Application of Fuzzy Decision Theory in Multi Objective Logistics Distribution Center Site Selection

Abstract

In this paper, a fuzzy multi-objective particle swarm optimization algorithm (FMOPSO) based on fuzzy set theory and multi-objective decision analysis is proposed, aiming at solving the uncertainty and multiobjective optimization challenges in the problem of logistics center location. By combining the fuzzy affiliation function and particle swarm optimization algorithm, FMOPSO is able to efficiently find the global optimal solution while dealing with data uncertainty. The design and implementation of the FMOPSO algorithm is detailed in the study, and its parameter sensitivity is tested to ensure the robustness and reliability of the algorithm. In order to verify the effectiveness of the FMOPSO algorithm, a series of computational experiments are conducted and its performance is compared with existing state-of-the-art methods such as genetic algorithm GA, differential evolution DE and multi-objective particle swarm optimization MOPSO. The experimental results show that FMOPSO exhibits significant advantages in terms of convergence speed, solution quality and uncertainty handling. Specifically, the fastest convergence time of FMOPSO is 23 seconds, the average convergence time is 35 seconds, and the slowest convergence time is 48 seconds, which is significantly better than other algorithms. In addition, FMOPSO also performs best in terms of average objective function error, maximum objective function error, and minimum objective function error, which are 0.012, 0.035, and 0.007, respectively. These quantitative results demonstrate the high efficiency and accuracy of FMOPSO in practical applications. By testing the parameter sensitivity, we found that FMOPSO has low sensitivity to parameter variations, leading to only 3.5% decrease in solution stability, which demonstrates its stability and reliability under different environmental conditions. In addition, FMOPSO shows unique advantages in handling high-dimensional data and complex constraints, and can better cope with real-world large-scale logistics center siting problems. In summary, this study demonstrates the superior performance of the FMOPSO algorithm in the logistics center location problem through detailed computational experiments and comparative analysis. The algorithm not only performs well in terms of numerical results, but also has a unique design and implementation approach, making it a powerful tool for solving such problems.

Headnote

Keywords: fuzzy decision theory, multi-objective logistics and distribution, distribution center site selection

Received: August 20, 2024

In this paper, a fuzzy multi-objective particle swarm optimization algorithm (FMOPSO) based on fuzzy set theory and multi-objective decision analysis is proposed, aiming at solving the uncertainty and multiobjective optimization challenges in the problem of logistics center location. By combining the fuzzy affiliation function and particle swarm optimization algorithm, FMOPSO is able to efficiently find the global optimal solution while dealing with data uncertainty. The design and implementation of the FMOPSO algorithm is detailed in the study, and its parameter sensitivity is tested to ensure the robustness and reliability of the algorithm. In order to verify the effectiveness of the FMOPSO algorithm, a series of computational experiments are conducted and its performance is compared with existing state-of-the-art methods such as genetic algorithm GA, differential evolution DE and multi-objective particle swarm optimization MOPSO. The experimental results show that FMOPSO exhibits significant advantages in terms of convergence speed, solution quality and uncertainty handling. Specifically, the fastest convergence time of FMOPSO is 23 seconds, the average convergence time is 35 seconds, and the slowest convergence time is 48 seconds, which is significantly better than other algorithms. In addition, FMOPSO also performs best in terms of average objective function error, maximum objective function error, and minimum objective function error, which are 0.012, 0.035, and 0.007, respectively. These quantitative results demonstrate the high efficiency and accuracy of FMOPSO in practical applications. By testing the parameter sensitivity, we found that FMOPSO has low sensitivity to parameter variations, leading to only 3.5% decrease in solution stability, which demonstrates its stability and reliability under different environmental conditions. In addition, FMOPSO shows unique advantages in handling high-dimensional data and complex constraints, and can better cope with real-world large-scale logistics center siting problems. In summary, this study demonstrates the superior performance of the FMOPSO algorithm in the logistics center location problem through detailed computational experiments and comparative analysis. The algorithm not only performs well in terms of numerical results, but also has a unique design and implementation approach, making it a powerful tool for solving such problems.

Povzetek: Raziskava predstavlja razvoj algoritma FMOPSO, ki zdruzuje teorijo mehkih mnoZic in optimizacijo roja delcev, za ucinkovito resevanje vecciljnih logisticnih problemov.

(ProQuest: ... denotes formulae omitted.)

1 Introduction

In the context of globalization and e-commerce development, logistics and distribution centers, as the key nodes of the supply chain, have a direct impact on the operating costs, customer satisfaction and market competitiveness of enterprises in terms of their site selection decisions. Logistics and distribution centers not only undertake the functions of commodity storage, sorting, packaging and transportation, but also have a significant pull effect on regional economic development. However, the site selection decision faces many complex factors, including transportation network conditions, land costs, market demand forecasts, environmental regulations, policy support, etc. These factors are often contradictory and full of uncertainty. In practice, companies often need to find a balance between multiple objectives, such as minimizing total cost, maximizing market coverage, reducing environmental pollution, and improving response time. Traditional site selection models usually only optimize a single objective and ignore realworld multi-objective conflicts. In addition, information such as market demand, traffic congestion level, and policy changes are often difficult to accurately quantify, which poses a great challenge for decision makers [1]. The emergence of fuzzy decision theory provides a powerful tool to solve the above problems.

