Application of Neural Networks to Heuristic Scheduling Algorithms

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<p>Computers &amp; Industrial Engineering 46 (2004) 679696www.elsevier.com/locate/dsw</p> <p>Application of neural networks to heuristic scheduling algorithmsDerya Eren Akyol*Department of Industrial Engineering, University of Dokuz Eylul, 35100 Bornova-Izmir, Turkey Available online 2 July 2004</p> <p>Abstract This paper considers the use of articial neural networks (ANNs) to model six different heuristic algorithms applied to the n job, m machine real owshop scheduling problem with the objective of minimizing makespan. The objective is to obtain six ANN models to be used for the prediction of the completion times for each job processed on each machine and to introduce the fuzziness of scheduling information into owshop scheduling. Fuzzy membership functions are generated for completion, job waiting and machine idle times. Different methods are proposed to obtain the fuzzy parameters. To model the functional relation between the input and output variables, multilayered feedforward networks (MFNs) trained with error backpropagation learning rule are used. The trained network is able to apply the learnt relationship to new problems. In this paper, an implementation alternative to the existing heuristic algorithms is provided. Once the network is trained adequately, it can provide an outcome (solution) faster than conventional iterative methods by its generalizing property. The results obtained from the study can be extended to solve the scheduling problems in the area of manufacturing. q 2004 Elsevier Ltd. All rights reserved.Keywords: Articial neural networks; Multilayered perceptron; Heuristic scheduling; Flowshop scheduling problems; Fuzzy membership functions</p> <p>1. Introduction The owshop scheduling problem is considered as one of the general production scheduling problems in which n different jobs must be processed by m machines in the same order. The problem can be considered as nding a scheme of allocation of tasks to a limited number of competing resources, with an objective of satisfying constraints and optimizing performance criteria. Much research literature addresses methods of minimizing performance measures such as makespan. The makespan minimization, within the general owshop scheduling domain, provides a useful area for analysis because it is an important model in scheduling theory and it is usually very difcult to nd its optimal solution (Jain &amp; Meeran, 2002).* Tel.: 90-232-3881047; fax: 90-232-3887864. E-mail address: derya.eren@deu.edu.tr (D.E. Akyol). 0360-8352/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2004.05.005</p> <p>680</p> <p>D.E. Akyol / Computers &amp; Industrial Engineering 46 (2004) 679696</p> <p>During the last 40 years, most of the work has been applied to the permutation owshop problem. In the permutation owshop, n different jobs have to be processed on m machines. Each job has one operation on each machine and all jobs have the same ordering sequence on each machine. At any time, each machine can process at most one job. Preemption is not allowed. The objective is to nd a permutation of jobs that minimizes the maximum completion time or makespan. This problem denoted by n=m=P=Cmax ; is found to be an NP-complete problem of combinatorial optimization problems. Complete enumeration, integer programming, branch and bound techniques can be used to nd the optimal sequences for small-size problems but they do not provide efcient solutions for large size problems. In view of the combinatorial complexity and time constraints, most of the large problems can be solved only by heuristic methods (Lee &amp; Shaw, 2000). Though, efcient heuristics cannot guarantee optimal solutions, they provide approximate solutions almost as good as the optimal solutions (Ho &amp; Chang, 1991). In recent years, the technological advancements in hardware and software have encouraged new application tools such as neural networks to be applied to combinatorially exploding NP-hard problems (Jain &amp; Meeran, 1998). They have emerged as efcient approaches in a variety of engineering applications where problems are difcult to formulate or awkwardly dened. They are computational structures that implement simplied models of biological processes which are preferred for their robustness, massive parallelism and ability to learn. They have proven to be more useful for complicated problems difcult to solve with conventional methods. The advantage of them lies in their resilience against distortions in the input data and their learning capabilities. With their learning capabilities, they avoid having to develop a mathematical model or acquiring the appropriate knowledge to solve a task. The ability to map and solve a number of problems motivated the proposal for using neural networks as a highly parallel model for general purpose computing. so that, they have been applied for solving scheduling and different combinatorial optimization problems. Finding the relationship between the