Design of Reverse Network for Recyclable Packaging Boxes under Uncertainties

Abstract

The increasing number of disposable packaging boxes has caused a huge waste of resources and environmental problems. In order to promote the effective use of sustainable express packaging, this study builds a reverse network of recyclable express packaging boxes for e-commerce businesses with self-established logistics. Considering the uncertainties of demand, return rate, retention rate and recovery rate, a stochastic programming model is proposed to determine the location of collection points and recovery centers in the supply chain network, as well as the flow of recycling boxes, with the objective of minimizing the total cost. Furthermore, an improved priority-based tabu search algorithm is proposed to solve the model. The feasibility and operability of the model are verified via a benchmark function experiment and numerical experiment. This study provides important support for the sustainable development of the express delivery industry and helps to protect the environment.

1. Introduction

Due to the e-commerce sector’s explosive growth in recent years, the express delivery market has seen an astonishing boom. Every year, the industry’s usage of disposable packaging results in the production of more than 1.8 million tons of plastic garbage and about 9 million tons of paper waste, significantly depleting Earth’s resources. More and more businesses are focusing on sustainability in order to achieve environmentally friendly improvements and attract customers who appreciate sustainability efforts [1]. To address this issue, delivery companies are considering the adoption of recyclable express packages, which can benefit businesses financially and lessen resource waste. However, there are several challenges in promoting and implementing recyclable packaging. Packages are sent from scattered warehouses, go through a series of distribution processes to reach widespread and densely populated customers, and are then followed by multiple layers of return to the warehouses. This complexity makes the reverse logistics network for recyclable packaging more extensive than a conventional network. Moreover, the uncertainties of demand, return rates, retention rates and recovery rates complicate the modeling. The high initial construction costs exert economic pressure on low-profit-margin express delivery companies. Consequently, it is crucial to design an effective reverse logistics network. Through this network, express packaging can be efficiently distributed and recovered, thereby fulfilling the dual role of environmental protection and long-term cost reduction for businesses.

Recyclable packaging, which can rapidly degrade or be reused without ecological disruption, has gained attention from scholars for its promotion. Lyu et al. optimized strategies for green express packaging recycling and sales on online platforms [2]. Bening et al. addressed economic and technological barriers associated with promoting recyclable packaging and proposed solutions [3]. Bates et al. explored green packaging with waste wood cellulose fiber substrates for offset printing [4]. Their research contributes to the promotion of recyclable express packaging. However, limited scholarly focus has been given to constructing reverse logistics networks for recyclable packaging. Designing a reverse logistics network incorporating recyclable packaging can assist express delivery companies in efficiently utilizing circular packaging.

The network design problem, which encompasses the long-term location decisions of network nodes and the short-term flow allocation decisions between network nodes, has been a subject of interest among scholars across various disciplines [5,6]. Wang et al. constructed an optimization model for the location path problem of an agricultural production waste recycling network [7]. Shokouhyar and Aalirezaei formulated a model for the waste recycling of electrical and electronic equipment (WEEE), while John et al. proposed a grade strategy to classify the returns of recycling WEEE [8,9]. Studies about reverse network design for medical waste, proton-exchange membrane fuel cell battery, construction and demolition waste, end-of-life vehicles and collaborative battery management for electric vehicles were conducted by Govindan et al., Alkahtani and Ziout, Jahangiri et al., Kusakci et al. and Wenzhu Liao et al. separately [10,11,12,13,14]. Shi et al. designed a circular symbiosis network for express packaging waste recycling based on an urban symbiosis strategy, which can save recycling costs and reduce carbon footprint [15]. These scholars focus on the recycling strategy of reverse logistics, ignoring the importance of building a high-level logistics network. The transportation routes and customers exposed to recyclable packaging are widely distributed and dense. This makes the reverse logistics network have the characteristics of multi-level and high density compared with general reverse logistics. Therefore, the reverse logistics design of recyclable express packaging cannot be equated with the general network design.

At the same time, the express delivery industry is characterized by high uncertainty. Parcel recycling is easily affected by social emergencies, such as shopping festivals and epidemic control, and the frequency and degree of demand uncertainty are higher than other industries [16]. Kunamaneni et al.’s research proved that the return rate of recyclable packaging affects the asset shrinkage rate of express companies and is an important factor in achieving sustainability [17]. Bradley et al. conducted a study on recyclable packaging, and the findings emphasized that customer retention rate is the key to long-term sustainability [18]. Yang et al. proved that the degree of mastery of recovery information greatly affected the expected cost [19]. However, few people combine the uncertainty of demand, return rate, retention rate and recovery rate. This uncertainty complicates the design of supply chain networks.

Generally speaking, scholars adopt fuzzy methods, robust methods and stochastic methods to solve uncertainty problems [20,21,22,23,24,25]. The stochastic method can simulate the uncertainty of recyclable packaging in reality well through the probability distribution function, which is less subjective than the fuzzy method. When the constraint conditions are satisfied, the optimal expected value of the objective function can be obtained through the stochastic method, which is less conservative than the robust method.

Table 1. Comparation of the three different algorithms.

Methods	Fuzzy Methods	Robust Methods	Stochastic Methods
Advantages	Flexibility, handling ambiguity.	Resilience to uncertainty, reduced sensitivity.	Less subjective and less conservative.
Disadvantages	Subjectivity in modeling.	Conservative solutions, increased complexity.	Computational complexity, limited insights into extreme events.

compares the three different methods to studying uncertainty. Therefore, a stochastic programming model accounting for unknown demand quantity, uncertain return rate, uncertain retention rate and uncertain recovery rate is established, aiming to optimize the expected objective value while satisfying the constraints.

