Original Article
The berth allocation problem: Optimizing vessel arrival time M i h a l i s M . G o l i a s a , *, G e o r g i o s K . S a h a r i d i s b , M a r i a B o i l e b , Sotirios Theofanisb and Marianthi G. Ierapetritoub a
PY
University of Memphis, 3815 Central Avenue, Memphis, Tennessee 38152, USA. E-mail:
[email protected] b Rutgers, The State University of New Jersey, 623 Bowser Road, Piscataway, New Jersey 08854, USA.
C O
*Corresponding author.
A
U
TH
O
R
A b s t r a c t The berth scheduling problem deals with the assignment of vessels to berths in a marine terminal, with the objective to maximize the ocean carriers’ satisfaction (minimize delays) and/or minimize the terminal operator’s costs. In the existing literature, two main assumptions are made regarding the status of a vessel: (a) either all vessels to be served are already in the port before the planning period starts, or (b) they are scheduled to arrive after the planning period starts. The latter case assumes an expected time of arrival for each vessel, which is a function of the departure time of the vessel from the previous port, the average operating speed and the distance between the two ports. Recent increases in fuel prices have forced ocean carriers to reduce current operating speeds, while stressing to terminal operators the need to maintain the integrity of their schedule. In addition, several collaborative efforts between industry and government agencies have been proposed, aiming to reduce emissions from marine vessels and port operations. In light of these issues, this article presents a berth-scheduling policy to minimize vessel delayed departures and indirectly reduce fuel consumption and emissions produced by the vessels while in idle mode. Vessel arrival times are considered as a variable and are optimized to accommodate the objectives of the proposed policy while providing ocean carriers with an optimized vessel speed. Example problems using real data show that the proposed policy reduces the amount of emissions produced by vessels at the port in idle mode, optimizes fuel consumption and waiting time at the port by reducing vessel operating speeds to optimal levels and minimizes the effects of late arrivals to the ocean carriers’ schedule. Maritime Economics & Logistics (2009) 11, 358–377. doi:10.1057/mel.2009.12
Keywords: planning and scheduling; resource allocation; container terminals; berth scheduling; emissions; optimization r 2009 Palgrave Macmillan 1479-2931 Maritime Economics & Logistics www.palgrave-journals.com/mel/
Vol. 11, 4, 358–377
Berth allocation problem
Introduction
A
U
TH
O
R
C O
PY
Congestion in container terminals, fierce terminal competition, the ever increasing role of the time factor in liner shipping (Notteboom, 2006) and the pressure by liner shipping operators for increased effectiveness and punctuality of service (berthing and vessel loading/unloading operations) exacerbate the need for improved container terminal seaside operations. The objective of terminal operators is to find vessel to berth assignments (also known as the berth scheduling problem (BSP)) to reduce vessel turnaround time, increase port throughput, and keep customer satisfaction at a desired level. The latter typically depends on contractual agreements through which terminal operators may provide differentiated services to customers with high container volumes or large number of vessels calling at the port. Cost is the most important parameter for the ocean carriers and is a key parameter when berth schedules are defined. Surprisingly, ocean carriers have calculated that fuel now accounts for more than 60 per cent of operating costs and have taken almost unprecedented immediate measures to slow ships to economic speeds of 20 knots from 25 knots. Moreover, ocean carriers are stressing the importance of schedule integrity and the ensuing benefits to the environment (Wacket, 2007). It is of little surprise, therefore, that CEOs and senior executives, such as Adolf Adrion (Hapag-Lloyd), Koji Miyahara (NYK Line), Eivind Kolding (Maersk Line) and Akimitsu Ashida (Mitsui OSK Lines), have slowed down their ships on several routes, sought for shorter waiting and handling times at terminals, setting fuel as one of their single biggest challenges in 2008 (Fossey, 2008). In addition to carriers’ profit concerns, rapid growth in container freight volumes increases the need to work on emission reduction and mitigation strategies.1 In certain regions, large container terminals have been singled out as the largest source of air pollution (for example, Port of Los Angeles and Long Beach2). The diesel engines that power almost all port activities, including container vessels, are concentrated sources of emissions, and they are often located near large urban centers