1. Introduction
Designing an optimal transportation network has always been a difficult yet strategic task for logistics firms [
1,
2]. While many network configurations are possible in theory and have been implemented in practice with various pros and cons, two contrasting networks are worthy of comparison. One is the traditional point-to-point (
P-P) network, which connects one node to all other nodes directly in the system and often assumes all nodes and links are similar in supply, demand, and capacity. While the
P-P network aims to serve all origin and destination pairs directly by avoiding trip transfers or stops, the
P-P network inherits some fundamental operational problems, such as low distribution efficiency, frequent sorting and storage at nodes, empty returns due to flow imbalance on links, and cost burdens and limited capacities on minor links and at small nodes. Moreover, the
P-P network normally requires greater total system costs, including fixed, operating, and maintenance costs, due to all the
O-D pairs being served while many small
O-D pairs may not have enough trips or some trips may not exceed a threshold. The other is the hub-and-spoke (
H-S) network, which recognizes the hierarchical nature of nodes, links, and paths and utilizes large important nodes and links as hubs and hub–hub (
H-H) links to better serve small nodes or non-hubs and hub–spoke (
H-S) links. While the
H-S network can support some point–point flow movement, its main feature is to move
O-D flows through hubs and
H-H links so that their scale economy effect can be realized to save costs. As such, fewer direct links and trips are needed while additional sorting, transfer, and stop-over capacities are needed at hubs and on
H-H links. Fewer direct connections and more hub transfers may result in higher overall system costs for
H-S networks. Therefore, the
H-S network can enable logistics firms to reduce costs and improve their business performance.
The current research on
H-S network design largely focuses on single-hub location and allocation for any
O-D pair; that is to say, the non-hub nodes in the network can only be connected to one hub node so the flows between non-hubs (
N-N) can only be transferred via the single hub node, hence avoiding direct
N-N links or flows while making flows concentrated at hubs and on
H-H links. In particular, the number of hubs and their locations in the network are the key factors affecting the operational efficiency of the entire
H-S network [
3]. At present, most of the logistics firms in China design their
H-S networks regarding the hub locations and non-hub allocations based on personal experience and local government regulations in transportation policies and/or urban planning. Compared with the increased transportation cost for operating more links and offering sorting at all nodes in a
P-P network, the flow delay costs in the
H-S network are often ignored. Today, as lean and green logistics operations are becoming the new normal, and in order to keep a competitive advantage, an increasing number of logistic firms have become more sensitive to transportation costs by strategically adopting the
H-S network. Lean implementation positively influences the implementation of sustainability practices for supplier selection and production [
4].
With respect to a traditional point-to-point (
P-P) network, a hub-and-spoke (
H-S) network not only uses a smaller number of links/paths but also utilizes the scale economy advantage on consolidated flows on hub–hub links and at hubs. However, the inevitable delays through hubs have always been a critical concern. The core research question of the paper is how to reduce the total cost of the logistics company by optimizing the
H-S network design while considering the flow delay cost and improving the operational efficiency of the logistics network. Following [
2], this paper introduces the concept of flow delay cost into the hub location and network design problem as an integral part of the total transportation cost and verifies its practical value through cases. The structure of this paper is as follows: after the introduction in
Section 1,
Section 2 provides a concise review of the relevant literature on the
H-S network.
Section 3 describes the concepts of the
H-S network.
Section 4 proposes an optimized
H-S network model considering the flow delay cost at hubs.
Section 5 applies the model to the SF firm’s highway network covering 13 prefecture-level cities in the Jiangsu Province, China, using the PSO algorithm to obtain the optimal
H-S network and compares the results against the SF’s existing network structure and outcome.
Section 6 summarizes and concludes this paper.
