Shipping Logistics Network Optimization with Stochastic Demands for Construction Waste Recycling: A Case Study in Shanghai, China

Abstract

In this paper, we introduce a shipping logistics network optimization method for construction waste recycling. In our case, construction waste is transported by a relay mode integrating land transportation, shipping transportation, and land transportation. Under the influence of urban economic life, the quantity (demand) of construction waste is uncertain and stochastic. Considering the randomness of construction waste generation, a two-stage stochastic integer programming model for the design of a shipping logistics network for construction waste recycling is proposed, and an accurate algorithm based on Benders decomposition is presented. Based on an actual case in Shanghai, numerical experiments are carried out to evaluate the efficacy of the proposed model and algorithm. Based on an actual case study in Shanghai, numerical experiments demonstrate that the proposed model can help to reduce transportation costs of construction waste. Sensitivity analysis highlights that factors like the penalty for untransported waste and capacity constraints play a crucial role in network optimization decisions. The findings provide valuable theoretical support for developing more efficient and sustainable logistics networks for construction waste recycling.

1. Introduction

As urbanization continues, both construction activities and the amounts of construction waste are increasing. Based on average annual outputs, construction waste accounts for between 30% and 40% of all urban waste, and this percentage is growing [1]. Moreover, a large amount of untreated construction waste is directly transported to landfills. Due to increasingly severe construction waste problems, the management of waste as a resource has become a major challenge for cities in China.

In many cities, the waste generated by rail transportation projects and other major projects is transported to disposal/backfill sites via either land transportation or a relay mode integrating land transportation and shipping transportation. For the second method, waste is first sent to loading wharves via land transportation; then, it is loaded onto cargo ships and transported to destination wharves near the disposal/backfill sites on rivers such as the Huangpu and Yangtze Rivers; finally, the waste is transported to disposal/backfill sites via land transportation. Compared to land transportation, shipping has a number of advantages, including a larger capacity, higher efficiency, and a weaker impact on the environment and city roads. Therefore, a shipping logistics network for construction waste recycling that integrates land transportation and shipping transportation could play an important role in reducing the traditional transportation impacts (air quality, noise, infrastructure impacts), traffic congestion, and road damage caused by waste transportation.

Optimizing this network is a typical strategic decision-making issue. The major decisions involved are as follows: (1) What changes can be made as part of the optimization of logistics processes? (2) How many berths need to be rented? (3) How should land transportation between construction sites and loading wharves be arranged? (4) How should shipping from loading wharves to destination wharves be arranged? To a significant extent, the amount of waste generated at construction sites determines the optimal shipping logistics network for construction waste recycling. Under the influence of an urban economy, the number of major municipal projects in the region has varied from year to year, implying that the quantity of construction waste also varies in an uncertain way and thus acts as a stochastic variable. Because the amount of construction waste is uncertain and stochastic, the shipping logistics network for construction waste recycling in the region must be optimized [2]. However, solving this issue is challenging. This paper constructs a two-stage model based on stochastic optimization that can optimize a shipping logistics network for construction waste recycling. In the first stage, the locations of the loading wharves and the berth leases are determined. In the second stage, the construction waste transportation arrangement is determined. To effectively solve the model, an accurate algorithm based on Benders decomposition is proposed. In addition, we use the Dongtan N1 warehouse area in Nanhui (under construction) of Shanghai as a case study. The numerical experimental results are derived through this case and used to evaluate the efficacy of the proposed model and algorithm.

2. Literature Review

In this section, we discuss the relevant literature on construction waste recycling.

2.1. Reverse Logistics

In recent years, there have been many studies on reverse logistics design, including qualitative analysis [3], case analysis [4], production planning and control [5], quantitative analysis, simulation [6], mathematical modeling [7], and inventory management [8]. At present, environmental protection is an important consideration [9]. Furthermore, there has been extensive research on reverse logistics, and related models are widely used in different research directions. The first study addressing network design problems was published more than 20 years ago [10]. With the development of logistics research, there has been an increasing number of papers addressing reverse logistic problems, and the number of publications has increased by approximately 25% in the past three years. The latest literature reviews include [11,12,13]. Reverse logistics has been applied to study product after-sales [14] and waste recycling. Reverse logistics is extensively applied across various industries, such as construction and demolition waste management [15], medical waste processing [16], battery and energy product recycling [17,18,19], environmental protection and waste management [20], electronic product recycling and remanufacturing [21], and high-tech manufacturing [22]. Therefore, reverse logistics network (RLN) design is a well-developed field that can serve as a reference for our research.

2.2. Waste Recycling Logistics

Waste recycling logistics encompasses the systematic planning, organization, implementation, and control of processes involved in waste generation, recycling, treatment, and reuse. Its objective is to ensure effective waste recovery and resource recycling, while considering economic, environmental, and social factors [11,23]. In the above research, waste recycling networks account for the majority of reverse logistics networks. Jens et al. [24] published a literature review to discuss waste recycling logistics. In the Philippines, a community-based solid waste management recovery network exists to promote the sustainability of river and coastal resources [25]. Chubarenko et al. [26] have discussed research questions related to the management and economically viable use of beach wrack. They found that for local economies within the Baltic Sea region, the organic component of beach cast (beach wrack and terrestrial debris) had reasonable economic prospects as a renewable natural resource. Moreover, many industries have developed waste recycling logistics systems, including the e-waste management industry [27,28] and the automobile industry [29]. Furthermore, the team who researched waste wood reverse logistics, i.e., Julien and Amin [30], addressed RLN design problems under environmental policies targeting recycled wood materials from the construction, renovation, and demolition (CRD) industry. A scenario-based analysis is performed to evaluate the impact of uncertainty on RLN design. In 2019, a two-stage stochastic programming model was presented for reverse logistics network design (RLND) under uncertainty with dynamic supply source locations [31]. Taís et al. [32] used a sustainability indicator matrix for municipal solid waste management and found six factors that caused the low sustainability of the management of municipal solid waste from Mata de São João; these findings could also be applied to other Brazilian municipalities. Then, in 2020, the models were further improved, as illustrated through a case study on wood waste recycling from the CRD industry in Quebec, Canada [33]. Katarina et al. [34] focused on recovery strategy optimization for waste materials and improved material efficiency in the iron and steel industry. By studying material supply, material storage, shipping systems, and shipping frequency, they developed a model that could identify a logistics solution to achieve a common recycling system.

