Multi-Objective Optimization of Municipal Solid Waste Collection Based on Adaptive Large Neighborhood Search

Abstract

To address the dual challenges of reducing carbon emissions and operating costs in municipal solid waste collection, this paper investigates the vehicle routing problem (VRP) for municipal solid waste collection and transportation, which significantly affects the transportation costs and efficiency. An analysis has shown substantial disparities in workload distribution across different routes, highlighting the need to consider both the workload balance and cost efficiency. Therefore, considering the factors of workload disparities and costs, a multi-objective VRP model is formulated, and a multi-objective adaptive large neighborhood search (MOALNS) algorithm based on balance remove and balance insert heuristics is proposed to solve the above problem. The proposed algorithm is compared with two classical multi-objective algorithms, and the results show its competitiveness. The designed model can effectively reflect the conflict between minimizing costs and balancing disparities in employee workloads.

1. Introduction

1.1. Background

Municipal waste generation has increased rapidly over the past few decades due to global industrial transformation. The increase in municipal solid waste (MSW) has placed significant pressure on management departments. A recent report indicates that cities worldwide generate approximately 1.47 billion tons of waste annually [1]. The management of MSW has become a challenge in both developing and developed countries [2]. MSW management includes the processes of waste sorting, collection and disposal. The cost of the waste collection process accounts for about 60–80% of the total waste treatment cost [3]. Designing efficient collection routes is essential to minimize unnecessary expenses.

Rising living standards and diverse consumption patterns have increased the variety of waste types in urban areas, complicating MSW management further. Inadequate waste collection can decrease resident satisfaction and may lead to various environmental and public health issues. The accumulation of waste can foster germ growth and increase the risk of disease spread. Moreover, waste buildup can detract from the aesthetic appeal of a city and lower the quality of life for its inhabitants. The issue of how to promptly address the accumulation of rubbish in cities while simultaneously reducing the costs associated with the transportation process has become a significant concern in MSW management. Given the complexity of urban transportation networks and the wide distribution of waste collection points, municipalities in practice usually establish designated transportation and collection routes for each vehicle. While this approach is simple and clearly delineated, it can lead to several problems, such as fixed-route collection and transportation, which make it difficult to manage the waste capacity. This could result in either the underutilization or overutilization of the capacity. When bins reach their capacity, they cannot be effectively emptied, which may lead to secondary environmental pollution. The waste of human resources and materials caused by this method should not be underestimated. Therefore, it is essential to develop a more efficient and flexible waste collection and transportation system.

1.2. Literature Review

To address the challenges posed by increasing municipal waste and the inefficiencies in the current waste collection and transportation systems, researchers have explored various strategies and technologies to optimize the process, including integrating intelligent information technologies and using algorithms to plan vehicle routes, etc. The introduction of these technologies aims to enhance the efficiency and responsiveness of waste management systems.

As the concept of smart cities continues to evolve, many countries are beginning to apply the Internet of Things [4] (IoT) to city management. Such applications enable the real-time monitoring of road traffic, environmental conditions and other critical data. The integration of smart cities with IoT technologies allows for the real-time monitoring of waste types across various locations, thereby facilitating the development of targeted cleanup strategies. These technologies are currently being utilized in several countries [5]. Digiesi et al. [6] developed a network technology-based vehicle collection system. The system collects and manages key data, including urban road information, available vehicles and waste fill levels. By analyzing and processing these data, the system develops optimized collection strategies. It has been implemented in a medium-sized city in Italy, resulting in significant cost reductions. Anagnostopoulos et al. [7] proposed an IoT-enabled system architecture and evaluated it using real-life data from St. Petersburg, Russia. Their model increased the vehicle load factors and reduced the number of vehicles required. Folianto et al. [8] designed an intelligent waste bin monitoring device that tracks the waste levels within bins, providing data to inform the development of targeted collection strategies. The intelligent MSW collection strategy differs from traditional waste collection methods in that it assesses the volume of waste at each site and then strategically plans the transport routes for vehicles. This approach improves the load utilization of transport vehicles and reduces the distance traveled.

Beliën et al. [9] conducted a review of studies on MSW collection, suggesting that the waste collection problem can be classified as a VRP. This problem is recognized as a classical NP-hard problem and is typically addressed using evolutionary algorithms. Ibrahim et al. [10] tackled the VRP with time windows by improving the crossover and mutation stages of the genetic algorithm. Sari et al. [11] employed the nearest neighbor method to generate an initial solution, using the tabu search (TS) algorithm for further optimization. Serkan [12] proposed a hybrid algorithm that combines a firefly algorithm based on local search with particle swarm optimization (PSO), addressing the issues of premature convergence and slow global optimization in the PSO algorithm.

In recent years, with the growing emphasis on environmental sustainability, research on greenhouse gas emissions, such as carbon dioxide, during route planning has increased. From an environmental perspective, the route with the shortest travel time is not necessarily the most fuel-efficient. The study by Masikos et al. [13] demonstrates that, in practical scenarios, the route with the lowest fuel consumption or emissions does not always correspond to the shortest distance. Testing conducted in Northern Virginia, USA, further reveals [14] that, on arterial roads, even with a 30% increase in the route length, the fuel consumption decreased by 18–23%, nitrogen oxide emissions dropped by 4% and carbon dioxide emissions were reduced by approximately 20% due to lower driving speeds.