The study of logistics and distribution center siting has developed into a mature field in the past decades. Early research focused on deterministic single-objective optimization models such as linear programming and integer programming. As the complexity of the problem increased, researchers began to explore stochastic programming, robust optimization, and fuzzy optimization to deal with uncertainty. In recent years, fuzzy decision theory has been increasingly used in the field of logistics. For example, the literature evaluates multiple alternatives for the location of logistics and distribution centers in India using Fuzzy Hierarchical Analysis (Fuzzy AHP) [2]; another study achieves an integrated location of logistics centers through the Fuzzy Comprehensive Evaluation (FCE) method, which takes into account the indicators of the three major dimensions, namely, economic, environmental, and social [3].

Despite the fact that fuzzy decision theory performs well in solving multi-objective problems, there are still some limitations in existing research. First, most models assume that the decision maker's preferences are fully rational and quantifiable, whereas in real situations, human judgments are often fuzzy and subjective. Second, existing fuzzy decision-making models may be less efficient when dealing with large-scale data and highdimensional features. Finally, many studies lack empirical validation, and the practicality and effectiveness of their models need to be further tested [4].

In view of the above research background and the problems revealed in the literature review, this study aims to develop a novel multi-objective logistics and distribution center siting model based on fuzzy decision theory, which will pay special attention to the following objectives:(1) Constructing a decision-making framework capable of handling uncertainty and multi-objective conflicts simultaneously to achieve a more comprehensive decision on logistics and distribution center siting. (2) Improve the efficiency and accuracy of the model in handling complex data sets by introducing more advanced data processing techniques and algorithm optimization. (3) Test the model using real-world datasets to evaluate its performance in different scenarios and conduct comparative analysis with existing models.

2 Theoretical framework

2.1 Multi-objective decision-making theory

Multi-Objective Decision Making (MADM) is an important branch of decision science, which deals with the problem of making optimal choices under multiple conflicting or complementary objectives [5]. In a typical MADM problem, the decision maker is faced with the problem of how to achieve the optimal combination of multiple objectives, which may be economic, social, environmental, or technological, under the condition of limited resources. Finding a single optimal solution is usually impossible due to the trade-offs among different objectives, so multi-objective decision theory focuses on generating a series of non-inferior solutions, i.e., Paretooptimal solutions, for the decision maker to make a final choice based on his/her personal preferences [6]. The application of multi-objective decision theory is especially critical in complex decision-making environments. For example, in the logistics distribution center location problem, decision makers must simultaneously consider multiple objectives such as cost minimization, service optimization, and environmental impact minimization. The literature proposes a site selection model based on fuzzy multi-objective planning, which effectively handles the multi-objective conflicts in logistics center location and demonstrates the potential of multi-objective decision theory in solving such problems [7].

2.2 Fuzzy set theory

Fuzzy set theory was proposed in 1965, which is an extension of classical set theory to deal with the ambiguities and uncertainties prevailing in the real world [8]. In fuzzy sets, the degree of affiliation of an element to a set is not simply "belongs" or "does not belong", but is a continuous value between 0 and 1, indicating the degree to which the element belongs to the set. This characteristic makes fuzzy set theory a powerful tool for dealing with fuzzy information and uncertainty [9]. The role of fuzzy decision making in dealing with uncertainty cannot be underestimated. In a complex problem such as logistics and distribution center location, many key parameters such as market demand, traffic flow, and land price are fuzzy or changing. A multi-objective genetic algorithm based on fuzzy sets has been proposed in the literature to successfully solve the decision-making problem containing fuzzy objectives and constraints. With fuzzy decision making, decision makers can make more robust and adaptive decisions in uncertain environments [10].

2.3 Fuzzy multi-objective decision-making methods

Combining fuzzy set theory with multi-objective decision-making theory can produce a series of effective fuzzy multi-objective decision-making methods, such as Fuzzy Analytic Hierarchy Process (FAHP) and Fuzzy Comprehensive Evaluation (FCE).FAHP quantifies the decision maker's preference for different options by constructing a fuzzy judgment matrix to quantify the decision maker's preference for different options, and then through fuzzy mathematical operations to derive the weight of each option, thus helping the decision maker to prioritize in a fuzzy environment. FCE is a comprehensive evaluation method, which is capable of transforming multiple fuzzy evaluation indexes into an overall evaluation result, and is suitable for decision-making problems involving multiple fuzzy criteria [11]. In the logistics distribution center location problem, the application of these methods can help decision makers find the best solution in the midst of uncertainty and multiobjective conflicts. For example, the literature uses FAHP to evaluate multiple factors of logistics center location, including transportation accessibility, market proximity, and environmental impacts, resulting in a more comprehensive and rational decision-making result [12].

