Constraint Satisfaction Problem (CSP) Applications and Job-Shop Scheduling

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Constraint Satisfaction Problem (CSP)
Applications and Job-Shop Scheduling
Factory Automation Lab. SNU. Nov. 18. 1999 Min,
Dai ki
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Contents

• Introduction
• Constraint Satisfaction Problem
• Algorithms
• Applications
• Evaluation
• Job shop scheduling using CSP
• paper review
• Conclusions

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Introduction

• Many combinatorial problems in OR has an
  exponential time requirement. (NP-hard)
• In CSPs, it may be sufficient to find a solution
  at a reasonable computational expense, that
  satisfies as many constraints as possible.
• The representation as a CSP is often much closer
  to the original problem.
• CHIP, ILOG Solver

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CSP Problem definition

• A CSP consists of
• a set of variables Xx1,,xn
• for each variable xi, a finite set Di of possible
  values (domain)
• a set of constraints restricting the values that
  the variables can simultaneously take.
• Constraint Cijk ? Di ? Dj ? Dk ? ...
• A solution is an assignment of a value from its
  domain to every variables.

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CSP consistency techniques

• Constraint graph (binary constraints)
• nodes (variables) and arcs (constraints)
• deterministic and pre-processing stage
• Node consistency
• unary constraint on variable
• Arc consistency
• binary constraints correspond to arc
• K-consistency (path consistency)
• constraint propagation

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CSP Search algorithm

• Simple backtracking
• the constraints b/w the current vars. and the
  past vars.
• Forward checking (by Haralick and Elliott in
  1980)
• the constraints b/w the current vars. and the
  future vars.
• Temporal assignment and removal
• Maintaining Arc Consistency (by Freuder in 1994)
• Look Ahead
• the constraints b/w the future vars.
• Note that it does more work

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CSP Variable and value ordering

• Variable ordering
• Static ordering vs. Dynamic ordering
• first-fail
• Value ordering
• impact on the time to find the first solution
• success first
• Variable and value ordering heuristics for the
  job shop scheduling constraint satisfaction
  problem Norman Sadeh

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CSP Applications

• Location
• variables
• yi whether facility is established or not at
  location i
• zj location of the facility that supplies
  customer j
• vj supply cost
• constraints
• vj czj,j
• yi0?zj? i
• ?fiyi?vj lt C

• Car sequencing
• variables
• set of cars
• constraints
• ratio constraint
• grouping constraint
• calendar constraint
• just-in-time constraint
• In some cases, the traditional variable and
  value ordering may not necessarily be best
• Succeed-first or fail-first
• smith (1996)

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CSP Applications

• Cutting stock
• variables
• cutting pattern
• constraints
• cost
• demand
• yield rate
• Integer linear programming and constraint
  programming approaches to a template design
  problem Proll(1998)

• Vehicle routing
• variables
• weather a vehicle travels directly from a
  customer to another
• constraints
• a vehicle travels from and travels to each
  customer
• all vehicles that leave the depot to return to
  the depot
• subtour elimination
• vehicle capacity
• There is a hybrid approach in which local search
  is combined

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CSP Applications

• Time tabling
• Rostering
• etc...

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CSP Applications in Scheduling

• Common obj. min. makespan
• Modeling
• variable start time (operation)
• domain predecessors and successors of an
  operation
• constraints precedence, disjunctive, capacity
• Prev. researches
• Thuriot et al.(1994)
• Nuijten and Aarts(1996), edge finding
• Baptiste and Le Pape(1995), ILOG shcedule
• alternative formulation Cheng and Smith (1997)

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CSP Evaluation

• CSP and BB
• tree search techniques.
• Constraint propagation vs. bounding scheme
• cost and tightness of the lower bound
• CSP and local search
• if CSP is used in a pure form, it is unlikely to
  be competitive with the best local search method.
• But the ideas from local search can be
  incorporated
• randomization
• restart procedures

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A computational study of constraint satisfaction
for multiple capacitated job shop scheduling

• W.P.M Nuijten a , E.H.L. Aarts b,c
• a ILOG S.A., France
• b Eindhoven Univ. of Tech., Dept. of Mathematics
  and Computing Science, Netherlands
• c Philips Research Lab., Netherlands
• E.J.O.R., Vol.90, 1996.

