Optimizing Route Planning via the Weighted Sum Method and Multi-Criteria Decision-Making

Abstract

Choosing the optimal path in planning is a complex task due to the numerous options and constraints; this is known as the trip design problem (TTDP). This study aims to achieve path optimization through the weighted sum method and multi-criteria decision analysis. Firstly, this paper proposes a weighted sum optimization method using a comprehensive evaluation model to address TTDP, a complex multi-objective optimization problem. The goal of the research is to balance experience, cost, and efficiency by using the Analytic Hierarchy Process (AHP) and Entropy Weight Method (EWM) to assign subjective and objective weights to indicators such as ratings, duration, and costs. These weights are optimized using the Lagrange multiplier method and integrated into the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) model. Additionally, a weighted sum optimization method within the Traveling Salesman Problem (TSP) framework is used to maximize ratings while minimizing costs and distances. Secondly, this study compares seven heuristic algorithms—the genetic algorithm (GA), particle swarm optimization (PSO), the tabu search (TS), genetic-particle swarm optimization (GA-PSO), the gray wolf optimizer (GWO), and ant colony optimization (ACO)—to solve the TOPSIS model, with GA-PSO performing the best. The study then introduces the Lagrange multiplier method to the algorithms, improving the solution quality of all seven heuristic algorithms, with an average solution quality improvement of 112.5% (from 0.16 to 0.34). The PSO algorithm achieves the best solution quality. Based on this, the study introduces a new variant of PSO, namely PSO with Laplace disturbance (PSO-LD), which incorporates a dynamic adaptive Laplace perturbation term to enhance global search capabilities, improving stability and convergence speed. The experimental results show that PSO-LD outperforms the baseline PSO and other algorithms, achieving higher solution quality and faster convergence speed. The Wilcoxon signed-rank test confirms significant statistical differences among the algorithms. This study provides an effective method for experience-oriented path optimization and offers insights into algorithm selection for complex TTDP problems.

1. Introduction

Decision-making constitutes an essential facet of human endeavors, particularly in contexts characterized by a multitude of options and constraints. Within the realm of tourism, travelers confront an extensive array of choices and limitations, rendering route planning a complex endeavor that necessitates the delicate balancing of factors such as cost, time, and personal preferences. Although various assistive tools are available, the task of planning optimal tourism routes remains formidable due to the need to reconcile multiple conflicting objectives and constraints, including budgetary limitations, temporal restrictions, and visa requirements, all while striving to ensure a seamless and enjoyable experience. Addressing these multifaceted challenges demands the application of advanced optimization techniques capable of effectively managing such intricacies.

Tourism route planning often parallels the Traveling Salesman Problem (TSP), a quintessential combinatorial optimization problem. The TSP entails identifying the shortest route to traverse a set of destinations while minimizing distance or time and has garnered extensive attention due to its NP-hard nature [1,2]. However, traditional optimization methods frequently falter in providing suitable solutions for such problems, particularly under complex scheduling constraints. With the advent of artificial intelligence, intelligent problem-solving methods have emerged, offering near-optimal solutions to intricate real-world optimization problems. Techniques such as heuristics, branch-and-bound methods, and metaheuristic algorithms (e.g., genetic algorithms, simulated annealing) have been employed to tackle the TSP, thereby advancing its practical applications in logistics and tourism [3,4,5]. The significance of metaheuristic algorithms is underscored by their capacity to swiftly identify near-optimal solutions, rendering them highly suitable for large-scale, time-sensitive problems [6].

The tourist trip design problem (TTDP) is a route-planning problem for tourists that aligns with their preferences and requirements while maximizing their entertainment, all within the confines of numerous constraints. Given the pivotal role of TTDP-related research in enhancing the experiences of tourists and bolstering the competitive advantages of tourism destinations, this research domain has garnered considerable attention [7,8].

Visa policies, such as China’s 72 h visa-free transit policy, introduce time-sensitive constraints, thereby inherently aligning the problem with the team orienteering problem with time windows (TOPTW). In TOPTW, time windows are treated as constraints, and similar visa policies influence travel behaviors by encouraging longer stays while requiring travelers to plan within limited time frames. These insights underscore the growing importance of addressing TTDP with advanced methods that incorporate time-sensitive constraints, as visa policies increasingly shape modern tourism dynamics [5,9].

Existing works have attempted to address some of these challenges. For instance, the TS algorithm can avoid the local optimal solution, but the running time is long [10]. The particle swarm optimization (PSO) algorithm has been used to solve the Traveling Salesman Problem [11]. Later, the ant colony optimization (ACO) method was proposed for estimating tourism revenue, and an expenditure approach was developed [12]. Then, a “multi-day tourism” optimization model employed a Monte Carlo-improved simulated annealing algorithm to design local tourist routes, with objectives such as the shortest path and lowest cost [13]. Additionally, a hybrid metaheuristic algorithm combining the genetic algorithm (GA) [14] and Variable Neighborhood Descent (VND) has been proposed [15]. In addition, in the hybrid GA-PSO algorithm, new individuals are created through GA operators—crossover and mutation as well as a mechanism of PSO to find the optimal and computational time [16]. The standard gray wolf optimization (GWO) method developed for numerical optimization is a new approach to solving TSP, inspired by the hunting and social behavior of gray wolf packs [17]. Moreover, the RL-SA leverages the whale optimization algorithm (WOA) to generate initial solutions for better sampling efficiency [18]. Another study, which integrates cluster hierarchy with heuristic algorithms, offers a structured approach to optimizing route planning under complex constraints [19]. While these methods provide valuable insights, they often rely on single-algorithm strategies and focus on narrow objectives, such as cost or path minimization. A summary of the above literature is shown in Table 1.

Table 1. Summary of the literature.

Citation	Research Objective	Key Findings	Conclusions	Relation to This Study
[1,2]	Study of the Traveling Salesman Problem (TSP), calculating the shortest path	TSP is an NP-hard problem, difficult to solve	Traditional optimization methods cannot effectively handle TSP problems with complex constraints	Provides background support for TSP, serving as the foundation for the optimization framework in this study
[3,4,5,6]	Study of metaheuristic algorithms in TSP	Metaheuristic algorithms can effectively handle TSP problems, suitable for large-scale problems	Metaheuristic algorithms perform excellently in time-sensitive problems	This study borrows metaheuristic algorithms and proposes a multi-algorithm evaluation framework
[7,8]	Study of the tourist trip design problem (TTDP), optimizing tourist routes	TTDP research helps enhance tourist experience and boosts destination competitiveness	Improved tourist experience, but methods rely on single-objective optimization	This study introduces multi-objective optimization to TTDP, enhancing algorithm adaptability
[5,9]	Study of the combination of visa policies and TOPTW in tourism planning	Visa policies introduce time constraints, affecting tourist behavior	The combination of visa policies and time constraints influences travel planning	This study incorporates time-sensitive visa policy constraints, proposing an improved optimization method
[10]	Study of the application of the tabu search (TS) algorithm	TS avoids local optima but has long running time	TS is useful for avoiding local optima, but its efficiency is low	This study improves the efficiency of TS by optimizing it through a hybrid algorithm
[11]	Study of the particle swarm optimization (PSO) algorithm in TSP	PSO performs well in solving TSP problems	PSO is suitable for solving large-scale TSP problems	This study improves PSO, enhancing its global search capability
[17]	Study of the application of the gray wolf optimization (GWO) algorithm	GWO is suitable for numerical optimization, inspired by gray wolf group behavior	GWO performs excellently in TSP problems	This study combines GWO with the TSP framework, proposing a new optimization approach
[18]	Study of the efficiency of combining the whale optimization algorithm (WOA) with Reinforcement Learning (RL)	WOA generates initial solutions, improving sampling efficiency	WOA improves sampling efficiency	This study incorporates multi-objective optimization, enhancing algorithm adaptability

These limitations impede adaptability to diverse conditions, such as evolving travel preferences, strict time constraints, or multi-objective requirements that necessitate a careful balance between cost, experience, and efficiency. Most existing approaches for determining criteria weights face challenges when used independently. For example, the Analytic Hierarchy Process (AHP), as a subjective weighting method, relies heavily on expert judgment for constructing its matrix [20]. Excessive subjectivity in expert evaluations can significantly influence the final results [21]. Conversely, the Entropy Weight Method (EWM), as an objective weighting method, provides strong mathematical rigor but often fails to incorporate the subjective priorities of decision-makers [22]. This highlights the need for a hybrid weighting approach that combines the strengths of both methods to effectively address their individual shortcomings. Furthermore, these approaches often lack benchmarking against alternative algorithms, which limits their ability to assess comparative effectiveness and scalability in diverse scenarios.

The structure of this paper is as follows. Section 1 introduces the complexity of tourism route planning, focusing on decision-making challenges under multiple objectives and constraints, reviewing the Traveling Salesman Problem (TSP) and its applications in tourism route optimization, and emphasizing the advantages of metaheuristic algorithms in large-scale combinatorial optimization. Section 2 discusses the data sources, data cleaning process, and the research framework. Section 3 constructs a TOPSIS comprehensive evaluation model based on weighted scores, achieving cost minimization and score maximization under the 144-h visa policy. Section 4 builds a weighted optimization model, considering both tourist experience and cost, and performs route planning based on scores, costs, and travel distances. Section 5 compares the performance of eight heuristic algorithms (GA, PSO, TS, GA-PSO, GWO, ACO, WOA, and PSO-LD) in path optimization. Section 6 conducts a parameter sensitivity analysis of the PSO-LD algorithm to verify its stability and adaptability. Section 7 discusses the advantages of the proposed method, combining multi-criteria decision-making and heuristic algorithms to optimize tourist experience, cost, and efficiency, with a focus on the weighting method and the role of the Lagrange multiplier method in optimizing weights. The PSO-LD algorithm performs excellently in multi-area planning, demonstrating high efficiency and scalability. Finally, Section 8 summarizes the main contributions of the study, presenting a framework that integrates weighted optimization methods, decision criteria, and heuristic algorithms, significantly improving solution quality and computational efficiency, especially with the PSO-LD algorithm, which improves solution quality by about 11% compared to traditional PSO.

Building on these developments, this research offers solutions for more effective tourism evaluation and route optimization. It introduces methods for evaluating tourist sites with advanced metrics, balancing cost and experience in route planning, and analyzing the strengths of different algorithms for complex TSP cases. Collectively, these contributions provide practical tools and insights to address challenges in tourism optimization. The primary contributions of this research are as follows:

(1)

This study introduces a TOPSIS comprehensive evaluation model [23] incorporating the Lagrange multiplier method to enhance the objectivity and rationality of decision-making. The model systematically evaluates complex tourism indicators, optimizing costs and travel experiences within the framework of the TSP, thereby supporting effective tourist route planning.