2.3. Construction Waste Recycling Logistic Network Optimization

In the reverse logistics field of the construction recycling industry, supply chain, economic benefits, and evaluation models were also studied early (

Table 1. Summary of closely related studies.

Author	Logistics Network Design	Optimization Under Uncertainty	Economic and Policy Implications
Nunes et al. (2009) [35]	√
Ma and Zhang (2020) [36]			√
Mohsen and Vahidreza (2018) [37]		√	√
Yang and Chen (2020) [38]		√
Mulvey and Ruszczynski (1995) [39]		√
Katarina et al. (2013) [34]	√

). Nunes et al. [35] introduced waste logistics and distribution channel networks in the Brazilian construction industry and analyzed some local C&D examples. Ma and Zhang [36] developed a dynamic evolutionary game model on construction waste recycling to analyze the symbiotic evolution between the behavior of construction enterprises and recycling enterprises in situations with or without government incentives, and they proposed a suggestion that is in accordance with the national condition of China.

There are several representative mathematical models. All the papers described above are unique. For example, Mohsen and Vahidreza [37] consider all aspects of sustainability and risk-aversion in a two-stage stochastic programming framework in the field of RLN optimization. They also consider on-site separation and off-site separation in the model, which might become a hot topic related to C&D waste recycling in the future. Yang and Chen [38] propose a robust optimization method, and their conclusions are comprehensive. They consider the variables important for multilayer strategic decision-making in the construction industry, and they consider the business plan in the current industrial environment. However, they leave room for optimization, as the study does not consider economies of scale in capacity acquisition and does assume that the robustness of the recycling network has the same effect on the decision utility of all decision-makers.

2.4. Research Gap

In summary, most existing research on construction recycling logistics has primarily focused on land transport, while little attention has been paid to optimizing water transport or road-river transport networks for construction waste recycling. This shortcoming is particularly critical given the increasing strain on urban traffic infrastructure and the environmental challenges posed by an over-reliance on land transportation. Although some studies have explored water transport optimization and road-river transport networks within the context of reverse logistics [34], these efforts are still in their early stages. Furthermore, they lack comprehensive models that account for factors like stochastic demand and multi-modal relay transport. This underscores the pressing need for a more robust and practical framework.

Therefore, this paper introduces a two-stage stochastic integer programming model that addresses risk aversion, stochastic demand, and sustainability concerns to optimize construction waste recycling logistics. The proposed model overcomes the limitations of previous research by incorporating a relay mode that seamlessly integrates land and water transportation, with the objective of minimizing expected capital investment and operational costs. To solve this problem efficiently, we have developed a precise algorithm based on Benders decomposition. These contributions offer new insights for future research on construction waste recycling logistics network optimization and pave the way for a more integrated land–water–land transport relay model.

3. Materials and Methods

3.1. Problem Definition

To reduce the negative impact of the long-distance land transportation of construction waste, such as dirt, on city roads and the environment and to improve the efficiency of construction waste transportation, some cities plan to adopt a relay mode that integrates land transportation and shipping transportation to effectively recycle construction waste. The shipping logistics network optimization of construction waste recycling is illustrated in Figure 1. Specifically, the recycling and transportation processes include the following steps:

(1)

Cargo trucks transport waste from waste-generating sites to loading wharves;

(2)

Waste is transported by cargo ships from loading wharves to destination wharves via the Huangpu River and Yangtze estuary channel;

(3)

Construction waste is unloaded at the destination wharves and transported to disposal/backfill sites by trucks.

In this paper, the shipping logistics network optimization for construction waste recycling involves strategic decision-making. This paper assumes that the government has selected the locations for the destination wharves and waste disposal/backfill sites and has constructed these facilities. As such, the major network optimization decisions include the selection of loading wharves, the selection of berths at the loading wharves, the arrangements for land transportation of waste between waste-generating sites and loading wharves, and the shipping arrangements between loading wharves and destination wharves. Factors to consider for the selection of loading wharves include geographic locations, traffic conditions, mooring conditions, site conditions, and wharf operations. Currently, there is an oversupply of berths at many cities’ ports; therefore, the existing berths at the loading wharves can be rented and reconstructed to suit construction waste transportation needs. The loading wharves come with a fixed cost that mainly includes the cost of renting the berths and the cost of reconstructing the rented berths to meet the needs of waste transportation (e.g., cost of specialized mooring equipment).

In contrast, the number of major projects, such as demolition of illegal buildings, varies by year; therefore, the quantity of construction waste generated in the future is a stochastic variable. Furthermore, decision-makers only have data on the quantity of construction waste generated in the past. Therefore, this paper assumes that the quantity of construction waste generated from a site (demand) is a random variable that follows a specific probability distribution. Without knowing exactly how much waste a construction site could generate in the future, decision-makers need to determine the construction waste recycling and transportation network that can minimize the expected capital investment and operating costs.

