A simulated annealing algorithm for solving the school bus

Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol.5, No.8, 2014
A Simulated Annealing Algorithm for Solving the School Bus
Routing Problem: A Case Study of Dar es Salaam
44
Denis M. Manumbu
1
, E. Mujuni
2
, Dmitry Kuznetsov
3
1
School of Computational and Communication Science and Engineering, TheNelson
Mandela African Institution of Science and Technology
(NM-AIST) P.O Box 447, TengeruArusha Tanzania
The research is financed by The Nelson Mandela African Institution of Science and Technology
Abstract
School bus routing is one of major problems facing many schools because student’s transportation system needs
to efficient, safe and reliable. Because of this, the school bus routing problem (SBRP) has continued to receive
considerable attention in the literature over the years. In short, 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 school while satisfying various constraints such as the maximum capacity of a bus, the maximum
transport cost, the maximum travelling time of students in buses, and the time window to reach at school. Since
school bus routing problems differ from one school to another, this paper aims to developing Simulated
Annealing (SA) heuristic algorithms for solving formulating a mathematical model for solving the student bus
routing problem. The objective of the model is to minimize amount of time students in the buses from the point
where they pickup to the school. We illustrate the developed model using data from five schools located at Dar
es salaam, Tanzania. We present a summary of results which indicates good performance of the model.
Keywords: bus stop, students, bus, simulated Annealing (SA), Objective function value, Current route, proposed
route.
1. Introduction
The School Bus Routing Problem (SBRP) seeks to plan an efficient schedule for a fleet of school buses that
pick up students from various bus stops and deliver them to the school by satisfying various constraints, such as
the bus capacity, all pupils are picked, a bus visiting a point also leaves that point, all buses visit the school, if a
bus picks up pupils at a point it must also visit that point (Li and Fu, 2002). Spasovic et al., (2001) added
constraints such as the time that students spend for travelling on the bus not exceed a given limit, the number of
students per bus must not exceed the number of seats available on bus, each bus stop is allocated to only one bus,
every route must have at least one stop, each stop is allocated to only one route and the number of buses leaving
the school must equal the number of buses returning to the school.
In SBRP, there are sets of buses; set of buses is containing all buses member in a single school. These buses
are assigned at the pick-up points to pick up students and deliver them to school. Each bus has a route to
transporting students during morning time by takes students from pick-up points and delivers them to school also
reversed this route during afternoon by transporting students from school and derivers them to bus stop nearly to
their home. There are sets of stops, these set containing all stops member in a single school. Because of students
to be scattered around a school , student is assigned to nearly pick-up point so can walk to nearby bus stop for
waiting a school bus every morning of school day. The route of a bus starting to pickup students at a pick-up
point that not visited by other bus then go to another pick-up point to picked students, when the students picked
up are full in all bus seats it gone to school. The other bus started another route by visited the pick-up point that
not visited by previous bus, it picked up students from pick-up points that not visited by previous bus and
delivers them to school, this situation continue up to a last bus, so here all students at the stops must picked up
by buses. The important thing is to assigned buses into pick-up points and scheduled routing of a bus in order to
minimize amount of time students spent in a bus to reach at school.
The bus routing problem varies among schools. For example, Schittekat et al., (2012) reports that in some
countries, students living within a certain distance to school are entitled by law free transportation to and from
school. A bus stop should be located at a maximum distance from home of each student (e.g. 750m). Hence a set
of potential bus stops is predefined in advance, from hierarchical point of view; one has to first select the bus
stops and assign the students to the bus stops and then defined the routes for the buses.
The mostly of schools management lacks scientific method to deal with student bus routing problem, these
it used school bus conductor to scheduled the routes of the buses, sometimes a route of bus is take more time
than expected for transporting students from pick-up points to school, because of this situation many parents
complaining that it takes their children much more time than expected to reach at school (Li and Fu, 2002).

Vol.5, No.8, 2014
The SBRP consists of smaller sub- problem: data preparation, bus stop selection (student assignment to
stops), bus route generation, school bell time adjustment and route scheduling, in the data preparation step, the
road network consisting of home, school, bus depot and the origin-destination (OD) matrix among them are
specified. For a given network, the bus stop selection step determines the location of stops and the students are
assigned to them. Thereafter, the bus routes for single school are generated in the bus route generation step
(Desrosier et al.1981, Park and Kim 2010).
SBRP has received considerable attention among researchers since it was introduced by Newton and
Thomas (1969). Below we present a brief survey on previous researches. Clearly, the school bus routing problem
is a generalization of the basic vehicle routing problem and therefore also is NP- hard problem (Schittekat et al.,
2013).