2.4 Current status of research on the location of logistics and distribution centers

Multi-objective optimization is one of the commonly used models in logistics and distribution center siting, Which considers multiple conflicting objectives, such as cost minimization, environmental impact minimization and customer satisfaction maximization. For example, the literature proposes a multi-objective optimization model based on fuzzy set theory, which is able to deal with uncertainties, such as fluctuations in demand and changes in transportation costs, and thus provide more flexible solutions for decision makers [13]. Geographic Information System (GIS) plays an important role in logistics and distribution center site selection, which can help analyze the geographic characteristics of potential locations, such as transportation networks, population density, and infrastructure conditions. The literature uses GIS technology combined with machine learning algorithms to develop an intelligent site selection system, which can automatically evaluate the comprehensive advantages of different locations and provide data-driven decision support for logistics and distribution center site selection. Considering that market demand and operating conditions change over time, dynamic optimization strategies have become a hot research topic [14]. Literature proposes a dynamic planning-based site selection method, which is able to adjust the location of distribution centers based on real-time data to cope with the challenges posed by unforeseen events (e.g., natural disasters or epidemics) and to ensure the flexibility and robustness of the logistics system [15]. As the global concern for sustainable development increases, the location of logistics and distribution centers also needs to consider its impact on the environment. Literature has investigated the problem of siting green logistics and distribution centers by building a multi-objective model that includes carbon emissions and energy consumption to find an optimal solution that reduces both cost and environmental burden [15]. Intelligent optimization algorithms such as genetic algorithm, simulated annealing, and ant colony algorithm are widely used in the siting of logistics and distribution centers to solve highdimensional complex problems. The literature uses deep reinforcement learning algorithms to develop an autonomous learning site selection model that can learn from historical data and predict future trends to achieve intelligent site selection decisions [16].

As shown in table 1, in the existing research, although numerous studies have attempted to address the logistics and distribution center location selection problem using various approaches, these methods often exhibit certain shortcomings. For instance, while methods based on fuzzy set theory are capable of handling uncertainties, they may not fully account for geographical characteristics. On the other hand, approaches that integrate GIS technology with machine learning can provide a comprehensive assessment of site advantages but might lack effective mechanisms to cope with dynamically changing environments. Moreover, although deep reinforcement learning demonstrates strong predictive and self-learning capabilities, its high computational costs limit its widespread application.

To tackle these challenges, this paper introduces a novel method that combines fuzzy decision theory with Particle Swarm Optimization (PSO) - FMOPSO. This approach not only enables effective multi-objective decision-making under uncertainty but also enhances the algorithm's ability to find optimal solutions through the incorporation of PSO. Specifically, the FMOPSO algorithm maintains high search efficiency while effectively balancing multiple conflicting objectives, thereby providing decision-makers with a series of Paretooptimal solutions. This makes our model particularly suitable for addressing real-world logistics location problems involving complex constraints and large datasets. Through specific case studies, we have validated that the proposed method outperforms traditional approaches in terms of convergence speed, solution quality, and uncertainty handling, demonstrating its effectiveness in solving such problems.

3 Methodology

3.1 Study design

In this study, a multi-objective optimization method based on fuzzy decision theory is used to solve the logistics distribution center location problem. Considering the uncertainties in site selection decisions, fuzzy set theory is used to quantify and deal with these uncertainties. In addition, multi-objective decision analysis was used to deal with conflicts between different objectives.

3.2 Data collection and pre-processing

The data collection phase involves obtaining information from public databases, industry reports and expert interviews on logistics demand forecasts, geographic information system (GIS) data, transportation costs, environmental impact indices and land use at potential sites. Data preprocessing includes cleaning, normalization, and fuzzification steps.

The data collection phase is a critical step in our study, involving the gathering of comprehensive information from multiple sources to ensure the accuracy and relevance of our analysis. We obtained data from public databases, industry reports, and expert interviews, focusing on logistics demand forecasts, geographic information system (GIS) data, transportation costs, environmental impact indices, and land use at potential sites. This diverse set of data provides a robust foundation for our multi-objective decision-making process.

Data Cleaning: Initially, we performed data cleaning to remove any inconsistencies, errors, or missing values. This step ensures that the data used for analysis is reliable and complete. Outliers were identified and either corrected or removed, depending on their nature and the overall context of the dataset.

Normalization: To facilitate meaningful comparisons across different datasets, we normalized the collected data. Normalization was carried out using standard techniques such as min-max scaling, which transforms the data to a common scale, typically between 0 and 1. This step is essential for ensuring that no single attribute disproportionately influences the final results due to its scale.

A key aspect of our approach is the fuzzification of data, which is necessary for incorporating uncertainty and imprecision into the decision-making process. Fuzzy set theory allows us to represent and handle these uncertainties more effectively. In this study, we used specific fuzzy membership functions to convert crisp data into fuzzy sets. For instance, triangular and trapezoidal membership functions were employed to quantify logistics demand forecasts and environmental impact indices.

We used a triangular membership function to model the logistics demand forecasts. The function is defined by three points: (a, b, с), where a and с are the lower and upper bounds of the forecast, and b is the most likely value. This function helps to capture the uncertainty in demand predictions, providing a more realistic representation of the data.

For environmental impact indices, a trapezoidal membership function was utilized. This function is defined by four points: (a, b, с, d), where a and d are the lower and upper bounds, and b and с are the points where the function reaches its maximum value. The trapezoidal function is particularly useful for representing ranges with a flat top, which is common in environmental data.