affected by pollution from other diesel-powered vehicles. Although trucks have been identified as the major source of emissions at container terminals, container vessels have their own contribution (Lazic, 2004) and it has been estimated that vessel hotelling3 emissions can make up a major portion of total port emissions. Several collaborative efforts between industry and government agencies demonstrating emission reduction options for marine vessels and port operations have been proposed that mainly focus on the use of alternative fueling sources for vessels in port. The interested reader is referred to the website of the Clean Ports USA for a number of relevant publications.4 r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
359
Golias et al
A
U
TH
O
R
C O
PY
In the berth scheduling problem, vessels arrive over time at a port and the terminal operator assigns them to berths for unloading and loading of containers based on several factors and considerations (Theofanis et al, 2009). Three broad classification schemes of the BSP may be specified: (a) the discrete versus continuous berthing space, (b) the static versus dynamic vessel arrivals, and (c) the static versus dynamic vessel handling time. The discrete problem (Imai et al, 1997, 2001, 2003, 2007; Hansen et al, 2008) considers the quay as a finite set of berths. In the continuous problem (Kim and Moon, 2003; Park and Kim, 2003; Guan and Cheung, 2004; Imai et al, 2005; Moorthy and Teo, 2006) vessels can berth anywhere along the quay. In the static arrival problem all the vessels to be served are already in the port, while in the dynamic arrival problem, adopted in the majority of the literature, not all vessels to be scheduled for berthing have arrived at the time scheduling begins, although estimated arrival times are known in advance and are used in the models. Finally, in the static handling time problem (Imai et al, 1997; Hansen et al, 2008) the vessel handling time is considered to be known, whereas in the dynamic (Park and Kim, 2003; Imai et al, 2008) it is a variable. For a detailed review and critical analysis of the current literature on berth scheduling we refer to Theofanis et al (2009). Defining the best berth scheduling policy for each port operator depends on several factors, including the type and function of the port (common user or dedicated facility, transshipment hub and so on), size, location, nearby competition, types of contractual agreements with the ocean carriers. All these factors have different effects on the desired objective, which could be the customer level of service (for example delayed departure) and/or the environmental effects of the vessel while at the port (for example emissions). Several berth scheduling policies and related factors have been explored in academic research but, to date, environmental considerations have not been included in BSP formulations. In light of the above, this article presents a berth scheduling policy, a model, and a resolution algorithm for the discrete berthing space and dynamic vessel arrival, where vessel arrival times are optimized. This is achieved by assuming that vessel arrival times vary within upper and lower bounds. The objective of the proposed policy is to reduce vessel fuel consumption and the amount of emissions produced by the vessels while at port, by minimizing vessel waiting time (an emissions mitigation operational strategy proposed by the Office of Natural and Human Environment, US Federal Highway Administration, in 2005). Reducing total port waiting time and allowing arrival times to vary could lead to very low levels of service (that is, jeopardize the integrity of the ocean carriers’ schedule). To account for this, the policy’s objective also includes minimization of vessels’ delayed departures, estimated from requested departure times as set by the carriers. The proposed berth 360
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
Berth allocation problem
PY
scheduling strategy has a positive effect not only on environmental aspects but is also beneficial to the ocean carriers, as described in more detail in the next section. To our knowledge this is the first berth scheduling study to appear in the literature that optimizes vessel arrival times while at the same time considering the environmental impacts of seaside operations at container terminals. Example problems using real data are presented to critically discuss the berth scheduling policy benefits against the constant vessel arrival time formulation. The rest of this article is structured as follows. The next section presents the model formulation while the subsequent section presents the solution approach. A number of numerical results are presented in the penultimate section, and the last section concludes the article.