Swersey and Ballard (1984) presented a work on scheduling of school buses. With the scheduling
situation considered here, a set of routes each associated with a particular school is given. A single bus is
assigned to each route to pickup students and arriving at their school within a specific time window. The
problem includes finding the fewest buses needed to cover all routes whiles meeting the time window
specifications. They presented two integer programming formulations of the scheduling problem and applied
them to actual data from New Heaven, Connecticut for two different years as well as to 30 randomly generated
problems. Linear programming relaxation of the integer programs was found to produce integer solutions more
than 75% of the time. In the remaining cases, they observed the few functional values can be adjusted to integer
values without increasing the number of buses needed. Their method reduces the number of buses needed by
about 25% compared to the manual solutions developed by the New Heaven school bus scheduler.
Bowerman, et al., (1995) proposed a new heuristic for urban school Bus Routing. The problem was
formulated as a multi-objective model and a heuristic based on this formulation is developed. The study
involves two interrelated problems. One has to do with the assignment of students to their respective bus stops
and the second has to do with routing of buses to the bus stops. A problem of these characteristics is a location-routing
problem. The nature of the formulation made it possible to organize their study into three layers, where
layer one is the school, layer two is the bus stops and layer three the students. School buses routes cause
interaction between layers one and two, while movements of student cause interaction between layers two and
three. The heuristic approach to this problem involves two algorithms which catered for the multi-objective
nature of the model. The first is a districting algorithm which groups students into clusters to be serviced by a
unique school bus route. The second is a routing algorithm, which generates a specific school bus route that
visits a sub set of potential bus stops sites.
Spasovic et al, (2001) presented a methodology for evaluating of school bus routing a case study of
Riverdale, New Jersey. The techniques were evaluated using the case study of Riverdale, New Jersey, the case
study involves a municipality with one elementary school and requires all of the buses to depart from and return
to the school, the routes and operating costs vary for each of the methodologies used.
Anderson et al, (2005) developed a method combining column generation with greedy heuristics, where the
objective is to minimization of costs. The problem was formulated as integer programming with constraints as
regarding the vehicle/ buses (the capacity of the fleet and individual load capacities of vehicles) and the bus
stops (which have to be visited in certain time interval, the time windows). Proposed the greedy heuristic
algorithm methods to solve the problem were implemented and evaluated. A comparison analysis showed that it
is possible to create plans with fewer vehicles as well as shorter driving distances than in the existing one.
Schittekat et al., (2006) presented another mathematical model for a school bus routing problem. The goal
of the model was to select a subset of stops that would actually be visited by the buses, determine which stop
each student should walk to and develop a set of tours that minimize the total distance travelled by all buses.
They develop an integer programming formulation for this problem, as well as a problem instance generator. It
was shown how the problem can be solved using a commercial integer programming solver.
Park and Kim, (2010) described five different sub-problems which are often treated separately in the SBRP
literature. The steps are data preparation, bus stop selection, bus route generation, school bell time adjustment
and bus scheduling. The SBRP in data preparation step, the road network consisting of home, school, bus depot
and the origin-destination matrix among them are specified. For a given network, the bus stop selection step
determines the location of stops, and the students are assigned to them. Thereafter, the bus routes for a single
school are generated in the bus route generation step. The school bell time adjustment and route scheduling steps
are necessary for the multi-school configuration when the school bus system is operated by the regional board.
Arias-Rojas et al., (2012) solved the school bus routing problem by ant colony optimization heuristic. They
considered a case study of SBRP for a school in Bogota, Colombia. Computational experiments that were
performed using real data, results lead to increased bus utilization and reduction in transportation time with on
time delivery to the school. The proposed decision aid tool has shown its usefulness for actual decision making
at the school, it outperforms current routing by reducing the total distance traveled by 8.3% and 21.4%
respectively in morning and in the afternoon.
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Taehyeong and Bum, (2013) formulated model for solving school bus problem as a mixed integer
programming problem, validated the model, by takes several random small network problems solved by using
the commercial optimization package CPLEX. Also a heuristic algorithm based on harmony search was
proposed to solve this problem. Computation results show that the solution by heuristic was exactly the same as
that of exact method using CPPLEX. But the heuristic produces the same results in a very short time. However,
there was a guarantee that the solution of harmony search is the global optimal as the size of network increases.
Ngonyani, ( 2013) formulated mathematical model in an integer programming such that the bus stops are
linearly ordered for a single school and the objective was to minimize the total travel time spent by pupils at all
point a case study of Dar es Salaam, they proposed heuristic algorithm which is tabu search for approximate
solution to SBRP, the algorithm has been programmed using Borland C++ 4.5 programming language and
implemented using secondary data from three school at Dar es salaam, they proposed new routes which reducing
students travelling time compare to current routes. The work of Ngonyani has inspired this research paper.
