Mathematical Problems in Engineering

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Modeling and Simulation in Transportation Engineering 2014

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Volume 2014 |Article ID 830632 |

Weiya Chen, Kaixian He, Xiaoping Fang, "Optimization of Hybrid Hub-and-Spoke Network Operation for Less-Than-Truckload Freight Transportation considering Incremental Quantity Discount", Mathematical Problems in Engineering, vol. 2014, Article ID 830632, 9 pages, 2014.

Optimization of Hybrid Hub-and-Spoke Network Operation for Less-Than-Truckload Freight Transportation considering Incremental Quantity Discount

Academic Editor: Huimin Niu
Received10 Jul 2014
Accepted16 Sep 2014
Published29 Sep 2014


This paper presents a mixed integer linear programming model (MILP) for optimizing the hybrid hub-and-spoke network operation for a less-than-truckload transportation service. The model aims to minimize the total operation costs (transportation cost and transfer cost), given the determined demand matrix, truck load capacity, and uncapacitated road transportation. The model also incorporates an incremental quantity discount function to solve the reversal of the total cost and the total demand. The model is applied to a real case of a Chinese transportation company engaged in nationwide freight transportation. The numerical example shows that, with uncapacitated road transportation, the total costs and the total vehicle trips of the hybrid hub-and-spoke network operation are, respectively, 8.0% and 15.3% less than those of the pure hub-and-spoke network operation, and the assumed capacity constraints in an extension model result in more target costs on the hybrid hub-and-spoke network. The two models can be used to support the decision making in network operations by transportation and logistics companies.

1. Introduction

Transportation companies usually have their own physical operation network with facilities located in different levels of nodes. Generally there are three kinds of physical operation network: direct transportation network, pure hub-and-spoke network, and hybrid hub-and-spoke network. It is a great challenge for transportation companies to make daily decisions about optimal network operations on an existing physical network, especially on a hub-and-spoke one.

A direct transportation network is a fully connected network which has the advantage of short transport distance and fast speed [1]. However, in terms of LTL (less-than-truckload) transportation, it is faced with the risk of losing economies of scale since not every origin-destination (OD) flow can stably satisfy the effective truck load capacity. The fact is that some small and unstable OD flows result in high operation cost, and even a high risk of partial loss. Overcoming the above disadvantages, the hub-and-spoke network makes use of economies of scale on trunk road transport by collecting goods to pivots [2, 3]. Although it inevitably sometimes produces bypass transportation and transfer processing costs, in order to cover a wider marketing, some transportation and logistics companies prefer to adopt the hub-and-spoke network [4].

To help with daily operational decision making, a lot of research has been done to optimize the hub-and-spoke network operation. In these research works, mathematical models and algorithms are used to optimize pure hub-and-spoke network operations in three aspects: hub location optimization, routing optimization, and both [512]. In a pure hub-and-network, all OD flows in between nonhub nodes need a transfer operation at least once. Demand flows between different OD pairs usually increase unevenly; however, some of the OD flows may achieve the effective scale of direct transport or need a transfer operation only once. This unnecessarily produces more transportation and transfer costs and delays the arrival time. Thus, a hybrid hub-and-spoke network is used to overcome these problems.

To optimize the hybrid hub-and-spoke network operation, different models, formulations, and algorithms have been addressed in numerous papers in the literature. Liu and colleagues investigated a mixed truck delivery system that allows both hub-and-spoke and direct shipment and used a heuristic algorithm to determine the delivery modes and vehicle routings [13]. Zapfel and Wasner presented a mathematical model to optimize the mixed hub-and-spoke network by minimizing the total costs of transportation cost and variable sorting cost based on a physical network [14]. Barcos and colleagues designed an LTL cargo operation network by minimizing the total transportation costs and transfer processing costs, guaranteeing a certain service level and allowing for vehicle routes with multiple unloading stations [15]. The latter two papers show that hybrid hub-and-spoke networks are more efficient and economical than pure hub-and-spoke networks as they do not consider the reconstruction and renovation cost of the physical network. More specifically, Adler investigated hub-spoke network operations in a competitive environment [16]. Elhedhli and Hu designed a hybrid hub-and-spoke network which took congestion into account [17]. Matsubayashi and colleagues studied the cost allocation problem in a hybrid hub-and-spoke network system [18]. These works showed that the hybrid hub-and-spoke network has advantages in terms of cost savings and equipment efficiency over the pure hub-and-spoke network.

