Schedule determination of a multiple route transit system
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The document describes the problem of determining an optimal schedule for a multiple route transit system to minimize passenger wait times. It presents a mathematical formulation of the problem as a mixed integer nonlinear program that minimizes total transfer time and waiting time. The objective function and some constraints are nonlinear. The problem is solved using genetic algorithms, which represent schedules as strings that are evolved over generations using selection, crossover and mutation operators. An example application for a network with 3 routes and a single transfer station is also presented.
Schedule determination of a multiple route transit system
- 1. Schedule Determination of a Multiple- Route Transit System By Pranamesh Chakraborty Sharath M N Department of Civil Engineering Indian Institute of Technology Kanpur India 1st of March, 2013
- 2. Introduction Transit system plays a very important role in handling the high travel demand and simultaneously reducing the problem of congestion and pollution in any urban area. A public transport system can be made efficient by scheduling it in such a manner so that the waiting time for passengers as well as transfer time from one route to another is minimized.
- 3. Problem Statement Route 1 Route 2 Route 4 Route 3 Route 6 Route 5 Fig 1. Typical Transit Network The solid lines represent the routes and the circled intersections represent transfer stations The problem is to determine the schedule such that waiting time for the passengers is minimized which includes Initial Waiting Time (IWT) as well as Transfer Time (TT).
- 4. Mathematical Formulation The objective function is ni nj aik,m aik,1 m m i ik, ,jl,m (d lj ,m aik,m )ik, j ,m j k 1 l 1 m i l vi ,k ,m (t )(aik,m aik,m1 t )dt 0 First term represents the total transfer time while the second term represents the total waiting time. aik,m : Arrival time of k th transit unit of i th route at m th transfer stop dik,m : Departure time of k th transit unit of the i th route from m th transfer stop vi ,k ,m : Arrival pattern of pasengers for k th transit unit of i th route at m th transfer stop ik, ,jl,m : A binary variable that takes the value of 1 if transfer from k th transit unit of i th route to lth transit unit of jth route at m th transfer point is ideal to passengers; 0 otherwise ik, j ,m : Number of passengers transferring from k th transit unit of i th route to jth route at m th transfer point
- 5. Constraints (Chakroborty, P., Das, A .,2003) The transit unit should stop for a certain minimum period as physical transfer of passengers takes some time and at the same time, the transit unit cannot stop at a stop for a very long time. The constraints g1 and g2 will take care of these aspects. Constraint g3 states that transferring of passengers cannot happen to transit units that have departed the transfer stations before the arrival of passenger at that transfer point. A person can transfer to only one transit unit and it is dictated by the constraint g4. . The time headway between any two successive vehicles must be positive and must be less than the policy headway. It is governed by constraints g5 and g6. No transferring passenger shall wait for more than a certain period of time and it is being dictated by g7. The arrival time of a transit unit is dependent on arrival time of a transit unit at a stop previous to transfer stop and also on the travel time between those two stops; constraints g8 and g9 state this. The constraint g10 states the minimum fleet size for each route while g11 states that the total fleet size of all the routes is to be equal to N.
- 6. Constraints g1 dik,m aik,m Simax i, m and k 1, 2,......., ni g 2 dik,m aik,m Simin i, m and k 1, 2,......., ni g3 d k,m aik,m M (1 ik, ,jl, m ) 0 j i, m, j i and k 1, 2,......., ni and l 1, 2,......., n j g 4 ik, ,jl,m 1 i, m, j i and k 1, 2,......., ni and l 1, 2,......., n j l g5 aik,m1 aik,m H i i, m and k 1, 2,......., ni g 6 aik,m1 aik,m 0 i, m and k 1, 2,......., ni g 7 (d lj ,m aik,m ) ik, ,jl,m T i, m, j i and k 1, 2,......., ni and l 1, 2, ......., n j g8 aik,m dik,m 1 timax 1,m ,m i, m and k 1, 2,......., ni g9 aik,m dik,m 1 timin1,m ,m i, m and k 1, 2,......., ni g10 n i nimin i g11 n i i N
- 7. where H i : Policy headway of i th route M : An arbitrary large number Simin : The minimum period for which the transit unit must stop at any stop on i th route Simax : The maximum period for which the transit unit can stop at any stop on i th route timinm2 : Minimum travel time of transit unit of i th route from stop m1 to stop m 2 , m1 timaxm2 : Maximum travel time of transit unit of i th route from stop m1 to stop m 2 , m1 T : Maximum transfer time for any transferring passenger N : Total fleet size n i : Fleet size of ith route
- 8. Solution Technique The scheduling problem formulated is a mixed integer nonlinear programming problem. The objective function and the constraint g7 are nonlinear whereas the variable is a 0-1 binary variable and the other variables are real. It is difficult to solve such a formulation problem using traditional optimization techniques. Hence Genetic Algorithms (GA) are used to solve such problems.
