Literature review

Studying impact of uncertainty in decision making has been done in a number of researchers to address hub location models (e.g., Alumur and Kara, 2008;

Campbell et al., 2002; Zanjirani Farahani et al., 2013). However, as we pointed out in the Section 6.1, the most of the recently published papers have mainly considered uncertainty in the spoke and link related parameters such as demand, time, etc. This

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includes Contreras et al. (2011), Alumur et al. (2012), and Mohammadi et al., 2014a, among others.

In this section, the most related literature of papers which have considered disruption in their model is reviewed. Most of the previous paper studying disruption in the network takes its roots in the classical p-median (Tansel et al., 1983) and the uncapacitated fixed-charge location problems (Nemhauser and Wolsey, 1988). Both these problems locate facilities and allocate customers to located facilities to minimize the total transportation cost while all facilities are assumed to be totally available and reliable. The first model of reliability facility location was proposed by Snyder and Daskin (2005), where the authors assume that some facilities are perfectly available while others are subject to failure and become unavailable with the same probability. In their model, each customer is allocated to a primary facility and a number of backup facilities, in which at least one facility must be available. If the current facility fails, the customer is served by the next available backup facility.

They proposed a linear integer programming model to formulate their problem and developed a Lagrangian relaxation solution method. No approximation algorithm has been proposed in Snyder and Daskin (2005).

A recent paper by Cui et al. (2010) relaxes the uniform failure probability assumption in Snyder and Daskin (2005) and allows the failure probabilities to be facility-specific. The authors proposed a compact mixed integer program formulation and a continuum approximation model to solve the model that seeks to minimize initial setup costs and expected transportation costs in normal and failure scenarios.

The continuum approximation model predicts the total system cost without details about facility locations and customer assignments, and it provides a fast heuristic to find near-optimum solutions. Their computational results show that for large-scale problems, the continuum approximation method is very effective algorithm, and it avoids prohibitively long running times.

Cui et al. (2010) presented two related models as reliable p-median problem and reliable uncapacitated fixed-charge location problem. Both models consider heterogeneous facility failure probabilities, one layer of supplier backup, and facility fortification within a finite budget. The authors formulated both model as nonlinear integer programming models and proved to be NP-hard. They also developed Lagrangian relaxation-based solution algorithms and demonstrate their computational efficiency. Similar to Cui et al. (2010), Aboolian et al. (2012) considered reliable facility location models in which facilities are subject to unexpected failures, and customers may be reassigned to facilities other than their regular facilities. Their main effort was to derive Lower bounds for reliable uncapacitated fixed-charge location problem (RUFLP) and introduce a class of efficient algorithms for solving the RUFLP problem.

Peng et al. (2011) introduced the p-robustness criterion so that the designed network performs well in both disrupted and normal conditions. The authors presented a mixed-integer programming model which tries to minimize the nominal cost (i.e., the cost when no disruptions occur) while reducing the disruption risk

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using the p-robustness criterion that bounds the cost in disruption scenarios. They proposed a hybrid meta-heuristic algorithm that is based on genetic algorithms, local improvement, and the shortest augmenting path method. They also proved the superiority of their heuristic algorithm comparing to CPLEX in terms of solution speed, while still delivering excellent solution quality. Shen et al. (2011) studied a reliable facility location problem wherein some facilities are subject to failure from time to time. If a facility fails, customers originally allocated to it have to be re- allocated to other (operational) facilities. They formulated this problem as a two- stage stochastic program and then as a nonlinear integer program. Several heuristics that can produce near-optimal solutions were proposed for this NP-hard problem.

For the special case where the probability of facility failure is constant (independent of the facility), they provided an approximation algorithm with a worst-case bound of 4. Li et al. (2013) presented two related models (i.e., reliable p-median and reliable uncapacitated fixed-charge location) for the design of reliable distribution networks.

Both models are formulated as nonlinear integer programming and considered heterogeneous facility failure probabilities, one layer of supplier backup, and facility fortification within a finite budget. The NP-hardness of the models was also proved.

To the best of our knowledge, there are just two studies dealing with the hub network design problem and taking hub disruptions into account. Parvaresh et al.

(2012) formulated a bi-level multiple allocation p-hub median problem under intentional disruptions by a bi-level model with bi-objective functions at an upper level and a single objective at a lower level. In their model, the leader aims at identifying the location of hubs so that normal and worst-case transportation costs are minimized while normal and failure conditions are taken into account. Finally, the worst-case scenario is modeled in a lower level, where the follower’s objective is to identify the hubs the loss of which would most diminish service efficiency.

Additionally, they developed two multi-objective meta-heuristics based on simulated annealing and tabu search to solve their proposed model. In a similar work, Parvaresh et al. (2013) developed a multiple allocation p-hub median problem under intentional disruptions using different definitions of a failure probability of the hub in comparison to their previous work. All of reviewed papers have considered complete disruption for facilities.

Regarding to studies dealing with spoke uncertainty, Mohammadi et al.

(2011a, 2015) and Sedehzadeh et al. (2014, 2015) studied a HLP with uncertain demand as a Poisson distribution, where limited number of flows can enter a hub.

They presented a M/M/c queuing system to handle the uncertainty of demands between each pair of O-D nodes. Similarly, Contreras et al. (2011) studied stochastic uncapacitated HLPs in which uncertainty is associated to demands and transportation costs. They showed that the stochastic problems with uncertain demands or dependent transportation costs are equivalent to their associated deterministic expected value problem, in which random variables are replaced by their expectations.

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Finally, to the best of our knowledge, this research is the first that has considered all hub, link and spoke uncertainties in designing a reliable hub-and- spoke network. we depart from common assumptions in the literature by considering complete and partial disruptions in the hubs, and partial disruptions in connection links.