AU2021100757A4 - Method for optimizing bus bunching based on the combinatorial scheduling of whole-zone bus and inter-zone bus - Google Patents
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Abstract
The present disclosure relates to a method for optimizing bus bunching based on the
combinatorial scheduling of the whole-zone bus and the inter-zone bus, comprising the
5 following steps: without changing the existing number of bus runs, changing some of the
whole-zone bus runs into inter-zone runs to reduce bunching by the combinatorial scheduling of
the whole-zone bus and the inter-zone bus; analyzing the historical IC card data and GPS data
of the bus route to determine the departure station and the number of runs for the inter-zone bus;
creating a mathematical model for the departure station of the inter-zone bus which is intended
10 to minimize the deviation between the departure time and the scheduled time of each bus run;
and transforming the created model into an MILP (Mixed-Integer Linear Programming) for
solution by the branch and bound algorithm, so as to acquire a combinatorial scheduling scheme
for the whole-zone and inter-zone buses. In the context of an increasingly severe problem of bus
bunching in urban areas, the distribution of transport capacity on different road segments is
15 adjusted by adopting the method of combinatorial scheduling of the whole-zone bus and the
inter-zone bus, so as to improve the service reliability and efficiency of bus routes. This method
is advantageous in reducing the times of bus bunching.
DRAWINGS
F-Actual bus interval : Actual bus interval
Scheduled bus interval Scheduled bus interval
~Bus route
Road congestion
FIG.1
Whole-zone bus
Inter-zone bus
Stop at first and Stop at Non-stop at first and Non-stop at
terminal stations intermediate station terminal stations intermediate station
FIG. 2
Description
F-Actual bus interval : Actual bus interval
Scheduled bus interval Scheduled bus interval
~Bus route
Road congestion
FIG.1
Whole-zone bus
Inter-zone bus
Stop at first and Stop at Non-stop at first and Non-stop at terminal stations intermediate station terminal stations intermediate station
FIG. 2
TECHNICAL FIELD The present disclosure provides a method for optimizing bus bunching based on the combinatorial scheduling of whole-zone bus and inter-zone bus, wherein some of the whole-zone bus runs in a bus route are changed into inter-zone runs to skip the road segment susceptible to congestion and the station where the number of passengers getting on and off the bus is less than a certain threshold, thereby reducing the times of bus bunching. The method pertains to the technical field of urban public transport, in which the historical IC card data and GPS data are utilized to determine the departure station and the number of runs of the inter-zone bus, and a mathematical model is created for the departure station of the inter-zone bus which is intended to minimize the deviation between the departure time and the scheduled time of each bus run so as to acquire respective departure times of the whole-zone and inter-zone bus runs.
BACKGROUND With the rapid development of urban modernization in China, the number of motor vehicles has increased sharply while the road resources are limited and the road traffic flow is increasingly saturated, resulting in the increasingly serious problem of urban traffic congestion, which has become one of the difficult problems that plague people's lives in large and medium cities in China, seriously confining the healthy development of cities and the further improvement of residents' living standards. As a large-capacity passenger transport mode, the ground bus can realize efficient utilization of resources l. Therefore, it is of great significance to strive to develop the ground bus for improving the urban traffic structure and building a resource-saving and environment-friendly society. However, in the practical operation of ground bus, the interval between vehicles often changes from uniform to non-uniform due to the dynamic fluctuation of road conditions, as shown in Fig. 1, even leading to the "bunching" phenomenon which seriously affects the service reliability and efficiency of the bus route. "Bunching" refers to the phenomenon that two or more buses arrive at a station at the same time or in a brief periodE23. Bunching causes many adverse effects on the bus route operation. First of all, bunching will lead to the change of the time headway between buses, which results in no bus arrival at some stations for a long time, lengthening the passengers waiting time and travel time; secondly, bunching will cause the seat occupancy imbalance on buses, decreasing the bus service level, wasting limited resources, and affecting passengers' travel experience. In addition, in the long run, bunching will affect passengers' perception of the bus route service reliability and thus impede the sustainable development of the ground bus. In one word, bunching makes the ground bus fail to provide fast, convenient and stable services, which has caused considerable losses to travelers, managers and the society. In order to solve the problem of bunching in the bus route, control strategies adopted in most traditional methods include holding buses at stops, skipping stops, and restricting passengers getting on and off, etc.2,,4, which, however, will seriously affect passengers' satisfaction and will be difficult to implement in the practical operation. References
[1] Daganzo C F, Pilachowski J. Reducing bunching with bus-to-bus cooperation. Transportation Research Part B Methodological, 2011, 45(1): 267-277.