While earlier studies have primarily focused on minimizing the total transportation costs, recent research has broadened the scope of the parameters considered, leading to more diverse optimization objectives. For example, Jammeli et al. [15] aimed to minimize both the transportation costs and the number of municipal waste bins. Zhang et al. [16] introduced the concept of resident satisfaction along with a soft time window. If a vehicle arrives at a collection site outside the time window, it results in varying degrees of decreased resident satisfaction. Valizadeh et al. [17] developed a bi-objective model that focuses on minimizing costs and reducing pollutant emissions, utilizing the NSGA-II [18] and multi-objective particle swarm optimization (MOPSO) algorithms to solve the problem.

Multi-objective optimization problems have found extensive application in various fields [19,20,21,22,23], and algorithms for the solution of such problems have been continuously developed. Multi-objective optimization algorithms can generally be divided into three categories: decomposition-based, such as MOEA/D [24]; dominance-based, such as NSGA-II; and indicator-based, such as HypE [25]. These algorithms adopt strategies such as crossover and mutation to generate offspring and complete the evolutionary process to ensure their generality. However, this strategy exhibits poor convergence in the context of the VRP. In the VRP, employing local search methods can accelerate the convergence speed and accuracy. These algorithms typically use different local search operators to perturb solutions to achieve the discovery of superior solutions. The issue of how to formulate the operators and select the perturbation nodes is critical to such algorithms [26,27,28].

Sivaramkumar et al. [29], through an analysis of the classic Solomon dataset in the context of the vehicle routing problem with time windows (VRPTW), found that minimizing the total driving distance of vehicles often leads to significant variations in the distances traveled and working hours across different routes. However, these workload variations can be reduced by balancing the distance and total time on different routes. In practical situations, such differences in employee workloads can have a significant impact on performance. A study by Inegbedion et al. [30] showed that employees’ perceptions of their workload affect job satisfaction. Employees are often dissatisfied when they compare their workload with that of their colleagues in similar positions at higher income levels and feel that their workload is greater. This negative perception may reduce their ability to do their job and lead to a decrease in efficiency [31]. This, in turn, affects the individual employee and the organization. Therefore, it is essential to address workload disparities in the workplace.

In summary, although some research has been conducted on the problem of MSW collection and transportation, most studies only focus on optimizing the total costs or total routes, neglecting the workload differences among different routes in practical situations. This can lead to significant disparities in workload across different routes and cause employee dissatisfaction. The algorithms used tend to have slow convergence speeds and low convergence accuracy. Therefore, this paper considers both the total costs and employee workload variations, establishes a multi-objective model and proposes an algorithm to solve the model.

1.3. Contributions

Although many studies have modeled and solved the MSW collection and transportation process as a VRP [32,33,34,35,36,37,38], it is still a challenge to design effective models and solve them with appropriate algorithms for two major reasons. First, most models use the total cost as the single objective function, while overlooking differences between routes. Second, the algorithms used in most studies to solve the VRP have low convergence accuracy and leave room for further optimization. To address these challenges, we attempt to establish a multi-objective model and design an algorithm specifically for this model. In the context of smart cities, this paper models and solves the MSW collection and transportation process. The main contributions are as follows.

• A multi-objective optimization model was developed to balance the costs and workload variations among employees involved in the vehicle collection process. Previous VRP models usually aim to minimize the cost and ignore differences in employee workload. However, when the cost is optimal, there may be significant differences between different routes. Neglecting these differences can lead to psychological imbalance among employees. Balancing workload differences can improve employee satisfaction and overall work efficiency.
• A multi-objective adaptive large neighborhood search algorithm based on the ALNS algorithm was designed. The MOALNS algorithm tackles the multi-objective problem in the proposed model by integrating balance removal and balance insertion heuristic strategies.
• To evaluate the effectiveness of the proposed MOALNS algorithm, test cases based on realistic models were designed and compared with four multi-objective algorithms. The results indicate that the MOALNS algorithm exhibits a distinct advantage within this model.

1.4. Paper Organization

The rest of this article is structured as follows. Section 2 introduces the model construction and elaborates on the objective function and constraint conditions in detail. Section 3 describes the design process of the MOALNS algorithm. Section 4 compares MOALNS with other multi-objective algorithms using test datasets and a real-world case. The final section summarizes this study.

2. Modeling and Analysis

2.1. Problem Description

The MSW collection problem involves planning vehicle routes based on the weights of various types of waste at each collection site. Vehicles then follow designated routes to collect the corresponding types of waste at each site. This process can be modeled as a classical multiple-depot capacitated vehicle routing problem (MDCVRP). In the MDCVRP, multiple depots dispatch vehicles to serve multiple collection sites, with each site having a specific demand for waste collection. The problem assumptions are as follows:

• During the collection process, vehicles visit each collection site only once;
• After completing the collection, vehicles must return to their originating depot;
• Each collection route is serviced by only one vehicle, and the capacity of the vehicles is limited.

2.2. Notations

Based on the described problem, the model is established using the following notations, as shown in

Table 1. Description of symbols.

