IBM ILOG CP Optimizer - u-bordeaux.frrsadykov/teaching/... · IBM ILOG CP Optimizer CP-AI-OR 2009 Masterclass 28/05/2009 Paul Shaw Constraint Programming Group [email protected]

IBM ILOG CP Optimizer

CP-AI-OR 2009 Masterclass

28/05/2009

Paul ShawConstraint Programming Group

[email protected]

28/05/2009 © ILOG, All rights reserved 2

What is CP Optimizer?

A Constraint Programming engine with an emphasis on modelling and automatic search

General feeling that CP toolkits were becoming too complicated, and the audience more limited

Models quickly became very hard to maintain

Simplify learning and use of the productIncrease model development speedQuickly evaluate CP technology on the problemYou can still program the search in the native engine

language (C++) if you wish


What kinds of problems?

CP Optimizer has a large emphasis on scheduling problems

However, we will not have time to cover interval-based modelling today

Look at examples beginning with “sched_”

Also addresses more traditional combinatorial problems

What we will do today


Getting at CP Optimizer

Available as a toolkit in C++, Java, .NETC++ is the native language and allows more

possibilities, like writing incremental custom constraints, and fully controlling the search process

Available as an engine inside OPLWhat we will use todayHigher level modelling and data manipulationCustom visualisation and scenarios via ODM


Getting Started with OPL Development Studio



Import Existing OPL 6.x projects





Browse projects



A project = - a set of model files (.mod) - a set of data files (.dat) - a set of settings files (.ops) - other files (e.g. .xls files …) - a set of rrrrrruuuuuunnnnnn ccccccoooooonnnnnnffffffiiiiiigggggguuuuuurrrrrraaaaaatitititititioooooonnnnnnssssss



Model edition: - data manipulation - expression - objective - constraint - script for pre- and post-processing - use Help-> Dynamic Help for information on keywords



Model navigation



Data file: - explicit data - connection to: * databases * Excel worksheets



Double-click for full-size window


Overview of an OPL Model

TopData manipulation and pre-processing

declarative (expressions) and/or imperative (script)

Variable declarations



CP Optimizer model



DDDDDDaaaaaattttttaaaaaa ssssssttttttrrrrrruuuuuuccccccttttttuuuuuurrrrrreeeeeessssss aaaaaannnnnndddddd dadadadadadattttttaaaaaa rrrrrreeeeeeaaaaaaddddddininininininggggggHere, read input grid



PrPrPrPrPrPrepreprepreprepreprococococococeeeeeessssssssssssiiiiiinnnnnnggggggHere: display grid



VVVVVVaaaaaarrrrrriiiiiiaaaaaabbbbbblllllle e e e e e DDDDDDeeeeeeccccccllllllaaaaaararararararattttttiiiiiionononononon


Some OPL Syntax: declarations

CP Optimizer has integer variables only (dvar int)No set variables, no floating point variables

BUT, floating point expressions are allowede.g. 0.3 * x + y / 13 <= 25.5;

Example declarationsint numColors = 3;

int numNodes = ...; // Comes from a data file

int capacity[1..4] = [5, 7, 2, 3];

range Nodes = 1..numNodes;

dvar int x[Nodes] in 1..numColors;

dvar int y[Nodes,Nodes] in 0..1;


Some OPL Syntax: declarations

CP Optimizer has integer variables only (dvar int) No set variables, no floating point variables

BUT, floating point expressions are allowede.g. 0.3 * x + y / 13 <= 25.5;

Example declarationsint numColors = 3;

int numNodes = ...; // Comes from a data file

int capacity[1..4] = [5, 7, 2, 3];

range Nodes = 1..numNodes;

dvar int x[Nodes] in 1..numColors;

dvar int y[Nodes,Nodes] in 0..1;


Some OPL Syntax: data preparation

int n = ...; // From data file

range R = 1..n;

int area[i in R] = width[i] * height[i];

int totalArea = sum(i in R) area[i];

{int} bigOnes = {i | i in R : area[i] >= 100};

tuple Pair { int p1; int p2; };

{Pair} sumTo100 = {

<a,b> | ordered a,b in R : area[a] + area[b] >= 100

};



int n = ...; // From data file

range R = 1..n;

int area[i in R] = width[i] * height[i];

int totalArea = sum(i in R) area[i];

{int} bigOnes = {i | i in R : area[i] >= 100};

tuple Pair { int p1; int p2; };

{Pair} sumTo100 = {

<a,b> | ordered a,b in R : area[a] + area[b] >= 100

};



An initial execution block can be useful for debugging and displaying input data

execute { writeln(sumTo100);}

gives...

