Research on Logistics Distribution Center Location Problem Based on Genetic Variation Firefly Algorithm

1. Introduction

The rapid growth of e-commerce and the logistics sector has made the efficiency and cost control of distribution systems a fundamental corporate competency [1,2]. As critical nodes within logistics networks, the placement of distribution centers significantly influences transportation costs, service efficiency, and overall system reliability. Traditional location methods, such as the center-of-gravity and analytic hierarchy process, often struggle to address the dynamics and complexities of contemporary logistics when tackling multi-objective, large-scale problems, primarily due to model oversimplification or limited solving capacity [3,4,5,6]. Therefore, the application of intelligent optimization algorithms for achieving efficient and precise location decisions is of considerable theoretical and practical importance. The firefly algorithm (FA), a swarm intelligence optimization technique that emulates the flashing and attraction behaviors of fireflies, was introduced by Professor Yang at Cambridge University in 2009 [7]. This algorithm progressively directs the population toward the global optimum by comparing individual brightness and updating positions, offering advantages such as a limited number of parameters, ease of operation, and favorable convergence properties. It has been successfully applied in areas such as image processing, path planning, and multi-objective optimization. However, the standard FA has notable limitations, including sensitivity to initial values and a propensity to converge to local optima when addressing high-dimensional, multimodal problems [8,9,10]. This study proposes enhancements to the FA tailored to the specific characteristics of the distribution center location problem and applies these improvements within this context.

1.1. Motivation

The location of logistics distribution centers represents a classic NP-hard combinatorial optimization problem. The primary objective is to minimize total construction and transportation costs while satisfying customer demand. As the distribution network expands, the solution space increases exponentially, complicating the ability of traditional optimization methods to yield satisfactory solutions within a reasonable timeframe. While intelligent optimization algorithms, such as ant colony and genetic algorithms, have been employed in this domain, they often encounter challenges, including slow convergence and limited solution accuracy. The Firefly Algorithm (FA), a relatively recent addition to swarm intelligence techniques, has exhibited strong global search capabilities in multi-objective optimization. However, its application to logistics distribution center location remains underexplored, lacking systematic model development and empirical validation.

1.2. Our Approach

This study develops a logistics distribution center location model aimed at minimizing total costs, which is addressed through an enhanced firefly algorithm that incorporates genetic mutation strategies (GVFA). To improve algorithm performance, we implemented a dynamic step-size mechanism alongside a genetic mutation strategy. The step-size factor adjusts according to the iteration count and the distance between individuals, thereby balancing global exploration with local exploitation. Additionally, a mutation operator is introduced to augment population diversity and mitigate the risk of premature convergence. Utilizing coordinate and logistics demand data from 159 Cainiao Post stations in Hengyang City as a sample, we established a normalized coordinate system and simulated the actual demand distribution, thereby validating the algorithm’s effectiveness and robustness in optimizing multi-center distribution locations.

2. Related Work

The positioning of logistics distribution centers is crucial for enhancing the efficiency, cost-effectiveness, and sustainability of logistics networks. As application scenarios become more complex, the development of tailored models that integrate domain-specific characteristics, along with the use of effective intelligent algorithms, has emerged as a key area of research.

Scholars have extensively investigated various constraints in domain-specific modeling. Tong (2022) developed a cost-oriented, two-layer location-routing coordination model specifically designed for the high-cost nature of maritime ship logistics [11]. In the context of regional logistics network optimization, Huang et al. (2022) employed a particle swarm algorithm to resolve a constrained p-median model aimed at achieving a balanced facility layout [12]. To tackle demand uncertainty and emergencies, Lei et al. (2023) proposed a hybrid decision-making model that addresses probabilistic uncertain linguistic information for the siting of medical logistics centers [13]. Additionally, Wang et al. (2023) constructed a multi-objective optimization model for the layout of large-scale distribution centers during the pandemic [14]. Emergency logistics prioritizes timeliness; thus, Li et al. (2023) and Cao et al. (2023) modified the jellyfish search algorithm and genetic algorithm, respectively, by incorporating time satisfaction as a critical factor in their models [15,16]. Moreover, the focus on green and sustainable practices is increasing, as evidenced by Zhang et al. (2024), who integrated carbon emission costs into cold-chain logistics location models [17].