Another noteworthy fact is that, in order to attract more order volumes, different price discount policies are used by transportation companies [19]. The most commonly used discount policy is unified discount. The larger the order volume is, the higher the price discount is. It allows customers to adjust their order volumes to reach a higher price discount point. One shortcoming of this pricing policy is that it leads to a reversal of the total cost and the total flow near some discount points, which often causes confusion or even conflict among transportation companies and their customers.

This paper aims to design and optimize the operation of a hybrid hub-and-spoke network by employing an existing physical hub-and-spoke network. Direct transport between different nodes is allowed if the OD flow reaches an effective scale. Otherwise the OD flows in between nontransfer nodes need to be transferred at hubs at least once. Vehicle routes are optimized and an incremental quantity discount function is embedded to make sure that the total cost is increasing strictly by total demand.

The remainder of this paper is organized as follows. Section 2 presents a mixed integer linear programming model (MILP) to optimize the LTL transportation operation of a hybrid hub-and-spoke network with uncapacitated road segments taking into account the incremental quantity discount. Section 3 shows an application of this model to the real case of a Chinese logistic company engaged in nationwide freight transportation. Section 4 formulates an extension model considering capacitated road segments, namely, links, and presents numerical results. Conclusions are drawn in Section 5.

2. Model Formulation

The model focuses on the optimization of an existing physical network in which less-than-truckload freight transportation services are provided by a transportation company. This physical network consists of a set of transfer nodes and a set of nontransfer nodes, namely, hubs and nonhub nodes in this paper. Let a complete graph be the physical network, where is the set of all nodes (specifically, is the set of hubs, which is a subset of ) and is the set of links. Given the OD matrix and a discount policy, the model aims to generate an optimal operational network based on the graph . Before the model is constructed, some assumptions are made as follows.(1)The OD flows are determined, splittable and allowed to be directly transported, and transferred once or twice.(2)The load capacity of each truck is fixed. The service frequency is determined by the goods flows and subject to a promised service level.(3)The total operation costs consist of transportation cost and transfer cost. Transportation cost is related to the flow, which is characterized by an incremental quantity discount function . Transfer cost at hubs usually consists of two parts: fixed transfer cost and variable transfer cost. In this paper, since we focus on the optimization problem of an existing physical network, the fixed transfer cost is ignored and the transfer cost is equal to the variable transfer cost, denoted by function .(4)The hubs and road segments are uncapacitated.

Other notations are as follows:: total flow (by weight) from node to node ;: unit transportation cost (per kilogram per kilometer) of link -;: flow of link - that adopts discount of interval , namely, link flow. The domain of the whole discount function is divided into intervals, ;: distance from node to node ;: flow transferred at node , namely, transfer flow;: unit transfer cost at node ;: switch function, valued as one or zero, respectively, means link uses discount of interval or not;: proportion of goods flows which take the route to all goods flows from node to . Thus, ---, ---, and --- are all equal to the direct route -. ---, ---, and --- are all equal to the route --.

In this study, the optimization target is to minimize the total operation costs , which can be denoted as the following objective function:

In the right side of (1), the first part is the total transportation cost, which is a function of the link flow ; the second part is the total transfer cost at hubs, which is a function of the transfer flow . In order to solve the reversal of the total cost and the total flow, an incremental quantity discount is used in this paper, which is distinct from threshold-based discounting [19]. The incremental quantity discount is decided by where and are the lower and upper bound of interval , respectively, and , ; is the discount rate of interval . Commonly, a decreasing discount rate is used to attract more freight volumes; thus let . Then we can obtain the average unit discount in

In (3), is the weighted average unit discount rate of one customer’s total freight volume. When and the discount interval is set properly, the larger the total volume is, the smaller the weighted average unit discount rate will be. Under this discounting policy, transportation or logistics companies can use quantity discounting to attract more order volume; while customers could adjust their order volume to obtain a self-satisfied average unit discount point to save cost, but their total cost will still rise along with the rising total volume.

