This document discusses dynamic modeling and nested tabu search approaches for berth and quay crane allocation problems. It presents a two-stage rolling horizon approximation model that minimizes sample average cost to effectively solve multistage problems with stochastic vessel arrival times. Computational results show tabu search heuristics find good solutions faster than CPLEX for dynamic berth allocation problems. The two-stage approach also outperforms one-stage models in optimizing total vessel dwell time.

1. Berth and Quay Crane Allocation
Problems: Dynamic Modeling and
Nested Tabu Search
Alan Erera1, Xiaole Han2
1 School of Industrial and Systems Engineering, Georgia Tech
2 Department of Mechanical Engineering, Shanghai Jiao Tong
TSL Workshop 2013

2. What to remember
1. Berth and quay crane allocation problems are
dynamic and stochastic
2. List-based multi-layer tabu search heuristics
fast and effective for dynamic problems
3. Multistage problems with stochastic arrival
times can be solved effectively with two-
stage rolling horizon approximation models
that minimize sample average cost

3. Costly, bottleneck resources
• Multiple vessel berthing areas

4. Costly, bottleneck resources
• Quay cranes

5. Berth and crane allocation
• Assign berthing time, berthing position,
and crane assignment to vessels
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
Time
Berth
Time period
Vessel
1
2
3

6. Berth and crane allocation
• Minimize
– Total dwell (flow) time
– Total vessel delay time
vessel arrival
berthed departed
scheduled depart
dwell (flow) time
delay (non-negative)

7. Dynamic BAP: Relative Position MIP
• Pack vessel rectangles in time and space, and
ensure no pairwise overlap

8. Deterministic Dynamic BAP
• Solving Relative Position MIP
– Difficulty arises when vessel arrival times clustered
and when berth length accommodates several
heterogeneous length vessels
– Short berth problems remain NP-hard: single
machine scheduling problem with release time
minimizing total completion (flow) time
– CPLEX for easier problems
– Nested Tabu Search (Ak, 2008) for harder problems

9. Nested Tabu Search for BAP
• Solution encoding
–L : Vessel priority list
–B : Berth location of vessels
(2,1,3,4,5)

10. Nested Tabu Search for BAP
• Fast search
– Default decoding: first-fit bin packing
– Second layer: berth position tweaking only for
primal non-tabu listsInitial
Solution
L
B
Move 1
Move 2
Iteration1
Move 1
Move 2
Iteration1
Move 1
Move 2
Iteration2
Move 1
Move 2
Iteration2
LAYER 1: PRIORITY LIST MOVES
LAYER 2:
BERTH MOVES
Best
Solution
First Fit
L P

11. Dynamic BAP Results
• Easier instances
– 10, 12, 14 vessels
– Berth length per avg vessel length: 3
– Arrival times in [0, 10], processing times in [1, 6]
• Harder instances
– 20, 25, 30 vessels
– Berth length per avg vessel length: 6
– Arrival times in [0, 20], processing times in [1, 6]

12. Dynamic BAP Results
• Easier instances
– 69% solved to optimality by CPLEX
– Tabu search finds equal or better solutions in every
instance
• Harder instances
– None solved to optimality by CPLEX
– Best lower bounds found by machine scheduling lower
bounding approach
– Tabu search improves initial solutions by ~ 70%, similar
to performance on easier instances

13. Dynamic BAP Results
• Tabu search is fast
– Computation time in seconds

14. Dynamic BQCAP Modeling
• Vessel processing times depend on number
of quay cranes assigned
– Assume crane assignment fixed for a shift
– Completion time for a vessel is equal to start
time plus required crane-time divided by cranes
assigned
– Fixed number of cranes available3
4
1
5
6 10
7
2
9
Time
Berth space
k-1 k k+1 k+2 k+3
8
11
12
13
Crane amount

15. Dynamic and Stochastic BQCAP
• Decision stages: shifts
• Information dynamics
– Vessel arrival times stochastic
– Known with certainty as arrival time nears
• Decision dynamics
– Berthing time, crane assignment adjustable each
shift
– Berth position fixed in advance, for effective
container yard operations

16. Dynamic and Stochastic BQCAP
• Assumptions
– Vessels arriving, and arrival times, during a shift known
– Berth position fixed one shift in advance
3
4
1
5
6 10
7
2
9
Time
Berth space
k-1 k k+1 k+2 k+3
8
11
12
13
A: berthed but unfinished at shift k
B: not berthed at shift k, but will arrive during k
C: uncertain arrival during k+1, k+2, …
num cranes
berth time*, num cranes**
berth pos, time**, cranes**
Decisions

