METAHEURISTIC ANALYSIS APPLIED TO THE BERTH ALLOCATION PROBLEM: CASE STUDY IN A PORT CONTAINER TERMINAL

Pereira, Elizangela Dias; Coelho, Antonio Sérgio; Longaray, André Andrade; Machado, Catia Maria dos Santos; Munhoz, Paulo Roberto

doi:10.1590/0101-7438.2018.038.02.0247

person Elizangela Dias Pereira

schoolUniversidade Federal de Santa Catarina - UFSC. Programa de Pós-Graduação em Engenharia de Produção - PPGEP, Santa Catarina, Brasil. E-mails: eliz math@yahoo.com.br; a.s.coelho@ufsc.br

person Antonio Sérgio Coelho

schoolUniversidade Federal de Santa Catarina - UFSC. Programa de Pós-Graduação em Engenharia de Produção - PPGEP, Santa Catarina, Brasil. E-mails: eliz math@yahoo.com.br; a.s.coelho@ufsc.br

person André Andrade Longaray *

schoolUniversidade Federal do Rio Grande - FURG. Programa de Pós-Graduação em Modelagem Computacional - PPGMC, Brasil. E-mails: andrelongaray@gmail.com; catiamachado.furg@gmail.com

person Catia Maria dos Santos Machado

schoolUniversidade Federal do Rio Grande - FURG. Programa de Pós-Graduação em Modelagem Computacional - PPGMC, Brasil. E-mails: andrelongaray@gmail.com; catiamachado.furg@gmail.com

person Paulo Roberto Munhoz

schoolUniversidade Federal do Rio Grande - FURG. Estação de Apoio Antártico - ESANTAR, Brasil. E-mail: paulorsmunhoz@gmail.com

person*Corresponding author.