Note that transportation cost also has a relation with the transportation distance. By incorporating the average unit discount rate , we can formulize the transportation cost as

For simplicity, the transfer cost is assumed to be proportional to the transfer flow at hubs. The transfer cost at node is formulized as

Thus, we can rewrite objective function (1) into subject to

Constraint (7) ensures that, if , the route --- will be calculated just once. Constraints (8) and (9) ensure that will be located in the interval . Constraint (11) ensures the equilibrium of all flows. Constraint (12) ensures that the goods flows are nonnegative. Constraint (13) ensures that the transferred flows can just be transferred at hubs. Constraints (10) and (14) ensure the consistency of the discount rate.

3. Model Application and Solution

3.1. Sample Data

The data come from a transportation company in China which is engaged in LTL freight transportation in 18 cities. Of those cities, five have transfer hubs and the rest are nonhub nodes. All nodes of cities in the physical network are shown in Table 1. We extract sample data covering all the business in the 18 cities for 2011, comprising 128018 records. The summed OD flows by weight are shown in Table 3. We take an average capacity of 28 tons for each truck in the following calculation.

HubsNonhub nodes

Beijing (BJ)Tianjin (TJ), Shijiazhuang (SJZ), Jinan (JN), Harbin (HEB), Changchun (CC), and Shenyang (SY)
Zhengzhou (ZZ)Taiyuan (TY) and Xi’an (XA)
Wuhan (WH)Changsha (CS)
Guangzhou (GZ)Chengdu (CD)
Shanghai (SH)Nanjing (NJ), Hangzhou (HZ), and Fuzhou (FZ)

The incremental quantity discount policy is given as follows: where means a very large value for running the calculations correctly. This incremental quantity discount policy is characterized by an incremental quantity of 20 tons with a decreasing discount rate of 0.03 with an exception of 0.05 for , which is adjusted according to the sample company’s marketing pricing strategy.

3.2. Calculations

The model of formulae (6) to (14) is a mixed integer linear programming model. An exponential algorithm in Lingo 11.0 can be used to solve this mixed integer programming problem. The computational results are shown in Table 4.

As regards the pure hub-and-spoke network and the hybrid hub-and-spoke network, respectively, the total costs are 8686950 and 7994138 Yuan, and the total trips are 145974 and 123601. Reductions of 8.0% and 15.3% in costs and trips apply to the hybrid hub-and-spoke network compared with the pure hub-and-spoke network. The optimized operation network is shown in Figure 1.

Lingo 11.0 is executed by a desktop with the CPU environment of AMD Athlon 642 Dual-Core Processor TK-53. The computing time increases greatly as the number of hubs increases. The results are listed in Table 2.



OriginDestination D

BJ346254 831170 305478 275138 1136480 200636 154463 348277 72556 227817 152150 493632 354951 336023 234325 277123 200491 504260
SH1492555 466158 357120 402294 1833265 164536 133084 372179 101944 177396 161269 577575 647424 714716 296518 381245 288024 589470
ZZ475739 332708 381684 190618 1438285 67613 123862 145397 94330 82627 54990 189129 197397 197312 121324 158656 236104 358152
WH282678 196287 285817 304963 714186 34587 40716 119363 69705 51688 30376 100588 119531 99469 64473 86642 137768 257214
GZ4405386 4074401 1920030 1659548 9512010 535071 482378 1359493 392251 619211 515248 1637551 1819369 1910156 1107434 1299065 1271186 3054502
CC123743 67176 26754 32738 171856 24343 26699 35862 10464 19247 11402 53625 32847 50201 29226 37981 42563 59479
HEB198882 71848 47824 35092 124474 25151 9844 57348 21085 17746 14381 41675 34057 44704 25894 35516 29361 37121
SY280709 157396 99170 69126 372673 54157 57124 87463 23343 53557 42415 117157 70343 95539 77923 87014 59094 91923
FZ52052 56433 28880 23172 133556 9053 10539 34382 2332 12552 19034 14823 28633 33675 8138 16069 11943 26865
SJZ237913 130914 44443 43326 318121 21383 22864 56088 16865 50852 81326 129901 43300 57619 48750 35515 35613 65823
TY110686 66410 58211 29351 232503 9443 9486 28800 12833 23171 15908 51046 27302 16565 21954 20741 17617 21868
TJ618265 372465 175975 166542 774788 68151 67585 172523 35276 150023 58796 198733 152156 146711 98751 144660 82602 204834
HZ489345 367531 157944 179272 942358 56066 32681 124304 32885 74322 46596 171123 133284 245613 108950 147298 102084 257641
NJ369900 248087 112739 116018 732371 37688 29480 92128 25210 49475 24487 126977 144194 154744 84170 88376 102316 132692
XA199660 134839 100116 71833 400062 25380 20820 40610 8119 22870 16162 129822 50443 58149 99564 48495 35683 117185
JN262089 211140 55830 69986 565681 21285 30595 79525 20483 26941 17833 93177 84442 69955 60601 161998 39682 117094
CS273589 155296 95777 96007 419551 19308 24785 35211 17302 45259 21968 48438 85111 88769 39535 50558 191007 139015
CD368194 295798 236696 394306 900767 76841 29225 85211 27424 82350 38769 125126 110637 153845 168064 194997 113667 500990