Notation	Explanation
$M=\{1,2,\cdots ,m\}$	Set of depot points
$U=\{m+1,m+2,\cdots ,m+n\}$	Set of waste sites
$V=\{1,2,\cdots ,K\}$	Set of vehicles
$m_k$	The depot point to which vehicle k belongs, $m_k\in M,k\in V$
$N=m_k\cup U$	The set of points accessible to vehicle k, $k\in V$
$d_{ij}$	Distance between node i and node j ($i,j\in M\cup U$)
$q_{ij}$	Load transported from node i to node j ($i,j\in M\cup U$)
${\rho }_0$	Fuel consumption rate when the vehicle is unloaded (L/km)
${\rho }^{*}$	Fuel consumption rate when the vehicle is fully loaded (L/km)
Q	Maximum load capacity of a vehicle (t)
$F_{ij}$	Fuel consumption between node i and node j (L)
$\epsilon$	Carbon emissions factor
$p_{carbon}$	Carbon tax price (CNY/kg)
$p_{fixed}$	Fixed cost per vehicle (CNY)
$p_f$	Fuel price per unit (CNY/L)
$C_{fixed}$	Total fixed cost for all vehicles (CNY)
$C_{fuel}$	Total fuel cost (CNY)
$C_{carbon}$	Total carbon tax cost (CNY)
$\mathbf{D}=[D_1,D_2,\cdots ,D_K]$	Distance vector; $D_i$ represents the route length of the i-th vehicle (km)
$\mathbf{W}=[W_1,W_2,\cdots ,W_K]$	Load vector; $W_i$ represents the total load of the i-th vehicle (t)
$x_{ijk}$	$\left\{\begin{array}{cc}1,\hfill & \mathrm{Vehicle}k\mathrm{picks}\mathrm{up}\mathrm{waste}\mathrm{from}\mathrm{node}i\mathrm{to}\mathrm{node}j\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
$y_{ij}$	$\left\{\begin{array}{cc}1,\hfill & \mathrm{Waste}\mathrm{collection}\mathrm{node}i\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{depot}j\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

.

2.3. Mathematical Model

Most studies focus on finding the solutions with the lowest cost, but this often overlooks the differences in routes caused by factors such as the distance and load, which may lead to imbalances in employee workloads. Previous studies [29] have shown that there is a conflict between minimizing the total costs and balancing employee workload differences, and it is necessary to construct objective functions that consider both goals simultaneously.

2.3.1. Objective Functions

The objective function consists of two parts: vehicle costs and workload variations. Vehicle costs are composed of fixed costs, variable costs and carbon emission costs.

• Fixed Costs
  The fixed vehicle cost includes expenses related to vehicle operation, such as insurance, taxes and employee wages. The total fixed cost for all vehicles in the model can be calculated using the following formula:

  $$C_{fixed}=\sum _{i\in M}\sum _{j\in U}\sum _{k\in V}{x}_{ijk}{p}_{fixed}$$

  (1)
• Variable Costs
  Variable costs include expenses related to fuel consumption and carbon emissions incurred during transportation between different nodes. The fuel consumption from node i to node j can be calculated using the fuel consumption model proposed by Xiao et al. [39], as follows:

  $$F_{ij}=({\rho }_0+{\displaystyle \frac{{\rho }^{*}-{\rho }_0}{Q}}·q_{ij})·d_{ij}$$

  (2)
  The total fuel consumption cost can be calculated by

  $$C_{fuel}=p_f\sum _{i\in N}\sum _{j\in N}\sum _{k\in V}{x}_{ijk}{F}_{ij}$$

  (3)
• Carbon Emission Costs
  With global climate change receiving increasing attention, carbon emissions have become a major concern. The most widely adopted method for the management of carbon emissions is the implementation of a carbon tax. The calculation for the carbon tax is outlined below:

  $$C_{carbon}=\epsilon p_{\mathrm{carbon}}\sum _{i\in N}\sum _{j\in N}\sum _{k\in V}{x}_{ijk}{F}_{ij}$$

  (4)
• Workload Variations
  In addition to optimizing the costs, it is essential to balance the workload across different routes. Balancing the workload among vehicles helps to ensure that employees feel that they are treated equitably. Workload disparities are measured using the variance in driving times and the variance in vehicle loads.

2.3.2. Model Formulated

Based on the description of the costs and workload, a mathematical model is proposed. Equation (5) aims to minimize both the total cost and workload variations. In $F_2$, $\alpha$ and $\beta$ are coefficients that control the different impacts of the distance and load. In this paper, $\alpha$ is set to 0.3 and $\beta$ is set to 0.7. Constraints (6) and (7) ensure that each waste collection site is visited by only one vehicle and that each route is traversed only once. Constraint (8) addresses subtour elimination, while Constraint (9) ensures that the number of vehicles entering a station equals the number leaving it. Constraint (10) stipulates that vehicles must return to their starting site after completing the collection. Finally, Constraint (11) ensures that the vehicle does not exceed its maximum load capacity during the collection process.

$$\left\{\begin{array}{c}min{F}_1={\displaystyle \sum _{i\in M}}{\displaystyle \sum _{j\in U}}{\displaystyle \sum _{k\in V}}{x}_{ijk}{p}_{fixed}+(p_f+\epsilon p_{carbon}){\displaystyle \sum _{i\in N}}{\displaystyle \sum _{j\in N}}{\displaystyle \sum _{k\in V}}{x}_{ijk}{F}_{ij}\hfill \\ min{F}_2=\alpha ·var\left(\mathbf{D}\right)+\beta ·var\left(\mathbf{W}\right)\hfill \end{array}\right.$$

(5)

Subject to

$$\sum _{i\in N}\sum _{k\in V}{x}_{ijk}=1,\forall j\in N$$

(6)

$$\sum _{j\in N}\sum _{k\in V}{x}_{ijk}=1,\forall i\in N$$

(7)

$$\sum _{i\in S}\sum _{j\in S}{x}_{ijk}\le \left|S\right|-1,\forall k\in V,S\subset N=\{m_k,m+1,\cdots ,m+n\},2\le \left|S\right|\le n$$