17. Modeling approach
• Multi-stage expected cost minimization
(dynamic program)
• Approximate solution approach
– Rolling horizon (roll-out)
– Two-stage sample-average stochastic program
• In each scenario, assume that all arrival
times are known at shift k+1
• Note that some actually some will indeed only become
known later: approximation!
C Type Vessel

18. cA
Two-stage approximation scheme
k k+1 k+m+1
AB ,bB
Parameter
AA, bA, sA
C
A
1st stage 2nd stage
uB, sB0, cB0
bC
B1 B1
,s c 
C C
,s c 
Certain info Uncertain scenarios
Fixed
decisions
Recourse decisions under
each scenario
executed adjustable by scenario
Fixed cost expected recourse cost
decisions

19. 3-Layer Nested Tabu Search
TS1 L
cA
uB
,sB0
,cB0
TS2
TS3
Decode
Re-decode
Pass 1
Pass 2
Supplement
Re-supplement
L
cA
,cB0
bC
cA
uB
,sB0
,cB0
Pass 1
Pass 2 bC
L Pass 2
Pass 2
AA
, AB
AA
, AB
B1 B1
C C
,
,
s c
s c
 
 
B1 B1
C C
,
,
s c
s c
 
 
2 ( )L 
CB1
,A A
B1 C
,A A
CB1
,A A
B1 C
,A A
0
k
0
1k 
k
k
k
1k 
Scenario decision
Executed decision
Best Fit
Sampled
scenarios
Certain
info &
expected
uncertain
info

20. 3-Layer Nested Tabu Search
• Ideas
– Greedy decisions (most cranes, earliest possible
berthing time) are used when decoding
– Neighborhoods all variations of swap
• Swap positions of two vessels in priority lists
• Swap one or more cranes between two vessels of the
same category that are simultaneously berthed

21. Dynamic BQCAP Results
• “Easier” instances than Ak (2008)
– Meisel & Bierwirth (2009)
– 1000m quay, 10m sections, 1 hr time buckets, 10
quay cranes
– 20, 30, 40 vessels per week
– Two weeks
– Shift length 24 hrs; horizon length 72 hrs
– Minor arrival time uncertainty: +/- 3 hrs

22. Dynamic BQCAP Results
• CPLEX solvable a posteriori
0
50
100
150
200
250
300
350
20 30 40
totaldwelltime(hr)
calling vessel # per week
posterior cplex
rolling cplex
one-sample dynamic

23. Dynamic BQCAP Results
• CPLEX solvable a posteriori
2883
10854
14671
107 94 2931 4 22
0
2000
4000
6000
8000
10000
12000
14000
16000
20 30 40
cpuruntime(s)
calling vessel # per week
posterior cplex
rolling cplex
one-sample dynamic

24. Dynamic BQCAP Results
• CPLEX solvable a posteriori
2883
10854
14671
107 94 2931 4 22
0
2000
4000
6000
8000
10000
12000
14000
16000
20 30 40
cpuruntime(s)
calling vessel # per week
posterior cplex
rolling cplex
one-sample dynamic

25. Two-stage outperforms one-stage
• CPLEX solvable a posteriori
1.49%
2.07%
5.31%
2.22%
5.07%
5.67%
0.67%
0.84%
1.70%
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
20 30 40
optimalitygapoftotaldwelltime
calling vessel # per week
rolling cplex-expect
rolling cplex-estimate
30-sample dynamic

26. Dynamic BQCAP Results
• CPLEX cannot solve a posteriori in 16 hours
– measure gap to best upper bound
3.31%
9.09%
1.90%
0.37%
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
10.00%
0 20000 40000 60000 80000
gapoftotaldwelltime
cpu run time (s)
rolling cplex-expect
rolling cplex-estimate
30-sample dynamic
one-sample dynamic
rolling cplex

27. What to remember
1. Berth and quay crane allocation problems are
dynamic and stochastic
2. List-based multi-layer tabu search heuristics
fast and effective for dynamic problems
3. Multistage problems with stochastic arrival
times can be solved effectively with two-
stage rolling horizon approximation models
that minimize sample average cost