International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia)

A Hybrid Genetic Algorithm-Gravitational Attraction Search algorithm (HYGAGA) to Solve Grid Task Scheduling Problem Amin. Jula, Narjes Khatoon. Naseri

between resources with respect to parameters of Quality of Service [3, 4]. On the other hand, memetic algorithms such as other methods are used to solve grid task scheduling problem in several cases [5, 6]. Memetic algorithms proved that they can reach more success in solving NP-hard problems compared with other applied algorithms [7, 8]. Since the local search has effective and basic role in highperformance behavior of memetic algorithms, a new local search algorithm is proposed in this paper to enhance performance of solving method. It will be shown that this local search algorithm increases speed of finding best solutions as well as finding better solutions in that time.

Abstract— Integrating distributed Heterogeneous resources to solve complicated scientific, commercial and industrial problems is the main purpose of grid computing. An efficient task scheduling system is a vital part to reach this scope. Dynamism and Heterogeny of resources in grid cause to need to a complicated scheduling system. Scheduling problem in such systems is resource assigning way to tasks compliance with quality of service parameters. In this paper Hybrid Genetic Algorithm-Gravitational Attraction Search, HYGAGA, proposed as a memetic algorithm to solve grid task scheduling problem. In this proposed approach gravitational attraction search as a local search algorithm has been associated to genetic algorithm to enhance its capability to search more intelligent in problem search space and achieve accurate response in less time. Comparing HYGAGA and genetic algorithm results asserts significant enhancement in the performance of search algorithm. In addition, HYGAGA could attain more appropriate solutions with comparing to other memetic algorithms like GA-HC, GA-SA and GA-TS implemented and analyzed to compare their ability in search in problem search spaces.

II. GRID TASK SCHEDULING Resources introduce themselves and save their specifications in grid information system when grid begins to work. Then, users transfer their task lists to grid and resource broker begin to assign transferred tasks to resources based on its scheduling algorithm [9, 10]. Scheduling process is shown in Fig 1.

Keywords— Grid Computing, Task Scheduling, Gravitational Attraction Search, Genetic Algorithm, Hybrid Algorithms. I. INTRODUCTION

G

RID computing is defined as “a hardware and software infrastructure that provides dependable, consistent, pervasive, and inexpensive access to high-end computational capabilities” [1]. Therefore, a grid computing communicates with a wide range of Heterogeneous resources include personal computers, workstations, super computers and clusters that each of these resources has different computational and configuration facilities and organized by various management policies [2]. Thus, resource scheduling and management is a huge challenge in such a system. Providing an appropriate scheduling algorithm to allocate tasks to resources is important and at the same time is very difficult. Hence, these systems are faced with a NP-hard problem to schedule their tasks and resources to obtain high-performance, accurate response time and maximal revenue. Resource scheduling in computational grids is considered as how to divide tasks

Fig. 1 Grid Resource Scheduling Process

Task list includes task details, task description and user requirements that transferred to grid broker through grid interface [11]. Resource broker converses with grid information system to obtain dynamic information of resources as number of available processor in the resource and process type after receiving task list. Resource broker schedules resources accordance with the needs of users and

Amin. Jula is with the Department of Computer Science, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran (e-mail: [email protected]). Narjes Khatoon. Naseri, is with Department of Computer Science, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran (e-mail: [email protected]).

158

International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia)

scheduling policy. Thence, due to the scheduling Task Dispatcher assign tasks to resources. Finally, results will be collected by Task Receptor. Today grid performance enhancement is a problem which needs appropriate and efficient scheduling. Grid task scheduling focused on task execute-time given the different type and computational load of tasks.

Fijd (t ) =

(2)

× ( x dj (t ) − xid (t ))

Rij (t ) = X i (t ), X j (t ) 2

A. Scheduling algorithms In the face of different conditions, grid requires various scheduling policies and algorithms as FCFS, Min-Min, MaxMin, RR, WQR and FPLTF. FCFS is the simplest scheduling algorithm which selects and assigns the first next resource that is consistent with the transferred task blindly [16]. FPLTF needs speed of resource processor and task rum-time. Tasks and resources will be sorted according to execute-time and speed of resource processor by scheduler. After that the bigger tasks will be assigned to faster or more efficient resources [14]. Tasks are placed in a queue in RR algorithm. Tasks execute respectively and every task has equal time with the others to execute. A task fall into tail of queue provided that it cannot be terminated after its prescribed time and waits for next turn. In WQR each task assign to several appropriate resource duplicative. Number of copies is specified by the user. When one of duplicated tasks terminates, remaining tasks aborted by the scheduler [15]. Some of schedulers apply tasks-prioritization system. This may be implemented using multi-layer queues with different priorities. When all of resources are ready to process, tasks in higher layer execute earlier. Distinct schedulers may be presented using this policy. Min-min algorithm places tasks which can be terminated earlier than others in higher priority. Then, assigns those tasks to resources can execute them in less time. Max-min, contrary to Min-min, locates tasks with higher execute-time in higher priority [17].

