G53CLP Constraint Logic Programming - Nottinghampszrq/files/11CLPOptimisation.pdfConstraint...

G53CLPConstraint Logic Programming

Constraint Optimisation Problems

Dr Rong Qu

Constraint Satisfaction Problems

So far All solutions are equally good

In some real world applications, we Not only want feasible solutions, but also good

solutions

We have different preferences on constraints

Problems are too constrained that there is no solution satisfying all constraints

G53CLP – Constraint Logic Programming Dr R. Qu


Real world problems present to be messy

In some cases a conflict-free solution is needed All constraints must be satisfied

In some cases preferences are given, rather than constraints Hard and soft constraints

In some cases, some constraints are more important than others Constraints with different weights (importance)



What are good solutions? Objective function

Problem specific function How much are constraints satisfied

Relatively new research Scheduling Timetabling Resource allocations etc



Find feasible solutions with the best value of the objective function

COP (C,f) Constraint Satisfaction Problem (C)

+

Objective function (f)(maps every solution to a numeric value)



A solution θ is preferred to solution θ’ If the value of objective function f under θ is less

(or larger in maximisation problem) than θ’

An optimal solution of COP (C,f) A solution θ* of C, such that no other solution of C

is preferred to θ*.



COP - examples

Graph colouring

C: different colour for adjacent vertices

f: Minimum number of colours used

Linear expression

C: constraints for variables in the expression

f: Minimise the result of linear expressions

G53CLP – Constraint Logic Programming

COP - examples

TSP with time windows

C Each city visited only once

Each city visited within a certain time window

f: shortest route

COP for TSP*

Constraint propagation to remove values of variable which lead to no better solution

*G. Pesant, M. Gendreau , J.Y. Potvin, J.M. Rousseau. An Exact Constraint Logic Programming Algorithm for the TravelingSalesman Problem with Time Windows. Transportation Science 32(1): 12 - 29, 1998*F. Focacci, A. Lodi, M. Milano, "A hybrid exact algorithm for the TSPTW", INFORMS Journal on Computing 14, 403-417, 2002

Branch and Bound (B&B)

General method for optimisation problems Systematically enumerate all candidate solutions Large subsets of fruitless candidates are discarded Use upper and lower estimated bounds of the

quantity being optimized

In COP most widely used CPLEX in ILOG Solver …



Based on depth first search Branches pruned during the search by a bound

Keep the best solution so far

During the search if partial solutions (nodes) are proved cannot improve the result Prune the branch under the node

All solutions in branches under the node are abandoned



Bestsolution

X

…

X

Bestsolution

…

No optimal solutionbelow these branches


B & B

Two important factors

A heuristic function, h

Estimated objective values for compound labels of partial solutions

A bound, b

Used to prune branches with no optimal solutions

Updated during the search


B & B

During the search in the search tree

Before labelling a variable, a value of the heuristic function is calculated

If the heuristic value is greater than* the bound The whole sub-tree under the node is pruned

If the heuristic value of a solution is less than the existing bound Update the bound Store the newly found solution

* For a minimisation problem


B & B – heuristic

Heuristic h

Function maps a partial solution to an estimate of the objective function value

Good estimate of the best values of all branches under the current node

If the best value under the branch is worse than the current bound, then there is no need to explore these branches

Problem specific


B & B – heuristic

Use heuristic to prune the search tree There is no solution in the sub-tree (under the

node)

All solutions in the sub-tree are not optimal

So that Solutions can be found earlier

The search space is reduced

Speed up the search


B & B – heuristic

A good heuristic h is the key to successful B&B

1. Must underestimate* Admissible Return the lower bound of the heuristic value Otherwise the optimal solution may be

pruned h’ is the actual cost, h <= h’

2. Good heuristics



B & B – heuristic

A good heuristic h is the key to successful B&B

1. Must underestimate

2. Good heuristics The closer the estimation of heuristic, the

larger the part of search tree pruned h’ is the actual cost, the closer h is to h’ the

better


B & B – bound

Usually set as infinite value* at the beginning

Updated during the search by recording the best so far heuristic value


infinite positive*


B & B – bound

Better bound, b Helps to find good solutions earlier

In practice, user can provide a bound

Satisfied even the solution is not optimal

…

v


SEND MORE MONEY - Problem

S E N D

+ M O R E

= M O N E Y

Cryptarithmetic problem: mathematical puzzles where digits are replaced by symbols