When the stochastic method is used to deal with uncertainty, multiple scenarios will increase the difficulty of solving the model. In particular, the reverse logistics network of recyclable packaging has the characteristics of multi-scene, multi-product and multi-level, and the solution complexity is high. Therefore, it is necessary to use heuristic algorithms, such as Tabu Search (TS) and Genetic Algorithm (GA), to solve this NP-hard problem. Khan et al. used combined TS and Simulated Annealing (SA) algorithms to solve a sustainable closed-loop supply chain that considers unqualified products [26]. Diarrassouba et al. and Gholizadeh et al. improved GA to avoid falling into local optimality [27,28]. To overcome these shortcomings, this study combines TS and GA. The proposed algorithm integrates TS’s framework with GA’s initialization, diversification and intensification mechanisms to achieve collaborative optimization among network echelons for large-scale problems. At the same time, we also use a priority-based coding approach to represent the proposed algorithm to avoid slower convergence due to additional repair or replacement processes.

This study considers the uncertainty of demand, return rate, retention rate and recovery rate of recyclable express packaging from the perspective of self-operated express companies, aiming to establish an efficient logistics network of recyclable express packages. A stochastic programming model with multiple scenarios, multiple echelons and multiple products is built for the network. The model is a supplement to the existing logistics network on the basis of the comprehensive investigation of reality.

The remainder of this paper is structured as follows. In Section 2, the problem description is presented. Section 3 formulates the mathematical model. An improved algorithm to solve this model is proposed and tested in Section 4. Section 5 shows the numerical experiments. Finally, we conclude the whole paper in Section 6.

2. Problem Description

The logistics and distribution of items are frequently given a lot of priority by self-operated e-commerce businesses in order to preserve their dependable brand image and regulate product quality. Self-operated e-commerce enterprises have formed their own express delivery companies to offer warehousing and distribution services for merchants on their online shopping platforms in an effort to increase distribution efficiency and offer consumers high-quality logistics services. Their logistics model for online order fulfillment is depicted in Figure 1. Based on demand projections, retailers keep products in warehouses. The warehouse, which is the place to create the order packages, is in charge of picking and packing the merchandise in accordance with online orders, as well as putting the merchant’s goods into storage. The order packages are subsequently delivered by the warehouse to the distribution center. Then, the distribution centers will sort, load, ship and dispatch the parcels to the corresponding pick-up points. Pick-up points will be arranged for couriers to deliver the parcels to the consumers. What’s more, self-operated express delivery companies (SEDCs) also provide a forward shipping service and after-sales service (returning or changing the goods) for consumers; thus, there is always a need to store packaging boxes at pick-up points.

Compared with other industries, the business orders of e-commerce enterprises are more random, and the distribution of consumers is wider. The e-commerce enterprises with self-built logistics have more network echelons, larger network scales and higher construction costs. Reverse network architecture that is based on an enterprise forward network can, therefore, effectively reduce fixed construction costs. Just adding collection and recovery duties at the original node is an easy way to raise their complication because the pick-up point and distribution center are required to carry out significant transfer jobs in the forward logistics. Such an arrangement is not workable. As a result, a multi-scenario, multi-echelon reverse network must be built. This study considers opening two types of collection points (CPs). The first type is the collection point (CCP) that is dedicated to collecting. The second type is opened as the joint collection/pick-up point (CDCP), which is located at the existing pick-up point with recycling and terminal distribution functions. Two types of recovery centers (RCs) are also considered. One is the recovery center (RRC), which is only used for recycling. The other is the integrated recovery/distribution center (RDRC) with recycling and distribution functions, and it is set up based on the forward distribution center. The reverse supply chain network design is shown in Figure 2. The reverse flows in the network are described as follows:

(1)

Returned packaging boxes from customers are collected by CPs, which includes CCPs and CDCPs.

(2)

Part of the returned packaging boxes that can be directly reused for the future forward services of sending packages from customers is stored in points with terminal distribution function. Since CCPs are only used for collection in the reverse flows, the retained parts in CCPs are shipped to CDCPs or pick-up points. The packaging boxes, excluding the retained parts in CPs (CCPs and CDCPs), are transported to RCs (RRCs and RDRCs) for further processing.

(3)

Packaging boxes in RCs are sorted into two parts; the recoverable parts that can be reused again after cleaning and repairing are transported to the enterprise’s warehouses for storage, and the unrecoverable parts are sent to landfills for disposal.