[2] Wang J, Sun L. Dynamic holding control to avoid bus bunching: A multi-agent deep reinforcement learning framework. Transportation Research Part C Emerging Technologies, 2020,116:102661.
[3] Sun A, Hickman M. The real-time stop-skipping problem. Journal of Intelligent Transportation Systems, 2005, 9(2): 91-109.
[4] Zhao S, Lu C, Liang S, Liu H. A self-adjusting method to resist bus bunching based on boarding limits. Mathematical Problems in Engineering, 2016, 746: 1-7.
SUMMARY Against current serious bus bunching which reduces service efficiency, a technical scheme provided in the present disclosure is: to provide a method for optimizing bus bunching based on the combinatorial scheduling of whole-zone buses and inter-zone buses by introducing an inter-zone bus scheduling mode to change the existing single whole-zone bus scheduling mode. As for a single bus route, the distribution of transport capacity on different road segments is adjusted by adopting the method of combinatorial scheduling of the whole-zone bus and the inter-zone bus, so as to improve the service reliability and efficiency of bus routes, which is
significantly advantageous in reducing times of bus bunching. A technical scheme provided in the present disclosure is: a method for optimizing bus bunching based on the combinatorial scheduling of whole-zone buses and inter-zone buses, which only changes some of the whole-zone bus runs into inter-zone runs without changing the existing number of bus runs, so as to reduce bunching by the combinatorial scheduling of the whole-zone bus and the inter-zone bus; analyzing the historical IC card data and GPS data of the bus route to determine the departure station and the number of runs for the inter-zone bus; creating a mathematical model for the departure station of the inter-zone bus which is intended to minimize the deviation between the departure time and the scheduled time of each bus run; and transforming the created model into an MILP (Mixed-Integer Linear Programming) for solution by the branch and bound algorithm, so as to acquire a combinatorial scheduling scheme for the whole-zone and inter-zone buses. The method is implemented by the following steps: (1) To collect the historical IC card data and GPS data of the bus route, analyze and screen the data to select road segments susceptible to congestion and bus stops with a few passengers, and then determine the departure station and the number of the inter-zone bus runs from the perspective of improving the operation efficiency and meeting the passenger demand. (2) After determining the departure station and the number of inter-zone bus runs, to specify the goal of the combinatorial scheduling of the whole-zone and inter-zone buses and corresponding constraints thereof; for the departure station of the inter-zone bus, to target the minimization of the deviation between the departure time and the scheduled time of each run, for which it is necessary to meet the constraints such as the number limit of inter-zone bus runs, and that two adjacent whole-zone bus runs cannot be both changed into inter-zone ones. (3) After specifying the optimization target and constraints, to create a mathematical model as follows: Symbol System: K is the set of all departure runs, i, k E K; k* is the number of inter-zone bus runs; ni is the departure station of the inter-zone bus runs; an, is the scheduled time that run i arrives at station n1 ; okn, is the actual time that run k arrives at station n, ;
Xk,, is the decision variable representing the optimized time run k departs from station
n;
Vk,, is the decision variable which equals to 1 if run k is changed to an inter-zone bus
run; otherwise 0;
Yk, is the decision variable which equals to 1 if run k is changed to run i at station n; otherwise 0;
For the departure station of the inter-zone bus, the target is to minimize the deviation between the departure time and the scheduled time of each run as follows: K K min I xn, - ai, Yki k=1 i=1
For ensuring basic services of the bus route, two adjacent whole-zone bus runs cannot be both changed into inter-zone ones, namely:
v, + Vk, 1, k =1, 2,A , K -1.