2. Literature Review
O’Kelly (1987) introduced the concept of a hub-and-spoke network and proposed a quadratic integer model to locate p hubs for single-hub location and allocation under total transportation cost minimization [
5]. Alumur et al. (2008) summarized the literature on single- and multiple-hub location and allocation and network design considering fixed costs, capacities, and coverage [
3]. Commemorating the twenty-five years of hub research, Campbell and O’Kelly (2012) provided a comprehensive review of the hub location and network design problem, including model formulations, scale economies, location and allocation schemes, solution algorithms, and application domains and issues [
2]. The review also pointed out important future hub research issues and flow delay cost is one of them.
First, different approaches can be taken to solve the same kind of problem. Along with many hub models and location and allocation schemes developed over the past three decades, various solution algorithms have also been developed and implemented. Zheng et al. (2017) proposed a mixed-integer linear programming model, factoring in ship-operating and container-handling costs, and conducted a numerical simulation to test the effectiveness of the model [
6]. Devika Kannan et al. (2023) proposed a multi-objective mixed-integer programming (MOMIP) model for configuring an RL network design; it incorporates multiple products, multiple recovery facilities, multiple processing technologies, and a selection of vehicle types [
7]. In solving the single-location–allocation model, Aloullal et al. (2023) considered time as a new dimension in the hub location-routing problem, and it employed a specially designed meta-heuristic that combines relax-and-fix, local branching, and variable neighborhood descent techniques for problem-solving [
8].
Second, when solving practical problems, there are many factors that need to be considered, and different scholars have different focuses. Alkaabneh et al. (2019) considered a hub-and-spoke network design problem with inter-hub economies of scale and hub congestion and proposed an optimal design of hub-and-spoke networks with nonlinear inter-hub economies of scale and congestion at hub locations [
9]. Zhou et al. (2022) designed an
H-S network with differentiated services, allowing clients to choose their preferred service levels, while considering multiple transportation modes, with environmental parameters and economies of scale incorporated into the modeling process [
10]. Zhou et al. (2023) proposed an
H-S network design (HSND) for container shipping in inland waterways based on the tree-like river structure [
11].
Third, the hub location and network design problem considering hub and
H-H link capacities has been studied as well with various models and/or solution methods reported [
12,
13,
14]. Guan et al. (2018) develop a learning-based probabilistic tabu search to solve the uncapacitated single allocation hub location problem (USAHLP) [
15]. Özgün-Kibiroğlu et al. (2019) used the particle swarm optimization (PSO) algorithm to solve the capacitated hub location and allocation model [
16]. Najy et al. (2020) considered a novel and more realistic variant of the uncapacitated hub location problem where both flow-dependent economies of scale and congestion considerations are incorporated into the multiple-allocation version of the problem [
17]. Daneshfar (2024) et al. proposed an improved version of the discrete laying chicken algorithm (IDLCA) that utilizes noun-based filtering to reduce the number of features and improve text classification performance [
18].
Fourth, variations of the hub location and network design model considering congestion, reliability, and other normal and abnormal specifics have also been explored in the literature, for example, flow congestions at the hubs by [
19,
20] and failure at one or more hubs by [
21,
22,
23]. Karimi-Mamaghan et al. (2020) modeled a single-allocation multi-commodity
H-S network problem through a bi-objective mathematical model, considering the congestion in both hubs and connection links [
24]. Bütün et al. (2021) tackled
H-S network design in the liner shipping sector, introducing a capacitated directed cycle hub location and cargo routing problem under congestion [
25]. The experiments show that the network design can be highly influenced by scale economies in mainline vs. feeder transportation costs, the port locations and hinterland flows, and congestion at the hub ports. The hub failure problem viewed from the system reliability perspective was considered by [
26,
27,
28]. Congestion is one of the important reasons for causing delays.
Finally, it is necessary to consider which kind of network structure should be chosen when designing the network. A point-to-point network also has the advantages that the
H-S logistics network cannot replace. Reza Lotfi et al. (2021) examine several logistics network designs and evaluate their performance for cost, quality, delivery, flexibility, and resilience [
29]. They point out that each network has its strengths: a hub-and-spoke network has economies of scale to reduce delivery costs and routing flexibility to mitigate the effects of disruptions; a cross-docking network provides lower inventory cost; and a pick-up and delivery network provides lower delivery times. They deem that considering a hybrid logistics model in situations where firms need to emphasize cost and resilience.
