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 = {S|s_il = 1}

Set of students who can be assigned to stop i

W_i = {S|q_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

Density-of-population 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 $j\left(\forall i,j\in N\right)$ , and 0 otherwise

Y_ik

1 if bus k visits stop i, 0 otherwise

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

if student l is delivered by bus k to school j, and 0 otherwise