Joint optimisation of regular and demand-responsive transit services

Figures & data

Figure 1. Topological graph of the joint service.

Table 1. Notation of key model parameters and variables.

Input variables
$I$	Set of demand areas
$i\in I$	Index of demand areas
$J$	Set of regular bus stops
$j\in J$	Index of regular bus stops
$L$	Set of regular bus lines
$l\in L$	Index of regular bus lines
$B$	Superscript $B$ indicates that the parameter is related with regular transit
$R$	Superscript $R$ indicates that the parameter is related with DRT
$n_l^B$	Initial fleet size of regular bus line $l$, vehicles
$N$	Fleet size limits, vehicles
$p_l$	Number of bus stops of bus line $l$
$J_l$	Set denoting the bus stops of line $l$ in a sequential order
$t_{j{j}^{\mathrm{\prime }}}^B$	Travel time of regular transit from bus stop $j$ to bus stop $j{}^{\mathrm{\prime }}$, h
$t_{ij}^C$	Travel time between demand area $i$ and bus stop $j$, h
$t_{i{i}^{\mathrm{\prime }}}^R$	Travel time of DRT between demand areas $i$ and $i{}^{\mathrm{\prime }}$, h
$t_r$	Penalty time for the transferring, h
$d_{i{i}^{\mathrm{\prime }}}$	Passenger demand from demand area $i$ to demand area $i{}^{\mathrm{\prime }}$, person/h
$K^B$, $K^R$	Vehicle capacity for regular transit and DRT, person
$\alpha$	Weight factor that convert the fleet size into units of time
${\gamma }_t$	Penalty factor for the waiting time against the in-vehicle time
${\gamma }_w$	Waiting time coefficient of the DRT
${\gamma }_c$	Fleet size coefficient of DRT
$\delta$	Threshold value of ridership upon which a bus stop can be considered to be removed, person/h
$\tau$	Threshold value of travel time between demand area and bus stop upon which a demand area can be considered to be served by the DRT, h
$\epsilon$	Threshold value of the stopping criteria in GA
Decision variables
$g_l^s$, $g_l^e$	Adjusted start and end bus stops of regular bus line $l$
${\beta }_i$	Binary variable for the demand area $i$ served by the DRT, 1 – yes, 0 – no
$m_l^B$	Adjusted fleet size of regular bus line $l$, vehicles
$m^R$	Fleet size of DRT, vehicles
Auxiliary variables
$Z$	Set of DRT routes
$z\in Z$	Index of possible DRT routes
$T$	Total travel time of passengers, h
$C$	Total fleet size of transit lines, vehicles
$S_l^s$, $S_l^e$	Set of potential adjusted start and end bus stops of bus line $l$
$S_r$	Set of potential removed bus stops
$S_d$	Set of potential demand areas served by the DRT
$T_{i{i}^{\mathrm{\prime }}}$	Travel time of passengers from demand area $i$ to demand area $i{}^{\mathrm{\prime }}$, h
$e_{j{j}^{\mathrm{\prime }}}$	Edge connecting bus stops
$e_{ij}$	Edge connecting demand areas and regular bus stops
$e_{i{i}^{\mathrm{\prime }}}$	Edge connecting demand areas
$w_{j{j}^{\mathrm{\prime }}}$	Weight of the edge connecting bus stops, h
$w_{ij}$	Weight of the edge connecting demand areas and regular bus stops, h
$w_{i{i}^{\mathrm{\prime }}}$	Weight of the edge connecting demand areas, h
$t_l^B$	Trip duration of regular bus line $l$, h
$t^R$	Trip duration of DRT line, h
$t_z^R$	Trip duration of DRT route $z$, h
$T$	Time period for average trip duration calculation, h
$P_i(A(T)\ge 1)$	Probability that the required traffic demand in demand area $i$ is no less than 1 in a time period $T$
$d_{lj{j}^{\mathrm{\prime }}}$	Ridership using regular bus line $l$ from bus stop $j$ to bus stop $j{}^{\mathrm{\prime }}$, person/h
$d_{zi{i}^{\mathrm{\prime }}}$	Ridership using DRT $z$ from zone $i$ to zone $i{}^{\mathrm{\prime }}$, person/h
$u_{j{j}^{\mathrm{\prime }}}^B$	Waiting time of regular transit at the bus stop $j$ for the trip from bus stop $j$ to bus stop $j{}^{\mathrm{\prime }}$, h
$u_z^R$	Waiting time of DRT $z$, h
$h_l^B$	Headway of regular transit line $l$, h
$h^R$	Headway of DRT line, h

Figure 2. Process of generating set of potential terminal bus stops for regular transit.

Figure 3. Process of generating set of potential serving areas for DRT.

Table 2. Parameters reflecting the difference between the two types of transit service.