Two important efforts can be found in the current hub location and network design research. One effort is directed to develop more efficient algorithms to solve larger
H-S problems faster. The other effort is on the hub model and network design for different transportation application domains. Few scholars, however, have studied the flow delays and the corresponding costs intrinsic to the
H-S network structure. Zhou et al. (2023) proposed a hub-and-spoke network design (HSND) for container shipping in inland waterways based on the tree-like river structure [
11]. Then, they determined the optimal hub location, branch port allocation, and fleet deployment, aiming to minimize the total cost of ships, transport, and transport. Considering a multimodal hub-and-spoke transportation network for emergency relief schedules, Li et al. (2023) established a mixed-integer nonlinear programming (MINLP) model considering multi-type emergency relief and multimodal transportation. The model is a bi-objective one that aims to minimize both transportation time consumption and transportation costs [
30].
Regarding congestion, there have been scholars conducting related studies but they were concentrated in certain fields, such as aviation and container shipping. Santos et al. (2017) suggested airline companies consider delay problems as routine operations [
31]. Sismanidou et al. (2022) revealed a significant correlation between delayed incoming flights and departure delays, offering valuable insights for policies aimed at mitigating airport and network congestion [
32]. Huang et al. (2022) investigated an extended container shipping hub-and-spoke network design problem (HSN), considering the failure and congestion of hubs; they developed a 0–1 nonlinear programming model for minimizing the transportation cost [
33]. Additionally, congestion and delay may affect service performance and the choice of network structure. Delay problems can be eased by re-designing network structures [
34,
35,
36,
37]. Lange et al. (2023) considered a location–allocation–routing problem, postulating that queueing problems result from limited capacities, where congestion occurs [
36]. Ashish (2018) pointed out that there were many causes of delay, which could be reduced according to the characteristics of the airport [
38]. Yazdi et al. (2017) demonstrated that a baggage fee policy may have a limited effect on alleviating the delays experienced by airlines. [
39]. Chen et al. (2021) extended the research on East Asian airports by emphasizing the importance of network attributes in determining flight delays [
40].
Few scholars considered flow delay problems when designing H-S logistics network structure, or focused studies on delay costs. Therefore, this paper starts from network structure, and focuses on studying the influence of delay problems on network structure, in the hope of realizing the purpose of improving earnings and enhancing network stability via optimizing network structure.
5. Algorithm and Application
There are various kinds of heuristic algorithms. This paper takes five heuristic algorithms as examples to find the solution algorithms suitable for this paper. For the merits of the five heuristics, the comparison results are shown in
Table 2.
Compared with other heuristic algorithms, particle swarm optimization (PSO) has the characteristics of faster convergence, fewer parameters to be adjusted, a simpler structure, and easier implementation. It can quickly and efficiently select the specified number of hubs. It should be noted that PSO is a heuristic algorithm that may fall into a local optimal solution, so parameter tuning and multiple runs according to the specific problem are required to increase the probability of finding the global optimal solution. In the process of network layout optimization, there are many parameters involved, the solution requires more cycles, and the particle swarm algorithm has the advantages of fast convergence, a fast solution, and it can give the optimization scheme quickly, so this paper chooses to use the particle swarm optimization algorithm to solve the model.
