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© International Journal of Combinatorial Optimization Problems and Informatics, Vol. 2, No. 1, Jan-April 2011, pp. 23-26. ISSN: 2007-1558.

School Bus Routing Problem Library-SBRPLIB Ocotlán Díaz-Parra1, Jorge A. Ruiz-Vanoye2, José C. Zavala-Díaz1 1 Universidad Autónoma del Estado de Morelos, Mexico. [email protected] 2 Universidad Popular Autónoma del Estado de Puebla, Mexico.

Abstract. This paper presents a set of test instances of the School Bus Routing Problem or SBRP. The test instances were created using an algorithm called SBRPGen. The depository of instances can be downloaded for others researchers for experimentation. Keywords: SBRPLIB, School Bus Routing Problem, SBRP.

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Introduction

School bus routing and scheduling is a transportation area with great impact on systems by improving service quality and reducing operating costs. The school bus routing problem consist of a set of students dispersed in a region needs travel to their schools every day. It´s necessary to assign students to bus stops, determine the bus stops locations, and route and schedule the buses to minimize the total operating cost while satisfying all requirements of the school. This paper presents a set of test instances of the School Bus Routing Problem by the School Bus Routing Problem Library or SBRPLIB.

2 School Bus Routing Problem The School Bus Routing Problem or SBRP is a significant problem in the management of school bus fleet for the transportation of students, each student must be assigned to a particular bus which must be routed in a efficient manner so as to pick up (or return home) each of these students [1]. The mathematical model of SBRP [2] is formed by the equations1-8: n

n

M

min z = ∑∑∑ Cij X ijk

(1)

i = 0 j = 0 k =1

M

k =1 j = 0

M

= 1 j = 1, 2,..., n

(3)

ijk

n

∑∑ X k =1 i = 0

M

= 1 i = 1, 2,..., n

(2)

ijk

n

∑∑ X

n

∑∑ X k =1 j = 0

M

0 jk

Received July 10, 2010 / Accepted Nov 12, 2010

n

= ∑∑ X i 0 k = M k =1 i = 0

(4)

Díaz-Parra et al. / School Bus Routing Problem Library-SBRPLIB. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 23-26 n

∑X j=0

(5)

n

ijk

= ∑ X jik ; i = 1, 2,..., n; k = 1, 2,..., M j =0

U ik + U jk + ( n − m + 1) X ijk ≤ ( n − M ) ;

(6)

1 ≤ i, j ≤ n, i ≠ j , k = 1, 2,..., M n

n

∑∑ X i =0 j = 0

n

ijk

n

∑∑ X i =1 j = 0

ijk

qi ≤ Q; k = 1, 2,..., M

(7)

gtij ≤ τ ; k = 1, 2,..., M

(8)

Where: School buses are centrally located and have collect waiting students at n pick-up points and to drive them to school. The number of students that wait in pick-up point i is qi , (qi > 0, i = 1, 2, …, n). The capacity of each bus is limited to Q students (qi ≤ Q). The objective function to the School Bus Problem is composing of two costs: a) cost incurred by the number of buses used, b) driving cost (fuel, maintenance, drivers salary, and others), subject to operational constraints, Cost a or b have to be minimized. The characteristics of SBRP [3] are: Number of School (single or multiple), surroundings of service (urban or rural), Problem scope (morning, afternoon, both), Mixed Load (allowed or no allowed), special-educations students (considered or not considered), Fleet mix (homogeneous fleet or heterogeneous fleet), Objectives (number of buses used, total bus travel distance or time, Total students riding distance or time, student walking distance, load balancing, maximum route length, Child´s time loss), Constraints (vehicle capacity, maximum riding time, school time windows, maximum walking time or distance, earliest pick-up time, minimum student number to create a route).

3 School Bus Routing Problem Library The School Bus Routing Problem Library-SBRPLIB is a depository of test instances of the School Bus Routing Problem. The depository of instances can be downloaded for others researchers for experimentation. In this section we present the parameters or characterization (Table 1) used for generating the SBRP instances.

Table 1. SBRP instances NS BS 0 … n

SS XCO x0 … xn

PS YCO Y0 … Yn

ML SN SN0 … SNn

SEE EPT EPT0 … EPTn

F DPT DPT0 … DPTn

VN MRT MRT0 … MRTn

VC MWT MWT0 … MWTn

Where NS: Number of School (single or multiple), SS: Surroundings of Service (urban or rural), PS: Problem scope (morning, afternoon, both), ML: Mixed Load (allowed or no allowed), SEE: special-educations students (considered or not considered), F: Fleet mix (homogeneous fleet or heterogeneous fleet), VN: Vehicle Number, VC: Vehicle Capacity, STWb: School Time Windows Begin, SWTdue: School Time Windows Due, BS: Bus Stop, XCO: X Coord., YCO: Y Coord., SN: Student Number, MRT: Maximum Riding time, EPT: earliest pick-up time, DPT: Due pick-up time, MWT: Maximum Walking Time or distance.

