0. Gavish, B. and Shlifer, E.: An approach for solving a class of transportation scheduling problems. European Journal of Operational Research, Vol. 3, No. 2. pp.122-134. 1979.

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.

1. Bowerman, R. et al. A multi-objective optimization approach to urban school bus routing: formulation and solution method. Transpn. Res.-A. Vol. 29A, No. 2, pp. 107-123, 1995.

The proposed technique first groups students into clusters using a multi-objective districting algorithm and then generates a school bus route and the bus stops for each cluster using a combination of a set covering algorithm and a traveling salesman problem algorithm. In school bus transportation services are provided through the public sector, the service must be evaluated by both efficiency and equity measures. The service involves considerations of efficiency (cost minimization) and user equity (fairness).

There are two main components to consider in the total cost of providing school bus transportation. One cost is the capital cost required to run one school bus for a school year. Components of this include payment of the bus driver as well as the costs of vehicle maintenance, purchasing and leasing. The other main cost is the incremental cost, or the cost of a school bus route per kilometre traveled.

Now, to improve equity is to load balance the routes serving an area so that each school bus route transports approximately an equivalent number of students. Other ways to improve equity are to length balance to avoid too large a variation in the route lengths of buses serving one school and to consider the trade-off between total student walking distance to bus stops and the length of the bus route with the objective of increasing public acceptability of a set of school bus routes.

This article considers the case of providing school bus transportation in urban areas (USBRP) and does not deal with school bus routing in rural areas. The USBRP, as just described, actually involves two interrelated problems. One problem is the assignment of students to their respective bus stops, and the

second problem is the routing of the bus to the bus stops. Problems with these characteristics are known as location-routing problems (LRP).

Optimization criteriaA number of optimization criteria are defined to evaluate the desirability of a particular set of school bus routes. From the preceding discussion, these criteria are as follows:1. Number of routes. Because the capital cost is significantly larger per bus than the incremental cost over the year, the number of routes generated should be held to a minimum.2. Total bus route length. This criterion reduces the total length of the school bus routes.3. Load balancing. Load balancing involves minimizing the variation in the number of students transported along each route.4. Length balancing. This criterion involves reducing the variation in route lengths.5. Student walking distance. This criterion balances the total distance that students walk from home to and from their bus stops against route length.Routing constraintsAs well as having objectives to evaluate the routes, several different constraints may be imposed on the school bus routes from board-specific policies and from the capacity of a school bus. These constraints include:1. An upper bound on the number of students on each route (bus capacity)2. An upper bound on the length (or travel time) on each route

2. LYO, Li and Z. Fu. The school bus routing problem: a case study. Journal of the Operational Research Society Vol. 53, pp. 552–558. 2002.

This paper describes a case study of the school bus routing problem. It is formulated as a multi-objective combinatorial optimization problem. The objectives considered include minimizing the total number of buses required, the total travel time spent by pupils at all pick-up points, which is what the school and parents are concerned with most, and the total bus travel time. It also aims at balancing the loads and travel times between buses.

Consider the following daily routine of school bus services in this real-world problem. In the morning a bus leaves its parking place (usually near the driver’s home), arrives at the first pick-up point where the route starts, and travels along a predetermined route, picking up the pupils at pick-up points and taking them to school. The morning problem will be considered. The afternoon problem is analogous, with minor modifications.

The school bus routing problem has received some attention in the last two decades. Of those mentioned in the operational research literature, each problem has its peculiarities and may also have different objectives and constraints (see Table 1).

3. Zong Woo Geem, Kang Seok Lee and Yongjin Park. Application of Harmony Search to Vehicle Routing. American Journal of Applied Sciences, Vol. 2. No. 12. pp.1552-1557. 2005.

A phenomenon-inspired meta-heuristic algorithm, harmony search, imitating music improvisation process, is introduced and applied to vehicle routing problem, then compared with one of the popular evolutionary algorithms, genetic algorithm. The harmony search algorithm conceptualized a group of

musicians together trying to search for better state of harmony. This algorithm was applied to a test traffic network composed of one bus depot, one school and ten bus stops with demand by commuting students. This school bus routing example is a multi-objective problem to minimize both the number of operating buses and the total travel time of all buses while satisfying bus capacity and time window constraints. Harmony search could find good solution within the reasonable amount of time and computation.

Harmony Search (HS) algorithm has been recently developed in an analogy with music improvisation process where musicians in an ensemble continue to polish their pitches in order to obtain better harmony. In order to demonstrate the searching ability of HS, it is applied to a school bus routing problem (SBRP) which is a practical optimization problem. From a school's perspective, the SBRP aims to provide students with an efficient and equitable transportation service.

The study network to be optimized consists of one bus depot, one school, and ten bus stops as shown in Fig. 6. Each bus stop is demanded by certain number of commuting students, and travel time (in minutes) between two stops is specified in the figure.

4. M.Fatih DEMİRAL, İbrahim GÜNGÖR, Kenan Oğuzhan ORUÇ. Optimization at service vehicle routing and a case study of Isparta, Turkey. 2007.

It is seen that SBRP has same characteristics with VRP in several ways; however, SBRP is different from VRP because of some properties. While a typical VRP mostly deals with the freight transportation, SBRP is related with student transportation. It can be said that the other differences are to provide human satisfaction, effectiveness while traveling. Also the service transportation should be executed with the public (students, parents, school board etc.). Because of those reasons, SBRP is more complicated problem thanVRP.

In this paper, SBRP is presented and as an application area, for the education year 2005-2006, Isparta Milli Piyango Anadolu High School was selected. The aim of the case study is to find optimal school bus routes for the selected school by using savings (time) algorithm. In this study, there is a feasible solution that how students are picked-up from their residencies and delivered to the school under the capacity and time constraints.