2. Student Transport Situation in Dar es salaam
Dar es salaam region is one of Tanzania 30 administration region. It is the largest in Tanzania, and Kumar and
Barret (2008) report that Dar es Salaam is among the rapid growing cities in Africa. Accordingly to the 2012
national census, the regional had a population of 4,364,541 which was much higher than the pre-census
projection of 3,270,255; the region 5.6 percent average annual population growth rate was the highest in the
country. It was also the most densely populated region with 3,133 people per square kilometer. The most
common form of transport in Dar es salaam are public buses called daladala which are often found at the many
bus terminal. (Census Report, 2012).
The student transport in Dar es Salaam is currently a big problem that faces primary and secondary students in
Dar es salaam city during the morning and evening where they are encountering various issues from daladala
(bus) operators and as a result, students reach their schools late. Masozi Nyirenda reported in the Guardian
Newspaper of 9th July 2012 that inadequate and unreliable transport for students in various cities and towns in
Tanzania has been one of the chronic problems. This hinders students’ academic progress. In addition, the
student transportation problem causes some other social problems such as poor academic performance, teen
pregnancies and other delinquencies such as students fighting with daladala conductors.
The school bus scheduling in Dar es Salaam is a challenge problem in many private schools, which provides the
transport to their students. In a school there are buses which picked up students at pick-up points and deliver
them to their school. The school management scheduled the bus routing by considering the number of pick-up
points, number of students at the pick-up points, number of buses which are available to the school, travelling
time between the bus stops and distance between the stops. Each school bus has a specific one route during the
morning session for transporting students from pick-up points and delivers them to school, and reversed this
route during afternoon for transporting students from school to stops nearly their home. The available stops are
generated, students are assigned to the stops, and the bus is assigned to stops in their existing route. The school
bus conductor scheduled the bus routing by take first bus and gives the stops with student required to picked by
that bus and deliver them to school, take second bus gives the stops which not visited by previous bus with
students required to picked by that bus and deliver them to school. This action is continue up to last bus, the
conductor uses experienced how to known the all area that can generates routes connecting the pick-up points
and ends to school, the conductor uses locally techniques to schedule the bus routing this arise in the route
students to spend more time within the bus than expected to reach at school and home during school day.
Unfortunately, most of school which provides the transport service to their students are lacking scientific
methods that can be used to route and schedule these school buses. This leads students spend much more
travelling time than expected to reach at school and home.
The heuristic algorithm proposed to solve the model was simulated annealing, the algorithm was implemented
using Borland C++ 4.5 programming language. The model was subjected to some constraints, so the penalty
function was inserted in the program to penalize the solution that violates the constraints. The program finds the
initial solution which is value of objective function for a existing route and continues to run by generated
iterations up to reached efficient value of objective function that is obtained when the initial temperature
decreases to the lowest one (called freezing point) accordingly the number of iterations assigned for. We
collected the data from five schools which allocated at Dar es salaam, Tanzania and validated in a program for
the solution. These data is 218 number of bus stops for five schools, number of students served by all buses is
915, time consumed to picked up student at a pick up point which is 0.5 minute, buses available to transport
students for five schools is 26 and Capacity of the buses for each school, in Hazina each bus served 40 students,
in Sahara each bus served 35 students, in Yemen each bus served 30 students, in African each bus served 40
students and in Atlas each bus served 60 students. Lastly we compare the value of objective function for a
current route and value of objective function for a proposed route, the aims is to get saved time in proposed
route. Since the proposed route reduced travelling time for the students within a bus at all bus stops compare to
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current route, we suggested this route to be used by school in order to minimize total travel time spent by
students within the bus in all bus stops to reach a school.
3. Presentation of the problem
The discussion in this section is also presented in Manumbu et al., (2014), and summarized here for clarity of
presentation.
Model Assumptions:
1. Each bus has only one route for transporting students to school.
2. The pickup points visited and picked students by bus are scattered and not necessary to be linearly
ordered.
3. If the bus visiting a point it must picks up all students at that point.
4. The time spend by students within the bus from one pick up point to another includes jams, road
condition, accident action and waiting time in traffic light.
5. Each pick up point is allocated to only one bus.
6. Each bus has only one route for transporting students to school in the morning and back to their home
after classes.
7. Each school bus picks up students at least on one pickup point.
Model Objective
The objective of the model is therefore to plan routes that will minimize the total travel time spent by students in
all bus stops by using the non- linear mixed integer programming model.
Sets: The following are the sets that are used in the model formulation.
1. a set of all bus stops where one or more students are picked up whereby N is the total
number of stops arranged scattered around the school and denotes the school.
2. a set of the available buses to be used where B is the total number of available
buses.
Parameters: Proposed model uses the following parameters;
1. represents the number of available buses for the school bus service.
