}$	U(1000, 1050)	U(1000, 1050)	${\pi }_p^H$	U(0.75, 1)	U(0.75, 1)
$f{c}_u$	U(900, 950)	U(900, 950)	$c{c}_m$	30	30
$f{c}_{lu}$	U(700, 750)	U(700, 750)	$c{c}_d$	30	30
$f{c}_{lu}$	U(900, 950)	U(900, 950)	$c{c}_o$	30	30
$f{c}_{lu}$	U(1000, 1050)	U(1000, 1050)	$c{c}_i$	30	30
$f{c}_e$	U(600, 650)	U(600, 650)	$c{c}_r$	30	30
$f{c}_{le}$	U(500, 550)	U(500, 550)	$c{c}_u$	30	30
$f{c}_{le}$	U(600, 650)	U(600, 650)	$c{c}_e$	30	30
$f{c}_{le}$	U(700, 750)	U(700, 750)

shows the carbon emissions generated by each link within the supply chain.

The multi-objective, multi-period, and robust models of CLSC networks with fixed and flexible facility capacities presented in this paper were solved using the GUROBI solver and integrated algorithms developed in this study on a computer equipped with an Intel(R) Core(TM) 13980HX CPU @5.20GHz and compiled via Matlab R2022b.

6.1. Uncertainty Parameter Analysis

This paper investigates the optimization of a CLSC network design under uncertain product market demand, recycling rate, and recycled product quality. The study assumes these uncertainties have bounded uncertainty. They have maximum deviations from their nominal values: demand coefficient up to 10%, recovery rate coefficient up to 8%, and reprocessed quality coefficient up to 15%. The impact of these uncertain parameters on the model is examined using a fixed facility capacity model with the objective of minimizing the economic cost incurred by the CLSC. Robust counterpart equations are applied to each uncertain parameter for each of the six uncertainty sets to solve the robust model for the corresponding uncertain environment. Robust models for ‘box set’, ‘polyhedral set’, and ‘interval + polyhedral set’ uncertainty environments are solved using an enhanced Benders decomposition algorithm. However, robust models for ‘ellipsoidal set’, ‘interval + ellipsoidal set’, and ‘interval + ellipsoidal set + polyhedral set’ environments use second-order cone programming with dual gaps, making them unsuitable for the Benders decomposition algorithm. The GUROBI solver is used for this task.

Table 7. The impact of uncertain parameters on the objective function’s value.

Example Size	Definied	Uncertainty
	Parameter	Parameter
		Demand	Recycling	Product	Demand, Recycling Rate
			Rate	Quality	and Product Quality
Example 1	22,899.05	24,273.47	26,677.393	26,379.71	27,089.58
Example 2	35,148.11	37,257.39	41,985.79	39,735.98	43,275.81
Example 3	57,381.13	60,824.75	69,682.9	66,226.55	71,117.38

presents the worst-case scenario outcomes for the box-set-induced robust model where $\Psi =1$.

The economic cost of the CLSC is significantly impacted by the recycling rate of recycled products and product quality, according to Table 7. This is evidenced by their objective function values, which are much higher compared to those affected only by demand uncertainty or free from uncertainty.

Figure 2, Figure 3, Figure 4 and Figure 5 display the objective and function values of the robust model under different sources of uncertainty: recycling rate, product quality, market demand, and all three factors simultaneously. When $\Psi$, $\Omega$, and $\Gamma$ are all equal to zero, the model represents the solution for the deterministic case, not subject to any uncertain factors. Each constraint includes more than one uncertain parameter ($J_i>1$) on the left-hand side of the equation, affecting both recovery and product quality. To derive adjustable parameter values for various types of uncertainties, equations based on $\Psi$ values are provided

$$\left\{\begin{array}{c}\Omega =\Psi \times \sqrt{J_i}\hfill \\ \Gamma =\Psi \times |J_i|\hfill \\ \Omega =\Psi |J_i|,\Gamma =0.5\times \Psi \sqrt{|J_i|}+0.5\times \Psi \left|J_i\right|\hfill \end{array}\right.$$