(3)

The imposed force on the particle i in direction of d

dimension d in time t, Fi (t ) , is equal to the total amount of the forces of other particles of the system imposed to it, equation 4.

Fi d (t ) =

m

∑r F

j =1, j ≠i

j

d ij

(t )

(4)

According to second Newton's law, each particle gains acceleration in direction of dimension d, and this acceleration is proportional to the imposed force on the particle in the same dimension divided by the particle gravitational mass. The particle acceleration in direction of dimension d in time t is d

shown by ai (t ) and it obtains from equation 5.

d Fi (t ) d ai (t ) = M i (t )

(5)

Velocity of each particle is equal to the sum of a coefficient of the present velocity of the particle and acceleration of the particle that is obtained from equation 6. The new position of the particle i in the dimension d is equal to the sum of its present position and its velocity that computes by equation 7.

Vi d (t + 1) = ri × Vi d (t ) + aid (t )

(6)

xid (t + 1) = xid (t ) + Vi d (t + 1)

(7)

ri and rj are the random numbers by uniform distribution

III. GRAVITATIONAL ATTRACTION SEARCH ALGORITHM AND MEMETIC ALGORITHM

in the interval of (0,1) which have been used to keep the random property of search. Equation 8 is used in order to adjust the gravitational constant. Gravitation constant tends to decline exponentially.

A. Gravitational Attraction Search Algorithm The search space is considered as a set of m particles. Position of each particle in search space is a point in space and is taken in account as a solution to the problem. This position is obtained from equation 1 in which position of particle i in

G (t ) = β

i

dimension d is shown by xd .

X i = ( xi1 ,..., xid ,..., xiD )

G (t ) × M i (t ) × M j (t ) Rij (t ) + ε

−α

1 T

(8)

B. Memetic Algorithms Memetic Algorithm is a kind of evolutionary Algorithm in which local heuristic searches are mixed with another evolutionary algorithm as an example Genetic Algorithm that explores the search space of problem in order to reduce the time of achieving the optimal solutions in NP-hard problems. Last mentioned algorithm which is the main search algorithm in a memetic algorithm is made to search in throughout searching spaces, while the local search explores neighborhood domain of any found solution by the means of the main Algorithms for better solutions. Choosing greater production operators in genetic part of a memetic algorithm 9B

(1)

In this system in time t, a force is imposed from particle I to each particle j in the direction of dimension d as much as

Fijd (t ) . The amount of this force is computed from equation 2 in which G (t ) is gravitation constant in time t, Rij (t ) is distance between two particles i and j in that time and ε is a small number. To obtain distance between the particles Euclidean distance has been used equation 3. 159

International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia)

and the type and the local search method used cause to quite different results [18, 19].

As the best solution, a solution with minimal time to execute tasks and in gravitational attraction search the best solution is solution with largest mass, we can subtract the calculated number for mass from a fixed number and thus it will be a solution with minimum time for execution has maximum amount of mass. Focused on gravitation search algorithm we realize that, due to being too improper solutions than suitable solutions, sum of gravitational masses of improper solutions exceeds the sum of gravitational masses of proper solutions, and consequently improper solutions attract proper solutions and proper solutions will be lost. HYGAGA suggested that for a limited number of ideal solutions explain a field named "Virtual Mass". The amount of virtual mass of each solution will be defined as the basis of its quality compared with other solutions and its rank in the ordered set of solutions. This increases the gravitational mass of solutions and as a result, improper solutions will be absorbed by them. However, determining the number of solutions involved in this is very important. Roulette Wheel algorithm has been used to dynamic determining the number of solutions that should have virtual mass. This algorithm generates a random number between 1 and sum of masses of all solutions. Then, to achieve this random number, algorithm attempts to calculate sum of masses of solutions starting from the best solution. Achieving the desired number in calculating the total condition will finish this process. On the other hand it is obvious that calculating attraction force of improper solutions leads to waste time and loss of proper solutions and has no effects to attain appropriate solutions. HYGAGA calculates the attraction force of K best solutions over other solutions and passes up this computation for other ones. This results keeping current appropriate solutions, absorbing other solutions toward them and search more around these best solutions. Value of K is very important and will be obtain using Roulette Wheel algorithm as already mentioned. The best local solution in ith solution neighborhood, it will be passed to genetic algorithm to be replaced it. After applying local search over all solution of generation, the solution located on first position in sorted population is the best solution. Finally, HYGAGA algorithm will be terminated after attaining desired conditions and presents the best obtained solution.