5.1. Particle Swarm Optimization Algorithm
The PSO is an evolutionary algorithm based on swarm intelligence. The PSO is derived from the study of the predatory behaviors of birds, which was proposed by Eberhart and Dr. Kennedy in 1995. The algorithm shares the information obtained by individuals through group activities, searches for solutions from disorder to order, and finally obtains an optimal solution. It has the advantages of a high rate of convergence and flexible parameter adjustment. Özgün-Kibiroğlu et al. (2019) used the particle swarm optimization (PSO) algorithm to solve the hub location problem and also proved the feasibility and advantage of the particle swarm optimization algorithm applied to the hub-and-spoke network structure [
16]. In the research range of this paper, the n-dimensional variable
is set to be the particles, and every particle represents a feasible program. Assuming the solution space is n-dimensional, every particle could be described in two conditions: location
and speed
. The detailed process of the PSO algorithm is as below:
The first step is to confirm the hinge node number p and all the node numbers n and to initialize the particle’s location and speed. This paper sets the initial = unifrnd (0, 1, varsize), where the variable varsize represents the matrix size of the decision variable and function unifrnd generates the random number group in continuous uniform distribution within a specified range. This paper sets the initial = zeros (varsize), and the function zeros creates an empty matrix with the specified size.
The second step is to calculate the particle fitness value for evaluation so as to confirm the particle’s individual optimal value and group optimal value. The evaluation standard of this paper is composite cost; with the purpose of minimizing the composite transport cost. The fitness function code is , wherein is a self-defining function, used to output and store the fitness value calculation result. , wherein the main file includes the expression of the fitting function and provides the output in the form of a function. According to the models in this paper, the fitness function expressions are and . The total transport cost is the sum of and , respectively, where i, j, k, and l are set up differently, with each node having a number, i < j, and k, l being hinge nodes. Every particle is moving towards the optimal solution according to its current speed and its experience.
The third step is to update the particle’s location and speed. The updated rules are as follows:
and
, with a cyclic update. According to article [
13], assume
,
, and
, wherein
,
, and
. If the number of iterations is reached, then output occurs; if it does not reach the iterations, it returns to the second step.
We fixed the number of initial particle swarms at 150 and the number of iterations at 100. Iterations and node numbers could be regulated voluntarily. In addition, in the software that runs the algorithm, the data module could be self-defined. Repeatable operations could be realized via data updates and the re-installation of parameters.
Table 3 shows the pseudo-codes of the improved PSO algorithm.
We have implemented the algorithm using the MatlabR2021b programming language due to its robust set of libraries for numerical computations and data analysis, which are particularly well suited for the complex calculations required in our study.
5.2. An Application
SF is a private express delivery company engaged in domestic and international packaging, customs declaration, inspection, and quarantine. In March 1993, it was established and headquartered in the Guangdong Province, China. During the early days of the company, it only provided express services in Hong Kong and Guangdong. With consistent quality service, the firm grew and expanded into provinces and cities in the Yangtze River Delta region, then into East China, North China, and Central China, and eventually provided services nationwide. Today, SF has 39 first-level branches and 2600 self-built outlets all over the country, covering 250 major cities and more than 900 county-level cities or towns.
In recent years, the rapid development of the express delivery industry has brought opportunities to the company’s fast growth, especially in Eastern China with several large cities, including megacities like Shanghai. SF built internal subsidiaries and implemented self-management. Taking the Jiangsu Province as an example, SF has opened nearly one thousand last-mile local business outlets and chose Huai’an, Nanjing, and Wuxi as the main transfer and distribution centers (i.e., hub nodes) for the northern, central, and southern regions of the Jiangsu Province. According to their geographical locations, these three hubs serve the whole Jiangsu Province. Other prefecture-level cities basically connect to their respective closest hub nodes to form SF’s three hub locations and single-hub allocation H-S network structure, in which Huai’an connects to “Xuzhou, Lianyungang, Suqian, Yancheng”; Nanjing connects to “Yangzhou and Zhenjiang”; and Wuxi connects to “Taizhou, Changzhou, Suzhou, and Nantong”. However, the flow delay costs, which have been ignored, have become a burden to the firm, especially with the increasing business volume. SF management felt it urgent to reconsider the cost-saving potential for its Eastern China regional operations using a better-designed multi-hub location and single-hub allocation H-S network structure. The central task is to locate more than one and up to four hubs.