C O
Problem Description
A
U
TH
O
R
Container vessels at marine terminals arrive over time, and terminal operators seek to assign them to berths to be served and depart as soon as possible. As discussed in the introductory section, carriers have two important objectives: (a) minimize fuel consumption, and (b) keep the integrity of their schedule (for example, minimize delayed departures). Until today, berth scheduling studies assumed the arrival time of a vessel to have a fixed and known value. This value was estimated from the vessel’s departure time from the previous port, its average operating speed and the distance between the two ports. As previously discussed, ocean carriers have been adjusting their operating speed to more fuel efficient levels. To capture their ability to do so, we introduce here vessel arrival time as a variable and assume that the carrier will provide the terminal operator with a range of vessel arrival times. The terminal operator will schedule all vessels using these ranges, optimizing for vessels’ arrival time. In this article we assume that the upper bound of the arrival time is equal to the estimated arrival time with an operating speed of 25 knots, which has been the practice so far, and the lower bound is equal to the estimated arrival time with an operating speed of 15 knots. Vessel arrival times are optimized to achieve two objectives: (a) minimize total waiting time of all vessels, thus reducing vessel fuel consumption and vessel emissions while in idle mode, and (b) minimize delayed departures of vessels. To accommodate an environmentally friendly berth scheduling policy, vessel waiting times are weighted using a coefficient that corresponds to the emissions produced hourly by the vessel in idle mode. The scheduling strategy described above is a win-win situation that can decrease pollution produced by vessels and at the same time provide fuel r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
361
Golias et al
PY
efficiency and schedule integrity to ocean carriers. The approach guarantees that the final solution will at worst be equal to or better than the solution obtained using constant arrival times (CATs). As the difference between the upper and lower bound of the variable arrival time (VAT) increases, the possibility of obtaining a better solution increases as well. We should point out that existing berth scheduling models cannot be modified to take variable times into account without affecting the structure of the problem. The introduction of the arrival time as a variable would lead to a nonlinear space in terms of the objective function and/or the constraints, thus complicating the model formulation and resolution approach. Formulation of the problem
R
C O
In the model presented here, we assume that the quay is divided into a number of berths and each berth can service one vessel at a time regardless of vessel size. We also assume that vessel handling time is proportional to vessel capacity, depending also on the assigned berth. Once a vessel is berthed, it will remain in its location until all cargo-handling operations are completed. The following notation is used in problem formulation:
O
Sets
Time when berth becomes idle for the first time in the planning horizon Handling time of vessel j at berth i Earliest arrival time of vessel j (Arrival time lower bound) Latest arrival time of vessel j (Arrival time upper bound) Departure time request of vessel j Amount of emissions produced hourly by vessel j in idle mode Large positive number
A
Si HTij Alj Auj DTRj wj M
U
Parameters
TH
Set of berths, i ¼ (1, y, I) Set of vessels, j ¼ (1, y, J) Set of service orders, k ¼ (1 ,y, JI þ 1)
iAB jAV kAK
Decision variables yijk xijk WTijk 362
Idle time of berth i between start of service of vessel j as the kth vessel, and the departure of its immediate predecessor 1 if vessel j is served at berth i as the kth vessel, and 0 otherwise Waiting time of vessel j served at berth i as the kth vessel
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
Berth allocation problem
Aj DDijk
Arrival time of vessel j parture delay of vessel j served at berth i as the kth vessel
The problem can then be formulated as follows: 0 1 X X X wj WTijk @ þ DDijk A min max ðw Þ j i2B j2V k2K
ð1Þ
j
Subject to:
XX
xijk ¼ 1
8j
ð2Þ
X
xijk p1
PY
i2B k2K
8i; k
j2V
X
X
ðHTim ximh þ yimh Þ
C O
yijk XðAj Si Þ
ð3Þ
m6¼j2V hok2K
Mð1 xijk Þ
8i; j; k
X
X
ð5Þ
ðHTim ximh þ yimh Þ þ Si Aj
O
WTijk X
8i; j; k
R
DDijk XWTijk þ HTij þ Aj DTRj Mð1 xijk Þ
ð4Þ
m6¼j2V hok2K
TH
Mð1 xijk Þ
8i; j; ðk41Þ
A
U
WTijk XðSi Aj Þ Mð1 xijk Þ
ð6Þ
8i; j; ðk ¼ 1Þ
ð7Þ
Auj XAj
8j
ð8Þ
Aj XAlj
8j
ð9Þ
xijk ; 2 f0; 1g
8i; j; k
ð10Þ
8i; j; k
ð11Þ
8j
ð12Þ
WTijk X0
8i; j; k
ð13Þ
DDijk X0
8i; j; k
ð14Þ
yijk X0 Aj X0
The objective function (1) minimizes the total waiting and delayed departure time for all vessels. Waiting time is weighted to account for the emissions produced by each vessel while in idle mode at the port, waiting for service. r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
363
Golias et al
Constraint set (2) ensures that vessels must be served once, while constraint set (3) ensures that each berth services one vessel at a time. Constraint set (4) ensures that each vessel is served after its arrival, while constraint set (5) estimate the delay of the departure of each vessel. Constraint sets (6) and (7) estimate the waiting time of each vessel. Constraints (8) through (14) define the decision (xijk, Aj) and auxiliary variables (WTijk, DDijk, yijk) and their range.