{<1,99>, <2,98>, <3,97>, <4,96>, <5,95>, <6,94>, ...





Variable declarationsMiddle

Declarative modelobjective (optional) and constraints



MMMMMModelodelodelodelodelodel::::::-VVVVVVaaaaaarrrrrriaiaiaiaiaiablblblblblbleeeeeessssss-EEEEEExxxxxxpppppprerererereressssssssssssionionionionionionssssss-OOOOOObjecbjecbjecbjecbjecbjecttttttiiiiiivvvvvveeeeee-CCCCCCononononononssssssttttttrrrrrraaaaaaiiiiiinnnnnnttttttssssss



Basic structure of middle section

[ minimize/maximize <expression> ]

subject to / constraints { <constraint>; <constraint>; ...}



For allffffffoooooorrrrrraaaaaallllllllllll (i in {1,3,5})

x[i] <= capacity[i];

For all (2)ffffffoooooorrrrrraaaaaallllllllllll (i,j in 1..100 : i < j)

...

Sum (likewise max, min, or)ssssssuuuuuummmmmm (i in 1..10) x[i] == 1;

AllallDifferent (aaaaaallllllllllll (i in 0..n-1) slot[i]);






...








...








...




Expressions and Constraints

Arithmetic constraints x + y, x – y, x * y, x / y, x div y, x % y min, max, abs, log, exp etc. Piecewise linear functions

Relational constraints x == y, x != y, x <= y, x < y, lb <= x <= ub

Logical constraints !c, c||d, c && d, c => d c and d are relational or conditional constraints













Reification Relational or logical constraints can be used in a value

context, where they evaluate to 0 or 1

Examples Arithmetic: max(0, abs(load[i] - cap)) Relational: wid * hei * depth * density <= maxLoad Logical: end[i] <= start[j] || start[j] <= end[i] Reification: spill == (load[i] > cap)



Reification Relational or logical constraints can be used in a value

context, where they evaluate to 0 or 1

Examples Arithmetic: max(0, abs(load[i] - cap)) Relational: wid * hei * depth * density <= maxLoad Logical: end[i] <= start[j] || start[j] <= end[i] Reification: spill == (load[i] > cap)



Count expression count(dvar int[] x, int c) Evaluates the number of variables in x with value c e.g. Count the number of nurses allocated to ward 5

count(wardAllocation, 5) >= 3

Element expression (int[] a)[dvar int x] OR (dvar int[] a)[dvar int x] Evaluates to the xth member of a e.g. travel == 2 * distFromPittsburgh[holidayTown] travel and holidayTown are variables



Count expression count(dvar int[] x, int c) Evaluates the number of variables in x with value c e.g. Count the number of nurses allocated to ward 5

count(wardAllocation, 5) >= 3

Element expression (int[] a)[dvar int x] OR (int var[] a)[dvar int x] Evaluates to the xth member of a e.g. travel == 2 * distFromPittsburgh[holidayTown] travel and holidayTown are variables



All Different allDifferent(dvar int[] x) All variables in x must take different values e.g. The rank (visit priority) of each city is different

allDifferent(rankOfVisit)

Allowed / Forbidden assignments allowedAssignments({<a,b,c>} A, dvar int[3] x) The assignments to x must fit with a tuple of A forbiddenAssignments is the negation of this e.g. Combinations of engine, gearbox, on promotion



All Different allDifferent(dvar int[] x) All variables in x must take different values e.g. The rank (visit priority) of each city is different

allDifferent(rankOfVisit)

Allowed / Forbidden assignments allowedAssignments({<a,b,c>} A, dvar int[3] x) The assignments to x must fit with a tuple of A forbiddenAssignments is the negation of this e.g. Combinations of engine, gearbox, on promotion



Bin packing constraint pack(dvar int[m] ld, dvar int[n]x, int[n] sz, dvar int c) ld[i] == sum(j ) (x[j] == i ) * sz[j]

c is the number of containers used

e.g. Length of ad breaks given the ad assignmentInverse constraint

inverse(dvar int[n] x, dvar int[n] y) x[i] == j <=> y[j] == i --- link primary and dual models