Authors and title	Studied Method(s)	Summarized Conclusions
Imai et al. (2001) and ^{Nishimura et al. (2001})	Lagrangian Relaxation (LR) and Genetic Algorithm (GA) applied to DBAP.	The calculations obtained with GA indicated the efficient use of 5 berths instead of 7 berths used in a Japanese public port. The tests (fictitious data) demonstrated that the quality of the solution obtained by the (GA) was the same/similar to the one obtained by the (LR).
Minimize the total waiting and handling times for every vessel.
^{Kim & Moon (2003})	Simulated Annealing applied to CBAP.	The unfeasibility of the mixed-integer program (MIP) technique with 7 vessels was demonstrated, and the Simulated Annealing algorithm was suggested.
Minimize the penalty cost resulting from delay and the additional handling costs resulting from non-optimal locations.	Simulated Annealing applied to CBAP.
^{Cordeau et al. (2005})	The Tabu search algorithm applied to DBAP.	It was demonstrated that the Tabu search algorithm outperformed both the first-come first-served rule and CPLEX.
Minimize the sum for each vessel of the service time.	The Tabu search algorithm applied to DBAP.
Imai et al. (2008)	Genetic Algorithm applied to DBAP.	The number of rejected vessels owing to lack of service within the maximum time limit was minimized. The study is useful for the efficient management of extremely busy container terminals in underdeveloped countries.
Minimize the total service time.	Genetic Algorithm applied to DBAP.
^{Theofanis et al. (2007})	Optimization based GA heuristic (OBGA) and GA for DBAP.	Sensitivity analysis for the parameters of the OBGA and GA heuristics was performed. Both heuristics proved to be efficient, with small changes in the value of the objective function.
Minimize the total weighted service time of all the vessels.	Optimization based GA heuristic (OBGA) and GA for DBAP.
^{Mauri et al. (2008b})	Minimization of the number of rejected vessels due to the lack of service within the maximum time limit established.	Population Training Algorithm (PAT) in combination with Linear Programming (LP) for Column Generation for DBAP.
Minimization of the number of rejected vessels due to the lack of service within the maximum time limit established.		The results demonstrated the potential of the proposed approach, where high quality solutions were obtained for relatively large problems, in significantly short processing time.
^{Buhrkal et al. (2011})	Generalized set-partitioning (GSPP) applied to DBAP.	The performance of five formulations of DBAPs was compared. It was claimed that the model proposed by ^{Cordeau et al. (2005}) is computationally advantageous. The results obtained with GSPP were superior to those obtained by (Cordeau et al. 2005).
Minimize the total waiting and handling times of every ship.	Generalized set-partitioning (GSPP) applied to DBAP.
^{Hu (2015})	The multi-objective GA (moGA) for the bi-objective DBAP.	The importance of the preference for daytime operations was demonstrated. Owing to the lack of research on the optimization of daytime shift allocation, the optimal performance of the algorithm should be studied in future work.
Minimize the work of cargo operated at nighttime.	The multi-objective GA (moGA) for the bi-objective DBAP.
^{Lalla-Ruiz et al. (2014})	Biased Random Key Genetic Algorithm (BRKGA) for DBAP.	The required computational time showed that, even in small-scale instances, the problem is difficult to solve by CPLEX. The use of BRKGA was justified for the solution of the tactical berth Allocation problem in realworld contexts; particularly, in those integrated designs where this problem frequently appears as a sub-problem. Therefore, the use of efficient procedures that provide nearoptimal solutions within short computational time is preferable.
Maximize the sum of the values of the chosen quay crane profiles and minimize the yard-related housekeeping cost.	Biased Random Key Genetic Algorithm (BRKGA) for DBAP.
^{Lalla-Ruiz et al. (2016b})	Simulated Annealing for the Waterway Ship Scheduling Problem (WSSP) for (BAP).	Computational experiments showed that the problem is difficult to solve, even in smallscale instances, using a general-purpose solver. Nevertheless, the proposed S approaches may provide high-quality solutions in short computational time. Moreover, the tests showed that the SA approachesare suitable for contemporary problems.
Minimize the total time required for the vessels to pass through the waterways.
^{Hsu (2016})	Hybrid Particle Swarm Optimization (HPSO) for BAP.	The proposed HPSO was compared with the two GA-based approaches. Among the three approaches, HPSO required the longest, albeit reasonable, computational time. Thus, it remains acceptable.
Minimize the total cost consisting of the sub-costs of waiting, delay and operation.	Hybrid Particle Swarm Optimization (HPSO) for BAP.
^{Mauri et al. (2016})	Adaptive Large Neighborhood Search (ALNS) DBAP and CBAP.	The computational tests indicated the relative superiority of the ALNS heuristic for solving DBAP and CBAP. It determined all known optimal solutions for DBAP within shorter computational time. Better solutions were obtained for CBAP.
Minimize the sum of the service times while the vessels stay into the port.	Adaptive Large Neighborhood Search (ALNS) DBAP and CBAP.
^{Venturini (2017})	CPLEX	A novel formulation was presented by integrating the BAP with vessel speed optimization for multiple ports under environmental considerations, in particular ship air emissions.
Minimize the cost of idleness, delay, handling and the fuel consumption.	CPLEX

1.	GIVEN (α, S Amax, T0eTc) DO
2.	GENERATE (a solution S using the distribution heuristic);
3.	EVALUATE (solution S using the programming heuristic);
4.	S← S*; {Optimal solution obtained so far}
5.	IterT ← 0; {Number of iterations at temperature T}
6.	T ← T0; {Current temperature}
7.	WHILE (T > Tc) DO
8.	WHILE (Iter T < S Amax) DO
9.	IterT ← IterT +1;
10.	GENERATE (a random neighbor S’ through one of the swap mov);
11.	EVALUATE (solution S’ using the programming heuristic);
12.	Δ ← f(S’) − f(S);
13.	IF (< 0)S ← S’;
14.	IF (f(S’) < f(S))S← S’; END IF
15.	ELSE
16.	TAKE (x ∈ [0, 1]);
17.	IF (x < e−Δ/T)S ← S’; END IF
18.	END IF
19.	END WHILE
20.	T ← α*T; IterT ← 0;
21.	END WHILE
22.	S ← S*;
23.	RETURN (S).