2. represents the travel time from to .
3. is the total number of bus stops available
4. denotes the capacity of bus
5. α is the average pick – up time of one student by bus at a pick-up point.
6. is the number of students at stop
7. is the set of pick-up point to be visited by bus
8. is represents the index number of pick-up point be visited by bus
The mathematical model to represent the problem is:
Σ Σ Σ Σ
47
Subject to;
1. Σ
2.
3. Σ
Σ Σ
4.
Constraints, (1) ensures that the sum of students picked up in all points by bus must not exceed the bus capacity;
(2) ensures that all buses finished their routes at a school; (3) ensures that all students are picked up; (4) the
number of students at each bus stop is nonnegative.
4. Implementation of Simulated Annealing Algorithm for SBRP
4.1 Simulated Annealing Algorithm
Simulated Annealing Algorithm is a compact and robust techniques, which provides excellent solutions to
single and multiple objective optimization problems with a substantial reduction in computation time when
metal cool and anneal, it is a method to obtain an optimal solution of a single objective optimization problem

Vol.5, No.8, 2014
and to obtain a pareto set of solutions for a multiobjective optimization problem, if a liquid metal is cooled
slowly, its atoms form a pure crystal corresponding to the state of minimum energy for the metal. The metal
reaches a state with higher energy if it is cooled quickly (Suman et al., 2006).
Simulated Annealing (SA) has received significant attention in many years ago to solve optimization problems,
where a desired global minimum is hidden among many poorer local minimum. Kirkpatrick et al, (1983) and
Cerny (1985) showed that a model for simulating the annealing of solids, proposed by Metropolis et al, (1953),
could be used for optimization of problems, where the objective function to be minimized corresponds to the
energy of states of the metal. Literature reviews deal with Simulated Annealing (SA) has listed as follows;
Dueck et al (1990), have used threshold accepting method, which is principally simpler than conventional
Simulated Annealing technique, they demonstrated their technique using TSP showed that threshold accepting
yields very near to optimum results for several known TSP.
Cryzzak et al (1998), they presented a multi-objective pareto Simulated Annealing approach, with the aim of
finding set of efficient solutions for multi-objective combinatorial optimization problems, in the overall
evolution of solutions, it employed objective weights.
Malek et al (1989), they discussed parallel Simulated Annealing approach and they tested this technique by using
several TSP from previously literature, they found out that the serial implementation of the SA is superior to
conventional SA for the solving of TSP.
Geng et al (2011), they discussed the solutions for TSP, they improved adaptive SA with greedy search and
introduced three different mutation strategies for the convergence generation of new solutions.
VanLaarhoven (1988) and Lundy et al (1986) they have been shown that SA works better than the descent
algorithm, Ingber et al (1992) they have proposed a very fast simulated annealing method that is efficient in its
search strategy and which statistically guarantees to find the global optima and Suman et al (2006) they have
proposed orthogonal SA, which combines SA with fractional factorial analysis and enhances the convergence to
accuracy of the solution
In this study we choose to use Simulated Annealing Algorithm, because it becomes one of the many heuristic
approaches designed to give a good, not necessarily optimal solution. It is simple to formulate, can handle ease
mixed discrete problem and takes less CPU time when used to solve optimization problems, since it finds the
optimal solution using point by point iteration rather than a search over population of individuals.
The mathematical model represented in this study is solved by a simulated annealing heuristic. Simulated
Annealing Algorithm is selected to find a solution for the minimization problem with solution space, in the
problem we have which is a finite set of all solution and the objective function , is a real valued function
defined for the members of . Simulated Annealing Algorithm attempted so as to avoid being trapped in a poor
local optimal by accepting probabilistically moves to worse solutions. The method initiates the physical
annealing process in metallurgy; starting from a randomly generated solution, a neighboring solution is sampled
and compared with the current one according to an appropriate probability function. The acceptance and
rejection of the worse move is controlled by a probability function. The probability of accepting a move, which
causes an increased in objective function , is called the acceptance function. It is normally set to
,
where is a control parameter, which corresponds to the temperature in analogy with the physical annealing.
This acceptance function implies that the small increase in objective function is more likely to be accepted
than a large increase in objective function . When is high most uphill moves are accepted, but as
approaches to zero, most uphill moves will be rejected. Therefore Simulated Annealing (SA) starts with a high
temperature to avoid being trapped in poor local optimal, the algorithm proceeds by attempting a certain number
of moves at each temperature and decreased the temperature. Thus, the configuration decisions in SA proceeds
in a logical order. The heuristic terminates when either the better minimal solution is obtained or the initial
temperature decreases to the lowest one (called freezing point).A pseudocode for Simulated Annealing for this
work is given in figure 1 below:
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49
{
{
}
}
}
Figure 1: Pseudocode for the Simulated Annealing heuristic.