In conclusion, a mathematical model is created as follows: K K min - ai, Yki k=1 i=1 K s.t. Yvkn = K* k-1
Vk 1+ Vk,,+1 1, k =1, 2,A, K -1, (Ok, - Xk,, X(-Vkn, )=0, k =1, 2,A K,
YY, = 1, i=1, 2,A , K,
yki = 1, k = 1, 2,A , K,
Xk, >0, k =1, 2,A ,K, Vk, E{0,1}, k =1, 2,A , K,
Yk E{0,1}, k,i=1, 2,A ,K. (4) The established mathematical model is a nonlinear programming because the objective function and the third constraint are nonlinear. In order to solve the model, the objective function and the third constraint are transformed into a linear form to obtain an MILP (Mixed-Integer Linear Programming) as follows: Step 1: Linearize the objective function of the created mathematical model by introducing
a class of dummy variables Zki, k, i =1, 2,A , K, so that:
Xk,, -a,,, YkiZki, k,i=1, 2,A , K
which is equivalent to:
- Zki - M (1- Yk ) Xk, -ain, zki + M (1- Yk, ), k,i =1, 2,A , K
wherein M is a very large positive number. Therefore, the objective function of the created mathematical model is equivalent to:
min Zk. S.t. -Z1-M(1-Yk, Xk,, -ain , Zki +M(1- Yk), k,i =1, 2,A ,K.
Step 2: Transform the third constraint of the created mathematical model into a linear one by introducing a very large positive number M', so that the third constraint is equivalent to:
- M' vk,! Okn, - xk,, !! M'v ,k,Ik = 1, 2,A , K.
Step 3: Equivalently transform the created mathematical model into the MILP as follows: K K
min Y zki k=1 i=1 S.t. -Zk -M(1- yk ) xk,, - ai, Zi +M(1- yk), + k,i =1, 2,A ,K, K
k=1 Yvkn = K*,
Vk, +Vk,, 1, k =1, 2,A ,K -1, -M'vk, !k, - xk, Xk M'vkn, k =1, 2,A ,K,
YYk =1, i=1, 2,A ,K,
yk, = 1, k =1, 2,A , K,
Xk, 0, k =1, 2,A ,K, Vk, E {0,1}, k =1, 2,A K, ykE {0,1}, k9i = 1, 2,A , K, Zk,0 k,i=1, 2,A ,K. (5) Solve the resultant MILP model by a branch and bound algorithm to acquire a combinatorial scheduling scheme for the whole-zone and inter-zone buses, including which whole-zone bus runs should be changed to the inter-zone runs, and a departure time of each run of the adjusted whole-zone and inter-zone buses as follows: (51) Solve the transformed MILP model by the branch and bound algorithm to obtain the values of decision variables vk,, k=1,2,A, K and x k = 1, 2,A , K.
(52) If v,, =1, run k should be changed to an inter-zone bus run and depart from
departure station ni of the inter-zone bus at the departure time of xk, ; and if vkn, =0,run k
should be a whole-zone bus run as it is and depart from the departure station of the bus route at the departure time of xkl.