3.2. Model Development

In this section, a two-stage stochastic integer programming model is proposed to optimize the shipping logistics network used for construction waste recycling [39]. The problem for the first stage is as follows: when the future quantity of construction waste from construction sites is unknown, decision-makers need to determine optimal wharf locations and the optimal number of berths to be rented to minimize investment. The problem for the second stage is as follows: when the quantity of waste at a construction site (i.e., a random variable) is generated, decision-makers need to optimize the routes used to transport waste from construction sites to disposal/backfill sites to minimize operating costs. Finally, the result of the logistics plan sends feedback to the upper-level model to achieve the recycling solution.

Assume that there are $n$ construction sites in the region that generate waste, and they form set $J$. Assume that there are $m$ candidate loading wharves for decision-makers to rent, and they form set $I$. For a given construction site, we use $\mathsf{\Omega }$ to denote the set of all random waste generation scenarios at the site during the year. To better describe the model, we define the following parameters:

• $K$: the set of destination wharves;
• ${\lambda }_i^{max}$: the maximum number of berths at candidate loading wharf $i$;
• $f_i$: the annual fixed cost for operating a berth at candidate loading wharf $i$ (including the cost to reconstruct the berth and the annual rental fee);
• $C_i$: the waste loading capacity per berth per year at candidate loading wharf $i$;
• $D_k$: the waste processing capacity per year at destination wharf $k$;
• $t_{ji}$: the unit cost for candidate loading wharf $i$ to serve waste-generating site $j$;
• $w_{ik}$: the unit waste transportation cost from candidate loading wharf $i$ to destination wharf $k$;
• $q_j$: the unit penalty cost when the demand for transporting waste from site $j$ is not met (e.g., waste is not transported from generating site $j$ to the disposal/backfill site);
• $p_s$: the probability that a random waste quantity scenario $s$ will occur; and,
• $d_j^s$: the waste quantity from waste-generating site $j$ under the random waste quantity scenario $s$.
• We then define the following decision variables:
• $x_i$: the number of berths rented at candidate loading wharf $i$;
• $y_{ji}$: the quantity of waste generated from site $j$ that is processed at candidate loading wharf $i$;
• ${\theta }_{ik}$: the quantity of waste shipped from candidate loading wharf $i$ to destination wharf $k$; and,
• $z_j$: the unshipped waste generated from site $j$.

3.2.1. Decision-Making in the First Stage

When precise information about future waste is unavailable, decision-makers still need to determine the optimal location of the loading wharves and the number of berths to rent to minimize expected costs, generalized costs (i.e., the annual fixed cost of operating a berth), and logistics costs (i.e., the cost to transport waste from the construction sites to the destination wharves) are introduced in the upper-level model, which can be expressed as an integer programming problem.

$$\underset{i\in I }{{}_{x\hspace{1em}}{}^{min}{\displaystyle \sum f_i{x}_i}}+Q(x)$$

(1)

In Formula (1), $Q\left(x\right)$ denotes the following: given decision $x$ made in the first stage, the optimal cost can be expected in the second stage of decision-making. $Q\left(x\right)$ can be interpreted as the expected cost for different random quantities of waste, namely, $Q(x)={\displaystyle \sum {}_{S\in \Omega }{P}_s{Q}_s(x)},$, where $Q_S\left(x\right)$ denotes the cost of the second-stage decision corresponding to scenario $s$. In this problem, the expected minimum cost includes berth rental costs at the wharves, berth reconstruction costs, and the logistics costs to transport waste from the construction sites to the destination wharves via the candidate loading wharves.

$$s.t.x_i\le {\lambda }_i^{\max },\forall i\in I$$

(2)

Constraints (2) indicate that the maximum number of berths rented at wharf $\mathrm{i}$ is limited to the total number of berths available for rent at the wharf.

$$x_i\ge 0 and x_i is integer,\forall i\in I$$

(3)

Constraints (3) limit the range of the decision variables.

3.2.2. Decision-Making in the Second Stage

After the number of berths is determined in the first stage, the lower decision-makers need to develop the optimal logistics plan for transporting construction waste, where the logistics route choice is based on stochastic user equilibrium, in which traffic flow is assigned to a network to minimize the cost of transporting waste from the construction sites to the destination wharves. For a given waste quantity scenario $s$, the optimal logistics for transportation can be expressed as the following decision problem:

$$Q_s(x)=\underset{y^s,z^s,{\theta }^s}{\min }{\displaystyle \sum _{j\in J}{\displaystyle \sum _{i\in I}{t}_{ji}{y}_{ji}^s}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{k\in K}{w}_{ik}{\theta }_{ik}^s}}+{\displaystyle \sum _{j\in J}{q}_j{z}_j^s}$$

(4)

In Formula (4), a superscript $S$ is added to each decision variable, indicating that the variable is a decision variable corresponding to random waste quantity scenario $S$. The objective function values include three components: the cost to transport construction waste from generating sites to loading wharves by land transportation, the cost to ship construction waste from loading wharves to destination wharves, and the penalty incurred when random waste quantities exceed the processing capacity at the loading wharves.

$$s.t.\hspace{1em}{\displaystyle \sum _{i\in I}{y}_{ji}^s+z_j^s\ge d_j^s,\hspace{1em}\forall j\in J},$$

(5)

Constraints (5) indicate that construction waste generated by site $j$ must be shipped through the loading wharves. If the loading wharf cannot process all the waste generated from site $j$ because of capacity limits, then unmet demand for waste processing $z_j^s$ arises and could lead to penalty $q_j{z}_j^s$.

$${\displaystyle \sum _{j\in J}{y}_{ji}^s}\le C_i{x}_i,\hspace{1em}\forall i\in I,$$

(6)

Constraints (6) indicate that the total quantity of construction waste shipped from a candidate loading wharf cannot exceed the total processing capacity of all rented berths at the wharf.

$${\displaystyle \sum _{i\in I}{\theta }_{ik}^s}\le D_k,\hspace{1em}\forall k\in K,$$

