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

Abstract

The selection of locations for logistics distribution centers poses a significant challenge in logistics network planning. Traditional methods often demonstrate limited accuracy in solutions and a tendency to become trapped in local optima when addressing large-scale, multi-constraint location models. To address these shortcomings, this study introduces a firefly algorithm enhanced by genetic mutation strategies (GVFA) to optimize the location of distribution centers. Within the framework of the standard firefly algorithm, we incorporate an adaptive step-size decay mechanism and a mutation operator. The movement step size adjusts dynamically based on iteration counts, while a mutation probability of 5% is implemented to maintain population diversity, effectively reducing the risk of premature convergence. A specialized boundary-handling strategy ensures that the search process remains within the feasible solution space, guiding the population toward the global optimum. Experiments were conducted using latitude–longitude coordinates and logistics demand data from 159 Cainiao Post stations in Hengyang City, resulting in the construction of a location model aimed at minimizing total costs. The findings confirm the efficiency and stability of our method in optimizing distribution center locations, thereby providing a novel intelligent optimization approach for the siting of logistics distribution centers.

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]. 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

In this study, aiming at minimizing the total cost of logistics, a location model for logistics distribution centers is constructed, and an improved glowworm swarm optimization algorithm GVFA integrating a genetic mutation strategy is proposed. Compared with existing variants of the glowworm swarm optimization algorithm, GVFA has two key improvements: First, a dual dynamic step-size adjustment mechanism combining number of iterations and individual spatial distance is designed, which addresses the limitation that traditional adaptive step size only depends on iteration adjustment, and better balances the global exploration and local exploitation capabilities of the algorithm. Second, the genetic mutation operation is introduced to maintain population diversity and alleviate the problems of premature convergence and falling into local optima that existing algorithms easily encounter when solving large-scale location problems.

To comprehensively verify the performance of GVFA, in addition to comparing it with existing glowworm swarm optimization variants, this study adds controlled experiments with the standard glowworm swarm optimization algorithm (SFA) and particle swarm optimization algorithm (PSO). In different scenarios where the number of distribution centers K ranges from one to seven, each algorithm runs independently 20 times and the average results are obtained. Experimental results show that GVFA significantly outperforms the two benchmark algorithms in three core indicators: optimal total cost, convergence speed and stability. Compared with existing methods, GVFA has better effectiveness and robustness in multi-center distribution location optimization, and can provide a more reliable decision scheme for practical logistics network planning.

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 [10]. 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 [11]. 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 [12]. Additionally, Wang et al. (2023) constructed a multi-objective optimization model for the layout of large-scale distribution centers during the pandemic [13]. 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 [14,15]. 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 [16].

Metaheuristic algorithms are extensively utilized for their effectiveness in addressing complex optimization problems, including particle swarm optimization [11], jellyfish search algorithm [14], genetic algorithm [15], and grey wolf optimizer [9]. 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 [17]. 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 [18].

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

We use 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:

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

(1)

where

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

(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).

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

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

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

$I=1,2,3,\dots ,n$: 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 [19], 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).

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

(3)

In this equation, $x_i\left(t\right)$ and $x_j\left(t\right)$ 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, and $rand$ 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: This comprises demand point coordinates, demand volumes, and 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), and then update firefly i’s position.

c. Apply boundary handling to the updated position.

d. Execute the 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 at K = 1 to 82,330.78 CNY at K = 7. The decline is driven primarily by shorter service distances. The most pronounced drop occurs between K = 1 and K = 3, where costs fall by nearly 70,000 CNY; thereafter the reduction becomes more gradual.

Figure 1. Comparison of transportation cost convergence curves for different K values.

Conversely, fixed costs rise linearly with each additional center. Assuming a fixed cost of 10,000 CNY per distribution center, the total cost (transportation plus fixed cost) exhibits a nonlinear trend, as shown in Figure 2. Total cost remains almost unchanged from K = 1 (227,720.25 CNY) to K = 2 (227,689.00 CNY), and then decreases sharply, reaching a minimum of 152,330.78 CNY at K = 7. This represents a 33.1% reduction relative to the single-center scenario. Notably, between K = 4 and K = 5, total cost experiences a slight rebound—from 163,862.85 CNY to 171,674.42 CNY which signals the beginning of diminishing marginal returns.

Figure 2. Comparison of total cost convergence curves for different K values (with Fixed Cost).

The change in cost per parcel, presented in Figure 3, further corroborates these observations. Based on a total parcel volume derived from the K = 1 case (217,720.25 CNY/3.4080 CNY per item ≈ 63,900 parcels), the per-parcel cost drops from 3.4080 CNY at K = 1 to 2.2832 CNY at K = 7, a 33.0% improvement. As shown in Figure 3, beyond K = 4 the reduction in per-parcel cost slows considerably, and it continues to undergo a slow decline thereafter. It should be noted that the slight rebound between K = 4 and K = 5 occurs in total cost (Figure 2) rather than in per-parcel cost. The overall pattern remains clear: after K = 4, further increases in K yield limited additional savings, confirming diminishing returns.

Figure 3. Comparison of per-parcel convergence curves for different K values (with fixed cost).

In terms of spatial efficiency, the maximum delivery distance decreases significantly as K increases, as summarized in Figure 4. For K = 1, the maximum distance is 40.88 km; at K = 3, it drops to 7.44 km—an 81.8% reduction. For K = 4 and above, the distance continues to decline but at a much slower pace, reaching 5.12 km at K = 7. This substantial reduction enhances endpoint service responsiveness and improves coverage balance across the entire region. The sharp initial drop between K = 1 and K = 3 suggests that even a moderate increase in the number of distribution centers yields considerable gains in spatial equity.

