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A simulated annealing algorithm for solving the school bus

A simulated annealing algorithm for solving the school bus

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A simulated annealing algorithm for solving the school bus

  1. 1. 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).
  2. 2. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) 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. 45
  3. 3. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol.5, No.8, 2014 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 46
  4. 4. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol.5, No.8, 2014 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 pick- up 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 pick- up 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
  5. 5. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) 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: 48
  6. 6. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol.5, No.8, 2014 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.
  7. 7. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) 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
  8. 8. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol.5, No.8, 2014 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. 51
  9. 9. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol.5, No.8, 2014 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)
  10. 10. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) 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 Bus Route Time Bus Route Time 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 Objective Function (Time in minutes)
  11. 11. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol.5, No.8, 2014 Table 4: results, in current routes before and proposed routes after Simulated Annealing Algorithm implementation School Current route Proposed route Bus Route Time Bus Route Time 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 Total 5 Buses 7814.5 5 Buses 5537 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. Table 5: results, in current routes before and proposed routes after Simulated Annealing Algorithm implementation School Current route Proposed route Bus Route Time Bus Route Time 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 Total 3 Buses 10060.5 3 Buses 5317 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 Objective Function (Time in minutes)
  12. 12. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) 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. Table 6: results, in current routes before and proposed routes after Simulated Annealing Algorithm implementation School Current route Proposed route Bus Route Time Bus Route Time 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 Objective Function (Time in minutes)
  13. 13. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) 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 Objective Function (Time in minutes)
  14. 14. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) 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. REFERENCES 1. Aarts, A., Jan, H.M.K., Laarhoven, P., Van, J.M., (1988). A Quantitative Analysis of the Simulated Annealing Algorithm, A case study for the travelling salesman problem, Journal of statistical physics, Vol.50, PP.1-2 2. Azencott, R., (1992). Sequential Simulated Annealing: Speed of convergence and acceleration techniques In: Simulated Annealing Penalization Techniques. Wiley. Newyork, P1. 3. Bayram, H., Sahin, R., (2013). A New Simulated Annealing Approach for Travelling Salesman Problem, Journal of Mathematical and Computational Applications, Vol.18, No.3, PP.313-322 4. Berger, J., Barkaoui, M., (2000). An improved hybrid genetic algorithm for the vehicle routing problem with time windows, International ICSC symposium on computational intelligence, part of the international ICSC congress on intelligent systems and application (ISA 2000), university of Wollongong, Wollongong, Australia. 5. Cerny , V., (1985). Thermo dynamical Approach to the Travelling Salesman Problem: An Efficient Simulation Algorithm, Journal of Optimization Theory and Applications 45(1), PP. 41-51 6. Chiang, W.C., Russel, R.A., (1996) simulated annealing met heuristics for vehicle routing problem with time windows. Annals of operations research 63, pp 3-27 7. Cordeau, J.F., Laporte, G., Mericier, A., (2000). 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  15. 15. Computer Engineering and Intelligent Systems www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol.5, No.8, 2014 15. Geng, X., Chen, Z., Yang, W., Shi, D., Zhao, K., (2011). Solving the Travelling Salesman Problem based on an adaptive Simulated Annealing algorithm with greedy search, Journal of Applied Soft Computing, Vol. 11, PP.3680-3689 16. Halse, K., (1992). Modeling and solving complex vehicle routing problems. PhD dissertation no.60, institute of mathematical statistics and operation research, technical university of Denmark, DK-2800 Lyngby, Denmark 17. Ingber, L., (1989). Very Fast Simulated re-annealing, Journal of Mathematical and Computer Modeling, Vol.12, PP.967-973 18. Ingber, L., Rosen, B., (1992). Genetic algorithms and very fast simulated annealing: a comparison, Journal of Math Comput Model Vol.16, PP.87-100 19. Kim, T., Park, B.-J., (2013). Model and algorithm for solving school bus problem, Journal of Emerging trends in computing and information sciences vol.4 no.8 20. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., (1983). Optimization by simulated annealing, Science, Newseries, vol.220 no.4598 pp 671-680 21. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., (1983). Optimization by Simulated Annealing, journal of Science 220, PP. 671-680 22. Kohl, N., Madsen, O.B.G., (1997). An optimization algorithm for the vehicle routing problem with time windows based on lagrangean relaxation, Opns.Res.45 pp 395-406 23. Kouvelis, P., Chiang, W., (1992). A Simulated Annealing procedure for single row layout problems in flexible manufacturing systems, Int J Product Res, Vol. 30, PP.717-732 24. Li, L., Fu, Z., (2002). The school bus routing problem: A case study, Journal of operational Research society 53, pp 552-558 25. Liujiang, K., Jianjun, W., Huijun, S., (2012). Using Simulated Annealing in a Bottleneck Optimization Model at Railway Stations, Journal of Transportation Engineering Vol. 138, PP. 1396-1462 26. Lundy, M., Mees, A., (1986). Convergence of an annealing algorithm, Journal of Mathematical programming, Vol. 34 PP.111-124 27. Malek, M., Guruswamy, M., Pandya, M., Owens, H., (1989). Serial and Parallel Simulated Annealing and Tabu Search Algorithms for the Travelling Salesman Problem, Journal of Annals of Operations Research, Vol.21, PP.59-84 28. Manumbu, D.M., Mujuni, E., Kuznetsov, D., (2014). Mathematical Formulation Model for a School Bus Routing Problem with Instance Data, Journal of Mathematical Theory and Modeling ISSN 2224- 5804(paper) ISSN 2225-0522(online) Vol 4, No 8(2014) pp121-132 29. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, E., (1953). Equations of State Calculations by fast Computing Machines, j. Chem.Phys 21, PP.1087-1092 30. Ngonyani, B., (2013). Optimizing bus scheduling for school bus routing problem: a case study of Dar es Salaam, Msc (mathematical modeling) dissertation university of Dar es Salaam, Tanzania 31. Park, J., Kim, B., (2010). The school bus routing problem: A review, European Journal of operation research, vol. 202 pp 311-319. 32. Rutenbar,R.A., (1989). Simulated Annealing Algorithm: an overview, IEEE circuits devices mag 5. 33. Saint, S.M., Youssef, H., (1999). Iterative Computer Algorithms with Applications in engineering. Press of IEEE Computer Society: silver spring MD. 34. Shaw, P., (1998). Using constraint programming and local search methods to solve vehicle routing problems, in: Principles and practice of constraint programming –CP 98, lecture notes in computer science, Maher, M., and Puget, J.-F (Eds), Springer-Verlag, Newyork, pp 417-431 35. Suman, B., Kumar, P., (2006). A Survey of Simulated Annealing as a tool for Single and Multiobjective Optimization, Journal of the Operation Research Society 57, PP.1143-1160 36. Szu, H., Hartley, R., (1987). Fast Simulated Annealing, journal of Physics Letters A 122, PP.157-162 37. Taillard, E., Badeau, P.,Gendreau, M., Guertin,F., Potvin, J.Y., (1997). A tabu search heuristic for the vehicle routing problem with soft time windows, Transportation science 31, pp 170-186 38. VanLaarhoven, P.J.M., (1988). Theoretical and Computational aspect of Simulated Annealing, PhD thesis, Erasmus University: Rotterdam. 39. Woch, M., Lebkowski, P., ( 2009). Sequential Simulated Annealing for the Vehicle Routing Problem with time windows, Journal of Decision Making in Manufacturing and Services Vol.3, No. 1-2, PP. 87-100. 58
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