(2)

Based on TOPSIS as a benchmark, multiple heuristic algorithms are evaluated to address the limitations of single algorithms in complex real-world applications further. By analyzing their performance in varying contexts, the most effective approach for optimizing tourist routes is identified. This multi-algorithm evaluation enhances adaptability and user experience, providing a robust framework for improving the efficiency and effectiveness of route optimization.

2. Dataset and Methods

The dataset for this study was sourced from https://www.qunar.com/ (accessed on 17 August 2023), a leading Chinese travel service platform offering comprehensive tourism-related data, including hotel bookings, flight reservations, and attraction information. A sample dataset is shown in Table 2. To ensure the scientific accuracy and reliability of the data, records with incomplete star ratings and scores were removed during the research process, resulting in a final selection of 617 tourist attractions across 10 provinces and autonomous regions. Based on the star rating, recommended visiting times for each attraction were determined, and the corresponding latitude and longitude information was obtained using the Gaode Map API. In addition, it is worth clarifying that the distances between these attractions are calculated using Euclidean distance, with a two-dimensional space used to reflect the actual geographical distance. The complete datasets can be accessed via the following link: https://github.com/GZHUone/excel (accessed on 23 August 2023).

Table 2. Examples of partial datasets.

City	Name	Duration	Rating	Price	Sales	Longitude	Latitude
Beijing	Badaling Great Wall	5	0.82	40	16,382	116.4806	40.6364
Shanxi	Huashan Scenic Area	5	1	160	8584	110.0868	34.5676
Sichuan	Dujiangyan	4	1	40	10,178	103.6451	30.99131
Chongqing	Heishan Valley	5	0.72	400	1792	107.0095	28.9148
Liaoning	Bingyu Valley	4	0.74	145	24	122.9607	39.67958

(1) For the quantitative comprehensive rating of tourist attractions, this research proposes an approach that uses the AHP, EWM, and Lagrange multiplier method to calculate weights and combines these with the TOPSIS model to construct a comprehensive evaluation model, resulting in an overall score for each attraction. More details can be seen in Section 3.

(2) For the purpose of balancing cost minimization and maximizing the comprehensive rating of tourist attractions under constraints, this study proposes a weighted sum programming model. With the basic parameters held constant, this study uses eight heuristic algorithms to find the optimal route-planning solution through iterative optimization. More details can be seen in Section 4 and Section 5.

To present the work of this study more intuitively, a detailed research framework is shown in Figure 1.

Figure 1. Research framework of this study.

3. TOPSIS Comprehensive Evaluation Model

This section provides a comprehensive overview of the structure of the TOPSIS comprehensive evaluation model (Section 3.1). Additionally, it introduces the AHP (Section 3.2), EWM (Section 3.3), and the Lagrange multiplier method (Section 3.4). Section 3.5 then elaborates on the integration of these methods into the proposed TOPSIS comprehensive evaluation model.

3.1. Structure of the TOPSIS Comprehensive Evaluation Model

This study constructs a comprehensive evaluation model to rank tourist attractions scientifically. The AHP is utilized to determine subjective weights for four key indicators: visit time, ratings, price, and sales volume. The EWM is employed to calculate objective weights based on data variability. To balance the influence of subjective and objective weights, the Lagrange multiplier method is introduced to optimize the combination of these weights. Finally, the optimized weights are applied to the TOPSIS comprehensive evaluation model to obtain the overall scores and rankings of the attractions. Figure 2 shows the steps in a comprehensive evaluation model flowchart.

Figure 2. Flowchart illustrating the TOPSIS comprehensive evaluation model.

3.2. The Analytic Hierarchy Process

To effectively evaluate and prioritize multiple criteria in decision-making, this study employs AHP, a widely used method in multi-criteria decision analysis, to determine the importance of each factor through pairwise comparisons [20,22]. The detailed steps are as follows:

First, a judgment matrix was constructed, which expresses the relative importance among the four indicators: visit time, visitor rating, attraction price, and attraction sales. To fully reflect the decision-maker’s experience and preferences, this research utilized Saaty’s 1–9 scale [22] to customize the judgment matrix, thereby aligning it more closely with real-world decision-making contexts. Additionally, the matrix can be dynamically adjusted to meet the personalized needs of users, allowing for real-time modifications based on factors such as changing preferences, seasonal trends, and external conditions. Below is an example of the constructed judgment matrix employed in the analysis:

$$\mathrm{A}=\left[\begin{array}{cccc}1& 2& 3& 1/5\\ 1/2& 1& 5& 1/3\\ 1/3& 1/5& 1& 1/7\\ 5& 3& 7& 1\end{array}\right]$$

(1)

Next, a consistency test to verify the logical coherence and consistency of the judgment matrix can be performed. This test is essential for ensuring the reliability of the matrix’s outcomes. The process involves calculating the Consistency Ratio (CR) [22], as detailed below:

$$\mathrm{CR}={\displaystyle \frac{\mathrm{CI}}{\mathrm{RI}}}$$

(2)

where the consistency index (CI) is calculated as follows:

$$\mathrm{CI}={\displaystyle \frac{{\lambda }_{\max }-\mathrm{n}}{\mathrm{n}-1}}$$

(3)

${\lambda }_{\max }$ is the maximum eigenvalue of the judgment matrix, n is the dimension of the judgment matrix, and the random consistency index RI is the empirical value based on the dimension of the matrix. Through this calculation, the maximum eigenvalue ${\lambda }_{\max }$ and a further CI and CR are obtained.

The consistency test standard is defined as CR < 0.1 [22]. With CR = 0.09856, this meets the consistency test standard, indicating that the judgment matrix has passed the consistency check. This demonstrates that the matrix exhibits good logical consistency in the subjective judgment process and can be used for subsequent weight calculations.

3.3. The Entropy Weight Method

To objectively allocate weights in a comprehensive evaluation and reasonably reflect the importance of each indicator, this research adopts the Entropy Weight Method (EWM) [24]. The method assesses the amount of information by analyzing the variation in indicator data: the smaller the variation, the less information it contains, and thus the lower the corresponding weight. This approach allows for an objective evaluation of each indicator’s importance. The specific calculation steps of the EWM are as follows:

First, the input weight matrix is normalized. Since the multiple indicator variables used to evaluate the comprehensive scores of cities have different attributes and units, direct comparisons between indicators are challenging. If TOPSIS is applied directly, the results may be biased. Therefore, to eliminate the dimensional differences among indicators, the original data should be normalized. Using the extreme value normalization method, the data for both positive and negative indicators are standardized, yielding the normalized data ${\mathrm{Y}}_{\mathrm{i}\mathrm{j}}$ [25,26]:

$$\left\{\begin{array}{c}\mathrm{P}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{I}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}: {\mathrm{Y}}_{\mathrm{ij} }={\displaystyle \frac{\left({\mathrm{X}}_{\mathrm{ij}}-\min \left\{{\mathrm{X}}_{\mathrm{ij}}\right\}\right)}{\left(\max \left\{{\mathrm{X}}_{\mathrm{ij}}\right\}-\min \left\{{\mathrm{X}}_{\mathrm{ij}}\right\}\right)}}\\ \mathrm{N}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{I}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}: {\mathrm{Y}}_{\mathrm{ij}}={\displaystyle \frac{\left(\max \left\{{\mathrm{X}}_{\mathrm{ij}}\right\}-{\mathrm{X}}_{\mathrm{ij}}\right)}{\left(\max \left\{{\mathrm{X}}_{\mathrm{ij}}\right\}-\min \left\{{\mathrm{X}}_{\mathrm{ij}}\right\}\right)}}\end{array}\right.$$

(4)

where $\max \left\{{\mathrm{X}}_{\mathrm{ij}}\right\} \mathrm{a}\mathrm{n}\mathrm{d} \min \left\{{\mathrm{X}}_{\mathrm{ij}}\right\}$ are the maximum and minimum values for each sample of attractions under the first indicator;${ \mathrm{Y}}_{\mathrm{ij}}$ reflects the level of satisfaction in achieving the goal.

The proportion of the ${\mathrm{i}}^{\mathrm{th}}$ sample under the ${\mathrm{j}}^{\mathrm{th}}$ indicator is calculated as follows:

$${\mathrm{K}}_{\mathrm{ij}}={\displaystyle \frac{{\mathrm{Y}}_{\mathrm{ij}}}{\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{Y}}_{\mathrm{ij}}}}$$

(5)

Then, calculate the information entropy for each indicator and then compute the information utility value. Finally, normalize the results to obtain the entropy weight for each indicator. For the ${\mathrm{j}}^{\mathrm{th}}$ indicator, the formula for calculating information entropy is as follows:

$${\mathrm{e}}_{\mathrm{j}}=-|{\displaystyle \frac{1}{\mathrm{lnn}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{K}}_{\mathrm{ij}}\ln \left({\mathrm{K}}_{\mathrm{ij}}\right)$$

(6)

The information utility value is

$${\mathrm{d}}_{\mathrm{j}}=1-{\mathrm{e}}_{\mathrm{j}}$$

(7)

where ${\mathrm{d}}_{\mathrm{j}}$ is the information utility value of each indicator of the city’s comprehensive score; the larger the information utility value, the more information it corresponds to. Then, the information utility value is normalized. This research can obtain the weight of each indicator:

$${\mathrm{W}}_{\mathrm{j}}={\displaystyle \frac{{\mathrm{d}}_{\mathrm{j}}}{\sum _{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{d}}_{\mathrm{j}}}}$$

(8)

where ${\mathrm{W}}_{\mathrm{j}}$ is the weight of each indicator for evaluating the city’s composite score.

3.4. Lagrange Multiplier Method for Determining Weights

The subjective weight reflects expert experience and knowledge but may introduce biases, while the objective weight, based on data analysis, offers quantitative advantages yet may overlook complex contextual factors. Therefore, combining subjective and objective weights is essential. This research employs the Lagrange multiplier method to obtain the optimal combined weight by minimizing the information entropy difference between the subjective and objective weights under constraint conditions.

