Production, Manufacturing, Transportation and Logistics
Alternate solution approaches for competitive hub location problems

https://doi.org/10.1016/j.ejor.2020.07.018Get rights and content

Highlights

  • We study the hub location problem faced by an airline entering a competitive market.

  • The problem is modeled as a non-linear integer program.

  • We propose four different approaches to solve it.

  • Our best performing method uses Kelley’s cutting plane within Lagrangian relaxation.

  • It is able to solve all the problem instances within 1% optimality gap in less than 10 minutes of CPU time.

Abstract

In this paper, we study the hub location problem of an entrant airline that tries to maximize its share in a market with already existing competing players. The problem is modeled as a non-linear integer program, which is intractable for off-the-shelf commercial solvers, like CPLEX and Gurobi, etc. Hence, we propose four alternate approaches to solve the problem. The first among them uses the Kelley’s cutting plane method, the second is based on a mixed integer second order conic program reformulation, the third uses the Kelley’s cutting plane method within Lagrangian relaxation, while the fourth uses second order conic program within Lagrangian relaxation. On the basis of extensive numerical tests on well-known datasets (CAB and AP), we conclude that the Kelley’s cutting plane within Lagrangian relaxation is computationally the best. It is able to solve all the problem instances of upto 50 nodes within 1% optimality gap in less than 10 minutes of CPU time.

Introduction

Hub and spoke network was pioneered in the airline industry by Delta Airlines in 1955 to compete with the low cost Eastern Airlines (Delta History, 1955). In a hub and spoke network, every origin–destination (O–D) pair is connected through one or more intermediate nodes, called hubs. Hubs are consolidation points, where traffic from various non-hub nodes, called spokes, is aggregated, thereby generating economies of scale. This leads to a lower operational cost, compared to alternative network configurations with direct O–D connections (Chen, 2007, Hamacher, Labbé, Nickel, Sonneborn, 2004, O’Kelly, 1986). The use of a hub and spoke network also results in a lower setup cost since it requires fewer links to connect various origins and destinations in the network. Several studies have documented the benefits, leading to competitive advantage, from the use of hub and spoke network (Bania, Bauer, Zlatoper, 1998, McShan, Windle, 1989, Oum, Zhang, Zhang, 1995, Martins de Sá, Contreras, Cordeau, Saraiva de Camargo, de Miranda, 2015). Since the deregulation of the US airline industry in 1978, hub and spoke networks have become almost a default choice for airline networks. Besides the airline industry, hub and spoke networks are also used in other industries, for instance, telecommunications (Klincewicz, 1998), energy (Lumsden, Dallari, & Ruggeri, 1999) and road transportation (Üster & Agrahari, 2011).

Designing a hub and spoke network calls for solving a hub location problem (HLP), which determines the optimal location of hubs and the path through some of those hubs between every O–D pair (Campbell & O’Kelly, 2012). In a multi-player setting, HLPs can be broadly classified either as cooperative games or non-cooperative games. Cooperative games focus on coalitions and joint actions among firms, thereby determining their collective payoffs, as opposed to non-cooperative games, where individual players focus on their own payoffs under competition. Lin and Lee (2010) study a cooperative game in freight services in an oligopolistic market. In a non-cooperative setting, an HLP is solved either by an entrant airline that intends to set up its network by locating the hubs, or by an incumbent airline that intends to revamp its existing network. In this paper, we focus on this network design problem faced by an entrant with the intention of setting up its hubs strategically.

Most of the existing literature on HLP has studied the above problem in a monopolistic setting, without accounting for the presence of competing firms in the market. The literature on the entrant’s problem in a competitive setting, as well is limited and can be broadly classified in the following two categories: (i) the best response of the entrant explicitly accounting for the competitors’ reaction (Mahmutogullari, Kara, 2016, Sasaki, 2005, Sasaki, Campbell, Krishnamoorthy, Ernst, 2014, Sasaki, Fukushima, 2001) (ii) the best response of the entrant without accounting for the reaction from the competitors (Eiselt, Marianov, 2009, Lüer-Villagra, Marianov, 2013, Marianov, Serra, ReVelle, 1999, Wagner, 2008). In this paper, we specifically focus on the problem of maximizing the market share of an entrant airline in the second category of competitive HLP. The problem results in a non-linear integer program (NLIP), which is challenging to solve using off-the-shelf solvers like CPLEX, Gurobi, etc. To the best of our knowledge, ours is the first study to solve the above problem exactly.

Through this paper, we make the following contributions to the related literature. We propose four alternate approaches to solve the problem. The first approach (CPA) is based on the Kelley’s cutting plane algorithm to address the non-linearity in the problem. The second approach (MISOCP) relies on a mixed integer second order conic program based reformulation of the problem. The third approach (LR-CPA) employs the Kelley’s cutting plane algorithm within Lagrangian relaxation, while the fourth approach (LR-SOCP) uses second order conic program within Lagrangian relaxation. Further, we compare the above four solution approaches based on extensive computational experiments using two well-known datasets (CAB and AP). Our analysis highlights the superiority of LR-CPA, using which we are able to solve all the AP test instances corresponding to 50 cities within an optimality gap of 1% in less than 10 minutes of CPU time.

The rest of the paper is organized as follows. In Section 2, we present a review of the literature on HLP and its variants in a competitive setting. The problem description, followed by its mathematical formulation, is presented in Section 3. In Section 4, we present our alternate solution approaches, followed by extensive computational results in Section 5. Finally, the conclusions and directions for future research are presented in Section 6.

Section snippets

Literature review

HLPs can be classified as p-median, p-center, covering or fixed-charge on the basis of their objective functions. The objective in the p-median HLP is to minimize the total transportation cost (Campbell, 1996, Ernst, Krishnamoorthy, 1996, O’Kelly, 1986, Skorin-Kapov, Skorin-Kapov, O’Kelly, 1996), whereas for a p-center HLP is to minimize the maximum transportation cost between any pair of nodes (Campbell, 1994, Kara, Tansel, 2000). A covering HLP can either be hub set-covering if the objective

Problem description and model formulation

Consider a network of cities, represented by a complete graph G=(N,A), in which a set C={1,2,,|C|} of airlines are competing for their respective market shares. There exists a given demand fij for each pair (i ∈ N, j ∈ N, i ≠ j) of cities. Each airline c ∈ C operates in a hub and spoke network with a set Hc ⊂ N of hubs, such that the traffic between any city pair (i, j) is routed via a maximum of two intermediate hubs (k, l ∈ Hc), so as to exploit the benefits arising from the economies of

Solution methods

In this section, we propose four alternate approaches to solve COHLP. The first approach is based on the linearization of the non-linear term in (3) using Kelley’s cutting plane approach (CPA) (Kelley, 1960). In the second approach, we reformulate COHLP into a mixed integer second order conic program (MISOCP), which can be solved efficiently using an off-the-shelf solver (Alizadeh & Goldfarb, 2003). The third and fourth approaches are based on Lagrangian relaxation of COHLP, which separates the

Numerical experiments

We first describe in Section 5.1 the relevant data used in our experiments, followed by a discussion of our results in Section 5.2.

Conclusions and directions for future research

In this paper, we studied a competitive hub location problem (COHLP), wherein an entrant is making a strategic decision of locating its hubs in a market with already existing competing players. The entrant tries to maximize its market share, which is a function of the utility that customers get from using its services. The resulting problem is an NLIP, which is computationally intractable for off-the-shelf solvers. Hence, papers that have studied similar problems (Eiselt, Marianov, 2009,

Acknowledgment

This research was supported by the Research fellowship provided to the first author by Indian Institute of Management, Ahmedabad.

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