IV. HYBRID GENETIC ALGORITHM-GRAVITATIONAL ATTRACTION SEARCH ALGORITHM (HYGAGA) Hybrid algorithm HYGAGA designed in order to achieve minimum response-time for the customers and maximum earning profit for the grid system by scheduling tasks. HYGAGA is obtained by incorporating Genetic Algorithm and Gravitational Attraction Search as a local search algorithm and creating a new memetic algorithm to reach appropriate solutions in less time. A solution in HYGAGA consists of an array for assigning resources to tasks, a fitness field which is called Mass and Acceleration and Velocity fields. Initially, each task be allocated to a resource randomly such a way that every resource has several tasks to execute. At this time, a solution has been generated for the problem. Thereafter, fitness of every solution will be calculated as mentioned in Figure 2 and put in its related field. fter generating the first population, solutions will be sorted ascending. Population currently is ready for applying genetic operations. A two-point crossover operator that its two points selected randomly is used to combine chromosomes and generate next generation. After generation each new solution a simple mutation operator with small probability 0.05 will be applied on it to prevent executing process from early congestion. At this time the solution is ready to apply local search algorithm. For this reason, SolNo different solutions will randomly be generated in the neighborhood zone of the solution. Neighborhood applied as assigning a resource to a task in solution i so that maximum difference between assigned resource number and prior resource number be 2. Then, fitness of each solution will be calculated and considered as its Mass as shown in Fig. 2. If Ti is required total time to terminate all assigned tasks in solution i, then the Mass of solution i is Ti . Finally, the solutions sort descending based on mass and the solution with the largest mass as in the equation 9 will be recognized as the best. (9) best (t ) = max fit (t ) j∈{1,...,m} j

V. EXPERIMENTAL RESULTS 4B

Genetic algorithm, GA-Hill Climbing, GA-Simulates Annealing, GA-Tabu Search and HYGAGA have been executed for three random-generated Grid Task Scheduling Problems which included 40, 100 and 180 tasks and 10, 20 and 30 processors respectively. Achieving more accurate results, every algorithm had been executed five times for each problem and the average of obtained results saved for comparison and graphing. Number of iterations was 5000 and number of chromosomes was 200 for executing genetic algorithm. Genetic part of HYGAGA included 200 chromosomes and 30 iterations and its local search involved 200 iterations and 100

Fig. 2 Calculating fitness of solutions

160

International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia)

particles too. These differences led to equality of runtime of two algorithms in used computer for simulation and better analysis of results. On the other hand, effective parameters in other algorithms are determined so that the execution time of all algorithms be equal. Because of much difference between genetic and HYGAGA algorithms in their execution iterations, their execution results graphs in two different diagrams to compare and analysis. The obtained results of five algorithms execution are shown in Fig. 3 and Table I. Analysis of two diagrams and the table shows that both algorithms had best solution with fitness 136. But after that HYGAGA diagram shows drastic reduction process and continues its declining with lower gradient while genetic algorithm has had a slope of a uniform reduction since its first steps. Finally, genetic algorithm shows its convergence in best solution fitness 121 whereas HYGAGA achieved 104 in the same time period and continues its reduction. Achieving better solutions by HYGAGA is also obvious compared with three different memetic algorithms.

VI. CONCLUSION Due to nature of genetic algorithm and its search way in problem search space it needs long time for exploring. Genetic algorithm and local search algorithms hybridization leads to achieving more accurate solutions in less time. Gravitational attraction search algorithm as a new high-performance local search method can lead to meet appropriate solutions in combination with genetic algorithm. In addition, applying virtual mass to particle structures in gravitational attraction search providing sensitive selection of number of particle to apply can help the algorithm to enhance its performance and purposeful movement in search space of problem. On the other hand, comparing hybridization of Genetic algorithm and Hill Climbing GA-HC, Genetic algorithm and Simulated Annealing GA-SA and Genetic algorithm and Tabu Search can reduce the time needed to achieve expected solutions and can lead to obtain better solutions in equal time with GA. Comparing these memetic algorithms with HYGAGA shows that HYGAGA can reach more appropriate solutions in same time significantly.

TABLE I. BEST SOLUTION FITNESS FOR FIRST PROBLEM

GA

GA-HC

GA-SA

GA-TS

HYGAGA

121

111

114

115

104

VII. REFERENCES [1] [2]

Fig. 4 shows execution results of second problem. HYGAGA and genetic algorithms obtained 150 and 193 respectively in their best solution fitness while genetic algorithm is converged and HYGAGA continues to find best solution with lower fitness. Comparing HYGAGA and other memetic algorithms' results for solving this problem shows obtaining more appropriate solutions. It is evident in results are inserted in Table II.

[3]

[4]

[5]

TABLE II. BEST SOLUTION FITNESS FOR SECOND PROBLEM

GA

GA-HC

GA-SA

GA-TS

HYGAGA

193

165

171

175

150

[6]

[7]

[8]

And eventually, it is convince that for the third problem as former results and diagrams, the difference between HYGAGA and genetic algorithms results is fully noticeable whereas HYGAGA decreasing results is obvious and attaining better results providing more iteration possibility is predictable. For this problem diagrams which are shown in Fig. 5 show 173 for amount of fitness of best HYGAGA solution fitness and 209 for genetic algorithm. Due to Table III. It is understandable that HYGAGA could reach better solutions for the problem in comparison to other implemented algorithms.

