Matrix Modelling

1
Matrix Modelling

• Pierre Flener (Uppsala)
• Alan M. Frisch (York)
• Brahim Hnich, Zeynep Kiziltan (Uppsala)
• Ian Miguel, and Toby Walsh (York)

2
What is matrix modelling?

• Constraint programs with one or more matrix of
  decision vars
• Common patterns in such models

3
Example warehouse location

• Which warehouses supply which stores?
• 0/1 matrices
• Open(warehouse)
• Supply(warehouse,store)
• Constraints
• Each store has a warehouse row sum on Supply1
• Warehouse capacity column sum on Supply lt ci
• Channelling from Supply to Open

4
Diversity of matrix models

• Combinatorial problems
• BIBDs, magic squares, projective planes,
• Design
• Rack configuration, template and slab design,
• Scheduling
• Classroom, social golfer,
• Assignment
• Warehouse location, progressive party,

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Why matrix model?

• Ease of problem statement
• Side constraints, variable indexing,
• Improved constraint propagation
• Symmetry breaking, indistinguishable values,
  linear models,
• We argue that matrix operations should become
  first class objects in constraint programming
  languages. MATLAB meet OPL?

6
Common constraint types

• Row or column sum
• Weighted row/column sum
• Single non-zero entry
• Matrix sum
• Scalar product
• Channelling

• This pretty much describes all the examples!
• These constraints should be provided as language
  primitives?
• Efficient and powerful propagators developed?

7
Ease of problem statement

• Steel mill slab design
• Nasty colour constraint
• Stops it being simple knapsack problem
• Channel into matrix model
• Colour constraint easily and efficiently stated
• Easy to combine models
• Multiple models

8
Improved propagation

• Warehouse location
• Either 1-d matrix, Supply(store)warehouse
• Or 2-d matrix, Supply(store,warehouse)0/1
• 2-d matrix is purely linear so can use LP solver

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Symmetry breaking

• Often rows or columns (or both) are symmetric
• All weeks (cols) can be permuted in a timetable
• All slabs (rows) of same size can be permuted
• Lex order rows/cols
• See our talk in Symmetry workshop

Alan Frisch
10
Indistinguishable values

• Values in problem can be indistinguishable
• In progressive party problem
• Assign(guest,period)host
• But host boats of same size are indistinguishable
• Channel into 0/1 matrix with extra dimension
• Assign3(guest,period,host)0/1
• Value symmetry gt variable symmetry

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Variable indexing

• Use variables to index into arrays
• E.g. channelling in progressive party problem
• Assign3(guest,period,Assign(guest,period))1
  compared to
• Assign3(guest,period,host)1 iff
  Assign(guest,period)host
• Reduces number of constraints from cubic to
  quadratic
• Hooker (and others) argue that such indexing is
  one of the significant advantages CP has over IP

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Conclusions

• Matrix models common
• Common types of constraints posted on matrices
• Row/column sum, symmetry breaking, channelling,
• Matrix operations should be made first-class
  objects in modelling languages
• MATLAB, EXCEL,