Berthing problem
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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

Berthing Problem

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

1NUS High School of Mathematics and Science, Singapore


Introduction

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.


Introduction1

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

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

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 scenario 1

Wait time scenario 1


Wait time scenario 11

Wait time scenario 1


Wait time case study

Wait time case study

  • When =1, LHS=RHS

  • Either order is the same.


Wait time scenario 2

Wait time scenario 2


Wait time scenario 21

Wait time scenario 2


Wait time case study1

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

First fit algorithm

  • Greedy algorithm.

  • Choose the lowest time possible.

  • Choose the lowest space possible.


Sample data

Sample data


Advantages

Advantages

  • Extremely fast to compute.

  • Give a reasonable good solution.

  • Modify the algorithm to meet realistic constraints.


Modified realistic algorithm

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

Discretization

  • Transform continuous data into discrete values.

  • Simplify the data input.

  • Scale the solution range.


Sample data1

Sample data


A novel approach

A novel approach

  • Apply graph theory.

  • Rigorous mathematical ground.


Berthing problem1

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 problem2

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

Solution space


Solution space1

Solution space

  • (ti, yi)

  • Every point in a solution space is a feasible solution.

  • Overlapping of solution spaces yields solution domain.


Solution space2

Solution space

  • Overlapping of rectangle <=> incompatible solution pairs.

  • Compatible solution pairs.

  • (ti, yi)  (tj, yj)


Solution space3

Solution space


Graph theory

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 theory1

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 theory2

Graph theory

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


Graph theory3

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 theory4

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 theory5

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

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.


Advantages1

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

Challenges

  • Large number of vertices and edges.

  • Complex relationship.


Conclusion

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

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

The end

  • Thank you for your attention.


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