(8)

$$\sum _{j\in N}{x}_{ijk}-\sum _{j\in N}{x}_{jik}=0,\forall i\in N,\forall k\in V$$

(9)

$$\sum _{j\in N}{x}_{m_{k}jk}=\sum _{i\in N}{x}_{m_{k}ik},\forall k\in V$$

(10)

$$\sum _{i\in N}\sum _{j\in N}{x}_{ijk}{q}_{ij}\le Q,\forall k\in V$$

(11)

3. Proposed Algorithm

3.1. Basic ALNS

The VRP is a classic NP-hard problem, which typically requires a significant amount of time to solve using exact algorithms. As a result, heuristic algorithms have become a mainstream approach to address the VRP. Traditional heuristic algorithms, such as PSO and others, are widely used to solve the VRP due to their general applicability, but they often suffer from slow convergence rates and reduced accuracy. In contrast, algorithms that incorporate neighborhood operations and local search strategies have shown improved performance in terms of both the convergence speed and accuracy when solving the VRP.

The ALNS algorithm, proposed by Ropke et al. [40], explores the neighborhood of solutions by applying various heuristic destruction and repair operations. Each operation i in segment j has an associated weight ${\omega }_{i,j}$. The ALNS algorithm initially selects a set of removal and insertion heuristics using a roulette selection method. It is important to note that the probabilities of choosing removal and insertion heuristics are independent of each other during this selection process. Once selected, the chosen removal and insertion heuristics are applied to the solution to generate a new candidate solution. The acceptance of the new candidate solution is determined according to the acceptance criteria of simulated annealing (SA) [41], as outlined in Algorithm 1.

Algorithm 1 SAAcceptanceCriteria

Input: Current solution S, new solution $S^{\prime }$, temperature T Output: Accepted solution  Compute $\Delta E=f\left(S^{\prime }\right)-f\left(S\right)$  if $\Delta E<0$ then   Accept $S^{\prime }$ as the new solution  else   Generate a random number $r\in [0,1]$   if $r<exp\left(-{\displaystyle \frac{\Delta E}{T}}\right)$ then    Accept $S^{\prime }$ as the new solution   else    Reject $S^{\prime }$ and keep S as the current solution   end if  end if

Here, $f\left(S^{\prime }\right)$ represents the fitness of the candidate solution, and $f\left(S\right)$ represents the fitness of the current solution. In each iteration, the selected removal and insertion heuristics will receive corresponding scores, denoted as ${\sigma }_i$, based on the quality of the solution generated in that iteration. The evaluation criteria and corresponding scores are shown in

Table 2. Score adjustment method for ALNS.

Parameter	Description
${\sigma }_1$	The selected remove–insert operation resulted in a new global best solution.
${\sigma }_2$	The selected remove–insert operation resulted in an unaccepted solution, and the cost of this solution is better than the current solution.
${\sigma }_3$	The selected remove–insert operation resulted in a solution that has not been accepted before. The cost of this new solution is worse than the current one, but it is still accepted.

. The obtained scores are then used to update the corresponding weights ${\omega }_{i,j}$.

In the iterative process of the ALNS algorithm, each iteration is divided into several segments. Ropke et al. set each segment to consist of 100 iterations. At the beginning of each segment, the scores of all heuristic methods are initialized to zero. At the end of each segment, the weights of the heuristic methods are adjusted according to Equation (12) to achieve the adaptive selection of the destruction and repair methods.

$${\omega }_{i,j+1}={\omega }_{i,j}\left(1-r\right)+r{\displaystyle \frac{{\pi }_i}{{\theta }_i}}$$

(12)

In the equation, ${\pi }_i$ represents the score obtained by heuristic method i during the previous segment, ${\theta }_i$ represents the number of times that heuristic method i was used during the previous segment, and r is the reaction factor that controls the speed at which the weights respond to changes in the effectiveness of the heuristic methods. The pseudocode of the ALNS algorithm is shown in Algorithm 2.

Algorithm 2 Basic ALNS

Input:  Maximum number of iterations $Max\_iter$, initial solution $S_0$, set of removal heuristics ${\sum }^{-}$, set of insertion heuristics ${\sum }^{+}$, cooling rate $\alpha$ Output:  Optimized solution S    1: $iter\leftarrow 0$    2: $S\leftarrow S_0$    3: while $iter<Max\_iter$ do    4:   if $iter\%100==0$ then    5:    Update weights by Equation (12).    6:    Set all elements of scores to 0.    7:   end if    8:   Randomly select both a removal heuristic ${\sigma }^{-}\in {\sum }^{-}$ and an insertion heuristic ${\sigma }^{+}\in {\sum }^{+}$.    9:   $S^{\prime }\leftarrow {\sigma }^{+}\left({\sigma }^{-}\left(S\right)\right)$  10:   $S\leftarrow SAAcceptanceCriteria(S,S^{\prime },T)$  11:   $T\leftarrow T\times \alpha$  12:   Update scores by Table 2.  13:   $iter\leftarrow iter+1$  14: end while

3.2. MOALNS

The basic ALNS algorithm mainly focuses on minimizing the route length or cost. When dealing with multi-objective optimization problems, previous studies have often combined the destruction and repair strategies of ALNS with the non-dominated sorting genetic algorithm II (NSGA-II). This approach allows the NSGA-II to remove and insert sites in routes, enhancing the algorithm’s convergence capabilities. However, this method still faces problems related to low convergence precision, and its runtime is generally longer compared to the ALNS algorithm. The advantages of the ALNS algorithm can be utilized more effectively by incorporating removal and insertion heuristics to achieve multi-objective optimization. For the problem addressed in this model, a balance removal and balance insertion heuristic are proposed to improve the algorithm’s ability to manage workload variations. The distances are the primary factor contributing to workload imbalances in MSW collection, and balance removal and insertion heuristics aim to minimize these imbalances by reducing the distance disparities between routes. In addition, the ALNS score adjustment method and acceptance criteria have been modified to better fit the proposed MSW model.

