BERTH SCHEDULING BY SIMULATED ANNEALING

By

Ananth

Kamala.V

Journal Title: Berth scheduling by simulated annealing

Authors: Kap Hwan Kim , Kyung Chan Moon

Journal Name: Transportation Research Part

Year: 2003

Volume No: 37

Page No:541-560

Container terminal logistics

Introduction

Literature Review

Research Gap

Methodology

Mathematical Modelling

Simulated Annealing

Conclusion

Quay Side

Yard Side

Land Side

*Christian Bierwirth , Frank Meisel A survey of berth allocation and quay crane scheduling problems in container terminals,European Journal of Operational Research 202 (2010) 615–627

Managers in a container terminals attempts to reduce costs by efficiently

utilizing resources, including human resources, berths, container yards and

various yard equipment.

Berths are the most important resource and good schedules of berths improve

customers satisfaction and increase port throughput, leading to higher

revenues of port.

Port managers usually schedule the usage of berths by an intuitive trial-and-

error method supported by a schedule board or a graphic-user-interface in a

computer system

S.No Authors Contribution Assumption & Methodology

1 Lai and Shih (1992)

compared three berth-allocation rules forcontainer vessel

• Wharf - continuous line• Simulation

2 Brown et al.(1994)

formulated an integer-programming modelfor assigning available sections of a wharf tovessels

• wharf – discrete berthing section• Integer Programming model

3 Lim (1998) minimizing the total length of a wharfoccupied by vessels.

• Wharf - continuous space• berthing times of all vessels -fixed• Heuristics

4 Li et al. (1998) as a scheduling problem with a singleprocessor through which multiple jobs can beprocessed simultaneously

• vessels had already arrived,

• Heuristics

5 Imai et al. (2001)

minimize the waiting time of vessels • Wharf –Discrete• mixed-integer-programming model

• Unlike in Lim’s study (1998), this study attempts to simultaneously determine the berthing

time and the berthing location.

• unlike in the studies by Brown et al. (1994), Imai et al. (2001), and Lai and Shih (1992), the

wharf is considered to be a continuous space rather than a collection of partitioned sections.

• unlike in the studies by Lai and Shih (1992) and Li et al. (1998) which suggested heuristic

rules, this study suggests an optimizing method

In this study

berth-scheduling problem, in which the berthing positions as well as the berthing times

must be determined,

Wharf - Continuous space,

MILP, Simulated Annealing

S.No Attribute Description

1. Spatial Disc - The quay is partitioned in discrete berthsCont - The quay is assumed to be a continuous lineHybr - The hybrid quay mixes up properties of discrete and continuous berthsdraft - Vessels with a draft exceeding a minimum water depth cannot be berthed

arbitrarily

2. Temporal stat - In static problems there are no restrictions on the berthing timesdyn - In dynamic problems arrival times restrict the earliest berthing timesdue - Due dates restrict the latest allowed departure times of vessels

3. Handling Time

fix - The handling time of a vessel is considered fixedpos - The handling time of a vessel depends on its berthing positionQCAP - The handling time of a vessel depends on the assignment of QCsQCSP - The handling time of a vessel depends on a QC operation scheme

4. Performance Measure

wait - Waiting time of a vesselhand - Handling time of a vesselcompl - Completion time of a vesseltard - Tardiness of a vessel against the given due datepos - Berthing of a vessel apart from its desired berthing position

*Christian Bierwirth , Frank Meisel A survey of berth allocation and quay crane scheduling problems in container terminals,European Journal of Operational Research 202 (2010) 615–627

S.No Attribute Description

1. Spatial Disc - The quay is partitioned in discrete berthsCont - The quay is assumed to be a continuous lineHybr - The hybrid quay mixes up properties of discrete and continuous berthsdraft - Vessels with a draft exceeding a minimum water depth cannot be berthed

arbitrarily

2. Temporal stat - In static problems there are no restrictions on the berthing timesdyn - In dynamic problems arrival times restrict the earliest berthing timesdue - Due dates restrict the latest allowed departure times of vessels

3. Handling Time

fix - The handling time of a vessel is considered fixedpos - The handling time of a vessel depends on its berthing positionQCAP - The handling time of a vessel depends on the assignment of QCsQCSP - The handling time of a vessel depends on a QC operation sche

4. Performance Measure

wait - Waiting time of a vesselhand - Handling time of a vesselcompl - Completion time of a vesselspeed - Speedup of a vessel to reach the terminal before the expected arrival timetard - Tardiness of a vessel against the given due datepos - Berthing of a vessel apart from its desired berthing position

*Christian Bierwirth , Frank Meisel A survey of berth allocation and quay crane scheduling problems in container terminals, European Journal of Operational Research 202 (2010) 615–627

Cont|dyn|fix| (w1tard + w2pos)

Mathematical formulation of the berth-

scheduling problem

Berthing times of all the

vessels are fixed, is known to be NP-hard

simulated annealing algorithm

• The quay is assumed to be a continuous line.

• Berthing time of all vessels is fixed.

• The handling time of a vessel is considered fixed.