By applying these membership functions, we transformed the crisp values into fuzzy sets, allowing for a more nuanced and flexible representation of the data. This fuzzification process is crucial for handling the inherent uncertainties and vagueness in the input data, ensuring that the subsequent analysis and decision-making are robust and reflective of real-world conditions.

3.3 Multi-objective fuzzy decision modeling framework and its application

In complex system decision making, multi-objective optimization problems are not only a hot topic in academic research, but also an important challenge in industry. Especially in the application scenario of logistics and distribution center siting, which involves numerous uncertainties, the multi-objective fuzzy decision theory is an ideal tool for solving this kind of problems because of its ability to deal with vagueness and uncertainty effectively. In this section, the details of constructing a multi-objective fuzzy decision-making model are discussed in depth. In the process of constructing a multiobjective fuzzy decision model, the first step is to fuzzify the objective function. Suppose we have m objectives, each objective function Z; (x) (1 = 1,2, ... т) needs to be converted to a fuzzy number 7,(х) to reflect the uncertainty of the objective values. For example, the objective function for cost minimization can be expressed as the fuzzy number Z, (x), while the objective function for service coverage maximization can be expressed as Reference [17].

For each objective function Z, (x), we define the corresponding affiliation function 447 (7), which is used to quantify the degree of goal achievement. The value of the affiliation function is in the interval [0, 1], and the closer the value is to 1, the higher the degree of goal attainment. The form of the affiliation function can be triangular, trapezoidal or other shapes, depending on the characteristics of the goal and the preferences of the decision maker.

The multi-objective fuzzy decision-making model can be formally expressed as Equation (1) [18].

...

Where, X is a vector of decision variables and Z, (x) is the defuzzified objective function for the ith objective, representing the fuzzy value of objective i at different decision points X .

The constraints also need to be fuzzified to adapt to the uncertainty environment. The fuzzified inequality constraints and equation constraints can be expressed as

...

where £ ,(X) and ho (x) are the fuzzified inequality and equation constraint functions, respectively.

3.4 Solution methods

In the decision-making problem of logistics distribution center location, it often faces a complex system with multiple objectives and constraints. Traditional optimization techniques may be difficult to capture the essence of the problem, especially when the objectives and constraints carry uncertainties. For this reason, we propose an innovative solution method, Fuzzy Multi-Objective Particle Swarm Optimization Algorithm (FMOPSO), which is an advanced algorithm that integrates Particle Swarm Optimization (PSO) and Fuzzy Logic. The particle swarm algorithm we used is shown in Fig. 1 [19].

The initialization phase is a key step in FMOPSO and involves the generation of particle swarms and the fuzzification of the objective function. First, we randomly generate a swarm of particles, each representing a possible decision scenario, i.e., the potential location of the distribution center. The position and velocity vectors of the particles are randomly initialized to ensure that the search space is fully explored. A fuzzy affiliation function is defined for each objective function to quantify the mapping of the objective function values to the interval [0, 1], reflecting the fuzzy nature of the degree of goal attainment. For example, if we have two objectives: minimize transportation cost and maximize customer satisfaction, we can define two fuzzy affiliation functions AX) and д, (x), where x denotes the particle location [20].

In the evaluation phase, the performance of each particle at its location is transformed into a series of fuzzy affiliation values to form a fuzzy target vector for that particle. For example, the fuzzy target vector of particle i ви, = [44 (X), 445 (x, )] . In addition, we need to check whether each particle satisfies all the constraints after fuzzification. This step ensures that all feasible solutions are reasonably taken into account. The core of FMOPSO is to update the individual optimal and global optimal positions. For each particle 1, if the fuzzy target vector of its current position is better than the fuzzy target vector of its historical optimal position pbest, , then pbest, will be updated to the current position. At the same time, updating the set of global optimal solutions using the Pareto dominance principle ensures that the algorithm recognizes all non-inferior solutions.

Velocity and position updates of particles are based on the basic rules of PSO, but fuzzy target vectors play a key role in this process. The velocity update formula is shown in Equation (4) [21,22].

...

where 1, у and Hg, are the personal optimal and global optimal fuzzy goal vectors of particle i, respectively, and M, (г) is the fuzzy goal vector of particle l's current position.

As shown in Fig. 2, in the FMOPSO algorithm, the multi-objective logistics and distribution center siting problem is transformed into fuzzy objective vectors by randomly initializing the particle swarm and defining the fuzzy affiliation function to ensure a comprehensive exploration of the solution space. In the algorithm iteration, the fuzzy objective performance of the particles to satisfy the constraints is evaluated, and the optimal solution is updated using the Pareto principle, which guides the adjustment of the particle speeds and positions until the convergence criteria are satisfied. The final output of the non-inferior solution set of the Pareto frontier provides decision makers with diverse choices based on the balance of cost and satisfaction [23, 24].

4 Case studies

4.1 Case background

In a hotbed of emerging markets in Asia, a major retailer is on the cusp of expansion, with plans to open a new Regional Distribution Center (RDC) within its service territory. As its business grows rapidly, the company is faced with an unprecedented challenge - how to maintain competitiveness while balancing economic efficiency with social and environmental responsibility. To this end, it set three core objectives as the cornerstones of its decision-making: first, to minimize total logistics costs, which include transportation costs, warehousing costs, and day-to-day operating expenses; second, to maximize customer service satisfaction, i.e., to ensure that goods are delivered to customers efficiently and on time, and to enhance brand loyalty; and third, to minimize environmental impact, and to work to reduce its carbon footprint and other forms of environmental damage in response to the global green initiatives.

