Table 3: Indices, sets, parameters, and decision variables used in the mathematical model.
Indices 

k 
Bus index 
i,j 
Node indices 
l 
Student index 
Set 

G 
Set of starting and ending depot locations (garage locations) 
K 
Set of buses 
s 
Set of students 
P^{+} 
Set of potential pickup locations (bus stop locations) 
P^{} 
Set of delivery locations (school locations) 
P = P^{} È P^{+} 
Set of stops and schools 
N = P È G 
Set of nodes 
Parameter 

c 
Bus capacity 
big M 
Large constant 
a_{i} 
Earliest arrival time at stop $i\in {P}^{+}$ 
b_{i} 
Latest arrival time at school $i\in {P}^{}$ 
ap 
Average pickup time at pickup points for each student 
ad 
Average delivery time at delivery points for each student 
C_{ij} 
Travel distance from node i to node $j\left(i,j\in N\right)$ 
t_{ij} 
Travel time from node i to $j\left(i,j\in N\right)$ 
s_{il} 
A parameter equal to 1 if student l can reach stop $i\in {P}^{+}$ , and 0 otherwise 
q_{il} 
A parameter equal to 1 if student is related to school $i\in {P}^{}$ , and 0 otherwise 
P_{g} 
Number of parking spaces at garage g 
ms 
Maximum number of allowable students for each stop 
O_{i} = {Ss_{il} = 1} 
Set of students who can be assigned to stop i 
W_{i }= {Sq_{il} = 1} 
Set of students who should be delivered to school i 
Tr 
Risk thrashold coefficient 
H_{i} 
Health risk factor in node i 
Pdr_{ij} 
Densityofpopulation risk factor from node i to node j 
Tv_{ij} 
Traffic volume risk factor from node i to node j 
Ar_{ij} 
Total risk factors from node i to node j: sumation of and Pdr_{ij }and Tv_{ij} 
Decision variables 

X_{ijk} 
1 if bus k traverses the arc from node i to $j\left(\forall i,\text{}j\in N\right)$ , and 0 otherwise 
Y_{ik} 
1 if bus k visits stop i, 0 otherwise 
$${Z}_{il}^{k}$$ 
1 if student l is picked up by bus k from stop i, and 0 otherwise 
T_{ik} 
Arrival time of bus k to node $i\left(\forall i\in N\right)$ 
L_{ik} 
Load of bus k after leaving node $i\left(\forall i\in P\right)$ 
h_{ik} 
1 if bus k visits school $i\in {P}^{}$ , and 0 otherwise 
$${D}_{jl}^{k}$$ 
if student l is delivered by bus k to school j, and 0 otherwise 