IOEMFG 543

1
IOE/MFG 543

• Chapter 14 General purpose procedures for
  scheduling in practice
• Sections 14.1-14.3 Dispatching rules and
  filtered beam search

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Motivation

• So far we have (mostly) discussed algorithms for
  obtaining an optimal solution for a specific
  problem
• Many scheduling problems are difficult to solve
  in practice
• Important to have general methods that can give
  good schedules in a relatively short amount of
  time

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Overview

• Dispatching (or priority) rules
• E.g., WSPT
• Composite dispatching rules
• Try to intelligently combine two or more
  dispatching rules
• Filtered beam search
• Heuristic implementation of branch and bound
• Local search
• Try to find a better schedule in a neighborhood
  of the current best schedule
• Methods Simulated annealing, tabu search,
  genetic algorithms

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Section 14.1Dispatching rules

• There are several ways to classify the
  dispatching rules
• Static vs. Dynamic
• WSPT is
• SRPT is
• Global vs. Local
• LAPT and Johnsons rules are
• WSPT is

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Applicability of elementary dispatching rules

• Dispatching rules can work well when there is
  only a single objective
• See Table 14.1 page 337 for a list of rules and
  in what environments they tend to work well
  (sometimes optimal)
• More sophisticated rules can often do better
• Minimizing a combination of objectives
• The objective can change with time and with the
  jobs waiting to be processed

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Section 14.2 Composite dispatching rules (CDR)

• A CDR is a ranking expression that combines two
  or more elementary dispatching rules
• An elementary rule is a function of attributes of
  jobs and/or machines
• Each elementary rule has a scaling parameter
• Depends on the attributes
• E.g., compute statistics for the set of jobs (are
  the due dates tight?)

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Composite rule for 1S wjTj

• Recall Problem 1S wjTj is NP-hard
• We developed a pseudopolynomial DP algorithm for
  1S Tj
• What rules could produce good schedules?
• If all due dates (and release dates) are zero?
• If due dates are loose and spread out?

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Apparant Tardiness Cost (ATC) rule for 1S wjTj

• Combines WSPT and minimum slack (MS)
• Slack of job j is max(dj-pj-t, 0)
• Ranking index for job j
• K is the scaling parameter
• p is the average of the processing time of the
  remaining jobs

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Apparant Tardiness Cost (ATC) rule for 1S wjTj
(2)

• How do we determine the value of K?
• Due date tightness factor
• Due date range factor
• R (dmax-dmin)/Cmax
• Empirical studies
• K 4.5 R if R0.5
• K 6 - 2R if Rgt0.5

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Apparant Tardiness Cost with setups (ATCS)

• Generalization of the ATC rule to take sequence
  dependent setup times into account
• sjk is the setup time of job k if it comes after
  job j on the machine
• s0k is the setup time if job k is scheduled first
  on the machine
• SST rule (Chapter 4)
• The job with the smallest setup time goes first
• ATCS combines WSPT, MS and SST

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Apparant Tardiness Cost with Setups (ATCS) (2)

• Ranking index for job j when job l has just
  completed its processing
• s is the average of the setup times of the jobs
  remaining to be scheduled

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Apparant Tardiness Cost with Setups (ATCS) (3)

• Choosing the scaling parameters
• Function of t, R and h s/p
• t is a function of Cmax
• Cmax is now schedule dependent
• Estimate Cmax as
• Cmax Spj ns
• K1 is computed the same way as K in ATC
• K2 t / (2vh)

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ATCS Example 14.2.1

• Job data Setup times

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Implementing a general composite rule

• Choose the elementary rules
• Compute the required job and/or machine
  statistics
• Use the statistics to compute the values of the
  scaling values
• Apply the dispatching rule to the set of jobs

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Section 14.3Filtered beam search (FBS)

• Enumerative branch and bound is one of the most
  widely used procedures for solving NP-hard
  problems
• It can be used to optimally solve any of the
  scheduling problems that we have considered so
  far
• Problem

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Branch and bound (BB)

• In branch and bound we can eliminate all nodes
  such that the lower bound is higher than the cost
  of the best feasible solution found so far
• Consequences
• If we can start off with a good solution then
  many nodes can potentially be eliminated
• Bad news
• The number of nodes can still be large

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Beam width

• FBS is a BB-based method in which only some
  nodes at any given level are evaluated
• Nodes that are not evaluated are discarded
  permanently
• The number of nodes that are retained is called
  the beam width

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Filter width

• Evaluating each node at a given level can be
  computationally expensive
• Instead, do a crude prediction of the quality
  of all nodes at a given level
• Evaluate a number of nodes thoroughly
• This number is called the filter width

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Crude prediction and thorough evaluation

• Example of a crude prediction
• Combine
• The contribution to the objective of the jobs
  already scheduled
• Some job statistic, e.g., due date factor
• Example of a thorough evaluation
• Schedule the remaining jobs according to a
  composite dispatching rule
• This gives an upper bound on the value at this
  node

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Example 14.3.1

• Solve 1 SwjTj using the following data
• Use beam width 2 and no filter