To begin with, the objective function is defined based on information entropy. The aim is to align the comprehensive weight ${\mathrm{W}}_{\mathrm{i}}$ as closely as possible with the subjective weight ${\mathrm{w}}_{1\mathrm{i}}$ and the objective weight ${\mathrm{w}}_{2\mathrm{i}}$, thereby finding the optimal weight distribution [25]:

$$\mathrm{E}=\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{W}}_{\mathrm{i}}\left(\ln {\displaystyle \frac{{\mathrm{W}}_{\mathrm{i}}}{{\mathrm{w}}_{1\mathrm{i}}}}\right)+\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{W}}_{\mathrm{i}}\left(\ln {\displaystyle \frac{{\mathrm{W}}_{\mathrm{i}}}{{\mathrm{w}}_{2\mathrm{i}}}}\right)$$

(9)

The Lagrange function is constructed as follows:

$$\mathrm{L}\left({\mathrm{W}}_{\mathrm{i}},\lambda \right)=\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{W}}_{\mathrm{i}}\left(\ln {\displaystyle \frac{{\mathrm{W}}_{\mathrm{i}}}{{\mathrm{w}}_{1\mathrm{i}}}}\right)+\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{W}}_{\mathrm{i}}\left(\ln {\displaystyle \frac{{\mathrm{W}}_{\mathrm{i}}}{{\mathrm{w}}_{2\mathrm{i}}}}\right)-\lambda \left(\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{W}}_{\mathrm{i}}- 1\right)$$

(10)

where the constraint $\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{W}}_{\mathrm{i}}=1$ ensures the normalization of the final weights, and $\lambda$ is the Lagrange multiplier. To find the optimal solution, take the partial derivatives of the Lagrange function with respect to ${\mathrm{W}}_{\mathrm{i}}$ and $\lambda$, and set them to zero [27,28].

Following these steps, the weights of each indicator in the comprehensive evaluation of city attractions can be determined. The calculation results are shown in Table 3.

Table 3. Results of the calculation of indicator weights.

Evaluation Metrics	Analytic Hierarchy Process	Entropy Weight Method	Lagrange Multiplier Method
Duration	0.19393507	0.425333	0.30963403
Rating	0.18645617	0.313875	0.25016559
Price	0.05719146	0.125119	0.09115523
Sales	0.5624173	0.135673	0.34904515

3.5. The TOPSIS Model Combined AHP-EWM

To facilitate comprehensive evaluation and optimization under complex conditions, the TOPSIS method constructs both optimal and worst-case solutions for the evaluation problem [29,30]. This method calculates the proximity of each feasible solution to the ideal, providing a comprehensive score for each attraction. The modeling steps of the TOPSIS method are as follows:

First, starting with the standardized matrix $\mathrm{Z}=\left[\begin{array}{cccc}{\mathrm{Y}}_{11}& {\mathrm{Y}}_{12}& ...& {\mathrm{Y}}_{1\mathrm{j}}\\ {\mathrm{Y}}_{21}& {\mathrm{Y}}_{22}& ...& {\mathrm{Y}}_{2\mathrm{j}}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{Y}}_{\mathrm{i}1}& {\mathrm{Y}}_{\mathrm{i}2}& ...& {\mathrm{Y}}_{\mathrm{ij}}\end{array}\right]$, define the optimal solution ${\mathrm{Z}}^{+}$ and the worst solution ${\mathrm{Z}}^{-}$ as follows:

$$\left\{\begin{array}{l}{\mathrm{Z}}_{\mathrm{j}}^{+}={\max }_{1\le \mathrm{i} \le \mathrm{m}}\left|{\mathrm{Z}}_{\mathrm{ij}}\right|\\ {\mathrm{Z}}_{\mathrm{j}}^{-}={\min }_{1\le \mathrm{i} \le \mathrm{m}}\left|{\mathrm{Z}}_{\mathrm{ij}}\right|\end{array}\right.$$

(11)

Next, compute the Euclidean distance between each selected tourism city’s comprehensive evaluation indicators and the above-mentioned optimal and worst solutions, as follows:

$$\left\{\begin{array}{l}{\mathrm{D}}_{\mathrm{i}}^{+}=\sqrt{{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{m}}}{\mathrm{W}}_{\mathrm{j}}{\left({\mathrm{Z}}_{\mathrm{j}}^{+}-{\mathrm{Y}}_{\mathrm{ij}}\right)}^2}\\ {\mathrm{D}}_{\mathrm{i}}^{-}=\sqrt{{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{m}}}{\mathrm{W}}_{\mathrm{j}}{\left({\mathrm{Z}}_{\mathrm{j}}^{-}-{\mathrm{Y}}_{\mathrm{ij}}\right)}^2}\end{array} \right.$$

(12)

where ${\mathrm{D}}_{\mathrm{i}}^{+}$ is the Euclidean distance between the ${\mathrm{i}}^{\mathrm{th}}$ city and the optimal solution, ${\mathrm{D}}_{\mathrm{i}}^{-}$ is the Euclidean distance between the ${\mathrm{i}}^{\mathrm{th}}$ city and the worst solution, and ${\mathrm{W}}_{\mathrm{j}}$ is the weight of each index for evaluating tourism cities [30].

The degree of each city’s closeness to the optimal solution ${\mathrm{S}}_{\mathrm{i}}$ is computed as follows:

$${\mathrm{S}}_{\mathrm{i}}={\displaystyle \frac{{\mathrm{D}}_{\mathrm{i}}^{-}}{{\mathrm{D}}_{\mathrm{i}}^{+}+{\mathrm{D}}_{\mathrm{i}}^{-}}}$$

(13)

Finally, the degree of closeness ${\mathrm{S}}_{\mathrm{i}}$ is normalized, with larger values indicating a higher overall score for the tourist city.

By solving the TOPSIS comprehensive evaluation model, the overall score for each attraction is determined. Table 4 presents the specific scores and ranking results for some attractions, providing an intuitive view of each attraction’s overall performance across different indicators.

Table 4. Top five attractions based on the overall TOPSIS evaluation.

Attraction Name	TOPSIS Score	Ranking
Chengdu Panda Base	0.691068072	1
The Forbidden City	0.684491317	2
Qin Mausoleum Museum	0.662509441	3
Wugong Mountain	0.535061157	4
Badaling Great Wall	0.526370538	5

4. Weighted Sum Planning Models

Since foreign tourists are typically constrained by a 144 h visa policy, their goal is to maximize their overall travel experience within this restricted time frame while minimizing costs. This scenario can be modeled as a weighted sum optimization problem. The model-building process is outlined below.

Numbering all attractions within the region is based on only arriving at the attraction versus not arriving at the attraction. Define ${\mathrm{x}}_{\mathrm{ijk}}$ as the number of the ${\mathrm{i}}^{\mathrm{th}}$ attraction, with $\mathrm{j}$ representing whether the ${\mathrm{i}}^{\mathrm{th}}$ attraction is chosen to be visited by the traveler or not and $\mathrm{k}$ representing whether the attraction is chosen to be visited.

In this study, to calculate the latitude and longitude of any two points on the Earth’s surface, the Haversine formula was used, for example, to compute the geographical distance between two cities [31]:

$$\mathrm{d}=2\mathrm{R}\cdot \arcsin \left(\sqrt{{\sin }^2\left({\displaystyle \frac{{\phi }_2-{\phi }_1}{2}}\right)+\cos \left({\phi }_1\right)\cdot \cos \left({\phi }_2\right)\cdot {\sin }^2\left({\displaystyle \frac{{\lambda }_2-{\lambda }_1}{2}}\right)}\right)$$

(14)

where ${\phi }_1$ and ${\phi }_2$ are the latitudes of the two points, ${\lambda }_1$ and ${\lambda }_2$ are the longitudes of the two points, and $\mathrm{R}$ is the radius of the Earth.

To calculate the total travel route, first identify all attractions where $\mathrm{j}=1$, representing the attractions selected for this tour. Next, arrange the attractions in the order of $\mathrm{k}$ from 1 to $\mathrm{M}$, and calculate the path between adjacent points using the Haversine formula to obtain the total distance ${\mathrm{D}}_{\mathrm{total}}$ [32]:

$${\mathrm{D}}_{\mathrm{total} }=\sum _{\mathrm{k}=1}^{\mathrm{M}-1}2\mathrm{R}\cdot \arcsin \left(\sqrt{{\sin }^2\left({\displaystyle \frac{{\phi }_{\mathrm{k}+1}-{\phi }_{\mathrm{k}}}{2}}\right)+\cos \left({\phi }_{\mathrm{k}}\right)\cdot \cos \left({\phi }_{\mathrm{k}+1}\right)\cdot {\sin }^2\left({\displaystyle \frac{{\lambda }_{\mathrm{k}+1}-{\lambda }_{\mathrm{k}}}{2}}\right)}\right)$$

(15)

For the $\mathrm{M}$ selected attractions, the total comprehensive score is calculated as follows:

$$\mathrm{score}=\sum _{\mathrm{i}=1|{\mathrm{x}}_{\mathrm{j}\ne 0}}^{\mathrm{M}}{\mathrm{score}}_{\mathrm{i}}$$

(16)

where ${\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}_i$ represents the score of the ${\mathrm{i}}^{\mathrm{th}}$ attraction, and ${\mathrm{x}}_{\mathrm{j}}$ is a binary variable, with 1 indicating the attraction is selected and 0 indicating it is not selected.

The total attraction cost is used to quantify the travel expenses. For the $\mathrm{M}$ selected attractions, the total cost is calculated using the following formula:

$$\mathrm{price}=\sum _{\mathrm{i}=1|{\mathrm{x}}_{\mathrm{j}\ne 0}}^{\mathrm{M}}{\mathrm{price}}_{\mathrm{i}}$$

(17)

where ${\mathrm{price}}_{\mathrm{i}}$ represents the cost of the ${\mathrm{i}}^{\mathrm{th}}$ attraction, and ${\mathrm{x}}_{\mathrm{j}}$ is a binary variable, with 1 indicating the attraction is selected and 0 indicating it is not selected.

The objective function should consider visitor preferences, cost-effectiveness, and travel efficiency. Therefore, this research constructs the objective function based on the total comprehensive score, total attraction cost, and total travel distance calculated above, providing a rational decision basis for the path planning problem in practical applications. When constructing the objective function, given that $\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}$, $\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e},$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$ have different dimensions, this research applies max–min normalization to incorporate all three into a single-objective function [33,34].

