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Scheduling problems using Neural network

Scheduling problems using Neural network. SNU MAI Lab Seminar 2000. 9. 29 Eoksu Sim(ses@ultra.snu.ac.kr). Contents. A neural network model for scheduling problems Network Application Sequencing jobs on a single machine: A neural network approach Network structure Application

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Scheduling problems using Neural network

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  1. Scheduling problemsusing Neural network SNU MAI Lab Seminar 2000. 9. 29 Eoksu Sim(ses@ultra.snu.ac.kr)

  2. Contents • A neural network model for scheduling problems • Network • Application • Sequencing jobs on a single machine: A neural network approach • Network structure • Application • Concluding remarks • References

  3. A neural network model for scheduling problems Ihsan Sabuncuoglu*, Burckaan Gurgun** *Dept. of Ind. Eng., Bilent Univ., Ankara, Turkey **Dept. of Ind. And Sys. Eng., Univ. of Florida, Gainesville, USA EJOR, Vol. 93, 1996, pp. 288-199

  4. Introduction • Various applications of ANNs • Classification(i.e., pattern recognition) • A variety of optimization problems( e.g. TSP, GPP) • The focus of this paper • On scheduling problems and their solution with neural networks • A survey of the ANN literature pertaining to scheduling • A new neural network model to solve two well known scheduling problems.

  5. Literature review(1/2) • Existing studies • Hopfield model and other optimizing networks • Single layered and fully interconnected NN model • Coding the objective function and hard constraints into a single energy function • Hopfield and Tank(1985) – TSP mapping • Foo and Takefuji(1988) – n|m job shop scheduling to mn by (mn+1) 2D neuron matrix • Precedence and resource constraints + the cost of total completion times of all jobs

  6. Literature review(2/2) • Competitive networks • Back propagation networks • Sabuncuoglu et. Al(92) – relationship between problem data and optimal schedule • Kim and Lee(93) – parameter of a job priority rule • Rabelo, Yih(93) - together with OR and AI tools in an integrated manner for real-time scheduling systems • Simulated annealing(SA) • Overcome local minimum of the conventional search methods • Osman and Potts(89) … - apply to scheduling problems

  7. The proposed network(1/2) • Competition property • The neurons(representing jobs in scheduling problems) are allowed to compete with each other to get the first available position in the sequence. • A sequence of jobs(tasks) on a given machine(resource) • An n x n neuron matrix  permutation matrix(ex. Single machine scheduling  Fig. 1)

  8. The proposed network(2/2) • The basic functions of the external processor • Sequentially selecting two random row during the interchange process • The normalization • Calculation of the expected cost as it can monitor the overall network

  9. Applications of the proposed approach(1/4) • Single machine mean tardiness problem • To minimize the mean tardiness • Proved to be NP-hard by Du and Leung(1990) • Optimization – Fisher, Schrage, Baker • The current limit on solvability of this problem is around 100 jobs • Heuristic – Panwalker, Potts, Wilkerson and Irwin • From simple dispatch rule to sophisticated algorithm

  10. Applications of the proposed approach(2/4) • External processor – expected mean tardiness, E[T ] 계산 E[Ci] : expected completion time of job i aij : the probability of assigning job i to the j th position

  11. Applications of the proposed approach(3/4) • Procedure • Step 1. Initialize the neuron matrix • Step 2. Pass the activation values of the jobs from the following sigmoid function: • Step 3. Normalize the neuron matrix for rows first and for columns as: • Step 4. Compute the value of the energy function(i.e., expected tardiness) as

  12. Applications of the proposed approach(4/4) • Step 5. Select two rows(jobs) randomly, interchange their activation values and compute the energy function again. • Step 6. If the energy function is improved accept the new state, else return it to the previous state • Step 7. Periodically(after a predetermined number of iterations), select a column beginning from the first position in the schedule. Assign the neuron with the highest activation value to 1 and make other neurons 0 in the selected column • Step 8. Normalize the neuron matrix again • Step 9. If the matrix is still infeasible go to step 2, else go to step 10 • Step 10. Even though, all positions of the neuron matrix are feasible repeat the step2 through 9 for some number of iterations and stop

  13. Experiment & results(1/2) • Performance evaluation and experimental results • With Wilkerson and Irwin(WI) algorithm • Problem generation using two problem parameters • TF(Tardiness Factor) – the rate of expected proportion of tardiness of jobs • RDD(Range of Due Date) – the range of due date • Two dimensional graph of data types

  14. Experiment & results(2/2) • Mean tardiness & computation time % of improvement ANN over WI • Linear-, linear+ • Tansel and Sabuncuoblu(94, 96) – hard data pattern