In order to achieve these goals, firms must overcome a number of constraints, including: (1) Geographic advantage: the location of the distribution center needs to be in close proximity to major transportation networks, such as highways, railroads, and ports, in order to allow for the rapid consolidation and distribution of goods. (2) Cost control: Land acquisition and construction costs must not exceed budget ceilings, and long-term operating costs must also be taken into account. (3) Future adaptability: the design and size of the distribution center should be flexible enough to cope with future business growth and avoid the need for another expansion in the short term [25, 26].

Objective 1: Minimize total logistics costs. The total logistics cost consists of transportation cost, warehousing cost and operation cost, which can be expressed as equation (5).

...

Where Crransport is the transportation cost, C,_, is the warehousing cost and С operation

Objective 2: Maximize customer service satisfaction. Customer service satisfaction can be measured by metrics such as on-time delivery rate, order completeness, and customer feedback, which can be expressed as a fuzzy affiliation function Equation 6.

...

Objective 3: Minimize environmental impact. The environmental impact can be evaluated by a combination of several indicators, such as carbon emissions, energy consumption and waste management, which is expressed by a fuzzy affiliation function as Equation 7 [27].

...

In order to ensure the feasibility of the siting decision, the firm must also satisfy the following constraints:

Constraint 1: Geographic Advantage. The distribution center should be located in a convenient area, which can be expressed as the distance from the nearest transportation hub is less than or equal to the maximum allowable distance D , which can be expressed as Equation 8 [28].

...

where D(x,hub) denotes the distance from location x to the nearest transportation hub.

Constraint 2: Cost control: land acquisition and construction costs should not exceed the budget ceiling Br which can be expressed as Cima + Construction < B max where Cima is the cost of land and C is the cost of construction.

Constraint 3: Future Adaptability. The capacity of the distribution center should meet the needs of future business growth, which can be expressed by the minimum inventory turnover rate R : R(x) > R the inventory turnover rate for location

x. 4.2 Implementation

process The first step in implementation is a detailed data collection and pre-processing process. The company sends a team of professionals to the potential site area to collect a series of key information on land prices, transportation infrastructure layout, population density distribution, local environmental regulations, and so on. These raw data are then imported into the data analysis platform for standardized processing to ensure that indicators of different scales can be fairly compared. What's more, considering the uncertainty and fuzziness of the data itself, a fuzzification method was adopted to process the data, converting quantitative data into fuzzy numbers by means of an affiliation function, as a way to accurately reflect the uncertainty of the information and to lay a solid foundation for the subsequent model construction.

Based on the preprocessed data, the company constructed a multi-objective fuzzy optimization model that encompasses three core objectives of cost minimization, customer service satisfaction maximization and environmental impact minimization, while strictly adhering to the constraints of geographic advantage, cost control and future adaptability. In order to explore the optimal solution in the complex decision space, the company adopts the innovative FMOPSO algorithm. This algorithm can effectively deal with the ambiguity and uncertainty in multi-objective decision making by searching in the ambiguity space of the objective function through group intelligence, and gradually approximating the Pareto optimal frontier, i.e., a series of non-inferiority solution sets, each of which represents a kind of equilibrium state between different objectives [29].

At the end of the model solving process, the decision makers of a company are faced with the task of analyzing in depth the set of Pareto optimal solutions generated by the FMOPSO algorithm. Decision makers not only need to evaluate the combined performance of each solution in terms of cost, service quality and environmental responsibility, but also need to consider its fit with the long-term strategic plan of the organization. The whole implementation process demonstrates how complex data analysis, advanced algorithms and strategic decisionmaking can be combined to solve the multi-objective optimization challenge of logistics and distribution center location in a scientific way. Through this case, the enterprise can not only improve operational efficiency and reduce costs, but also establish a responsible corporate image in the competitive market and win wide recognition from consumers and society [30,31].

4.3 Analysis of results

After completing the site selection decision process, we compiled a series of key data and analysis results to provide a more intuitive understanding of the advantages and disadvantages of different site selection options. Below are four summary tables showing the cost-benefit analysis, environmental impact assessment, customer service quality metrics, and overall score comparison.

As shown in Table 2, we can observe the differences in the performance of different addresses in terms of costeffectiveness. Address А-01 has a relatively high net present value (NPV) and return on investment (ROI) despite having a low total cost, suggesting that location is a better investment. Address A-02 has the highest total cost, but also the highest expected revenue, resulting in its NPV and ROI being the highest of the three, showing strong profitability and investment attractiveness.

As shown in Table 3, the EIA reflects the potential environmental impacts of the different siting options. The А-01 address has relatively high CO: emissions and energy consumption, but low wastewater emissions and solid waste generation. The A-02 address exhibits low values for all environmental indicators, suggesting that the location has relatively low environmental impacts, which is conducive to the enterprise's ability to achieve sustainable development. The A-03 address has generally higher environmental impact indicators, especially CO: emissions and energy consumption, which may increase the environmental liabilities and operational costs of the enterprise. Therefore, the environmental friendliness of the A-02 address is more in line with the requirements of green development in site selection.