(7)

Constraints (7) indicate that the total construction waste unloaded at a destination wharf cannot exceed its processing capacity.

$${\displaystyle \sum _{j\in J}{y}_{ji}^s}={\displaystyle \sum _{k\in K}{\theta }_{ik}^s},\hspace{1em}\forall i\in I,$$

(8)

Constraints (8) indicate that the quantity of construction waste processed at the loading wharf equals the quantity processed at the destination wharf.

$$y_{ji}^s\ge 0,\hspace{1em}\forall i\in I,\forall j\in J,$$

(9)

$$z_j^s\ge 0,\hspace{1em}\forall j\in J,$$

(10)

$${\theta }_{ik}^s\ge 0,\hspace{1em}\forall i\in I,\forall k\in K.$$

(11)

Constraints (9)–(11) set the value ranges for the decision variables.

3.3. Benders Decomposition Algorithm

To solve the two-stage stochastic integer programming model for the shipping logistics network for recycling construction waste developed in Section 3, this paper proposes an iterative, exact algorithm based on Benders decomposition. The approach aims to reduce the complexity of problem solving by dividing the problem into a master problem and a subproblem. Based on each random construction waste quantity, the subproblem is defined. The Benders cut is then developed by solving the subproblem. The master problem and the subproblem are coupled through the Benders cut to solve the problem [3,4,5]. We discuss the detailed steps of the algorithm in the following sections.

Assume that ${\psi }_S$ is the lower bound of the value range of the objective function that corresponds to random waste quantity scenario $s$ in the second stage. The algorithm is an iterative process. During iteration $t$, the algorithm first solves the relaxed master problem as follows:

$$M{P}_t:\hspace{1em}\underset{x,\psi }{\min }\hspace{1em}{\displaystyle \sum _{i\in I}{f}_i}{x}_i+{\displaystyle \sum _{s\in \mathsf{\Omega }}{p}_s{\psi }_{\mathrm{s}}}$$

(12)

Relaxed master problem $M{P}_t$ is a mixed integer programming problem.

$$s.t.\hspace{1em}{x}_i\le {\lambda }_i^{max},\hspace{1em}\forall i\in I$$

(13)

$$x_i\ge 0 and x_i is integer,\forall i\in I$$

(14)

Constraints (13) represent the set of Benders cuts added in the iterative process. The Benders cut is also referred to as the optimality cut. The set of optimality cuts is updated during the iteration. $M{P}_0$ does not include the Benders cut. Next, we discuss how to derive the optimality cut based on the solution to the master problem.

$$S{P}^s(\widehat{x}): Q_s(\widehat{x})={\min }_{y^s,z^s,{\theta }^s}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{i\in I}{t}_{ji}}}{y}_{ji}^s+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{k\in K}{w}_{ik}}}{\theta }_{ik}^s+{\displaystyle \sum _{j\in J}{q}_j}{z}_j^s$$

(15)

Given the solution in the first stage $\widehat{x}$, for each random waste quantity scenario $s\in \Omega$, we define a restrictive problem, $S{P}^s\left(\widehat{x}\right)$.

$$s.t.\hspace{1em}{\displaystyle \sum {}_{j\in J}y_{ji}^s}+z_j^s\ge d_j^s,\hspace{1em}\forall j\in J$$

(16)

$${{\displaystyle \sum }}_{j\in J}{y}_{ji}^s\le C_i{\widehat{x}}_i,\hspace{1em}\in I$$

(17)

$${{\displaystyle \sum }}_{i\in I}{\theta }_{ik}^s\le D_k,\hspace{1em}\forall k\in K$$

(18)

$${{\displaystyle \sum }}_{j\in J}{y}_{ji}^s={{\displaystyle \sum }}_{k\in K}{\theta }_{ik}^s,\hspace{1em}\forall i\in I$$

(19)

$$y_{ji}^s\ge 0,\hspace{1em}\forall i\in I,\forall j\in J$$

(20)

$${\theta }_{ik}^s\ge 0,\hspace{1em}\forall i\in I,\forall k\in K$$

(21)

$$z_j^s\ge 0,\hspace{1em}\forall j\in J$$

(22)

Let ${\pi }_j$, ${\varpi }_i$, ${\gamma }_k$, and ${\tau }_i$ be dual variables of constraints (16)–(19), respectively.

$$DS{P}^s(\widehat{x}):\hspace{1em}\underset{\pi ,\varpi ,\gamma }{\max }{\displaystyle \sum {}_{j\in J}d_j^s{\pi }_j}+{\displaystyle \sum {}_{i\in I}C_i{\widehat{x}}_i{\varpi }_i}+{\displaystyle \sum {}_{k\in K}{D}_k{\gamma }_k}$$

(23)

Using the dual variables, we dualize the restrictive problem $S{P}^s\left(\widehat{x}\right)$ into $DS{P}^s\left(\widehat{x}\right)$.

$$s.t.\hspace{1em}{\pi }_j+{\varpi }_i+{\tau }_i\le t_{ji},\forall i\in I,\forall j\in J,$$

(24)

$${\gamma }_k-{\tau }_i\le w_{ik},\hspace{1em}\forall k\in K,\forall i\in I$$

(25)

$$0\le {\pi }_j\le q_j,\hspace{1em}\forall j\in J$$

(26)

$${\varpi }_i\le 0,\hspace{1em}\forall i\in I$$

(27)

$${\gamma }_k\le 0,\hspace{1em}\forall k\in K$$

(28)

$${\tau }_i is \mathrm{free} \mathrm{variable},\forall i\in I.$$

(29)