As ${\mathrm{score}}_{\max },{\mathrm{score}}_{\min },{\mathrm{price}}_{\max },{\mathrm{price}}_{\min }, {{\mathrm{D}}_{\max }, \mathrm{and} \mathrm{D}}_{\min }$ are obtained, the final objective function can be defined as follows:

$$\max \mathrm{f}\left(\mathrm{x},\mathrm{s}\right)=\alpha \cdot {\displaystyle \frac{\sum _{\mathrm{i}=1|{\mathrm{x}}_{\mathrm{j}}\ne 0}^{\mathrm{M}}{\mathrm{score}}_{\mathrm{i}}-{\mathrm{score}}_{\min }}{{\mathrm{score}}_{\max }-{\mathrm{score}}_{\min }}}-\beta \cdot {\displaystyle \frac{\sum _{\mathrm{i}=1|{\mathrm{x}}_{\mathrm{j}}\ne 0}^{\mathrm{M}}{\mathrm{price}}_{\mathrm{i}}-{\mathrm{price}}_{\min }}{{\mathrm{price}}_{\max }-{\mathrm{price}}_{\min }}}-\gamma \cdot {\displaystyle \frac{{\mathrm{D}}_{\mathrm{total}}- {\mathrm{D}}_{\min }}{{{\mathrm{D}}_{\max } - \mathrm{D}}_{\min }}}$$

(18)

where $\alpha$ = 1, $\beta$ = 0.5, and $\gamma$ = 0.5 represent the weights of the sum of composite scores, the sum of attraction spending, and the sum of paths, respectively.

The constraints are established as follows: For foreign tourists, the total travel time is limited to 144 h. When planning the travel route, both the visiting time for attractions and the travel time between attractions need to be considered. The travel time is calculated based on the assumption that the car travels at a constant speed of 80 km/h. This speed is chosen by referencing typical speed limits for urban and suburban roads, ensuring that the planning results align with real-world traffic conditions. Therefore, the following expression can be obtained:

$$\left\{\begin{array}{c}{\mathrm{t}}_{\mathrm{total} }={\displaystyle \sum _{\mathrm{i}=1|{\mathrm{x}}_{\mathrm{j}}\ne 0}^{\mathrm{M}}}{\mathrm{time}}_{\mathrm{i} }+{\displaystyle \frac{{\mathrm{D}}_{\mathrm{total}}}{80}} \\ {\mathrm{t}}_{\mathrm{total} }\le 144\end{array}\right.$$

(19)

where $\sum _{\mathrm{i}=1|{\mathrm{x}}_{\mathrm{j}}\ne 0}^{\mathrm{M}}{\mathrm{time}}_{\mathrm{i} }$ is the total time to visit the selected attraction, and ${\displaystyle \frac{{\mathrm{D}}_{\mathrm{total}}}{80}}$ is the traveling time.

5. Heuristic Algorithms

Using our TOPSIS evaluation model as a benchmark, this study compares the performance of eight heuristic algorithms. This section provides a comprehensive overview of these algorithms (Section 5.1): the genetic algorithm (GA) leverages crossover and mutation to explore diverse solutions; particle swarm optimization (PSO) achieves a balance between exploration and exploitation for global optimization; the tabu search (TS) focuses on a local search while avoiding revisiting previous solutions; and the hybrid GA-PSO combines GA’s solution diversity with PSO’s optimization efficiency for enhanced performance. The gray wolf optimizer (GWO) mimics wolf pack hierarchy and cooperative hunting mechanisms to balance global exploration with precise local exploitation; ant colony optimization (ACO) utilizes pheromone-guided pathfinding and positive feedback mechanisms for efficient combinatorial optimization in discrete search spaces; and the whale optimization algorithm (WOA) emulates bubble-net feeding behavior and spiral updating strategies to dynamically transition between exploration phases and exploitation refinement. Furthermore, this study introduces a novel variant of PSO, namely PSO with Laplace disturbance (PSO-LD), which incorporates dynamic adaptive Laplace perturbation to enhance global search capabilities and improve stability and convergence speed. Additionally, this section outlines the parameter settings, ensuring consistent basic parameters, such as population size and iterations, across algorithms (Section 5.2), and presents both the quantitative and qualitative results of the solutions (Section 5.3).

5.1. Introduction of Heuristic Algorithms

The traditional traversal method exhibits high computational complexity when addressing the constrained TSP, often necessitating substantial time and computational resources [35]. Therefore, in weighted sum optimization tasks, it is crucial to approach the problem from diverse perspectives and employ various strategies. This study utilizes eight heuristic algorithms—GA, PSO, TS, GA-PSO, PSO-LD, GWO, ACO, and WOA—each employing iterative optimization to identify the optimal combination of travel routes, thereby enhancing both the quality and stability of the solution. This study describes eight algorithms and discusses their convergence. Figure 3 illustrates the specific processes of the eight algorithms.

Figure 3. Flowcharts of the five heuristic algorithms: (a) genetic algorithm; (b) particle swarm optimization algorithm; (c) tabu search algorithm; (d) genetic-particle swarm algorithm; (e) particle swarm Laplace disturbance algorithm; (f) gray wolf optimizer; (g) ant colony optimization; (h) whale optimization algorithm.

(a) GA [36]: This algorithm simulates natural selection by representing route combinations as individuals in a population. It uses crossover and mutation operations to create new routes, with a fitness function evaluating each combination. In the initial stage, the high diversity of the population allows it to explore a wide range of solutions. Through iterative evolution, the population converges toward the optimal solution. Figure 3a shows GA.

(b) PSO [37]: This algorithm models particle movements influenced by personal best and global best positions. Each particle adjusts its position iteratively, enabling the swarm to converge on the optimal route. Figure 3b shows PSO.

(c) TS [38]: This algorithm converges by continuously exploring neighboring solutions while avoiding revisiting previous solutions through a tabu list, thereby avoiding local optima. Through iterations, the solution is refined to approach optimality. Figure 3c shows TS.

(d) GA-PSO [39]: This algorithm combines GA’s genetic diversity with PSO’s global search ability. It converges more efficiently under complex constraints. PSO guides the search direction, while GA’s crossover and mutation maintain diversity, leading to robust optimization. Figure 3d shows GA-PSO.

(e) PSO-LD: This algorithm extends particle swarm optimization (PSO) by integrating a dynamic Laplace perturbation term. PSO-LD enhances global search capabilities and escapes local optima more effectively, which also affects its convergence. The Laplace perturbation, characterized by its sharp peak and heavy tails, introduces variability that aids in both local refinement and global exploration. The scale parameter adjusts dynamically, balancing exploration and exploitation across iterations. Figure 3e depicts PSO-LD.

(f) GWO [40]: This algorithm mirrors the hierarchy and hunting strategies of wolf packs. Wolves are characterized by their decision-making and hunting behaviors. By simulating the hunting behavior of gray wolves, including tracking, encircling, and attacking prey, GWO balances exploration and exploitation. The algorithm updates the positions of the wolves iteratively to converge toward the optimal solution. Figure 3f shows GWO.

(g) ACO [2]: This algorithm imitates the foraging behavior of ants, particularly their use of pheromone trails. Through iterative updates of pheromone trails and path selection, the algorithm efficiently explores the search space and converges on optimal or near-optimal solutions. Figure 3g shows ACO.

(h) WOA [41]: This algorithm is a simple, robust, and swarm-based stochastic optimization algorithm. The population-based WOA has the ability to avoid local optima and achieve a global optimal solution. These advantages make WOA an appropriate algorithm for solving different constrained or unconstrained optimization problems for practical applications without structural reformation in the algorithm. Figure 3h shows the WOA.

5.2. Parameter Settings

In this study, to ensure a fair comparison, the performances of GA, PSO, TS, GA-PSO, PSO-LD, GWO, ACO, and WOA were evaluated using identical parameter settings, including population size and the number of iterations. These settings are listed in Table 5.

Table 5. Parameter settings for the heuristic algorithms.

Heuristic Algorithms	Attributes	Value
GA	Population Size	200
	Iterations	10,000
	Mutation Rate	0.1
	Elite Count	5
PSO	Particle Count	200
	Iterations	10,000
	Cognitive Factor	1.5
	Social Factor	1.5
	Inertia Weight	0.8
TS	Initial Point Count	200
	Iterations	10,000
	Initial Tabu List Size	50
	Maximum Tabu List Size	200
	Mutation Rate	0.1
	Tabu Breaking Threshold	0.05
GA-PSO	Particle Count	200
	Iterations	10,000
	Cognitive Factor	1.5
	Social Factor	1.5
	Inertia Weight	0.8
	Mutation Rate	0.1
PSO-LD	Particle Count	200
	Iterations	10,000
	Cognitive Factor	1.5
	Social Factor	1.5
	Laplace Perturbation Scale Max	5
	Decay Rate	5
GWO	num_wolves	200
	Iterations	10,000
	Selection Threshold	0.6
	Convergence Factor	2
ACO	num_ants	200
	Iterations	10,000
	Pheromone Importance	1
	Heuristic Importance	2
	Pheromone Evaporation Rate	0.1
	Pheromone Deposit Constant	1
WOA	num_whales	200
	Iterations	10,000
	Spiral Shape Constant	1
	Convergence Factor	2

5.3. Results of the Heuristic Algorithm Solutions

5.3.1. Quantitative Results

Performance Enhancement of Heuristic Algorithms with the Lagrange Multiplier

Through comparative analysis of the experimental results, the total score [41] metric was employed to measure the solution quality of different heuristic algorithms, where a higher total score indicates superior solution quality in a given region. Additionally, the attraction score reflects the number and quality of selected attractions based on their star ratings. Table 6 shows, under the same baseline parameter settings, the solution quality of PSO, GA, TS, GA-PSO, GWO, WOA, and ACO across all 14 regions.

Table 6. Table of heuristic algorithm solution results without the Lagrange multiplier.