4.2 Simulated Annealing Implementation
4.2.1 The initial solution
Most of the researchers in the world are introduced ways be used to selected initial solution as follows;
Suman et al (2006), in their Simulated Annealing method for solved single objective optimization, started with a
randomly generated initial solution vector, and use it to generated the objective function value .
Woch et al (2009) , in their Simulated Annealing method for solved vehicle routing problem with time windows,
started with current route to inserted in the place in order to use it to generated the objective function value
which is selected as initial solution.
Bayran etal (2013), in their developed simulated annealing method for solved travelling salesman problem, they
started and generated a random initial solution
Liujiang et al 2012, in their developed Simulated Annealing method for solved railway station problem, they
started and generated a randomly feasible solution , and use it to calculate the objective function value
and display as initial solution.

Vol.5, No.8, 2014
In this study Simulated Annealing Algorithm started by a selected current route of the buses used by school to
transported students from pickup points to their school, and uses it to calculated objective function value, this
value is started as initial solution. Also is the first solution created before any search for other solution in the
algorithm. The initial solution for a School Bus Routing Problem is computed from sets of available buses to
served students at the pickup points allocated to each bus for the bus current routes. The order of buses visited
the pickup points is that; the first bus starts to picked students at the first pickup point and then to next pickup
point until it is full, with assumption condition that the pickup points visited by bus are not linearly ordered. The
next bus starts to non-visited pickup points followed the same way until all students picked up by buses.
50
4.2.2 The Initial Value of Temperature (T)
In the literature review the way can be used to selected initial value of temperature is presented by researchers as
follows;
Dowsland (1995) they introduced various methods for finding the appropriate starting temperature have been
developed, they subsisted to quickly raise the temperature of the system initially up to the point where a certain
percentage of the worst solutions is acceptable and after that point, a gradual decrement of temperature is
proposed.
Laarhoven et al (1988) have proposed a method to select the initial temperature based on the initial acceptance
ratio , and the average increase in the objective function, :
Where is defined as the
number of accepted bad moves divided by the number of attempted bad moves.
Saint et al (1999) with the only difference being in the definition of , they have defined as the number of
accepted moves divided by the number of attempted moves.
Kouvelis et al (1992) have proposed a simple way of selecting initial temperature; they selected the initial
temperature by the formula
where is the initial average probability of acceptance and is taken
in the range of
But in this study temperature is chosen such that it can capture the entire solution space. We choice a very high
initial temperature as it increases the solution space. However, at a high initial temperature, Simulated Annealing
performs a large number of iterations, which may be giving better results. Therefore, the initial temperature
chosen in this experimentation is and the range of change is in the value of the objective
function with different moves up to lowest one . The initial value of temperature should be
considerably larger than the largest .
4.2.3 The neighborhood structure
Some researchers they introduced the way to generated new solutions known as neighborhood states from
solution space as follows;
Ngonyani (2013), in school bus routing problem, they introduced the exchange move involved in exchanging
pickup point from one bus route to another bus route, the new solution of the route that formed is known as
neighborhood solution. Their exchange move a pickup point is removed from its original route and is inserted in
a random selected route to generated neighborhood solutions.
Bayram et al (2013), in travelling salesman problem, they introduced the exchange move of city from one array
of city to another array of city, for generation of the neighbor solutions. They introduced the simplest
representation encoding; in permutation encoding the order of the numbers in the array represents the visiting
order of the cities.
In this study, the Simulated Annealing Algorithm due to implementation it searches new solutions from set of
feasible solution space. The new solution for the bus routes it is used to generate objective function value after
given an initial solution for the objective function value which is computed from existing bus routes (bus
routes used by school). The new solutions for the objective function generated after moves pickup points from
one bus route to another bus route due to available bus routes at a school, it is form neighbor solutions with some
different pickup points compare to current solutions before swamping process done, and used it to compute the
objective function values for the neighbor solutions. The searches solution space (Neighborhood search) is
deal with pickup point moves on exchanging to the bus routes from set of feasible solution, an objective function
values for the neighbor solution at each iteration and readable as value of iteration.
In this School Bus Routing Problem, exchanging moves of pickup points from one route of a bus to another
route of a bus, the option can be used to generated neighborhood solution are

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Option, this means one pickup point exchanging moves from first bus route to second bus route, the
neighbor solution is when the second route added the pickup point, use it to computed objective function
values for the neighborhood solution. General the bus routes before exchanging moves of pickup point is known
as current solution and the bus route formed after pickup point exchanging moves are neighbor solution.