3.2.1. Balance Remove

Let $g(i,k)$ denote the distance reduction after removing collection site k from route i. The balance remove first calculates the total distance traveled by each route and then selects the route i with the greatest distance. Next, it finds the collection site k in route i with the maximum $g(i,k)$ and removes it from the route. The balance remove is similar to the worst removal strategy in the basic ALNS algorithm, focusing on the collection point with the maximum $g(i,k)$ across all routes.

3.2.2. Balance Insert

The core idea of the balance insertion method, corresponding to the balance removal method, is to make the distances of all routes as similar as possible. First, the distances of all routes in the candidate solution are calculated, and the shortest route is selected. Next, a random unassigned collection site is chosen and inserted at the position that results in the smallest increase in distance. If insertion is not possible due to workload constraints, a new empty route is created to accommodate the collection site. The balance insertion method not only helps to minimize workload variations but also contributes to cost optimization through its greedy strategy.

3.2.3. Score Adjustment Method

The proposed MOALNS retains the Pareto optimal set throughout the iteration. After each remove and insert heuristic, the candidate solutions are compared with those in the Pareto optimal set, rather than with the current solution or the global optimal solution. The scoring method is detailed in

Table 3. Score adjustment method for ALNS.

Parameter	Description
${\sigma }_1$	The chosen remove–insert operation resulted in a solution that dominates the solutions in the Pareto optimal set.
${\sigma }_2$	The chosen remove–insert operation resulted in a non-dominated solution.
${\sigma }_3$	The chosen remove–insert operation resulted in a previously unaccepted solution, but this solution is dominated by solutions in the Pareto optimal set.

.

3.2.4. Acceptance Criteria

The ALNS algorithm using simulated annealing criteria may accept solutions that are significantly worse than the current one. However, tests conducted by Santini et al. [42] on various ALNS acceptance criteria indicate that the linear Record-to-Record Travel (RRT) method [43] performs better than the simulated annealing approach. When a candidate solution is worse than the current solution, the RRT criterion accepts it if the absolute difference between the current and candidate solutions is below a threshold value. This threshold T is updated in each iteration based on a step size $\gamma$, but it will not drop below the end threshold $T_{end}$.

$$T=max\{T_{end},T-\gamma \}$$

(13)

In the MOALNS algorithm, different step sizes are defined for the two objectives to better adjust the search space for each. A candidate solution is accepted if it is either non-dominated or dominates the current solution. If the candidate solution is dominated but the absolute differences between the two objectives are both smaller than their respective thresholds, the solution will still be accepted. Otherwise, the candidate solution is rejected. The pseudocode of the MOALNS algorithm is shown in Algorithm 3.

Algorithm 3 MOALNS

Input:  Maximum number of iterations $Max\_iter$, initial solution $S_0$, set of removal heuristics ${\sum }^{-}$, set of insertion heuristics ${\sum }^{+}$, temperature T, step size $\gamma$ Output:  Optimized solution S    1: $iter\leftarrow 0$    2: $S\leftarrow S_0$    3: $T\leftarrow f\left(S_0\right)$    4: $PF\leftarrow \varnothing$    5: while $iter<Max\_iter$ do    6:    if $iter\%100==0$ then    7:     Update weights by Equation (12).    8:     Set all elements of scores to 0.    9:    end if  10:    Randomly select both a removal heuristic ${\sigma }^{-}\in {\sum }^{-}$ and an insertion heuristic ${\sigma }^{+}\in {\sum }^{+}$.  11:    $S^{\prime }\leftarrow {\sigma }^{+}\left({\sigma }^{-}\left(S\right)\right)$  12:    $S\leftarrow RRTcceptanceCriteria(S,S^{\prime },T)$  13:    $T\leftarrow max\{T_{end},T\times \gamma \}$  14:    Update scores by Table 3.  15:    $PF\leftarrow PF\cup S$  16:    Remove non-dominated solutions from $PF$.  17:    $iter\leftarrow iter+1$  18: end while

4. Computational Experiments

This section compares the proposed MOALNS with four multi-objective algorithms, NSGA-II, MOEA/D, UNSGA-III [44] and MultiGPO [45], on both the test datasets and real-world cases. NSGA-II, MOEA/D and UNSGA-III were implemented using pymoo [46]. The Wilcoxon rank-sum test was used to compare the results of the algorithms under comparison at a significance level of 0.05 over 10 runs. The symbols “+”, “−”, and “≈” indicate that the compared algorithm outperforms MOALNS, the compared algorithm underperforms MOALNS and there is no significant difference in performance between the two algorithms, respectively. The HV (hypervolume) indicator was used to measure the performance of the algorithms.

