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 2 3 11111 2 1 Slab Sizes: {1, 3, 4} ( = 3) Orders: {o a, …, o i } (j = 9) Colours: {red, green, blue, orange, brown} (k = 5) p = 2 abcdefghi 2 3 11 1 1 1 1 2 Solution:

5. Model A – Redundant Variables Number of slabs is not fixed. –Assume highest order weight does not exceed maximum slab size. Slab variables: {s 1, …, s j }. –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 s i. –If s i is not necessary to solve the problem, s i = 0. Slab variables are indistinguishable. So model A suffers from symmetry: –Counteract with binary symmetry-breaking constraints: s 1 s 2, s 2 s 3, etc.

7. Model A Order Matrix oaoa obob ococ odod s1s1 0011 s2s2 0100 s3s3 1000 s4s4 0000 Slab variables assigned the same size are indistinguishable. When s i = s i+1 : Corresponding rows of order A are lexicographically ordered. E.g. 1001 0110.

8. Model A Colour Matrix RedGreenBlueOrange s1s1 0011 s2s2 0100 s3s3 1000 s4s4 0000 Channelling:

9. A Solution: Model A 2 3 11111 2 1 oaoa obob ococ odod oeoe ofof ogog ohoh oioi order oaoa obob ococ odod oeoe ofof ogog ohoh oioi s 1 =4000000111 s 2 =3101000000 s 3 =3010000000 s 4 =3000111000 …000000000 colourRedGreenBlueOrangeBrown s1s1 00011 s2s2 11000 s3s3 01000 s4s4 00110 …00000

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) assWt i is the weight of orders assigned to s i. –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) waste i = s i – assWt i 1. 2. (under conditions 1, 2).

13. Model B – Abstraction 2-phase approach: 1.Construct/solve an abstraction of the problem. 2.Solve independent sub-problems, assigning a subset of the orders to slabs of a common size. Phase 1: –Slab size variables, {z 1, z 2, …}. –Domains: {0, …, j} number of slabs of corresponding sized used. –Solution quality:

14. Model B, Phase 1 Order Matrices oaoa obob ococ odod z1z1 0011 z3z3 0100 z4z4 1000 RedGreenBlueOrange z1z1 0011 z3z3 0100 z4z4 1000 Channelling:

15. A Solution: Model B, Phase 1 2 3 11111 2 1 oaoa obob ococ odod oeoe ofof ogog ohoh oioi oaoa obob ococ odod oeoe ofof ogog ohoh oioi z 1 =0000000000 Z 3 =3111111000 Z 4 =4000000111 RedGreenBlueOrangeBrown z1z1 00000 z3z3 11110 z4z4 00011

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 2 3 11111 2 1 oaoa obob ococ odod oeoe ofof ogog ohoh oioi oaoa obob ococ odod oeoe ofof s1s1 101000 s2s2 010000 s3s3 000111 3 Slabs of size 3 1 Slab of size 4 ogog ohoh oioi s1s1 111

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. o a = 4 o b = 4 o c = 4 o d = 4 z 4 > 2 3 3 11 oaoa obob ococ odod oaoa obob ococ odod s1s1 ???? s2s2 ???? 2 Slabs of size 4 Slab Sizes: {4}, p = 1

20. A Dual Model A/B Model A and model B, phase 1. –Explicit slab variables (s i ) and slab-size variables (z i ). –Order matrices referring to explicit slabs (order A ) and to slab-sizes (order B ). –Both types of colour matrix. Channelling constraints between the models maintain consistency, aid pruning. –Number of occurrences of i in {s 1, …, s j } = z i. –order A [h, i] = 1 order B [h, s i ] = 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 OrdersOptimalModel AModel AB 159295: 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 1699107: 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 17103107: 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 18110119: 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.