data (i.e. processing times, due dates, etc.) and schedules, determining the optimal sequence for the jobs to be processed, identifying best dispatching strategies (i.e. scheduling rules), etc. are some of the application areas of neural networks in the scheduling literature (Sabuncuoglu &amp; Gurgun, 1996). Sabuncuoglu (1998) presented a detailed review of the literature in the area of scheduling, and study of Smith (1999) involves a review of the research works on the use of NNs in combinatorial optimization. Neural networks as learning tools, have demonstrated their ability to capture the general relationship between variables that are difcult or impossible to relate to each other analytically by learning, recalling and generalizing from training patterns as data (Shiue &amp; Su, 2002). In other words, they are universal function approximators and are therefore attractive for automatically learning the (nonlinear) functional relation between the input and output variables (Raaymakers &amp; Weijters, 2003). In this study, a scheduling problem in a real permutation owshop environment is considered. Using the information of production orders for 1 month and the global operation recipe, the best sequence of ve different products are found by six different heuristic algorithms. For each sequence found by six different heuristic algorithms, the completion time of each job on each machine, job waiting times and machine idle times are computed and read into the system. To model these six different heuristic scheduling algorithms, one of the most popular neural network architectures, the multilayered perceptron (MLP) neural network is used. In order to develop a neural network, the Backpack Neural Network System Version 4.0 (by Z Solutions) is used and some necessary steps are followed. For each of the heuristic algorithms, the neural network model is used for estimating the makespan of ve jobs processed on 43 machines. By this way, we presented a neural network based implementation alternative</p> <p>D.E. Akyol / Computers &amp; Industrial Engineering 46 (2004) 679696</p> <p>681</p> <p>to the existing heuristic algorithms. The proposed method is simple and straightforward. An MLP neural network is trained on data from a real world problem to learn the functional relationship between the input and output variables. After the training process is completed, MLP can provide outputs of adequate accuracy over a limited range of input conditions, having the advantage of requiring a lot less computation than other modeling methods (Feng, Li, Cen, &amp; Huang, 2003). In other words, the neural networks computational speed permits fast solutions to problems not seen previously by the network (El-Bouri, Balakrishnan, &amp; Pooplewell, 2000). This paper is organized as follows. Section 2 presents a mathematical formulation of the permutation owshop scheduling problem with the makespan objective. In Section 3, we give information about the heuristic procedures considered in this study. Steps of developing a backpropagation network are explained in Section 4. Section 5 includes the experimental results. Finally, Section 6 provides conclusions. 2. The permutation owshop scheduling problem with the makespan criterion In a permutation owshop scheduling problem, there are a set of jobs I {1; 2; 3; ; n} and a set of machines J {1; 2; 3; ; m}: Each of the n jobs has to be processed on m machines 1,2,m in the order given by the indexing of the machines. Thus the job i; i [ I consists of a sequence of m operations, each of which must be processed on machine j for an uninterrupted processing time pij : Each machine j; j [ J; can process at most one job at a time, each job can be processed on at most one machine at a time and once an operation is started, it must be completed without interruption (Baker, 1974). Let Cij be the completion time of operation i on machine j: The makespan Cmax is the maximum completion among all jobs. In the permutation owshop problem with the makespan objective, the goal is to nd a permutation schedule that minimizes the makespan Cmax ; where a permutation schedule for a owshop instance is a schedule in which each machine processes the jobs in the same order. In order to provide a formal mathematical model of the problem, we apply the notion of job processing order represented by a permutation P {p1 ; p2 ; ; pn } on the set I; where pi denotes the element of I which is in position i in P (Nowicki, 1999). Then we calculate the completion time of the partial schedule pi on machine j denoted by Cpi ; j as follows: Cp1 ; 1 pp1 ; 1 Cpi ; 1 Cpi21 ; 1 ppi ; 1 Cp1 ; j Cp1 ; j 2 1 pp1 ; j for i 2; ; n for j 2; ; m for i 2; ; n; j 2; ; m: 1 2 3 4</p> <p>Cpi ; j max{Cpi21 ; j; Cpi ; j 2 1} ppi ; j Finally, we dene the makespan as Cmax p Cpn ; m:</p> <p>5</p> <p>The permutation owshop scheduling problem is then to nd a permutation pp in the set of all permutations P such that (Rajendran &amp; Chaudri, 1991) Cmax pp # Cmax p ;p [ P: 6</p> <p>682</p> <p>D.E. Akyol / Computers &amp; Industrial Engineering 46 (2004) 679696</p> <p>In this study, we consider a owshop consisting of M machines, each with unlimited buffer space. There is no additional restriction that the processing of each job has to be continuous so that, there may be waiting times between the processing of any consecutive tasks of each job. The main assumptions for this problem are: a set of n multiple-operation jobs is available for processing at time zero (each job requires m operations and each operation requires a different machine). the set up times for the operations are sequence-independent and are included in the processing times. m different machines are continuously available. individual operations are not preemptable.