Resolution Approach
TH
O
R
C O
PY
Due to the complexity of the formulation, it is unlikely that the problem can be solved by commercial mathematical programming solvers (that is CPLEX, LINDO). Results have shown that the use of exact optimization tools is computationally time-consuming and requires several hours to define optimal berth schedules, even in simple berth scheduling policies (that is minimization of total service time), because of the large-scale model that has to be built to describe a real case example. In order to reduce the CPU resolution time, a Genetic Algorithms (GA)-based heuristic is developed based on a previously presented GA heuristic (Golias, 2007). Furthermore, in order to further improve on the efficiency of the resolution approach a series of valid inequalities are introduced to take advantage of the problem’s physical properties. They are presented in the following subsection. GA-based heuristic: Representation, genetic operations and selection
A
U
A two-layer integer chromosomal representation is used. Figure 1 shows a small example of a chromosome for a problem with two berths and five vessels. The first dimension of the chromosome determines vessel arrival time while the second determines the berth assignment and service order for each vessel. Initial values for the chromosomes were obtained using the mean value of the upper and lower bound of vessel arrival time, and a first come first served policy. We experiment with four different types of mutation: insert, swap, invert and scramble mutations (Figure 2) that are applied to all the chromosomes at each generation. Computational experiments show that when all four mutations are applied, the GA heuristic converges at a faster rate and there is significant
Figure 1: Chromosome representation. 364
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
C O
PY
Berth allocation problem
Figure 2: Schematic representation of different mutation types.
A
U
TH
O
R
improvement in the value of the objective function. Thus in our algorithm, we employ all four mutation types, but as the GA progresses, the weight shifts from the invert and scramble mutation to the insert and swap one. This way, in the beginning of the search, the heuristic performs large jumps and as the objective function improves the heuristic searches in an increasingly smaller region. Mutation is applied using a two-stage procedure. At the first stage the arrival of each vessel is determined by sampling from a uniform distribution with upper and lower bounds equal to the earliest and latest arrival (provided by the minimum and maximum allowable speed of each vessel and the estimated departure from the port of origin). At the second stage, the service order and berth assignment are determined, using the four mutation types previously discussed, and two mutation rules described in the following subsection. The problem formulation presented in the previous subsection belongs to the category of the unrelated machine scheduling problem, known to be NP-hard or NP-complete (Steenken et al, 2004; Pinedo, 2008), and cannot be solved to optimality with analytical methods, even for small instances. By exploiting the special characteristics of the physical problem, we introduce a class of valid inequalities to the proposed formulation to construct improved approximations of the feasible space and reduce the size of the search tree (Hooker, 1998). The valid inequalities presented here can also be used in other berth scheduling policies as they are not based on the particularities of the objective function but rather on the structure of the constraint sets whose r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
365
Golias et al
general formulation was presented in the previous section. In this article we introduce two sets of valid inequalities based on the problems’ structure, which are described below. Priority mutation rule Assume that vessel aAV arrives at time Aa and vessel bAV arrives at time Ab. Let ka and kb be the order of service of each vessel at its assigned berth. It is then easy to prove (Golias, 2007) that irrespective of the objective function, Cia þ Aa oCib þ Ab , ka okb
8i 2 B; a 6¼ b 2 B
ð15Þ
C O
PY
In other words, if both vessels are served at the same berth and the left side of equation 15 is valid then vessel b will always be served after vessel a and vice versa. This rule is included into the insert and swap mutation operations.
TH
O
R
Tabu mutation rule In real-world conditions, it is unlikely that a vessel arriving at the end of the planning horizon will be served before a vessel arriving at the beginning of the planning horizon. To capture this property of the problem, we apply a tabu mutation rule by restricting certain mutation operations. This approach reduces the feasible search space of the GA heuristic improving on the speed of the resolution. This rule is presented in equation (16). Aa oAb þ t , ka okb
8i 2 B; a 6¼ b 2 B
ð16Þ
A
U
where t is a predefined time period limit. An example of the tabu mutation is shown in Figure 3, for the insert and inverse mutation. Assume that vessel 8 arrives at time A8 while another vessel (vessel 5) at time A5. The tabu rule will forbid a mutation that will allow vessel 5 to be served before vessel 8 during the insert mutation or invert the vessels after vessel 8 and before vessel 5 during the invert mutation, if A5 exceeds a certain limit, and allow it if it does not exceed that time limit. The same rule is applied when selecting the cells for the invert and scramble mutation, that is, all the vessels within the selected cells should have arrived within the same period not exceeding t. In practice, vessels will not wait for longer than 2 days for service and thus in this article the time limit of 48 hours was adopted. The problem is a minimization problem; thus the smaller the value of the objective function, the higher the fitness value will be. In order to avoid trapping the algorithm at local optimal locations of the solution space, a number of medium and low fitness solutions are selected probabilistically at each generation, using the route wheel selection algorithm (Goldberg, 1989). 366
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
C O
PY
Berth allocation problem
O
R
Figure 3: Tabu mutation rule example (insert and invert).