Regions	Heuristic Algorithm	Total Score	Attraction Score	Total Price	Total Distance	Total Time
Average	GA	0.110591711	0.819934047	3775.174428	4263.650483	140.1822977
	PSO	0.223699792	0.867052393	4409.662567	4811.586231	142.2314945
	TS	0.104824523	0.727486732	4010.367822	4677.155828	138.7111145
	GA-PSO	0.223996294	0.873713939	4413.300728	4774.151686	141.9202294
	GWO	0.246372199	0.696389342	3387.4214	4891.844113	139.3028646
	ACO	0.123976916	0.948399372	3178.038898	2754.626207	142.5017808
	WOA	0.174279642	0.760213368	4677.886506	4599.537102	138.4263412
Chongqing	GA	−0.2409773	0.404	3620.932	4444.987	135.5290018
	PSO	−0.1791882	0.376	4076.958	4870.783	137.5514554
	TS	−0.2842986	0.419	3688.706	4319.119	129.9889875
	GA-PSO	−0.1832796	0.368	4079.770	4782.378	137.9463931
	GWO	0.37107107	0.349730734	1793.2	4929.296753	134.6162094
	ACO	0.20927	0.68656	3423.54771	2411.06181	143.13827
	WOA	0.26375	0.37791	4307.5166	4198.05999	135.47575
Beijing	GA	0.04108256	1.355	2254.629	1771.043	139.3380383
	PSO	0.19313777	1.451	2776.859	2205.518	142.2689709
	TS	0.03514477	1.143	2412.204	2061.644	134.4705542
	GA-PSO	0.19358917	1.475	2795.174	2189.538	141.9358962
	GWO	0.02413	1.0794631	3848.5	3421.98167	142.787271
	ACO	0.18732	1.49147	1549.065	501.56558	142.26957
	WOA	0.05408	1.34238	3218.24506	1773.0083	140.1626
Fujian	GA	0.22514772	0.590	3637.826	4579.250	142.2406257
	PSO	0.28325903	0.570	3913.100	5072.394	143.4049205
	TS	0.20638095	0.571	3878.496	4835.857	142.5148742
	GA-PSO	0.28217874	0.574	4015.743	5149.098	142.7637285
	GWO	0.175213	0.6904693	3546.73	5237.965926	142.4745741
	ACO	0.0881	0.55828	3118.80161	3013.93706	140.67421
	WOA	0.24423	0.5894	3969.74674	4894.99465	143.18743
Guangdong	GA	−0.0182155	0.773	3726.528	3890.209	139.7609521
	PSO	0.20443682	0.847	5099.786	5321.917	143.623968
	TS	0.07819436	0.689	4267.939	5055.408	140.9925964
	GA-PSO	0.19191621	0.868	4924.046	5102.463	143.4807898
	GWO	0.07642	0.529106633	2413.4	5494.184909	139.6773114
	ACO	0.01171	0.93869	2813.96189	2635.5113	143.94389
	WOA	0.08631	0.90093	5400.70997	4434.6065	140.43258
Henan	GA	0.03165401	0.545	3368.298	4092.461	140.822433
	PSO	0.1737905	0.625	4085.756	4587.971	143.3829755
	TS	0.07302559	0.511	3659.571	4629.002	139.1291955
	GA-PSO	0.17435136	0.632	3991.971	4448.920	141.8448306
	GWO	0.14465	0.4756831	3489.42	4248.275944	142.6228449
	ACO	0.148942621	0.770005605	3299.449583	3630.342338	140.8817316
	WOA	0.07757	0.53026	5025.86857	4643.06323	142.03829
Hubei	GA	0.32561949	0.737	3960.658	4540.024	141.3503044
	PSO	0.44341662	0.732	4829.252	5508.681	143.4251846
	TS	0.31185214	0.638	4302.333	5251.857	142.4815486
	GA-PSO	0.4425353	0.761	4764.457	5264.542	143.1067773
	GWO	0.335696	0.577853453	4735.92	4919.142248	142.7301778
	ACO	0.09014	0.73359	3270.89874	2754.0963	143.4262
	WOA	0.32754	0.385152194	4629.4214	4920.452728	143.0093099
Liaoning	GA	0.22707764	0.762	4526.120	4936.105	141.6013069
	PSO	0.30033789	0.793	5019.892	5299.013	143.3043345
	TS	0.21273809	0.693	4684.906	5266.900	142.4362506
	GA-PSO	0.31364074	0.859	5252.908	5413.817	143.3727105
	GWO	0.2366516	0.66942011	4912.39	5278.033327	143.7512917
	ACO	0.165936542	0.719808117	3619.304888	3859.977924	141.4368164
	WOA	0.13588612	0.803536619	4932.725	5549.337234	143.1917154
Shanxi	GA	0.06694821	0.968	4047.072	4696.883	140.7777041
	PSO	0.15143889	1.032	4567.632	5183.789	141.0973677
	TS	0.01522404	0.820	4280.944	5004.750	140.9927127
	GA-PSO	0.14965432	1.016	4441.090	5061.885	142.1069006
	GWO	0.4374191	0.764998972	3767.254	4313.144202	143.9137643
	ACO	0.12805	1.26367	3369.59206	2531.29845	143.64123
	WOA	0.1342813	0.233596359	3727.224	5912.481336	140.9060167
Sichuan	GA	0.29636834	1.563	4462.515	4508.532	141.5566519
	PSO	0.4093752	1.739	5190.265	4865.166	142.6479077
	TS	0.26092581	1.328	4807.577	5125.814	142.2393422
	GA-PSO	0.41521569	1.697	5200.512	5095.747	143.3968373
	GWO	0.312411	1.4832594	1051.8	5838.707521	143.983844
	ACO	0.16679	1.73444	3727.02255	2623.80746	142.79759
	WOA	0.29775	1.58897	6245.01672	4432.65036	139.40813
Yunnan	GA	0.15121193	0.502	4147.166	5177.010	138.8459589
	PSO	0.25699343	0.506	4537.127	5200.629	141.6078605
	TS	0.13905806	0.463	4121.003	5221.207	131.8650833
	GA-PSO	0.26016103	0.488	4667.335	5233.128	139.2474333
	GWO	0.35006022	0.343908615	4315.6	5237.708628	116.4713578
	ACO	0.04351	0.58748	3588.74495	3584.66385	142.8083
	WOA	0.121399	0.849998505	5322.391	5236.716691	116.4515896

Note: The best-performing heuristic algorithms are highlighted in bold.

To validate the impact of the Lagrange multiplier method on the solution quality in the comprehensive evaluation model, this study independently ran 30 experiments for each of the 14 regions, both with and without the Lagrange multiplier method. The results allow us to assess the effectiveness of introducing the Lagrange multiplier method in improving solution quality. According to Table 6, the average solution quality of the seven algorithms—GA, PSO, TS, GA-PSO, GWO, WOA, and ACO—is 0.11, 0.22, 0.10, 0.22, 0.24, 0.17, and 0.12, respectively, with an overall average solution quality of 0.16. The GWO algorithm demonstrates the best performance. In Table 7, the average solution quality for GA, PSO, TS, GA-PSO, GWO, WOA, and ACO is 0.32, 0.52, 0.28, 0.51, 0.31, 0.26, and 0.18, respectively, with an overall average solution quality of 0.34. The PSO algorithm achieves the best solution quality. The experimental results indicate that the introduction of the Lagrange multiplier method improves the solution quality of all seven heuristic algorithms, thereby confirming the effectiveness of the Lagrange multiplier method.

Table 7. Heuristic algorithm solution results with the Lagrange multiplier.

Regions	Heuristic Algorithm	Total Score	Attraction Score	Total Price	Total Distance	Total Time
Average	GA	0.322178	1.286521	3587.533895	3953.797915	139.5926595
	PSO	0.516070	1.488647	4255.794908	4437.428319	142.0211873
	TS	0.275704	1.114073	3829.256913	4337.805595	138.3759033
	GA-PSO	0.508757	1.471750	4221.335275	4365.732308	142.5949872
	PSO-LD	0.573223	1.517661	4693.792445	4604.835046	143.210438
	GWO	0.314600	1.24696	3269.271475	4687.873074	142.3470281
	ACO	0.187983	1.43824	3209.850322	2889.86823	142.4782551
	WOA	0.261681	1.11985	4592.304133	4316.428602	142.104248
Chongqing	GA	−0.270373	0.5394144	3429.641719	4141.647626	128.6424247
	PSO	−0.180834	0.4939792	4222.172682	4877.11915	137.130656
	TS	−0.281946	0.4916011	3667.159743	4309.469524	129.1350357
	GA-PSO	−0.179522	0.5100561	4202.856713	4918.551989	142.2818999
	PSO-LD	0.029990	0.6154620	4960.241142	4903.510069	142.293876
	GWO	0.374297	0.46719	2751.73	4929.799021	137.62249
	ACO	−0.41524	0.87813	3484.46871	2124.04707	143.55059
	WOA	−0.2753	0.50204	5238.93679	4624.97835	143.81223
Beijing	GA	0.384573	2.554607	2212.259580	1601.286197	141.149411
	PSO	0.598734	2.938980	2746.666748	1949.049860	140.163123
	TS	0.279542	2.016655	2262.845423	1843.104519	134.205473
	GA-PSO	0.600229	2.839160	2703.647904	1992.752302	142.609404
	PSO-LD	0.665692	2.889716	2882.597641	2143.055150	142.421523
	GWO	0.2441	2.73339	1978.2	2464.06244	140.80078
	ACO	0.17395	2.51034	1641.78556	1647.73044	143.08966
	WOA	0.3124	1.63231	3541.91016	2688.37731	142.00472
Fujian	GA	0.531964	0.954620	3521.110483	4328.519918	142.691956
	PSO	0.596072	0.929261	3790.227579	4864.455047	143.105688
	TS	0.480412	0.882013	3803.271538	4693.139134	141.664239
	GA-PSO	0.598953	0.948503	3739.960691	4799.782554	143.663949
	PSO-LD	0.608167	0.944115	3986.643754	4749.743860	143.371798
	GWO	0.379270	0.84871	3596.712	5238.678316	142.48348
	ACO	0.30302	0.98839	3220.0485	3080.57131	143.50714
	WOA	0.24155	0.93505	3456.5377	4697.11099	143.21389
Guangdong	GA	0.191454	1.277165	3694.493547	3573.563192	142.759773
	PSO	0.580721	1.920536	4501.281092	3998.897965	142.352891
	TS	0.284078	1.273907	4045.067295	4339.585729	138.078155
	GA-PSO	0.569503	1.868134	4339.508174	3853.390449	141.700714
	PSO-LD	0.660199	2.057983	4729.486954	4106.535990	142.331700
	GWO	0.4213	1.57092	4493.76	5359.958377	143.99948
	ACO	0.34266	1.72628	3018.27976	2507.34387	142.34180
	WOA	0.41379	1.69509	6098.30179	3692.78982	140.15987
Henan	GA	0.260709	0.922805	3251.513215	3742.518932	137.011013
	PSO	0.519148	1.139095	3838.084707	4229.849247	142.306449
	TS	0.256105	0.804810	3613.979525	4422.425450	141.480318
	GA-PSO	0.501995	1.132567	3694.892615	4015.206472	143.056748
	PSO-LD	0.568605	1.163118	4049.914186	4239.367518	143.992094
	GWO	0.209650	0.17384	3543.4	3587.953943	142.21551
	ACO	0.37485	1.04335	3058.53375	3581.69467	136.77118
	WOA	0.34236	0.89641	3542.212	4142.90839	143.78635
Hubei	GA	0.488349	0.935730	3425.649541	4133.381761	138.769343
	PSO	0.690466	1.056581	4606.237396	4903.503928	142.793799
	TS	0.457141	0.882143	3703.522707	4502.212088	142.177651
	GA-PSO	0.681646	1.055116	4400.932758	4647.966270	142.032912
	PSO-LD	0.750173	1.049517	4937.608339	5435.265037	143.807480
	GWO	0.43318	0.98421	2869.91	5679.857075	143.99821
	ACO	0.02823	0.75069	3617.81015	2719.52484	143.99406
	WOA	0.3967	0.32204	5426.8389	4712.59525	140.76996
Liaoning	GA	0.482333	1.148816	4130.981870	4401.587766	142.504302
	PSO	0.638487	1.202988	4776.567819	4926.335223	143.379190
	TS	0.386229	0.942706	4399.628486	4940.500250	140.589586
	GA-PSO	0.620727	1.232774	4739.839950	4701.972595	143.141324
	PSO-LD	0.658352	1.223738	5121.062512	5091.577888	143.644724
	GWO	0.42668	1.16191	3948.22	5439.081667	143.98852
	ACO	0.248813	1.14844	3606.715001	3837.845429	142.21631
	WOA	0.309148	1.01482	4226.8123	4321.01016	141.17525
Shanxi	GA	0.391012	1.874504	3902.026583	4152.240300	139.517677
	PSO	0.508645	1.946172	4347.713354	4763.207684	142.806763
	TS	0.265351	1.576359	4026.282831	4324.026499	141.116998
	GA-PSO	0.498398	1.879460	4391.284745	4862.051494	141.842310
	PSO-LD	0.527258	1.909319	4583.369569	5110.605850	142.882573
	GWO	0.231457	1.09921	2931.5485	4599.42873	143.99267
	ACO	0.22787	2.08402	3384.45864	2467.80105	143.84751
	WOA	0.37634	1.75271	4965.51255	4462.31565	142.77895
Sichuan	GA	0.445382	1.957627	4110.874179	4265.924883	140.110375
	PSO	0.676469	2.429643	5063.932190	4598.927088	142.886589
	TS	0.355302	1.652804	4643.428224	4809.412390	141.750988
	GA-PSO	0.671780	2.440860	5485.669038	4664.091320	143.167808
	PSO-LD	0.711608	2.480286	6985.482857	5038.988290	143.987354
	GWO	0.23754	2.78022	3125.6	5258.911991	143.89989
	ACO	0.34209	1.96537	3431.46501	3046.99182	143.08740
	WOA	0.21122	1.73452	4124.9259	4352.06452	142.97602
Yunnan	GA	0.316375	0.699917	4196.788231	5197.308576	142.770320
	PSO	0.532787	0.829239	4665.065514	5262.938001	143.286725
	TS	0.274829	0.617735	4127.383356	5194.180365	133.560588
	GA-PSO	0.523864	0.810870	4514.760158	5201.557637	142.452804
	PSO-LD	0.552190	0.843359	4701.517495	5229.700812	143.371260
	GWO	0.18853	0.65004	3453.645	4232.554834	140.46924
	ACO	0.253588	1.28749	3634.93814	3885.1318	142.37690
	WOA	0.2886	0.713607	5301.05324	5470.13558	140.36524