The Jiangsu Province is an economically developed region in Eastern China with excellent transportation services and capacities. Each of its major cities is one of the top 100 cities in China with unique strength and can serve as an alternative hub. The company’s express package delivery services are balanced, and the demand is high in all directions without many empty trucks returning. Therefore, each of the 13 prefecture-level cities in the province can be regarded as an alternative hub, namely Nanjing, Wuxi, Xuzhou, Changzhou, Suzhou, Nantong, Lianyungang, Huaian, Yancheng, Yangzhou, Zhenjiang, Taizhou, and Suqian, which are, respectively, labeled as I1 to I13 in
Figure 3.
The number of hubs and their locations in an
H-S network directly influence the effects of flow concentration on
H-H links and their economies of scale. If there are too many hubs, they may dilute the express traffic, resulting in a decrease in the
H-H flow discount effects. Otherwise, if there are too few hubs, a high flow concentration may occur at hubs and on
H-H links, leading to large delay cost losses and heavy hub operation pressures. Therefore, determining the optimal number of hubs and their locations is an important part of the
H-S network design. Currently, there is no single best way to determine the number of hubs for an
H-S network. According to Boland et al. (2004), when n = 13,
p can be 1, 2, 3, or 4 [
43]. When
p = 5 or 6, the number of hubs is excessive largely due to the large fixed and operating costs of hubs and, hence, becomes contradictory to SF’s original idea of cost saving [
39]. When
p ≥ 7, at least one non-hub node connects two hub nodes for multi-hub allocation, which is inconsistent with this study’s assumptions. When
p = 1, the hub must be Nanjing, because Nanjing is the capital city of the Jiangsu Province. Therefore, the model application below only considers the cases when
p = 2, 3, or 4.
Table 4 and
Table 5 show the average daily
O-D flows and distance matrices for the 13 prefecture-level cities, respectively. The data were obtained from the research department of SF Company. The
O-D flow matrix provides a detailed breakdown of the average daily shipment flows between each pair of the 13 prefecture cities within the network. Each cell in the matrix represents the average volume of goods (express packages) that are transported daily from one city to another. The matrix is symmetrical, as the flow from city A to city B is the same as from city B to city A, assuming bidirectional traffic. The distance matrix outlines the average distances between each pair of the 13 cities. Each cell in the matrix indicates the average distance that must be traveled from one city to another, based on the most commonly used transportation routes.
Considering the minor variations of roads in the Jiangsu Province, the transportation rate between any two cities in the Jiangsu Province is set at C level. According to SF’s express delivery cost schedules in the Jiangsu Province, the average transportation cost rate at level is C = 0.03 yuan/kg*km.
6. Results and Analysis
6.1. Main Results
Applying the data in
Table 3 to the modified model with the improved PSO algorithm and following the steps in
Table 2, we obtained the optimal results corresponding to different
p (
2, 3, 4) and
values in
Table 6.
The first column shows the number of cities (n = 13) as nodes. The second column shows the number of hubs to be located, p. The third and fourth columns provide the optimal hub locations selected. The fifth column lists the values of discount factor α. Columns six–eight list the optimal O-D transportation costs Y1, the flow delay cost Y2, and the total optimal cost . For example, for p = 4 and α = 0.8, the optimal objective function value is = CNY 3,288,575 and the four selected hubs are located at I4, I9, I11, and I13, or in cities of Changzhou, Yancheng, Zhenjiang, and Suqian.
To simulate the SF’s current operations, the three hub cities are prefixed at I1, I2, and I8 with the corresponding cities being Nanjing, Wuxi, and Huai’an. The non-hub nodes connecting the hub node Nanjing (I1) are Yangzhou (I10) and Zhenjiang (I11); those connecting the hub node Wuxi (I2) are Changzhou (I4), Suzhou (I5), Nantong (I6), and Taizhou (I12); and those connecting the hub node Huai’an (I8) are Xuzhou (I3), Lianyungang (I7), Yancheng (I9), and Suqian (I13). The overall
H-S network configuration is shown in
Figure 4d. Simulating current operations using the final model and data in
Table 3 and with the SF’s past operations, we set the discount factor at
α = 0.8 and applied the PSO algorithm to solve the model. The objective function value is CNY 4,120,494, which is much larger than the optimal objective function value of CNY 3,540,886 when
p = 3 and
α = 0.8.