Find unique digits the letters represent, satisfying the above constraints


SEND MORE MONEY - Model

Variables

S, E, N, D, M, O, R, Y

Domain

{0, …, 9}


SEND MORE MONEY - Model

Constraints

Distinct variables, S ≠ E, M ≠ S, …

S*1000 + E*100 + N*10 + D

+

M*1000 + O*100 + R*10 + E

=

M*10000 + O*1000 + N*100 + E*10 + Y


SEND MORE MONEY – How?

How would you solve the problem using CP techniques? Search tree with backtracking

Constraint propagation

Forward & backward checking

Combination of above?

Different problems may find different techniques more appropriate


SEND MORE MONEY - Solution

9 5 6 7

+ 1 0 8 5

= 1 0 6 5 2

Is this the only solution?

Sometimes we want to maximise an objective

S E N D

+ M O R E

= M O N E Y


SEND MOST MONEY - Problem

S E N D

+ M O S T

= M O N E Y

Objective: we now want to maximise MONEY


SEND MOST MONEY - Problem

Modelling

What does “best” mean

How to find best solution

Search

Assign scores for proposed solution, h

Update the bound, b


COP – real world examples

Resource allocation Nurse rostering systems in hospitals BT services

Timetabling University course/exam scheduling systems

Transportation Flight scheduling (ECLiPSe) and Aircraft allocation (ILOG) at

BA ILOG vehicle routing

Scheduling Job shop scheduling


Summary

Constraint optimisation problem

Branch and bound (B&B)

Heuristic h

Bound b

Examples

SEND + MORE = MONEY (CSP)

SEND + MOST = MONEY (COP)



Constraint Optimisation Problems – Demos in OPL Studio

SEND MORE MONEY - model

S E N D

+ M O R E

= M O N E Y

Variables:

Domain:



//.mod file//declaration//variablesenum Letters {S, E, N, D, M, O, R, Y};

//domainvar int l[Letters] in 0..9;

…



//.mod file//declaration…//problem - constraintssolve {

alldifferent(l) onDomain;l[S] <> 0; l[M] <> 0; ...

};



//.mod file//declaration…//problem - constraintssolve {

...1000*l[S] + 100*l[E] + 10*l[N] + l[D] + 1000*l[M] + 100*l[O] + 10*l[R] + l[E]= 10000*l[M] + 1000*l[O] + 100*l[N] + 10*l[E] + l[Y];

};


G52AIP – AI Programming


G52AIP – AI Programming

SEND MORE MONEY – decision tree

SEND MOST MONEY – model

S E N D

+ M O S T

= M O N E Y

Variables:

Domain:


Smuggler’s Knapsack Problem

A smuggler has a knapsack of capacity 9

He can only make one trip

He can smuggle Bottles of whisky size 4 profit 15

Bottles of perfume size 3 profit 10

Cartons of cigarettes size 2 profit 7

He wants At least a profit of 30

Maximise the profit


Smuggler’s Knapsack Problem

Variables W, P, C

Domain {0, …, 9}

Constraints 4*W + 3*P + 2*C <= 9;

15*W + 10*P + 7*C >=30;

Bottles of whisky size 4 profit 15Bottles of perfume size 3 profit 10Cartons of cigarettes size 2 profit 7


Knapsack Problems

Which items should be put in the bag to maximise the profit


Knapsack Problems

General problems

Given a number of items with associated costs and values

Determine the collection of items so that

Total cost is minimised, or within a limit

Total value is maximised


Simple Assignment Problem

In a factory 4 workers

4 products

Profits of assigning workers to products

Each worker has one product

Each product is assigned to one worker

At least total profit >= 19

p1 p2 p3 p4

w1 7 1 3 4

w2 8 2 5 1

w3 4 3 7 2

w4 3 1 6 3


Simple Assignment Problem

Variables

li,j: worker i assigned to product j

Domain

0, 1

Constraints

sum (li) = 1, i={..4}

sum (weighti,j * li,j) >= 19


Lab Session Next Week

In IBM ILOG OPL IDE

Build COP models


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