3. Mathematical Model

In this section, a stochastic programming model is proposed to make decisions for the openings of CPs and RCs and the allocation of the flows in the network. Sets, parameters, decision variables and the mathematical model are as follows:

3.1. Sets

$K$: Set of customers, ($1\cdots k\cdots \left|K\right|$);

$I_1$: Set of potential CCPs, ($1\cdots i_1\cdots {|I}_1|$);

$I_2$: Set of potential CDCPs, that is set of the existing pick-up points, ($1\cdots i_2\cdots {|I}_2|$);

$I$: Set of potential CPs that include CCPs and CDCPs, $I=I_1\cup I_2$, ($1\cdots i\cdots \left|I\right|$);

$J_1$: Set of potential RRCs, ($1\cdots j_1\cdots \left|J_1\right|$);

$J_2$: Set of potential RDRCs, ($1\cdots j_2\cdots \left|J_2\right|$);

$J$: Set of potential RCs that include RRCs and RDRCs, ($1\cdots j\cdots \left|J\right|$);

$W$: Set of warehouses, ($1\cdots w\cdots \left|W\right|$);

$L$: Set of landfills, ($1\cdots l\cdots \left|L\right|$);

$A$, $A^{\prime }$: Set of all the potential nodes in the network, $A{=A}^{\prime }=K\cup I\cup J\cup W\cup L$, ($1\cdots a\cdots \left|A\right|$), ($1\cdots a^{\prime }\cdots {|A}^{\prime }|$);

$S$: set of scenarios, ($1\cdots s\cdots \left|S\right|$);

$N$: set of types of the recyclable packaging boxes, ($1\cdots n\cdots \left|N\right|$).

3.2. Parameters

$d_{kns}$: Demand for packaging box $n$ of customer $k$ in scenario $s$;

${dis}_{a{a}^{\prime }}$: Distance (in km) between node $a$ and node $a^{\prime }$;

$c_{a{a}^{\prime }n}$: Transportation cost (in Yuan) of unit packaging box $\mathrm{n}$ from node $\mathrm{a}$ to node ${\mathrm{a}}^{\prime }$;

$h_{in}$: Collection cost (in Yuan) at candidate CP $i$ for unit packaging box $n$;

$h_{jn}$: Recovery cost (in Yuan) at candidate RC $j$ for unit packaging box $n$;

$h_{ln}$: Disposal cost (in Yuan) at landfill $l$ for unit packaging box $n$;

$g_{i_{2}n}$: Storage cost (in Yuan) at pick-up point $i_2$ for unit packaging box $n$;

$g_{wn}$: Storage cost (in Yuan) at warehouse $w$ for unit packaging box $n$;

$F_i$: Fixed opening cost (in Yuan) of CP $i$;

$F_j$: Fixed opening cost (in Yuan) of RC $j$;

$U_{i_{2}n}$: Storage capacity at pick-up point $i_2$ for unit packaging box $n$;

$U_{wn}$: Storage capacity at warehouse $w$ for unit packaging box $n$;

$V_{in}$: Collection capacity at potential CP $i$ for unit packaging box $n$;

$V_{jn}$: Recovery capacity at potential RC $j$ for unit packaging box $n$;

${\theta }_{ns}$: Return ratio of packaging box $n$ in scenario $s$;

${\delta }_{ns}$: Retention ratio of packaging box $n$ in scenario $s$;

${\alpha }_{ns}$: Recovery ratio of packaging box $n$ in scenario $s$;

$p_s$: Probability of occurrence of scenario $s$.

3.3. Decision Variables

$x_{ki}^{ns}$: Quantity of returned packaging box $n$ from customer $k$ to CP $i$ in scenario $s$;

$x_{i_1{i}_2}^{ns}$: Quantity of retained packaging box $n$ from CCP $i_1$ to pick-up point (potential CDCP) $i_2$ in scenario $s$;

$x_{ij}^{ns}$: Quantity of packaging box $n$ from CP $i$ to RC $j$ in scenario $s$;

$x_{jw}^{ns}$: Quantity of recoverable packaging box $n$ from RC $j$ to warehouse $w$ in scenario $s$;

$x_{jl}^{ns}$: Quantity of unrecoverable packaging box $n$ from RC $j$ to landfill $l$ in scenario $s$;

$Y_i$: Binary variable, equals to 1 if a CP is located at potential site $i$; 0, otherwise;

$Y_j$: Binary variable, equals to 1 if a RC is located at potential site $j$; 0, otherwise.

3.4. Stochastic Programming Model

$$\begin{array}{c}Minimize F={ TC}_1+{ TC}_{2 }+{TC}_3+{ TC}_4\end{array}$$

(1)

$$\begin{array}{c}{TC}_1=p_s{\displaystyle \sum _s}({\displaystyle \sum _k}{\displaystyle \sum _i}{\displaystyle \sum _n}{x}_{ki}^{ns}{dis}_{ki}{c}_{kin}+{\displaystyle \sum _{i_1}}{\displaystyle \sum _{i_2}}{\displaystyle \sum _n}{x}_{i_1{i}_2}^{ns}{dis}_{i_1{i}_2}{c}_{i_1{i}_{2}n}\\ +{\displaystyle \sum _i}{\displaystyle \sum _j}{\displaystyle \sum _n}{x}_{ij}^{ns}{dis}_{ij}{c}_{ijn}+{\displaystyle \sum _j}{\displaystyle \sum _w}{\displaystyle \sum _n}{x}_{jw}^{ns}{dis}_{jw}{c}_{jwn}+{\displaystyle \sum _j}{\displaystyle \sum _l}{\displaystyle \sum _n}{x}_{jl}^{ns}{dis}_{jl}{c}_{jln})\end{array}$$

(2)

$$\begin{array}{c}{TC}_2=p_s{\displaystyle \sum _s}({\displaystyle \sum _k}{\displaystyle \sum _i}{\displaystyle \sum _n}{x}_{ki}^{ns}{h}_{in}+{\displaystyle \sum _i}{\displaystyle \sum _j}{\displaystyle \sum _n}{x}_{ij}^{ns}{h}_{jn}+{\displaystyle \sum _j}{\displaystyle \sum _l}{\displaystyle \sum _n}{x}_{jl}^{ns}{h}_{ln})\end{array}$$

