1. Introduction
With the rapid growth of online shopping, the demand in the logistics and transportation industry is continuously increasing. The number of parcels shipped in the United States reached a record 21.5 billion in 2021. Express delivery plays a vital role in our daily life, and the express service market is highly competitive [
1]. Express service providers must improve the reliability of their delivery services while minimizing total costs [
2]. The increasing number of parcels has become a challenge for the express delivery service system. Developing decision-making tools for service providers is necessary to improve this situation and stand out from the competition, since manual decisions for large-scale network planning have become intractable. To mitigate the energy crisis and carbon emissions, effectively addressing service network design issues can also contribute to environmental and energy sustainability. This work develops an integer programming model for the express service network design (SND) problem and proposes a layer-based relaxation method for large-scale applications.
In an express service network, parcels are sorted through a sequence of sorting centres before arriving at their destinations, as shown in
Figure 1. Optimal planning can be achieved by solving the consolidation-based SND problem [
3], which determines the services and the delivery paths. SND designates decision-making tools to schedule the activities and assets of a transportation system, aiming to satisfy customers with a high-quality standard. In SND, carriers’ most severe challenge is how to reduce costs while continuously improving service quality. The cost is mainly composed of the cost of transportation fuel. Therefore, while increasing the loading rate, choosing the optimal route and vehicle model minimizes the total cost while reducing carbon emissions and air pollution. Carriers must efficiently manage limited transportation assets that they own [
4,
5]. Researchers proposed models with solution methodologies that explicitly examine resource management issues in the context of SND [
6,
7]. In particular, design-balance constraints are introduced in SND models for resource management. Design-balance constraints guarantee the mass balance of transportation assets entering and leaving every time–space node [
8]. Researchers explicitly considered the resource-availability issue in an uncertain situation [
9]. SND is particularly relevant in a consolidation-based delivery system, an umbrella term for service providers that group and deliver parcels for customers within the exact vehicle, aiming for a balance between high service standards for customers and economy-of-scale-based costs [
10]. There is still a lack of efficient and practical algorithms for solving large-scale SND problems with heterogeneous vehicles for express delivery service. As the express delivery business grows, SND plays a vital role in planning time-critical express processes.
This paper addresses the consolidation-based SND problem with a heterogeneous fleet of vehicles in which vehicle repositioning could happen within multiple service cycles. The contributions are threefold: First, we propose a mixed-integer programming model for the variant of the SND problem, and the formulation is arc-based, with which it is easier to model the consolidation-based operations. Second, a layer-based relaxation algorithm is introduced to solve large-scale applications; the computational experiments demonstrate that the proposed approach efficiently solves the problem with high-quality solutions. Third, the numerical results confirm that the total costs could be reduced significantly for service providers by introducing a heterogeneous fleet of vehicles.
The rest of this work is as follows. In
Section 2, a literature review on SND problems is elaborated. The problem and the model formulation are presented in
Section 3.
Section 4 provides the framework of the solution methodology, and
Section 5 includes the numerical experiments. Conclusions are provided in
Section 6.
2. Literature Review
SND arises in the transportation sector when there is a requirement to determine cost-effective routes and schedules to meet customer demand with a specified service level. SND methodologies are extensively used in many fields, including telecommunication, logistics, transportation, and manufacturing systems [
11,
12,
13]. In transportation systems, SND was successfully applied to maritime transportation [
14,
15], airport transportation [
16], express shipment [
17,
18,
19], rail transportation [
20,
21,
22], air traffic flow management [
23,
24], and multimodal transportation systems [
25,
26,
27]. In addition, different network structures were addressed in the literature, such as multimodal networks [
28], long-haul and local transportation structures [
29], autonomous fleets [
30], and heterogeneous fleets [
31]. Furthermore, SND methodologies have attracted much scientific research attention in developing green logistics and green supply-chain systems: for example, the green closed-loop supply chain network design for ventilators during the COVID-19 epidemic [
32], the multitier supply network design methodology in reducing cost and carbon emissions [
33], the intermodal green p-hub median problem [
34], an approach to evaluating the coal transportation chain to reduce carbon emissions and pollution [
35], reducing the negative environmental impact in freight transportation by understanding service buyers [
36], effective green transit network design methodologies under urban expansion [
37,
38], an electrical transportation network design method to reduce congestion and carbon emissions [
39,
40], and a collaborative fleet routing methodology for sustainable seaport transportation [
41].