4.2.4 The Cooling Schedule
Large number of researchers have introduced the way used to determined cooling schedule or temperature
decrement functional form of the change in temperature required in Simulated Annealing Algorithm as follows;
Azencott (1992) they introduced the three important cooling schedule are logarithmic, Cauchy and exponential.
Geman et al (1984) they introduced that SA converges to the global minimum of the cost function if temperature
change is governed by a logarithmic schedule in which the temperature at step is given by
.
Szu et al (1987), proposed a fast SA approaches such that the SA is inversely linear in time, showed that the
cooling strategy is superior to the conventional SA technique, a faster schedule is the Cauchy schedule in which
converges to the global minimum when moves are drawn from a Cauchy distribution.
Ingber et al (1989) they studied very fast SA, they introduced a new exponentially or geometric schedule in
which where C is a constant, but to reach global optimum it is require good heuristic
arguments for its convergence have been made for a system in which annealing state variables are bounded.
Azencott (1992) they introduced the three important cooling schedule are logarithmic, Cauchy and exponential.
Aarts et al (1988) introduced way of the temperature cooling for the success of the simulated annealing
algorithm; they suggested the following way to decrement the temperature:
where is a positive
constant, an alternative is the geometric relation. Where parameter , is a constant near 1, in effect
its typical values range between 0.8 and 0.99
In this study the annealing schedules can be based on the analogy with physical annealing; therefore we are set
initial temperature high enough to accept all processes, which means heating up substances till all the metal are
randomly arranged in liquid. A proportional temperature is used, that is Where is constant known as
the cooling factor, it varies from finally when temperature becomes very small one
(frozen state) and it does not search any smaller energy level.
4.2.5 The Stopping criterion to terminate the algorithm
Surveys on the stopping criterion to terminate the algorithm have been performed by researchers as follows:
Rutenbar, (1989) have introduced the Simulated Annealing Algorithm terminates when the cost improvement
across three temperatures is very small.
Suman et at (2006) they introduced the termination of Simulated Annealing Algorithm, stopping criteria have
been developed with time as temperature closed to zero at very low temperature (frozen point) has been given
due implementation of a SA. In implemented this algorithm number of iteration to move at each temperature
have been produced, this criteria leads to higher or low computation time without much update in objective
function and sometimes it may lead the global optimum due to less number of iteration before stopped of
algorithm.
In our study the Simulated Annealing Algorithm stopped at final iteration when final temperature becomes very
small one (frozen point) which is and gives objective function to lead the global optimum (final
solution) either due to less number of iteration or to final iteration.
5 Experimental and Result Analyses
The algorithm was tested on data taken from three schools in Dar es Salaam, Tanzania. The schools are Atlas
primary school, African nursery and primary school and Yemen DYCCC secondary school. The algorithm was
implemented using Borland C++ Version 4.5. We ran the algorithm on a 2GHz machine with 1.87 GB RAM and
Windows 7. The size of the is given in Table 1.
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52
Table 1: Size of input data
School Number of Buses Number of Bus Stops Number of Students
Atlas PS 9 68 445
African N&PS 7 65 197
Yemen SS 5 39 113
Sahara P&PS 3 27 95
Hazina PS 2 19 65
Results at Atlas Primary School
It is discussed and compared the results produced before and after implementation of SA Algorithm of each bus
for all 9 buses. In this part, the trend of objective function values from initial solution to final solution its shown
bellow by graph with objective function value in minutes against iterations.
Table 2: Discussed results, in current routes before and proposed routes after Simulated Annealing Algorithm
implementation.
School Current route Proposed route
Bus Route Time Bus Route Time
Atlas
P/School
Bus 1
Bus 2
Bus 3
Bus 4
Bus 5
Bus 6
Bus 7
Bus 8
Bus 9
46,65,29,10,31,48,67,2,S
23,40,58,22,3,57,39,68,S
49,32,1,30,12,66,47,4,59,S
41,24,7,44,63,15,33,50,S
5,26,43,61,42,25,S
56,38,21,35,52,17,9,S
64,45,28,11,53,36,19,55,62,S
60,8,37,14,51,20,S
13,34,16,27,18,54,6,S
5004.5
3680.5
5058
2862
3693
3246.5
3314.5
3205.5
2819
Bus 1
Bus 2
Bus 3
Bus 4
Bus 5
Bus 6
Bus 7
Bus 8
Bus 9
67,30,10,31,48,65,46,20,S
21,35,68,22,58,57,S
59,47,4,12,30,1,32,S
40,39,6,63,15,33,24,41,S
25,2,61,43,14,26,5,S
36,52,56,42,38,17,23,44,3,S
66,55,19,45,53,28,11,50,S
64,49,51,8,37,60,S
62,9,54,13,27,16,18,34,7,S
3013
2087
2702
2128
2365
2092
2189
1434.5
2446
Total 9 Buses 32883.5 9 Buses 20456.5
6000
5000
4000
3000
2000
1000
(a) : Current and Proposed routes of 9 buses for Atlas PS
3.4
3.2
3
2.8
2.6
2.4
2.2
4 Trend of the objective Function value for Atlas P/School
(b) : Trend of objective function values for Atlas PS
Figure 1: Summary results for Atlas Primary School
0
Bus 1
Bus 2
Bus 3
Bus 4
Bus 5
Bus 6
Bus 7
Bus 8
Bus 9
Time in minutes
Time values on bus current and proposed
routes for Atlas P/School.