[9]

[10]

[11]

[12]

TABLE III. BEST SOLUTION FITNESS FOR THIRD PROBLEM

GA

GA-HC

GA-SA

GA-TS

HYGAGA

209

186

192

192

173

[13]

161

Foster and C. Kesselman, "The Grid: Blueprint for a Future Computing Infrastructure", Morgan Kaufmann Publishers, 1999, USA. Ran Zheng, Hai Jin, "An Integrated Management and Scheduling Scheme for Computational Grid", Grid and Cooperative Computing, Lecture Notes in Computer Science Volume 3033, 2004, pp 48-56. Arora, S.K. Das, R. Biswas, "A Decentralized Scheduling and Load Balancing Algorithm for Heterogeneous Grid Environments", M. International Conference on Parallel Processing Workshops, 2002; pp:499–505, Vancouver, British Columbia Canada. Fangpeng Dong, Selim G. Akl, "Scheduling Algorithms for Grid Computing: State of the Art and Open Problems", Technical Report, 2006, No. 504. O. Rossi-Doria, C. Blum, J. Knowles, M. Samples, K. Socha, B. Paechter, "A local search for the timetabling problem", Proceedings of the 4th international conference on the Practice and Theory of Automated Timetabling, 2002; pp.124–127. Gent, Belgium. E. Burke and J. Newall, "A multistage evolutionary algorithm for the timetable problem", IEEE Trans. Evol.Comput, 1999; vol. 3, no. 1, pp. 63–74. K. Burke, G. Kendall, and E. Soubeiga, "A tabu search hyperheuristic for timetabling and rostering", EJournal of Heuristics, 2003; vol. 9, no. 6. A memetic algorithm for multiobjective optimization, Knowles J.D, Corne D.W, M-PAES, Proceedings of the congress on evolutionary computation, 2000; IEEE press, pp. 325–332. R. Buyya and M. Murshed, "GridSim: a toolkit for the modelling and simulation of distributed resource management and scheduling for Grid computing", Concurrency and Computation: Practice and Experience, 2002, vol.14, pp.1175-1220. L. Hao, "Implement of Computational Grid Application Scheduling Simulation with GridSim Toolkit", Journal of Jilin Normal University (Nature Science Edition), 2003, vol.3, pp.63-64. Z. Huifu, Z. Zude, L. Fangmin, "Research on Interface Model of Manufacturing Resource Sharing Grid", China Mechanical Engineering, 2005; vol. 16, pp. 424-427. N. Muthuvelu, J. Liu, and N. L. Soe, "A dynamic job grouping-based scheduling for deploying applications with fine- grained tasks on global grids", Proceedings of the 3rd Australasian Workshop on Grid Computing and e-Research, 2005, Newcastle, New South Wales, Australia, pp. 41-48. C. Wei, Y. Shoubao, and S. Kai, Journal of Huazhong University of Science & Technology (Nature Science Edition), vol. 34, pp. 148-151, 2006.

International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia) [17] Amin Jula, Narjes Khatoon Naseri and Amir Masood Rahmani, "Gravitational Attraction Search with Virtual Mass (GASVM) to solve Static Grid Job scheduling Problem", The Journal of Mathematics and Computer Science, 2010; Vol .1 No.4, pp.305-312. [18] Amin Jula, Narjes Khatoon Naseri, "Using CMAC to Obtain Dynamic Mutation Rate in a Metaheuristic Memetic Algorithm to Solve University Timetabling Problem", European Journal of Scientific Research, 2012, Vol.63 No.2, pp. 172-181. [19] Krasnogor, Jim Smith, "A Tutorial for Competent Memetic Algorithms: Model, Taxonomy, and Design", Natalio IEEE Transactions on Evolutionary Computation, 2005; Vol. 9, No. 5.

[14] Abraham Silberschatz, Peter Baer Galvin, Greg Gagne, "Grouped task scheduling design for coarse-grained grid application", Operating System Concepts, 2005; 7th edition, John Wiley & Sons. [15] M. Maheswaran, S. Ali, H.J. Siegel, D. Hensgen, R. Freund , "Dynamic matching and scheduling of a class of independent tasks onto heterogeneous computing system", Journal of Parallel and Distributed Computing, 1999, 59, pp.107–131. [16] Barry Lynn, "Solving Combinatorial Optimization Problems Using a New Algorithm Based on Gravitational Attraction", Barry Lynn Website, A dissertation submitted to the College of Engineering at Florida Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science.