(3)

$$\begin{array}{c}{TC}_3=p_s{\displaystyle \sum _s}({\displaystyle \sum _{i_1}}{\displaystyle \sum _{i_2}}{\displaystyle \sum _n}{x}_{i_1{i}_2}^{ns}{g}_{i_{2}n}+{\displaystyle \sum _j}{\displaystyle \sum _w}{\displaystyle \sum _n}{x}_{jw}^{ns}{g}_{wn})\end{array}$$

(4)

$$\begin{array}{c}{TC}_4={\displaystyle \sum _i}{Y}_i{F}_i+{\displaystyle \sum _j}{Y}_j{F}_j\end{array}$$

(5)

Subject to:

$$\begin{array}{c}{\theta }_{ns}{d}_{kns}-\frac{1}{2} \le {\displaystyle \sum _i}{x}_{ki}^{ns}\le \frac{1}{2}+{\theta }_{ns}{d}_{kns}, \forall k\in K,\forall n\in N,\forall s\in S\end{array}$$

(6)

$$\begin{array}{c}\left(1-{\delta }_{ns}\right){\displaystyle \sum _k}{x}_{ki}^{ns}-\frac{1}{2} \le {\displaystyle \sum _j}{x}_{ij}^{ns} \le \frac{1}{2}+\left(1-{\delta }_{ns}\right){\displaystyle \sum _k}{x}_{ki}^{ns}, \forall i\in I, \forall n\in N,\forall s\in S\end{array}$$

(7)

$$\begin{array}{c}{\displaystyle \sum _{i_2}}{x}_{i_1{i}_2}^{ns}+{\displaystyle \sum _j}{x}_{i_{1}j}^{ns}={\displaystyle \sum _k}{x}_{k{i}_1}^{ns}, \forall i_1\in I_1, \forall n\in N,\forall s\in S\end{array}$$

(8)

$$\begin{array}{c}{\alpha }_{ns}{\displaystyle \sum _i}{x}_{ij}^{ns}-\frac{1}{2} \le {\displaystyle \sum _w}{x}_{jw}^{ns} \le \frac{1}{2}+{\alpha }_{ns}{\displaystyle \sum _i}{x}_{ij}^{ns}, \forall j\in J, \forall n\in N,\forall s\in S\end{array}$$

(9)

$$\begin{array}{c}{\displaystyle \sum _w}{x}_{jw}^{ns}+{\displaystyle \sum _l}{x}_{jl}^{ns}={\displaystyle \sum _i}{x}_{ij}^{ns}, \forall j\in J, \forall n\in N,\forall s\in S\end{array}$$

(10)

$$\begin{array}{c}{\displaystyle \sum _k}{x}_{ki}^{ns}\le Y_i.V_{in}, \forall i\in I, \forall n\in N,\forall s\in S\end{array}$$

(11)

$$\begin{array}{c}{\displaystyle \sum _i}{x}_{ij}^{ns}\le Y_j.V_{jn}, \forall j\in J, \forall n\in N,\forall s\in S\end{array}$$

(12)

$$\begin{array}{c}\left({\displaystyle \sum _k}{x}_{k{i}_2}^{ns}-{\displaystyle \sum _j}{x}_{i_{2}j}^{ns}\right)+{\displaystyle \sum _{i_1}}{x}_{i_1{i}_2}^{ns}\le U_{i_{2}n}, \forall i_2\in I_2, \forall n\in N,\forall s\in S\end{array}$$

(13)

$$\begin{array}{c}{\displaystyle \sum _j}{x}_{jw}^{ns}\le U_{wn}, \forall w\in W, \forall n\in N,\forall s\in S\end{array}$$

(14)

$$\begin{array}{c}{Y}_i,Y_j\in \left\{\mathrm{0,1}\right\}, \forall i\in I,j\in J\end{array}$$

(15)

$$\begin{array}{c}{x}_{ki}^{ns},x_{i_1{i}_2}^{ns},x_{ij}^{ns},x_{jw}^{ns},x_{jl}^{ns}\ge 0, \forall n\in N,i\in I,i_1\in I_1,i_2\in I_2,j\in J,w\in W,l\in L,s\in S\end{array}$$

(16)

Objective function (1) represents the minimal total cost of the logistics system consisting of four parts, which represent the transportation costs between nodes (2), costs of collection, recovery and disposal (3), the storage cost of distribution network and the warehouse distribution center (4) and the fixed opening costs of the collection points and recovery centers (5) accordingly. Constraints (6–9) use a stochastic way to express the uncertainty of demand quantity, return rate, retention rate and recovery rate. These uncertainties are crucial to the study of reverse logistics networks for recyclable packaging and are important factors affecting the expected costs. Constraint (6) ensures that all the returned recyclable packages are from sent express packages and also uses ${\theta }_{ns}$ to indicate uncertainty in the return rate. Constraint (7) states that CPs will transport the remaining recyclable packages, which eliminate the retained part to RCs. It uses ${\delta }_{ns}$ to describe uncertainty in the retention rate. Constraint (8) ensures that the retained packaging boxes in CCPs are shipped to pick-up points (potential CDCPs) for storage. Constraint (9) illustrates that the recoverable part of packaging boxes in RCs is stored in warehouses, while the unrecoverable ones are transported to landfills, which is restricted in constraint (10). It uses ${\alpha }_{ns}$ to describe the uncertainty of the recovery rate. Constraints (11) and (12) ensure that the number of recyclable packaging boxes assigned to CPs and CRs cannot surpass the capacity. Constraints (13) and (14) ensure that the number of recyclable packaging boxes stored in pick-up points and warehouses cannot exceed the capacity. Constraint (15) is the binary restriction for decision variables. Constraint (16) is the non-negative restriction for decision variables.

