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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 case study
Wait time case study
  • When =1, LHS=RHS
  • Either order is the same.
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.
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.
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 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)
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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