(105)

Figure 2, Figure 3 and Figure 4 indicate that for the ellipsoidal-based robust model, the solutions for both the ellipsoidal set and the “interval + ellipsoidal” set are the same when $\Omega \le 1$. When $\Omega \le 1$, both the ellipsoidal set-based robust model and the “interval + ellipsoidal” set-based robust model have the same solutions because of the identical uncertainty sets. For $\Omega \ge 1$, the solution of the robust model in the “interval + ellipsoidal” set does not get better and reaches the worst-case scenario because the uncertainty set in the “interval + ellipsoidal” set is an interval of $\Psi =1$ with no variation. When $\Gamma \le 1$, the identical solution can be observed for both robust models induced by the polyhedral set and the “interval + polyhedral” set; when $\Gamma \ge 1$, the solution lies with the “interval + polyhedral” set-based robust model. When $\Gamma \ge 1$, the solution obtained from the “interval + polyhedral” set-based robust model reaches the worst-case scenario, and the objective function ceases to increase. These findings suggest that combining the uncertainty set with the interval will eliminate conservative solutions when faced with bounded uncertainty.

Next, compare the solutions of the robust models based on the “interval + ellipsoidal” set and the “interval + polyhedral” set in Figure 2, Figure 3 and Figure 4. When $\Gamma =\Omega \left(\right|J_i{\left|\right)}^{1/2}$, the solution of the interval + polyhedral set-based model is always worse than the solution of the interval + ellipsoidal set-based model. This is because the ‘interval + polyhedral’ uncertainty set is larger and completely covers the ‘interval + ellipsoidal’ set. Assuming $\Gamma =\Omega$, the solution based on the ‘interval + polyhedral’ set model outperforms the solution based on the ‘interval + ellipsoidal’ set model since the uncertainty set of the ‘interval + polyhedral’ set is larger. Meanwhile, the uncertainty set of the ‘interval + polyhedral’ set is smaller than the ‘interval + ellipsoidal’ set and completely contained by it. Moreover, by comparing the ellipsoidal set and the ‘interval + ellipsoidal’ set in Figure 2, Figure 3 and Figure 4, it is evident that the solution provided by the robust model for the ellipsoidal set is superior to or equivalent to the solution based on the ‘interval + ellipsoidal’ set. However, the solution may be inferior when compared to the ‘interval + ellipsoidal’ set. Based on the similar analysis, comparable outcomes can be obtained for the polyhedral set and the ‘interval + polyhedral’ set. Combining the ellipsoidal or polyhedral set with the interval set results in a smaller uncertainty set, facilitating a less conservative solution for the model.

The uncertain parameters in Figure 5 are derived from the uncertain market demand for the right-hand term of the constraint. This condition is a special case where the number of uncertain parameters per relevant constraint $J_i$ = 1. Figure 5 indicates that the solutions of the robust model are identical for all uncertainty sets when $\Psi \le 1$. This finding is consistent with the definition of the corresponding uncertainty set, as at this point $\Omega =\Gamma \le 1$. Thus, these different types of uncertainty sets are equivalent to the same interval. When $\Psi \ge 1$, the uncertainty sets for the “interval + ellipsoidal set”, “interval + polyhedral set”, and “interval + ellipsoidal set + polyhedral set” are interval enclosures. At this point, the solution of the robust model reaches the worst case and does not increase. To increase the risk resilience of the supply chain during uncertainty, decision-makers within the supply chain need to adapt to their own risk preferences and select the appropriate adjustable parameters under the uncertainty set for decision-making. To enable the CLSC model established in this paper to be resilient to the uncertain environment, it is necessary to consider decision makers not making excessively conservative decisions. Therefore, a comparative analysis of the flexible facility capacity strategy and government subsidies is conducted based on an “interval + ellipsoidal” model with a $\Psi =0.4$ set robust model.