OriginDestination D


4. Model Extension: Capacitated Hybrid Hub-and-Spoke Network Operation for LTL Freight Transportation

When goods flow is increasing, the capacity constraint should be taken into consideration. We take the road capacity, hub capacity, and truck capacity as one unified capacity constraint, namely, link capacity on a graph, because these kinds of capacity in a physical network finally restrain the flows in the link. Consequently, an extension model based on the model in Section 2 is constructed. Mathematically, we obtain the following extension model: subject to

In (18), is the upper limit of the flow in link -. Since the road segments are not subject to any capacity constraint in the sample enterprise, all the segments are assumed to have an upper limit of 3.1 million tons per month. The optimized network is presented in Figure 2.

Compared with the uncapacitated model, when the flow exceeds the link capacity and these kinds of links are few, the operational network does not change very much. The goods flows sent by each link computed by the two models are listed in Tables 4 and 5. However, the total costs have changed. The total costs are 8065921 Yuan, 0.90% higher than those of the uncapacitated network. The main reason is that the goods flow exceeding the road capacity is sent by other links because of insufficient capacity. In the example, each link has an upper capacity of 3.1 million tons per month, and only four links are fully used in the optimized network (see the red links in Figure 2).

OriginDestination D


5. Conclusions

Operation optimization is critical for reducing operation costs for transportation companies engaged in nationwide LTL freight transportation in an existing physical hub-and-spoke network. In this paper, a MILP model for LTL transportation operation optimization of a hybrid hub-and-spoke network is proposed which takes into account an incremental quantity discount function in order to solve the reversal of the total costs and the total flow. In fact, for a company engaged in nationwide transportation in an existing hub-and spoke network, when the OD flows are not big enough for direct transport, a hub-and-spoke system can help to obtain economies of scale. The results of the model application for the sample enterprise indicate that 56.21% of the goods flows are sent by direct transport, 32.15% of which are sent from one depot to another; 27.95% of the goods flows are sent and transferred once at a hub, and only 15.84% are transferred twice; the hybrid hub-and-spoke network operation saves 8.0% more on total costs and 15.3% more on total trips than the pure hub-and-spoke network does. Furthermore, the incremental quantity discount price policy can eliminate the reversal of the total cost and the total transport volume.

In daily management, trucks are seldom dispatched between different pairs of OD except in case of emergency or by request. Thus, we have to take capacity constraint into account. In the extension model in this paper, the assumed capacity constraint increases the total costs. The main reason is that the excess OD flow has to be sent via bypass transport, at higher cost. This extension model can be used to support business owners in making decisions on optimizing facilities.

The proposed models can be applied to logistic companies and cargo transportation companies with an existing physical hub-and-spoke network. The models adapt well to different numbers of hubs and the OD flow matrix. They can be used to support daily decision-making in the case of changing OD flow matrices.


See Tables 3, 4, and 5.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


This research was supported by the National Natural Science Foundation of China (NSFC) under Grant no. 61203162 and the National Social Science Foundation of China under Grant no. 11CGL032. This research is also supported in part by the Research and Development Funds of Science and Technology of China Railway Corporation (2013X008) and Social Science Foundation of China Education Ministry (11YJC790143).


