Distributionally robust cluster-based hierarchical hub location problem for the integration of urban and rural public transport system

Highlights

• A cluster-based hierarchical hub location problem for integration of urban and rural public transport system is introduced.

• A distributionally robust optimization is proposed to model this problem formally.

• Two heuristics based on a variable neighborhood search algorithm and a population-and-searching algorithm are developed.

• Two sets of numerical experiments based on a case study and a randomly generated example are conducted.

Abstract

This paper introduces a distributionally robust cluster-based hierarchical hub location problem for the integration of urban and rural public transport system at the strategic level. Lacking complete information on the true probability distributions of construction cost and travel time, a distributionally robust optimization (DRO) model is proposed to formulate this problem formally. The proposed DRO model is demonstrated to be semi-infinite and computationally hard but admits a safe approximation and an equivalent under two kinds of ambiguity sets: bounded perturbations with zero mean and Gaussian perturbations, respectively. Since the resulting formulations can be solved to optimality by the CPLEX software for only small instances, a variable neighborhood search algorithm and a population-and-searching based heuristic algorithm are designed to handle the realistic-sized instances. To validate the superiority of the DRO model and the efficiency of the proposed methods, two sets of numerical experiments are implemented on a case study of Guangrao in Shandong Province of PR China and a randomly generated example, respectively. Suggestions for possible extensions of the original problem are also discussed.

Introduction

In recent years, many countries have been dedicated to developing and improving the integration of urban and rural public transport system (IURPTS), which is to make more effective use of land and infrastructures for regional economic development. For example, China aims to promote the integration of urban and rural passenger transport services, improve the equalization level of public services, and ensure that urban and rural passengers can enjoy a better trip, according to a document jointly released by the Communist Party of China Central Committee and the State Council. In the IURPTS, directly connecting all villages and towns with a city is cost inefficiency, because the passengers of villages and towns are scattered and travel less frequently than their urban counterparts. Therefore, the first motivation of this study is to model the IURPTS through the use of hub-and-spoke (H&S) typed networks, wherein the passengers between origin and destination (O-D) pair nodes are transported via at least one hub node to save in the cost and time due to economies of scale.

On the one hand, the classic H&S networks comprise only one type of hub and two layers: one between hubs and demand nodes (spokes) and the other among hubs. However, the transportation network for the IURPTS requires two types of hubs: urban transport hubs and central-town transport hubs; and three layers: the top layer connects urban transport hubs, the second layer connects urban transport hubs and central-town transport hubs, and the third layer connects hubs (urban or central-town transport hubs) and spokes. On the other hand, the traditional H&S networks do not consider the realistic geographical and administrative features, resulting in hubs usually located within most of the well-developed areas because of their concentrated demands. However, the transportation network for the IURPTS is grouped into clusters in advance based on the geographic and administrative features of urban and rural, and public transport hubs are then distributed in each cluster, resulting in providing better service for the passengers and eventually improving the overall IURPTS performance. For the second motivation, we present a cluster-based hierarchical hub location modeling approach to design the transportation network for the IURPTS.

Since the design transportation network for the IURPTS is made a priori in the strategic decision process, it is essential to take stochastic uncertainty into account when designing it. If decision-makers lack the information on the distribution of underlying random variables, it might be a reasonable option to make an optimal decision under the worst-case in a certain range by adopting a robust optimization (RO) method, generally called an over-conservative solution. An alternative but possibly less conservative RO method is the distributionally robust optimization (DRO), which can be regarded as a generalized version of the classic RO. In practical terms, decision-makers can construct ambiguity sets of parameter distributions with historical data by containing the partial true distribution information rather than fixed probability distributions. In addition to the flexibility of distribution, the charm of DRO is its ability to yield computationally tractable models (Chen, Sim, & Xv, 2019). By combining these aforementioned aspects, we develop a DRO model to design transportation networks for the IURPTS under the uncertainty of travel time and construction cost in this research. As a result, it can avoid over-conservative solutions by incorporating partial probability distribution information ignored by the conventional RO method. Therefore, the third motivation pertains to revealing that the proposed DRO model outperforms the RO model, demonstrating that the DRO model provides a practical decision-making tool to design transportation networks for the IURPTS.

