Discrete Optimization
Heuristics and augmented neural networks for task scheduling with non-identical machines

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Abstract

We propose new heuristics along with an augmented-neural-network (AugNN) formulation for solving the makespan minimization task-scheduling problem for the non-identical machine environment. We explore four task and three machine-priority rules, resulting in 12 combinations of single-pass heuristics. The task-priority rules are Highest-Level-First (HLF), Highest-Total-Remaining-Processing-Time-First (HTRPTF), Smallest-Latest-Finish-Time-First (SLFTF) and Minimum-Slack-First (MSF). For machine priority, we propose a greedy rule called Fastest-Available-Machine-First (FAMF), and two non-greedy rules: (1) Fastest-Available-Machine-First-With-Conditional-Wait-1 (FAMF-CW-1), and (2) Fastest-Available-Machine-First-With-Conditional-Wait-2 (FMF-CW-2). The AugNN approach integrates neural-network learning with domain and problem specific knowledge through heuristics, to produce good results. A single-pass heuristic is embedded in a neural network designed specifically for each problem. We give the AugNN formulation for each of the 12 heuristics and show computational results on 100 randomly generated problems of sizes ranging from 20 to 70 tasks and 2 to 5 machines. Results demonstrate that AugNN provides significant improvement over single-pass heuristics. The reduction in the gap between the obtained solution and the lower-bound due to AugNN over single-pass heuristics ranged from 24.4% to 50%. As far as heuristics, the non-greedy machine-priority rules performed significantly better than the greedy rule. The average gaps for the non-greedy rules ranged from 16.1% to 23.5% compared to 33.7% to 40.4% for greedy. For AugNN, for non-greedy rules, the gap ranged from 11.5% to 15.5% compared to 18.4% to 25.0% for greedy. The HLF and HTRPTF task priority rules performed better than the other two. The HTRPTF/FAMF-CW-1 combination gave the best results, closely followed by HLF/FAMF-CW-2 and HTRPTF/FAMF-CW-2 combinations.

Introduction

The problem of optimally scheduling non-preemptive tasks with precedence constraints, on non-identical machines (or processors) belongs to the class of NP-Hard problems (De and Morton, 1980). The problem occurs in many computing environments. For example, in a multiprocessor environment, a computing job is broken down into smaller tasks, to be performed on different, and often, non-identical processors (Ibarra and Kim, 1977, El-Rewini et al., 1995). In network computing environment, a job may be processed on several idle processors available on a computer network (Saltzman, 1995). Processors on a network are likely to be non-identical. The problem also appears in an n-tier client–server environment where a job is distributed across servers of varying speeds. Similarly, on the Internet, file transfer occurs in packets, where packets go through various paths having processors of different speeds. Traditionally, the scheduling problem with non-identical machines is found in the manufacturing environment (Graves, 1981). There is therefore a need for good scheduling techniques for scheduling in the non-identical machine environment. Any improvements in the solution procedures for scheduling for this type of problem will have implications in both computing and manufacturing environments.

We consider such a task-scheduling problem in which a set of non-preemptive tasks with precedence constraints is to be scheduled on non-identical and non-uniform machines, to minimize makespan. Each task needs to be processed on only one machine and a machine can handle only one task at a time. The distinguishing feature of this problem is that the processing times of tasks depend on the machine. We assume non-uniformity of machines, i.e., a fast machine is not always faster than a slower machine for all tasks. For example, assume that machine A has a higher processing speed than machine B, but that machine B has higher memory. A CPU intensive task would run faster on machine A, while an I/O intensive task would run faster on machine B. The uniform machine case would be a special case of the non-identical machine case.

Due to its NP-Hard nature, there are not likely to be any polynomial-time algorithms to optimally solve this problem. Numerous heuristics are available for various scheduling problems involving the use of different dispatching rules for scheduling unexecuted ready tasks to machines or processors (Panwalker and Iskander, 1977, El-Rewini et al., 1995).

Several heuristics and other iterative search techniques exist for the task-scheduling problem for the case of identical machines (Kasahara and Narita, 1984, Agarwal et al., 2003, Agarwal et al., in press), or uniform machines (Epstein and Sgall, 2000, Gonzalez and Sahni, 1978), but, to the best of our knowledge, none for non-identical machines. Heuristics exist for non-identical machine scheduling problem for the case where all tasks are ready at time zero, i.e., the tasks do not follow precedence constraints (De and Morton, 1980). For the problem we consider in this paper, a heuristic would be a combination of a task-priority rule and a machine priority rule. While task-priority rules have been well studied, machine priority rules have not.

One of the difficulties with the non-identical machine problem is finding good quality lower-bounds for comparison of results. In the identical-machine case, relaxing the resource constraint gives a quick critical-path based lower-bound of fairly good quality. For the non-identical machine case, finding a lower-bound requires a second assumption that all tasks are assigned to the fastest machine. If the differences in processing times between a faster and a slower machine are substantial, a likely scenario, this second assumption leads to poor quality lower bounds. Therefore, the gaps between the obtained solution and the lower-bound are likely to be much wider for the non-identical machine case than those for the identical-machine case. In other words, the performance ratio (solution to lower bound) for search techniques for this type of problem will inherently be poor. Further, it will be hard to show the optimality of any reasonably sized problem. The only way to show the effectiveness of an iterative search technique would be to show significant reductions in the gaps, compared to single-pass heuristics.

