# Using GPS Trajectories to Adaptively Plan Bus Lanes
## 2 Problem Formulation
### Constraints  
Upon comprehensively reviewing the present literature on transportation planning [12, 17, 18, 19, 20, 21], four constraints are considered.  
(1) Road conditions (C1). Building a bus lane partially occupies the road, and the original road needs to meet the transformation standard, which generally considers the number of lanes and road width [22]. Candidate roads are therefore filtered according to the road type through a process mathematically expressed as shown in Equation (1):  
\[\forall i\in E^{\prime},t_{i}\in T \tag{1}\]  
where \(E^{\prime}\) represents all the candidate road segments of the bus lane, \(i\) is one candidate of road segments, \(t_{i}\) is the road type for road segment \(i\), and \(T\) is the appropriate road type set.  
(2) Traffic flow (C2). Generally, road segments with low traffic flows are not needed for reconstruction into bus lanes. In practice, some cities have also introduced corresponding traffic flow standards for bus lanes. For example, the bus flow limitation of bus lane planning is 90 vehicles/h for three-lane roads and 120 vehicles/h for two-lane roads in Shenzhen, China. A bus flow threshold is therefore established for each candidate road segment, and it is formulated as shown in Equation (2):  
\[\forall i\in E^{\prime},f_{i}\geq F \tag{2}\]  
where \(f_{i}\) is the bus flow of candidate road segment \(i\) and \(F\) is the bus flow threshold. In model implementation, with GPS trajectories matched to road segments, the traffic flow on each road segment can be directly specified.  
(3) Bus lane continuity (C3). [PERSON] et al. [23] and [PERSON] et al. [24] suggest that short-distance or discrete bus lanes can lead to the further deterioration of traffic conditions. Continuous bus lanes are also relatively advantageous in terms of construction, passenger comfort, and safety [19, 25]. To accommodate continuous constraints on bus lanes, the numbers of continuous candidate road segments and continuous noncandidate road segments are restricted in our model. \(E_{c}\) is a set of connected candidate road segments for bus lanes form continuous candidate road segments. \(E_{cnon}\) is a set of continuous noncandidate road segments with a minimum number of noncandidate road segments between two disconnected segments \(E_{c}\). There is a limitation on the number of road segments for every \(E_{c}\) and \(E_{cnon}\). These restrictions are denoted as shown in Equations (3) and (4):  
\[E_{c}\geq k_{c} \tag{3}\]  
\[E_{cnon}\geq k_{cnon} \tag{4}\]  
where \(k_{c}\) and \(k_{cnon}\) are the corresponding thresholds of numbers of continuous candidate road segments and continuous noncandidate road segments.  
(4) Construction budget (C4). The construction budget of bus lanes is often limited. In our model, for any road segment \(i\), there is a construction cost \(c_{i}\) for the bus lane. Correspondingly, there is a total construction budget \(B\). In practice, because the cost \(c_{i}\) of each road segment is generally correlated with the road length, the length constraint is therefore used to replace the construction budget constraints in our model. The problem is formulated as Equation (5):  
\[\sum_{i\in E^{\prime}}c_{i}\leq B \tag{5}\]