Table IV. Best Solution Fitness for three different size grid task scheduling problems GA

GA-HC

GA-SA

GA-TS

HYGAGA

40 tasks and 30 Processors

121

111

114

115

104

100 tasks and 30 Processors

193

165

171

175

150

180 tasks and 30 Processors

209

186

192

192

173

Fig. 3 Best solution fitness for a grid task scheduling problem with 40 tasks and 10 processors

Fig. 4 Best solution fitness for a grid task scheduling problem with 100 tasks and 20 processors

Fig. 5 Best solution fitness for a grid task scheduling problem with 180 tasks and 30 processors

162

A Hybrid Genetic Algorithm-Gravitational Attraction Search algorithm (HYGAGA) to Solve Grid Task Scheduling Problem Amin. Jula, Narjes Khatoon. Naseri

between resources with respect to parameters of Quality of Service [3, 4]. On the other hand, memetic algorithms such as other methods are used to solve grid task scheduling problem in several cases [5, 6]. Memetic algorithms proved that they can reach more success in solving NP-hard problems compared with other applied algorithms [7, 8]. Since the local search has effective and basic role in highperformance behavior of memetic algorithms, a new local search algorithm is proposed in this paper to enhance performance of solving method. It will be shown that this local search algorithm increases speed of finding best solutions as well as finding better solutions in that time.

Abstract— Integrating distributed Heterogeneous resources to solve complicated scientific, commercial and industrial problems is the main purpose of grid computing. An efficient task scheduling system is a vital part to reach this scope. Dynamism and Heterogeny of resources in grid cause to need to a complicated scheduling system. Scheduling problem in such systems is resource assigning way to tasks compliance with quality of service parameters. In this paper Hybrid Genetic Algorithm-Gravitational Attraction Search, HYGAGA, proposed as a memetic algorithm to solve grid task scheduling problem. In this proposed approach gravitational attraction search as a local search algorithm has been associated to genetic algorithm to enhance its capability to search more intelligent in problem search space and achieve accurate response in less time. Comparing HYGAGA and genetic algorithm results asserts significant enhancement in the performance of search algorithm. In addition, HYGAGA could attain more appropriate solutions with comparing to other memetic algorithms like GA-HC, GA-SA and GA-TS implemented and analyzed to compare their ability in search in problem search spaces.

II. GRID TASK SCHEDULING Resources introduce themselves and save their specifications in grid information system when grid begins to work. Then, users transfer their task lists to grid and resource broker begin to assign transferred tasks to resources based on its scheduling algorithm [9, 10]. Scheduling process is shown in Fig 1.

Keywords— Grid Computing, Task Scheduling, Gravitational Attraction Search, Genetic Algorithm, Hybrid Algorithms. I. INTRODUCTION

G

RID computing is defined as “a hardware and software infrastructure that provides dependable, consistent, pervasive, and inexpensive access to high-end computational capabilities” [1]. Therefore, a grid computing communicates with a wide range of Heterogeneous resources include personal computers, workstations, super computers and clusters that each of these resources has different computational and configuration facilities and organized by various management policies [2]. Thus, resource scheduling and management is a huge challenge in such a system. Providing an appropriate scheduling algorithm to allocate tasks to resources is important and at the same time is very difficult. Hence, these systems are faced with a NP-hard problem to schedule their tasks and resources to obtain high-performance, accurate response time and maximal revenue. Resource scheduling in computational grids is considered as how to divide tasks

Fig. 1 Grid Resource Scheduling Process

Task list includes task details, task description and user requirements that transferred to grid broker through grid interface [11]. Resource broker converses with grid information system to obtain dynamic information of resources as number of available processor in the resource and process type after receiving task list. Resource broker schedules resources accordance with the needs of users and

Amin. Jula is with the Department of Computer Science, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran (e-mail: [email protected]). Narjes Khatoon. Naseri, is with Department of Computer Science, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran (e-mail: [email protected]).

158

International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia)

scheduling policy. Thence, due to the scheduling Task Dispatcher assign tasks to resources. Finally, results will be collected by Task Receptor. Today grid performance enhancement is a problem which needs appropriate and efficient scheduling. Grid task scheduling focused on task execute-time given the different type and computational load of tasks.

Fijd (t ) =

(2)

× ( x dj (t ) − xid (t ))

Rij (t ) = X i (t ), X j (t ) 2

A. Scheduling algorithms In the face of different conditions, grid requires various scheduling policies and algorithms as FCFS, Min-Min, MaxMin, RR, WQR and FPLTF. FCFS is the simplest scheduling algorithm which selects and assigns the first next resource that is consistent with the transferred task blindly [16]. FPLTF needs speed of resource processor and task rum-time. Tasks and resources will be sorted according to execute-time and speed of resource processor by scheduler. After that the bigger tasks will be assigned to faster or more efficient resources [14]. Tasks are placed in a queue in RR algorithm. Tasks execute respectively and every task has equal time with the others to execute. A task fall into tail of queue provided that it cannot be terminated after its prescribed time and waits for next turn. In WQR each task assign to several appropriate resource duplicative. Number of copies is specified by the user. When one of duplicated tasks terminates, remaining tasks aborted by the scheduler [15]. Some of schedulers apply tasks-prioritization system. This may be implemented using multi-layer queues with different priorities. When all of resources are ready to process, tasks in higher layer execute earlier. Distinct schedulers may be presented using this policy. Min-min algorithm places tasks which can be terminated earlier than others in higher priority. Then, assigns those tasks to resources can execute them in less time. Max-min, contrary to Min-min, locates tasks with higher execute-time in higher priority [17].