In comparison to the prior art, the present disclosure provides the following beneficial effects: (1) Considering the complexity of actual road conditions, the prior method proposed for bus bunching optimization cannot be applied in practice. Without changing the existing number of bus runs, the present disclosure changes some of the whole-zone bus runs into inter-zone runs to address the bunching problem by the combinatorial scheduling of the whole-zone bus and the inter-zone bus, which is of practical significance. (2) The method can minimize the deviation between the departure time and the scheduled time of each run at the departure station of the inter-zone bus by deciding the departure time of each inter-bus run, and can reduce the change to the existing bus route schedule as much as possible. (3) The present disclosure provides a method for optimizing bus bunching based on the combinatorial scheduling of whole-zone bus and inter-zone bus, wherein some of the whole-zone bus runs in a bus route are changed into inter-zone runs to skip the road segment susceptible to congestion and with a small passenger traffic, thereby reducing the times of bus bunching. The method pertains to the technical field of urban public transport, in which the historical IC card data and GPS data are utilized to determine the departure station and the number of runs of the inter-zone bus, and a mathematical model is created for the departure station of the inter-zone bus which is intended to minimize the deviation between the departure time and the scheduled time of each bus run so as to acquire a combinatorial scheduling scheme for the whole-zone and inter-zone buses, as well as a departure time of each inter-zone bus run. For the created mathematical model, a method of transforming the model into an MILP (Mixed-Integer Linear Programming) is designed and in turn solved by using the branch and bound algorithm to acquire a global optimal solution of the model, thus preventing the problem that the global optimal solution cannot be ensured when the nonlinear model is directly solved. (4) In the context of an increasingly severe problem of bus bunching in urban areas, the distribution of transport capacity on different road segments is adjusted by adopting the method of combinatorial scheduling of the whole-zone bus and the inter-zone bus, so as to improve the service reliability and efficiency of bus routes. When all runs are whole-zone bus runs, intervals between vehicles in the second half of the bus route will become non-uniform, causing bunching among some vehicles; after changing some of the whole-zone bus runs to inter-zone ones, intervals between the buses in the second half of the bus route will remain relatively uniform, which is not susceptible to bunching. Therefore, this present disclosure is advantageous in reducing the times of bus bunching.
BRIEF DESCRIPTION OF THE DRAWINGS Fig. 1 is a schematic diagram showing bus bunching; Fig. 2 is a schematic diagram of the whole-zone bus scheduling mode and the inter-zone bus scheduling mode; Fig. 3 is a schematic diagram of bus operation in the whole-zone bus scheduling mode; Fig. 4 is an implementation flow chart of the method in the present disclosure;
Fig. 5 is a schematic diagram of determining the departure station of the inter-zone bus; Fig. 6 is a schematic diagram of bus operation in the combinatorial scheduling mode of the whole-zone bus and the inter-zone bus.
DETAILED DESCRIPTION The present disclosure will be further described with reference to embodiments below. Nowadays, bus routes generally use the single scheduling mode of the whole-zone bus that departs from the first station and returns after arriving at the terminal, as shown in Fig. 2. Being subjected to some congested road segments, buses that depart at uniform intervals become non-uniform in interval during driving, and will run into bunching in severe cases, as shown in Fig. 3. In view of this situation, the present disclosure introduces the inter-zone bus scheduling mode into the bus route to address the problem of "bunching" during the bus driving. As shown in Fig. 2, the inter-zone bus scheduling mode refers to that the bus departs from the first station, turns around and returns after arriving at a certain intermediate station, or directly departs from the intermediate station to the terminal. As shown in Fig. 4, without changing the existing number of bus runs, the present disclosure changes some of the whole-zone bus runs into inter-zone ones and proposes a mixed scheduling mode of the whole-zone bus and the inter-zone bus. First, as shown in Fig. 5, the historical IC card swiping data and GPS data of the bus route are utilized to screen and select road segments susceptible to congestion and bus stops with a small passenger traffic, and then to determine the departure station and the number of the inter-zone bus runs. Then, a mathematical model is created for the departure station of the inter-zone bus which is intended to minimize the deviation between the departure time and the scheduled time of each bus run. The created model is transformed into an MILP (Mixed-Integer Linear Programming) for solution by the branch and bound algorithm, so as to acquire a combinatorial scheduling scheme for the whole-zone and inter-zone buses and the departure time of each inter-zone bus. As shown in Fig. 6, intervals between buses becomes relatively uniform through the mixed scheduling of the inter-zone bus and the whole-zone bus, which effectively reduces potential "bunching". Given a specific bus route with 49 stops, a total of 22 runs depart during the time period from 17: 25 to 22: 25 at an interval of 8 minutes between every two adjacent runs. However, during the actual operation, as the road from the 4th stop to the 15th stop belongs to segments susceptible to congestion, all the bus runs actually arrive at the 15th stop at very nonuniform intervals (as shown in Table 1), with very small arrival intervals between the 10th, 11th, and 12th runs, and between the 20th, 21st and 22nd runs, thereby resulting in the occurrence of bunching.
