Introduction

• The berthing assignment problem requires that a detailed time-and-space-schedule be planned for incoming ships, with the goal of minimizing the total waiting time.

• The problem can be graphically represented in a two dimensional space:

The Berth Problem

• Each ship arrives to the port at a time T. However, in our planning, each ship enters the berth at a different time, tThe problem, therefore, is to find an arrangement which yields:

0

minn

i ii

t T

Where n is the number of total ships

• As this problem is NP – hard, it is impossible to computationally reach an optimal solution. Local heuristics often fail to utilize the entire data set; hence a global algorithm which constantly takes into consideration all the ships is preferred. We propose a method based on Bohr’s quantum atom model.

Theory

• In 1913, Niels Bohr first introduced his model of the atom. It is characterized by three main traits:

1) The atom is composed of several electrons circling a nucleus

2) These electrons have well defined discrete (quantized) orbits

3) An atom may emit or absorb energy by means of photons, which causes the electrons to go to lower or higher orbits (lower or higher energy levels), respectively.

Bohr’s Atom Model

• By creating our own variant of the atom model, we are able to satisfy the unique demands of the problem: harnessing a global scope, yet retaining the individual properties of each ship, thus allowing to minimize waiting times and tackle delays.

Since we deal with a circle, we prefer to use polar coordinates, with the center of axes set in the center of the disc. Stretching must be taken into consideration.

• In order to apply Bohr’s orbiting electron model to the problem, a stretching transformation from a rectangular plane to a circular disc is needed.

Graphic representation Application

0

0

1t

xr

r t r

The stretching factor was found by interpolating between t = 0 and t = H, and is applied to every line in the form of t = t0

x

Ht1 – second arrival time

R0

Rn

t1 → R1

• Each ship is represented by an electron-like-creature, (called “Skapho”) orbiting a nucleus which is positioned in the center of the disc. Unlike an electron, however, the skapho has width and height.

• The energy levels of the original atom model, are represented by discrete time orbits – each orbit is a circle, which represents an hour; the skapho’s orbiting distance (r) therefore denotes at what time it reaches the port, whereas the position on the orbit indicates where it is on the berth ().

• Achieving minimal waiting time is parallel to having all the skaphos as close to the center as possible – minimal energy. A skapho cannot go below its arrival time orbit as this would be the equivalent of arriving early (this doesn’t happen in real life…).

DefinitionsApplication

• We start with an “empty” atom – a single nucleus.

• The skaphos are released one by one into the atom – each skapho goes to its arrival time orbit (this is its minimal orbit allowed); the order of releasing is from the highest energy skapho to the closest skapho.

• Inevitably, the orbits will start to get clustered. Each skapho tries to find an empty place on its orbit, but collisions will occur, as skaphos each span over several orbits. When this happens, we treat the atom as if it has been hit by a photon – one of the skaphos in this orbit becomes excited (going to a further orbit).

• The excited skapho is chosen by a weighted distribution – each skapho has a chance to get excited depending on its size – the smaller the area, the higher the chances of excitation.

AlgorithmApplication

1

1

1

0

1

i i

j j

n

tot ii

t li n

t lj

p p

p

• An excited skapho will jump to the next orbit, as well as moving a certain amount dl to one of the sides, chosen randomly, in order to decrease collisions.

• If a skapho in an excited state still collides with another, the process is repeated.

• Skaphos do not stay in excited states for long – they try to go back to their original positions (the equivalent of atomic level decay). If a skapho can decay without colliding, it does so automatically. Otherwise, it has a certain chance of decaying each turn, with smaller skaphos decaying more quickly than larger ones.

• The process is stopped once a stable state is achieved. This means that for several consecutive turns, no excitation and no decay has occurred.

Algorithm cont.Application

• Each step takes O(n2), where n is the number of skaphos in the atom.

• Careful consideration is needed when unfolding the atom back into a rectangle.

• Our algorithm is adaptive; in case of delay, only the surrounding area is affected.

• There is also a need to parameterize factors (e.g the decaying time, dl, etc )

Analysis