(3)

The imposed force on the particle i in direction of d

dimension d in time t, Fi (t ) , is equal to the total amount of the forces of other particles of the system imposed to it, equation 4.

Fi d (t ) =

m

∑r F

j =1, j ≠i

j

d ij

(t )

(4)

According to second Newton's law, each particle gains acceleration in direction of dimension d, and this acceleration is proportional to the imposed force on the particle in the same dimension divided by the particle gravitational mass. The particle acceleration in direction of dimension d in time t is d

shown by ai (t ) and it obtains from equation 5.

d Fi (t ) d ai (t ) = M i (t )

(5)

Velocity of each particle is equal to the sum of a coefficient of the present velocity of the particle and acceleration of the particle that is obtained from equation 6. The new position of the particle i in the dimension d is equal to the sum of its present position and its velocity that computes by equation 7.

Vi d (t + 1) = ri × Vi d (t ) + aid (t )

(6)

xid (t + 1) = xid (t ) + Vi d (t + 1)

(7)

ri and rj are the random numbers by uniform distribution

III. GRAVITATIONAL ATTRACTION SEARCH ALGORITHM AND MEMETIC ALGORITHM

in the interval of (0,1) which have been used to keep the random property of search. Equation 8 is used in order to adjust the gravitational constant. Gravitation constant tends to decline exponentially.

A. Gravitational Attraction Search Algorithm The search space is considered as a set of m particles. Position of each particle in search space is a point in space and is taken in account as a solution to the problem. This position is obtained from equation 1 in which position of particle i in

G (t ) = β

i

dimension d is shown by xd .

X i = ( xi1 ,..., xid ,..., xiD )

G (t ) × M i (t ) × M j (t ) Rij (t ) + ε

−α

1 T

(8)

B. Memetic Algorithms Memetic Algorithm is a kind of evolutionary Algorithm in which local heuristic searches are mixed with another evolutionary algorithm as an example Genetic Algorithm that explores the search space of problem in order to reduce the time of achieving the optimal solutions in NP-hard problems. Last mentioned algorithm which is the main search algorithm in a memetic algorithm is made to search in throughout searching spaces, while the local search explores neighborhood domain of any found solution by the means of the main Algorithms for better solutions. Choosing greater production operators in genetic part of a memetic algorithm 9B

(1)

In this system in time t, a force is imposed from particle I to each particle j in the direction of dimension d as much as

Fijd (t ) . The amount of this force is computed from equation 2 in which G (t ) is gravitation constant in time t, Rij (t ) is distance between two particles i and j in that time and ε is a small number. To obtain distance between the particles Euclidean distance has been used equation 3. 159

International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia)

and the type and the local search method used cause to quite different results [18, 19].

As the best solution, a solution with minimal time to execute tasks and in gravitational attraction search the best solution is solution with largest mass, we can subtract the calculated number for mass from a fixed number and thus it will be a solution with minimum time for execution has maximum amount of mass. Focused on gravitation search algorithm we realize that, due to being too improper solutions than suitable solutions, sum of gravitational masses of improper solutions exceeds the sum of gravitational masses of proper solutions, and consequently improper solutions attract proper solutions and proper solutions will be lost. HYGAGA suggested that for a limited number of ideal solutions explain a field named "Virtual Mass". The amount of virtual mass of each solution will be defined as the basis of its quality compared with other solutions and its rank in the ordered set of solutions. This increases the gravitational mass of solutions and as a result, improper solutions will be absorbed by them. However, determining the number of solutions involved in this is very important. Roulette Wheel algorithm has been used to dynamic determining the number of solutions that should have virtual mass. This algorithm generates a random number between 1 and sum of masses of all solutions. Then, to achieve this random number, algorithm attempts to calculate sum of masses of solutions starting from the best solution. Achieving the desired number in calculating the total condition will finish this process. On the other hand it is obvious that calculating attraction force of improper solutions leads to waste time and loss of proper solutions and has no effects to attain appropriate solutions. HYGAGA calculates the attraction force of K best solutions over other solutions and passes up this computation for other ones. This results keeping current appropriate solutions, absorbing other solutions toward them and search more around these best solutions. Value of K is very important and will be obtain using Roulette Wheel algorithm as already mentioned. The best local solution in ith solution neighborhood, it will be passed to genetic algorithm to be replaced it. After applying local search over all solution of generation, the solution located on first position in sorted population is the best solution. Finally, HYGAGA algorithm will be terminated after attaining desired conditions and presents the best obtained solution.