Table 1 Bus Operation Schedule before Optimization
Run Stop 1 Stop 4 Stop 15 Stop 49 Run Stop 1 Stop 4 Stop 15 Stop 49 1 17:25 17:32 18:16 19:26 12 18:53 19:00 19:58 21:08 2 17:33 17:40 18:28 19:38 13 19:01 19:08 20:03 21:13 3 17:41 17:48 18:42 19:52 14 19:09 19:16 20:08 21:18 4 17:49 17:56 19:05 20:15 15 19:17 19:24 20:15 21:25 5 17:57 18:04 19:11 20:21 16 19:25 19:32 20:26 21:36 6 18:05 18:12 19:28 20:38 17 19:33 19:40 20:33 21:43 7 18:13 18:20 19:41 20:51 18 19:41 19:48 20:42 21:52 8 18:21 18:28 19:44 20:54 19 19:49 19:56 20:53 22:03 9 18:29 18:36 19:49 20:59 20 19:57 20:04 21:03 22:13 10 18:37 18:44 19:56 21:06 21 20:05 20:12 21:04 22:14 11 18:45 18:52 19:55 21:05 22 20:13 20:20 21:04 22:14
For the above bus route, without changing the existing number of bus runs, the present disclosure changes some of the whole-zone bus runs into inter-zone runs to reduce bunching. The specific steps are as follows: (1) Analyzing the historical IC card data and GPS data of the bus route to determine the departure station and the number of runs for the inter-zone bus; for this bus route, analyzing the historical IC card data and GPS data to set the departure station of the inter-zone bus as the 15 stop, the terminal as the 49 stop, and the number of the inter-zone bus runs as 6; (2) For the departure station of the inter-zone bus, targeting the minimization of the deviation between the departure time and the scheduled time of each run, for which it is necessary to meet the constraints such as the number limit of inter-zone bus runs, and that two adjacent whole-zone bus runs cannot be both changed into inter-zone ones, with a mathematical model to be created as follows:
22 22 min x -a,, Yki k=I i=1 22 s.t. v 6, k-I
Vk +Vk 1, k=1,2,A ,21,
(Oky, - -Xk,, V1,1 0, k=1, 2,A 22, 22
Yk =1 k =1, 2,A ,22,
xk,, 0, k =1, 2,A ,22,
v e {0,1}, k =1, 2,A ,22, Yke {0,1}, k,i=1, 2,A ,22.
(3) Linearizing the objective function of the created mathematical model by introducing a
class of dummy variables Zki, k,i =1, 2,A ,22, so that:
x,,, -a,,,Yki Zki, k,i=1, 2,A ,22
which is equivalent to:
- Zk - M(1)- y,) xk,, - a,,, zk, + M(1- Yk,), k,i=1, 2,A ,22,
wherein M is a very large positive number. Therefore, the objective function of the created mathematical model is equivalent to:
mi 'k s.t. -Zk - M(1-yk ) Xk,, - ai,, Zki M(1- Yki), k,i =1, 2,A ,22.
(4) Transforming the third constraint of the created mathematical model into a linear one by introducing a very large positive number M', so that the third constraint is equivalent to:
- M', , :! 0kn, - xkn, " M' vkn,, Ik = 1, 2,A ,22.
