Discrete Dynamics in Nature and Society

Volume 2019, Article ID 8651728, 13 pages

https://doi.org/10.1155/2019/8651728

## A Combined Average-Case and Worst-Case Analysis for an Integrated Hub Location and Revenue Management Problem

^{1}School of Economics and Management, Tongji University, Shanghai 200092, China^{2}Laboratory IBISC, Univ Evry, Université Paris-Saclay, Evry 91025, France^{3}School of Traffic and Transportation Engineering, Hefei University of Technology, Hefei 230009, China^{4}Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China

Correspondence should be addressed to Yan-Ting Hou; nc.ude.ijgnot@tyuoh4102

Received 28 September 2018; Revised 16 December 2018; Accepted 13 January 2019; Published 12 March 2019

Academic Editor: Leonid Shaikhet

Copyright © 2019 Jia-Zhen Huo 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.

#### Abstract

This paper investigates joint decisions on airline network design and capacity allocation by integrating an uncapacitated single allocation p-hub median location problem into a revenue management problem. For the situation in which uncertain demand can be captured by a finite set of scenarios, we extend this integrated problem with average profit maximization to a combined average-case and worst-case analysis of this integration. We formulate this problem as a two-stage stochastic programming framework to maximize the profit, including the cost of installing the hubs and a weighted sum of average and worst case transportation cost and the revenue from tickets over all scenarios. This model can give flexible decisions by putting the emphasis on the importance of average and worst case profits. To solve this problem, a genetic algorithm is applied. Computational results demonstrate the outperformance of the proposed formulation.

#### 1. Introduction

Since the airline deregulation act enacted in 1978, airlines introduce the concept of revenue management, restructure airline network and develop centralized airline control centers to establish and sustain a competitive edge in this market-driven environment [1]. Hence, hub location and revenue management play significant roles in boosting the development of the airline industry. New challenges remain in these two disciplines.

Hub location problem by locating the hubs and connecting fewer links from the hubs to the non-hubs makes an airline network obtain more traffic flows and largely reduce the cost. In the area of revenue management, the flows can be performed as air tickets which can be effectively allocated to different segments of customers to obtain more revenue. Hence, these two disciplines are closely related. Until now, only two publications [2, 3] have considered about integrating the hub location into the revenue management to maximize the airline profit on the one hand. These studies consider the average profit, which is difficult to identify the underlying distribution probability. On the other hand, our research is also inspired by a weight-based combined consideration of the average-case and worst-case values in [4]. They provide a flexible decision for an emergency response network design problem by putting relative emphasis on average-case and worst-case cost. Consequently, our aim is to maximize the profit of integrating a hub location problem into a revenue management problem and to provide less conservative and restrictive solutions by a weight-based combination of average-case and worst-case profits.

The main contribution of our research is developing a new mathematical model and proposing flexible decisions for an integrated problem of hub location and revenue management. In this paper, we present a weight-based two-stage stochastic programming framework formulation by combining average-case and worst-case profits. To reduce the computation time, a genetic algorithm (GA) has been applied. The results show that our formulation outperforms the other one only considering the average case. Accordingly, the effects of the weight value, the network configuration and the discount factor on the profit are discussed.

The organization of this paper is as follows: Section 2 presents a brief literature review. Section 3 describes an integrated problem of hub location and revenue management. Section 4 details the solution methodology. Section 5 discusses the results. Section 6 summarizes our conclusions.

#### 2. Literature Review

Since [5] introduces the concept of revenue management, the research on revenue management has received tremendous attention in both academia and industry. The main problems in revenue management can be categorized: pricing, capacity control, overbooking, auctions and forecasting [6]. This paper focuses on capacity control problem by allocating the capacities on all itineraries of an airline network to different booking classes segmented by customer demand over a fixed period from departure to maximize the airline revenue. Early publications provide an effective analysis of revenue management on single flight leg without consideration of network effects ([5, 7]). Progress in this area has moved forward to network revenue management problem which is proposed by [8]. The main difference between these two problems is that dramatic expansion of the airline network makes the itineraries largely increase. The research about network topology in network revenue management can be classified by: on the one hand, for multiple legs without a specific network structure, some research ([9–14]) decomposes the multi-leg problem into many single leg problems and then makes a decision on allocating the capacities to every single leg. The research on the hub-and-spoke network also has received significant attention on the other hand. [15] considers the network revenue management in a hub-and-spoke network. In their assumptions, a single hub can serve spokes. [16] studies the capacity allocation in a two-airline alliance under competitions in a hub-and-spoke network. [17] studies the airline revenue management with consideration of a competitor’s behavior under simultaneous price and capacity competition in a hub-and-spoke network. [18] explores the impact of structural properties on a hub-to-hub network revenue management problem. As seen from the above literature, the research of airline revenue is closely relevant to airline network structure. Until now, only two papers ([2, 3]) focus on integrated research of hub location and revenue management. Our paper continues to consider this integrated problem. For reviews on revenue management, we refer the readers to e.g., [19, 20].