Lexicographic ordering constraint lex(dvar int[n] x, dvar int[n] y) --- break symmetries



Bin packing constraint pack(dvar int[m] ld, dvar int[n]x, int[n] sz, dvar int c) ld[i] == sum(j ) (x[j] == i ) * sz[j]

c is the number of containers used

e.g. Length of ad breaks given the ad assignmentInverse constraint

inverse(dvar int[n] x, dvar int[n] y) x[i] == j <=> y[j] == i --- link primary and dual models

Lexicographic ordering constraint lex(dvar int[n] x, dvar int[n] y) --- break symmetries





Variable declarationsMiddle

Declarative modelobjective (optional) and constraints

BottomPost-processing of solutions

Declarative (expressions) and/or imperative (script)



PosPosPosPosPosPosttttttprprprprprprococococococeeeeeessssssssssssiiiiiinnnnnnggggggHere: display solution grid


Some OPL Syntax: post-processing

dvar int day[Jobs] in Days;

constraints { ... }

// .........................

{int} jobsOnDay[d in Days] = { j | j in Jobs : day[j] == d };

execute {for (var d in Days) {

write(“Day “ + d + “: “);for (j in jobsOnDay[d])

write(“ “ + j);

writeln(); }

}


Some OPL Syntax: post-processing

dvar int day[Jobs] in Days;

constraints { ... }

// .........................

{int} jobsOnDay[d in Days] = { j | j in Jobs : day[j] == d };

execute {for (var d in Days) {

write(“Day “ + d + “: “);for (j in jobsOnDay[d])

write(“ “ + j);

writeln(); }

}



A run configuration = A variation of a project for execution purposes: - at least one model file (.mod) - a set of data files (.dat) - a set of settings files (.ops)



Running a configuration:Right-click Run this



Engine log



Engine statistics for performance analysis



Script output



Solution and data inspection


Search in CP OptimizerAutomatic search is emphasised

Simpler, more maintainable, benefit from upgradesSearch Phases

What group of variables to assign first(optionally) define instantiation strategy

ParametersInference levels and search control parameters

Problem still hard?Improve modelSimplify or relax specificationDecompose: CPLEX often useful here


TTTTTTyyyyyypipipipipipiccccccaaaaaallllll uuuuuusssssseeeeee ooooooffffff CCCCCCPPPPPP OOOOOOptptptptptptiiiiiimmmmmmiiiiiizzzzzzeeeeeerrrrrr

UUUUUUse se se se se se CCCCCCP P P P P P OOOOOOptptptptptptiiiiiimmmmmmiiiiiizzzzzzeeeeeerrrrrr’’’’’’ssssssaaaaaauuuuuuttttttomomomomomomaaaaaattttttiiiiiicccccc sssssseeeeeeaaaaaarcrcrcrcrcrchhhhhh

RRRRRReeeeeevvvvvviewiewiewiewiewiewmmmmmmodelodelodelodelodelodel((((((rrrrrreeeeeedudududududunnnnnndadadadadadannnnnnttttttccccccononononononssssssttttttrrrrrraaaaaaiiiiiinnnnnnttttttssssss,,,,,,dedededededeccccccisisisisisisiiiiiiononononononvvvvvvaaaaaarrrrrriiiiiiaaaaaablblblblblbleeeeeessssss eeeeeettttttcccccc......))))))

MMMMMModelodelodelodelodelodel prprprprprproblobloblobloblobleeeeeemmmmmmiiiiiinnnnnn OOOOOOPLPLPLPLPLPL

Satisfied?

UUUUUUsssssseeeeee ““““““sssssseeeeeeaaaaaarrrrrrcccccchhhhhhphphphphphphaaaaaasssssseeeeeessssss”””””” tttttto o o o o o aaaaaadddddddddddd ssssssomomomomomomeeeeeedomdomdomdomdomdomaaaaaaiiiiiinnnnnnkkkkkknnnnnnoooooowlwlwlwlwlwledgedgedgedgedgedgeeeeeetttttto o o o o o tttttthhhhhheeeeee sssssseeeeeeaaaaaarrrrrrcccccchhhhhhprprprprprprococococococeeeeeessssssssssss

CCCCCChhhhhhaaaaaannnnnnggggggeeeeeepapapapapaparrrrrraaaaaammmmmmeteteteteteteeeeeerrrrrrssssssorororororor seaseaseaseaseasearrrrrrcccccchhhhhhmmmmmmeeeeeetttttthhhhhhodododododod

YES NO

DDDDDDeeeeeeccccccomomomomomomposposposposposposeeeeeemmmmmmooooooddddddeeeeeellllll

CCCCCCuuuuuussssssttttttomomomomomom ggggggoaoaoaoaoaoallllllorororororor ccccccononononononssssssttttttrrrrrraaaaaaininininininttttttssssss((((((CCCCCC++++++++++++))))))





Satisfied?Satisfied?