</p> <p>3. Heuristics Six owshop heuristics are considered in this study. One can nd the explanations of these methods in Aksoy (1980), Campbell, Dudek, and Smith (1970), Koulamas (1998) and Nawaz, Enscore, and Ham (1983). The other two heuristic algorithms are new and are presented below. 3.1. Aslans frequency algorithm This algorithm is developed by Aslan (1999) with the objective of minimizing makespan and works as follows: Step 1 Take the operation times of each job on each machine, generate an n m dimensional problem. Step 2 Consider all of the combinations of all jobs, produce nn 2 1 pairs (two by two). Step 3 Calculate the partial makespan of all pairs by loading the jobs on the machines. Pair i; j and pair j; i are compared. For the pairs which have smaller completion time, the rst job takes the frequency value of 1, the other job 0. By these comparisons nn 2 1=2 frequency values are obtained. If the completion time of the pairs are the same then both jobs take the frequency value of 1. Step 4 Sum up the frequency values of all jobs and sort them in decreasing order. (This method sequences the jobs in decreasing frequency value order). Step 5 If the jobs have equal frequency values, consider alternative sequences and the sequence which results in less total completion time is the nal sequence.</p> <p>D.E. Akyol / Computers &amp; Industrial Engineering 46 (2004) 679696</p> <p>683</p> <p>3.1.1. Improvement phase The job pairs which have equal total completion times are evaluated and the dominant pairs are found. The pairs which have less total machine idleness are considered as dominant. A frequency value of 1 is added to the rst job and a frequency value of 1 is subtracted from the other job and a new sequence is generated. 3.1.2. A numerical example to Aslans frequency (dual sequencing) algorithm Consider a owshop with 5 machines. There are 4 jobs to be scheduled and their processing times are as shown in Table 1. We compare the calculated total completion times for pairs ij and ji for all of the combinations of the jobs. Pairs (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) . First, pair (1,2) and (2,1) are compared. Since pair (1,2) results in less partial makespan, job1 takes the frequency value of 1 and job 2 takes the frequency value of 0. By executing these nn 2 1=2 comparisons, we obtain the frequency values for all jobs. As it is seen in Table 2, we assign frequency value of 1 to jobs 2 and 4 because pair (2,4) and pair (4,2) have both total completion times of 39. Then the frequency values of all jobs are summed. The frequency values for each job are obtained as follows: Frequency for Frequency for Frequency for Frequency for job 1:1 job 2:1 job 3:3 job 4:2 Completion times 41 46 41 41 39 38 Pairs (2,1): (3,1): (4,1): (3,2): (4,2): (4,3): Completion times 42 42 40 40 39 41</p> <p>Table 1 Processing times for 4 job 5 machine problem Job Machine M1 J1 J2 J3 J4 5 9 9 4 M2 9 3 4 8 M3 8 10 5 8 M4 10 1 8 7 M5 1 8 6 2</p> <p>684</p> <p>D.E. Akyol / Computers &amp; Industrial Engineering 46 (2004) 679696</p> <p>Table 2 Frequency values of each job at each comparison Comparison Job J1 1 2 3 4 5 6 1 0 0 J2 0 1 1 0 1 1 1 1 0 J3 J4</p> <p>The frequency values of all jobs are sorted in decreasing frequency value order. The method yields the sequence 3-4-2-1 or 3-4-1-2. The makespan of these two sequences are compared and it is found that 3-4-1-2 sequence with Cmax 57 is better than the sequence 3-4-2-1 with Cmax 58: 3.1.3. Improvement phase The job pairs (2,4) and (4,2) have equal total completion times. The dominant pairs are investigated. Because pair (2,4) and pair (4,2) have equal total completion times of 39, we compare these two pairs and try to decide which one is dominant. For each pair, except for the rst machine, we calculate how much the subsequent machine waits the preceding machine. In Fig. 1, the numbers on the upper side of each circle indicate the starting time of each job on each machine and the numbers below the circle indicate the execution time of each job on each machine. For the above example, for pair (2,4), in order to start operation, fourth machine waits for the third machine 7 min, fth machine waits for the fourth machine 6 min so for pair (2,4), the total delayed time is 6 7 13 min. For pair (4,2), in order to start operation, fourth machine w...</p>

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