# Berthing Problem - PowerPoint PPT Presentation

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Berthing Problem. Chen Fang Yew Nicholas 1 , Gani Zhi Hao Terry 1 , Vo Thanh Minh Tue 1 1 NUS High School of Mathematics and Science, Singapore. Introduction. The given problem is very complex and arises in daily management of the Port of Singapore .

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Berthing Problem

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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.