Research on the combinatorial optimization of EBs departure interval and vehicle configuration based on uncertain bi-level programming

Figures & data

Figure 1. The framework of this study.

Figure 2. The operation of the multi-EBs and passenger.

Table 1. Symbol definition.

Symbol	Definition	Dimension
$\mathrm{C}\mathrm{P}$	Total cost of passenger travel	yuan
$\mathrm{C}\mathrm{B}$	Total cost of EBs operation	yuan
$C_w$	Passenger waiting cost	yuan
$C_r$	Passenger travel cost	yuan
$C_s$	EBs company employee cost	yuan
$C_o$	Vehicle running cost	yuan
$C_m$	Vehicle maintenance cost	yuan
$\omega$	Waiting cost per passenger time	yuan (person min)^−1
$\mathrm{{\rm Y}}$	Travel cost per passenger time traveling	yuan (person min)^−1
${\theta }_1^i$	EBs driver cost per hour	yuan h^−1
${\theta }_2^i$	Cost per mile of public transport	yuan km^−1
${\theta }_3^i$	EBs idle cost per unit time	yuan h^−1
${\epsilon }_w$	Passenger waiting time	min
${\epsilon }_z$	Interstation travel time	min
$\psi$	Departure interval	min
$i$	EBs type	–
$x_j^i$	0–1 variable: if EBs type is i of car number j, take 1; otherwise, take 0	–
$\mathrm{n}$	Number of EBs stations	station
${\mathrm{Q}}_j^n$	The number of passengers on EBs number j at station n	person
${\mathrm{H}}_j^n$	The section passenger flow of number j from station n to n + 1	person
$\mathrm{j}$	The departure frequency	sequence•h^−1
$\mu ,\sigma$	Load factor parameters	–
$\delta$	Perception coefficient of EBs load factor	–
$U_i$	Capacity of model i	person
$u_i$	Number of seats for model i	person
$D_{n,n+1}$	Distance from station n to station n + 1	km
$T_{max}$	Maximum departure interval during investigation period	min
$T_{min}$	Minimum departure interval during investigation period	min
$maxH$	Cumulative passenger flow of maximum section	person

‘ – ’ means dimensionless.

Figure 3. Fitting distribution of waiting time of passengers with different departure intervals.

Figure 4. Distribution of running time between stations of different models.

Figure 5. Flow chart and iterative diagram of genetic algorithm.

Table 2. Parameters of model.

Symbol	Definition	Dimension	Value
$\omega$	Waiting cost per passenger time	yuan (person h)^−1	50
$\mathrm{{\rm Y}}$	Traveling cost per passenger time	yuan (person h) ^−1	22
${\theta }_1^1$	B bus driver cost per hour	yuan h^−1	53
${\theta }_1^2$	M bus driver cost per hour	yuan h^−1	42
${\theta }_1^3$	S bus driver cost per hour	yuan h^−1	35
${\theta }_2^1$	B bus vehicle cost per kilometer	yuan km^−1	1.2
${\theta }_2^2$	M bus vehicle cost per kilometer	yuan km^−1	0.9
${\theta }_2^3$	S bus vehicle cost per kilometer	yuan km^−1	0.7
${\theta }_3^1$	B bus vehicle idle cost per unit time	yuan h^−1	1.15
${\theta }_3^2$	M bus vehicle idle cost per unit time	yuan• h^−1	0.92
${\theta }_3^3$	S bus vehicle idle cost per unit time	yuan h^−1	0.57
$\mu$	Load factor influence degree parameter	–	0.15
$\sigma$	Load factor influence degree parameter	–	2.00
${\delta }_1$	Influence degree of B bus load factor when full	–	1.08
${\delta }_2$	Influence degree of M bus load factor when full	–	1.11
${\delta }_3$	Influence degree of S bus load factor when full	–	1.16

Table 3. Section passenger flow and passenger flow of Nanchang bus No. 211 in each period.

The number of station	Travel time
	9:00–10:00		17:30–18:30
	Passenger flow	Section passenger flow	Passenger flow	Section passenger flow
1	59	59	85	85
2	22		37
3	70	81	100	123
4	33	107	52	156
5	22	118	37	171
6	30	100	47	147
7	78	59	109	94
8	63	89	90	132
9	37	111	57	161
10	89	85	124	127
11	81	115	114	166
12	100	104	138	151
13	56	115	81	166
14	104	89	143	132
15	59	115	85	166
16	89	115	124	166
17	52	137	76	195
18	67	96	95	142
19	85	78	119	118
20	93	100	129	147
21	107	118	148	171
22	74	144	105	204
23	96	111	133	161
24	63	111	90	161
25	59	93	85	137
26	104	93	143	137
27	59	133	85	190
28	67	107	95	156

Figure 6. EBs scheduling scheme with certainty and uncertainty.

Figure 7. Pareto optimal set and comparison between schemes.

Figure 8. The temporal and spatial distribution of average travel time of passengers.

Table 4. Results of four schemes.

	Average departure interval (min)	The number of EBs	EBs cost (yuan)	Passenger cost (yuan)	Energy cost (yuan)	Average delay (s)
CP	12	26	6761	2981	3542	218
CF	15	11	4344	2463	2207	174
UP	10	20	5953	2122	3075	201
UF	17	10	4191	1925	1616	162