Modelling a Steel Mill Slab Design Problem

1
Modelling a Steel Mill Slab Design Problem

• Alan Frisch, Ian Miguel, Toby Walsh
• AI Group
• University of York

2
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

3
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.

4
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
5
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

6
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.

7
Model A Order Matrix
oa ob oc od
s1 0 0 1 1
s2 0 1 0 0
s3 1 0 0 0
s4 0 0 0 0

• Slab variables assigned the same
• size are indistinguishable.
• When si si1
• Corresponding rows of orderA are
  lexicographically ordered.
• E.g. 1001 ? 0110.

8
Model A Colour Matrix
Red Green Blue Orange
s1 0 0 1 1
s2 0 1 0 0
s3 1 0 0 0
s4 0 0 0 0
Channelling
9
A Solution Model A
3
2
2
1
1
1
1
1
1
oa
ob
oc
od
oe
of
og
oh
oi
order oa ob oc od oe of og oh oi
s14 0 0 0 0 0 0 1 1 1
s23 1 0 1 0 0 0 0 0 0
s33 0 1 0 0 0 0 0 0 0
s43 0 0 0 1 1 1 0 0 0
0 0 0 0 0 0 0 0 0
colour Red Green Blue Orange Brown
s1 0 0 0 1 1
s2 1 1 0 0 0
s3 0 1 0 0 0
s4 0 0 1 1 0
0 0 0 0 0
10
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.

11
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

12
Model A Implied Constraints (3)

• wastei si assWti

(under conditions 1, 2).

13
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

14
Model B, Phase 1 Order Matrices
oa ob oc od
z1 0 0 1 1
z3 0 1 0 0
z4 1 0 0 0
Red Green Blue Orange
z1 0 0 1 1
z3 0 1 0 0
z4 1 0 0 0
Channelling
15
A Solution Model B, Phase 1
3
2
2
1
1
1
1
1
1
oa
ob
oc
od
oe
of
og
oh
oi
oa ob oc od oe of og oh oi
z10 0 0 0 0 0 0 0 0 0
Z33 1 1 1 1 1 1 0 0 0
Z44 0 0 0 0 0 0 1 1 1
Red Green Blue Orange Brown
z1 0 0 0 0 0
z3 1 1 1 1 0
z4 0 0 0 1 1
16
Model B Implied Constraints

• Unary constraints on order matrix

17
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.

18
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

oa ob oc od oe of
s1 1 0 1 0 0 0
s2 0 1 0 0 0 0
s3 0 0 0 1 1 1
og oh oi
s1 1 1 1
19
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 gt 2

3
3
1
1
oa
ob
oc
od
Slab Sizes 4, p 1

• 2 Slabs of size 4

oa ob oc od
s1 ? ? ? ?
s2 ? ? ? ?
20
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.
• orderAh, i 1 ? orderBh, si 1.

21
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.

22
Results
Orders Optimal Model A Model AB
15 92 95 21, 0.1s 94 5108, 1.1s 93 5619, 1.2s 92 17734, 3.6s 95 21, 0.2s 94 4529, 1.2s 93 4948, 1.4s 92 15983, 4.6s
16 99 107 17, 0.1s 101 5112, 0.9s 100 5305, 0.9s 99 92441, 17.8s 107 17, 0.1s 101 2934, 0.8s 100 3103, 0.9s 99 78548, 23.5s
17 103 107 23, 0.1s 105 13074, 2.6s 104 26757, 5.5s 103 237290, 50.2s 107 23, 0.1s 105 11201, 2.9s 104 21580, 6.4s 103 204513, 67.1s
18 110 119 19, 0.2s 111 1012, 0.4s 110 1179281, 253.4s 119 19. 0.2s 111 988 0.4s 110 1014092, 350.3s
23
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.

24
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.