Metaheuristic algorithms are extensively utilized for their effectiveness in addressing complex optimization problems, including particle swarm optimization [12], jellyfish search algorithm [15], genetic algorithm [16], and grey wolf optimizer [10]. The complexity of model structures is also increasing; for example, Wang et al. (2024) implemented a bi-level programming model that captures hierarchical corporate decision-making [18]. A prominent trend is the integration of traditional optimization algorithms with artificial intelligence techniques, as demonstrated by Liu et al. (2024), who combined deep reinforcement learning with swarm intelligence to tackle multi-objective agricultural logistics optimization [19].

A review of the current literature indicates that, despite significant advancements in model specialization and algorithm application, the majority of research emphasizes either the direct application or singular modification of existing algorithms. This highlights a need for greater integration and innovation within algorithmic search mechanisms, particularly in achieving a balance between global and local search capabilities and enhancing robustness for high-dimensional, complex models. This identified gap motivates our development of a novel hybrid algorithm that combines genetic mutation mechanisms with the attraction dynamics of the firefly algorithm.

3. Proposed Methodology

3.1. Problem Description

Using the Cainiao Post stations in Hengyang City as a case study, the city comprises N station sites that serve dual functions as demand points—receiving parcels from outside the city—and supply points—dispatching parcels outward. The objective is to select M stations from these existing sites to serve as distribution centers, which will be responsible for collecting and dispatching parcels within their designated regions. This study focuses on the selection of several sites from the existing stations to establish logistics distribution centers, with the aim of minimizing two key objectives: total transportation cost and maximum response time, the latter measured by the longest service distance. The following constraints are imposed:

• Each station can be served by only one distribution center.
• The distance between sites is computed using Euclidean distance, under the assumption of a fixed traffic network.
• Transportation costs are directly proportional to distance, with variations in parcel weight disregarded.
• The construction cost and daily fixed operating cost for each distribution center remain constant.

3.2. Mathematical Model

Based on the preceding description, the problem is formulated as a continuous-space location-allocation optimization model aimed at minimizing total costs. The mathematical model is defined as follows:

$$MinZ=M·F+\alpha ·{\displaystyle \frac{Z_1}{Z_1^{*}}}+(1-\alpha ·{\displaystyle \frac{Z_2}{Z_2^{*}}})$$

(1)

where:

$$\begin{array}{cc}\hfill Z_1& =C·\sum _{i\in I}·\sum _{j\in J}·d_{ij}·w_i·y_{ij}\hfill \\ \hfill Z_2& =T_{max}\hfill \\ \hfill Z_1^{*}& =Minimize{Z}_1\hfill \\ \hfill Z_2^{*}& =Minimize{Z}_2\hfill \end{array}$$

(2)

Z: Target value for final delivery optimization.

M: Number of distribution centers.

F: Construction cost and average daily fixed operating cost per distribution center.

$Z_1$: Total delivery transportation cost.

$Z_2$: Maximum service distance (response time metric).

$Z_1^{*}$: Optimal value for minimizing transportation cost.

$Z_2^{*}$: Optimal value for minimizing maximum distance.

$\alpha \in [0,1]$: Weighting coefficient reflecting priority of cost objectives.

$I=1,2,3,...,n$: Set of all Cainiao Stations.

$J=I$: Set of candidate distribution centers (identical to the station set).

c: Transportation cost coefficient per unit distance per unit parcel.

$d_{ij}$: Euclidean distance from station to candidate center.

$w_i$: Daily average total parcel processing volume of station.

$y_{ij}\in 0,1$: Whether station is served by distribution center.

$T_{max}=ma{x}_{i\in I}\{mi{n}_{j\in k|x_k=1}·d_{ij}\}$: Maximum distance from all stations to their serving distribution centers.

$x_k$: Indicator for establishing a distribution center at site k.

3.3. Algorithm Implementation

The standard firefly algorithm optimizes by simulating the luminous attraction behavior among individual fireflies. To enhance its performance for the location problem [20], we introduce two improvement mechanisms: an adaptive step-size strategy and a diversity-preserving mutation, resulting in an improved firefly algorithm integrated with genetic variation strategies (GVFA).