TH
Computational Experiments
A
U
Problems used in the experiments were generated randomly but systematically. When creating the experimental data the focus was on obtaining computationally challenging data sets that portray real-life conditions. The random generation process was based on data from two existing container terminals with similar terminal operating systems (one in Europe and one in the United States). The range of variables and parameters considered were chosen according to the data obtained from these two container terminals. We developed 30 problem instances where vessels were served with various handling volumes at a multi-user container terminal with five berths. The first 10 problem instances were generated assuming an exponential distribution with average vessel inter-arrival time of 3 hours, the next 10 with an average arrival of 4 hours, and the last 10 with an average arrival of 5 hours. In the data sets used in the experiments, vessel handling volumes (loading and unloading) and capacities (total TEUs a vessel can carry) ranged from 250 to 4000 (TEU/vessel), based on a uniform distribution pattern. The handling time of a vessel was assumed to depend on the berth assigned, and the number of cranes that may be assigned based on the size and handling volume of the vessel. We considered that one to r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
367
Golias et al
TH
O
R
C O
PY
three cranes operate on small-sized vessels (o2000 TEU of handling volume), two to four cranes on medium-sized vessels (o3000 TEU of handling volume), and three to six on large mother vessels (o4000 TEU of handling volume). The average crane productivity was assumed to be 25 TEU/hour. Based on these considerations the number of quay cranes assigned to each vessel was calculated using a uniform probability. The minimum handling time of a vessel was calculated by dividing the handling volume by the average productivity of a crane multiplied by the number of cranes operating on the vessel. The handling time of vessels at the other berths was generated in relation to the berth with the minimum handling time. Availability of berths was calculated using a uniform probability with a minimum of 0 and a maximum of 10 hours. Upper and lower bounds for the vessel arrival time were estimated based on the expected vessel departure from the port of origin and the stated maximum and minimum speed of each vessel while in transit. Weights for the vessels were generated based on the total capacity of each vessel, assuming that the larger the capacity of the vessel the larger the amount of emissions, while idle in the port. Using these data we perform two different types of analysis. The first one focuses on the performance of the developed algorithm for the resolution of the problem and the second on the comparison of results obtained from the proposed formulation to results obtained from assuming a constant vessel arrival time. The constant vessel arrival time was considered to be the mean of the upper and lower bound of the vessels’ arrival time in the original problem. Genetic algorithms performance
A
U
In this section we present a series of numerical results where we investigate the performance of the GA-based heuristic presented in the previous section, and compare it to solutions obtained from CPLEX, for VAT. Figure 4 shows the convergence of the objective function for different (randomly selected) instances used in the computational experiments. Similar graphs for the rest of the data sets were obtained, but are omitted here for space considerations. The y-axis represents the objective function value (in hours) while the x-axis the iterations. Figure 4 also shows the convergence pattern of the objective function as obtained from CPLEX, for the same problem instances. Both resolution algorithms were terminated after 2 hours although the GA-based heuristic converged on average after the first 5 min. We should mention that the optimality gaps as reported from CPLEX at the point of termination exceed 90 per cent. Results of the actual objective function values obtained from the GA-based heuristic and CPLEX are omitted, as the differences exceeded 100 per cent. Based on the results obtained by the 30 examples, we observe 368
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
O
R
C O
PY
Berth allocation problem
TH
Figure 4: Genetic Algorithm versus CPLEX objective function convergence patterns.