Note: The best-performing heuristic algorithms are highlighted in bold.

Introduction and Superiority of the PSO-LD Algorithm

As evidenced in Table 6, PSO achieves the highest average total score (0.52) with Lagrange multipliers, demonstrating its superior ability to balance solution quality under complex constraints. Yet struggles persist in adapting to dynamic constraints and regional variations (e.g., attraction density disparities). To overcome these limitations, this paper proposes an enhanced variant: particle swarm optimization with Lagrange dynamics (PSO-LD), incorporating dynamic parameter adaptation and constraint enhancement mechanisms. The following section details its design and empirically validates its performance advantages.

The Laplace distribution is a continuous probability distribution with a sharp peak and heavy-tailed characteristics. It is suitable for modeling noise with sudden changes or high uncertainty. Compared to traditional disturbances, its peakedness ensures a higher concentration around the center, increasing the probability of small-range perturbations and aiding local optimization. In addition, the slowly decaying tail allows larger perturbations, helping escape local optima and enhancing the global search capability. Its probability density function is defined as follows:

$$\mathrm{f}\left(\mathrm{x}|\mu ,\mathrm{b}\right)={\displaystyle \frac{1}{2\mathrm{b}}}\exp (-{\displaystyle \frac{\left|\mathrm{x}-\mu \right|}{\mathrm{b}}})$$

(20)

where μ is the mean of the distribution, and b is the scale parameter. The larger the scale parameter is, the greater the perturbation generated by the Laplacian perturbation term.

Based on the standard particle swarm optimization (PSO) velocity update formula, an additional Laplacian perturbation term $\eta$ is incorporated:

$${\mathrm{v}}_{\mathrm{i}}^{\mathrm{t}+1}=\omega {\mathrm{v}}_{\mathrm{i}}^{\mathrm{t}}+{\mathrm{c}}_1{\mathrm{r}}_1\left({\mathrm{p}}_{\mathrm{i}}^{\mathrm{best}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{t}}\right)+{\mathrm{c}}_2{\mathrm{r}}_2\left({\mathrm{g}}^{\mathrm{best} }{- \mathrm{x}}_{\mathrm{i}}^{\mathrm{t}}\right)+\eta$$

(21)

where the perturbation term $\eta$ follows a Laplace distribution with a mean of 0 and a scale parameter of b:

$$\eta ~\mathrm{L}\mathrm{a}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{e}\left(0,{\mathrm{b}}_{\mathrm{t}}\right)$$

(22)

The scale parameter $b_t$ is initialized with an exponential decay strategy and dynamically adjusted according to the number of iterations:

$$b_t=b_0{\mathrm{e}}^{-\alpha {\displaystyle \frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}}$$

(23)

where $b_0$ denotes the initial scale parameter, which is set to 5 to ensure sufficiently large initial perturbations. The decay parameter α is configured as 5 to maintain a reasonable descent rate for the scale parameter. Here, t and $t_{max}$ represent the current iteration number and maximum iteration count, respectively. The Laplace perturbation adaptive adjustment mechanism functions as follows: During the initial phase, it maintains substantial perturbations to enhance particle randomness, thereby facilitating global exploration while progressively reducing perturbation magnitudes in the middle and later stages to stabilize particle movements and strengthen local search capabilities.

This study recorded the Laplace perturbation terms generated in each iteration and plotted corresponding line, which is shown in Figure 4. The visual representation clearly demonstrates that the perturbation terms maintain relatively large magnitudes during the early search phase while remaining confined within a small fluctuation range during the latter search stages.

Figure 4. The value of Laplace perturbation.

Further comparative experiments can be conducted to comprehensively and quantitatively validate the superiority of the PSO-LD algorithm in terms of solution quality, convergence rate, and stability.

To systematically assess the differences in performance metrics among the algorithms, this study independently ran 30 experiments for each of the 10 regions and calculated the mean results for each group, as shown in Table 8. This paper applied the Wilcoxon signed-rank test to perform statistical significance analysis for each pair of the seven algorithms. A p-value less than 0.05 indicates a significant difference between the two algorithms; otherwise, no significant difference is observed. The p-values are presented in Table 8, with the results visualized in Figure 5. The findings show that all p-values are less than 0.05, indicating significant differences between each pair of the eight algorithms.

Table 8. Wilcoxon test comparisons.

Comparison	p-Value
GA vs. PSO	0.001953125
GA vs. TS	0.037109375
GA vs. GA-PSO	0.001953125
PSO vs. TS	0.001953125
PSO vs. GA-PSO	0.02734375
TS vs. GA-PSO	0.001953125
PSO-LD vs. PSO	0.001953125
PSO-LD vs. TS	0.001953125
PSO-LD vs. GA	0.001953125
PSO-LD vs. GA-PSO	0.001953125
GWO vs. ACO	0.02734375
GWO vs. WOA	0.19335937
GWO vs. PSO	0.083984375
GWO vs. GA	1.0
GWO vs. TS	0.083984375
GWO vs. GA-PSO	0.083984375
GWO vs. PSO-LD	0.009765625
ACO vs. WOA	0.00390625
ACO vs. PSO	0.037109375
ACO vs. TS	0.625
ACO vs. GA	0.232421875
ACO vs. GA-PSO	0.048828125
ACO vs. PSO-LD	0.00390625
WOA vs. PSO	0.064453125
WOA vs. GA-PSO	0.064453125
WOA vs. PSO-LD	0.064453125
WOA vs. GA	0.130859375
WOA vs. TS	0.064453125

Figure 5. Wilcoxon test p-value heatmap of GA, PSO, TS, GA-PSO, PSO-LD, GWO, ACO, and WOA.

The total scores of GWO, ACO, and WOA are consistently lower than those of GA, PSO, GA-PSO, and PSO-LD. This performance gap may stem from the severe convergence of individuals in the later stages of the GWO, ACO, and WOA algorithms, which reduces population diversity and diminishes their optimization capability. Although GWO, ACO, and WOA achieve higher total scores than TS, the Wilcoxon test reveals no statistically significant difference between their results and those of GA, PSO, GA-PSO, TS, and PSO-LD. This suggests that GWO, ACO, and WOA exhibit similar characteristics to these algorithms in certain aspects, indicating that further parameter tuning or structural modifications may be necessary to enhance their effectiveness in solving the TTDP problem. Given the limited existing research on this issue, this study focuses on conducting a more in-depth analysis of four widely used algorithms, GA, PSO, TS, and GA-PSO, along with our proposed innovative algorithm PSO-LD.