Recently, SND with resource management has attracted much scientific attention [
42], especially regarding the application of design-balanced constraints in SND to improve the utilization of transportation assets [
43,
44,
45]. The design-balanced constraints guarantee that the fleet of vehicles are used cyclically to keep a high utilization under a fixed fleet size. Four models for the SND with asset management were introduced, and the numerical results demonstrated that the arc-based model formulation is computationally better than the others [
46]. This paper offers an arc-based SND formulation with design-balance constraints, since the express service network is consolidation-based. The arc-based formulation allows for consolidation operations at each node. An early express SND model was developed in 2002 [
47]. The authors studied SND application for a next-day air network in a multimodal express package-delivery problem; however, it was not a consolidation-based network. There is still a lack of academic research on express service networks based on heterogeneous vehicles.
There are many heuristic approaches for solving SND problems [
48,
49]. Two common heuristic approaches frequently appearing in the literature are local search [
50,
51], and slope scaling [
52]. However, the solution quality of these methods cannot be guaranteed in large-scale applications. So far, none of these heuristics has been able to solve our problem directly, because our problem has incorporated a heterogeneous fleet and allowed for the vehicle to continuously serve multiple service cycles. For the exact algorithm, a branch-and-price algorithm is proposed to effectively solve the cycle-based formulation [
53]. Since the vehicle does not necessarily return within the service period, this exact solution method is not suitable for our problem. In this work, we propose an efficient algorithm based on the large-scale application of a layer-based relaxation algorithm. The computational experiments demonstrate that the solution’s quality and the algorithm’s efficiency can significantly improve.
3. Problem Formulation
In SND models, nodes represent terminals, and arcs correspond to services that can be executed by service providers. Assets are generally vehicles that are employed to transport a commodity with origin–destination (O–D). Considering the time–space network and a set of O–D commodities, service providers must offer transportation services to meet all demands at a lower total cost while ensuring service quality for a long cycle.
This work assumes that services must be repeated within the planning horizon. That is, services operate repetitively and cyclically. Let’s consider that the planning horizon
is a set of discrete time periods, e.g.,
. Services can be scheduled with these periods. The vehicles that execute services would travel through a circular path in the time-space network. The network is depicted in
Figure 2.
Let be the set of physical terminals in the transportation network where service providers can perform consolidation-based operations. Each node corresponds to a tuple in the time-space network, including a terminal and a time index. Demand is a total volume to be moved between its origin and destination terminals within a certain time limit. Let be a commodity with source and destination. denotes the set of all commodities. Each commodity k has a total volume that must be transported from its origin o to its destination d.
We modeled a carrier’s operations over a time–space network,
, where
is the set of time–space nodes, and
is the time–space arc set. Each time–space node
includes a terminal
with a time period
. Each arc represents a service between the two time–space nodes. Arc set
contains service, waiting, and rotation arcs. Let
,
, and
represent service, waiting, and rotation arcs, respectively. The waiting arcs are the horizontal arcs that connect nodes with consecutive time periods of an identical terminal from
to
, as depicted in
Figure 2. Blue arcs are service arcs, and oblique arcs in
Figure 2 are rotation arcs. One service arc denotes service operation at a certain point in time between two terminals. Each time-space arc
has a service time
, a fixed traveling cost
, and a volume flow cost
.
The model formulation is as follows.
subject to:
The objective function is to minimize the combination of vehicle operating and commodity transportation costs. Constraints (
2) are the mass balance constraints for vehicle routing. Constraints (
3), (
4) and (
5) ensure the mass balance for the commodity flow. The supply and demand at the origin and destination node for commodity
k are
, and inflow equals outflow on other time–space nodes. Constraints (
6) guarantee that the total volumes of all commodities on arc
cannot exceed the total vehicle capacity. Lastly, decision variable restrictions appear in Constraints (
7) and (
8).