Table 4 Customer Service Quality Indicators: as shown in Table 3, customer service quality indicators are an important measure of the impact of the location options on customer service. a-01 addresses are longer in average delivery time, but have higher customer satisfaction and order accuracy and lower out-of-stock rates. a-02 addresses perform well on all customer service quality indicators, especially on delivery time and customer satisfaction, which suggests that the location is able to provide more efficient and customer-satisfying services. A-03 addresses have the longest average delivery time, the lowest customer satisfaction, and relatively low order accuracy, which may affect the firm's competitiveness in the market. Therefore, A-02 addresses have a clear advantage in improving the quality of customer service.

Table 5 Comprehensive Score Comparison: as shown in Table 4, the comprehensive score comparison demonstrates the combined performance of the site options across multiple dimensions. address A-02 scored first in all three dimensions of cost-effectiveness, environmental friendliness, and quality of service and had the highest composite score, suggesting that the location has strengths in multiple dimensions, making it an allaround excellent site option. address A-01, although it performed well in some individual categories, had a composite score was slightly lower than A-02. address A03 had the lowest scores in all areas and the lowest composite score, indicating that the site performed the worst in the overall evaluation. Therefore, based on the composite score, address A-02 is the best site option.

4.4 Comparative analysis of models

In order to visualize the results of the Fuzzy MultiObjective Particle Swarm Optimization Algorithm (FMOPSO) used in this study in comparison with similar models, the following four tables will be used to make detailed comparisons in terms of four aspects: speed of convergence, quality of solution, parameter sensitivity, and the ability to deal with uncertainty, respectively.

As shown in Fig. 3, our algorithm has an advantage in convergence speed. Convergence speed is an important indicator for evaluating the efficiency of optimization algorithms, as shown in Table 6. The FMOPSO algorithm outperforms the other models in terms of the fastest, average, and slowest convergence times, showing that it has a higher computational efficiency in the solution process. This means that the FMOPSO algorithm is able to find a satisfactory solution in a shorter period of time, which is a significant advantage for real-world problems that require fast decision making. In contrast, other models such as MOPSO, GA, DE and FLS-GA perform slightly worse in convergence speed, which may increase the solution time and computational cost.

The quality of the solution is a key factor in measuring the performance of the optimization algorithm, as shown in Table 7. The FMOPSO algorithm is lower than the other models in terms of average objective function error, maximum objective function error, and minimum objective function error, suggesting that it is able to provide a more accurate and higher quality solution. This accuracy is critical for site selection decisions as it ensures the reliability and utility of the results obtained. The other models, while also providing reasonable solutions, are not as accurate as the FMOPSO algorithm, which may affect the degree of optimization of the final decision.

As shown in Fig. 4, the parameter sensitivity reflects the algorithm's ability to adapt to parameter changes. The FMOPSO algorithm has the lowest percentage decrease in solution stability due to parameter changes, indicating that it is insensitive to changes in parameter settings and has good robustness. This property makes the FMOPSO algorithm more reliable in practical applications because it does not require frequent parameter adjustments to maintain good performance. In contrast, other models have poorer stability of the solution when the parameters change, and may require finer parameter tuning to ensure the quality of the solution.

As shown in Table 8, uncertainty handling ability is an important indicator for evaluating the stability of an algorithm in maintaining stability in the face of uncertainty. The FMOPSO algorithm has the smallest fluctuation in the solution under uncertainty scenarios, indicating that it has a high degree of stability in dealing with uncertain information, which is crucial for risk management and strategy development in site selection decisions. Other models can also handle uncertainty, but the magnitude of fluctuation is large, which may increase the risk of decision making. Therefore, the FMOPSO algorithm is more advantageous in dealing with complex and uncertain site selection problems.

Through a comprehensive assessment of key data and analysis results from the site selection decision process, we found that the A-02 address demonstrated significant advantages in a number of dimensions, including costeffectiveness, environmental impact, and customer service quality. Specifically, although the A-02 address has a higher total cost, it has the highest expected revenue, net present value and return on investment, indicating that it has good economic benefits. In terms of environmental impact, the A-02 address has relatively low emissions and energy consumption, which is conducive to the realization of green development of enterprises. Meanwhile, A-02 address also performs well in customer service quality indicators, which can win the competitive advantage for the enterprise in the market. Through the comparative analysis of the fuzzy multi-objective particle swarm optimization algorithm (FMOPSO) and other similar models, we find that the FMOPSO algorithm outperforms the other models in terms of convergence speed, solution quality, parameter sensitivity, and the ability to deal with uncertainty. The fast convergence and high solution quality of the FMOPSO algorithm help to improve the efficiency and reliability of the siting decision, and its low sensitivity to parameter changes and strong uncertainty handling ability make the algorithm more robust and practical in practical applications. The A-02 address as a siting option has a high overall score and is the best siting choice in this study. Meanwhile, the application of FMOPSO algorithm in the site selection decision model shows excellent performance and provides powerful decision support for enterprises in the complex and changing site selection environment.