4. Solution Method

In this study, an improved priority-based tabu search (ITS) algorithm is developed to solve the model. This paper adopts a priority-based encoding method to represent solutions for the network design, and a corresponding decoding method is proposed. To overcome the dependence of the tabu search algorithm on the initial solution in a large search space, an initialization mechanism is designed. An intensification mechanism is proposed to accelerate the speed of convergence. Furthermore, a diversification mechanism is designed to avoid the algorithm trap in local optimum.

4.1. Hybrid Genetic Algorithm–Tabu Search Algorithm

The tabu search algorithm was proposed by Professor Glover in 1986. The principle of the algorithm is to start from the “initial feasible solution”, use the neighborhood search mechanism to generate a candidate solution set based on the “current optimal solution”, select the best solution among them and update it to the “current optimal solution”. The process is repeated until the termination condition of the algorithm is met, and the “historical optimal solution” is output [29]. The genetic algorithm was proposed by Professor Holland in 1975. The principle of the algorithm is to start from an initial population containing multiple solutions. Based on the fitness function, the individual solutions with a high fitness value are retained, and those with a low fitness value are eliminated, so as to constantly update and iterate until the termination conditions of the algorithm are met to obtain the optimal population and the final solution of the algorithm [30]. Combining the local search ability and the global search ability of the two algorithms can effectively overcome the shortcomings of the two algorithms and develop a hybrid algorithm with high solving efficiency, the tabu genetic hybrid algorithm.

Due to the comprehensive consideration of multiple levels, multiple products and multiple stochastic scenarios in the mixed-integer stochastic programming model constructed in this paper, the difficulty of solving the problem itself is higher than that of general network optimization design problems. The integration of the research problem in this paper with the adopted algorithm through the selection of an appropriate encoding scheme is a crucial factor influencing the efficiency of the algorithm’s solution. Spanning-tree-based representations have garnered attention in network design problems [31]. Among the prevalent encoding methods, Prüfer number-based and priority-based encodings have shown promise. While Prüfer number-based encoding effectively represents connections between K sources and J depots with a concise solution string length of |K|+|J|−2, it may violate flow conservation and require additional repair or replace mechanisms [32]. On the other hand, priority-based encoding, utilizing |K|+|J| digits, avoids infeasible solutions, as demonstrated by researchers, such as Roghanian and Pazhoheshfar and Gen et al., in supply chain design [33,34]. This paper also adopts the priority-based encoding method for the representation of the proposed algorithm to escape the slower speed of convergence caused by additional repair or replace procedures.

4.2. Encoding and Decoding Procedures

A permutation string $v\left(t\right)$ is used as the encoding representation for a solution of the proposed model. The length of the string is equal to the number $B$, which is the sum of the number of potential RCs and potential CPs ($B=\left|J\right|+\left|I\right|$). A solution is generated by assigning $B$ values from 1 to $B$ to $B$ bits of the array. The scheme array is divided into two sub-scheme arrays, $v_1\left(j\right)$ and $v_2\left(i\right)$, according to the level of processing center and the level of recovery dot, which are used to design the decoding mode. Figure 3 illustrates an encoding that includes three RCs ($J=3$) and five CPs ($I=5$). At this time, the length of the scheme array $v\left(t\right)$ is 8. The values on the bits of the array elements correspond to the priorities assigned to each site selection node.

The flow allocation of a supply chain network can be considered as a combination of multiple transportation trees between two echelons that are sources and depots. With respect to the capacity constraint of the depots, products of the sources are transported to depots. Based on that, decoding algorithm 1 and decoding algorithm 2 are proposed for obtaining the values of location and allocation decision variables from the permutation string, respectively. The overall decoding procedure for a permutation string composes three stages according to the flow direction of recycling packaging boxes from customers to warehouses and landfills. The stages are illustrated as follows:

(1)

For the first stage, based on sub-string $v_2\left(i\right)$, location decisions of CPs are obtained by taking advantage of the decoding algorithm 1, the transportation tree among customers and CPs, and CCPs and pick-up points (potential CDCPs) are determined by adopting the decoding algorithm 2.

(2)

For the second stage, sub-string $v_1\left(j\right)$ is used to obtain the location decisions of RCs by means of the decoding algorithm 1. Permutation string v(t) is used to obtain the transportation tree from CPs to RCs.

(3)

For the third stage, according to the priority information contained in sub-string $v_1\left(j\right)$, transportation trees between RCs and warehouses, RCs and landfills are determined through the decoding algorithm 2.

A flowchart of the decoding procedure for a permutation string is shown in

Table 2. Decoding procedure for a permutation string.