current route
proposed route
0 50 100 150 200 250 300 350 400
2
x 10
Iterations
Objective Function (Time in minutes)

Vol.5, No.8, 2014
Results at African Nursery and Primary School
It is the results produced before and after implementation of Simulated Annealing Algorithm of each bus for all
7 buses. In this part, the trend of objective function value from initial solution to final solution its shown bellow
by graph with objective function value in minutes against iterations.
Table 3: results, in current routes before and proposed routes after Simulated Annealing Algorithm
implementation
School Current route Current route
53
African
N&PS
Bus 1
Bus 2
Bus 3
Bus 4
Bus 5
Bus 6
Bus 7
12,28,46,63,1,20,38,55,14,32,S
49,65,31,16,51,34,3,21,39,S
6,23,41,58,15,33,50,48,S
62,45,27,11,2,57,4,22,40,S
60,43,25,8,42,7,24,59,5,S
19,37,54,13,30,64,47,29,18,53,S
61,9,44,26,36,52,17,35,10,56,S
2507
3101
749
2243.5
3882
1571.5
3304.5
Bus 1
Bus 2
Bus 3
Bus 4
Bus 5
Bus 6
Bus 7
38,63,12,11,14,13,1,35,32,S
21,39,34,51,30,44,55,27,60,S
10,7,58,6,15,48,43,65,23,S
29,53,17,41,22,40,57,62,16,S
4,8,59,5,24,19,61,42,25,37,36,S
18,64,28,54,47,20,3,2,26,S
56,33,52,49,9,46,31,50,45,S
1246.5
1312
850
1163
1660
1063
845.5
Total 7 Buses 17358.5 7 Buses 8140
5000
4000
3000
2000
1000
0
Bus
1
Bus
2
Bus
3
Bus
4
Bus
5
Bus
6
Bus
7
(a) : Current and Proposed routes of 7 buses for African N& PS
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
4 Trend of the objective Function value for African NP/School
(b) : Trend of objective function values for African N&
PS
Figure 2: Summary results for Atlas Primary School
Results at Yemen Secondary School
It is discussed and compared the results produced before and after implementation of SA algorithm of each
bus for all 5 buses. In this part, the trend of objective function value from initial solution to final solution its
shown bellow by graph with objective function value in minutes against iterations.
Time in minutes
Time value on bus current and proposed
routes for African NP/ School.
Current route
Proposed route
0 50 100 150 200 250 300 350 400
0.8
x 10
Iterations

Vol.5, No.8, 2014
implementation
54
Yemen
SS
Bus 1
Bus 2
Bus 3
Bus 4
Bus 5
7,12,1,26,18,15,23,20,24,S
4,6,8,14,17,19,22,25,5,S
3,9,11,21,16,13,2,10,S
31,38,29,36,32,34,30,39,27,S
37,35,28,33,S
1119
2021
2183
1854
637.5
Bus 1
Bus 2
Bus 3
Bus 4
Bus 5
26,7,15,23,18,1,5,21,24,S
12,31,14,38,4,6,8,22,S
13,16,11,9,3,27,10,S
20,34,36,30,28,35,32,S
29,37,17,19,2,39,25,33,S
812
1392.5
1185
834.5
1313
2500
2000
1500
1000
500
0
Bus 1 Bus 2 Bus 3 Bus 4 Bus 5
(a) : Current and Proposed routes of 5 buses for Yemen SS
8000
7500
7000
6500
6000
(b) : Trend of objective function values for Yemen SS
Figure 3: Summary results for Yemen Secondary School
Results at Sahara Nursery and Primary School
It is discussed and compared the results produced before and after implementation of SA algorithm of each bus
for all 3 buses. In this part, the trend of objective function value from initial solution to final solution its shown
bellow by graph with objective function value in minutes against iterations.