4.4 Comparative analysis of models

To evaluate the robustness of the Fuzzy MultiObjective Particle Swarm Optimization (FMOPSO) algorithm, we conducted a parameter sensitivity analysis focusing on key parameters such as the inertia weight, cognitive coefficient, and social coefficient. The inertia weight controls the balance between global and local search, while the cognitive and social coefficients influence the particles" movement towards their personal best and the global best, respectively.

The inertia weight w was varied from 0.2 to 0.9 in increments of 0.1. A lower inertia weight encourages more exploration, whereas a higher value promotes exploitation. Our results indicate that an inertia weight of 0.7 provided the best trade-off between exploration and exploitation, leading to faster convergence and higher solution quality. However, the FMOPSO algorithm showed relatively low sensitivity to changes in the inertia weight, with only a 5% decrease in solution stability when the weight was set to extreme values (0.2 or 0.9).

The cognitive coefficient was tested in the range of 1.0 to 2.5. This parameter affects how much a particle is influenced by its own best position. A higher value leads to more individualistic behavior, while a lower value promotes more collective behavior. We found that a cognitive coefficient of 1.5 yielded the most stable and high-quality solutions. The sensitivity analysis revealed that the FMOPSO algorithm's performance was relatively insensitive to variations in , with a 3% decrease in solution stability at the extremes of the tested range.

The social coefficient was also tested in the range of 1.0 to 2.5. This parameter determines the influence of the global best position on the particle's movement. A higher value encourages more collective behavior, while a lower value allows for more individual exploration. An optimal value of 2.0 was identified, providing a good balance between exploration and exploitation. The FMOPSO algorithm showed a 4% decrease in solution stability when was set to the extremes of the tested range.

Overall, the FMOPSO algorithm demonstrated robust performance across a wide range of parameter settings, indicating its reliability and adaptability in realworld applications. This insensitivity to parameter changes is a significant advantage, as it reduces the need for fine-tuning and ensures consistent performance even under varying conditions.

The computational complexity of the FMOPSO algorithm is an important consideration, especially given the large search space and multiple objectives involved in logistics and distribution center location problems. The time complexity of the standard PSO algorithm 15 generally O(T-N- D) > where T is the number of iterations, N is the population size, and D is the dimensionality of the problem. In the case of FMOPSO, the additional fuzzy logic operations introduce a constant factor, but the overall complexity remains linear with respect to these parameters.

For practical implementation, the choice of N and T is crucial. In our study, we used a population size of 50 and a maximum of 100 iterations, which provided a good balance between computational efficiency and solution quality. As the problem dimensionality increases, the computational cost grows linearly, making the algorithm scalable to larger problems. However, for very highdimensional problems, the number of iterations may need to be increased to ensure adequate exploration of the search space, which can lead to higher computational costs.

To address potential scalability issues, parallel computing techniques can be employed to distribute the computational load. Additionally, adaptive strategies, such as dynamically adjusting the population size and the number of iterations based on the problem's characteristics, can further enhance the algorithm's efficiency. These considerations are essential for ensuring that the FMOPSO algorithm remains practical and efficient, even as the complexity of the optimization problem increases.

The implementation of the FMOPSO algorithm involves several key steps, each of which is critical for achieving high-quality solutions. Here, we provide more detailed information on the specific implementation details, including the selection of initial parameters and their impact on the results.

Population Size N: We initialized the population with 50 particles. This size was chosen based on a trade-off between computational efficiency and the diversity of the search. A larger population can explore the search space more thoroughly but at the cost of increased computational time.

Number of Iterations T: The algorithm was run for a maximum of 100 iterations. This number was determined through preliminary experiments, which showed that 100 iterations were sufficient to achieve convergence for the majority of test cases.

Inertia Weight w: Set to 0.7, which balances exploration and exploitation effectively.

Cognitive Coefficient: Set to 1.5, promoting a balanced individual and collective behavior.

Social Coefficient: Set to 2.0, encouraging more collective behavior and convergence towards the global best.

Population Size: Increasing the population size can improve the diversity of the search, potentially leading to better solutions. However, this comes at the cost of increased computational time. For our problem, a population size of 50 was found to be a good compromise.

Number of Iterations: A higher number of iterations can improve the quality of the solution, but it also increases the computational cost. The 100 iterations used in our study were sufficient to achieve stable and highquality solutions. For more complex problems, a higher number of iterations may be necessary.

Triangular Membership Function for Logistics Demand Forecasts: The triangular membership function was defined with points (a, b, с) to represent the lower bound, most likely value, and upper bound of the demand forecasts. This function effectively captures the uncertainty in demand predictions.

The trapezoidal membership function was defined with points (a, b, с, d) to represent the lower and upper bounds and the flat top of the environmental data. This function is particularly suitable for representing ranges with a flat top, common in environmental indices.

4.5 Discussion

In this section, we will provide a detailed discussion on the comparison of the FMOPSO algorithm with existing state-of-the-art solutions, such as Genetic Algorithms (GA), Differential Evolution (DE), and MultiObjective Particle Swarm Optimization (MOPSO), in terms of convergence speed, solution quality, and uncertainty handling. Through these comparisons, we will highlight the advantages of the FMOPSO algorithm and its suitability for logistics center location problems under uncertain conditions.

