# International Airline Alliance Network Design with Uncertainty
## 1 Introduction
By redesigning and adjusting the route network, the transportation costs of airlines are reduced, resulting in a risk-resistant model for providing reliable services and promising strategies for airline expansion in the future. In particular, the hub-and-spoke (HS) network is a promising network type for formulating route networks due to its feasibility and efficiency in practice. Several research works have been written to address the airline network optimization problem with the HS network. The authors of related articles ([PERSON] and [PERSON] [10]; [PERSON] and [PERSON] [11]) examined single allocation, multiple allocation, as well as capacitated, incapacitated, strict, and non-strict models with extensions. However, the works presented above only work for the common alliance problem, and they assume deterministic alliance models with few constraints introduced [12], which leads to the proposition to exploit uncertain models with complex factors for designing international alliance networks. The common alliance model neglects the uncertainty, which represents unknown situations and may be encountered in practical operations, as well as the flexible parameters that are challenging to estimate. For those flexible parameters, a reasonable interval range or possible probability distribution is obtained by statistics, such as the airport capacity distribution, airline capacity, demands among airports, and unit transportation costs. Furthermore, finding the worst-case scenario with respect to a determined uncertainty set ([PERSON], [PERSON], and [PERSON] [13]) is another gap that has not been sufficiently investigated in previous research. This paper explores the methods for narrowing the gaps in airline network optimization with uncertainties. We applied the uncertainty optimization method to optimize the design of an alliance route network, and we present an international route network with both economy and risk resistance for airlines. Specifically, we propose an approach that is able to use the complementary alliance model with the HS route method for the purpose of optimization. An incapacitated multiple p-hub median problem (UMpHMP)-based alliance model is proposed, along with the integration of uncertain factors. The network optimization problem was addressed by developing a three-subscript model and was measured on the worst-case scenario. Compared with typical network model structures [12], the proposed three-subscript optimization model is applicable for more than two airline alliances and multiple gateway numbers, and the uncertain factors can be integrated to meet the practical operation requirements. The rest of this paper is organized as follows. Section 2 provides a brief review of the literature on airline alliances and HS networks. Section 3 details the construction of an alliance route network based on an HS network and proposes a mathematical model to formulate the alliance route network design problem with uncertain costs. A case is provided in Section 4 to illustrate the proposed model. Section 5 presents the conclusion and suggestions for future research.