The transportation network design problem for the IURPTS can be regarded as a branch of general hub location problems (HLPs), which is also referred to as the H&S network design problem. Following seminal works of O’Kelly, 1986, O’Kelly, 1987, the HLP has been well-studied in the literature (Tan and Kara, 2007, Alumur and Kara, 2009, Campbell, 2009, Ernst et al., 2009, Contreras et al., 2012, Perio et al., 2017, Alumur, 2019). Considering the cost and time objective functions, the HLP can be divided into the hub median problem and the hub center problem, respectively. The interested reader may refer to the survey of the HLPs in Alumur and Kara, 2008, Campbell and O’Kelly, 2012, Farahani, Hekmatfar, Arabani, and Nikbakhsh (2013) and Contreras (2015). Three paradigms in HLPs will be considered in this study: hierarchical HLPs, robust HLPs and heuristic solutions for HLPs.

Hierarchical HLPs. Prior literature concerning the HLPs assumes only one kind of hub and associated two layers. Relaxing this assumption, Chou (1990) introduced a hierarchical HLP model, and applied it to airline networks. Elmastas (2006) presented a hierarchical HLP model for cargo delivery systems in Turkey with the star-mesh-star network structure and the objective of minimizing the total establishment cost. Yaman (2009) was the first to study the hierarchical HLP in the cargo delivery systems concerning the complete-star-star network structure, aiming to minimize the total routing cost. Khameseh and Mohamadi (2014) considered the hierarchical HLP in an incomplete network environment. Dukkanci and Kara (2017) considered a ring-star-star network for the hierarchical HLP, in which the ground and airport transportation modes were integrated into these multimodal networks. Zhong, Juan, Zong, and Su (2018) built a star-complete-complete network structure for the hierarchical HLP in the urban and rural public transport system. More literature on the hierarchical HLP can be referred to Sahin and Sural (2007) and Torkestani, Seyedhosseini, Makui, and Shahanaghi (2016). In addition to constructing different hierarchical network structures, some researches considered other practical constraints. For example, Alumur, Yaman, and Kara (2012b) studied the hierarchical HLP with time-definite deliveries aiming to minimize the total transportation cost and operational cost. Yaman and Elloumi (2012) designed a three-layer star network for the hierarchical HLP to optimize the poorest service quality. Karimi, Eydi, and Korani (2014) formulated a mathematical model by integrating the capacity limitation of hubs into the hierarchical HLP.

Robust HLPs. The significance of uncertainty has prompted some researchers to address the robust HLPs. For example, Alumur, Nickel, and Saldanha-da-Gama (2012a) developed a generic model to capture the data uncertainty, in which robust optimization was adopted to consider the uncertainty in set-up costs. Shahabi and Unnikrishnan (2014) presented robust formulations for the single and multiple assignment HLP with uncertain demand, in which a mixed-integer nonlinear program was formulated and then transformed into a mixed-integer conic quadratic program. Ghaderi and Rahmaniani (2016) designed a robust H&S network for the extension to the classic single assignment HLP under uncertainty of demands and travel times. Habibzadeh-Boukani, Farhang-Moghaddam, and Saman-Pishvaee (2016) used the RO approach to deal with the uncertainty of fixed setup cost and capacity of hubs for the capacitated single assignment and multiple assignment HLP. Zetina, Contreras, Cordeau, and Nikbakhsh (2017) applied the RO technique to incorporate the uncertainties in three cases: demand, transportation cost, and both simultaneously. It is widely known that the RO aims to optimize the worst-case when lacking any information about uncertain variables, which may lead to over-conservative results. Given that, the DRO can overcome the inherent shortcoming of the RO method (Popescu, 2007, Goh and Sim, 2010, Duzgun, 2012, Gabrel et al., 2014). Recently, two works employed the DRO method to study the HLP with uncertainty. Yin, Chen, Song, and Liu (2019) addressed the uncertain carbon emissions for the HLP with the ambiguous chance constraint by the DRO technique. Wang, Chen, and Liu (2020) used the tool of DRO to study the uncapacitated and capacitated HLP with the uncertainty of demand and cost.