IV. HYBRID GENETIC ALGORITHM-GRAVITATIONAL ATTRACTION SEARCH ALGORITHM (HYGAGA) Hybrid algorithm HYGAGA designed in order to achieve minimum response-time for the customers and maximum earning profit for the grid system by scheduling tasks. HYGAGA is obtained by incorporating Genetic Algorithm and Gravitational Attraction Search as a local search algorithm and creating a new memetic algorithm to reach appropriate solutions in less time. A solution in HYGAGA consists of an array for assigning resources to tasks, a fitness field which is called Mass and Acceleration and Velocity fields. Initially, each task be allocated to a resource randomly such a way that every resource has several tasks to execute. At this time, a solution has been generated for the problem. Thereafter, fitness of every solution will be calculated as mentioned in Figure 2 and put in its related field. fter generating the first population, solutions will be sorted ascending. Population currently is ready for applying genetic operations. A two-point crossover operator that its two points selected randomly is used to combine chromosomes and generate next generation. After generation each new solution a simple mutation operator with small probability 0.05 will be applied on it to prevent executing process from early congestion. At this time the solution is ready to apply local search algorithm. For this reason, SolNo different solutions will randomly be generated in the neighborhood zone of the solution. Neighborhood applied as assigning a resource to a task in solution i so that maximum difference between assigned resource number and prior resource number be 2. Then, fitness of each solution will be calculated and considered as its Mass as shown in Fig. 2. If Ti is required total time to terminate all assigned tasks in solution i, then the Mass of solution i is Ti . Finally, the solutions sort descending based on mass and the solution with the largest mass as in the equation 9 will be recognized as the best. (9) best (t ) = max fit (t ) j∈{1,...,m} j

V. EXPERIMENTAL RESULTS 4B

Genetic algorithm, GA-Hill Climbing, GA-Simulates Annealing, GA-Tabu Search and HYGAGA have been executed for three random-generated Grid Task Scheduling Problems which included 40, 100 and 180 tasks and 10, 20 and 30 processors respectively. Achieving more accurate results, every algorithm had been executed five times for each problem and the average of obtained results saved for comparison and graphing. Number of iterations was 5000 and number of chromosomes was 200 for executing genetic algorithm. Genetic part of HYGAGA included 200 chromosomes and 30 iterations and its local search involved 200 iterations and 100

Fig. 2 Calculating fitness of solutions

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International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia)

particles too. These differences led to equality of runtime of two algorithms in used computer for simulation and better analysis of results. On the other hand, effective parameters in other algorithms are determined so that the execution time of all algorithms be equal. Because of much difference between genetic and HYGAGA algorithms in their execution iterations, their execution results graphs in two different diagrams to compare and analysis. The obtained results of five algorithms execution are shown in Fig. 3 and Table I. Analysis of two diagrams and the table shows that both algorithms had best solution with fitness 136. But after that HYGAGA diagram shows drastic reduction process and continues its declining with lower gradient while genetic algorithm has had a slope of a uniform reduction since its first steps. Finally, genetic algorithm shows its convergence in best solution fitness 121 whereas HYGAGA achieved 104 in the same time period and continues its reduction. Achieving better solutions by HYGAGA is also obvious compared with three different memetic algorithms.

VI. CONCLUSION Due to nature of genetic algorithm and its search way in problem search space it needs long time for exploring. Genetic algorithm and local search algorithms hybridization leads to achieving more accurate solutions in less time. Gravitational attraction search algorithm as a new high-performance local search method can lead to meet appropriate solutions in combination with genetic algorithm. In addition, applying virtual mass to particle structures in gravitational attraction search providing sensitive selection of number of particle to apply can help the algorithm to enhance its performance and purposeful movement in search space of problem. On the other hand, comparing hybridization of Genetic algorithm and Hill Climbing GA-HC, Genetic algorithm and Simulated Annealing GA-SA and Genetic algorithm and Tabu Search can reduce the time needed to achieve expected solutions and can lead to obtain better solutions in equal time with GA. Comparing these memetic algorithms with HYGAGA shows that HYGAGA can reach more appropriate solutions in same time significantly.

TABLE I. BEST SOLUTION FITNESS FOR FIRST PROBLEM

GA

GA-HC

GA-SA

GA-TS

HYGAGA

121

111

114

115

104

VII. REFERENCES [1] [2]

Fig. 4 shows execution results of second problem. HYGAGA and genetic algorithms obtained 150 and 193 respectively in their best solution fitness while genetic algorithm is converged and HYGAGA continues to find best solution with lower fitness. Comparing HYGAGA and other memetic algorithms' results for solving this problem shows obtaining more appropriate solutions. It is evident in results are inserted in Table II.

[3]

[4]

[5]

TABLE II. BEST SOLUTION FITNESS FOR SECOND PROBLEM

GA

GA-HC

GA-SA

GA-TS

HYGAGA

193

165

171

175

150

[6]

[7]

[8]

And eventually, it is convince that for the third problem as former results and diagrams, the difference between HYGAGA and genetic algorithms results is fully noticeable whereas HYGAGA decreasing results is obvious and attaining better results providing more iteration possibility is predictable. For this problem diagrams which are shown in Fig. 5 show 173 for amount of fitness of best HYGAGA solution fitness and 209 for genetic algorithm. Due to Table III. It is understandable that HYGAGA could reach better solutions for the problem in comparison to other implemented algorithms.

