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 ai Earliest arrival time at stop $i\in {P}^{+}$ bi 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 Cij Travel distance from node i to node $j\left(i,j\in N\right)$ tij Travel time from node i to $j\left(i,j\in N\right)$ sil A parameter equal to 1 if student l can reach stop $i\in {P}^{+}$ , and 0 otherwise qil A parameter equal to 1 if student  is related to school $i\in {P}^{-}$ , and 0 otherwise Pg Number of parking spaces at garage g ms Maximum number of allowable students for each stop Oi = {S|sil = 1} Set of students who can be assigned to stop i Wi = {S|qil = 1} Set of students who should be delivered to school i Tr Risk thrashold coefficient Hi Health risk factor in node i Pdrij Density-of-population risk factor from node i to node j Tvij Traffic volume risk factor from node i to node j Arij Total risk factors from node i to node j: sumation of and Pdrij and Tvij Decision variables Xijk 1 if bus k traverses the arc from node i to , and 0 otherwise Yik 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 Tik Arrival time of bus k to node $i\left(\forall i\in N\right)$ Lik Load of bus k after leaving node $i\left(\forall i\in P\right)$ hik 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