4. Solution Approach
The SND problem is NP-hard [
43]; in this section, we designed a layer-based relaxation algorithm to efficiently solve large-scale instances. First, we separated the time–space nodes into layer-based sets according to the time index, as shown in
Figure 3.
Then, we chose the service arcs outgoing from nodes in one layer into an arc layer set. In
Figure 4, the arcs in red are in Layers 1 and 2, respectively. In the solution process of the algorithm, in each iteration, we selected an arc layer and released integer decision variables
for arcs
in other arc layers. We denoted the problem as the layer-based linear programming relaxation of SND (LLPRSND). LLRPSND can be formulated as follows.
Notations:
Arc set of layer l.
Then, we solved LLPRSND from Layer 1 to Layer , and with each iteration, we deleted arc in layer if was 0, and fixed the backbone arcs of that layer. Then, we solved the SND model in the backbone network.
Figure 5 depicts the working mechanism of the layer-based relaxation algorithm with a flow chart. Solving LLRPSND at each layer deletes service arcs with a fleet size that does not meet the constraints, and the network shrinks, fixing the backbone of each layer’s service arcs. Then, the SND was solved on the reconstructed network to obtain the optimal solution.
5. Numerical Study
In this section, we assess the computational performance of the layer-based relaxation algorithm. The algorithm was coded in Python, integrating Gurobi v9.1.1 as the optimization solver. The computational tests were performed using a workstation with 32 processors, 2.9 GHz, and 256 GB with the Windows 10 operating system.
This work generated two types of instances with simulated data on the basis of an express service company: medium- and large-scale instances. The medium instances were 10 terminals with 10–20 time periods, and the large-scale instances were 21–30 terminals with 15 periods. Three vehicle models were considered in this case. The dimensionality of all instances is demonstrated in
Table 1.
Columns 2–3 show the number of terminals and the time periods. Columns 4–7 present the cardinality of the service-arc, waiting-arc, rotation-arc, and commodity sets. We refer to Instances 1–11 as medium, and 12–21 as large. The large-scale instances are capable of reflecting the practical situations of the express service system.
For medium-scale instances, the time limit was 1000 s, and we set the MIP gap as 5%; if the feasible solution reached any bound, the solver stopped. For large-scale instances, the time limit was 3600 s, and we set 10% as the MIP gap.
5.1. Numerical Results for Medium-Scale Instances
Table 2,
Table 3 and
Table 4 reports the numerical results of medium-scale instances with three types of vehicle models. To show the difficulty of solving SND instances, we first employed Gurobi to solve the original model for each instance. In these tables, Columns 2 and 3 show the objective value and the running time for obtaining the best feasible integer solutions after termination. Column 4 gives each instance’s mixed integer programming (MIP) gap. Columns 5–7 are numerical results obtained with the layer-based relaxation algorithm. Another advantage of the proposed approach is that it took substantially less computing time to achieve a better solution. The computational results demonstrate that, regarding these 11 medium-scale instances, the objective value received by the proposed algorithm was considerably better than that solved directly by the commercial solver. The average computational time for obtaining the best integer solutions was about 552 s, which was only 27.92% of that solved by Gurobi. The numerical results show that the computational time was reduced, while the solution quality was improved significantly. All these results confirm the effectiveness and efficiency of the layer-based relaxation algorithm.
The objective value comparison for the medium-scale instances is depicted in
Figure 6, and M indicates the number of vehicle models. The objective value obtained by the proposed algorithm was better in the same setting. Another trend is that the objective value decreased with the number of vehicle models because a heterogeneous fleet increases the flexibility of cost reduction. The cost reduction by introducing more vehicle models reached about 26.67%.
Figure 7 demonstrates the running time comparison between the original and proposed algorithms. The time needed by the layer-based relaxation algorithm was remarkably shorter.
