Modelling a steel mill slab design problem
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Modelling a Steel Mill Slab Design Problem. Alan Frisch, Ian Miguel, Toby Walsh AI Group University of York. Background/Motivation. Many problems exhibit some structural flexibility. E.g. the number required of a certain type of variable .

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Modelling a Steel Mill Slab Design Problem

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Modelling a steel mill slab design problem

Modelling a Steel Mill Slab Design Problem

Alan Frisch, Ian Miguel, Toby Walsh

AI Group

University of York


Background motivation

Background/Motivation

  • Many problems exhibit some structural flexibility.

    • E.g. the number required of a certain type of variable.

  • Flexibility must be resolved during the solution process.

  • Slab design representative of this type of problem.

  • Dawande et al. ” Variable Sized Bin Packing with Color Constraints”.

    • Approximation algorithms guaranteed to be within some bound of an optimal solution


The slab design problem

The Slab Design Problem

  • The mill can make  different slab sizes.

  • Given j input orders with:

    • A colour (route through the mill).

    • A weight.

  • Pack orders onto slabs, minimising total slab capacity. Constraints:

    • Capacity: Total weight of orders assigned to a slab cannot exceed slab capacity.

    • Colour: Each slab can contain at most p of k total colours.


An example

An Example

  • Slab Sizes: {1, 3, 4} ( = 3)

  • Orders: {oa, …, oi} (j = 9)

  • Colours: {red, green, blue, orange, brown} (k = 5)

  • p = 2

2

1

3

Solution:

2

1

1

1

1

3

1

2

2

1

1

1

1

1

1

a

b

c

d

e

f

g

h

i


Model a redundant variables

Model A – Redundant Variables

  • Number of slabs is not fixed.

    • Assume highest order weight does not exceed maximum slab size.

  • Slab variables: {s1, …, sj}.

    • Value is size of slab.

  • Solution quality:


Slab variable redundancy symmetry

Slab Variable Redundancy/Symmetry

  • Some slab variables may be redundant:

    • 0 is added to the domain of each si.

    • If si is not necessary to solve the problem, si = 0.

  • Slab variables are indistinguishable.

  • So model A suffers from symmetry:

    • Counteract with binary symmetry-breaking constraints: s1s2, s2s3, etc.


Model a order matrix

Model A Order Matrix

  • Slab variables assigned the same

  • size are indistinguishable.

  • When si = si+1:

    • Corresponding rows of orderAare lexicographically ordered.

  • E.g. 1001  0110.


Model a colour matrix

Model A Colour Matrix

Channelling:


Modelling a steel mill slab design problem

A Solution: Model A

3

2

2

1

1

1

1

1

1

oa

ob

oc

od

oe

of

og

oh

oi


Model a implied constraints

Model A Implied Constraints

  • Combined weight of input orders is a lower bound on optimisation variable:

  • Lower bound on number of slabs required:

  • With symmetry-breaking constraints, decomposes

  • into unary constraints on slab variables.


Model a implied constraints 2

Model A Implied Constraints (2)

  • assWti is the weight of orders assigned to si.

    • Prune domains by reasoning about reachable values via dynamic programming [Trick, 2001].

    • Incorporate both size and colour information.

    • More powerful if done during search (future work).

  • Minimum number of slabs required:


Model a implied constraints 3

Model A Implied Constraints (3)

  • wastei = si – assWti

(under conditions 1, 2).


Model b abstraction

Model B – Abstraction

  • 2-phase approach:

    • Construct/solve an abstraction of the problem.

    • Solve independent sub-problems, assigning a subset of the orders to slabs of a common size.

  • Phase 1:

    • Slab size variables, {z1, z2, …}.

    • Domains: {0, …, j} number of slabs of corresponding sized used.

    • Solution quality:


Model b phase 1 order matrices

Model B, Phase 1 Order Matrices

Channelling:


Modelling a steel mill slab design problem

A Solution: Model B, Phase 1

3

2

2

1

1

1

1

1

1

oa

ob

oc

od

oe

of

og

oh

oi


Modelling a steel mill slab design problem

Model B Implied Constraints

  • Unary constraints on order matrix:


Model b phase 2

Model B, Phase 2

  • Model B, Phase 1 is ambiguous.

  • A Phase 1 solution does provide:

    • Number and sizes of slabs required.

    • Size of slab each order is assigned to.

    • Quality of final solution.

  • Phase 1 solution used to construct much simpler, independent, phase 2 sub-problems.


Modelling a steel mill slab design problem

Model B, Phase 2 Sub-problems

3

2

2

1

1

1

1

1

1

oa

ob

oc

od

oe

of

og

oh

oi

  • 3 Slabs of size 3

  • 1 Slab of size 4


The price of ambiguity

The Price of Ambiguity

  • Phase 2 sub-problems may be inconsistent.

    • Isolate reasons for failure.

    • Post constraints at phase 1.

    • Solve phase 1 again.

  • E.g.

    oa = 4 ob = 4 oc = 4 

    od = 4 z4 > 2

3

3

1

1

oa

ob

oc

od

Slab Sizes: {4}, p = 1

  • 2 Slabs of size 4


A dual model a b

A Dual Model A/B

  • Model A and model B, phase 1.

    • Explicit slab variables (si) and slab-size variables (zi).

    • Order matrices referring to explicit slabs (orderA) and to slab-sizes (orderB).

    • Both types of colour matrix.

  • Channelling constraints between the models maintain consistency, aid pruning.

    • Number of occurrences of i in {s1, …, sj} = zi.

    • orderA[h, i] = 1orderB[h, si] = 1.


A b search strategies

A/B Search Strategies

  • Instantiate model A variables first:

    • Channelling constraints ensure model B variables instantiated.

    • Analogous to pure model A approach.

  • Instantiate model B variables first:

    • Channelling constraints constrain model A variables.

    • Analogous to pure model B approach.

  • Interleaved Strategy:

    • Obtain most efficient pruning of the search space.


Results

Results


Model b results

Model B Results?

  • On these problems, many solutions at phase 1.

  • Cycle is therefore lengthy.

  • Improve efficiency:

    • Model phase 1 as a dynamic CSP.

    • Reduce arity of recorded constraints.

    • Phase 1 heuristics.

    • Use dynamic programming information.


Conclusions

Conclusions

  • Results only on small instances.

  • All models need further development:

    • More implied constraints.

    • Better heuristics

  • Set variable model:

    • Each represents a slab

    • Domain is set of orders assigned.

  • Activity DCSP model:

    • Model A slab variables `activated’ according to remaining capacity of open slabs.


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