Items	Difference in travel time	Related parameters
In-vehicle travel time	It depends on the setting of bus lines and their travel time.	$t_{j{j}^{\mathrm{\prime }}}^B$, $t_{i{i}^{\mathrm{\prime }}}^R$
Walking time	Regular transit passengers need to walk between the demand areas and bus stops while DRT passengers have negligible walking times.	$t_{ij}^C$
Waiting time	Regular transit has a longer waiting time at bus stops under the same service frequency conditions.	${\gamma }_w$

Figure 4. Main idea of the optimisation procedure.

Figure 5. Flowchart for the solution procedure.

Figure 6. String of the solution.

Figure 7. Passenger demand zones and original transit network of the case study.

Table 3. Input data of the case study.

Input parameters	Values
Set of demand areas	$I=\{0,1,\dots ,35\}$
Set of regular bus stops	$J=\{36,37,\dots ,60\}$
Set of regular bus lines	$L=\{1,2,3,4,5\}$
Initial fleet size of regular bus line $l$, vehicles	$n_l^B=\{5,5,5,5,5\},\mathrm{\forall }l\in L$
Fleet size limits, vehicles	$N=50$
Set of bus stops of line $l$ in a sequential order	$J_1=\{36,\dots ,40\}$, $J_2=\{41,\dots 45\}$, $J_3=\{46,\dots ,50\}$, $J_4=\{51,\dots ,55\}$, $J_5=\{56,\dots ,60\}$
Travel time of regular transit from bus stop $j$ to bus stop $j{}^{\mathrm{\prime }}$, h	$t_{j{j}^{\mathrm{\prime }}}^B=1/12$ for the neighbouring bus stops
Travel time between demand area $i$ and bus stop $j$, h	$t_{ij}^C=1/10$ for the bus stop $j$ besides the demand area $i$, otherwise, 1/4 h are added for crossing one passenger demand zone
Travel time of DRT between demand areas $i$ and $i{}^{\mathrm{\prime }}$, h	$t_{i{i}^{\mathrm{\prime }}}^R=1/15$ for crossing one demand zone
Penalty time for the transferring	$t_r=1/120$
Passenger demand from demand area $i$ to demand area $i{}^{\mathrm{\prime }}$, person/h	$d_{i{i}^{\mathrm{\prime }}}=10,\mathrm{\forall }i,i^{\mathrm{\prime }}\in I\mathrm{\setminus }\{5,11,17,23,29,35\};i\ne i^{\mathrm{\prime }}$$d_{i{i}^{\mathrm{\prime }}}=1\mathrm{\forall }i\in \{5,11,17,23,29,35\},i^{\mathrm{\prime }}\in I;i\ne i^{\mathrm{\prime }}$$d_{i{i}^{\mathrm{\prime }}}=0,\mathrm{\forall }i{i}^{\mathrm{\prime }}\in I;i=i^{\mathrm{\prime }}$
Vehicle capacity for regular transit and DRT, person/veh	$K^B=60$, $K^R=30$
Weight factor that convert the fleet size into units of time	$\alpha =20$
Penalty factor for the waiting time	${\gamma }_t=1.1$
Equivalent factor that convert the fleet size of DRT into that of the regular transit	${\gamma }_c=0.5$
Threshold upon which a bus stop can be considered to be removed, person/h	$\delta =60$
Threshold upon which a demand area can be considered to be served by the DRT, h	$\tau =1/4$
Threshold value of the stopping criteria in GA	$\epsilon =0.001$

Figure 8. Original passenger demand flow distribution on regular bus lines.

Table 4. Potential terminal bus stops and service zones.

Items	Results
Set of potential start bus stops for regular transit	$S_l^s=[36,41,46,51,56]$
Set of potential end bus stops for regular transit	$S_l^e=[39,40,44,45,49,50,55,60]$
Set of potential service zone for demand-response transit	$S_d=[4,5,10,11,16,17,22,23,28,29,34,35]$

Figure 9. Design schemes at each optimisation step.

Figure 10. Convergence and computational time of the case study.

Table 5. Comparison with the optimisation method without the preparation step.

Method	Objective value (h)	CPU time (s)
With preparation (proposed model)	4216	1667
Without preparation	4216	14,153

Table 6. Comparison with the optimisation method without boundary-start searching step.

Method	Objective value (h)	CPU time (s)
With boundary-start searching (proposed model)	4216	1667
Without boundary-start searching	4216	4145

Table 7. Comparison with two single-side optimisations.

Method	Objective value (h)	Travel time (h)	Fleet size (vehicles)
Joint optimisation	4216	3806	20.5
Only regular transit	4496	4146	17.5
Only DRT	4557	4047	25.5

Figure 11. Effect of the time-fleet size coefficient $\alpha$.

Figure 12. Effect of the waiting time coefficient for DRT ${\gamma }_w$.

Figure 13. Effect of the threshold $\delta$.

Figure 14. Effect of the travel time of DRT.