As shown in Figure 3 and Table 9, the FMOPSO algorithm significantly outperforms other algorithms in terms of convergence speed. Specifically, FMOPSO exhibits the best performance in the fastest, average, and slowest convergence times. This advantage is mainly attributed to the combination of fuzzy decision theory and particle swarm optimization, which allows FMOPSO to maintain global search while quickly finding local optimal solutions. In contrast, although GA, DE, and MOPSO can also find high-quality solutions, their slower convergence speeds may increase computational costs and time.

According to the results in Table 6, the FMOPSO algorithm also excels in solution quality. Whether in terms of average objective function error, maximum objective function error, or minimum objective function error, FMOPSO consistently outperforms the other algorithms. This means that FMOPSO can provide more accurate and higher-quality solutions. This advantage stems from the effective handling of data uncertainty and fuzziness through the fuzzification process, enhancing the robustness and reliability of the model. While GA, DE, and MOPSO can also provide reasonable solutions, they are slightly less accurate.

The FMOPSO algorithm demonstrates superior performance in handling uncertainties. As shown in Figure 4 and Table 10, FMOPSO exhibits low sensitivity to parameter changes, indicating that it can maintain stable performance under varying environmental conditions. This makes FMOPSO more reliable in practical applications, as it does not require frequent parameter adjustments to maintain good performance. In contrast, GA, DE, and MOPSO show poorer stability when parameters change and may require more fine-tuning to ensure solution quality.

As show in table 11, FMOPSO shows unique advantages in handling high-dimensional data and complex constraints. By employing fuzzy membership functions, FMOPSO can better handle uncertainties and fuzziness in the data, leading to the discovery of better solutions in high-dimensional spaces. Additionally, the adaptive mechanisms in FMOPSO enable effective searching under complex constraints without getting stuck in local optima. This capability is particularly important for solving large-scale logistics center location problems, Which often involve multiple conflicting objectives and complex constraints.

In summary, the FMOPSO algorithm outperforms existing classic algorithms such as GA, DE, and MOPSO in terms of convergence speed, solution quality, and uncertainty handling. These advantages are not only reflected in the numerical results but also in the unique design and implementation methods. Specifically, by integrating fuzzy decision theory and particle swarm optimization, FMOPSO can better handle uncertainties and complex constraints, making it particularly suitable for real-world logistics center location problems. Additionally, FMOPSO's strong capabilities in handling high-dimensional data and large datasets further enhance its practicality and effectiveness. These features make FMOPSO a powerful tool for solving logistics and distribution center location problems.

5 Conclusion

Facing the dual challenges of global competition and environmental sustainability, the site selection decision of logistics and distribution centers has become a key link in determining the success or failure of enterprises. In this context, this study introduces a multi-objective optimization method based on fuzzy set theory and integrates the particle swarm optimization algorithm to develop the FMOPSO model for the multi-objective and uncertainty problems inherent in the site selection decision. Through systematic research design, including data collection, preprocessing and fuzzification, as well as model construction and solving, we successfully applied FMOPSO to solve the difficult problem of site selection decision-making in the case of a retail enterprise in Asia. During the research process, we first conducted extensive data collection, covering key information such as logistics demand forecasts, GIS data, transportation costs, environmental impact indices, and land use. In the data preprocessing stage, we performed data cleaning, standardization and fuzzification to ensure the accuracy and consistency of model inputs. Subsequently, a multiobjective fuzzy decision-making model was constructed with cost minimization, customer service satisfaction maximization, and environmental impact minimization as the objective functions, and solved by the FMOPSO algorithm to generate a series of Pareto optimal solutions. The case study reveals that the A-02 address not only excels in economic benefits, but also achieves a high level of environmental responsibility and customer satisfaction, making it the optimal siting solution. Meanwhile, the FMOPSO algorithm exceeds the traditional algorithm in convergence speed, solution quality, parameter sensitivity and ability to deal with uncertainty, which highlights its advantages in solving complex decision problems.

Funding

This study was supported by 1) Fujian Provincial Department of Education Postgraduate Education Reform Project "Teaching Practice Exploration of 'Studio' System for Professional Master of Map Situation" (Project No. FBJG20190046); Undergraduate Education reform project of Fujian Normal University "Information Resource Construction" Curriculum Innovation and Practice Reform Exploration (Project No. 1201812013), Closing item, main participant.

2) Fujian Normal University Postgraduate Education Reform Project: "Research on the Cultivation Mode of China's Professional Master in Mapping (MLIS)" (No. JZ160282) and Research topic on the Educational reform and Practice of Professional Master in Mapping (MLIS) under the "Studio" mode, Conclusion, main participants. 3) The 2022 National Social Science Fund Project "Research on the Development Strategy of Chinese Library Science Subject towards a Cultural Powerful Country" (Project No. 22BTQ035) is under research and is the main participant. 4) Key scientific research Project of colleges and universities in Henan Province: "Research on Rural E-commerce Live Streaming to Help Rural Revitalization and Development-A Case study of Wenxian, Henan Province" (No. 22B790010), Closing item, main participants.

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