# A Multi-Objective Performance Evaluation in Grid Task Scheduling using Evolutionary Algorithms

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3 =,,, (1),,,,, (2),,, :,,, (3) t, r, t, r, (4) Where,,,,,, =1,, (5),, r, (6) s= t r,,t r,,..,t r,, s (7) 1,,, 1,,, 1,,, 1,, and 1,, (8) Equation (2) indicates that all the attributes requirements of a task have to be satisfied under hard constraints. Equation (3) ensures that available resources meet the resource requirements for a task and equation (4) is the access rights constrain of an user to resources to execute their tasks. Equations (5) and (6) show the sets and attributes used in the mathematical model and equation (7) indicates a feasible allocation of tasks to resources, which should be part of the set ( of all feasible solutions that satisfy the problem constraints and meet the definition of the Pareto Front [11]. 4 NSGA-II for Grid Task Scheduling In year 2000, Kalyanmoy Deb et. al. in [12] presented the Non-dominated Sorting Genetic Algorithm II (NSGA-II), which is a review of the NSGA algorithm of MOEA s first generation. NSGA-II allows complexity reduction, incorporates an elitism operator and eliminates the parameters on the diversity operator, allowing for greater transparency in the algorithm. 4.1 Evolutionary Algorithm Operators The process of convergence of the solutions set to the Pareto Front is highly linked to the use of evolutionary operators. Now, the configuration parameters and operators designed for the NSGA-II applied to the Job Scheduling Problem are described. The algorithm 1 shows NSGA-II pseudo code applied in task scheduling and Figure 1 show the algorithm s operation. Fig. 1 NSGA-II Operation Chromosome, Comparison and Selection The chromosome encoding is done indirectly. Given a chromosome of length n, the position of each gene in the chromosome is the task number identifier and the allele value (value the gene can take) m is the resource to which the task is assigned. If n is the number of task to be scheduled in the batch and m the amount of resources available, an instance, with =6 and =3 indicates that the chromosome that represents a solution is encoded as a vector of six positions, where each position can take the values 1, 2 and 3. The crowding comparison operator is used to choose the solution set of the last front population trying to enter the next generation population. Binary tournament operator is used into the creation of child population Crossover and Mutation The crossover operator extends the search space to generate new solutions from the existing ones. The selected crossover strategy is the single point crossover [8], which showed a good performance to solve this problem. ISBN:

4 The mutation operation provides Metaheuristics with local search functions and solution diversity. The mutation process acts in three different ways: a) Fast Load Balancing between the most loaded (greater makespan) and the least loaded (less makespan). This heuristic selects the most loaded resource and the less loaded of a chromosome and exchanges two task. b) Random migration of jobs between resources. Given a chromosome, two genes are randomly selected and the value of their alleles is exchanged. c) Min-Min strategy. Given a random point in the chromosome, the Min-Min heuristics generate the remaining values of the chromosome. d) Incorporation of new random population. Incorporates new solutions in a random fashion to extend the search space which must meet the problem constrains. The strategy of mutation has a low complexity and increases the capabilities of the NSGA-II to find the real Pareto Front. The procedures a), b) and c) apply to (90%) of the individuals to be mutated and the remaining (10%) are eliminated and replaced by the procedure d). 4.2 Evolutionary Algorithm Parameters The initial population in our algorithm is mostly randomly generated meeting the problem constraints, but maintaining a low complexity in the algorithm. On the other hand, a few individuals are generated with fast Metaheuristics/Heuristics, enabling greater convergence and higher speed of the algorithm. The proposed initial Population, uses a size of 100 individuals, where 95 (95%) are randomly generated, and of the remaining 5 (5%), four are generated by genetic algorithms (G.A) where one of the individuals minimizes the makespan and the other minimizes the flowtime. The last individual is generated by the Min-Min heuristic. The termination criterion used is the number of maximum generations, which was set at 500. This value was experimentally chosen because any increase upon this value, offered no further improvements in the solutions. At last, the relevant objective function at Grid according to [10] can be classified according to the stakeholders involved in the Grid (users, resource providers and administrators of the VO). For the model, we selected two of these functions to observe their behavior in the performance tests of the Multi-Objective Algorithm: load balancing (makespan minimization) and job completion time minimization (flowtime minimization). 5 Evaluations and Results One of the best ways to measure the quality of a Multi-Objective algorithm is using metrics to evaluate it against theoretical solutions of the models. In this work, we built a real Pareto Front evaluating the mathematical problem with the method of -constrains and using the deterministic/exact algorithm of B&B. The obtain results allow us first to use the metric of generational distance (GD) [11] to evaluate the Pareto front convergence found with our proposal against the real Pareto front found by the exact algorithm. On the other hand, it is necessary to evaluate the ability to generate well-distributed solutions through the real Pareto, so the metric spacing (S) [11] is use. The DG and S equation are presented in equation 9 and 10. The algorithm was run 20 times and on each run the GD and S metrics were obtained. Subsequently their average and standard deviation are obtained. The ε-constrains method configuration parameters are presented in table 1. Options Integer Problem Relaxation Solution Search Heuristic Value Simplex-Dual Pure Depth First Quality Integer solution for Acceptance 0.05% Max Time Exploration for Solution 3600 seconds ε values calculated 30 Table 1. -constrains method configuration parameters. DG= (9) = S= d -d (10) 5.1 Pareto Front, Spacing and GD Results In this evaluation, three instances of the problem were used to calculate the real Pareto Front and the one generated by our proposal. Being one instance a pair like (m, n), where m is number of machines and n the number of jobs. Then the instances used for evaluation were: small instance (3,13), which generates a search space of 1,594,323 solutions, medium instance (5,100) that generates a space of 7.88x10 solutions and a big instance (50,300) with a search space of solutions. The obtained Pareto Fronts are presented in the figures 2, 3 and 4 and spacing and generational distance values in table 2. ISBN:

5 Fig. 2 Pareto Front in the instance (3,13) Fig. 3 Pareto Front in the instance (5,100) The instance s (3,13) results show that all the points of the real Pareto were found by the MOEA, thus when computing its spacing and the generational distance, the values are below the unity because of the precision on the decimal values of the exact algorithm. These results confirm the convergence of the algorithm towards the real Pareto front. Furthermore, both the generational distance and the spacing held a low standard deviation, which shows that the algorithm converges always in every execution. Instance Metric Average Standard deviation (3,13) GD 0,091 0,038 Spacing (5,100) GD 1,29 0,75 Spacing 1,44 1,18 No calculated because ε-constrains (50,300) GD method don t work (the memory RAM was exceeded) Spacing 1,787 1,3217 Table 2. Generational Distance and Spacing metrics results Fig. 4 Pareto Front in the instance (50,300) In the (5,100) instance, the algorithm s efficiency was demonstrated again, even in bigger search spaces. The calculated metrics show the high quality and diversity of the Pareto s front non-dominated solutions found by the proposed algorithm. Although the spacing and the GD both increased, they are still low in scale of the objective functions. Finally, in the instance (large-scale combinatory problem), the program used to calculate the optimum values of the real Pareto did not work. The space and computational complexity of the mathematical programming model turned the deterministic algorithm of the classic method useless. The calculated spacing and its deviation showed that through each of the algorithm s execution, the Pareto s distribution almost always stayed constant. This indicates that the generated fronts on each execution were similar one to another, meaning convergence to the same optimum values. 5.3 Complexity Analysis The computational complexity of NSGA-II algorithm is polynomial and equal to for each G generation, where is the number of objective functions and the individuals in the population, meaning its implementation complexity is and this is independent of the number of resources and task. On the other hand, B&B has a complexity equal to fully evaluate the search space of combinatorial problem, and it is concluded that this algorithm is, in the worst case, of exponential complexity, where is the number of resource and is the number of task. In a practical way, the execution times of our proposal and of the traditional method were measured. The table 3 shows the practical results of the execution time in both algorithms. This table shows the inefficiency of the deterministic algorithms in NP-Complete problems. In a small problem instance, the difference, in terms of execution time elapsed, between algorithms is low. However, in large problem instance the deterministic algorithm have very high execution therefore the optimization results should be obtained as a gap respect to optimization problem relaxed solution. The last instance (50,300) the ISBN:

6 algorithm deterministic do not finalize because its spatial complexity beyond the capabilities of the machine used for assessments (Dual Core Xeon Processor and 4 GB of RAM). Instance NSGA-II for Grid Scheduling (seconds) ε-constrains using Branch and Bound (seconds) >> ,36 62, , ,27 << ,28 Intractable Table 3. Execution Time of NSGA-II and Branch and Bound 7 Conclusions and Future Work It is important to note that the proposed algorithm finds a set of optimal solutions (close to the Optimal Pareto Front). The charts presented and supported by the GD and spacing metrics, showed that the use of MOEA algorithms is highly recommended to tackle Multi-Objective task scheduling problems which are of combinatory nature and generate a large scale and/or non-convex search space. In the medium-large scale size Grid scheduling problem (5-100 y ), the difficulties suffered by the traditional deterministic-algorithm method when trying to find the exact Pareto front (because of its computational and spatial complexity) were noted. In the instances, the computational complexity caused the B&B algorithm s execution to be very slow to find the ε values used to generate the real Pareto front using the MO classic method. In the instances, the B&B algorithm failed its execution because the optimizer software doesn t work because the machine memory capacity is surpassed. The problem restriction matrix used above 4GB of RAM. It is worthwhile to remember that HTC s task scheduling Problem includes scenarios of hundreds of thousands of resources/task that must be schedule quickly, where deterministic algorithms solutions cannot be used and therefore, it necessary to use alternatives such as the one suggested here, which showed to be efficient and effective when tackling the problem. As a future work, it is proposed to compare the algorithm with other MOEA on large-scale instances. Furthermore, it is also suggested to begin evaluating other evolutionary operators and incorporating new heuristics on the mutation operator to improve a faster the algorithm s convergence to the real Pareto. References: [1] Lei, Zhou and Yun, Zhifeng, Allen, Gabrielle., "Grid Resource Allocation." [ed.] Lizhe Wang, Wei Jie and Jinjun Chen. Grid Computing: Infraestructure, Service, and Applications. Boca Raton : CRC Press, 2009, 7, pp [2] Braun, Tracy D., et al., A comparison of eleven static heuristics for mapping a class of independent tasks onto heterogeneous distributed computing systems. s.l. : Journal of Parallel and Distributed Computing, Academic Press, Inc, pp Vol. 61. ISSN: [3] Izakian, Hesam, Abraham, Ajith and Snasel, Václav., "Comparison of Heuristics for Scheduling Independent Tasks on Heterogeneous Distributed Enviroments." Proceedings of the 2009 International Joint Conference on Computational Sciences and Optimization. s.l. : IEEE Computer Society, Vol. 1, pp [4] Xhafa, Fatos and Abraham, Ajith., Metaheuristics for Scheduling in Distributed Computing Environments ISBN: [5] Bäck, Thomas., Evolutionary Algorithms in Theory and Practice. New York : Oxford University Press, Inc, ISBN [6] Pinedo, Miguel L., Scheduling: Theory, Algorithms, and Systems. Fifth Edition. s.l. : Springer, p ISBN [7] Ye, Guangchang, Rao, Ruonan and Li, Minglu., A Multiobjective Resources Scheduling Approach Based on Genetic Algorithms in Grid Environment. Hunan, China : Fifth International Conference on Grid and Cooperative Computing Workshops, ISBN: [8] Deb, Kalyanmoy., Multi-Objective Optimization using Evolutionary Algorithms. New York : John Wiley & Sons, Ltd., ISBN: X. [9] Grimme, Christian, Lepping, Joachim and Papaspyrou, Alexander., "Discovering Performance Bounds for Grid Scheduling by using Evolutionary Multiobjective Optimization." Atlanta, GA, USA : Proceedings of the 10th annual conference on Genetic and evolutionary computation, ACM, 2008, pp [10] WG, OGSA HPC Profile., "HPC Job Scheduling: Base Case and Common Cases." [Online] [Cited: 06 10, 2010.] [11] Deb, K., et al., "A fast elitist non-dominated sorting genetic algorithm for multi-objective optimisation: NSGA-II." PPSN VI: Proceedings of the 6th International Conference on Parallel Problem Solving from Nature, London, UK : Springer Verlag, 2000, pp [12] Open Grid Forum - Group Scheduling Working., "Ten Actions When SuperScheduling." [Online] ISBN:

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