5.2. Computational Experience for Large-Scale Instances
The numerical results for large-scale instances with heterogeneous vehicle models are shown in
Table 5. For these instances, the number of O–D commodities and service arcs exponentially increased, which was computationally expensive.
The computational results show the complexity and challenge of handling large-scale SND instances. Within the computing time limit, the original algorithm could find feasible solutions for all medium instances within the MIP gap requirement; however, only 18 out of 30 cases were solved with feasible solutions for large-scale instances. The layer-based relaxation algorithm could efficiently solve all instances and find better solutions, as depicted in
Figure 8. The average computing times for our algorithm were 233, 1632, and 1540 s. The computational times of primal formulation were several times longer, as shown in
Figure 9. The computational results demonstrate that our approach is more efficient.
6. Conclusions
We studied one variant of the SND problem for express services in this work and proposed a mixed-integer optimization model. The model jointly considers vehicle dispatching, heterogeneous fleet sizing, and commodity flow. A distinguishing feature of the model is that there is no restriction of each vehicle having to return in one cycle, a realistic rule that prevents existing algorithms from directly solving the problem. Due to the problem’s NP-hard nature and the lack of efficient algorithms, we propose a layer-based relaxation algorithm for solving large-scale instances. The computational results indicate that our algorithm outperforms solving directly by a state-of-the-art commercial solver on computational time and solution quality. We also found that the heterogeneous fleets could reduce the total cost.
As the demand for transportation continues to grow, the design algorithms for service networks are becoming increasingly important. Solving service network design problems with high quality can not only improve the competitiveness of service providers, but also help in reducing energy consumption and carbon emissions in terms of sustainable development. Although the algorithm is very efficient, it is heuristic. Theoretical proofs do not guarantee the quality of the solution. Therefore, future research focuses on developing efficient and precise algorithms to solve the proposed formula, i.e., a branch-cut-and-price framework for large fleet sizes. The branch-and-bound process is NP-hard and determines the overall efficiency of the entire branch-and-cut-and-price algorithm. To enhance the algorithm, we employed reinforcement learning techniques to improve the efficiency of the branch-and-bound process. The next research direction is to search for valid inequalities via mathematical structure-property to obtain a tight model formulation. The valid inequalities narrow the feasible region and facilitate finding the optimal solution. In future research on sustainability, we will also try to explore the modelling and solution methods of electric vehicles in express service network design.
Author Contributions
Conceptualization, X.D. and X.J.; methodology, X.D.; software, A.G.; validation, A.G. and X.J.; writing—original draft preparation, X.D. and A.G.; writing—review and editing, H.C.; funding acquisition, X.D. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by the Shenzhen Polytechnic Research Fund (6022312034K). Innovation Team by Department of Education of Guangdong Province, P.R.China (2020KCXTD041).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Informed consent was obtained from all subjects involved in the study.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