  1. C. C. Lin, Y. J. Lin, and D. Y. Lin, “The economic effects of center-to-center directs on hub-and-spoke networks for air express common carriers,” Journal of Air Transport Management, vol. 9, no. 4, pp. 255–265, 2003. View at: Publisher Site | Google Scholar
  2. M. E. O'Kelly and D. L. Bryan, “Hub location with flow economies of scale,” Transportation Research Part B: Methodological, vol. 32, no. 8, pp. 605–616, 1998. View at: Google Scholar
  3. M. W. Horner and M. E. O'Kelly, “Embedding economies of scale concepts for hub network design,” Journal of Transport Geography, vol. 9, no. 4, pp. 255–265, 2001. View at: Publisher Site | Google Scholar
  4. G. Nero and J. A. Black, “Hub-and-spoke networks in the inclusion of environmental costs on airport pricing,” Transportation Research Part D: Transport and Environment, vol. 3, no. 5, pp. 275–296, 1998. View at: Publisher Site | Google Scholar
  5. J. F. Campbell, “Integer programming formulations of discrete hub location problems,” European Journal of Operational Research, vol. 72, no. 2, pp. 387–405, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. J. Ebery, M. Krishnamoorthy, A. Ernst, and N. Boland, “Capacitated multiple allocation hub location problem: formulations and algorithms,” European Journal of Operational Research, vol. 120, no. 3, pp. 614–631, 2000. View at: Publisher Site | Google Scholar
  7. A. Marín, “Formulating and solving splittable capacitated multiple allocation hub location problems,” Computers and Operations Research, vol. 32, no. 12, pp. 3093–3109, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. H. Yaman, B. Y. Kara, and B. Ç. Tansel, “The latest arrival hub location problem for cargo delivery systems with stopovers,” Transportation Research Part B: Methodological, vol. 41, no. 8, pp. 906–919, 2007. View at: Publisher Site | Google Scholar
  9. A. T. Ernst, H. Hamacher, H. Jiang, M. Krishnamoorthy, and G. Woeginger, “Uncapacitated single and multiple allocation p-hub center problems,” Computers and Operations Research, vol. 36, no. 7, pp. 2230–2241, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. R. Z. Farahani, M. Hekmatfar, A. B. Arabani, and E. Nikbakhsh, “Hub location problems: a review of models, classification, solution techniques, and applications,” Computers and Industrial Engineering, vol. 64, no. 4, pp. 1096–1109, 2013. View at: Publisher Site | Google Scholar
  11. S. Melkote and M. S. Daskin, “An integrated model of facility location and transportation network design,” Transportation Research Part A: Policy and Practice, vol. 35, no. 6, pp. 515–538, 2001. View at: Publisher Site | Google Scholar
  12. D. Sun and L. Elefteriadou, “Research and implementation of lane-changing model based on driver behavior,” Transportation Research Record, no. 2161, pp. 1–10, 2010. View at: Publisher Site | Google Scholar
  13. J. Liu, C.-L. Li, and C.-Y. Chan, “Mixed truck delivery systems with both hub-and-spoke and direct shipment,” Transportation Research Part E: Logistics and Transportation Review, vol. 39, no. 4, pp. 325–339, 2003. View at: Publisher Site | Google Scholar
  14. G. Zapfel and M. Wasner, “Planning and optimization of hub-and-spoke transportation networks of cooperative third-party logistics providers,” International Journal of Production Economics, vol. 78, no. 2, pp. 207–220, 2002. View at: Publisher Site | Google Scholar
  15. L. Barcos, V. M. Rodriguez, M. J. Alvarez, and F. Robuste, “Routing design for less-than-truckload motor carriers using ant colony techniques,” Workings Paper, Business Economics Series 14, Universidad Carlos III, 2004. View at: Google Scholar
  16. N. Adler, “Hub-spoke network choice under competition with an application to Western Europe,” Transportation Science, vol. 39, no. 1, pp. 58–72, 2005. View at: Publisher Site | Google Scholar
  17. S. Elhedhli and F. X. Hu, “Hub-and-spoke network design with congestion,” Computers and Operations Research, vol. 32, no. 6, pp. 1615–1632, 2005. View at: Publisher Site | Google Scholar
  18. N. Matsubayashi, M. Umezawa, Y. Masuda, and H. Nishino, “A cost allocation problem arising in hub-spoke network systems,” European Journal of Operational Research, vol. 160, no. 3, pp. 821–838, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. H. Podnar, J. Skorin-Kapov, and D. Skorin-Kapov, “Network cost minimization using threshold-based discounting,” European Journal of Operational Research, vol. 137, no. 2, pp. 371–386, 2002. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2014 Weiya Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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