4.1. Performance Indicator

This paper uses the HV to evaluate the algorithms’ performance. The HV is a comprehensive metric used to assess both the convergence and diversity of algorithms. A larger HV value indicates that the algorithm’s approximate front occupies a larger volume in the objective space, indicating the better overall performance of the algorithm. The advantage of using the HV metric is that there is no need to know the Pareto front (PF); only a reference point is needed for evaluation.

The formula for the calculation of the HV is as follows:

$$HV=\delta \left(\bigcup _{i=1}^{\left|S\right|}{v}_i\right)$$

(14)

where $\delta$ is the Lebesgue measure [47], $\left|S\right|$ is the number of non-dominated solutions and $v_i$ represents the hypervolume formed by the reference point and the i-th non-dominated solution.

4.2. Sensitivity Analysis of Scoring Strategies

The design of the scoring strategy in MOALNS has a significant impact on the algorithm’s performance. To develop a reasonable scoring strategy, four different scoring strategies were tested on five problem sets of varying sizes, and the HV indicator was calculated. In the “Parameter” row, four different parameter settings are displayed: [25, 5, 1], [25, 5, 0], [25, 50, 1] and [25, 10, 1]. The results are presented in

Table 4. Score adjustment method for ALNS.

Instance	Parameter
	[25, 5, 1]	[25, 5, 0]	[25, 50, 1]	[25, 10, 1]
P-n23-k8	0.754 (0.139)	0.694 (0.136)	0.672 (0.175)	0.771 (0.084)
P-n101-k4	0.743 (0.187)	0.681 (0.166)	0.663 (0.229)	0.702 (0.200)
X-n237-k14	0.626 (0.082)	0.548 (0.101)	0.513 (0.075)	0.543 (0.107)
X-n367-k17	0.763 (0.171)	0.775 (0.155)	0.573 (0.154)	0.630 (0.089)
X-n401-k29	0.754 (0.078)	0.678 (0.187)	0.701 (0.152)	0.715 (0.132)

, showing the mean values and standard deviations. The bolded values indicate the smallest mean results. These settings represent the values of [${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$].

It can be observed that, when the score is set to [25, 5, 1], the HV indicator performs excellently and yields consistent results across problems of different scales.

The population size of NSGA-II, MOEA/D, UNSGA-III and MultiGPO was set to 100 and the genetic operators used were simulated binary crossover [48] and polynomial mutation [49]. The distribution indexes ${\eta }_c$, ${\eta }_m$ for crossover and mutation were both set to 20, and the crossover probability $p_c$ was set to 0.9. The scores of MOALNS were set to [25, 5, 1], and the step size $\gamma$ was set to 0.02. All algorithms’ maximum number of iterations was set to 2000.

4.3. Complexity Analysis

Due to MOALNS dynamically adjusting the usage probabilities of various heuristic operators through weight control during its evolution, conducting a theoretical analysis of its complexity becomes challenging. Therefore, we opt for an experimental approach to demonstrate and evaluate the complexity characteristics of MOALNS. The experiments were carried out on the P-n50-k10 dataset, from which it can be observed in Figure 1 that the complexity of MOALNS is not high. This is primarily because, when decoding a single path and calculating its cost, the complexity of other algorithms is $O\left(n\right)$, while MOALNS only needs to recalculate the altered nodes of the path, resulting in complexity of $O\left(1\right)$.

4.4. Experiments on the Test Dataset

Since most problems and algorithms focus on the single-depot CVRP, multi-depot public datasets are relatively scarce and have limited impact. Therefore, this paper modifies classic datasets from the CVRP [50,51] to increase the number of depots, creating a multi-depot dataset.

Table 5. Experimental results from the five algorithms on test instances.

