# Hub-and-Spoke Network Design Considering Congestion and Flow-Based Cost Function
## 2 Literature Review
### Utility-Based Demand Calculation for Hub Location Problems  
In this work, we formulate the problem for an airline transportation system, which also envisions the perceived utility of users to decide upon the spoke locations. In the literature, utility is also referred to as \"attractiveness\". In this line, [PERSON] and [PERSON] [30] developed a new model for a new airline entering the market. They formulated the problem as a p-hub location, considering a demand function, based on the competitive nature of the airline industry. They consider attractiveness factors to represent the demand, which is based on the cost and travel time of a route, in addition to the attractiveness of the hub as perceived by each passenger. The authors utilize heuristic concentration to solve the problem in two phases: (1) reducing the number of hub location candidates to a much smaller set that they state to be local optima, and (2) solving the two-vertex substitution problem and applying it to the Australian Postal dataset.  
The proximity approach has the subtle assumption that customers choose a facility merely based on spatial factors: they only choose the closest facility. Such an assumption leads to an \"All or Nothing\" result: the customers either choose a facility and the whole demand uses the chosen facility, or they do not choose a facility (and the demand is lost). This is not a true representation of the real world. While distance is an important factor for customers as to whether or not to choose a facility, there are several other factors in this decision-making procedure, which support the use of the utility approach:  
If a facility is located in a slightly different location, it would lose the whole demand and will not be chosen by customers.  
There might be some other nearby facilities with higher attraction for customers, and the customers prefer to drive a longer distance there.  
It is not ensured that all customers measure the distance from a facility exactly and precisely, which leads them to choose different facilities, which are not necessarily the closest facility to the customer.  
As discussed by [PERSON] et al. (2018) and [PERSON] et al. (2019), the type C error in aggregation may happen, which is as follows: the facilities are not a spot on a map; instead, they are locations with surface area. Thus, a facility may be closest to one customer who resides in a segment of that area while for another customer that case is not true.  
In addition to all the analytical reasons behind choosing a facility, shopping and touristic trips may be other aspects for choosing a facility which rely on other purposes for the trip ([PERSON] et al., 2019). Therefore, customers may prefer to drive a longer distance to reach a facility with a different arrangement of shopping and recreational activities.  
In this research, we not only find locations for hub nodes but also select the spoke nodes (domestic facilities) to design the hub-and-spoke network together. Based on an earlier discussion, the attractiveness (utility) of a facility should be considered while selecting the spoke nodes.  
In all of the above-mentioned articles, the hub locations are chosen from a set of given nodes. It is important to find out the origin and method to obtain the initial set of nodes. In a recent study, [PERSON] et al. (2019) propose a profit maximization hub network problem that decides upon not only the hub locations but also the spoke facilities, with the goal of profit maximization. In another study, [PERSON] and [PERSON] (2019) propose a hub-and-spoke location model, using the gravity function to choose a set of potential facilities based on some attractiveness criteria--[PERSON] (2019); [PERSON] and [PERSON] (2019); [PERSON] and [PERSON] (2020). The model then decides upon a subset of nodes to be upgraded to be hub facilities. This model, on the other hand, does not handle neither congestion of the hub nodes nor the variable cost function of the hub network.