Notations:
| |
| Vehicle model set. |
| Time period set. |
| Terminal set. |
| Time–space node set. |
| O–D commodity set. |
| Commodity k with origin o and destination d, |
| Arc set of the time-space network, |
| Service arc set. |
| Waiting arc set. |
| Rotation arc set. |
| Arc set for commodity transportation. |
| Arc set for vehicle model . |
| Arc set for delivering commodity k. |
| Volume of commodity k. |
| Operational cost for vehicle model to select arc . |
| Unit flow cost of arc for commodity k. |
| Capacity of vehicle model . |
Decision variables:
| |
| The quantity of vehicle model v on arc , |
| The volume of O–D commodity k on arc . |
References
- Hewitt, M. The Flexible Scheduled Service Network Design Problem. Transp. Sci. 2022, 56, 799–1110. [Google Scholar] [CrossRef]
- Wieberneit, N. Service network design for freight transportation: A review. OR Spectr. 2008, 30, 77–112. [Google Scholar] [CrossRef]
- Lanza, G.; Crainic, T.G.; Rei, W.; Ricciardi, N. Scheduled service network design with quality targets and stochastic travel times. Eur. J. Oper. Res. 2021, 288, 30–46. [Google Scholar] [CrossRef]
- Li, X.; Ding, Y.; Pan, K.; Jiang, D.; Aneja, Y.P. Single-path service network design problem with resource constraints. Transp. Res. Part E Logist. Transp. Rev. 2020, 140, 101945. [Google Scholar] [CrossRef]
- Chouman, M.; Crainic, T.G. Freight Railroad Service Network Design. In Network Design with Applications to Transportation and Logistics; Springer: Cham, Switzerland, 2021; pp. 383–426. [Google Scholar]
- Fontaine, P.; Crainic, T.G.; Jabali, O.; Rei, W. Scheduled service network design with resource management for two-tier multimodal city logistics. Eur. J. Oper. Res. 2021, 294, 558–570. [Google Scholar] [CrossRef]
- Crainic, T.G.; Hewitt, M.; Toulouse, M.; Vu, D.M. Service network design with resource constraints. Transp. Sci. 2016, 50, 1380–1393. [Google Scholar] [CrossRef] [Green Version]
- Pedersen, M.B.; Crainic, T.G.; Madsen, O.B. Models and tabu search metaheuristics for service network design with asset-balance requirements. Transp. Sci. 2009, 43, 158–177. [Google Scholar] [CrossRef]
- Hewitt, M.; Crainic, T.G.; Nowak, M.; Rei, W. Scheduled service network design with resource acquisition and management under uncertainty. Transp. Res. Part B Methodol. 2019, 128, 324–343. [Google Scholar] [CrossRef]
- Crainic, T.G.; Hewitt, M. Service network design. In Network Design with Applications to Transportation and Logistics; Springer: Cham, Switzerland, 2021; pp. 347–382. [Google Scholar]
- Wu, X.; Lü, Z.; Glover, Z. A matheuristic for a telecommunication network design problem with traffic grooming. Omega 2020, 90, 102003. [Google Scholar] [CrossRef]
- Belieres, S.; Hewitt, M.; Jozefowiez, N.; Semet, F.; Van Woensel, T. A Benders decomposition-based approach for logistics service network design. Eur. J. Oper. Res. 2020, 286, 523–537. [Google Scholar] [CrossRef]
- Niari, M.R.; Eshgi, K.; Valilai, O.F. Topology analysis of manufacturing service supply–demand hyper-network considering QoS properties in the cloud manufacturing system. Robot. Comput. Integr. Manuf. 2021, 72, 102205. [Google Scholar] [CrossRef]
- Aslaksen, I.E.; Svanberg, E.; Fagerholt, K.; Johnsen, L.C.; Meisel, F. Ferry Service Network Design for Kiel fjord. In Proceedings of the 11th International Conference on Computational Logistics, Enschede, The Netherlands, 28–30 September 2020; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Chu, X.; Shao, S.; Xu, S.X.; Kang, K. Data-driven ferry network design with candidate service arcs: The case of Zhuhai Islands in China. Marit. Policy Manag. 2020, 47, 598–614. [Google Scholar] [CrossRef]