6.2. Flexible Facility Capacity Strategy Analysis

Using the numerical arithmetic examples and relevant parameters given in this paper, we solve the multi-objective model constructed in this paper using a combination of an augmented $ϵ$-constraint approach and an enhanced Benders decomposition algorithm. Due to the multi-objective property of the model, the recorded results are plotted on a two-dimensional plane of CO₂ emissions and total economic costs to assess the effectiveness of the flexible facility capacity strategy.

The Pareto frontiers for the fixed and flexible facility capacity models are depicted in Figure 6. From Figure 6, it is evident that the performance of the fixed facility capacity model is dominated by that of the flexible facility capacity model. A CLSC decision with a flexible facility capacity strategy will result in lower total cost than a decision with fixed facility capacity for the same level of carbon emissions. Conversely, for the same level of total cost, the flexible facility capacity strategy will generate lower carbon emissions than the fixed facility capacity strategy. Adopting a flexible facility capacity strategy within a CLSC can enhance decision effectiveness. The choice of location operations and capacity levels for manufacturers, inspections centers, remanufacturing centers, recycling centers, and energy centers vary in three phases, as illustrated in

Table 8. Results of site selection for flexible facility capacity strategy.

Facility Type	t = 1	t = 2	t = 3
Manufacturers	[$l_2$ 0 $l_3$ 0 0 ]	[$l_3$ $l_3$ $l_1$ 0 0]	[$l_3$ $l_2$ $l_3$ 0 $l_2$]
Inspection Center	[ 0 $l_3$ 0 $l_1$ 0 ]	[0 $l_3$ 0 $l_3$ $l_1$]	[$l_2$ $l_3$ 0 $l_3$ $l_3$]
Remanufacturing Center	[$l_2$ 0 0 $l_1$]	[$l_2$ 0 0 $l_3$]	[$l_3$ 0 $l_2$ $l_3$]
Recycling Center	[$l_1$ 0 $l_3$ 0]	[$l_3$ 0 $l_3$ 0]	[$l_2$ $l_3$ $l_2$ 0]
Energy Center	[$l_3$ 0 0 0]	[$l_2$ 0 0 $l_2$]	[$l_3$ $l_1$ 0 $l_2$]

and

Table 9. Results of site selection for fixed facility capacity strategy.

Facility Type	t = 1	t = 2	t = 3
Manufacturers	[1 0 1 0 0]	[1 1 1 0 0]	[1 1 1 0 1]
Inspection Center	[0 1 0 1 0]	[0 1 0 1 1]	[1 1 0 1 1]
Remanufacturing Center	[1 0 0 1]	[1 0 0 1]	[1 0 1 1]
Recycling Center	[1 0 1 0]	[1 0 1 1]	[1 0 1 1]
Energy Center	[1 0 0 0]	[1 0 0 1]	[1 1 0 1]

, for the flexible facility capacity and fixed facility capacity models, correspondingly. For instance, in the flexible facility capacity model, the capacity level can either be raised (such as manufacturer 1 increasing its capacity level to high during the second period), lowered (such as energy center 1 reducing its capacity level to medium in the second period), or kept constant throughout all periods (such as inspection 2 operating at high capacity level during all three periods). Location options are also available under the fixed facility capacity model. For example, period centers 1 and 3 are designated for construction during the first period, while period center 4 is chosen to be constructed during the second period.

The changes in facility capacity levels from the first to the third period under the flexible facility capacity model can be found in

Table 10. Modifications in the capacity levels of facilities using the flexible facility capacity strategy.

Facility Type	Number of Capacity Level Changes
Manufacturer	[2 1 2 0 0]
Inspection Center	[0 0 0 1 1]
Remanufacturing Center	[1 0 0 1]
Recycling Center	[2 0 1 0]
Energy Center	[2 0 0 0]

. For example, manufacturing producers 1 and 3 undergo two capacity level changes in each of the three periods, and inspections 4 and 5 make one capacity level change in each of the three periods. The flexible facility capacity strategy allows for changes in facility location decisions when compared to the fixed facility capacity model. For instance, Recycling Center 2 is located in the third period in the flexible facility capacity model, but not in the fixed facility capacity model.