[9]

[10]

[11]

[12]

TABLE III. BEST SOLUTION FITNESS FOR THIRD PROBLEM

GA

GA-HC

GA-SA

GA-TS

HYGAGA

209

186

192

192

173

[13]

161

Foster and C. Kesselman, "The Grid: Blueprint for a Future Computing Infrastructure", Morgan Kaufmann Publishers, 1999, USA. Ran Zheng, Hai Jin, "An Integrated Management and Scheduling Scheme for Computational Grid", Grid and Cooperative Computing, Lecture Notes in Computer Science Volume 3033, 2004, pp 48-56. Arora, S.K. Das, R. Biswas, "A Decentralized Scheduling and Load Balancing Algorithm for Heterogeneous Grid Environments", M. International Conference on Parallel Processing Workshops, 2002; pp:499–505, Vancouver, British Columbia Canada. Fangpeng Dong, Selim G. Akl, "Scheduling Algorithms for Grid Computing: State of the Art and Open Problems", Technical Report, 2006, No. 504. O. Rossi-Doria, C. Blum, J. Knowles, M. Samples, K. Socha, B. Paechter, "A local search for the timetabling problem", Proceedings of the 4th international conference on the Practice and Theory of Automated Timetabling, 2002; pp.124–127. Gent, Belgium. E. Burke and J. Newall, "A multistage evolutionary algorithm for the timetable problem", IEEE Trans. Evol.Comput, 1999; vol. 3, no. 1, pp. 63–74. K. Burke, G. Kendall, and E. Soubeiga, "A tabu search hyperheuristic for timetabling and rostering", EJournal of Heuristics, 2003; vol. 9, no. 6. A memetic algorithm for multiobjective optimization, Knowles J.D, Corne D.W, M-PAES, Proceedings of the congress on evolutionary computation, 2000; IEEE press, pp. 325–332. R. Buyya and M. Murshed, "GridSim: a toolkit for the modelling and simulation of distributed resource management and scheduling for Grid computing", Concurrency and Computation: Practice and Experience, 2002, vol.14, pp.1175-1220. L. Hao, "Implement of Computational Grid Application Scheduling Simulation with GridSim Toolkit", Journal of Jilin Normal University (Nature Science Edition), 2003, vol.3, pp.63-64. Z. Huifu, Z. Zude, L. Fangmin, "Research on Interface Model of Manufacturing Resource Sharing Grid", China Mechanical Engineering, 2005; vol. 16, pp. 424-427. N. Muthuvelu, J. Liu, and N. L. Soe, "A dynamic job grouping-based scheduling for deploying applications with fine- grained tasks on global grids", Proceedings of the 3rd Australasian Workshop on Grid Computing and e-Research, 2005, Newcastle, New South Wales, Australia, pp. 41-48. C. Wei, Y. Shoubao, and S. Kai, Journal of Huazhong University of Science & Technology (Nature Science Edition), vol. 34, pp. 148-151, 2006.

International Conference on Soft Computing and its Applications(ICSCA'2012) August 25-26, 2012 Kuala Lumpur (Malaysia) [17] Amin Jula, Narjes Khatoon Naseri and Amir Masood Rahmani, "Gravitational Attraction Search with Virtual Mass (GASVM) to solve Static Grid Job scheduling Problem", The Journal of Mathematics and Computer Science, 2010; Vol .1 No.4, pp.305-312. [18] Amin Jula, Narjes Khatoon Naseri, "Using CMAC to Obtain Dynamic Mutation Rate in a Metaheuristic Memetic Algorithm to Solve University Timetabling Problem", European Journal of Scientific Research, 2012, Vol.63 No.2, pp. 172-181. [19] Krasnogor, Jim Smith, "A Tutorial for Competent Memetic Algorithms: Model, Taxonomy, and Design", Natalio IEEE Transactions on Evolutionary Computation, 2005; Vol. 9, No. 5.

[14] Abraham Silberschatz, Peter Baer Galvin, Greg Gagne, "Grouped task scheduling design for coarse-grained grid application", Operating System Concepts, 2005; 7th edition, John Wiley & Sons. [15] M. Maheswaran, S. Ali, H.J. Siegel, D. Hensgen, R. Freund , "Dynamic matching and scheduling of a class of independent tasks onto heterogeneous computing system", Journal of Parallel and Distributed Computing, 1999, 59, pp.107–131. [16] Barry Lynn, "Solving Combinatorial Optimization Problems Using a New Algorithm Based on Gravitational Attraction", Barry Lynn Website, A dissertation submitted to the College of Engineering at Florida Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science.

Table IV. Best Solution Fitness for three different size grid task scheduling problems GA

GA-HC

GA-SA

GA-TS

HYGAGA

40 tasks and 30 Processors

121

111

114

115

104

100 tasks and 30 Processors

193

165

171

175

150

180 tasks and 30 Processors

209

186

192

192

173

Fig. 3 Best solution fitness for a grid task scheduling problem with 40 tasks and 10 processors

Fig. 4 Best solution fitness for a grid task scheduling problem with 100 tasks and 20 processors

Fig. 5 Best solution fitness for a grid task scheduling problem with 180 tasks and 30 processors

162