A
U
that the classical exact resolution approach provided by CPLEX fails to converge to a good solution because of the large scale of the problem. As the research space is large, complex and poorly understood, a heuristic approach as the GA presented in the previous section seems to be a good alternative. In any evolutionary-based heuristic, a metric is needed to evaluate the heuristic in terms of its consistency in obtaining similar solutions when different starting points are used. For the purpose of this article the following commonly used metric was applied (Taboada, 2007). For each of the problems tested, the ratio of the range of the objective function values for five trials (different starting populations, different number of generations and population) to the lowest objective value, which can be expressed by the highest objective function value during the five trials divided by the lowest objective function value during the five trials, was calculated for each objective function. The average ratio varied from 2 per cent for the small instances to 5 per cent for the large instances, and thus we can conclude that results obtained from the GA heuristic are consistent for different trials. r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
369
Golias et al
Optimized versus constant vessel arrival time
C O
PY
The problem presented in the second section was reformulated (see Appendix A) using a constant vessel arrival time, set equal to the mean of the upper and lower bound of VAT. The GA-based heuristic was also applied as the resolution algorithm of the problem with CATs. Tables 1–3 show the differences for the variable and CAT instances between the values of the different objective function components. The first table shows the differences in the values of the objective function, the second of the waiting time (and thus the weighted emissions differences), and the third shows differences in the delay time. In each table the first column contains the example number while the second, third and fourth column show the value of the difference (in percentages) for the 3, 4, and 5 hour vessel inter-arrival time instances, respectively. Positive values mean
Table 1: Objective function difference – optimized versus constant arrival time Instance
Objective function
9 15 21 1 35 75 6 49 20 29
TH
O
1 2 3 4 5 6 7 8 9 10
4 hours (%)
5 hours (%)
41 41 0 0 37 0 28 47 0 80
60 0 80 0 32 0 0 15 65 33
A
U
Example Example Example Example Example Example Example Example Example Example
R
3 hours (%)
Table 2: Waiting time difference – optimized versus constant arrival time total Instance
Example Example Example Example Example Example Example Example Example Example 370
Waiting time reduction
1 2 3 4 5 6 7 8 9 10
3 hours (%)
4 hours (%)
5 hours (%)
10 1 19 12 28 10 14 2 11 49
35 14 0 0 43 0 14 56 0 61
63 0 14 0 33 0 0 9 32 60
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
Berth allocation problem
Table 3: Total delay time difference – optimized versus constant arrival time Instance
1 2 3 4 5 6 7 8 9 10
3 hours (%)
4 hours (%)
5 hours (%)
12 24 21 8 37 76 0 61 27 12
56 49 0 0 0 0 28 31 0 82
56 0 91 0 30 0 0 26 77 21
PY
Example Example Example Example Example Example Example Example Example Example
Delay time reduction
A
U
TH
O
R
C O
that the optimized arrival time solution outperforms CAT solution. As expected the proposed policy outperforms in almost every example CAT policy. VAT gives a better solution for the port (or the system under study) than CAT as shown in Table 1. In cases where the percentage is equal to zero the two approaches provide exactly the same solution. Attention should be paid to the fact that, in some cases, the differences are large (that is, example 10 with 4 hours of vessel inter-arrival time) showing that the later approach (CAT) could be much less beneficial for the port, the ocean carrier and the environment. Interesting results are also obtained with respect to the examination of the objective functions’ components. Tables 2 and 3 present the numerical results for each part of the objective function, that is, the weighted waiting time that also corresponds to the total emissions and the delay time. Table 2 shows the differences (in percentages) between the CAT and VAT values obtained for the weighted waiting time and Table 3 the same differences for the delay time. The first column in each table corresponds to the instance number while the second, third, and fourth columns to the three different inter-arrival time patterns. Again positive values mean that VAT outperforms CAT. Analyzing the results presented in Table 2 we observe that the waiting time in most cases has improved when the arrival time is optimized and in some cases the benefit is really high (for example, example 10 with 3 and 5 hours of vessel inter-arrival time). There are some examples where the use of CAT gives better solutions than the relaxation of it to a variable. Considering example 8 with 3 hours of vessel inter-arrival time, we observe a benefit of only 2 per cent using the CAT, but examining the equivalent result for the delay departure in Table 3 we find that the benefit is actually quite high as regards delay of vessels (76 per cent) and substantially high for the overall objective (49 per cent). This example shows exactly our interest to optimize the arrival time given the two objectives. Some r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
371
Golias et al
PY
of the vessels will decrease their operating speed and wait for a small additional time (as compared to the waiting time if the CAT schedule was adopted) but will be served to a more preferential berth and leave earlier than if they choose to come in at the scheduled time and then served at a berth that might delay them more. In general though we cannot exclude the event where a vessel will have to wait more and also depart with a larger delay in the VAT case than in the CAT case, but as the results show the overall benefits (total delay, total waiting time, total emissions) of the system are optimized. To evaluate the influence of the weights we increased the value of the waiting time coefficients by 50 per cent for all the instances, thus giving more influence to the reduction of emissions. The same results as in Tables 1–3 are shown in Tables 4–6. For the objective function, the same pattern as previously is observed. As expected, the increase in the waiting time coefficient forced the
C O
Table 4: Objective function difference – optimized versus constant arrival time (50 per cent increase of wait time weights) Instance
Objective function
55 49 54 69 67 68 53 81 55 68
TH
O
1 2 3 4 5 6 7 8 9 10
4 hours (%)
5 hours (%)
86 60 80 90 79 64 67 77 77 92
91 95 69 98 77 98 95 66 92 86
A
U
Example Example Example Example Example Example Example Example Example Example
R
3 hours (%)
Table 5: Total waiting time difference – optimized versus constant arrival time (50 per cent increase of wait time weights) Instance
Example Example Example Example Example Example Example Example Example Example 372
Waiting time reduction
1 2 3 4 5 6 7 8 9 10
3 hours (%)
4 hours (%)
5 hours (%)
21 15 30 38 23 27 25 47 30 46
83 12 10 0 53 32 38 48 53 0
48 100 49 0 30 50 0 27 86 61
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
Berth allocation problem
Table 6: Total delay time difference – optimized versus constant arrival time (50 per cent increase of wait time weights) Instance 3 hours (%)
4 hours (%)
5 hours (%)
76 75 70 83 85 83 70 89 71 76
86 79 94 95 86 80 79 85 85 97
100 95 78 98 90 99 100 84 93 91
1 2 3 4 5 6 7 8 9 10
PY
Example Example Example Example Example Example Example Example Example Example
Delay time reduction
R
C O
model to give better results than previously. The interesting result though is that the delay time reduction still remained in the positive range. Future research should focus in a more detailed sensitivity analysis of the weights or the application of an approach that would not require the use of weights. This issue is left for future research.
