Scheduling problems using Neural network

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Scheduling problemsusing Neural network

• SNU MAI Lab Seminar
• 2000. 9. 29
• Eoksu Sim(ses_at_ultra.snu.ac.kr)

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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

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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

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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.

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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) nm job shop scheduling
  to mn by (mn1) 2D neuron matrix
• Precedence and resource constraints the cost of
  total completion times of all jobs

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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

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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)

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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

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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

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Applications of the proposed approach(2/4)

• External processor expected mean tardiness, ET
  ??

• ECi expected completion time of job i

• aij the probability of assigning job i to the j
  th position

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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

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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

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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

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Experiment results(2/2)

• Mean tardiness computation time

of improvement ANN over WI

• Linear-, linear
• Tansel and Sabuncuoblu(94, 96) hard data pattern

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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

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Sequencing jobs on a single machineA 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

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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

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Problem statement

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

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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
maxPi
Latest due date of the n jobs maxdi
Largest slack for the n jobs maxSLi
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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

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A NN for single machine sequencing(3/3)

• The n-job example problems are generated randomly
• pi U1, 100, di UP(1-TF-RDD/2),
  P(1-TFRDD/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
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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(TF0.6, RDD0.4)

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Experiment results(2/4)

• Job sequence 4-3-6-7-5-1-2
• Optimal sequence 4-3-7-6-5-1-2

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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.

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Experiment results(4/4)

• Mean job tardiness(or total tardiness) NP-hard
• Sequences are found by dynamic programming

• Wilkerson-Irwin algorithm?? ??? ?? ? ? ??? ?? ?
  ??.

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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

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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
• BB, SA, TS, Heuristic.

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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