Heuristic solutions for HLPs. The HLPs are known to be NP-hard, so an indispensable topic is how to solve them effectively. For small-sized instances, the HLPs can be solved to optimality by the general optimization software, e.g., Gurobi (Alumur, Kara, & Karasan, 2012c), GAMS and CPLEX (Yaman, 2009). Since the software could not solve the larger-sized instances within limited computational time, some popular heuristic or meta-heuristic algorithms have been developed, such as genetic algorithm (Damgacioglu, Dinler, Ozdemirel, & Iyigun, 2015), variable neighborhood search (Brimberg, Mladenovic, Todosijevic, & Urosevic, 2017), particle swarm optimization (Ozgun-Kibiroglu, Serarslan, & Topcu, 2019), tabu search (Abyazi-Sani & Ghanbari, 2016), memetic algorithm (Shang, Yang, & Wang, 2020), and so on. Due to additional complexity of the hierarchical HLPs, some researchers developed several effective heuristic algorithms to obtain near-optimal solutions, such as, subgradient-based heuristic algorithm (Dukkanci & Kara, 2017), tabu search heuristic (Chen, 2010), simulated annealing and iterated local search algorithms (Fazel-Zarandi, Davari, & Haddad-Sisakht, 2015). To get a comprehensive review of the solutions for HLPs, the interested readers can refer to the works provided by Kartal, Krishnamoorthy, and Ernst (2019).

Previous studies on the HLPs neglect the realistic features of geographical and administrative, which are very important to make the constructed H&S network in line with the area development. Hence, the cluster-based policy is introduced for a better designing of the H&S network in the transportation and communication network. For example, Sung and Jin (2001) considered a hybrid H&S network design problem (allowing the direct links between spokes), in which the network service was divided into the predetermined clusters. Yu, Liu, Chang, and Yang (2008) studied the urban transit HLP using a cluster-based optimization approach. Subsequently, the problem was extended to considering the hierarchical hub network (Yu, Liu, Chang, & Ma, 2009) and the multimodal transportation network (Yuan & Yu, 2018). However, the review of the literature indicates that three features including the hierarchical H&S typed network, the cluster-based policy and the distributionally robust optimization method are separately considered or partially integrated for HLPs in the area of the IURPTS.

To understand the focuses of this research clearly, the characteristics of the proposed model and solution method are mainly compared with those of closely related studies listed in Table 1. Based on this table and the above references, it is the first study to integrate the hierarchical H&S typed network, the cluster-based policy and the DRO method for the IURPTS simultaneously. It is worth noting that two recent studies on the distributionally robust HLPs were carried out independently by Yin et al. (2019) and Wang et al. (2020). Twofold differences made by this paper relative to Yin et al. (2019) and Wang et al. (2020) are emphasized here. Firstly, given the studied problems, this paper builds a hierarchical H&S network with three layers for the IURPTS which is distinctly different from the problem proposed by Yin et al. (2019) and Wang et al. (2020) that consider the classic two-layer H&S network. Secondly, in terms of the solution methods, this paper develops two heuristics to obtain high-quality solutions for the realistic-sized instances, while Yin et al. (2019) and Wang et al. (2020) used CPLEX and Gurobi to solve the relatively small instances, respectively.

To fill these above-mentioned gaps, this paper contributes to the literature in the following ways:

• Using the hierarchical H&S typed network to model the IURPTS by considering two types of hubs and three layers, and adopting a cluster-based policy to capture real-world geographical and administrative features of the IURPTS.

• Developing a DRO model to formulate the network design problem for the IURPTS that minimizes the total construction costs in a worst-case VaR criterion under the ambiguous chance constraints concerning the travel time requirements.

• Deriving the tractable forms for the ambiguous chance constraints in two special ambiguity sets: bounded perturbations with zero mean and Gaussian perturbations, and further designing a variable neighborhood search algorithm and a population-and-searching based heuristic algorithm to solve the problem efficiently.

• Conducting the numerical experiments on a real case study of Guangrao in Shandong Province of PR China and larger-sized instances, and computational results demonstrate the superiority and application of the DRO model and the efficiency of the proposed algorithms.

• Extending the original model by incorporating the multiple assignment policy into the third layer of the hierarchical H&S network and considering a bi-objective optimization simultaneously.

The remainder of the paper is organized as follows. Section 2 describes a modeling framework and presents the deterministic and DRO models for the IURPTS. Section 3 designs a variable neighborhood search algorithm and a population-and-searching based heuristic algorithm to solve the tractable formulations of the DRO model. Section 4 conducts a series of numerical experiments to show the superiority of the proposed models and the effectiveness of the proposed methods. Section 5 provides the model extensions and discussions. Finally, Section 6 presents the conclusions and future directions.