INPUT

• Length of vessel• Arrival time• Handling time• Departure time• Lowest cost

location• Penalty cost per

hour• Additional

handling cost

BAP

• Mixed integer

programming

(MIP)

• Simulated

Annealing (SA)

OUTPUT

• Berth planning

1. Assumption

2. Parameters

3. Decision variables

4. Objective function and constraint

N The total number of vessels

L The length of wharf

𝑝𝑖 The lowest cost berthing location of vessel i

𝑎𝑖 estimated arrival time of vessel i

𝑑𝑖 departure time requested by vessel I

𝑐1𝑖 The additional travel cost per unit distance for delivering the containers for

vessel i, resulting from non-optimal berthing location relative to the lowest-

cost point.

𝑐2𝑖 The penalty cost per unit time of vessel i, resulting from a departure delayed

beyond the requested due time.

𝑙𝑖 The length of vessels i

𝑥𝑖 The berthing position of vessel i

𝑦𝑖 The berthing time of vessel I

𝑧𝑖𝑗𝑥 1 : if vessel i is located to the left of vessel j on wharf

0 : otherwise

𝑧𝑖𝑗𝑦

1 : if vessel i is berthed before vessel j in time

0 : otherwise

!Minimise the Penalty cost and additional cost;

SETS:

VESSELS:LENGTH,OPERATION_TIME,ARRIVAL_TIME,DEPARTURE_TIME,LOWESTCOST_LOCAT

ION,PENALTY_COST,HANDLING_COST,BERTH_POSITION,BERTHING_TIME,BPP,BPN,BTP,BT

N,L,VTOA,VTOT;

ENDSETS

DATA:

VESSELS,LENGTH,OPERATION_TIME,ARRIVAL_TIME,DEPARTURE_TIME,LOWESTCOST_LOCAT

ION,PENALTY_COST,HANDLING_COST = @OLE('E:\\Book1.XLSX');

L=1200;

M=100000;

ENDDATA

1) Minimise the Penalty cost and handling cost

Min ∑{ c1i│xi - pi│ + c2i( yi + bi-di)+}

where x+ = max{0,x}

Min ∑{c1i(α1+ + α2

+) + c2iβi+}

100 BPP1 + 100 BPN1 + 2000 BTP1 + 2000 BTN1 + 100BPP2 + 100 BPN2 + 2000 BTP2 + 2000 BTN2 + 100 BPP3+ 100 BPN3 + 2000 BTP3 + 2000 BTN3

MIN=@SUM(VESSELS(I):(HANDLING_COST*(BP

P+BPN)+PENALTY_COST*(BTP)));

2) Related to additional handling cost due

non optimal berthing location

xi - pi = αi+ - αi

- , for all i

3) Related to penalty cost resulting from the

delay in the departure of vessels

yi + bi – di = βi+ - βi

- for all I

4) position of the rightmost end of vessel i

is restricted by the length of the wharf

xi + li <= L for all i

BERTHT1 - BTP1 + BTN1 = 13BERTHT2 - BTP2 + BTN2 = 24BERTHT3 - BTP3 + BTN3 = 40

BERTH1 <= 378

BERTH2 <= 265BERTH3 <= 266

BERTH1 - BPP1 + BPN1 = 20BERTH2 - BPP2 + BPN2 = 300BERTH3 - BPP3 + BPN3 = 550

5-6) excludes the case where two vessels are in conflict with each other with respect to the berthing time and the berthing position.

xi + li <= xj + M( 1 – zijx) for all i and j , i ≠ j and

a large positive number M

yi + bi <= yj + M( 1 – zijy) for all i and j , i ≠

j and a large positive number M

BERTH1 + 100000 VTOA1 - BERTH2 <= 99818BERTH1 +100000 VTOA2 - BERTH3 <= 99818BERTH2 +100000 VTOA3- BERTH1 <= 99705BERTH2 +100000 VTOA4 - BERTH3 <= 99705-BERTH1 + BERTH3 +100000 VTOA5 <= 99706-BERTH2 + BERTH3 +100000 VTOA6 <= 99706

BERTHT1 + 100000 VTOT1 - BERTHT2 <= 99993BERTHT1 +100000 VTOT2- BERTHT3 <= 99993

- BERTHT1 + BERTHT2 +100000 VTOT3 <= 99978BERTHT2 +100000 VTOT4 - BERTHT3 <= 99978

- BERTHT1 + BERTHT3 + 100000 VTOT5 <= 99987- BERTHT2 + BERTHT3 +100000 VTOT6 <= 99987

7) excludes the case ,in which case the rectangles

representing the schedules for vessel i and j

overlap

zijx + zji

x + zijy + zji

y >= 1 for all i and j , i ≠ j

8) a vessel cannot berth before she arrives at a

port

yi >= ai for all i

9)αi+ , αi , βi

+ , βi- , xi >= 0 for all i

zijx, zij

y 0/1 integer for all i and j , i ≠j

VTOA1+VTOA3+VTOT1+VTOT3>=1VTOA2+VTOA5+VTOT2+VTOT5>=1 VTOA4+VTOA6+VTOT4+VTOT6>=1

BERTHT1>=12BERTHT2>=22BERTHT3>=27

Vessels Length (m)Loading time (h)