(5) Equivalently transforming the created mathematical model into the MILP as follows:
22 22
min YYzki k=1 i=1 S.t. -Z - M(1-yk ) Xk,, -ain, 1 Zki + M(1- Yki), k,i =1, 2,A ,22, K
YVkn, =6, k=1
Vk, +Vkn,,1 1, k=1,2,A ,21, - M'vkn O 0k,, - x , M'vk, , k =1, 2,A ,22,
YY, = 1, i = 1, 2,A ,22,
Yk, =1, k =1, 2,A ,22,
Xk, 0, k =1, 2,A ,22, Vk, E0,1, k = 1, 2,A ,22, yk E 10,11, k, i = 1, 2,A ,22, Zki0, k,i=1, 2,A ,22. (6) Solving the transformed model by the branch and bound algorithm. According to the results, the 6 original whole-zone bus runs including the 7th, 9th, 11thq13th, 20th and 22nd runs should be changed to inter-zone bus runs, and the departure times of the 6 inter-zone bus runs from the 15th stop should be 18:24, 18:48, 18:56, 19:20, 19:36, and 20:48 respectively, as shown in Table 2. On the optimized schedule, the departure intervals from the 15th stop are very uniform, indicating that the effectiveness of the present disclosure in addressing the bus bunching problem. Table 2 Bus Operation Schedule after Optimization Run Stop 1 Stop 4 Stop 15 Stop 49 Run Stop 1 Stop 4 Stop 15 Stop 49 1 17:25 17:32 18:16 19:26 12 18:53 19:00 19:58 21:08 2 17:33 17:40 18:28 19:38 13 - - 18:48 19:58 3 17:41 17:48 18:42 19:52 14 19:09 19:16 20:08 21:18 4 17:49 17:56 19:05 20:15 15 19:17 19:24 20:15 21:25 5 17:57 18:04 19:11 20:21 16 19:25 19:32 20:26 21:36 6 18:05 18:12 19:28 20:38 17 19:33 19:40 20:33 21:43 7 - - 20:48 21:58 18 19:41 19:48 20:42 21:52 8 18:21 18:28 19:44 20:54 19 19:49 19:56 20:53 22:03 9 - - 18:24 19:34 20 - - 19:36 20:46 10 18:37 18:44 19:56 21:06 21 20:05 20:12 21:04 22:14 11 - - 19:20 20:30 22 - - 18:56 20:06
The above calculation examples illustrate the effectiveness and advantages of the present disclosure, and it is concluded that the combinatorial scheduling of the whole-zone bus and the inter-zone bus proposed by the present disclosure can reduce the deviation between the departure time and the scheduled time of each run at the departure station of the inter-zone bus, thereby effectively reducing the times of bunching in the bus route.
Claims (3)
1. A method for optimizing bus bunching based on combinatorial scheduling of a whole-zone bus and an inter-zone bus, comprising:
Step 1, collecting historical IC card data and GPS data of a bus route, analyzing and screening the data to select road segments susceptible to congestion and bus stops where a number of passengers getting on and off is less than a threshold Num, and then determining a departure station and a number of inter-zone bus runs from the perspective of improving the operation efficiency and meeting the passenger demand;
Step 2, based on the determined departure station and the number of inter-zone bus runs, specifying the goal of the combinatorial scheduling of the whole-zone and inter-zone buses and corresponding constraints thereof; for the departure station of the inter-zone bus, targeting the minimization of the deviation between the departure time and the scheduled time of each run, for which it is necessary to meet the constraints such as the number limit of inter-zone bus runs, and that two adjacent whole-zone bus runs cannot be both changed into inter-zone ones;
Step 3, according to the optimization target and constraints, creating a mathematical model as follows:
K K min x -ai, Yki k=1 i=1 K S.t. v =K k=1
Vk +Vk 1, k =1, 2,A ,K -1, (Ok, - Xk, -Vk, )= 0, k =1, 2,A ,K,
YY, = 1, i=1, 2,A , K,
Yki =1, k =1, 2,A ,K,
Xk, >0, k =1, 2,A , K, v E {,1}, k =1, 2,A , K, ykE {0,1}, k, i = 1, 2,A , K.