Decoding Procedure for a Permutation String
Inputting the permutation string $v\left(t\right)$.
Stage 1: using sub-string $v_2\left(t\right)$
Decoding algorithm 1 → location decisions for CPs         Decoding algorithm 2 → allocation decisions from customers to CPs         Decoding algorithm 2 → allocation decisions from CCPs to pick-up points (potential CDCPs)
Stage 2: using sub-string $v_1\left(j\right)$ & permutation string $v\left(t\right)$
$v_1\left(j\right)$-decoding algorithm 1 → location decisions for RCs $v\left(t\right)$-decoding algorithm 2 → allocation decisions from CPs to RCs Stage 3: using sub-string $v_1\left(j\right)$         Decoding algorithm 2 → allocation decisions from RCs to warehouses         Decoding algorithm 2 → allocation decisions from RCs to landfills
Substituting the values of all the decision variables into the proposed formula to obtain the value of the objective function.

. The pseudo-codes of the decoding algorithm 1 and the decoding algorithm 2 are detailed in Figure 4 and Figure 5.

4.3. Initialization Mechanism

To realize the acceleration of convergence, GA is utilized to generate an initial solution with high quality for ITS. Three greedy strategies are proposed. According to the strategies, three chromosomes in the initial population of GA are generated, and the rest of the chromosomes are generated randomly. After iterating for given times, the optimal solution found in GA is input into ITS as the initial solution. The pseudo-code of GA is outlined in Figure 6, and the proposed greedy strategies are:

(1)

Greedy strategy 1: Listing CPs and RCs in an ascending order of their collection or process capacity, assigning the priorities to the listed CPS and RCs in descending order.

(2)

Greedy strategy 2: Listing CPs and RCs in increasing order of their fixed opening cost, assigning the priorities to the listed CPs and RCs in a decreasing order.

(3)

Greedy strategy 3: Listing CPs and RCs in an ascending order of their unit collection or process cost (fixed opening cost/collection or process capacity), assigning the priorities to the listed CPs and RCs in descending order.

4.4. Intensification and Diversification Mechanism

The reverse supply chain network constructed in this study requires making location decisions at the two echelons of CPs and RCs. The priority information of the two echelons is combined to generate a permutation string that can be decoded as a solution. When a permutation string is able to obtain the optimal location decisions for both of the echelons at the same time through the decoding procedures, the collaborative optimization between CPs and RCs is realized, and the total cost of the system can be effectively optimized. For the above consideration, an intensification mechanism is developed. When the global optimal solution of the algorithm is iterated to m times without any improvement, several pairs of different permutation strings stored in the tabu list are randomly selected; then, for the permutation strings in each pair, the priority information of echelon CPs and echelon RCs is exchanged to generate two new strings. The best one of these newly generated permutation strings is set as the intensified solution. If the intensified solution performs better than the current optimal solution and is not tabu, the intensified solution is used as the current optimal solution, and the algorithm continues based on the replacement.

In addition, in order to avoid ITS from being trapped in the local optimal solution, a GA-based diversification mechanism is developed. When the global optimal solution is consecutively not updated for o times, GA is applied to obtain a diversified solution. If the diversified solution jumps out of the current local optimum, the remaining iterations of ITS are continued based on the diversified solution, which is viewed as the current optimal solution. A flowchart of the proposed ITS is presented in Figure 7.

4.5. Benchmark Function Tests for ITS

To demonstrate the improved performance of ITS, the widely used Sphere and Griewank benchmark functions are chosen to carry out experiments. The Sphere function is a unimodal function, and the Griewank function is a multimodal function. Information on the functions is listed in

Table 3. Formulas and parameter settings of the test functions.

Name	Function	Dimensions	Range
Sphere	$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{D}}}{{\mathrm{x}}_{\mathrm{i}}}^2$	30	[−100, 100]
Griewank	$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{4000}{\displaystyle \sum _{\mathrm{i}}^{\mathrm{D}}}{{\mathrm{x}}_{\mathrm{i}}}^2-{\displaystyle \prod _{\mathrm{i}}^{\mathrm{D}}}\cos \left(\frac{{\mathrm{x}}_{\mathrm{i}}}{\sqrt{\mathrm{i}}}\right)+1$	30	[−500, 500]

. A comparison is made among ITS, basic tabu search algorithm (BTS) and basic genetic algorithm (BGA) to validate the evolutionary efficiency and global optimization ability of ITS. The maximum iteration of the three algorithms is set as 1000, and each algorithm runs 10 times for each function. Figure 8 outlines the evolution process of the average objective value of Sphere and Girewank in 10 runs separately.

Table 4. The average optimal objective value and the worst objective value of text functions in 10 runs.

	The Average Optimal Objective Value			The Worst Objective Value
	BTS	BGA	ITS	BTS	BGA	ITS
Sphere	$7.07{\times 10}^{-8}$	$9.02{\times 10}^{-8}$	$5.65{\times 10}^{-8}$	$8.75{\times 10}^{-8}$	$4.50{\times 10}^{-7}$	$8.07{\times 10}^{-8}$
Girewank	$9.42{\times 10}^{-2}$	$4.35{\times 10}^{-4}$	$6.26{\times 10}^{-6}$	$6.05{\times 10}^{-1}$	$1.09{\times 10}^{-3}$	$7.54{\times 10}^{-6}$

records the average optimal objective value and the worst objective value of Sphere and Girewank in 10 runs.