O
Conclusions
A
U
TH
In this article we studied the discrete space and dynamic BSP where vessel arrival time is optimized to account for the minimization of port-related emissions, waiting time of vessels and delayed departures. The problem was formulated as a mixed integer optimization problem, and a GA-based heuristic was used to solve the resulting problem. Computational results using real data showed that the proposed berth scheduling policy, where the terminal operator decides the vessel arrival time, can be more beneficial both for the carrier and the terminal operator. Future research should focus on formulating the proposed or similar policies as a multi-objective problem (single or multi-level) to take advantage of the benefits of the multi-objective approach (Taboada, 2007), as the multi-objective optimization could be a good approach for the sensitivity analysis. The proposed GA presented here can be easily modified and used for the estimation of the Pareto front of the multi-objective formulations (Appendix B).
Acknowledgements This article is based on work supported by the National Science Foundation under grants No. 0538901 and No. 0625515, and by the Environmental r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
373
Golias et al
Bioinformatics and Computational Toxicology Center under grant GAD R832721-010. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors would like to thank the anonymous referees for their constructive comments.
Notes
C O
PY
1 http://www.northeastdiesel.org/ports.htm. 2 http://www.ens-newswire.com/ens/jun2004/2004-06-21-04.asp. 3 The term hotelling refers to vessels using their auxiliary engines while docked in order to provide electrical power to the ship for climate control, lighting, cargo refrigeration, on-board cargo handling equipment, and other uses. 4 http://www.epa.gov/cleandiesel/ports/publications.htm.
References
A
U
TH
O
R
Fossey, J. (2008) In the bunker. CI-Online, http://www.ci-online.co.uk/, accessed 19 September 2008. Goldberg, D.E. (1989) Genetic Algorithms in Search, Optimization, and Machine-Learning. Reading, MA: Addison-Wesley. Golias, M. (2007) The discrete and dynamic berth allocation problem: Models and algorithms. PhD Dissertation, Rutgers University, NJ. Guan, Y. and Cheung, R.K. (2004) The berth allocation problem: Models and solution methods. OR Spectrum 26: 75–92. Hansen, P., Oguz, C. and Mladenovic, N. (2008) Variable neighborhood search for minimum cost berth allocation. European Journal of Operational Research 131: 636–649. Hooker, N.J. (1998) Constraint satisfaction methods for generating valid cuts, http://Citeseer.Ist .Psu.Edu/163317.html, accessed 19 September 2008. Imai, A., Chen, H.C., Nishimura, E. and Papadimitriou, S. (2008) The simultaneous berth and quay crane allocation problem. Transportation Research Part E 44(5): 900–920. Imai, A., Nagaiwa, K. and Tat, C.-W. (1997) Efficient planning of berth allocation for container terminals in Asia. Journal of Advanced Transportation 31: 75–94. Imai, A., Nishimura, E. and Papadimitriou, S. (2001) The dynamic berth allocation problem for a container port. Transportation Research Part B 35: 401–417. Imai, A., Nishimura, E. and Papadimitriou, S. (2003) Berth allocation with service priority. Transportation Research Part B 37: 437–457. Imai, A., Sun, X., Nishimura, E. and Papadimitriou, S. (2005) Berth allocation in a container port: Using continuous location space approach. Transportation Research Part B 39: 199–221. Imai, A., Zhang, J.