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### Berthing Problem

Chen Fang Yew Nicholas1, Gani Zhi Hao Terry1, Vo Thanh Minh Tue1

1NUS High School of Mathematics and Science, Singapore

Introduction

- The given problem is very complex and arises in daily management of the Port of Singapore.
- The port success depends on a robust and efficient berthing plan.
- The program must compute on demand.

Introduction

- We propose a greedy first fit algorithm to solve the berthing problem.
- We also introduce graph theory as a possible approach to solve the problem.

The problem

- Ships arrive at various times and have different lengths and berthing times.
- Our task is to devise a berthing plan that minimizes the waiting time for all ships.

Objective

- Minimize the waiting time function
- Where is the time the ship is berthed,
Is the arrival time, and is the waiting constant

Wait time case study

- When =1, LHS=RHS
- Either order is the same.

Wait time case study

- Second option incurs more waiting time
- Minimizing total height equates to minimizing total j, waiting time.
- First fit algorithm.

First fit algorithm

- Greedy algorithm.
- Choose the lowest time possible.
- Choose the lowest space possible.

Advantages

- Extremely fast to compute.
- Give a reasonable good solution.
- Modify the algorithm to meet realistic constraints.

Modified realistic algorithm

- “port stay times are often delayed beyond the estimated values”
- Takes into account delay time
- Models after realistic conditions
- Allows for inter-clearance distance between ships

Discretization

- Transform continuous data into discrete values.
- Simplify the data input.
- Scale the solution range.

A novel approach

- Apply graph theory.
- Rigorous mathematical ground.

Berthing problem

- Variables
- ai: Arrival time of ith ship.
- hi: Processing time.
- Li: Length of the ship.
- Waiting penalty for ith ship: (yi – ai )α

Berthing problem

- Solution: { (t1, y1), (t2, y2), …, (tn, yn) }
- ti: Time in which ith ship starts to dock.
- yi : Lower y-coordinate of ith ship.
- Objective function.

Solution space

- (ti, yi)
- Every point in a solution space is a feasible solution.
- Overlapping of solution spaces yields solution domain.

Solution space

- Overlapping of rectangle <=> incompatible solution pairs.
- Compatible solution pairs.
- (ti, yi) (tj, yj)

Graph theory

- Each solution (ti, yi, i) can be assigned as a vertex.
- Compatible solutions are joined by an edge.

Vertex for ith ship: (ti, yi)

E = (ti – ai)α + (tj – aj)α

Vertex for jth ship: (tj, yj)

Graph theory

- Consider six ships A to F with solution pairs SA to SF
- The vertices A to F and their edges form a complete graph
- A cycle is created as all the vertices join to each other once.

AF

A

F

AB

EF

E

B

BC

DE

CD

D

C

Graph theory

- By adding the weight of each edge of the cycle together, the overall delay time can be calculated.

Graph theory

- A swap in position of vertices within the cycle does not change the overall delay time.

AF

AF

A

F

F

A

AB

BF

EF

AE

=

E

E

B

B

BC

DE

BC

DE

CD

CD

Graph theory

- Consider another vertex not within the cycle
- If weight of new edge < weight of old edge, replace old vertex with new vertex

AF

A

F

FG

AB

EF

G

E

B

BC

DE

DG

CD

Graph theory

- For example,
- If wFG + wDG < wEF + wDE, then replace E with G

AF

A

F

FG

AB

EF

G

E

B

BC

DE

DG

CD

The algorithm

- Start with the earliest ship.
- Depth-first-search for a possible complete sub graph.
- Check all vertices in the graph if improvement possible.
- If no improvement possible, then the program terminates.

Advantages

- Greedy algorithm can give a reasonable good solution.
- By transforming from a geometrical problem to a graph problem, we can handle constraints more easily.

Challenges

- Large number of vertices and edges.
- Complex relationship.

Conclusion

- Justified using first fit algorithm
- Devised an algorithm to solve the problem
- Improved on it to take into account realistic conditions
- Suggested a novel approach using graph theory to solve this problem

References

- Dai, J., Lin, W., Moorthy, R., & Teo, C.-P. (2003). Berth Allocation Planning Optimization in Container Terminal. 1-33.
- Duin, C. W., & Sluis, E. v. (n.d.). On the complexity of Adjacent Resource Scheduling.
- Lim, A. (n.d.). An Effective Ship Berthing Algorithm.

The end

- Thank you for your attention.

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