- Zhao, R.; Liu, W.; Zhang, F.; Koo, T.T.; Lodewijks, G. Passenger shuttle service network design in an airport. Transp. B Transp. Dyn. 2022, 10, 1099–1125. [Google Scholar] [CrossRef]
- Martin, F.; Hemmelmayr, V.C.; Wakolbinger, T. Integrated express shipment service network design with customer choice and endogenous delivery time restrictions. Eur. J. Oper. Res. 2021, 294, 590–603. [Google Scholar] [CrossRef]
- Liu, S.; Lin, B.; Wu, J.; Zhao, Y. Modeling the service network design problem in railway express shipment delivery. Symmetry 2018, 10, 391. [Google Scholar] [CrossRef] [Green Version]
- Pérez, J.M.Q.; Lange, J.C.; Tancrez, J.S. A multi-hub express shipment service network design model with flexible hub assignment. Transp. Res. Part E Logist. Transp. Rev. 2018, 120, 116–131. [Google Scholar] [CrossRef]
- Laaziz, E.H.; Sbihi, N. A service network design model for an intermodal rail-road freight forwarder. Int. J. Logist. Syst. Manag. 2019, 32, 465–482. [Google Scholar] [CrossRef]
- Lulli, G.; Pietropaoli, U.; Ricciardi, N. Service network design for freight railway transportation: The Italian case. J. Oper. Res. Soc. 2011, 62, 2107–2119. [Google Scholar] [CrossRef]
- Zhu, E.; Crainic, T.G.; Gendreau, M. Scheduled service network design for freight rail transportation. Oper. Res. 2014, 62, 383–400. [Google Scholar] [CrossRef] [Green Version]
- Diao, X.; Chen, C.-H. A sequence model for air traffic flow management rerouting problem. Transp. Res. Part E Logist. Transp. Rev. 2018, 110, 15–30. [Google Scholar] [CrossRef]
- Diao, X.; Lu, S. Optimization approach to data-driven air traffic flow management. Transp. Res. Rec. 2022, 3, 398–404. [Google Scholar] [CrossRef]
- Inghels, D.; Dullaert, W.; Vigo, D. A service network design model for multimodal municipal solid waste transport. Eur. J. Oper. Res. 2016, 254, 68–79. [Google Scholar] [CrossRef]
- Elbert, R.; Rentschler, J.; Schwarz, J. Combined Hub Location and Service Network Design Problem: A Case Study for an Intermodal Rail Operator and Structural Analysis. Transp. Res. Rec. 2022, 03611981221101391. [Google Scholar] [CrossRef]
- Tawfik, C.; Limbourg, S. Scenario-based analysis for intermodal transport in the context of service network design models. Transp. Res. Interdiscip. Perspect. 2019, 2, 100036. [Google Scholar] [CrossRef]
- van Riessen, B.; Negenborn, R.; Dekker, R.; Lodewijks, G. Service Network Design for an Intermodal Container Network with Flexible due Dates/Times and the Possibility of Using Subcontracted Transport. No. EI2013-17; Erasmus School of Economics: Rotterdam, The Netherlands, 2013. [Google Scholar]
- Medina, J.; Hewitt, M.; Lehuede, F.; Peton, O. Integrating long-haul and local transportation planning: The service network design and routing problem. EURO J. Transp. Logist. 2019, 8, 119–145. [Google Scholar] [CrossRef]
- Scherr, Y.O.; Saavedra, B.A.N.; Hewitt, M.; Mattfeld, D.C. Service network design with mixed autonomous fleets. Transp. Res. Part E Logist. Transp. Rev. 2019, 124, 40–55. [Google Scholar] [CrossRef]
- Wang, Z.; Qi, M.; Cheng, C.; Zhang, C. A hybrid algorithm for large-scale service network design considering a heterogeneous fleet. Eur. J. Oper. Res. 2019, 276, 483–494. [Google Scholar] [CrossRef]
- Poursoltan, L.; Seyed-Hosseini, S.-M.; Jabbarzadeh, A. Green Closed-Loop Supply Chain Network under the COVID-19 Pandemic. Sustainability 2021, 13, 9407. [Google Scholar] [CrossRef]
- Sirilertsuwan, P.; Thomassey, S.; Zeng, X. A Strategic Location Decision-Making Approach for Multi-Tier Supply Chain Sustainability. Sustainability 2020, 12, 8340. [Google Scholar] [CrossRef]
- Ibnoulouafi, E.M.; Oudani, M.; Aouam, T.; Ghogho, M. Intermodal Green p-Hub Median Problem with Incomplete Hub-Network. Sustainability 2022, 14, 11714. [Google Scholar] [CrossRef]
