Matrix Modelling Alan M Frisch and Ian Miguel

Matrix Modelling Alan M. Frisch and Ian Miguel (York) Brahim Hnich, Zeynep Kiziltan (Uppsala) Toby Walsh (Cork)

What is matrix modelling? w Constraint programs with one or more matrix of decision vars w Common patterns in such models n Extend modelling languages & solvers to exploit them

Obvious matrix models w Often the problem has a matrix in the specification w Many timetabling and scheduling problems n Mike Trick schedules the College baseball league using a matrix model!

Sports scheduling. Regin’s Round Robin Tournament model Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Period 1 0 vs 2 4 vs 7 3 vs 6 3 vs 7 1 vs 5 2 vs 4 Period 2 2 vs 3 1 vs 7 0 vs 3 5 vs 7 1 vs 4 0 vs 6 5 vs 6 Period 3 4 vs 5 3 vs 5 1 vs 6 0 vs 4 2 vs 6 2 vs 7 0 vs 7 Period 4 6 vs 7 4 vs 6 2 vs 5 1 vs 2 0 vs 5 3 vs 4 1 vs 3

Matrix modelling w Matrix models are very common in other applications n Even if the matrix is not so obvious in the problem specification

Example: warehouse location w Supply chain management w How many warehouses do we build and where do we build them? n n One in the middle of the US One on each coast … w Problem (and solution) taken from OPL manual

Matrix model: variables Warehouse Supply Store 0. . 1 0. . 1 0. . 1 Warehouse Open 0. . 1

Matrix model: constraints Warehouse Supply § Each store has a warehouse: row sum Supply=1 § Warehouse capacity: column sum Supply <= ci § Channelling constraints: Supply[i, j]=1 implies Open[i]=1 § Optimization variable: Cost = matrix sum Supply[i, j]*distcost[i, j] + opencost * row sum Open Store 0. . 1 0. . 1 0. . 1 Warehouse Open 0. . 1

Matrix model: spot the difference! Warehouse Supply Store 0. . 1 0. . 1 <c 2 … 0. . 1 Open 0. . 1 =1 =1 …

Matrix model: spot the difference! Periods Sched Course 0. . 1 0. . 1 l< <d … 0. . 1 Load 0. . l 0. . 1 =1 =1 …

Balance academic curriculum problem w Very similar matrix model n n Channelling now to weighted sum not 0/1 var Optimization var changes w Matrix modelling n n Exploiting these patterns Provide high level primitives l l To state such problems compactly To reason effectively

Diversity of matrix models w Combinatorial problems n BIBDs, magic squares, projective planes, … w Design n Rack configuration, template and slab design, … w Scheduling n Classroom, social golfer, … w Assignment n Warehouse location, progressive party, …

Why matrix model? w Ease of problem statement n Side constraints, variable indexing, … w Improved constraint propagation n Symmetry breaking, indistinguishable values, linear models, … Suggests two lines of attack 1. Matrix operations should become first class objects in constraint programming languages. 2. MATLAB should be extended so it is a constraint modelling language

Common constraint types w Row or column sum w Weighted row/column sum w Single non-zero entry w Matrix sum w Scalar product w Channelling w Ordering rows/matrices This pretty much describes all the examples! These constraints should be provided as language primitives? Efficient and powerful propagators developed?

Ease of problem statement w Steel mill slab design w Nasty “colour” constraint n Stops it being simple knapsack problem w Channel into matrix model n Colour constraint easily and efficiently stated

Steel mill slab design • 3 possible slab sizes (1, 3 or 4), 9 orders • 5 colours, but only 2 allowed on any one slab 3 2 a 2 b 1 1 1 c d e f g 1 h i

Example solution 2 2 (size 4) h 1 a 3 b 1 1 (size 3) e f g 1 (size 1) c 1 (size 1) d 1 (size 1) i

Matrix model for the slab design problem 3 2 oa 1 oc ob 1 od 1 oe 1 of s 1 s 2 1 og Red oa ob oc od oe of og oh oi 2 oh Green Blue Orange 1 oi Brown s 1 1 … Col sum=1, row sum<slab size s 2 1 … Col sum=1, row sum<2

Example solution 3 2 oa 1 oc ob 1 od 1 oe 1 of 2 oh 1 og 1 oi s 1 = 4, s 2 = 3, s 3 = 3, s 4 = 3 Red Green Blue Orange Brown s 1 0 0 0 1 1 oa ob oc od oe of og oh oi s 1 0 0 0 1 1 1 s 2 1 0 0 0 0 s 2 1 1 0 0 0 s 3 0 1 0 0 0 s 4 0 0 0 1 1 1 0 0 0 s 4 0 0 1 1 0 … 0 0 0 0 0 … 0 0 0

Improved propagation w Warehouse location n n Could have used 1 -d matrix, Supply(store)=warehouse But then warehouse capacity constraint l Difficult to state w Need to sum demand from all stores such that Supply(store)=warehouse l Difficult to propagate

Improved propagation w Warehouse location n n Instead of 1 -d matrix, Supply(store)=warehouse We used 2 -d matrix, Supply(store, warehouse)=0/1 2 -d matrix allows us to use purely linear constraints => can even use LP solver

Symmetry breaking w Often rows or columns (or both) are symmetric n n All weeks (cols) can be permuted in a timetable All slabs (rows) of same size can be permuted w Lex order rows/cols n n Never work with children or animals Can do both if row and cols are both symmetric Efficient O(n) GAC algorithm, patent pending

Symmetry-breaking • Lexicographically ordering rows breaks all row symmetry • Lexicographically ordering cols breaks all col symmetry

Symmetry-breaking problem • Lexicographically ordering both rows & cols does not break all row & col symmetry: 0 1 1 0 Other techniques for dealing with symmetry like SBDS also have problems as there are n! symmetries

Indistinguishable values w Values in problem can be indistinguishable In progressive party problem: Assign(guest, period)=host But host boats of same size are indistinguishable n w Channel into 0/1 matrix with extra dimension Assign 3(guest, period, host)=0/1 n Value symmetry => variable symmetry

Transforming value symmetry into variable symmetry w Suppose all values are indistinguishable a c {b, d} a b c d 1 0 0 0 1 0 1 Now the rows are indistinguishable So we are in a previously solved case!

Variable indexing w Use variables to index into arrays n E. g. channelling in progressive party problem Assign 3(guest, period, Assign(guest, period))=1 compared to Assign 3(guest, period, host)=1 iff Assign(guest, period)=host n 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

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