Instance	NSGA-II	MOEA/D	UNSGA-III	Multi-GPO	MOALNS	Best Solution	BKS
P-n23-k8	0.905 (0.099) +	0.575 (0.221) ≈	0.862 (0.108) +	0.821 (0.083) +	0.417 (0.165)	346.5	529
P-n40-k5	0.792 (0.120) −	0.569 (0.130) −	0.743 (0.064) −	0.768 (0.038) ≈	0.827 (0.048)	403.7	458
P-n45-k5	0.633 (0.138) −	0.458 (0.127) −	0.787 (0.066) −	0.765 (0.064) −	0.922 (0.038)	482.4	510
P-n50-k7	0.651 (0.084) −	0.488 (0.105) −	0.789 (0.090) −	0.775 (0.053) −	0.902 (0.042)	515.1	554
P-n50-k8	0.723 (0.093) −	0.600 (0.060) −	0.762 (0.067) −	0.858 (0.058) ≈	0.897 (0.062)	613.2	631
P-n50-k10	0.767 (0.054) −	0.589 (0.072) −	0.750 (0.096) −	0.828 (0.055) −	0.902 (0.083)	659.1	696
P-n51-k10	0.743 (0.059) −	0.508 (0.049) −	0.727 (0.063) −	0.732 (0.074) −	0.825 (0.068)	682.8	741
P-n55-k7	0.734 (0.072) −	0.605 (0.083) −	0.659 (0.163) −	0.748 (0.060) −	0.938 (0.029)	537.2	568
P-n55-k15	0.752 (0.057) −	0.454 (0.102) −	0.783 (0.153) ≈	0.781 (0.140) ≈	0.798 (0.132)	831.8	989
P-n60-k10	0.672 (0.088) −	0.437 (0.097) −	0.807 (0.077) ≈	0.838 (0.076) ≈	0.829 (0.089)	705.5	744
P-n65-k10	0.640 (0.130) −	0.495 (0.123) −	0.651 (0.090) −	0.689 (0.130) −	0.902 (0.042)	750.9	792
P-n70-k10	0.656 (0.072) −	0.556 (0.103) −	0.705 (0.077) −	0.752 (0.116) −	0.897 (0.075)	797.4	827
P-n76-k5	0.729 (0.082) −	0.642 (0.074) −	0.740 (0.096) −	0.827 (0.103) −	0.947 (0.038)	590.9	627
P-n101-k4	0.420 (0.097) −	0.259 (0.114) −	0.347 (0.083) −	0.518 (0.125) −	0.956 (0.023)	616.3	681
A-n32-k5	0.851 (0.044) ≈	0.623 (0.082) −	0.844 (0.051) ≈	0.867 (0.059) ≈	0.842 (0.062)	648.5	784
A-n39-k6	0.758 (0.112) −	0.514 (0.124) −	0.824 (0.093) −	0.841 (0.102) −	0.916 (0.027)	787.3	831
A-n44-k6	0.775 (0.120) −	0.452 (0.165) −	0.801 (0.081) −	0.814 (0.092) −	0.926 (0.049)	858.8	937
A-n46-k7	0.724 (0.114) −	0.490 (0.098) −	0.784 (0.103) −	0.733 (0.129) −	0.858 (0.075)	874.7	914
A-n48-k7	0.745 (0.069) −	0.479 (0.119) −	0.755 (0.074) −	0.774 (0.088) −	0.901 (0.055)	831.9	1073
A-n55-k9	0.702 (0.056) −	0.479 (0.126) −	0.781 (0.089) −	0.745 (0.082) −	0.912 (0.084)	935.8	1073
A-n60-k9	0.737 (0.089) −	0.608 (0.082) −	0.763 (0.078) −	0.734 (0.101) −	0.922 (0.035)	1022	1354
A-n61-k9	0.683 (0.068) −	0.501 (0.092) −	0.736 (0.104) −	0.764 (0.102) −	0.871 (0.104)	883.7	1034
A-n62-k8	0.651 (0.102) −	0.504 (0.113) −	0.673 (0.095) −	0.687 (0.084) −	0.875 (0.084)	899.7	1288
A-n63-k9	0.720 (0.076) −	0.495 (0.153) −	0.774 (0.061) −	0.748 (0.068) −	0.924 (0.043)	996.9	1616
A-n63-k10	0.657 (0.069) −	0.436 (0.114) −	0.653 (0.095) −	0.588 (0.078) −	0.834 (0.059)	1015.7	1314
A-n64-k9	0.690 (0.085) −	0.551 (0.121) −	0.728 (0.081) −	0.739 (0.101) −	0.923 (0.063)	1021	1401
A-n65-k9	0.659 (0.068) −	0.483 (0.118) −	0.737 (0.068) −	0.722 (0.064) −	0.905 (0.083)	1028	1174
A-n69-k9	0.661 (0.076) −	0.517 (0.111) −	0.711 (0.101) −	0.669 (0.096) −	0.934 (0.072)	968.1	1159
A-n80-k10	0.576 (0.065) −	0.432 (0.147) −	0.631 (0.081) −	0.645 (0.086) −	0.875 (0.053)	1204	1763
X-n106-k14	0.602 (0.140) −	0.401 (0.073) −	0.658 (0.074) −	0.683 (0.104) −	0.797 (0.090)	9960.8	26,362
X-n125-k30	0.681 (0.089) −	0.430 (0.065) −	0.697 (0.072) −	0.666 (0.102) −	0.781 (0.059)	18,475.3	55,539
X-n134-k13	0.517 (0.083) −	0.534 (0.084) −	0.557 (0.077) −	0.479 (0.073) −	0.912 (0.037)	7498.5	10,916
X-n148-k46	0.528 (0.080) −	0.167 (0.070) −	0.563 (0.116) −	0.517 (0.121) −	0.894 (0.063)	21,804.1	43,448
X-n153-k22	0.576 (0.090) −	0.294 (0.190) −	0.638 (0.096) −	0.609 (0.102) −	0.731 (0.061)	12,155.7	21,220
X-n162-k11	0.535 (0.095) −	0.608 (0.073) −	0.385 (0.186) −	0.513 (0.084) −	0.790 (0.044)	11,959.1	14,138
X-n181-k23	0.561 (0.079) −	0.494 (0.083) −	0.536 (0.088) −	0.627 (0.085) −	0.929 (0.023)	12,595.7	24,145
X-n200-k36	0.451 (0.049) −	0.403 (0.092) −	0.543 (0.089) −	0.580 (0.074) −	0.805 (0.082)	26,108.8	58,578
X-n204-k19	0.474 (0.067) −	0.364 (0.150) −	0.492 (0.042) −	0.439 (0.061) −	0.761 (0.106)	17,997.6	19,565
+/−/≈	1/36/1	0/37/1	1/34/3	1/32/5

shows the HV values of the five algorithms on the test dataset. The best solution is the shortest distance obtained by MOALNS, while the best-known solution (BKS) refers to the best-known optimal solution for this instance. It is worth noting that the BKS represents the optimal solution for a single-depot scenario. When the dataset is modified to a multi-depot version, the best solution is compared with the BKS to verify the effectiveness of the algorithm’s convergence.