The fixed step size in the standard FA can include oscillations near the optimum in later iterations, thereby impairing convergence precision, while an excessively small step size can hinder convergence. To address this issue, we propose a dynamic, variable step size: the step size decreases as the iteration count and the distance between fireflies increase. For example, substituting the constant step size factor $\alpha$ with $k·l_{ij}$ facilitates rapid convergence initially and stabilization near the optimum later, effectively balancing convergence speed and precision. The displacement formula for the FA with a dynamic variable step size is given by Equation (3).

$$x_i(t+1)=x_i(t)+\beta ·(x_j(t)-x_i(t))+k·l_{ij}·(rand-{\displaystyle \frac{1}{2}})$$

(3)

In this equation, $x_i(t)$ and $x_j(t)$ denote the current spatial positions of fireflies i and j at time t, $\beta$ is the attractiveness of firefly, k is a constant, $l_{ij}$ is the Euclidean distance between fireflies i and j, $rand$ is a random number uniformly distributed on [0,1].

To enhance population diversity, a mutation operation with probability $p_m=0.05$ is applied to selected individuals. For each chosen individual, its position vector is reinitialized to a random value within the feasible boundaries. This mechanism ensures that a segment of the population preserves advantageous positions while permitting new individuals to explore alternative regions, thereby maintaining a balance between exploitation and exploration.

The algorithm procedure is as follows:

Step 1. Input: Demand point coordinates, demand volumes, algorithm parameters (population size S, maximum iterations $G_{max}$, ${\alpha }_0$, ${\beta }_0$, $\gamma$, $p_m$, etc.).

Step 2. Data Preprocessing: Normalize demand point coordinates.

Step 3. Initialization: Randomly generate initial positions for S fireflies. Calculate each firefly’s objective function value (total cost) and brightness.

Step 4. Iterative Optimization:

a. Rank fireflies by brightness.

b. For each firefly i, traverse all brighter fireflies j. Calculate attraction and movement step size using Equation (3), then update firefly i’s position.

c. Apply boundary handling to the updated position.

d. Execute mutation operation with probability $p_m$.

e. Recalculate all fireflies’ objective function values and brightness.

f. Record the current best solution.

Step 5. Output: After iterations conclude, output the global best solution (optimal distribution center locations) and the corresponding minimum total cost.

4. Experiments and Results Analysis

To assess the effectiveness of the proposed multi-objective distribution center location model and the enhanced Genetic Variation Firefly Algorithm (GVFA) in a real-world urban logistics context, systematic simulations were performed utilizing empirical data from 159 Cainiao Post stations in Hengyang City, China, which handle a total daily parcel volume of 66,820 items. The experimental environment was established using VMware Workstation Pro virtual machine software, operating on an Ubuntu 64-bit OS with 16 GB of RAM. Python was utilized as the programming language within the PyCharm platform. Algorithm parameters were set as follows: firefly population size = 60, light absorption coefficient $\gamma$ = 0.12, initial step-size factor ${\alpha }_0$ = 0.6, base attractiveness ${\beta }_0$ = 0.15, mutation probability $p_m$ = 0.05, and maximum iterations $G_{max}$ = 200. To ensure robustness, each scenario for different numbers of distribution centers (K = 1 – 7) was independently run 20 times, with results averaged.

The optimization results across varying numbers of distribution centers (K) illustrate a distinct trade-off between cost and efficiency. As depicted in Figure 1, transportation costs decrease markedly with an increase in K, falling from 217,720.25 CNY for K=1 to 82,330.78 CNY for K=7, primarily due to shorter service distances. Conversely, the fixed costs associated with distribution centers rise linearly, leading to a nonlinear trend in total costs Figure 2. Notably, total costs reach a minimum of 152,330.78 CNY at K=7, reflecting a reduction of approximately 33.1% compared to the single-center scenario (K=1). The change in cost per parcel further corroborates this observation Figure 3: at K=7, the per-parcel cost is 2.2832 CNY/item, representing the most substantial improvement relative to the single-center scheme (3.4080 CNY/item). Beyond K=4, the pace of cost reduction diminishes, with a slight rebound noted between K=4 and K=5, indicating diminishing marginal returns.