- 9. Genetic Algorithms Genetic algorithms are optimization algorithms based on the principles of natural genetics and natural selection. GAs are generally used to solve maximization problems. For minimization problems, the objective function (referred as Fitness function Ƒ(x) in GA problems) can be transformed into an equivalent function to be maximized. .
- 10. Genetic Algorithms The design variables are first coded randomly in string structures. The length of the string depends on desired level of accuracy. Then the fitness value of each string is evaluated. The population is then operated by three main operators- reproduction, crossover and mutation thus forming a new population set.
- 11. Genetic Algorithms Reproduction The good strings are selected and reproduced with a probability (pi) proportional to its fitness value. Crossover Two strings are chosen at random from the population and some portion of the strings is exchanged between the pair. The crossover is done with a small probability of pc (i.e. 100pc % of the strings are used in crossover). Mutation The mutation operator changes randomly a particular bit of a string from 1 to 0 or vice versa
- 12. Revised Formulation (considering only one transfer stop) k The new decision variables hik , si and ni are introduced where, hik = Headway between the kth and the (k—l)th bus of the ith route; sik = Stopping time of the kth bus of the ith route; The bounds for the variable sik is provided by constraint g1 and g2. The upper bound for hik is provided by g5, lower bound by g6. The k ,l variables is also not required because the schedule i, j encoded in a string is known and thus it is simple to find which transfers were made. Thus constraints g3 and g4 are also eliminated. The constraint g11 is used to eliminate one of the ni variables. Thus we are left with only g7. This revised formulation can be used to determine the optimal solution using Penalty methods in satisfying the constraints (Chakroborty, Deb, & Sharma, 2001).
- 13. String A string is concatenation of several substrings There will be as many substrings as number of variables Number of bits in a string is governed by level of accuracy required The upper bound and lower bound of substrings can be defined
- 14. String n1 is the first substring s1 is the second substring k The third substring is h1 k si = s for all k i k Number of h1 variables is n1max 1 r is the number of routes No substring for n r is required
- 15. Evaluation of Strings k ni , si and hi strings are decoded ni is checked for its range k Only first ni number of hi are decoded and rest are ignored Last head way for each route is computed based on sum of all headway which should be equal to scheduling period and checked if it is in permissible range Thus the entire schedule can be determined The objective function and rest of constraints can be calculated If a constraint is not satisfied, a heavy penalty is assigned to the string
- 16. Example (Chakroborty, Deb, & Sharma, 2001) Scheduling is done for a single transfer station Number of routes, r=3 Total fleet size, N=30 Scheduling time period, H=240 minutes Minimum fleet size=5 Policy headway is assumed to be proportional to demand of total demand Percentage Policy headway (mins) 5-15 56 15-25 47 25-45 40 45-60 35 >60 31
- 17. Example (contd.) Minimum stopping time=2mins Maximum stopping time=5mins Headway range=32mins Maximum transfer time=30mins ik, j=300, for all routes Arrival pattern is triangular with =0.8
- 18. Example (contd.) GA parameters Population size =800 Maximum number of generations =800 Cross over probability = 0.95 Mutation probability = 0.005 Variable Number of bits Fleet size 4 Headway 5 Stopping time 2
- 19. Example (contd.) Objective function with only IWT term Case 1 Equal Demand on all routes Theoretical optimum fleet size for each route is 10 Theoretical global optimum objective function value=3456mins Source: (Chakroborty, Deb, & Sharma, 2001)
- 20. Example (contd.) Case 2 Demand ratio among routes=1:4:9 Theoretical fleet size allocation should be in the ratio 1:2:3 Optimum objective function value=13825mins Source: (Chakroborty, Deb, & Sharma, 2001)
- 21. Example (contd.) Objective function with both IWT and TT This problem can not be solved by using traditional optimisation techniques Case 1 Demand ratio is 1:4:4 Source: (Chakroborty, Deb, & Sharma, 2001)
- 22. Example (contd.) Case 2 Demand ratio 1:4:9 Source: (Chakroborty, Deb, & Sharma, 2001)
- 23. Example (contd.) Comparison for Demand ratio 1:4:4 Variables IWT+TT IWT only Fleet size distribution 8:11:11 7:12:11 Number of matches 8 1 Comparison for Demand ratio 1:4:9 Variables IWT+TT IWT only Fleet size distribution 7:11:12 6:10:14 Number of matches 7 2
- 24. References Chakroborty P, Das. A. (2003). Principles of Transportation Engineering. New Delhi: Prentice Hall of India. Chakroborty, P., Deb, K., & Sharma, R. K. (2001). Optimal fleet size distribution and scheduling of transit systems using genetic algorithms. Transportation Planning and Technology , 24.3, 209-225. Chakroborty, P., Deb, K., & Subrahmanyam, P. S. (1995). Optimal Scheduling of Urban Transit Systems using Genetic Algoritms. Journal of Transportation Engineering , 544-553. Deb, K. (2006). Optimization for Engineering Design. New Delhi, India: Prentice Hall of India Private Limited.
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