wherein:
K is the set of all departure runs, i, kE K;
k* is the number of inter-zone bus runs; ni is the departure station of the inter-zone bus, n, E N ; a, is the scheduled time that run i arrives at station n1;
Okn, is the actual time that run k arrives at station n1;
Xk,, is the decision variable representing the optimized time run k departs from station
n 1;
Vk,, is the decision variable which equals to 1 if run k is changed to an inter-zone bus run;
otherwise 0;
Yki is the decision variable which equals to 1 if run k is changed to run i at station n1 ;
otherwise 0;
For the departure station of the inter-zone bus, the target is to minimize the deviation between the departure time and the scheduled time of each run as follows:
K K min x1 -- ain, Yk; k=1 i=1
For ensuring basic services of the bus route, two adjacent whole-zone bus runs cannot be both changed into inter-zone ones, namely:
vkn, + vkn +1:! 1, k = 1, 2,A , K -1I;
Step 4, solving the mathematical model by transforming the objective function and the third constraint into a linear form to obtain an MILP (Mixed-Integer Linear Programming);
Step 5, solving the resultant MILP model by a branch and bound algorithm to acquire a combinatorial scheduling scheme for the whole-zone and inter-zone buses, namely which whole-zone bus runs should be changed to the inter-zone runs, and the departure time of each run of the adjusted whole-zone and inter-zone buses.
2. The method for optimizing bus bunching based on the combinatorial scheduling of whole-zone buses and inter-zone buses according to claim 1, wherein: in Step 4, the mathematical model is solved as follows:
(1) Linearizing the objective function of the created mathematical model by introducing a
class of dummy variables Zki, k,i =1, 2,A , K , so that:
xk, -- ai,, ), Yi :zk, k9 i =1, 2,A , K , which is equivalent to:
- Zki - M (1- yk,) Xk,, -ain, ! Zki +M (1 - Yk) k,i =1, 2,A , K
wherein M =maxlai,, ie K f-mintain,,iE , the objective function of the created
mathematical model is equivalent to:
min Zki
.t. -Zk-i M(1-yk,) Xk,, -ain, 1 i z +AM(1- yki), k,i =1, 2,A, K
(2) Transforming the third constraint of the created mathematical model into a linear one by introducing a positive number M'= maxOkl,, an, , k, i E- KJ- mink, , a,,, k, i E- KI so that the
third constraint is equivalent to:
- M' vkn :! 0kn, - xkn, " M' vkn,, Ik = 1, 2,A ,K.
(3) Equivalently transforming the created mathematical model into the MILP as follows:
K K min YYzki k=1 i=1 s.t. - Zi- M(I1- yki )' xkn, - ain, : Zki + M (1 - yki ), k, i =1, 2,A ,K, K Vkl = K* k=1
Vk, +Vk In 1, k =1, 2,A , K -1, - M'vk, O, - x,, k M'vk,, k =1, 2,A , K,
YY, = 1, i =1, 2,A , K,
Yk =1, k =1, 2,A ,K,
Xk, 0, k =1, 2,A ,K, Vk, E {0,1}, k =1, 2,A ,K, ykE {0,1}, k9 i= 1,2,A , K, Zk,0 k,i=1, 2,A ,K.
wherein:
K is the set of all departure runs, ikEK.
k* is the number of inter-zone bus runs;
n1 is the departure station of the inter-zone bus , ni E N.
ai", is the scheduled time that run i arrives at station n1;
Okn, is the actual time that run k arrives at station n;
M is the positive number;
M' is the positive number;
x,,, is the decision variable representing the optimized time run k departs from station
n;
Vk,, is the decision variable which equals to 1 if run k is changed to an inter-zone bus run;
otherwise 0;
Yki is the decision variable which equals to 1 if run k is changed to run i at station n1 ; otherwise 0;
Zki is the auxiliary decision variable.
3. The method for optimizing bus bunching based on the combinatorial scheduling of whole-zone buses and inter-zone buses according to claim 1, wherein: in Step 5, the resultant MILP model is solved by using a branch and bound algorithm to acquire a combinatorial scheduling scheme for the whole-zone and inter-zone buses, comprising which whole-zone bus runs should be changed to the inter-zone runs and the departure time of each run of the adjusted whole-zone and inter-zone buses as follows:
(1) Solving the transformed MILP model by the branch and bound algorithm to obtain the values of decision variables vkl,, k=1,2,A, K and x k =1, 2,A , K;
(2) If vk,, =1, run k should be changed to an inter-zone bus run and depart from
departure station ni of the inter-zone bus at the departure time of x k, ; and If vkn, =0, run k
should be a whole-zone bus run as it is and depart from the departure station of the bus route at the departure time of Xkn
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