This paper presents a sensitivity analysis of the facility capacity change costs under the model to further validate the effectiveness of the flexible facility capacity strategy. The capacity level change cost was originally 300,000 and has been tested with low and high-level change costs of 100,000 and 1,000,000, respectively. Figure 7 shows the Pareto frontiers. Figure 7a,b demonstrate that the Pareto frontier of the flexible capacity model approaches the Pareto frontier of the fixed capacity model as the capacity level cost increases. Also, the flexible capacity model outperforms the fixed capacity model for different capacity level change costs. Thus, this analysis concludes that by providing flexibility in facility capacity levels in a CLSC, which results in a higher degree of freedom in decision-making, the overall performance of the CLSC network can be improved.

As presented in this paper, the flexible facility capacity model is superior to the fixed facility capacity model. Therefore, the following analysis will focus on the effects of the flexible facility capacity model. The first effect to be analyzed is the impact of transport costs on the flexible facility capacity model. Figure 8 demonstrates the Pareto frontiers variation of the flexible facility capacity model across three transport cost cases: (1) the first case is the base case for the model parameters, with transport costs decreasing from the first to the third period; (2) transport costs remain constant across all periods, including the second and third, where they are consistent with the first period; and (3) uniformly-distributed low transport cost parameters at [0.15, 0.3] are applied over the three periods. Figure 8 illustrates that keeping transport costs low outperforms the other two cases. The case of keeping transport costs constant exhibits the lowest level of performance. Keeping transport costs low and decreasing them both leads to a reduction in carbon emissions for the same costs as a CLSC. This reduction is brought about through the decreased costs of the CLSC. In the supply chain, distribution serves as the link connecting the different parts of the CLSC, and charges for distribution and transport represent a significant proportion of the overall supply chain.

This section examines the effect of processing costs on the flexible facility capacity model. Three scenarios are considered: (1) decreasing processing costs over time, (2) constant processing costs, and (3) low processing costs. Low processing costs are defined as having parameters for remanufacturing costs that are uniformly distributed within U[20,23], reprocessing costs that are uniformly distributed within U[22,25], and energy recovery costs that are uniformly distributed within U[12,15]. As shown in Figure 9, the scenario with low reprocessing costs performs better than the other two scenarios. Lower processing costs allow for more efficient operation of the CLSC for the same level of carbon emissions. However, as processing costs increase, the performance of each objective in multi-objective optimization decreases.

6.3. Government Subsidy Factor Analysis

Government subsidies play a significant role in developing reverse logistics in CLSC networks and promoting innovation in the remanufacturing and reprocessing sectors. This section analyzes the impact of government subsidies on CLSCs and explores four types of subsidies provided by the government: subsidies for the remanufacturing segment, subsidies for the reprocessing segment, subsidies for the energy recovery segment, and subsidies for all three sectors simultaneously.

This study analyzes the impact of government subsidies on the remanufacturing link in the CLSC. Changes in the Pareto frontiers of the flexible facility capacity model for three scenarios of government subsidies are depicted in Figure 10. These scenarios include: (1) no government subsidies for the remanufacturing link; (2) constant government subsidies for the remanufacturing link per unit of product over three periods, consistent with the first period; and (3) decreasing government subsidies for the remanufacturing segment, which tends to decrease in the first to third periods as set in the original model parameters. It can be observed from Figure 10. Compared to scenarios without government subsidies, subsidizing the remanufacturing chain can enhance the performance of the CLSC while also reducing the cost of the supply chain. This can stimulate market dynamics, facilitate technological innovation, and promote the industry’s transition to a circular economy. An analysis of Figure 10 reveals that the Pareto frontiers of the model resemble those of the scenario with constant government subsidies. Maintaining constant subsidies by the government does not considerably enhance the efficiency of the CLSC model. However, it has the potential to increase the financial load on the government. Therefore, additional studies are necessary to develop a rational mechanism for subsidies in the remanufacturing segment through government initiatives.