According to Figure 8a, compared with BTS and BGA, ITS converges to the optimal value of the test functions with fewer iterations, indicating that the initialization mechanism and the intensification mechanism improve the evolutionary efficiency of ITS. Figure 8b shows that for the multimodal function, BTS and BGA fail to converge to the optimal value in 500 iterations. It means within the limited iterations, BTS and BGA are likely to be trapped in local optimum. However, with the diversification mechanism, the ITS algorithm can avoid local optimum, and it possesses a higher efficiency of convergence. It can be clearly seen from Table 4 that for the average optimal objective value, ITS outperforms BTS and BGA for the two test functions, and the difference between the average optimal objective value and the worst objective value of ITS is smaller than BTS’s and BGA’s, which implies that the ITS algorithm performs better in terms of accuracy and stability.

5. Numerical Experiments and Result Analysis

In this section, to verify the effectiveness of the proposed model and the efficiency of ITS when applied to the reverse supply chain network design problem, numerical experiments are conducted on a PC (Intel core i5, 1.60 Ghz; Memory 8 G) and for small-scale experiments. The mathematical model is implemented by CPLEX 12.5.1 (Visual Studio 2015, C #).

5.1. Experiment Settings

As indicated in

Table 5. Settings of parameters.

Parameter	Distribution	Parameter	Distribution
${\mathrm{d}}_{\mathrm{kns}}$	$\mathrm{N}(200,20)$	${\mathrm{F}}_{\mathrm{j}}$	$\mathrm{U}\left(\mathrm{150,000},\mathrm{250,000}\right), \mathrm{j}\mathsf{ϵ}{\mathrm{J}}_1$
$\mathrm{di}{\mathrm{s}}_{{\mathrm{aa}}^{’}}$	$\mathrm{U}(3,40)$		$\mathrm{U}\left(\mathrm{80,000},\mathrm{100,000}\right), \mathrm{j}\mathsf{ϵ}{\mathrm{J}}_2$
${\mathrm{c}}_{{\mathrm{aa}}^{’}}$	$\mathrm{U}\left(0.03,0.06\right)$	${\mathrm{U}}_{{\mathrm{i}}_2\mathrm{n}}$	$\mathrm{U}\left(100,200\right)$
${\mathrm{h}}_{\mathrm{in}}$	$\mathrm{U}\left(0.02,0.04\right)$	${\mathrm{V}}_{\mathrm{in}}$	$\mathrm{U}\left(400,600\right), \mathrm{i}\mathsf{ϵ}{\mathrm{I}}_1$$;\mathrm{U}\left(200,300\right), \mathrm{i}\mathsf{ϵ}{\mathrm{I}}_2$
${\mathrm{h}}_{\mathrm{jn}}$	$\mathrm{U}\left(0.03,0.06\right)$	${\mathrm{V}}_{\mathrm{jn}}$	$\mathrm{U}\left(4000,6000\right), \mathrm{j}\mathsf{ϵ}{\mathrm{J}}_1$
${\mathrm{h}}_{\ln }$	$\mathrm{U}\left(0.01,0.03\right)$		$\mathrm{U}\left(2000,3000\right), \mathrm{j}\mathsf{ϵ}{\mathrm{J}}_2$
${\mathrm{g}}_{{\mathrm{i}}_2\mathrm{n}}$	$\mathrm{U}\left(0.01,0.03\right)$	${\mathrm{U}}_{\mathrm{wn}}$	$\mathrm{U}\left(2000,5000\right)$
${\mathrm{g}}_{\mathrm{wn}}$	$\mathrm{U}\left(0.02,0.04\right)$	${\mathsf{\theta }}_{\mathrm{ns}}$	$\mathrm{N}\left(0.9,0.04\right)$
${\mathrm{F}}_{\mathrm{i}}$	$\mathrm{U}\left(\mathrm{10,000},\mathrm{16,000}\right), \mathrm{i}\mathsf{ϵ}{\mathrm{I}}_1$	${\mathsf{\delta }}_{\mathrm{ns}}$	$\mathrm{N}\left(0.2,0.01\right)$
	$\mathrm{U}\left(5000,8000\right), \mathrm{i}\mathsf{ϵ}{\mathrm{I}}_2$	${\mathsf{\alpha }}_{\mathrm{ns}}$	$\mathrm{N}\left(0.8,0.03\right)$

, sets of parameters are generated in accordance with random probability distributions in order to be realistic and to demonstrate the broad application of the proposed model and algorithm.

5.2. Small-Scale Experiments

In general, CPLEX demonstrates superiority in solving small-scale numerical experiments and is capable of obtaining optimal solutions easily. Therefore, comparing the results obtained from CPLEX to those of the ITS algorithm serves as an effective approach to validate the effectiveness of the ITS algorithm. In this section, both CPLEX and the ITS method are applied to conduct small-scale numerical experiments on the mixed-integer programming model. Each experiment consists of different scenario quantities, namely 10, 30 and 50, generated randomly for each group of test instances. The obtained results in terms of optimal solutions and computational time are compared and presented in

Table 6. Comparison of the results between ITS and CPLEX solver.

