# Hub-and-Spoke Network Design Considering Congestion and Flow-Based Cost Function
## 7 Conclusions  
In this paper, we present a hub-and-spoke location/transportation problem and several alternative formulations of the problem with additional considerations, such as perceived utility function, hub congestion, and variable hub link cost function. We propose a new approach to include congestion and add congestion balancing constraints as opposed to congestion minimization. To formulate the problem, we develop a nonlinear hub-and-spoke location model. We apply linearization techniques to linearize the model and propose some tightening rules to further skim the model and produce a tighter formulation to aid the solution procedure. As in many hub-and-spoke problems, the model is not solvable to optimality for larger instances (it is NP hard), but we were able to run small sample sizes, either to optimality or with less than a 5% gap. We then develop an enhanced Benders algorithm to solve larger instances of the problem and successfully solve instances up to 40 nodes. To tackle the larger instances of the problem, we develop a heuristic algorithm that we run and check the efficiency of. The results demonstrate the following:  
* Considering the congestion in the model would significantly increase the total cost objective but would comply with post-pandemic regulations on limited congestion in each facility. It would also increase the computational time for the problem but would increase customer satisfaction when traveling through less-congested facilities. * Using the variable value of \(\alpha\) would result in a more realistic solution. * The smaller the values of \(\alpha\) (hub link transportation cost coefficient), the larger the number of hub facilities would be. This, in addition, would result in a longer solution time but better objective function value. * Modeling and solving the integrated model, in which both spoke-and-hub locations should be determined, results in a better solution, compared to the case where the number and/or location of the spoke facilities are given. This requires more computational effort and time. Managerial insights for high-level decision-making problems, such as location/transportation decisions, are categorized as strategic-level decisions. From the numerical experiments, we understand that considering a flow-based inter-hub transportation discount not only brings a more robust solution but the obtained solution is closer to real-world costs. It is recommended to choose the spoke locations, together with hub locations, to obtain higher-quality solutions (comparing local versus global optimal). Further research can be extended in the following directions:  
* In this work, we try to evenly distribute the congestion among different hub locations. We also determine the passing flow of each node, based on some utility functions. This can be extended in two directions:
* To model the congestion as a separate objective function and then solve the multi-objective model. * To model the effect of congestion as a part of the utility function, which would affect the utility of facilities and may change the favorite hub locations for passengers to pass through. * To get closer to the real-world decision-making procedure, it is recommended to include a time dimension into the formulation. It is obvious that hub location decisions are not planned for short-term utilization but are a strategic decision with long-term effects. During this time period, several initial parameters (such as demand, which is only an estimation at the time of decision-making on hub-and-spoke locations) would change which would impact the \"current optimal\" or \"near-optimal\" solution. Considering this, a multi-period, or any other time-included, formulation seems to be a better fit for such strategic mid-level decision-making. In this work, as in other research projects, we have experienced limitations. The first limitation is not having the exact customer demand before and after locating the facilities.