Compared with the results from the simulated SF operations when p = 3 and α = 0.2–1.0, we obtained hub cities at I4, I11, and I13 and the corresponding cities are Changzhou, Zhenjiang, and Suqian. The non-hub nodes connecting to the hub node Changzhou (I4) are Wuxi (I2), Suzhou (I5), and Nantong (I6); those connecting to the hub node Zhenjiang (I11) are Nanjing (I1), Yangzhou (I10), Taizhou (I12), and Yancheng (I9); and those connecting to the hub node Suqian (I13) are Xuzhou (I3), Lianyungang (I7), and Huai’an (I8). The optimal objective function values range from CNY 2,482,684 to CNY 3,893,620.
The optimal
H-S configurations corresponding to the above results and the rest of the optimal
H-S networks are summarized in
Figure 4a–c, in which the yellow square represents the hub nodes, and the circles represent the non-hub nodes. Also, the
N-H connections are linked by colored lines and the
H-H connections are shown by bold black lines.
6.2. Main Analysis
When
p = 3 and α = 0.8, the total cost simulated for current operations for SF is CNY 4,120,494 per day on average, while the total cost obtained using the optimized model is CNY 3,540,886 per day, leading to CNY 579,608 savings per day or a 14.1% total cost reduction.
Table 6 provides the cost-saving breakdowns for the transportation cost, flow delay cost, and total cost. Savings are in CNY amount and percentages also were summarized in
Table 7.
In comparing simulated results for current operations and modeled optimal results in
Table 6, we can find that (i) the modeled results slightly reduced transportation costs by 3.2% and significantly reduced the flow delay cost by 55.5%, which together led to a 14.1% total cost saving; (ii) the modeled results substantially lowered the flow delay cost from 20.75% to 10.74% while not increasing the transport costs much; (iii) further breakdowns of the cost savings show that transportation costs and flow delay costs contribute 18.1% and 81.9%, respectively. In other words, of the 14.1% of the total cost saving, 2.55% was realized through the transportation cost and 11.55% was through the flow delay cost. Please note that the cost saving here is daily and on average. Annual and long-term savings would be tremendous for SF. In general, it is of paramount importance to consider flow delay cost in planning or designing
H-S networks.
The optimal results in
Table 5 show that if other conditions remain unchanged when the number of hubs
p increases up to four, the optimal values of the objective function or the total transportation cost decrease, even with different discount factor
α values. This is largely caused by the fixed cost factor yet to be considered. The total fixed cost factor can be easily modeled by adding it to the objective function in the model (13)–(14) as
sum over
k. The fixed cost is often considered in the literature to include the hub construction cost and the hub trans-shipment cost. Normally, when the total fixed cost estimate is known, the number of hubs can be roughly estimated. The reverse holds as well.
Based on the results in
Table 5, if
α = 0.8 and
, the total cost is 3,971,306 + 2
when
p = 2, 3,540,886 + 3
when
p = 3, and 3,288,575 + 4
when
p = 4. When the fixed cost satisfies
with
p = 4, the total cost is always the lowest;
if , the best hub number is
p = 3; when
, the optimal hub number is
p = 2. Here, the lower and upper bounds are based on actual resource planning for hub fix costs.
Figure 5 shows the relations of these bounds and values.
Also, if other conditions remain unchanged, when the discount factor
α for
H-H flows increases in a certain range, the total transportation cost increases correspondingly as shown in
Table 5. However, the value selection of
α does not affect the number of hubs and the hub locations, but it does affect the total optimal cost in a simple proportional way.
