I am trying to model traffic in a city, $(i,j)$ represents a road in a city.

There are $H$ vehicles in a city they have some prescheduled set of destinations to visit, $A_{j,h}$ denotes arrival time of vehicle $h$ at node $j$ and $D_{j,h}$ is departure time at node $j$.

$b_{i,j,h}$ represents vehicle $h$ has road $(i,j)$ in its route, i.e., it travels from node $i$ to $j$. (Only one incoming and outgoing edge at each node)

My Attempt to model traffic:

There are constants for arrival time which is not required here so I am only discussing departure time constants.

If there was no traffic then:

\begin{equation} dt_{j,h} = at_{j,h} \end{equation}

departure time at any node is equal to arrival time at that node no delay no traffic.

To check for traffic I used a binary variable $T_{j,h, h^{\prime}}$ representing if there is traffic delay at node $j$ for vehicle $h$ due to vehicle $h^{\prime}$

My current idea is if two-vehicle arrives at any node within the time window of $[-\tau, +\tau]$ then there will be some delay introduced because of that, and each node has a constant value called traffic penalty $TP_{j}$ (for example 10 min or 5min penalty)

\begin{equation} T_{j,h, h^{\prime}} = \sum_{\forall i \in V} b_{i,j,h} \cdot \sum_{\forall i \in V} b_{i,j,h^{\prime}} \quad \text{ if } \quad at_{j,h} - at_{j,h^\prime} \in [-\tau, +\tau] \end{equation}

$ \sum\limits_{\forall i \in V} b_{i,j,h} = 1$ if vehicle $h$ has node $j$ in its route.

\begin{equation} dt_{j,h} = at_{j,h} + \sum_{\forall h^\prime in H } T_{j,h, h^{\prime}} \cdot TP_{j} \end{equation}

The problem with this model is this is not a "realistic model", I am looking for some suggestion to make it more accurate.



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