# Hub-and-Spoke Network Design Considering Congestion and Flow-Based Cost Function
## 3 Mathematical Formulation of the Model
### Model with Balanced Congestion  
Among the earliest works on hub congestion, [PERSON] and [PERSON] [43] define the congestion for a single hub location problem as follows:  
\[\sum_{i}\sum_{j>i}\sum_{m}w_{ij}x_{ijkm}=g_{k}\forall k\in K \tag{21}\]  
where \(g_{k}\) is the congestion of the hub facility k. Later, [PERSON] et al. [44] extended the definition for the congestion of a multi-allocation hub location, as the sum of incoming and outgoing flow from that hub:  
\[\sum_{i}\sum_{j}\sum_{l}w_{ijkl}+\sum_{i}\sum_{j}\sum_{l\
eq k}w_{ijlk}=g_{k} \forall k\in K \tag{22}\]  
[PERSON] et al. [44] define the congestion function as \(\tau_{k}(g_{k})=e\cdot g_{k}^{b}\), where e and b are positive constants. To have an increasing function b \(\geq 1\) as considered by [PERSON] and [PERSON] [43].  
In the mentioned articles, as well as other published hub location research, the aim is to minimize the congestion. This can be argued as a counterproductive objective since while a highly congested facility is not desirable, the flow/congestion of a hub facility is important to make a profit.  
We also believe that it is not an easy task to put monetary value (penalty) on the congestion of a facility and then of course find the correct value of this cost factor. In this regard, congestion balance across the network seems to be an alternative substitution to the minimization of congestion. This ensures not having highly congested hubs on one side and low congested hubs (underutilized) on the other side. Thus, we calculate the deviation between the most and least congested facilities and model it as a constraint to distribute the flow in a fairly balanced way through the network. We believe such a model is a more realistic representation of a decision-makers' concern, as the congestion might not be seen necessarily as a cost or even a negative factor (which needs to be directly minimized). In contrast, we believe it needs to be well managed and distributed among different facilities in the network as evenly as possible.  
To mathematically formulate this concept, we first need to calculate the congestion of a hub (as reflected in (23)). We then find the hubs with maximum and minimum flow/congestion (24) and then limit the difference between the two (25). The following terminology is used for the congestion of a hub facility:  
\(g_{k}\): the congestion of a hub facility k.  
\(g^{Max}\): maximum congestion among all the hub facilities across the network.  
\(g^{Min}\): minimum congestion among all the hub facilities across the network.  
\(\
abla\): maximum allowed dispersion among hub congestions across the network.  
The final mathematical model (denoted as HSLPT-02) is the extension of model HSLPT-01 by additional constraints (23)-(26).  
\[\begin{array}{c}\sum\limits_{i}\sum\limits_{j\
eq i}\sum\limits_{l\
eq k}w_{ij} x_{ijkl}+\sum\limits_{i}\sum\limits_{j\
eq i}\sum\limits_{l\
eq k}w_{ij}x_{ijlk}+ \sum\limits_{i}\sum\limits_{k}\sum\limits_{j\
eq i}w_{ij}a_{ijk}+\sum\limits_{ i\
eq k}\sum\limits_{j\
eq i}w_{ij}b_{ijk}\\ \hskip 14.226378 pt+\sum\limits_{i\
eq k}2w_{ik}e_{ik}+\sum\limits_{i\
eq k}2w_ {ki}e_{ik}+\sum\limits_{i}\sum\limits_{j\
eq i}\sum\limits_{l\
eq k}w_{ij}x_{ ijkl}\cdot h_{i}\cdot h_{j}\\ \hskip 14.226378 pt+\sum\limits_{i}\sum\limits_{j\
eq i}\sum\limits_{l\
eq k}w_{ ij}x_{ijlk}\cdot h_{i}\cdot h_{j}=2\cdot g_{k}\forall k\in K\end{array} \tag{23}\]  
\[\begin{array}{c}g^{Max}=\max\limits_{k}(g_{k}),g^{Min}=\min\limits_{k}(g_{k} ),\forall k\in K\\ \hskip 14.226378 ptg^{Max}\leq\
abla\cdot g^{Min}\\ \hskip 14.226378 ptg_{k}\in R^{+}\end{array} \tag{24}\]