When calculating the HV indicator, the PF solutions of each algorithm are first normalized, mapping the values of the two objective functions to the interval [0, 1]. Then, the reference point is set to (1, 1).

The results show that the HV indicator of MOALNS is better than that of the other algorithms across 37 instances, only performing poorly on one small-scale instance. This may be due to the fact that, when the dataset is small, all algorithms can find shorter path solutions, but MOALNS’s performance in searching for solutions under workload variations is not as effective as that of the other algorithms. Figure 2 shows the Non-dominated solution sets of the five algorithms on the test dataset. From the figure, it can be observed that MOALNS consistently demonstrates superior convergence in terms of the cost (primarily controlled by the distance) compared to the other algorithms.

However, MOALNS may exhibit poor convergence effects regarding the objective of work balance and result in the uneven distribution of the Pareto solution set. This could be due to MOALNS primarily focusing on minimizing the total route distance. When the dataset is small, it tends to concentrate more on searching for shorter distances, leading to less effective convergence for the work balance compared to other algorithms.

4.5. Experiments on a Real Case

The proposed MOALNS algorithm was applied to a practical case study in a city in Hunan Province, featuring an MSW collection scenario with waste sorting. The case involved 112 primary waste collection sites and two waste processing centers, as illustrated in Figure 3. Due to the complexity of the road network and other factors, the model established in this paper considers only the primary collection points, which imposes certain limitations. In practice, waste is transported to these primary collection points from various locations through different means.

To meet the waste sorting requirements, routes were calculated for recyclable waste, kitchen waste and other types of waste. All waste collection vehicles were identical. The result is shown in

Table 6. Results of the real case study.

Type of Waste	NSGA-II	MOEAD	UNSGA-III	AGE-MOEA2	MOALNS
Recyclable waste	0.632 (0.091) −	0.501 (0.083) −	0.620 (0.096) −	0.609 (0.103) −	0.743 (0.129)
Kitchen waste	0.548 (0.141) −	0.363 (0.118) −	0.621 (0.091) ≈	0.669 (0.128) ≈	0.652 (0.914)
Other types of waste	0.455 (0.071) −	0.362 (0.086) −	0.481 (0.814) −	0.504 (0.719) −	0.975 (0.028)
+/−/≈	0/3/0	0/3/0	0/2/1	0/2/1

. MOALNS achieved the best results when compared with the other four algorithms. Moreover, it outperformed the the other algorithms in the remaining scenarios. This experiment confirms that the proposed MOALNS algorithm delivers better optimization results for this model compared to traditional metaheuristic algorithms. Moreover, it can be observed in Figure 4 that MOALNS achieved excellent results in the collection of recyclable waste and other types of waste. MOALNS demonstrated very good convergence effects, both in the total cost and workload variance. For the kitchen waste issue, MOALNS also achieved a convergence effect on the total costs that was not reached by the other algorithms. This indicates that MOALNS fully leverages the advantage of ALNS in achieving better convergence over metaheuristic algorithms in the VRP.

MOALNS also exhibits limitations, as the proposed balance remove and balance insert heuristic operators are specifically tailored to this model. For other multi-objective models, different operators would need to be designed, indicating that MOALNS’s general applicability is not as strong as that of other heuristic algorithms.

4.6. Decision-Making in Real Case

After obtaining the Pareto set through an algorithm, it is still necessary to determine one solution from the set; this step is also known as the decision-making step. For the problem in this model, the following methods can be employed for decision-making.

Weight-based method: By assigning corresponding weights to each objective and then calculating the weighted sum of each solution, this approach can simply and quickly yield a final result. However, it requires the decision-maker to have a clear preference for the importance of each objective.

Ideal point method: In cases where the true Pareto set is unknown, the optimal value of each objective within the obtained solution set can be combined to define an ideal point. Then, we calculate the distance (such as the Euclidean distance or weighted distance) between each solution and the ideal point, ultimately selecting the solution that is closest to the ideal point.

5. Conclusions

This paper proposes an MSW collection model designed within the context of smart cities. The model plans vehicle routes based on the waste volume at each collection point. This model not only considers conventional transportation costs but also introduces carbon emissions as a factor of influence, aiming to promote the city’s carbon neutrality and sustainable development by reducing greenhouse gas emissions. The model optimizes two primary objectives: first, minimizing the total costs, including transportation expenses and carbon emission costs; second, minimizing workload variations to ensure that employees have as equal workloads as possible.

To address the multi-objective problem in the proposed model, the MOALNS algorithm is introduced. MOALNS leverages the advantages of ALNS in solving the VRP, including its fast convergence and high precision. The proposed algorithm is first compared with other multi-objective algorithms on benchmark datasets. The experimental results demonstrate that MOALNS not only provides superior cost solutions for the VRP but also exhibits outstanding capabilities in handling multi-objective problems within the model. Furthermore, to validate the effectiveness and practicality of MOALNS in real-world applications, it was tested in a real-life case study. MOALNS achieved excellent waste collection and transportation routes for various types of waste, significantly reducing the operational costs and effectively balancing workload variance among employees. This application example not only further proves the efficiency of the MOALNS algorithm in practical scenarios but also provides new insights and technical support for waste management in smart cities.

Despite the demonstrated advantages of MOALNS under this model, we acknowledge certain limitations in the established model and the proposed algorithm. These include insufficient convergence capabilities for workload disparities and the constraints of the heuristic operators tailored to this model. Future research will focus on developing models that better reflect real-world conditions and exploring more generalized algorithms to enhance the performance.