implementation
Sahara
N&PS
Bus 1
Bus 2
Bus 3
7,21,24,5,3,15,9,12,S
1,11,14,26,18,20,23,16,4,S
2,6,8,10,13,17,19,22,25,27,S
2562.5
4218.5
3279.5
Bus 1
Bus 2
Bus 3
24,21,3,9,12,15,13,7,5,S
4,26,20,23,14,18,16,11,1,S
19,10,17,22,27,25,2,6,8,S
1342.5
1967.5
2007
Time in minutes
Time value on bus current and proposed
routes for Yemen S/School
Current route
Proposed route
0 50 100 150 200 250 300 350 400
5500
Trend of the objective Function value for Yemen S/School
Iterations

Vol.5, No.8, 2014
55
5000
4000
3000
2000
1000
0
Bus 1 Bus 2 Bus 3
(a) : Current and Proposed routes of 3 buses for Sahara N&PS
11000
10000
9000
8000
7000
6000
(b) : Trend of objective function values for Sahara N&PS
Figure 4: Summary results for Sahara Nursery and Primary School
Results at Hazina Secondary School
It is the results produced before and after implementation of Simulated Annealing Algorithm of each bus for all
2 buses. In this part, the trend of objective function value from initial solution to final solution its shown bellow
by graph with objective function value in minutes against iterations.
implementation
Hazina
SS
Bus 1
Bus 2
18,14,8,3,12,5,11,19,1,S
7,2,10,17,4,15,9,13,6,16,S
1517.5
1235.5
Bus 1
Bus 2
8,1,5,11,12,3,14,18,16,S
4,7,2,10,17,15,9,13,19,6,S
1169.5
1041.5
Total 2 Buses 2753 2 Buses 2211
Time in minutes
Time value on bus current and proposed routes
for Sahara PN/School
Current route
Proposed route
0 50 100 150 200 250 300 350 400
5000
Trend of the objective Function value for Sahara NP/School
Iterations

Vol.5, No.8, 2014
Yemen
S/School
4200
4000
3800
3600
3400
3200
3000
2800
2600
2400
Sahara
NP/School
56
Overall Performance
35000
30000
25000
20000
15000
10000
5000
0
Atlas
P/School
African
NP/School
Hazina
P/School
Figure 6: Comparison between total travel time in current and proposed routes for each school
1600
1400
1200
1000
800
600
400
200
Table 7: The saved time by proposed routes from current routes in percentage of each school for five schools
School Current routes (Cr)-
travel time in minutes
Proposed routes (Pr)-
travel time in minutes
Saved time (Cr-Pr) Saved from Cr (%)
Atlas P/School 32883.5 20456.5 12427 37.8%
African
17358.5 8140 9218.5 53.1%
NP/School
Yemen
S/School
7814.5 5537 2277.5 29.2%
Sahara
NP/School
10060.5 5317 4743.5 47.2%
Hazina
P/School
2753 2211 542 19.7%
Time in minutes
Objective function value for the current and proposed
routes for 5 schools.
Current route
Proposed route
(a) : Current and Proposed routes of 2 buses for Hazina SS
(b) : Trend of objective function values for Hazina SS
Figure 5: Summary results for Hazina Primary School
0
Bus
1
Bus
2
Time in minutes
Time value on bus current and
Proposed routes Hazina P/School
Current route
Proposed route
0 50 100 150 200 250 300 350 400
2200
Trend of the objective Function value for Hazina P/School
Iterations

Vol.5, No.8, 2014
6. CONCLUSION AND FUTURE WORK
In this study, Simulated Annealing Algorithm is proposed to solve the mathematical model presented for the
school bus routing problem. The model objective is to minimize the time spent by students within the bus at all
pickup points to their school, since the model formulated to minimize the time so its combinatorial optimization
problem. Simulated Annealing algorithm solved model, where a desired global minimum is hidden among
many local minimum. These Simulated Annealing methods have attractive and it’s faster to reach final solution
compare with other optimization technique. The reasons that its attractive are, a solution does not get trapped in
local minimum by sometimes it is accepted even the worse move and configuration decision proceed in a logical
manner in simulated annealing. The paper provides pseudocode of Simulated Annealing for making a solution of
presented mathematical mode, implementation of SA is shown clearly in this paper, also the SA algorithms
should suggested to use to generate a larger set of optimal solutions giving a wider choice to the decision maker.
The annealing schedule is the essential part of Simulated Annealing as help to determine the performance of the
method. The good performance of this method is tested by data’s input in the program for results, data’s
collected from five schools located in Dar es salaam, Tanzania. From analyses of the results it shows that the
school management and students benefits by this study when should use proposed routes. The proposed routes
reduce total time spent by students within the bus at all pickup points to their school compared to current routes.
The Borland C++ 4.5 programming language used to write the codes for simulated annealing algorithm is simple
to understand and if run it gives the better solution in a short time. In future the researchers should improve the
quality of data collected by measure time from one stop to another not take approximation data from drivers and
conductor of school buses, also should added the constraints in the model such as time windows and the buses to
serves malt schools instead of single school.
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