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Díaz-Parra et al. / School Bus Routing Problem Library-SBRPLIB. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 23-26

We generated 24 instance set for the School Bus Routing Problem (Table 2). Where NS: Number of School, SS: Surroundings of Service, PS: Problem scope, F: Fleet mix. The urban means routes inside the city, rural means routes outside the city, morning means 7:00-9:00, afternoon means 13:00-16:00, both means morning and afternoon, homogeneous fleet is the vehicle capacity equal in all the fleet, heterogeneous is the vehicle capacity different in all the fleet. Table 2. SBRP instances sets.

Instance Set S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

NS

SS

PS

F

1 1 1 1 1 1 1 1 1 1 1 1 2-700 2-700 2-700 2-700 2-700 2-700 2-700 2-700 2-700 2-700 2-700 2-700

Urban Urban Urban Urban Urban Urban Rural Rural Rural Rural Rural Rural Urban Urban Urban Urban Urban Urban Rural Rural Rural Rural Rural Rural

Morning Morning Afternoon Afternoon Both Both Morning Morning Afternoon Afternoon Both Both Morning Morning Afternoon Afternoon Both Both Morning Morning Afternoon Afternoon Both Both

Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous Homogeneous Heterogeneous

Instances number 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

The instances can be downloading from the SBRPLIB site. (http://diazparra.net/SBRPLIB.aspx).

4 Conclusions We plan to continuously extend the library with characteristics similar to the ones already presented. The extension depends on the progress made in the development of (meta) heuristic and exact procedures to the School Bus Routing Problem.

Acknowledgements The first author UAEMOR-PTC-231 would like to thank the support to SEP-PROMEP (Mexico) through grant PROMEP/103.5/10/4453.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13]

Newton, R.M.: Bus routing in a multi-school system, Computers & Operations Research, Vol. 1, No. 2 (1974) 213222. Gavish, B. and Shlifer, E.: An approach for solving a class of transportation scheduling problems. European Journal of Operational Research, Vol. 3, No. 2 (1979) 122-134. Park, J. and Kim, B-I.: The school bus routing problem: A review. European Journal of Operational Research, Vol. 202, No. 2 (2010) 311-319. Angel, R.D., Caudle, W.L., Noonan, R. and Whinston, A.: Computer-Assisted School Bus Scheduling. Management Science B, Vol. 18 (1972) 279-288. Newton, R.M. and Thomas, W.H.: Design of school Bus Routes by computer. Socio-economic planning science, Vol. 3 (1969) 75-85 Schittekat, P., Sevaux, M., Sorense, K. and Springael, J.: A metaheuristic for the School Bus Routing Problem. 22nd European Conference on Operational Research EURO XXII, Prague (2007). Bodin, L. and Berman, L.: Routing and scheduling of school buses by computer. Transportation Science, Vol. 13 (1979) 113-129. Bennet, B. and Gazis, D.: School Bus Routing by computer. Transportation Research, Vol. 6 (1972) 317-326. Desrosiers, J., Ferland, J., Rousseau, J.-M., Lapalme, G. and Chapleau, L.: An overview of a school busing system. Scientific Management of Transport Systems Vol. IX, International Conference on Transportation, New Delhi (1981) 235-243. Newton, R.M.: Bus routing in a multi-school system, Computers & Operations Research, Vol. 1, No. 2 (1974) 213222. Desrosiers, J., Ferland, J.A., Rousseau, J.-M., Lapalme, G. and Chapleau, L.: TRANSCOL: A multi-period school bus routing and scheduling systems. Management sciences, Vol. 22 (1986) 47-71. Swersey, A.J. and Ballard, W.: Scheduling School Buses. Management Science, Vol. 30 (1984) 844-853. Thangiah, S.R., Fergany, A., Wilson, B., Pitluga, A. and Mennel, W.: School Bus Routing in Rural Regions. M. Hickman, P. Mirchandani, and S. Voss (eds.), Computer-Aided Systems in Public Transport, Lecture Notes in Economics and Mathematical Systems 600, Computer-Aided Systems in Public Transport, Heidelberg: Springer. (2008).

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