We refer to the duality problem $DS{P}^s\left(\widehat{x}\right)$ as the subproblem corresponding to the random waste quantity scenario $s$. Given the solution $\widehat{x}$ to the first stage of decision-making, an optimal solution must exist; therefore, the duality problem $DS{P}^s\left(\widehat{x}\right)$ has an optimal solution. Assume that ${\mathcal{H}}^{\ast }=\left({\pi }^{\ast },{\varpi }^{\ast },{\gamma }^{\ast },{\tau }^{\ast }\right)$ is the optimal solution to the duality problem $DS{P}^s\left(\widehat{x}\right)$. Then, we have ${\psi }_s\ge {{\displaystyle \sum }}_{i\in I}{C}_i{\varpi }_i^{\ast }{x}_i+{{\displaystyle \sum }}_{j\in J}{d}_j^s{\pi }_j^{\ast }+{{\displaystyle \sum }}_{k\in K}{D}_k{\gamma }_k^{\ast }$. Let $Q_s^{\ast }\left(x,{\mathcal{H}}^{\ast }\right)={{\displaystyle \sum }}_{j\in J}{d}_j^s{\pi }_j^{\ast }+{{\displaystyle \sum }}_{i\in I}{C}_i{\varpi }_i^{\ast }{x}_i+{{\displaystyle \sum }}_{k\in K}{D}_k{\gamma }_k^{\ast }$ and the following Benders cuts: ${\psi }_s\ge Q_s^{\ast }\left(x,{\mathcal{H}}^{\ast }\right)$.

These Benders cuts must satisfy any feasible value of $\widehat{x}$, which is the solution to the master problem. The pseudocode of the algorithm is shown in Algorithm 1. In the algorithm, $ϵ$ is a very small number.

Algorithm 1: Benders Decomposition Algorithm

Step 1. Parameters and model initialization Set $t=1$, upper bound $UB=+\infty$, and lower bound $LB=-\infty$ Initialize $M{P}_o$ Step 2. When $UB-LB>ϵ$, execute the following steps; otherwise, go to Step 3 Step 2.1 Set $M{P}_t=M{P}_{t-1}$ and solve for $M{P}_t$ to obtain optimal solutions $\widehat{x}$ and $\widehat{\psi }$ Step 2.2 Set $LB={{\displaystyle \sum }}_{i\in I}{f}_i{\widehat{x}}_i+{{\displaystyle \sum }}_{s\in \Omega }{p}_s{\widehat{\psi }}_s$ Step 2.3 Set $\mathrm{U} \mathrm{B} \prime ={{\displaystyle \sum }}_{i\in I}{f}_i{\widehat{x}}_{i}DS{P}^s\left(\widehat{x}\right)$ Step 2.4 The optimality cut is derived for each demand scenario $s\in \Omega$, execute Solve for subproblem to generate optimal solution ${\mathcal{H}}^{\ast }=\left({\pi }^{\ast },{\varpi }^{\ast },{\gamma }^{\ast },{\tau }^{\ast }\right)$ Set $U{B}^{\prime }=U{B}^{\prime }+p_s\left({{\displaystyle \sum }}_{j\in J}{d}_j^s{\pi }_j^{\ast }+{{\displaystyle \sum }}_{i\in I}{C}_i{\widehat{x}}_i{\varpi }_i^{\ast }+{{\displaystyle \sum }}_{k\in K}{D}_k{\gamma }_k^{\ast }\right)$ Develop the optimality cut: ${\psi }_s\ge Q_s^{\ast }\left(x,{\mathcal{H}}^{\ast }\right)$ Introduce the optimality cut ${\psi }_s\ge Q_s^{\ast }\left(x,{\mathcal{H}}^{\ast }\right)$ to the master problem $M{P}_t$ end for Step 2.5 If $UB>U{B}^{\prime }$, then Set $UB=U{B}^{\prime }$’ Update optimal solution $x^{\ast }=\widehat{x}$ Step 2.6 Set $t=t+1$ Step 3. Optimal solution $x^{\ast }$ and its objective function value $UB$ are returned

4. Numerical Analysis

Shanghai is the center of economy, finance, trade, shipping, and technological innovation in China. The demolition of illegal buildings and the construction of key projects, such as rail transportation projects and cross-river channels, has dramatically increased construction waste in recent years. According to statistics from the Shanghai Municipal Bureau of Ecology and Environment, between 2011 and 2019, applications for the disposal of construction waste grew each year. In 2018, construction waste amounted to 65.26 million tons [40]. Currently, multiple rail transportation projects and cross-river channels are underway in Shanghai, which could generate a large amount of construction waste. The estimate is that, during the next several years, the construction waste generated from rail transportation projects and cross-river channels in Shanghai could amount to 40 million tons or more annually. Because of the limited space for disposal sites and the slow progress in converting waste into resources, the conflict between construction waste generation and disposal has become increasingly severe. In July 2016, construction waste from Shanghai was illegally dumped on Xishan Island of Lake Tai of Suzhou and in Haimen of Nantong. To prevent illegal dumping of dirt and construction waste during transportation, Shanghai Municipality has decided to process and dispose of all construction waste within the city’s borders. In consideration of the new requirements, Shanghai has carried out a series of projects to build construction waste disposal/backfill sites and related wharves, including the Dongtan N1 warehouse area in Nanhui. These facilities are mainly for the disposal of construction waste generated by major municipal projects, such as rail transportation projects and cross-river channels.

In this section, we use a numerical experiment to evaluate the effectiveness of the decision-making model and the algorithm developed in the previous sections. The results of the numerical analysis are expected to provide a reference for shipping logistics network optimization for construction waste recycling. All programs used in the experiment are written using the C language and are run on personal computers (with a 2.4 GHz Intel CPU and 12 GB RAM).