-T., Nishimura, E. and Papadimitriou, S. (2007) The berth allocation problem with service time and delay time objectives. Maritime Economics & Logistics 9(4): 269–290. Kim, K.H. and Moon, K.C. (2003) Berth scheduling by simulated annealing. Transportation Research Part B 37: 541–560. Lazic, B.M. (2004) Container terminal operation emissions. West Coast Region Conference on Marine Port Air Quality Impacts. Seattle Tacoma. Moorthy, R. and Teo, C.-P. (2006) Berth management in container terminal: The template design problem. OR Spectrum 28: 495–518. 374
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
Berth allocation problem
PY
Park, M.Y. and Kim, H.K.A. (2003) A scheduling method for berth and quay cranes. OR Spectrum 25: 1–23. Pinedo, M. (2008) Scheduling: Theory, Algorithms, and Systems. Englewood Cliffs, NJ: Prentice Hall. Steenken, D., Voss, S. and Stahlbock, R. (2004) Container terminal operation and operations research – A classification and literature review. OR Spectrum 26: 3–49. Taboada, H. (2007) Multi-objective optimization algorithms considering objective preferences and solution clusters. PhD Dissertation, Rutgers University, NJ. Theofanis, S., Boile, M. and Golias, M.M. (2009) Container Terminal Berth Planning: Critical Review of Research Approaches and Practical Challenges. Marine Transportation and Port Operations 2009: Transportation Research Board 2100, pp. 22–28. Wacket, M. (2007) A rollercoaster year in 2007: A white-knuckle ride in 2008? CI-Online, http:// www.ci-online.co.uk/, accessed 19 September 2008.
Appendix A
i2B j2V
0
C O
min
1 w WT ijk @ j þ DDijk A max ðw Þ j k2K
XXX
ðA1Þ
j
Subject to:
xijk ¼ 1
8j
R
XX
ðA2Þ
O
i2B k2K
TH
X
xijk p1
8i; k
ðA3Þ
j2V
U
yijk XðAj Si Þ
X
X
ðHTim ximh þ yimh Þ
m6¼j2V hok2K
8i; j; k
ðA4Þ
A
Mð1 xijk Þ
DDijk XWTijk þ HTij þ Aj DTRj Mð1 xijk Þ WTijk X
X
X
8i; j; k
ðA5Þ
ðHTim ximh þ yimh Þ þ Si Aj
m6¼j2V hok2K
Mð1 xijk Þ
8i; j; ðk41Þ
WTijk XðSi Aj Þ Mð1 xijk Þ xijk ; 2 f0; 1g yijk X0 r 2009 Palgrave Macmillan 1479-2931
ðA6Þ
8i; j; ðk ¼ 1Þ
8i; j; k
ðA8Þ
8i; j; k
Maritime Economics & Logistics
ðA7Þ
ðA9Þ Vol. 11, 4, 358–377
375
Golias et al
WTijk X0
8i; j; k
ðA10Þ
DDijk X0
8i; j; k
ðA11Þ
where Aj ¼
ðAuj Alj Þ 2
þ Alj
8j
min
XXX
WTijk
i2B j2V k2K
C O
XXX
PY
Appendix B
min
ðB1Þ
DDijk
ðB2Þ
wj WTijk
ðB3Þ
i2B j2V k2K
XXX
R
min
O
i2B j2V k2K
Subject to:
TH
XX
xijk ¼ 1
8j
ðB4Þ
i2B k2K
A
U
X
xijk p1
8i; k
ðB5Þ
j2V
yijk XðAj Si Þ
X
X
ðHTim ximh þ yimh Þ
m6¼j2V hok2K
Mð1 xijk Þ
8i; j; k
DDijk XWTijk þ HTij þ Aj DTRj Mð1 xijk Þ WTijk X
X
X
ðB6Þ
8i; j; k
ðB7Þ
ðHTim ximh þ yimh Þ þ Si Aj
m6¼j2V hok2K
Mð1 xijk Þ 376
r 2009 Palgrave Macmillan 1479-2931
8i; j; ðk41Þ
Maritime Economics & Logistics
ðB8Þ Vol. 11, 4, 358–377
Berth allocation problem
WTijk XðSi Aj Þ Mð1 xijk Þ xijk ; 2 f0; 1g
8i; j; ðk ¼ 1Þ
8i; j; k
ðB9Þ ðB10Þ
8i; j; k
ðB11Þ
WTijk X0
8i; j; k
ðB12Þ
DDijk X0
8i; j; k
ðB13Þ
A
U
TH
O
R
C O
PY
yijk X0
r 2009 Palgrave Macmillan 1479-2931
Maritime Economics & Logistics
Vol. 11, 4, 358–377
377