Arrival Time Departure Time

Lowest cost berthing location Penalty Cost

Handling Cost

1 182 7 12 20 20 2000 100

2 295 22 22 46 300 4000 200

3 294 13 27 53 550 2000 100

4 185 6 28 36 900 2000 100

5 263 14 11 34 50 2000 100

6 184 7 21 32 600 2000 100

7 262 18 5 30 900 2000 100

Berth Position/TIME Location BERTH_POSITION( 1) 20.00000 0.000000BERTH_POSITION( 2) 300.0000 0.000000BERTH_POSITION( 3) 595.0000 0.000000BERTH_POSITION( 4) 900.0000 0.000000BERTH_POSITION( 5) 37.00000 0.000000BERTH_POSITION( 6) 600.0000 0.000000BERTH_POSITION( 7) 900.0000 0.000000BERTHING_TIME( 1) 12.00000 0.000000BERTHING_TIME( 2) 22.00000 0.000000BERTHING_TIME( 3) 28.00000 0.000000BERTHING_TIME( 4) 28.00000 0.000000BERTHING_TIME( 5) 19.00000 0.000000BERTHING_TIME( 6) 21.00000 0.000000BERTHING_TIME( 7) 5.00000 0.000000

0

50

100

150

200

0 8 16 24 32 40

No of Vessels

Com

puta

tional

Tim

e (

sec)

Original Paper Introducing the Concept

–Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., “Optimization by Simulated Annealing,”

Science, Volume 220, Number 4598, 13 May 1983, pp. 671-680.

Simulated Annealing strategy

Physical Analogy

Simulated Annealing Algorithm

Parameters

• SA is based on neighbourhood search

• Neighbourhood of a solution x is a set of solutions that can be reached from

x by a simple operation (move)

SA is a strategy which occasionally allows uphill moves.

Uphill moves in SA are applied in a controlled manner

• A solid material is heated past its melting point and then cooled back into a solid state

(annealing). The final structure depends on how the cooling is performed

slow cooling → large crystal (low energy)

fast cooling → imperfections (high energy)

• Metropolis’ algorithm simulates the change in energy of the system when subjected to the

cooling process; the system converges to a final “frozen” state of a certain energy.

Thermodynamic simulation SA Optimization

System states Feasible solutions

Energy Cost

Change of state Neighbouring solution

Temperature Control parameter

Frozen state Solution (close to optimal)

• TI must be high enough - in order the finalsolution to be independent of the startingone.

• Temperature length (TL): number ofiterations at a given temperature

• Cooling ratio (f): rate at which temperatureis reduced

• A given minimum value of the temperaturehas been reached

• A specified number of total iterations hasbeen executed

• Generated randomly.

• Good solution (possibly produced by another heuristics); in this case the starting temperature should be lower.

Initial temperature (TI)

Temperature length (TL)

cooling ratio (function f)

stopping criterion

starting solution

Initial temperatures: 10, 40, 70, 100, and 130 C

Temperature length: N(N-1)/2

Cooling ratios:0.55, 0.65, 0.75, 0.85, 0.95, and 0.99

Initial temperature (TI)

Temperature length (TL)

cooling ratio (function f)

The minimum average objectivevalues were found when the initialtemperature was 40 and thecooling rate was 0.65

1. A solution whose objective value is zero is found.

2. Temperature becomes less than 1.

3. The best value of the objective function has not

been changed during the previous five consecutive

external loops.

4.The initial sequence of vessels is in the increasing

order

∝𝑝𝑖

𝐿+

1−∝ 𝑑𝑖

𝑚𝑎𝑥𝑑𝑖Where0<∝< 1.

5. Pairwise exchanges (swapping the positions of two

randomly selected elements) are used to obtain

neighbours

Initial temperature (TI)

Temperature length (TL)

cooling ratio (function f)

stopping criterion

starting solution

space of feasible solutions

and neighborhood structure

Real data for a berth and sizes of vessels were collected from the Pusan East

Container Terminal (PECT), which is one of the largest container terminals in Korea.

The length of the wharf at PECT is 1200 m

The arrival times and the requested departure times were generated randomly

pi was also generated randomly within the range of the wharf

T = 40,r=0.65,

Additional travel cost = $10 li /230

Penalty cost = $2000 li /230

S.No Factors Number of Levels # Levels

1. No of vessels L1-7, L2- [11–16], L2-[26–29], and L3-[38–40] 4

2. Problem Instance per Configuration 5

Total Problem Instances 4*5= 20

optimal solutions

were obtained for 18 of

the 20 example

problems.

A linear integer program was formulated for the berth-scheduling problem and the problem

was solved using the LINDO package.

The computational time of LINDO increased rapidly when the number of vessels became

higher than 7

Simulated annealing algorithm for the berth-scheduling problem was suggested.

It was found that the simulated annealing algorithm results in near-optimal solutions and that

the computational time was within the limits of practical usage

• Many planners simultaneously consider the schedule of quay cranes and that of the berth.

• This study assumed a wharf in the shape of a straight line. However, there are practical

examples of wharfs in different shapes or there may be multiple wharfs in the same area