- Dong, M.; Li, Y.; Xu, X.; Zha, Y. A Practical Accessibility Evaluation Method for Port-Centric Coal Transportation Chains: Considering the Environment and Operational Adaptability. Sustainability 2022, 14, 11619. [Google Scholar] [CrossRef]
- Dahlgren, S.; Ammenberg, J. Environmental Considerations Regarding Freight Transport among Buyers of Transport Services in Sweden. Sustainability 2022, 14, 11244. [Google Scholar] [CrossRef]
- Liu, Y.; Zhang, H.; Xu, T.; Chen, Y. A Heuristic Algorithm Based on Travel Demand for Transit Network Design. Sustainability 2022, 14, 11097. [Google Scholar] [CrossRef]
- Li, J.; He, Z.; Zhong, J. The Multi-Type Demands Oriented Framework for Flex-Route Transit Design. Sustainability 2022, 14, 9727. [Google Scholar] [CrossRef]
- Barraza, O.; Estrada, M. Battery Electric Bus Network: Efficient Design and Cost Comparison of Different Powertrains. Sustainability 2021, 13, 4745. [Google Scholar] [CrossRef]
- Neufville, R.; Abdalla, H.; Abbas, A. Potential of Connected Fully Autonomous Vehicles in Reducing Congestion and Associated Carbon Emissions. Sustainability 2022, 14, 6910. [Google Scholar] [CrossRef]
- Ypsilantis, P.; Zuidwijk, R. Collaborative Fleet Deployment and Routing for Sustainable Transport. Sustainability 2019, 11, 5666. [Google Scholar] [CrossRef] [Green Version]
- Vu, D.M.; Crainic, T.G.; Toulouse, M. A three-phase matheuristic for capacitated multi-commodity fixed-cost network design with design-balance constraints. J. Heuristics 2013, 19, 757–795. [Google Scholar] [CrossRef]
- Li, X.; Wei, K.; Aneja, Y.P.; Tian, P.; Cui, Y. Matheuristics for the single-path design-balanced service network design problem. Comput. Oper. Res. 2017, 77, 141–153. [Google Scholar] [CrossRef]
- Chouman, M.; Crainic, T.G. Cutting-plane matheuristic for service network design with design-balanced requirements. Transp. Sci. 2015, 49, 99–113. [Google Scholar] [CrossRef]
- Katayama, N. MIP neighborhood search heuristics for a service network design problem with design-balanced requirements. J. Heuristics 2020, 26, 475–502. [Google Scholar] [CrossRef]
- Andersen, J.; Crainic, T.G.; Christiansen, M. Service network design with asset management: Formulations and comparative analyses. Transp. Res. Part C Emerg. Technol. 2009, 17, 197–207. [Google Scholar] [CrossRef]
- Barnhart, C.; Krishnan, N.; Kim, D.; Ware, K. Network design for express shipment delivery. Comput. Optim. Appl. 2002, 21, 239–262. [Google Scholar] [CrossRef]
- Gao, A.; Jin, X.; Diao, X. An Iterative Backbone Algorithm for Service Network Design Problems. Processes 2022, 10, 1373. [Google Scholar] [CrossRef]
- Tawfik, C.; Gendron, B.; Limbourg, S. An iterative two-stage heuristic algorithm for a bilevel service network design and pricing model. Eur. J. Oper. Res. 2022, 300, 512–526. [Google Scholar] [CrossRef]
- Ghamlouche, I.; Crainic, T.G.; Gendreau, M. Cycle-based neighbourhoods for fixed-charge capacitated multicommodity network design. Oper. Res. 2003, 51, 655–667. [Google Scholar] [CrossRef] [Green Version]
- Ghamlouche, I.; Crainic, T.G.; Gendreau, M. Path relinking, cycle-based neighbourhoods and capacitated multicommodity network design. Ann. Oper. Res. 2004, 131, 109–133. [Google Scholar] [CrossRef]
- Jarrah, A.I.; Johnson, E.; Neubert, L.C. Large-scale, less-than-truckload service network design. Oper. Res. 2009, 57, 609–625. [Google Scholar] [CrossRef]
- Andersen, J.; Christiansen, M.; Crainic, T.G.; Gronhaug, R. Branch and price for service network design with asset management constraints. Transp. Sci. 2011, 45, 33–49. [Google Scholar] [CrossRef]
| Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).