4.1. Experiment Settings

In this section, we describe the detailed settings of the experiment. Currently, Shanghai is constructing the Dongtan N1 warehouse area in Nanhui as a disposal site for construction waste. To prepare the data for the experiment, the waste disposal site at the Dongtan N1 warehouse area in Nanhui is referred to, and some simulation data are generated. The baseline settings are as follows.

(1)

Waste-generating sites and the quantity of construction waste

The construction waste-generating sites in 14 representative administrative districts of Shanghai are used in the model (

Table A1. Construction waste generation in 14 administrative districts of Shanghai (2017–2019).

No.	Administrative District	2017	2018	2019
1	Pudong New Area	2778.90	2552.64	2068.00
2	Minhang District	778.04	714.69	579.00
3	Baoshan District	455.54	418.45	339.00
4	Putuo District	645.01	592.49	480.00
5	Jiading District	290.25	266.62	216.00
6	Yangpu District	215.00	197.50	160.00
7	Xuhui District	201.56	185.15	150.00
8	Hongkou District	0.00	0.00	0.00
9	Chongming District	129.00	118.50	96.00
10	Changning District	120.94	111.09	90.00
11	Jing’an District	107.50	98.75	80.00
12	Huangpu District	161.25	148.12	120.00
13	Songjiang District	43.00	39.50	32.00
14	Qingpu District	43.00	39.50	32.00
Total		5969.00	5483.00	4442.00

). Because the sites are dynamic and continuously change within each district, we cannot pinpoint their locations. To simplify the simulation without loss of generality, we assume that the waste-generating site is at the geographic center of each respective administrative district. This paper produces results based only on data simulation. The actual locations of the waste-generating sites affect the results when solving the problem. If the model and algorithm developed in this paper are to be applied in practice, then the actual locations of the waste-generating sites need to be determined.

Currently, only three years of data (2017–2019) on construction waste disposal are available at the Shanghai Construction Waste Comprehensive Service Monitoring Platform. We assume that the quantities of construction waste from the 14 administrative districts follow the same distribution and are independent of each other. To facilitate further analyses, we use the Monte Carlo simulation method to randomly create 5000 samples based on the statistics of the construction waste quantities generated from the 14 districts during the past three years (42 items). We use histograms to determine the probability distribution of the waste quantities [6]. Finally, we create a variety of random construction waste quantity scenarios for evaluating the algorithm.

Based on data analysis, expert opinions, and relevant literature, if the construction waste from a site is not transported to the disposal/backfill site, then we set a unit penalty of $q_j=\mathrm{50,000,000}$ RMB. We further analyze the impact of the penalty on decision-making via sensitivity analysis.

(2)

Settings for candidate loading wharves

We select six candidate loading wharves for the numerical experiment; they are in the administrative districts of Yangpu, Minhang, and Pudong New Area. The information on the candidate loading wharves used in the numerical experiment is summarized in

Table 2. Information on candidate loading wharves.

Number	Wharf	Number of Berths	Fixed Cost Per Berth (10,000 CNY/Year)	Waste Loading Capacity Per Berth (10,000 Tons/Year)
1	Nenjiang Road Wharf	6	542	177
2	Wharf of Zhongnong Wujing Agricultural Trade Limited Company, located in Shanghai, China	7	549	160
3	Wharf of Changjiang Shipping Co. Ltd. Minnan Shipyard, located in Shanghai, China	7	568	149
4	Guangang Warehouse Wharf	4	536	177
5	Wharf of Shanghai Ocean Petroleum Bureau No. 3 Ocean Geology Exploration Brigade	4	521	179
6	Wharf of Shanghai Pudong Gas Manufacturing Co., Ltd, located in Shanghai, China	5	536	178

.

Distances from the 14 waste-generating sites to the loading wharves were obtained from a map of Baidu, and they are presented in

Table 3. Distances between waste-generating sites and loading wharves (kilometers).

Waste-Generating Sites	Loading Wharves
	1	2	3	4	5	6
1	16.7	32.9	35	21.5	24.2	11.8
2	38.7	17.3	18.4	11.3	47.9	41.3
3	15	54.9	56.9	48	13.1	24.9
4	21.4	28.9	31	22.1	30.7	24
5	36.4	50.8	50.8	43.5	34.6	37.5
6	9.4	43.6	45.3	32.8	26.1	10.8
7	26.5	22.8	22.9	16	36.1	29.1
8	10.2	36.4	38.5	24.7	22.2	12.8
9	83	117.9	122.6	110	84.9	79.7
10	20.4	27.8	29.9	20.9	30	23
11	17.4	28.3	30.3	21.2	27.1	20
12	18.5	31.2	33.2	19.5	28.1	21.1
13	58.5	36.6	37.7	30.6	72.7	71
14	62.6	56.5	57	43.1	66.5	65.2

.

(3)

Settings of destination wharves

Shanghai has completed the construction of the Dongtan N1 warehouse area wharf in Nanhui, which is mainly used to dispose of waste generated from rail transportation projects and cross-river channels. Our numerical experiments set only one destination wharf and assume that it has a waste processing capacity of 15 million tons per year. Distances between loading wharves and the destination wharf are shown in

Table 4. Distances between loading wharves and destination wharf.

Loading Wharf	Distance Between Loading Wharf and Destination Wharf (Kilometers)
1	63.4
2	103.0
3	109.0
4	95.7
5	45.6
6	66.0

.

(4)

Transportation cost settings

Based on real market information on land and shipping transportation in Shanghai, we set the unit transportation cost between waste-generating sites and loading wharves to 10,000 RMB/10,000 ton·kilometer, and the unit shipping transportation cost between loading wharves and the destination wharf is 500 RMB/10,000 ton·kilometer.

4.2. Analysis of Experiment Results

In this section, we present the results of the analysis.