Figure 4. Final weighted objective, total cost, final cost, and final maximum distance versus K value.

To more comprehensively verify the performance advantages of the proposed GVFA, we added a comparative analysis with other benchmark algorithms. Specifically, in the algorithm performance analysis, we extended the experimental design by including direct comparisons with the standard firefly algorithm (SFA) and particle swarm optimization (PSO). The experimental settings are identical to those described above: for different K values (K = 1 to 7), 20 independent runs were conducted and averaged, using the same fitness function and evaluation metrics. The newly added Table 1 presents the performance of the three algorithms on three core metrics: optimal total cost, average convergence generation, and algorithm stability. As can be seen from Table 1, GVFA exhibits significant advantages in both solution quality and convergence speed, as well as better stability compared to SFA and PSO.

Table 1. Performance comparison of different algorithms.

Algorithm	Average Best Total Cost (CNY)	Average Number of Generations to Converge	Algorithm Stability
GVFA (Our)	152,777.45	86	Excellent
SFA	155,830.31	1	poor
PSO	155,506.09	120	poor

Regarding algorithm performance, Figure 5 presents the convergence curves of the weighted objective function for K = 1 to K = 7. For all tested K values, the curves stabilize well within 50 generations, demonstrating fast and robust convergence of the optimization procedure. Among the configurations, K = 3 attains the lowest weighted objective value of 1.0189, closely followed by K = 2 (1.0174) and K = 4 (1.0885). Values for K = 5, K = 6, and K = 7 are 1.1290, 1.2444, and 1.3593, respectively, indicating a clear increase beyond K = 4. The minimum at K = 3 suggests that this configuration optimally balances the three competing terms in the weighted objective: transportation cost, fixed cost, and delivery distance. Adding more centers beyond K = 3 worsens the overall weighted score, despite further reductions in transportation cost and maximum distance, due to the linearly rising fixed cost and diminishing returns.

Figure 5. Comparison of weighted objective function convergence curves for different K values (with fixed cost).

As reported in Table 2, when the number of distribution centers K increases from one to six, the algorithm’s running time grows only from 22.3 s to 22.8 s—a mere 0.5 s increase despite a sixfold rise in K. The average increment per additional center is only 0.1 s. This marginal increase demonstrates the algorithm’s computational efficiency and scalability. The near constancy of runtime with respect to problem complexity suggests that the method is well suited for large-scale logistics optimization, avoiding exponential time overhead.

Table 2. Algorithm running time comparison for different K values.

Number of Distribution Centers (K)	1	2	3	4	5	6
Running Time (s)	22.3	22.5	22.5	22.7	22.7	22.8
Time Increase from Previous K (s)	-	+0.2	0	+0.2	0	+0.1
Cumulative Time Increase (s)	0.0	0.2	0.2	0.4	0.4	0.5

To elucidate the intrinsic mechanisms underlying the superior performance of GVFA, we attribute its performance mainly to the adaptive step-size decay mechanism and the genetic mutation operator. The adaptive step-size decay mechanism balances global exploration and local exploitation throughout the optimization process, while the genetic mutation operator effectively prevents the premature loss of population diversity and helps the algorithm escape local optima. Consequently, GVFA achieves better solution quality, faster convergence, and higher robustness, providing solid quantitative and comparative support for the contribution of our algorithm.

A comprehensive analysis of the optimal configuration (K = 3, Figure 6) reveals a highly uneven allocation of stations among the three distribution centers. Center 1 serves 72 stations with a daily volume of 26,840 parcels, Center 2 serves only a single station processing 320 parcels per day, and Center 3 covers 86 stations handling 39,660 parcels daily. This imbalance reflects the underlying spatial distribution of demand points. The total number of stations served is 159, and the overall average delivery distance is 2.82 km, indicating reasonably efficient routing.

Figure 6. Distribution center allocation map (K = 3, with fixed cost).

In this configuration, the transportation cost is 148,247.44 yuan, while the fixed cost (30,000 yuan for three centers) accounts for only 16.8% of total expenses (178,247.44 yuan). Transportation costs thus dominate at 83.2%, which suggests that operational expenditures drive the total cost, and infrastructure investment remains manageable. The cost per parcel equals 2.6676 yuan, and the weighted objective value reaches 1.006058—the lowest among all tested K values, as shown previously in Figure 5.

Moreover, the maximum delivery distance is 7.44 km, which stays within a practical range for last-mile logistics. This distance, combined with the low fixed cost share and favorable weighted objective, demonstrates that the K = 3 configuration effectively balances service accessibility, vehicle routing efficiency, and overall cost. The single station center (Center 2) does not significantly degrade performance, as its small volume allows flexible routing without extending the maximum distance.

In summary, through systematic simulation experiments under multi-objective constraints that explicitly account for both fixed and transportation costs, this study identifies the optimal number of distribution centers for the Cainiao Post station network in Hengyang City as three. The K = 3 configuration yields a transportation cost of 148,247.44 yuan, a total cost (including 30,000 yuan fixed cost) of 178,247.44 yuan, and a weighted objective value of 1.006058, the lowest among all tested K values (K = 1 to K = 7). The maximum delivery distance is 7.44 km, and the average distance is 2.82 km, indicating good service accessibility without excessive routing detours. Fixed costs account for only 16.8% of total expenses, meaning that operational expenditures dominate the cost structure. This balance between economic efficiency and spatial performance supports the choice of three centers as the most favorable trade-off. Furthermore, the weighted objective function converges within 50 generations across all configurations, demonstrating fast and robust optimization behavior. The results provide empirical evidence and decision support for the planning and design of similar urban last-mile logistics networks, particularly those facing analogous trade-offs between infrastructure investment and delivery efficiency. 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.