Application area, Isparta city center is divided into 37 sub-center areas with the 37 subcenter points. Sub-center points were determined, in general, at the intersections of the main roads on the city map of Isparta.

In the education year 2005-2006, there were 540 students attending to school and 255 of them took service transportation. Necessary datum for the problem was obtained with talking to school board and transportation company, “Sertur”. The datum is as follows:

1-Traveling time, approximately “30-40” minutes,2-Velocities of vehicles, for the normal and heavy traffic, “30 km/h (500 m/min.) and 50 km/s (833 m/min.) ”3-Capacity and other characteristics of vehicles (shown in Table 3.1. below)4-Number of students at stops (shown in Table 3.2.) and5-Addresses of students

5. Schittekat, Patrick et al. A metaheuristic for solving large instances of the School Bus Routing Problem. MIC 2007: The Seventh Metaheuristics International Conference. Montreal, Canada, June 25–29, 2007.

To efficiently solve large instances of the SBRP we develop an efficient GRASP (constructive heuristic based saving algorithm) +VND metaheuristic. Our method can be called a matheuristic because it uses an exact algorithm to optimally solve the subproblem of assigning students to stops and to routes. The results of our matheuristic approach on 112 artificially generated instances.

The basic SBRP is a generalization of the basic vehicle routing problem (VRP) and therefore also NP-hard.

6. Armin Fugenschuh. Solving a school bus scheduling problem with integer programming. European Journal of Operational Research Vol. 193. pp. 867–884. 2009.

In many rural areas in Germany pupils on the way to school are a large if not the largest group of customers in public transport. If all schools start more or less at the same time then the bus companies need a high number of vehicles to serve the customer peak in the morning rush hours. In this article, we present an integer programming model for the integrated coordination of the school starting times and the public bus services. We discuss preprocessing techniques, model reformulations, and cutting planes that can be incorporated into a branch-and-cut algorithm.

7. Junhyuk Park, Byung-In Kim. The school bus routing problem: A review. European Journal of Operational Research. Vol. 202. pp. 311–319. 2010.

This paper aims to provide a comprehensive review of the school bus routing problem (SBRP). SBRP seeks to plan an efficient schedule for a fleet of school buses where each bus picks up students from various bus stops and delivers them to their designated schools while satisfying various constraints such as the maximum capacity of a bus, the maximum riding time of a student in a bus, and the time window of a school. This class of problem consists of different sub-problems involving data preparation, bus stop selection, bus route generation, school bell time adjustment, and bus scheduling.

The combined problem of bus stop selection and bus route generation falls into the class of location-routing problems (LRPs). LRP includes determining the location of the facilities (in SBRP, bus stops) serving more than one customer and the optimal set of routes for a fleet of vehicles (Min et al., 1998).

The data preparation sub-problem prepares the data for the other sub-problems. In this sub-problem, the road network is specified, and four types of data for SBRP are prepared: students, schools, vehicles, and OD matrix. The data for students include the location (address) of their homes, the destination school of a student, and type of student. The type of student is either general or handicapped (for details, see Section 3.5).

School data contain information about the location of the schools, the starting and ending time of schools for bus arrivals, and the maximum riding time of a student in a bus. In most studies, the starting and ending time of schools for bus arrivals are given. However, several available studies assume that the starting and ending time can be determined in school bell time adjustment as summarized in Section 2.4.

In bus route generation, the school routes are constructed. The algorithms used in bus route generation can be classified into either the ‘‘route-first, cluster-second” approach or the ‘‘cluster-first,route-second” approach (Bodin and Berman, 1979). The ‘‘routefirst, cluster-second” approach builds a large route by a traveling salesman problem algorithm that considers all the stops, and partitions it into smaller routes considering the constraints. Newton and Thomas (1969) and Bodin and Berman (1979) implemented this approach. The ‘‘cluster-first, route-second” approach groups the students into clusters so that each cluster can be served as a route satisfying the constraints that exist. Dulac et al. (1980), Chapleau et al. (1985) and Bowerman et al. (1995) applied this approach to SBRP. For additional information on these two approaches, see the works of Dulac et al. (1980) and Laporte and Semet (2002).

After the school routes are generated, improvement heuristics can be applied on the routes. There are plenty of improvement heuristics and metaheuristic approaches. The heuristic approach suggested by Lin (1965) dubbed as k-opt algorithm is widely adopted for VRP studies. This algorithm deletes k edges

from the route then forms a new feasible tour by adding k edges. The idea of k-opt algorithm is adopted for several SBRP studies. Newton and Thomas (1969), Dulac et al. (1980), Chapleau et al. (1985)and Desrosiers et al. (1986a) applied 2-opt algorithms to improve solution. Bennett and Gazis (1972) and Bodin and Berman (1979) adopted 3-opt algorithms.

2.4. School bell time adjustmentIn most studies, the starting and ending time of schools are constraints. However, there are a number of works that consider the times as decision variables and attempts to find the optimal starting and ending times to maximize the number of routes that can be served sequentially by the same bus and to reduce the number of buses used.

2.5. Route schedulingRoute scheduling specifies the exact starting and ending time of each route and forms a chain of routes that can be executed successively by the same bus.

3. Classification based on problem characteristicsIn this section, we attempt to classify SBRP from its problem characteristics. Although there are a number of ways for classification, we mainly focus on the practical aspects of the problems. Table 3 shows the classification criteria from the problem perspective. The details of these criteria are discussed in the following sub-sections. Table 4 summarizes the SBRP literature with regard to the classification criteria.

SBRPLIB.

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.

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