Instance ID	Num. of Scenarios	CPLEX		ITS		Gap
		obj1	Time(s)	obj2	Time(s)
6−4(2+2)−2(1+1)−2−2−1	10	132,782	1	132,792	1	0.01%
	30	138,266	6	138,291	1	0.02%
	50	121,008	16	121,028	1	0.02%
6−4(2+2)−2(1+1)−2−2−2	10	136,023	1	136,132	2	0.08%
	30	141,954	230	142,082	5	0.09%
	50	145,015	1152	145,142	8	0.09%
8−6(2+4)−2(1+1)−2−2−1	10	136,788	3	136,796	1	0.01%
	30	124,808	333	124,873	2	0.05%
	50	129,184	2227	129,278	2	0.07%
8−6(2+4)−2(1+1)−2−2−2	10	144,506	3	144,623	3	0.08%
	30	146,022	960	146,083	6	0.04%
	50	128,590	4712	128,681	10	0.07%
10−7(2+5)−2(1+1)−2−2−1	10	163,239	3	163,307	2	0.04%
	30	156,011	171	156,076	3	0.04%
	50	151,611	2258	151,753	4	0.09%
10−7(2+5)−2(1+1)−2−2−2	10	153,031	3	153,169	4	0.09%
	30	150,601	2062	150,869	8	0.18%
	50	137,057	9378	137,455	12	0.29%
Avg. Gap						0.08%

Notes: (1) The numbers of the instance id (e.g., 6-4(2+2)-2(1+1)-2-2-1) denotes 6 customers, 4 CPs which includes 2 CCPs and 2 CDCPs, 2 RCs that contains 1 RRC and 1RDRC, 2 warehouses, 2 landfills and 1 type of packaging box; (2) Gap = (obj² − obj¹)$/$obj¹.

.

The results of the experiments in Table 6 demonstrate that when the variety of packaging box kinds and scenario possibilities rises, the CPLEX solver’s computation time increases rapidly while the computing time for ITS grows relatively slower. When the instance size is increased to 50 scenarios, 2 types of packaging box, 10 customers, 7 CPs, and 2 RCs, the computing time of CPLEX reaches 9378 s while the computing time of ITS is only 12 s, which indicates that ITS outperforms the CPLEX solver in terms of the optimization speed. In addition, the average gap of the objective value between ITS and the CPLEX solver is only 0.08%, which means ITS can obtain solutions that approximate the optimum. That is, ITS has a high degree of accuracy for solving the proposed reverse supply chain network design problem.

At the same time, it can be seen from the table that as the number of scenarios increases, the total target cost continues to decrease. When the number of customers and products doubled, the total cost increased by only about 15%. This proves that the logistics network proposed in this paper is effective in solving a multi-scene, multi-product and multi-echelon logistics network. The model in this paper can help enterprises reduce costs.

5.3. Large-Scale Experiment

With the increase in the number of network nodes, the number of scenarios and the number of packaging box types, the solution space of the proposed problem expands exponentially, and the CPLEX solver cannot find the optimal solution in a reasonable time. To validate the solvability of ITS for the proposed problem with high-dimensional decision variables, large-scale experiments are conducted. The experimental results are shown in

Table 7. The computing time under different scales.

Instance ID	Num. of Scenarios	Num. of Variables	Time(s)
20-17(7+10)-4(2+2)-2-2	150	148,221	224
20-17(7+10)-4(2+2)-2-3	150	222,321	738
30-25(9+16)-4(2+2)-2-2	150	303,029	833
30-25(9+16)-4(2+2)-2-3	150	454,529	2399
40-31(11+20)-4(2+2)-2-2	150	480,035	2531
40-31(11+20)-4(2+2)-2-3	150	720,035	7323

.

Table 7 shows that when the size of the instance increases and the number of decision variables reaches 720,035, ITS can solve the problem in about 2 h, which implies that the algorithm proposed in this study can effectively solve the large-scale and high-dimension complex reverse network design problem for recycling express packaging boxes.

The reverse logistics network established in this study heavily relies on the existing logistics network of the self-operated express delivery company. Despite the objective of reducing the overall cost of the reverse network for recyclable packaging in this study, it still requires the company to have substantial initial construction costs. Furthermore, the establishment of this network necessitates a well-developed infrastructure in society. Therefore, the network and model established in this study are applicable to economically developed regions, but their application in underdeveloped regions is limited.

6. Conclusions and Future Research

In the context of significant resource wastage and environmental pollution caused by disposable express packaging, promoting recyclable packaging boxes holds immense potential. This study designs a reverse network for recyclable packaging boxes in the express service industry. A stochastic programming model is proposed to address the network problem, encompassing the decisions of collection point and recovery center locations, as well as flow allocation decisions between network nodes. In summary, the contributions of this research can be summarized as follows:

(1)

From the perspective of the self-operated express delivery industry, this study considers the limited previous literature that combines recyclable packaging with the design of reverse logistics networks. It investigates a stochastic programming model for the design of reverse networks for recyclable packaging boxes, considering uncertainty in demand, return rates, retention rates and recovery rates. Moreover, the model incorporates multiple scenarios, products and tiers, making the problem more complex yet closer to reality and ensuring the robustness of network design.

(2)

An improved priority-based tabu search algorithm is proposed to solve the model. Benchmark experiments demonstrate that the improved algorithm outperforms the basic tabu search and basic genetic algorithms in terms of both local search efficiency and global search efficiency. The algorithm exhibits generality and provides valuable insights for researching network design models.

The reverse logistics network of recyclable express packaging established in this paper can benefit society and practitioners. From a social point of view, the network helps reduce the use of disposable express packaging. It helps to save resources and protect the environment and promotes the sustainability of the entire social and economic operation. From the practitioners’ point of view, the network helps self-owned delivery companies save costs in the long run. The examples in this paper prove that the more products and gradients involved in the network, the more obvious the cost control. At the same time, the environmental protection behavior of enterprises can bring a good corporate image for enterprises and promote the sustainable development of enterprises.