DDDDDDeeeeeeccccccomomomomomomposposposposposposeeeeeemmmmmmooooooddddddeeeeeellllllDDDDDDeeeeeeccccccomomomomomomposposposposposposeeeeeemmmmmmooooooddddddeeeeeellllll

CCCCCCuuuuuussssssttttttomomomomomom ggggggoaoaoaoaoaoallllllorororororor ccccccononononononssssssttttttrrrrrraaaaaaininininininttttttssssss((((((CCCCCC++++++++++++))))))


Automatic Search in CP OptimizerCP Optimizer's search is based on restarts

Search is can be restarted at the top of the search tree, allowing initial bad decisions to be undone

Search can be make complete by iteratively increasing the time between restarts

This technique is only really effective if a different search path is taken at each restart

…


Following Different Search Paths

Impact-based strategiesBy observing constraint propagation during previous

restarts, search learns about the problem and modifies its strategy accordingly

RandomisationDecisions are slightly randomised, for example as a tie-

breaking techniqueNo-goods

The explored part of the search tree is represented in CNF and maintained as a global constraint


Impact-based Strategies

An implementation of the general principles1/ Pick a variable which leaves least future choice2/ Choose a value which leaves the most future choice

Idea is to measure the remaining search space when an assignment is made

Impact is the proportion of search space reduction arising from an assignment

The impact for a particular assignment x = a is learned by averaging individual impacts over previous assignments


Search Phases

In CP Optimizer, search phases are simple search directives to help the automatic search process

Most common useDefine decision variables, or the most important onesDefine priority-ordered groups of decision variables

Less common useDefine decision variables and how to instantiate those

variablesFor this, there are various pre-defined evaluators


Search Phases in OPL

In OPL, search phases are created in an executeblock before the main constraint block

dvar int x[1..n] in 1..n;

// ... other declarations ...

execute {

var f = cp.factory;

cp.setSearchPhases(f.searchPhase(x));

}



Instantiate variables in x, then those in y


dvar int y[1..n] in 1..n;

// ...other declarations...

execute {

var f = cp.factory;

cp.setSearchPhases(f.searchPhase(x), f.searchPhase(y));

}



Instantiate variables in x using minimum domain size and maximum value


// ...other declarations...

execute {

var f = cp.factory;

cp.setSearchPhases(

f.searchPhase(x, f.selectSmallest(f.domainSize()),

f.selectLargest(f.value()))

);

}


Search Control Parameters

Controlling inference strengthGlobal, or per constraint type

Set limits on searchRun time, branches, fails, etc.

Optimality toleranceRelative and/or absolute gap

Controlling restartsControl the growth factor

Search LogVerbosity, frequency



Settings file -

© ILOG, All rights reserved63

Worked Examples

CP-AI-OR 2009 Masterclass

28/05/2009

Paul ShawConstraint Programming Group

[email protected]


CSPLIB 38: Steel mill slab designSlabs size 44

Slabs size 32

Slabs size 23

...(20 different sizes available)

16

16

16

14

14

14

...(100+ orders, with weightand build process identified by color)

1216 16

16 14

(loss 2)

•Max 2 colors per slab•Process all orders•Minimize total loss

© ILOG, All rights reserved


Addition chains

© ILOG, All rights reserved

1

2

3

5

10

20

40

60

100

Build a minimal length sequence of numbers leading to a particular target, beginning with 1

Each number is built as the sum as two smaller numbers

The first two numbers are necessarily 1 and 2

© ILOG, All rights reserved66

End of Part One

Thank You

Documents

IBM ILOG CP Optimizer - u-bordeaux.frrsadykov/teaching/... · IBM ILOG CP Optimizer CP-AI-OR 2009 Masterclass 28/05/2009 Paul Shaw Constraint Programming Group [email protected]