In this paper, we propose 12 heuristics and augmented neural networks (AugNN) for solving this non-identical machine scheduling problem. As mentioned earlier, while heuristics for the identical-machine case involve only task-priority rules, those for the non-identical-machine case are a composite of a task-priority rule and a machine-priority rule. We explore four rules for task priority and three for machine priority. The task-priority rules we consider are routinely found in scheduling literature (Hu, 1961, Panwalker and Iskander, 1977, Agarwal et al., 2003, Agarwal et al., in press). Same heuristics often appear in different studies with different names or acronyms. We consider Highest-Level-First (HLF), Highest-Total-Remaining-Processing-Time-First (HTRPTF), Smallest-Latest-Finish-Time-First (SLFTF) and Minimum-Slack-First (MSF). Machine-priority rules for the non-identical machine case are not found in the literature. We propose three rules, one greedy and two non-greedy. The greedy heuristic is called Fastest-Available-Machine-First (FAMF). The non-greedy rules are (1) Fastest-Available-Machine-First-with-Conditional-Wait-1 (FAMF-CW-1), and (2) Fastest-Available-Machine-First-With-Conditional-Wait-2 (FAMF-CW-2). Twelve heuristics result from various combinations of these four task and three machine priority rules.

We also formulate the AugNN approach for this problem and show significant improvements over each of the single-pass heuristics. The AugNN approach is shown to have worked well on the task-scheduling problem with identical machines (Agarwal et al., 2003, Agarwal et al., in press). In this approach, a given scheduling problem is framed as a neural network, where the task and machine nodes act as processing elements of a neural network. Input, output and activation functions for these nodes are defined in a manner that the constraints of the problem are enforced and a heuristic is applied to produce a feasible solution. There are weights associated with the connections between the task and machine nodes. These weights are the same for all the links during the first iteration, therefore the first-iteration solution is no different from a single-pass heuristic solution. In subsequent iterations, however, the weights are modified using a certain learning algorithm to produce different solutions in the search space. The weight change essentially amounts to a perturbation, and thus a non-deterministic local neighborhood search mechanism. We propose input, output and activation functions for the task and machine nodes and also propose a learning strategy. We apply the AugNN approach in conjunction with each of the twelve heuristics and report results on randomly created problems of various sizes.1

In Section 2, we discuss the literature briefly. Section 3 describes the various task and machine priority rules with the help of two example problems. Section 4 describes the AugNN framework. The exact mathematical formulation appears in Appendix 1. Computational results are reported and discussed in Section 5. We conclude the paper with summary and conclusions in Section 6.

Section snippets

Relevant literature

Research on the use of neural networks for optimization started with the seminal paper by Hopfield and Tank (1985), who first applied neural networks to the traveling-salesman problem. Haldun et al. (1994) provide a review of machine-learning techniques used in scheduling problems. They explore the use of expert systems, rote learning, inductive learning and case-based learning. They only briefly discuss neural-network learning. Sabuncuoglu (1998) has focused his review paper on the use of

Priority rules with examples

For purposes of discussing the priority rules we will use a small example problem. Fig. 1 shows a sample task graph of nine tasks (T1–T9) and four non-identical machines (M1–M4). The nine tasks follow precedence constraints, represented by the given acyclic directed graph. Each task can be processed on any of the four machines, but needs to be processed on only one of them. The tasks are non-preemptive. The objective is to minimize the makespan or the completion time of the last task in the

Augmented neural network formulation

In the AugNN formulation, the task graph is converted into a neural network of processing elements (PEs). Each task node in the task graph becomes a PE and each of the task-node PEs is connected to a machine-node PE, one for each machine. The task-node PEs of AugNN maintain the precedence relation of the original task graph. See Fig. 6 for an AugNN representation of the task graph of Fig. 1. We then add an initial node and a final node for ease of formulation. Following each task-node PE are

Problem generation and experimental setting

Problems were generated for sizes 20–70 tasks and 2–5 machines. For each task size, four precedence relationships were developed at random. For problem sizes of 20, 25 and 30 tasks, 2–4 machines were used. For larger problems, 2–5 machines were used. The processing times were generated randomly from uniform distribution in the range of 1–50. A total of 100 problems were thus generated. This approach of generating problems is consistent with Kasahara and Narita’s (1984) approach, and adequately

Summary and conclusions

In this paper, the augmented-neural-network (AugNN) approach is applied to the task-scheduling problem with non-identical machines. The AugNN approach was first applied to task-scheduling problem for the identical machines case (Agarwal et al., 2003). This approach applies a base single-pass heuristic and tries to improve upon the single-pass solution through a weight modification learning strategy using neural-network principles. We propose 12 base heuristics for this problem and show

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