PSO-LD outperformed all the baseline algorithms, achieving the highest total score (0.573 vs. PSO’s 0.516 and GA-PSO’s 0.509) and attraction score (1.518 vs. PSO’s 1.489) across regions (Table 6), with statistical significance (p < 0.002 for all comparisons). Its superiority stems from the integration of dynamic parameter adaptation and Lagrange multiplier-driven constraints, enabling robust global exploration while maintaining precise local refinement. Unlike PSO, which balances resource efficiency (total price: CNY 4256) and runtime (142.0 s), PSO-LD prioritizes the solution quality despite higher resource costs (price: CNY 4694; distance: 4605). GA-PSO, while consistent due to its hybrid GA-PSO structure, lagged behind PSO-LD in both the total score and computational efficiency (runtime: 256.5 s vs. 143.2 s), reflecting the limitations of static hybrid mechanisms in dynamic environments. GA excelled in speed (139.6 s) and low resource use (price: CNY 3588) but delivered subpar optimization quality (total score: 0.322; p < 0.05). TS remained the weakest performer (total score: 0.276; runtime: 1084.5 s), highlighting its incompatibility with modern multi-objective tasks.

PSO-LD is optimal for quality-critical scenarios requiring adaptability to dynamic constraints (e.g., tourism planning with spatial heterogeneity). PSO is ideal for balanced quality-speed needs, offering an efficient global search with moderate resource use. GA is suitable for time-sensitive tasks with relaxed quality thresholds. TS should be avoided due to its prohibitive slowness and ineffectiveness compared to modern alternatives. This analysis underscores PSO-LD’s unmatched optimization capability, positioning it as the premier choice for high-stakes applications demanding precision and adaptability.

The statistical significance analysis using the Wilcoxon signed-rank test further confirms the performance differences among the algorithms. All pairwise comparisons involving PSO-LD showed p-values of less than 0.002, indicating highly significant differences. Similarly, significant differences were observed between other algorithm pairs, with p-values consistently below 0.05. This statistical evidence reinforces the reliability of the performance rankings and the validity of the trade-off analysis presented.

5.3.2. Qualitative Results

The research analyzed the planned routes generated by GA, PSO, TS, GA-PSO, and PSO-LD across four regions. The results showed that GA-PSO and PSO-LD generally selected more tourist spots and achieved higher total scores in most regions. In Chongqing and Sichuan, GA-PSO performed the best in terms of the number of spots selected. In Shanxi, GA-PSO and PSO-LD had similar high scores. In Fujian, PSO-LD outperformed others in both the number of spots and total score. TS and GA usually selected fewer spots and had lower scores. These results highlight the performance differences among the algorithms in specific problem instances.

In Figure 6, histograms for only four regions—Chongqing, Shaanxi, Sichuan, and Fujian—are displayed, with visualizations for the remaining six regions included in Appendix A.

Figure 6. Histogram of selected attraction counts using different heuristic algorithms.

Figure 7 illustrates the path planning results for four regions, while the results for the remaining six regions are available in Appendix B, with detailed route information provided in Appendix C, offering a comprehensive comparative analysis of different algorithms in path planning.

Figure 7. Path planning results for selected regions using different heuristic algorithms. Introduction to each section: (a) GA; (b) PSO; (c) TS; (d) GA-PSO; (e) PSO-LD. Each red point represents a selected attraction, with larger points indicating higher star ratings. The size of the points highlights the quality of the attractions included in the optimized routes for the selected regions.

In Figure 7, the blue lines represent the planned paths, green dots indicate all attractions, red dots denote the selected attractions, and the yellow dot marks the starting point. The size of each attraction’s circle reflects its TOPSIS comprehensive score, with higher scores represented by larger circles, visually demonstrating the priority selection by each algorithm. Figure 7a shows the path planning result using GA; Figure 7b displays the result using PSO; Figure 7c shows the result using TS; Figure 7d shows the path planning result using GA-PSO; and Figure 7e shows the path planning result using PSO-LD.

PSO-LD shows the best performance in path planning, with highly optimized paths focusing on high-score attractions and excellent path efficiency. GA-PSO also performs well, generating well-optimized paths that cover broad areas and include many high-score attractions. In contrast, GA and TS generally have lower path efficiency, with GA often producing scattered paths and TS showing a more balanced but less optimized approach. PSO demonstrates better path efficiency than GA by focusing on central or key areas but is still outperformed by PSO-LD and GA-PSO. Overall, PSO-LD stands out as the most effective algorithm for optimizing paths for high-score attractions.

6. Sensitivity Analysis

To evaluate the robustness of the heuristic algorithms, this study conducted a parameter sensitivity analysis. Population size and iterations were selected as the analysis parameters based on their significance in previous studies [41]. The initial values were set to 200 for population size and 10,000 for iterations. These parameters were adjusted within a ±30% range of their initial values, using a step size of 3%. Robustness was assessed through both quantitative and qualitative methods.

6.1. Sensitivity Analysis of Population Size

Figure 8 illustrates the solution results for the scenic spot planning problem in four cities, highlighting the sensitivity of five heuristic algorithms to variations in the population size parameter. Additional results for the remaining six regions are provided in Appendix D. The figure demonstrates that fitness values fluctuate to varying degrees in response to changes in the population size parameter, reflecting differences in the sensitivity and robustness of the algorithms.

Figure 8. Line chart of the sensitivity analysis of population size. Introduction to each section: (a) GA; (b) PSO; (c) TS; (d) GA-PSO; (e) PSO-LD.

Among the algorithms, TS shows the largest fluctuations in its sensitivity curves, indicating a high sensitivity to the population size parameter and relatively low robustness. In contrast, PSO exhibits consistently smooth sensitivity curves across all four cities, with minimal changes in fitness values despite adjustments to the population size parameter. This behavior underscores the higher robustness of the PSO algorithm in comparison to the other methods.

To quantitatively assess the sensitivity of the algorithms to the population size parameter, the standard deviation of fitness values was used as a metric to measure dispersion. By evaluating the variation in fitness values as the population size parameter changes, the sensitivity and robustness of the five algorithms were analyzed. Table 9 summarizes the average standard deviation of each algorithm across ten regions, along with the standard deviation for individual regions.

Table 9. Table of the population size sensitivity analysis results of the heuristic algorithms.

Regions	Heuristic Algorithm	Standard Deviation of Total Score
Average	GA	0.025131815
	PSO	0.010396969
	TS	0.055339984
	GA-PSO	0.011262257
	PSO-LD	0.009141917
Chongqing	GA	0.006517431
	PSO	0.004080315
	TS	0.028811662
	GA-PSO	0.006822374
	PSO-LD	0.006856436
Beijing	GA	0.030421694
	PSO	0.012549434
	TS	0.057717018
	GA-PSO	0.015671883
	PSO-LD	0.007785523
Fujian	GA	0.008746289
	PSO	0.003380016
	TS	0.051442043
	GA-PSO	0.00490045
	PSO-LD	0.006869849
Guangdong	GA	0.068290794
	PSO	0.016860248
	TS	0.046145321
	GA-PSO	0.024564914
	PSO-LD	0.016232611
Henan	GA	0.030064013
	PSO	0.009458381
	TS	0.034397698
	GA-PSO	0.013851158
	PSO-LD	0.011134416
Hubei	GA	0.029953996
	PSO	0.011290682
	TS	0.059597174
	GA-PSO	0.012481333
	PSO-LD	0.009331745
Liaoning	GA	0.024259181
	PSO	0.008509057
	TS	0.047126347
	GA-PSO	0.009076379
	PSO-LD	0.009329457
Shanxi	GA	0.01779858
	PSO	0.008817328
	TS	0.042793592
	GA-PSO	0.00940341
	PSO-LD	0.008387251
Sichuan	GA	0.032366996
	PSO	0.012333474
	TS	0.043860678
	GA-PSO	0.009681206
	PSO-LD	0.008387251
Yunnan	GA	0.031365843
	PSO	0.008104632
	TS	0.052952877
	GA-PSO	0.016860248
	PSO-LD	0.007104632

Note: The most robust heuristic algorithms are highlighted in bold.

The results in Table 9 reveal that PSO-LD demonstrates the most consistent performance with the lowest standard deviations, both on average and in most regions, indicating high stability and reliability. GA and TS generally have higher standard deviations, suggesting less stable performance. PSO and GA-PSO fall in between. For example, in Chongqing, PSO-LD has a standard deviation of 0.006856436, lower than GA (0.006517431), PSO (0.004080315), TS (0.028811662), and GA-PSO (0.006822374). This pattern is repeated in other regions like Fujian, Guangdong, and Henan, where PSO-LD maintains lower standard deviations than other algorithms. Overall, PSO-LD is the most stable algorithm for this task.

6.2. Sensitivity Analysis of Iterations

Figure 9 presents the fitness response of the GA, PSO, TS, GA-PSO, and PSO-LD algorithms to iterations in four regions: Chongqing, Fujian, Guangdong, and Henan. Detailed results for the other six regions are provided in Appendix E.

Figure 9. Line chart of the sensitivity analysis of iterations. Introduction to each section: (a) GA; (b) PSO; (c) TS; (d) GA-PSO; (e) PSO-LD.

The results indicate that PSO and PSO-LD exhibit the most stable curves among all the algorithms. Their fitness values show a gradual convergence trend as the number of iterations increases, demonstrating superior robustness. Notably, PSO-LD exhibits a slightly faster convergence rate than PSO in some regions, but both maintain consistent overall trends. GA-PSO follows closely behind, with relatively small fluctuations in fitness values during iterations. In contrast, GA and TS display more significant fluctuations, with TS showing notable fitness oscillations in the four regions presented in Figure 9, indicating the lowest robustness.

In the robustness analysis of the iterations parameter presented in Table 10, PSO-LD demonstrates superior stability, achieving the smallest standard deviation of fitness values in seven out of ten regions. This indicates its strong robustness and reliability in handling complex optimization problems.

Table 10. Table of the iterations sensitivity analysis results of the heuristic algorithms.