Genetic Algorithm			Simulated Annealing		Simulated Annealing + Reheating Technique
Crossover operation	O.F.	Time	O.F.	55,842.80	O.F.	55,212.83
OX	55,212.83	4’	Time	11’25”	Time	43’09”
CX	55,212.83	3’58”
PMX	55,212.83	3’15”
HX	55,212.83	9’12”

Genetic Algorithm			Simulated Annealing		Simulated Annealing + Reheating Technique
Crossover operation	O.F.	Time	O.F.	55,292.80	O.F.	54,672.83
OX	54,672.83	4’08”	Time	8’27”	Time	22’40”
CX	54,672.83	3’24”
PMX	54,672.83	3’22”
HX	54,672.83	7’06”

Genetic Algorithm			Simulated Annealing
Crossover operation	O.F.	Time	O.F.	134,334.70
OX	129,489.00	6’30”	Time	20’35”
CX	129,489.00	6’26”
PMX	129,489.00	6’30”
HX	129,489.00	16’27”

Genetic Algorithm			Simulated Annealing
Crossover operation	O.F.	Time	O.F.	127,149.80
OX	128,137.83	6’44”	Time	14’25”
CX	127,689.81	6’50”
PMX	126,534.83	5’43”
HX	126,534.83	11’45”

Genetic Algorithm			Simulated Annealing
Crossover operation	O.F.	Time	O.F.	149,324.40
OX	147,539.67	6’40”	Time	22’52”
CX	148,759.67	6’25”
PMX	147,539.67	6’07”
HX	147,539.67	15’23”

Genetic Algorithm			Simulated Annealing
Crossover operation	O.F.	Time	O.F.	147,160.50
OX	144,585.50	8’25”	Time	18’32”
CX	144,585.50	7’54”
PMX	143,885.83	8’31”
HX	143,195.53	16’30”

Estimations	Metaheuristics
Estimations	Simulated Annealing	Genetic Algorithm
Average Time ${\bar{x}}_{t}$	29’04”	20’50”
Final OF Average ${\bar{x}}_{f}$	194,073.97	185,951.42
Standard Deviation s	3,267.26	2,2023.42
Coefficient of Variation C _v	1.68351	1.08814

Date	Berth 1	Berth 2	Berth 3	O.F./day
12/01/2012
SA	XIN YAN TIAN	AL MARACANA	ER BERLIN	24.289,50
AG	ER BERLIN	XIN YAN TIAN	CAP MELVILLE	22.179,50
			AL MARACANA
13/01/2012
SA	CAP MELVILLE	LEXA MAERSK	SANTA CLARA	1.335,00
	MSC GEMMA
AG	MSC GEMMA	LEXA MAERSK	SANTA CLARA	0,00
16/01/2012
SA	LOG IN PANT		MSC ADRIATIC	14.030,33
			MESSOLOGI
AG	MESSOLOGI	LOG IN PANT		12.695,50
	MSC ADRIATIC

Minimize \begin{matrix}  \end{matrix} Z^{*} = \sum_{i \in N} \sum_{k \in M} v_{i} (T_{i}^{k} - a_{i} + t_{i}^{k} \sum_{j \in N \cup {d (k)}} x_{i j}^{k})

\begin{matrix}  \end{matrix} \sum_{j \in N \cup {d (k)}} x_{i j}^{k} - \sum_{j \in N \cup {o (k)}} x_{j i}^{k} = 0, \begin{matrix}  \end{matrix} \forall k \in M, \forall i \in N

{Minimize Z}^{*} = w_{0} \sum_{i \in N} \sum_{k \in M} v_{i} (T_{i}^{k} - a_{i} + t_{i}^{k} \sum_{j \in N \cup {d (k)}} x_{i j}^{k})

\begin{matrix}  \end{matrix} + w_{1} \sum_{i \in N} \sum_{k \in M} \sum_{j \in N \cup {d (k)}} x_{i j}^{k} (\max (0, a_{i} - T_{i}^{k}) + \max (0, T_{i}^{k} + t_{i}^{k} - b_{i}))