(1)

The results of shipping logistics network optimization under different demand scenarios

We first use the Benders decomposition algorithm to solve the model under different random waste quantity scenarios. In the numerical experiment, we consider eight scenarios (

Table A2. Quantity of construction waste generated in each administrative region across 8 scenarios in the case.

Random Waste Quantity Scenario	100	300	500	800	1000	2000	3000	5000
Pudong New Area	2552.64	2778.90	2552.64	2552.64	2068.00	2068.00	2552.64	2552.64
Minhang District	579.00	579.00	714.69	579.00	579.00	579.00	579.00	778.04
Baoshan District	455.54	339.00	339.00	339.00	339.00	339.00	455.54	418.45
Putuo District	592.49	645.01	480.00	645.01	480.00	645.01	592.49	480.00
Jiading District	216.00	216.00	266.62	266.62	266.62	290.25	290.25	266.62
Yangpu District	160.00	160.00	215.00	160.00	197.50	215.00	215.00	160.00
Xuhui District	201.56	201.56	150.00	185.15	150.00	150.00	150.00	201.56
Hongkou District	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Chongming District	129.00	118.50	96.00	118.50	118.50	118.50	129.00	129.00
Changning District	90.00	120.94	120.94	120.94	111.09	90.00	90.00	90.00
Jing’an District	98.75	98.75	107.50	107.50	98.75	107.50	107.50	98.75
Huangpu District	120.00	120.00	148.12	148.12	148.12	120.00	161.25	161.25
Songjiang District	39.50	32.00	32.00	39.50	43.00	43.00	43.00	43.00
Qingpu District	39.50	43.00	32.00	39.50	39.50	39.50	43.00	39.50
Total	5273.98	5452.66	5254.52	5301.48	4639.08	4804.76	5408.68	5418.82

), each corresponding to a distinct random waste quantity sampled from the histogram distribution derived from the previous analysis. The results of the model under the eight scenarios are summarized in

Table 5. Results of model-solving.

No.	Random Waste Quantity Scenario	Number of Rented Berths	Berth Rental and Reconstruction Cost (CNY 10,000)	Waste Transportation Cost (CNY 10,000)	Unshipped Waste (10,000 tons)	Ratio of Scenarios in Which Waste is not Transported to the Total Number of Scenarios
1	100	8	4172	53,322.35	0	0.00
2	300	8	4208	52,402.26	0	0.00
3	500	9	4796	54,342.87	0	0.00
4	800	9	4700	53,829.65	2	0.13
5	1000	9	4724	53,019.1	0	0.00
6	2000	8	4244	54,153.1	59	0.05
7	3000	8	4244	53,540.34	126	0.07
8	5000	9	4711	53,539.56	30	0.02

. Columns 2–4 show the number of berths rented, the cost of reconstruction, the annual berth rental cost, and the expected waste transportation cost (including the land transportation cost from waste-generating sites to loading wharves and the shipping transportation cost from the loading wharves to the destination wharf). Column 5 presents the waste quantity that is not transported from the construction sites. Column 6 provides the ratio of the scenarios in which waste is not transported to the total number of scenarios.

The results in Table 5 indicate that, under different scenarios with different random waste quantities, the optimal decision provided by the model varies, as evidenced in the number of berths rented, the berth rental and reconstruction cost, and the expected total transportation cost. The results indicate that the randomness of the quantity of construction waste generated in Shanghai has a significant impact on shipping logistics network optimization for waste recycling. Additionally, when the quantity of waste is large, the construction waste may not be shipped to disposal/backfill sites under certain random waste quantity scenarios, mainly because, to obtain an optimal solution for each scenario, the model attempts to balance the penalty costs from unshipped waste and the operating costs from berth rental at the wharves. The computational results also indicate that, when optimizing the shipping logistics network for waste recycling, decision-makers need to thoroughly research waste-generating patterns and determine a precise random distribution for the quantities of waste generated.

(2)

Analysis of the impact of the penalty factor for unshipped waste

The results in

Table 6. Demand penalty factor effect.

Item	Penalty Factor $\mathit{\delta }$
	0.08	0.1	0.3	0.5	1	5	10	20
Expected total cost (CNY 10,000)	57,507.74	57,615.64	57,971.03	58,107.83	58,250.56	58,253.23	58,253.23	58,253.23
Cost of berth rental and reconstruction (CNY 10,000)	3692	3692	4244	4244	4711	4760	4760	4760
Waste transportation cost (CNY 10,000)	53,815.74	53,923.64	53,727.03	53,863.83	53,539.56	53,493.23	53,493.23	53,493.23
Number of berths rented	7	7	8	8	9	9	9	9

indicate that the penalty cost from unshipped construction waste affects the model’s optimization. In this section, we use sensitivity analyses to examine the impact on the model’s solution of the penalty factor for unshipped waste. Based on the baseline penalty assumed in Section 3.2, we define a different penalty factor $\delta$ and calculate the model’s optimal results. For example, when the penalty factor $\delta =3$, the unit penalty when the construction waste is not shipped from the construction site to the disposal/backfill sites is $q_j=\mathrm{50,000,000}\delta =\mathrm{50,000,000}\ast 3=\mathrm{150,000,000}$ RMB. We use 5000 as the random waste quantity to conduct sensitivity analyses, and the results are summarized in Table 6. Figure 2 shows the statistical trend of unshipped waste without a penalty factor.

The results in Table 6 and Figure 2 indicate that the penalty factor has a significant impact on the decisions made by the model. When the penalty factor is small, the operating cost of renting berths is significantly greater than the penalty from unshipped waste. The optimal decision is to rent fewer berths to reduce costs; however, this decision results in a large amount of construction waste not being shipped to disposal/backfill sites. The results further indicate that, given an increase in the penalty factor, the optimal decision is to rent more berths to ship more construction waste to the disposal/backfill sites.