  15. Concluding remarks • ANN scheduling literature review • A new neural network model • Better solutions than WI for the single machine problem • Hybrid OR/ANN methodology that incorporates both qualitative and quantitative aspects of scheduling problems

  16. Sequencing jobs on a single machine:A neural network approach Ahmed El-Bouri, Subramaniam Balakrishnan, Neil Popplewell Dept. of Mechanical Eng., Univ. of Manitoba, Winnipeg, Manitoba, Canada EJOR, Vol. 126, 2000, pp. 474-490

  17. Introduction • The purpose of this paper • to present a novel approach for single machine sequencing that is based on ANNs. • It is motivated by the desire for speed and flexibility in producing a solution • Computational speed – large number of subproblems • Flexibility – no proven sequencing methods may be readily available • An Artificial neural network • functional relationship between a set of single machine example problems and the corresponding job sequences that optimize the stated performance criterion

  18. Problem statement • Performances measures • Mean flowtime • Mean weighted flowtime • Mean tardiness • Minimum cost function • To minimize a weighted combination of job tardiness and flowtime

  19. A neural network for single machine sequencing(1/3) • 11-9-1 network • Input layer – information for each of the n jobs • Hidden layer • Output layer – one unit • values that are in the range of 0.1-0.9 • the magnitude being an indication of where the job represented at the input layer should desirably lie in the sequence Slack for job i=(di-pi) Longest processing time among the n jobs = max[Pi] Latest due date of the n jobs = max[di] Largest slack for the n jobs = max[SLi]

  20. A NN for single machine sequencing(2/3) • Methodology • The target value Gi for the job holding the i th position in the optimal sequence • The steps for training and employing the neural network form the single machine sequencing problem • (a) Generate a random set of example problems • (b) Find the optimal solutions for the example problems • (c) Select the input-output training patterns form the solved problems • (d) Train the neural network by using backpropagation • (e) Use the trained neural network to solve new problems

  21. A NN for single machine sequencing(3/3) • The n-job example problems are generated randomly • pi ~ U[1, 100], di ~U[P(1-TF-RDD/2), P(1-TF+RDD/2)] • RDD – range of due dates, TF – tardiness factor ~ [0.1, 1.0] • 5000 training patterns • Test – evaluation is based on monitoring the average ‘positioning error’ • The position error indicates how closely the neural network is able to position the job represented by pattern q to the position that the job should occupy in the optimal sequence Output response when pattern q is presented at the input layer Target response for the test pattern q

  22. Experiment & results(1/4) • Example problem • To minimize a cost function that combines tardiness and flowtime measures • 2500 example problems are solved • Training by simulation program written in the DESIRE/NEUNET matrix language • A 7-job problem as an example(TF=0.6, RDD=0.4)

  23. Experiment & results(2/4) • Job sequence : 4-3-6-7-5-1-2 • Optimal sequence : 4-3-7-6-5-1-2

  24. Experiment & results(3/4) • Performance for different criteria • Mean flowtime • Weighted mean flowtime • Maximum job tardiness • The neural network can deduce • Well structured rule such as SPT or EDD sorting, • The neural network can apply • the training is performed on information from only 12-job problems • n much higher that 12.

  25. Experiment & results(4/4) • Mean job tardiness(or total tardiness) – NP-hard • Sequences are found by dynamic programming • Wilkerson-Irwin algorithm과의 비교를 하는 게 더 의미가 있을 것 같다.

  26. Discussion and conclusions • Some instances in production and service industries • A number of jobs need to be sequenced in an order that optimizes a performance criterion which is not common • No algorithms are known beforehand • Quick and dirty sorting heuristics • A lengthy process of deducing an algorithm for the particular problem • NN approach, a middle ground between these two extremes • Speed and quality • When objectives change frequently • When good solutions are required without the effort of developing detailed and problem-specific algorithms

  27. Concluding remarks • Need for time and effort to make the network • Parameters dependent sequences • Lack of explainability for the results • Sequence using neural network as an initial solution to the optimal sequence • B&B, SA, TS, Heuristic….

  28. References • C.N. Potts, L.N. Van Wassenhove, Single machine tardiness sequencing heuristics. IIE Transactions 23 (1991), pp. 346-354 • S.K. Sim, K.T. Yeo, W.H. Lee, An expert neural network system for dynamic job shop scheduling, International Journal of Production Research 32 (2) 1994, pp. 1759-1773 • H.C. Zhang, S.H. Huang. Applications of neural networks in manufacturing: a state-of-the-art survey, International Journal of Production Research 33 (3) (1995), pp. 705-728

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