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Contents

• The multiple capacitated job shop scheduling
• A constraint satisfaction approach
• Consistency checking algorithm
• Forward checking
• Arc consistent
• Sequencing checking
• Checking remaining capacity
• Computational results
• Conclusion

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Multiple capacitated job shop problem

• General job shop scheduling problem
• Variable operation o??
• Domain start time D(o) 0, D-pt(o)
• Constraints
• precedence constraint
• co,o(s) ? s(o) pt(o) ? s(o)
• capacity constraint c?, machine ? ? m

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Constraint satisfaction approach

• Operation and start time selection
• randomization
• Dead end handling
• chronological backtracking
• complete restart of the search

• while not solved and not infeasible do
• check consistency
• if a dead end is detected then
• try to escape from dead end
• else
• select variable
• select value
• end_if
• end_while

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Consistency checking Forward Checking

• capacity constraint

• Theorem
• all start times in (a1 - pt(o),b1 ? ? (ax -
  pt(o), bx
• are inconsistent for o.

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Consistency checking Arc consistency

• capacity constraint

• Theorem
• co,o(s) ? s(o) pt(o) ? s(o) ? s(o)pt(o)
  ? s(o).
• Then current domain ?(o) is arc consistent with
  ?(o) for
• co,o, if and only if ?(o)?(lst(o)-pt(o),
  ect(o))?

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Consistency checking Sequencing Checking

• Three different bounds
• lower bound on the earliest start time
• upper bound on the latest completion time
• Time complexity
• LBest(o),UBlct(o) O(??2)
• LB2est(o),UB2lct(o) LB3est(o),UB3lct(o)
  O(??3)
• Nuijten et al.1993
• an operation must be scheduled before or after a
  specific subset of operations on the same machine
  for machines with capacity 1.

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Consistency checking Sequencing Checking
continue

• LBest(o) earliest start time of operation o
• UBlct(o) latest completion time of operation o

b)
a)
b)
a)
rest(S,j)A(S)-(C(S)-e(S)) ?(cp(?)-j)
A(S) ?o?Spt(o)sz(o)
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Consistency checking Checking
remaining capa.

• the capacity o uses in time

• Theorem
• Va1,b1?...?an,bn?N be such that
• then all start times in (a1-pt(o),b1
  ?...?(an-pt(o),bn are
• inconsistent for o.

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The algorithm CheckConsistency

• Proc CheckConsistency
• Forward_Check
• while domain have changed do
• 2-ConsCheck
• Sequencing_Check
• RCP_Check
• end_while
• end_Proc

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Computational results Problem Sets

• Job shop scheduling
• 40 instances of the JSSP from Lawrence (1984)
• 3 instances from Fisher and Thompson (1963)
• Multiple capacitated job shop scheduling
• 30 randomly generated instances with 5-10
  machines and 100-200 operations

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Computational results Job shop scheduling
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Computational results Multiple capacitated
job shop scheduling
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Computational results Varying the
consistency checking
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Conclusions

• Present an algorithm based on constraint
  satisfaction techniques.
• Performs well both on instances of the JSSP and
  the MCJSSP.
• Shown the extensive consistency checking algorithm

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References

• A computational study of constraint satisfaction
  for multiple capacitated job shop scheduling,
  W.P.M. Nuijten, E.H.L. Aarts, EJOR Vol.90(1996).
• Edge-finding constraint propagation algorithm for
  disjunctive and cumulative scheduling, Phillip
  Baptiste and Clause Le Pape, Proceeding of the
  15th workshop of the U.K. Planning Special
  Interest Group, 1996
• Guide to Constraint Programming, Roman Bart, 1998
• http//kti.msmff.cuni.cz/bartak/constraints/
  intro.html
• etc...