An Overview of Solving Job Shop Scheduling Problem by Local Search Techniques

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An Overview of Solving Job Shop Scheduling
Problem by Local Search Techniques

• Chiu Wo Chio

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Job Shop Scheduling Problem (JSSP)

• Operations
• Machines
• Jobs
• The processing time of an operation
• G (O, A, E)

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Schedule

• Schedule
• Feasible Schedule
• Makespan of a schedule

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Orientation

• Orientation
• Complete/partial orientation
• Feasible orientation (no cycles)
• Feasible complete orientation ? unique feasible
  schedule

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Local Search

• Iterative
• Making small changes
• Set of feasible solutions F
• Cost function
• Neighbor function N
• Execution of search is walk in F
• Local minimum

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A Local Search Algorithm

• Defines
• A solution representation
• A cost function
• A neighborhood function
• For JSSP
• Start times of operation
• makespan

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Some Definitions Useful for Defining
Neighborhoods

• js(v), jp(v), mpS(v), msS(v)
• Block
• Internal operation of a block

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Properties Useful in Obtaining Good Neighborhoods

• Given a feasible orientation, reversing an
  oriented edge on a longest path in the
  corresponding diagraph results again in a
  feasible orientation
• If reversing an oriented edge of a feasible
  orientation that is not part of the longest path
  results in another feasible orientation, the new
  makespan is at least as large as the previous one

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Properties Useful in Obtaining Good Neighborhoods

• 3. Given a feasible orientation, reversing an
  oriented edge between two internal operations of
  a block results in a feasible schedule with a
  makespan at least as long the previous one
• 4. Given a feasible schedule reversing the two
  first operations of the first block results in a
  feasible schedule with makespan at least as long
  as the previous one. The same is true if we
  reverse the last two operations of the last block.

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Threshold Algorithms

• Accepts a neighbor of a solution if the cost
  difference is below a certain treshold
• Threshold updated at every iteration if necessary
  
• Basic threshold algorithms
• Iterative improvement
• Threshold accepting
• Simulated annealing

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Iterative Improvement

• Threshold is set to 0
• Employs a multi-start strategy
• N1 interchanges two adjacent operations of a
  block
• N2 interchanges
• two adjacent operations of a block except when
  they are both internal
• jp(w) and mp(jp(w))
• js(v) and ms(js(v)), if such edges exists

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Threshold Accepting

• Threshold is set to large positive number
• Threshold increased gradually to 0 at every
  iteration
• but general rules lacking ? determined
  empirically
• Was tested with N1

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Simulated annealing

• Used positive and stochastic thresholds
• -T ln u , where T is a control parameter
• Values gradually decrease according to a cooling
  schedule
• u is drawn from a uniform distribution from (0,1
  at each iteration

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Tabu Search Algorithms

• Selects a solution of minimum cost from a subset
  of neighbors which is not on the tabu list
• OR satisfies a certain aspiration criteria
• tabu list
• defined in terms of forbidden moves from the
  current solution to a neighbor
• recomputed at every iteration

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DellAmico and Trubian

• Was tested with N3
• For any two adjacent operations of a block v, w
  (except when they are both internal), a
  neighboring orientation is obtained by permuting
  mp(v), v and w
• Or by permuting v, w and ms(w)
• Such that v and w are interchanged and a feasible
  orientation results
• For any block of size two, a neighbor is obtained
  by interchanging its operation if feasible
• Otherwise v is removed to the right or the left
  as long as the orientation remains feasible

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DellAmico and Trubian

• The length of the list varies according to
  whether the current solution is better than the
  one before and the best one or not.
• The minimal and maximal allowable length of the
  list changes over iterations
• Aspiration is moving to a better solution
• When no allowable neighbor exists a random one is
  chosen

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Tabu Search with Backtracking

• Use N4
• has the same interchanges as N1 except those
  involving internal operations
• Disallows the interchange of the first two
  operation of the first block and the two last
  operations of the last block when size of the
  block is greater than 2
• Neither a schedule of with only one block nor one
  with only blocks of size 1 has neighbors
• It only allows reorientations of arcs on a
  single shortest path

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Tabu Search with Backtracking

• Length of tabu list is 8
• If no allowable neighbor found
• If there is only one neighbor that is tabu,, this
  one becomes the new schedule
• else the oldest items on the tabu list is removed
  until one neighbor becomes non-tabu

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Tabu Search with Backtracking

• Backtracking resumes the tabu search from
  unvisited neighbors of solutions previously
  generated
• Suppose S is a new best solution
• R(S) the set of feasible arc orientation in S
• R the next orientation
• If R(S) gt2, (S,R(S)\r,T) is stored on a list
  with max length 5
• When R(S) becomes 1, the triple is deleted,
  otherwise its orientation will exclude the
  orientation made in the next iteration

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Results and Observations

• Among the threshold algorithms, simulated
  annealing perform the best given the same amount
  of processing time
• Tabu search produce even better solutions in a
  reasonable time even without specific information
  on the problem structure
• Tabu search with backtracking is one of the
  champions for solving JSSP.