The second aspect is the impact of government subsidies on the reprocessing chain within the CLSC. Similar to Figure 10, Figure 11 demonstrates the modifications to the Pareto frontiers of the flexible facility capacity model for three different government subsidy scenarios: (1) No subsidy from the government for the reprocessing segment. (2) The government grants subsidies for the reprocessing segment, and the amount of subsidy per unit of product stays stable for three periods, akin to the first period. (3) The initial model defines the parameters, and the government subsidy gradually decreases. As demonstrated in Figure 11, the government supplies subsidies to the reprocessing segment to enhance the performance of the CLSC. Reducing subsidies raises the cost of the CLSC while keeping carbon emissions constant. Conversely, reducing subsidies increases carbon emissions in the CLSC while holding costs constant. Figure 12 illustrates the impact of government subsidies on the energy recovery chain in the CLSC, divided into three cases. The analysis parallels the above findings that government subsidies for energy recovery enhance the performance of the CLSC. It is important to note that minimizing the cost of the CLSC, while keeping government subsidies constant over time, can result in a substantial rise in the supply chain’s carbon emissions. If a decision maker with the aforementioned preference chooses to implement a CLSC system, it could undermine the sustainability principles and the objectives of the CLSC network. As such, the government should enforce carbon emissions controls via carbon taxes when providing energy recovery subsidies to prevent extreme situations.

Figure 13 illustrates the impact of government subsidies on all three segments of the CLSC. The analysis reveals that transitioning to CLSCs is a key step towards sustainable development and climate change mitigation. However, while subsidies can enhance the economic and environmental performance of CLSCs, they also present challenges such as increased government financial burden, rent-seeking behaviors, excessive investment by local governments, and information asymmetry.

To address these issues, subsidies should be targeted and adaptable, taking into account the varying operational objectives of different industries. Policymakers should establish specific subsidy mechanisms supported by legal frameworks to prevent fraudulent activities and ensure the precision and appropriateness of subsidies. Furthermore, setting achievable carbon emission reduction goals can encourage enterprises to transition to a CLSC network. Enhancing legislation and regulation of the carbon trading market, establishing carbon trading organizations, implementing enterprise responsibility systems, and fulfilling supervisory duties can provide a solid foundation for promoting CLSC development.

7. Conclusions

This paper presents a comprehensive study of CLSC network design under uncertainty, taking into account uncertainties in demand, recycling rate, and recycled product quality. The study developed a model with multiple objectives, periods, and products. The model incorporates both fixed and flexible facility capacities and includes a government subsidy factor to minimize economic costs and carbon emissions. The proposed model considers uncertainties in product market demand, recycled product quality, and recycling rate. It utilizes robust optimization techniques to construct a multi-objective robust model amidst six distinct sets of uncertainties. To improve the solution efficiency, the augmented $ϵ$-constraint method is used to deal with multi-objectives, and a three-stage approach is introduced to design the Benders decomposition algorithm to solve the model.

Through numerical case studies, this paper concludes that the CLSC cost is significantly affected by the recycling rate and the quality of recycled products. Subsequently, the paper investigates the impact of uncertainty level changes on the CLSC cost. As the level of uncertainty increases, the CLSC incurs higher costs. Secondly, implementing a flexible facility capacity strategy can lower the cost and carbon emissions of the CLSC while improving its resilience to uncertain environmental conditions. Finally, an examination of the impact of government subsidies on CLSCs is conducted under various circumstances. Government subsidies can effectively reduce costs and carbon emissions in CLSCs. However, it is important to consider certain limitations and implement well-designed policies to avoid extreme effects.

This paper offers valuable guidance for companies, policymakers, and governments that strive to build sustainable supply chain centers. Future research could explore additional uncertain environments or factors that may impact the costs and carbon emissions of CLSCs. This inquiry could also evaluate the efficacy of various government subsidies or regulations in advancing sustainable supply chain practices. Furthermore, the model could benefit from the inclusion of environmental impact factors beyond the scope of carbon emissions, such as water consumption and waste generation. Subsequently, incorporating additional sustainability indicators would lead to a more comprehensive understanding of the environmental impacts of supply chain centers. Additionally, future research could investigate the effects of advanced technologies, such as blockchain and artificial intelligence.