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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

- 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 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.

- 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

- 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:

- 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: s1s2, s2s3, etc.

- Slab variables assigned the same
- size are indistinguishable.
- When si = si+1:
- Corresponding rows of orderAare lexicographically ordered.

- E.g. 1001 0110.

Channelling:

A Solution: Model A

3

2

2

1

1

1

1

1

1

oa

ob

oc

od

oe

of

og

oh

oi

- 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.

- 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:

- wastei = si – assWti

(under conditions 1, 2).

- 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:

Channelling:

A Solution: Model B, Phase 1

3

2

2

1

1

1

1

1

1

oa

ob

oc

od

oe

of

og

oh

oi

Model B Implied Constraints

- Unary constraints on order matrix:

- 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.

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

- 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

- 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] = 1orderB[h, si] = 1.

- 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.

- 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.

- 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.