Moreover, the trans-shipping nature of hubs and the positive scale economy of flow consolidation on H-H links may well offset the flow delay cost to a certain extent, making the H-S network superior to the point-to-point network. This is the single most important reason that most urban logistics firms adopt operations under a full or semi-hub-and-spoke network configuration. However, it is worthwhile here to quickly compare the P-P and H-S networks to gain a general understanding of major influencing factors.
Figure 6 is the
P-P network for the 13-city SF logistics operation. Its total system cost conceptually can be thought of as consisting of transportation costs and fixed costs, which can be expressed as
Assuming , with c > 1, a simple calculation yields the P-P transportation at CNY 3,963,642. Therefore, when (1) , the highest total cost is , and it is best to set In the meantime, when , , the H-S is superior to the P-P; when , the relative advantage of the two network typologies depends on values of the discount factor and the fixed cost; (2) when , the highest total cost is , the lowest total cost is over , and the best number of hubs is . If , , the H-S network is better than the P-P; however, when , the H-S advantage cannot be determined due to a lack of conditions. (3) When , the lowest total cost is over and the optimal number of hubs . Because , the relative advantage of the two network typologies depends on the values of the discount factor and the fixed cost.
7. Discussion
The model can be improved in many ways, which may be grouped into two broad directions. One includes the technical improvements of the model, including adding fixed cost directly into the objective function, allowing for more than four hubs to be located and/or only one hub to be located, and expanding the discount factor range beyond 0.2–1.0. In each of these improvements, the total H-S costs can be compared with the P-P costs. Also, capacities for city–city links or streets and trans-shipping time or cost at hubs may be considered. Moreover, reliability on links and at hubs and other practical considerations such as speed limits, congestions, situations for less truck-load or empty-truck load, etc., can be built into the model to make it closer to reality. The other direction for improvements may be on operational and policy aspects to take account of the rapid development of the urban logistics industry, especially on the location, construction, safety, and sustainability of the hub nodes and on the long-term, dynamic, and systematic process of building a complex yet efficient H-S network meeting the business needs of the enterprise and the industry, functional planning requirements of the local cities, and economic and social benefits of the society.
8. Conclusions
This research developed a hub-and-spoke network design model based on the seminal quadratic hub location and allocation model by O’Kelly (1987), considering flow delay cost, scale economy, and multiple hub locations with single-allocation [
5]. The model was linearized using the algebraic approach suggested by Shen (1996) and solved using the particle swarm optimization for a local urban express delivery company SF in the Jiangsu Province of Eastern China [
42]. In addition to the modeled optimal results and corresponding optimal
H-S configurations, exploratory discussions on cost savings with respect to SF’s current operations and the effects of hub fixed cost, discount factor, the number of hubs, and
P-P configuration are also presented.
The results show that first, the simple algebraic approach in linearizing the quadratic term worked well. The modeled results showed sizable daily savings (14.1%) in the total cost over the simulated results with the hub locations and other parameters from the current operations of SF. The long-term cost savings are considerable. Second, the model can locate up to four hubs at variable locations, with location 11, or Zhenjiang, selected as a hub regardless of the number of hubs to be located (p = 2, 3, or 4), followed by locations 4 and 13, or Changzhou and Suqian, when p = 3 or 4. Locations 8 and 9, or Huanan or Yancheng, were only selected once as a hub when p = 2 or 4, respectively. In general, when the number of hubs increases, the hubs selected are further apart when fewer hubs are to be located and closer when more hubs are to be located. Also, along with more hubs to be located, the total transportation cost decreases. However, the discount factor seems to have no effect on hub locations but it does proportionally affect the total costs. The flow delay cost has a greater impact on the locations of hubs with larger flow trans-shipments or more allocations by smaller and nearby connecting cities. Given that the hub location, construction, maintenance, and flow concentration are the most important parts of any H-S network, considering flow delay cost in H-S planning is highly recommended.