(3)

Impact of wharf processing capacity

We then use sensitivity analyses to examine how the annual processing capacity of the wharves affects the model’s decision. To this end, we define the capacity factor as $\chi$ and use the baseline capacity. For example, suppose Wharf No. 1 has a baseline processing capacity of 1.77 million tons; if the capacity factor is 2, then the new capacity of the berths in the wharf is 1.77χ = 1.77 × 2 = 3.54 million tons. The computational results are summarized in

Table 7. Capacity factor effect.

Item	Capacity Factor $\mathit{\chi }$
	0.1	0.3	0.5	1	1.2	1.5	1.8	2	2.5
Expected total cost (CNY 10,000)	841,731.74	72,194.15	64,578.03	58,250.56	57,194.60	56,350.17	55,797.22	55,761.18	55,248.62
Cost of berth rental and reconstruction cost (CNY 10,000)	17,150	13,994	9021	4711	3728	3176	2611	2133	2084
Waste transportation cost (CNY 10,000)	824,581.74	58,200.15	55,557.03	53,539.56	53,466.60	53,174.17	53,186.22	53,628.18	53,164.62
Unshipped waste (10,000 tons)	778,547.4	947.8	156	30	81.8	0	9.6	260	0
Number of berths rented	32	26	17	9	7	6	5	4	4

, and Figure 3 shows the trend in the total expected cost under different capacity factors.

The results in Table 7 and Figure 3 indicate that the capacity of the candidate wharves has a significant impact on the decisions derived from the model. When the factor is small, say 0.1, because of the small capacity of the berths, a large amount of waste cannot be shipped to the disposal/backfill sites. Thus, 32 berths need to be rented, significantly increasing the total expected cost. Given the increase in capacity factor $\chi$, the expected cost decreases significantly, primarily as a result of the reduction in rental and reconstruction costs. Meanwhile, when the capacity factor is large, some waste may still exist that cannot be shipped to disposal/backfill sites, mainly because the extra costs to rent more berths exceed the penalty for unshipped waste, given the processing capacity of the berths. When the capacity factor $\chi$ is 2.5, the expected cost is the lowest, and the number of rented berths is the smallest.

Thus, based on the above research, we put forward the following management strategic suggestions:

(1)

When optimizing the shipping logistics network for waste recycling, decision-makers need to thoroughly research waste-generating patterns and determine a precise random distribution for the quantities of waste generated, especially based on the construction plans for major subways, river crossings, channels, and other projects. The big data analysis and prediction of construction waste generation in the future can then be carried out based on this information.

(2)

To prevent a large amount of construction waste from failing to be shipped to disposal/backfill sites, more berths should be rented to provide the necessary capacity for receiving construction waste prior to its final transport to disposal/backfill sites. However, to avoid the waste of berth capacity, the berth scheme should be dynamically adjusted, such as by closing some berths and opening new berths, according to the changes in the volume of construction waste produced and the distribution of disposal/backfill sites.

(3)

Improving the capacity of source terminal wharves has a significant effect on reducing logistics costs. Therefore, the capacity of source terminal wharves should be improved through informatization, intelligence, improving loading and unloading operations, streamlining processes, and other measures.

5. Conclusions

In many cities, the conflict between the quantity of construction waste generated and the waste processing capacity within the city boundary is becoming increasingly severe. The traditional waste recycling method that relies on land transportation has many drawbacks, such as low efficiency and ever-increasing negative impacts on the environment and traffic. This paper proposes a shipping logistics network optimization method based on a relay mode that integrates land transportation and shipping transportation. Given the influences of social–economic development, the quantity of construction waste generated within many cities varies in a stochastic way. Decision makers are unable to know exactly how much waste a construction site could generate in the future but still need to determine the construction waste recycling and transportation network that could minimize the expected capital investment and operating costs. In this paper, a two-stage stochastic integer programming model is proposed for the shipping logistics network optimization for construction waste recycling. To effectively solve the problem, an accurate algorithm based on Benders decomposition is proposed. These ways will strengthen the model’s adaptability to complex real-world conditions and provide more robust theoretical support and practical guidance for building efficient and sustainable construction waste recycling logistics networks.

Based on a practical case study in Shanghai, numerical experiments and sensitivity analyses demonstrate that the proposed two-stage stochastic integer programming model effectively optimizes the shipping logistics network for construction waste. Solving the model under various stochastic demand scenarios significantly reduces transportation costs and the volume of untreated waste. Sensitivity analysis shows that the penalty factor for untransported waste and the capacity factor are crucial factors influencing network optimization decisions. The results show that as the penalty factor for unshipped waste increases, the number of rented berths rises, leading to a significant reduction in unshipped waste, which eventually approaches zero. Similarly, as capacity factor increases, both the number of rented berths and the total costs decrease, with unshipped waste also approaching zero. These findings offer valuable theoretical support for developing a more efficient and sustainable logistics network for construction waste recycling.

The primary focus of this study was on the preliminary optimization of the transportation network’s overall structure and strategies, emphasizing network design under stochastic demand. This study focused on preliminarily optimizing the transportation network structure under stochastic demand. Future work will refine the model by integrating waste classification, time constraints for operation efficiency, and dynamic adjustments to waste generation point locations, enhancing the model’s practicality and adaptability. This will involve incorporating waste classification to address the varying handling and transportation requirements of different waste types, thereby improving resource recovery efficiency. Additionally, time constraints and minimization objectives will be integrated to optimize logistics operations, ensuring they are completed efficiently within specified timeframes. The model will also account for the dynamic changes in the locations of waste generation points, enabling real-time adjustments in transportation routes and resource allocation.