1. A Constraint Programming Approach to Solving a Ship Loading Problem Andreas Jordan Professor Markus Stumptner

2. 2 Background Currently, load plans for amphibious ships are created by ‘dragging and dropping’ representative size boxes into a trim and stability software program. There is a requirement for some automated support when planning loads for both the amphibious ‘mother ships’ and associated watercraft conducting landings / withdrawals.

3. 3 Objectives The primary objective minimise the space used by the loads => minimising the number of amphibious ships and watercraft to conduct operations. Near-optimal solutions preferred within the order of seconds The load must also produce a stable load plan

4. 4 Problem Description Belongs to the general class of “Bin Packing Problems” Description: The packing of a finite number of rectangles into a number of larger sized bins where: No items are allowed to extend past the edges of the bin nor any overlap permitted. The challenge is to assign all the items, with no overlap, to the minimum number of bins with their edges parallel to those of the bins. Problem is NP-Hard (ie. cannot be solved in polynomial time) Exhaustive search impractical. Therefore, heuristical methods are required for most practical problems.

5. 5 Sourcing Software Description of problem readily lends itself to modelling as a Constraint Optimisation Problem (COP) Possible choices:- Matlab Eclipse Prolog ILOG CPLEX® Comet Selection criteria Acquisition costs Optimised libraries with pre-built constraint-reasoning functionality Availability of Global Constraints ( eg. selectMin) => Comet

6. 6 Modelling as a Constraint Optimisation Problem Determine the finite domain decision variables Packing – establishes a relationship between an item and a bin. eg. Load – tracks the current load of each bin X – represents x-positions within each bin Y – represents y-positions within each bin Domains Items – represents the number of items to be packed Bins – total number of bins Constraints Constrain load of bin Constrain placement of items within bin Minimise the number bins used to pack items

7. 7 Packing Heuristic Selection Several packing algorithms exist, including:- Finite-First-Fit Bottom-Left-Fill …and many more Most not suitable for this application:- they typically require items to be sorted in non-decreasing order no advantage taken of additional attributes (eg. weight) Key requirements, including:- Prioritisation of items – Strict, Flexible and None Stability of packing (lateral and longitudinal) Can make no assumptions about the distribution of item sizes

8. 8 Nine barrier types C, RL, LR, LCR, CLR, LRC, RCL, RLC, CRL Barrier composed of (up to) 5 segments Pack item into lowest barrier LX, LY, CX, RX, RY represent cartesian barrier coordinates Packing Barrier (Heidelberg et al,1998) RLCLR CLR RCL LRCLCR RLCCRL LX LY RY CXRX Packed items

9. 9 Representing Variables in Comet Example var {int} packing[Items](cp, Bins); Explanation Var - declaration for a finite domain decision variable {int} – declares that the domain comprises discrete integers packing[Items] – name of variable where “Items” represent the range of variables associated with ‘packing’ (cp, Bins) - range of valid integer values that each variable can take. This representation allows the constraint solver to reason packing[1] = 1..12 … packing[100] = 1..12

10. 10 Representing Constraints in Comet Solver cp(); //declare the variables Solve { //post the constraints //Constraint which ensures that the load of any bin is less than or equal to a bins capacity forall(b in Bins) cp.post((load[b] == sum(i in Items) (packing[i]==b)*itemWeight[i]) && load[b] <= binCapacity_) ; } using{ //non deterministic search }

11. 11 Preliminary Results 4 items ==== Post-processing Results ==== Solution found with: 2 bins load[43800,42000,0,0,0,0,0,0,0,0] packing[1,2,1,1] x[0,0,34,44] y[0,0,0,0] #Choices = 8 #fail = 0 Time: 0 seconds ==== End ==== 8 items ==== Post-processing Results ==== Solution found with: 2 bins load[45400,44000,0,0,0,0,0,0,0,0] packing[1,2,1,1,1,2,2] x[0,0,34,0,18,0,15] y[0,0,0,96,96,96,96] #Choices = 39728 #fail = 158149 Time: 4 seconds ==== End ==== >8 items (using Windows machine with 2GB RAM) runs out of virtual ram after approximately 3 hours

12. 12 Further Work Implement item placement based on multi-tiered priorities Incorporate centre-of-gravity calculations to build solutions already favouring a stable load Automatic barrier adjustment after failing to insert any items Adjust upper bound on item length to be uppermost adjacent level plus/minus a tolerance, t. Incorporate No-Go zones

13. 13 Thankyou Questions? Source: http://www.tomw.net.au/technology/transport/amphibious.shtml …different watercraft and ship types should be considered Source: Wikipedia

14. 14 References Dyckhoff, H 1990, 'A typology of cutting and packing problems', European Journal of Operational Research, vol. 44, no. 2, pp. 145-159. Sexton, J, et al. 2004, Efficient Solutions in Load Planning, DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION SALISBURY (AUSTRALIA) SYSTEMS SCIENCES LAB.