A 2-dimentional Crossover for Multi-objective Evolutionary Scheduling of Soft Real-time Tasks

A 2-dimensional Xover-based non-dominated sorting (2DXNS) genetic algorithm is proposed for multi-objective scheduling of static soft real-time tasks on heterogeneous computing systems. The objectives of scheduling problems are generally to minimize makespan, total processor idle time, and the number of processors as well as maximize guarantee ratio. In contrast to earlier approaches that commonly convert the problem to a single objective form; we propose an inherently multi-objective approach that aims to make appropriate tradeoffs among all of the above objectives. A new 2-dimensional structure is also proposed for crossover/mutation operator, in comparison to the more common crossover operators. This operator promotes better recombination due to processor based task decomposition. The simultaneous operation of this crossover operator across these ‘smaller’ problems increases better mating/combination, while trying to maintain more ‘sections’ of the parent chromosomes. This leads to better results with fewer generation passes. Finally, we compare the proposed algorithm with HEFT, a well-known list scheduling algorithm. Experiments on a real world application as well as three sets of random DAGs demonstrate the high efficiency of the proposed 2DXNS in multiprocessor task scheduling. KEYWORDSDirected Acyclic Task Graph (DAG), Heterogeneous Multiprocessor Systems, Load Balancing, Multi-objective Optimization, Task Scheduling, Genetic Algorithm Multiprocessor task scheduling is a challenging problem both in terms of its complexity in decision making and optimization as well as its importance in terms of applicability to modern systems. More specifically, from the perspective of decision making and optimization, this problem can be seen at the crossroads of three important fields of research, i.e. multiprocessor systems, real time decision making, and multi-objective optimization, each of which present a challenging area with its own significant set of complexities [1]. Furthermore, from an application perspective, many real world problems such as aircraft[2], networks [3], electronics and semiconductor manufacturing[4] require tradeoffs to be made among multiple objectives in order to optimize the overall performance of the system. One specific real world application of multiprocessor task scheduling is project management. Each processor corresponds to a resource (human or non-human) and each task corresponds to a sub-project that must be performed withina specific parallel order so that total time of project is minimized. Obviously, the multi-objective scheduling problems are more complex than their traditional single objective problems due to inconsistency, confluence or even contradiction among objectives [5]. Multiprocessor systems present their own set of complexities where inappropriate scheduling of tasks can lead to deteriorated performance of the system in terms of either excessive communication overhead or underutilization of resources [6, 7]. In a heterogeneous computing system, for instance, if one processor executes a given task faster than other processors, it does not necessarily mean that it executes all tasks fastest, due to the differences in internal architecture of different processor. Finally, from a real-time system perspective, additional soft or hard time constraints must be satisfied. In hard real-time systems, the violation of timing constraints of a certain task can be dangerous. Some examples of hard real-time systems are patient monitoring systems and nuclear plant control. While in the soft real time system, the violation of timing constraints of a task results in decreasing usefulness of results over time after the deadline expires without causing any damage to the controlled environment. Some examples of soft real-time systems are telephone switching systems and image processing applications [8]. Task scheduling problem is considered for two different multiprocessor systems: multiprocessor systems with fully connected topology [8-14] and arbitrary topology [6, 15-20]. In its general form, multiprocessor task scheduling is an NP-complete problem [9, 21]. There are two well-known special cases for this problem where optimal polynomial-time solutions are known. These SEDAGHAT ET AL.: A 2-DIMENTIONAL CROSSOVER FOR MULTI-OBJECTIVE EVOLUTIONARY SCHEDULING OF... Indian J.Sci.Res. 4 (3): 318-334, 2014 cases are scheduling tree-structured task graphs with identical computation costs on an arbitrary number of processors and scheduling arbitrary task graphs with identical computation costs on two processors. In these two cases, no communication is assumed among the tasks of the parallel program [9]. Hence, while many researchers have to date proposed various algorithms, their solutions generally simplify the actual problem and fail in one or more ways to consider the true complexity of an actual real time multiprocessor task scheduling problem. Task scheduling problem is inherently multiobjective fashion. Because of the complexity of the scheduling problem, many researchers use evolutionary algorithms in its single objective form [8, 10, 11, 13, 14, 22-24]. Converting multi-objective problem to single objective problem has to make hard tradeoffs among objectives that can lead to reduced performance. Here, we propose a new 2-dimensional Xover-based nondominated sorting (2DXNS) genetic algorithm that takes benefit from the pioneering works of [25] on nondominated sorting genetic algorithm II (NSGA-II) for multi-objective scheduling of static soft real-time tasks on heterogeneous computing systems. Real-world constraints including the precedence relationship between tasks as well as communication costs between the processors are considered. In contrast to the existing literature, the proposed multi-objective approach simultaneously considers several main objectives in scheduling problem such as length of scheduling (makespan), total tardiness (for real time tasks), number of processors as well as guarantee ratio. Furthermore, a new 2-dimensional Xover/mutation operator is proposed that, in comparison to the more common crossover operators such as cycle [10] and one-point [8, 11, 14] crossovers, promotes better recombination in the task scheduling problem. The 2-D representation decomposes the list (vector) of tasks to a task matrix with each row corresponding to a processor; hence it essentially decomposes the large task scheduling problem to several smaller problems. Simultaneous crossover operation across this matrix leads to better mating/combination while larger segments of each parent’s solutions are maintained. As a benchmark for comparison, the proposed 2DXNS is compared with HEFT (heterogeneous earliest finish time)[26], a well-known list scheduling algorithm. In these experiments, the performance of the 2DXNS is examined under different conditions: varying number of tasks, varying communication to computation ratio (CCR) and varying Sparsity of directed acyclic graph (DAG). The rest of the paper is organized as follows. In Section 2, the multiprocessor task scheduling problem is formulated. Section 3 presents a brief review of the prior relevant works. 2DXNS is presented in Section 4. Experimental results are illustrated in Section 5. Finally, Section 6 concludes the paper and suggests several ideas for further development.