Regions	Heuristic Algorithm	Standard Deviation of Total Score
Average	GA	0.027978482
	PSO	0.009538357
	TS	0.046484441
	GA-PSO	0.012331335
	PSO-LD	0.006717210
Chongqing	GA	0.009537532
	PSO	0.007511732
	TS	0.023387923
	GA-PSO	0.008050469
	PSO-LD	0.003233831
Beijing	GA	0.034179029
	PSO	0.016072372
	TS	0.044615311
	GA-PSO	0.013991921
	PSO-LD	0.006835266
Fujian	GA	0.009812209
	PSO	0.003537793
	TS	0.039848350
	GA-PSO	0.004986591
	PSO-LD	0.003611949
Guangdong	GA	0.052006900
	PSO	0.013175710
	TS	0.069766161
	GA-PSO	0.019649286
	PSO-LD	0.007566793
Henan	GA	0.030326817
	PSO	0.011325019
	TS	0.061378017
	GA-PSO	0.011206841
	PSO-LD	0.010673853
Hubei	GA	0.026742293
	PSO	0.008000299
	TS	0.080815773
	GA-PSO	0.011558958
	PSO-LD	0.004181048
Liaoning	GA	0.018601269
	PSO	0.008000299
	TS	0.067504483
	GA-PSO	0.006287184
	PSO-LD	0.006627069
Shanxi	GA	0.011885994
	PSO	0.007976459
	TS	0.042793592
	GA-PSO	0.007691356
	PSO-LD	0.005127771
Sichuan	GA	0.030165131
	PSO	0.012297636
	TS	0.061602403
	GA-PSO	0.013067360
	PSO-LD	0.005273514
Yunnan	GA	0.028060975
	PSO	0.016072372
	TS	0.061687824
	GA-PSO	0.016132606
	PSO-LD	0.014034173

Note: The most robust heuristic algorithms are highlighted in bold.

Compared to other algorithms, PSO-LD outperforms TS, which has the largest fitness value fluctuations and poorest stability. It also surpasses GA and GA-PSO, whose stability is inferior to that of PSO-LD in most regions. PSO-LD’s enhanced performance stems from its incorporation of local search mechanisms into the PSO framework, allowing for greater local optima refinement while preserving PSO’s global search capabilities. This makes PSO-LD particularly effective in complex optimization scenarios, where both global and local search capabilities are crucial.

7. Discussion

This study proposes a comprehensive approach to the tourist trip design problem (TTDP), integrating multi-criteria decision-making and heuristic algorithms to balance tourist experience, cost, and efficiency. The key contribution of this approach lies in combining subjective and objective weighting methods, incorporating expert judgment and data variability for weight distribution, resulting in a more balanced and scientific weighting process. Additionally, the introduction of the Lagrange multiplier method enhances the robustness of the approach by optimizing the weights under constraints, ensuring consistency and fairness in the evaluation of tourist destinations. The experimental results demonstrate that the PSO-LD algorithm performs excellently in multi-area planning, providing optimal solutions in most regions and leading other algorithms in both overall scores and individual attraction scores, highlighting the effectiveness of PSO-LD in comprehensive route optimization under complex constraints.

The strategy proposed in this study consists of three main components: the integrated evaluation model optimized by the Lagrange multiplier method, the weighted and optimization model, and the PSO-LD algorithm. The time complexity analysis of each component is as follows:

• Integrated Evaluation Model: This model involves solving small-scale systems of equations, with a time complexity of O(1), meaning the computation is constant in nature.
• Weighted and Optimization Model: During each fitness evaluation, all decision variables need to be traversed, resulting in a time complexity of O(n), where n represents the number of decision variables.
• PSO-LD Algorithm: In each generation, P particles need to be updated, and the computation of each particle’s update and fitness evaluation is linearly related to the problem size n. The total computation in a single iteration is P times the problem size n, and the algorithm iterates T times, resulting in an overall time complexity of O(PTn).

By combining these three parts, we find that the O(1) complexity of the integrated evaluation model and the O(n) complexity of the weighted and optimization model are relatively small, while the O(PTn) complexity of the PSO-LD algorithm is the dominant factor determining the overall computational load. Despite this, the multi-area planning optimization method based on the PSO-LD algorithm remains highly efficient and scalable in practical applications.

The integration of multi-criteria weighting methods with heuristic algorithms marks a significant advancement in TTDP research. Compared to existing approaches, this study not only compares multiple algorithms but also enhances the adaptability, scalability, and robustness of the method, providing valuable insights for real-world applications, particularly in multi-day travel planning under time or cost constraints.

To further substantiate the effectiveness of the heuristic algorithms employed, Table 11 below compares the characteristics of each algorithm used in this study based on various criteria, such as global search ability, computational cost, convergence speed, and suitability for different scenarios.

Table 11. Comparison of the heuristic algorithms.

Algorithm	Key Features	Applicable Scenarios	Advantages	Disadvantages
GA	Simulates natural selection, uses crossover and mutation for path optimization	Suitable for large-scale optimization problems like TSP and route planning	Good global search ability, avoids local optima	Slow convergence, may get stuck in local optima
PSO	Particles adjust their positions based on individual best and global best positions	Continuous optimization problems, suitable for finding optimal solutions	Faster convergence, effective for complex high-dimensional problems	Prone to local optima, depends on initial particle distribution, may require parameter adjustments
TS	Explores neighboring solutions and uses a tabu list to avoid revisiting previous solutions, preventing local optima	Suitable for combinatorial optimization problems like TSP and other route planning	Effectively avoids local optima, adapts well to complex constraints	High computational cost, may take long to converge in some problems
GA-PSO	Combines GA’s genetic diversity with PSO’s global search capability	Suitable for path optimization problems with complex constraints	Balances global and local search, robust optimization	Complex parameter tuning, may require fine-tuning for specific problems
PSO-LD	Extends PSO by adding a dynamic Laplace perturbation, enhancing global search ability	Suitable for high-dimensional, complex problems, especially in dynamic environments	Efficient global search, avoids local optima, faster convergence	Increased computational complexity, requires dynamic adjustment of perturbation parameters
GWO	Simulates the hunting strategies of gray wolves, with decision-making and hunting behaviors	Suitable for combinatorial optimization problems, especially hierarchical problems	Effective global exploration, avoids local optima	Slower convergence, susceptible to local optima, heavily influenced by parameters
ACO	Imitates the foraging behavior of ants, especially their use of pheromone trails	Suitable for path planning, TSP, and other discrete problems	Strong path search capability, suitable for discrete problems	High computational cost, sensitive to parameters like ant numbers and pheromone evaporation rate
WOA	Simulates the hunting and social behaviors of whales, optimizing through encircling and attacking	Suitable for solving constrained or unconstrained optimization problems	Avoids local optima, strong adaptability, low computational cost	Performance may degrade in high-dimensional problems, requires parameter tuning to avoid premature convergence

As shown in Table 11, each algorithm has distinct features and is suited to different types of optimization problems. For example, GA is highly effective for large-scale optimization problems, while PSO excels in continuous optimization and is quicker to converge compared to TS, which is better at avoiding local optima. PSO-LD, with its integration of a dynamic Laplace perturbation, outperforms the others, achieving the best overall results in most regions due to its enhanced global exploration and faster convergence. This algorithm’s superior performance underscores its effectiveness in solving the TTDP under complex constraints.

Despite these achievements, this study has certain limitations. For example, the dataset is limited to tourism data from China, restricting the generalizability of the findings to regions with different cultural, logistical, and economic characteristics. Furthermore, due to the limitations of the dataset, the model has not been fully validated for its adaptability to dynamic factors such as real-time pricing and weather changes. While dynamic parameter adjustments can address these factors, direct integration with real-time data streams still requires further development. Lastly, the robustness of the TOPSIS framework in different optimization domains has not been sufficiently explored, limiting its broader applicability across various fields.

Future research will validate the model’s adaptability and scalability using more diverse regional datasets and enhance its dynamic response capability by integrating real-time data streams via APIs and incorporating online learning techniques. Additionally, combining this framework with recommendation algorithms will further optimize the weighting and parameters, improving model performance. Through application validation in other fields, the robustness and improvements of the TOPSIS framework will be further confirmed, ensuring its broader applicability and providing more effective solutions for complex multi-objective optimization problems.

8. Conclusions

This study presents a comprehensive framework for tourism route planning by integrating a weighted sum optimization method with multi-criteria decision-making and heuristic algorithms. The method combines subjective and objective evaluation approaches for balanced and scientific analysis, using AHP for subjective weights based on expert judgment and EWM for objective weights based on data variability. The Lagrange multiplier method optimizes the combination of these weights, ensuring balanced and robust weighting, which are then applied in the TOPSIS model to generate fair rankings for tourist sites.

The study evaluates eight heuristic algorithms—GA, PSO, TS, GA-PSO, GWO, ACO, WOA, and PSO-LD—using the TOPSIS model across multiple regions. One innovation is that this study introduces the Lagrange multiplier method to solve GA, PSO, TS, GA-PSO, GWO, WOA, and ACO. The experimental results show that the introduction of the Lagrange multiplier method improves the solution quality of all seven heuristic algorithms, the quality of the average solution has been improved by 112.5% (from 0.16 to 0.34), and the PSO algorithm can achieve the best solution quality. Based on this, this study improves the PSO algorithm.

A key contribution is the development of the PSO-LD algorithm, which incorporates a dynamic adaptive Laplace perturbation into the PSO framework, enhancing its performance. PSO-LD outperforms other algorithms with the highest scores in most regions, demonstrating superior stability and robustness in sensitivity analyses. Compared with the traditional PSO algorithm, the total score obtained by the PSO-LD algorithm is improved by about 11% on average, achieving higher solution quality and faster convergence.

PSO-LD outperformed all baseline algorithms, achieving the highest total score (0.573 vs. PSO’s 0.516 and GA-PSO’s 0.509) and attraction score (1.518 vs. PSO’s 1.489) across regions, with statistical significance (p < 0.002 for all comparisons). Its superiority stems from the integration of dynamic parameter adaptation and Lagrange multiplier-driven constraints, enabling robust global exploration while maintaining precise local refinement. Unlike PSO, which balances resource efficiency (total price: CNY 4256) and runtime (142.0 s), PSO-LD prioritizes solution quality despite higher resource costs (price: CNY 4694; distance: 4605). GA-PSO, while consistent due to its hybrid GA-PSO structure, lagged behind PSO-LD in both total score and computational efficiency (runtime: 256.5 s vs. 143.2 s), reflecting the limitations of static hybrid mechanisms in dynamic environments. GA excelled in speed (139.6 s) and low resource use (price: CNY 3588) but delivered subpar optimization quality (total score: 0.322; p < 0.05). TS remained the weakest performer (total score: 0.276; runtime: 1084.5 s), highlighting its incompatibility with modern multi-objective tasks.

The results show that GA is suitable for time-sensitive tasks with relaxed quality thresholds. PSO performs well in multi-regional planning under specific constraints, achieving high solution quality and efficiency. GA-PSO is competitive in scenarios requiring more attractions due to its balanced exploration and exploitation. TS, however, shows limited performance under constraints. In the same way, GWO, ACO, and WOA should be avoided because their total scores are consistently lower than those of GA, PSO, GA-PSO, and PSO-LD. This performance gap may stem from the severe convergence of individuals in their later stages.