In terms of spatial efficiency, the maximum delivery distance decreases significantly as K increases Figure 4. For K=1, the maximum distance is 40.88 km; however, for K=3, it reduces to 7.44 km, reflecting an 81.8% reduction. This substantial decrease enhances both end-point service responsiveness and coverage balance. Regarding algorithm performance, the convergence curves of the weighted objective function Figure 5 stabilize within 50 iterations across all K values, with K=3 yielding the lowest weighted objective value of 1.0189. This result indicates optimal performance in balancing transportation costs, fixed costs, and delivery distances. The runtime for each scenario ranges from approximately 23.5 to 23 seconds Table 1, underscoring the algorithm’s computational efficiency and scalability.

A comprehensive analysis of the optimal configuration (K=3, Figure 6) reveals that the three distribution centers serve 186, and 72 stations, processing daily volumes of 32,039,660, and 26,840 parcels, respectively. The overall average delivery distance is 2.82 km. In this configuration, fixed costs account for only 16.8% of the total expenses, while transportation costs dominate at 83.2%, thereby optimizing operational expenditures while managing infrastructure investments. Moreover, the maximum delivery distance of 7.44 km remains within a reasonable range, effectively balancing service accessibility with vehicle routing efficiency.

In summary, through systematic simulation experiments, this study demonstrates that, under multi-objective constraints that consider both fixed and transportation costs, the optimal number of distribution centers for the Cainiao Post station network in Hengyang City is three. This configuration achieves an optimal balance between economic efficiency and spatial performance, providing empirical evidence and decision support for the planning and design of similar urban logistics networks. Future research could incorporate more complex factors, such as dynamic demand and time-varying traffic network characteristics, to enhance the model’s applicability in real-world scenarios.

5. Conclusions

This study addresses the intricate optimization challenge of distribution center location within urban logistics networks by proposing an enhanced firefly algorithm integrated with genetic mutation strategies (GVFA). The primary innovation involves the incorporation of an adaptive, dynamically decaying movement step size alongside a low-probability mutation operator within the standard firefly algorithm framework. These enhancements effectively balance global exploration with local exploitation capabilities, thereby significantly reducing the risk of premature convergence. Additionally, a specially designed boundary-handling mechanism ensures that the search process remains confined within the feasible solution space, thereby improving optimization stability and solution quality.

Empirical research utilizing actual coordinates and parcel volume data from 159 Cainiao Post stations in Hengyang City facilitated the construction and resolution of a multi-objective location model. This model aims to minimize both transportation and fixed costs while adhering to service distance constraints. Simulation results demonstrate that the proposed GVFA exhibits commendable convergence speed and robustness in addressing this large-scale discrete optimization problem, efficiently approximating the Pareto front. A systematic analysis of cost structures and spatial efficiency across varying numbers of distribution centers (K=1 to 7) revealed that the system achieves optimal comprehensive performance at K=3. This configuration results in the lowest total cost (178,247.44 CNY), reduces per-parcel distribution costs by 21.7% compared to the single-center scheme, and decreases the maximum delivery distance by 81.8%, thereby achieving an optimal balance between operational economy and service accessibility.

The primary contributions of this work are threefold. First, it presents an efficient and stable novel intelligent framework for addressing logistics distribution center location problems. Second, through a comprehensive empirical case analysis, it clarifies the nonlinear trade-off between fixed and transportation costs, as well as the law of diminishing marginal returns related to the number of distribution centers, thereby offering a quantitative basis for decision-making in logistics network planning. Third, it demonstrates the practical value of enhanced heuristic algorithms in solving complex location problems constrained by real-world factors.

Future research may advance in several directions. First, dynamic and time-varying customer demands, along with traffic conditions, could be integrated into the model to improve the real-time adaptability of scheduling schemes. Second, the model could be expanded to include multiple sustainability objectives, such as environmental impact and energy consumption, thereby creating a more comprehensive evaluation system. Finally, at the algorithmic level, integrating machine learning methods could utilize historical data to predict optimal parameter configurations, further enhancing optimization efficiency and intelligence. The model and methodology proposed in this study offer robust theoretical and practical support for the refined planning and operational management of smart city logistics systems.