The hub is used to largely reduce the links between the origins and the destinations in an airline network [21]. Hub location problem is a discipline which focuses on locating the hubs from a set of nodes and routing the links from the hubs to the non-hubs. Since [22, 23] contribute the first mathematical formulation and the solution method to the hub location problem, this research has received constant attention. Four fundamental hub location problems are [24]: (1) -hub median, (2) -hub center, (3) uncapacitated or capacitated -hub location, and (4) hub covering. Diverse network topologies also play a role in the development of these four problems, e.g. tree network, star-star network, cycle network and hub line network. This paper focuses on a star-star -hub location problem. [25] studies a single allocation -hub median problem in a star-star network structure. [26] minimizes the routing cost between the hubs and the non-hubs in a star-star network. [27] formulates two separated problems on a star-star network. One is -hub center problem, which is to minimize the maximum length of the paths between different nodes. The other is -hub median problem with bounded path length to minimize the cost. Then they analyze the performances of these two problems. [28] explores the hardness of a star -hub center problem. For poor service quality incurred by minimizing the routing cost, [29] proposes -star single-allocation -hub center problem. A min-max criterion is introduced by maximizing the service quality and minimizing the cost. These above research with a cost-based objective assume that every origin or destination node should be served. However, if some origin & destination (O&D) pairs are not profitable, the airline isn’t necessary to operate them. This assumption is relaxed by [30]. They measure the tradeoff between the profits of the commodities served on some O&D pairs and the cost of hub location and routing between the hubs and the non-hubs. Our paper also considers the hub location problem from the perspective of optimizing the profit. The overviews on the hub location problem ([21, 31]) provide a detailed treatment.

This paper is based on a very recent work [3]. They integrate an uncapacitated single allocation -hub median problem and a revenue management problem in a star-star network. A two-stage stochastic programming formulation is presented to maximize the profit. In the first stage, the hub location, the link between the hub and the non-hub and the protection level of tickets for different booking classes are determined. The booking limit of tickets can be obtained in the second stage. The demand is captured in a set of discrete scenarios under the average case. The average-case analysis is effective when a probability distribution is known. However, demand information is always hard to obtain which makes the results more sensitive and less accurate. To address this concern, robust optimization is proposed as an alternative method to handle data uncertainty. Instead of considering distributional information, robust optimization assumes a defined set of values that any uncertainty can be realized. The solution is obtained under the realization of most unfavorable uncertainty in this set. That is to say, robust optimization analyzes the worst-case value. But choosing an uncertain parameter set makes the result over-conservative. For more information regarding the average-case and worst-case analysis, the readers are referred to [32, 33]. To overcome these shortcomings, the research on the complementary effects of worst-case and average-case analysis is appeared ([34–38]). In addition, there is relatively sparse research regarding the average-case or worst-case analysis for a two-stage optimization problem. [39] explores the performance between the worst-case cost of two-stage robust optimization and the expected cost of two-stage stochastic programming. [40] discusses the performance between a two-stage scenario-based stochastic optimization model and a two-stage robust optimization formulation in response management for residential appliances under real-time price-based demand. The above research still explores the pros and the cons between these two methods rather than providing a more flexible solution. The recent advance in the research is proposed by [4]. This paper formulates a two-stage stochastic programming framework for an emergency response network design problem to minimize a weighted sum of average and worst case costs. This contribution can give the flexibility of network configuration by putting relative emphasis on the average-case and the worst-case costs. In our paper, this research is introduced into an integrated problem of hub location and revenue management.

#### 3. Problem Description

We consider an integrated problem of an uncapacitated single allocation hub median location problem and a network revenue management problem. In this star-star network, a central hub 0 is predefined. The hubs must be connected to the central hub 0. The number of hub is and these hubs are chosen from a set of nodes . The non-hub refers to the node that hasn’t been chosen as a hub. Consequently, two stars are formed: (1) the links between the hubs and the non-hubs, and (2) the other links between the hubs and the central hub. Here, the O&D pairs in this network can traverse at least one hub and at most two hubs (not including the central hub). That is to say, all these O&D pairs can be made of these two basic links: (1) is the link between non-hub and hub ; (2) is used for connecting hub to central hub 0. Figure 1 illustrates the underlying star-star network structure in this paper.