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Simulation is a very powerful tool for the analysis and planning of seaport operations. A well-designed and calibrated simulation model can provide useful insights about complex port operations that analytical models cannot capture accurately. An imporÂtant part of any port operations simulator is the module that produces the ship traffic. This paper presents a ship traffic modeling methodology based on statistical analysis of containership traffic and cargo data obtained from a port in the United States. Implementation of the described procedure led to the creation of a simulation algorithm that captured ship traffic characteristics well. Functional relationships are also developed between ship length and ship draft and between ship length and cargo capacity. The relationship between the average number of handled containers and the number of cranes used is described. The information and conclusions provided here are intended to give guidance on achieving time efficiency and accuracy in the modeling of ship traffic and calibration of container port simulation models.
General introduction regarding the optimization methods and techniques
Simulation models have been used extensively in the planning and analysis of port operations. Many different simulators exist of varying complexity and objectives; some study bulk terminals UNCTAD 1969; Park and Noh 1997, and others study container Kozan 1995; Ramani 1996; Shabayek and Yeung 2002 or miliÂtary Nevins et al. 1998 terminals. The algorithm that generates the ship traffic that goes through the simulation model is a very important part of the simulation process, as it provides the input to the port simulation model.
The existing literature includes two approaches for simulating ship traffic. In the first UNCTAD 1969 the characteristics of the ships calling at the port are collected in a database, ship arrivals are generated as a Poisson process, and then ship characteristics are randomly assigned from the database using the empirical freÂquencies with which the ships call at the port. In the second and most frequently used approach Hayuth et al. 1994; Ramani 1996; Park and Noh 1997; Turner 2000; Shabayek and Yeung 2002 the ship traffic is divided into many classes according to ship size or capacity, and either averages or samples from empirical and fitted distributions are used to generate the loaded and discharged cargo. The published papers lack detail and transparency because most of the models are proprietary Holguin-Veras and Walton 1995 the traffic data on which these models are based are also proprietary and therefore not presented in great detail. Lacking ture. In simulating port operations and revenues, however, much more detail is required, such as ship lengths, drafts, and cargo distributions. As a result, each time a new simulation model is created, a new analysis of port operations and traffic data has to be performed for model calibration and verification. In cases where simulation is performed at the planning stage, no data are available with regard to ship specifications and distributions of loaded and discharged cargo, but at best only estimates of the yearly number of ships expected to call at the port and the number of containers expected to pass through the port facilities.
This paper describes a comprehensive methodology for modÂeling ship traffic in the detail necessary to achieve accurate repÂresentation of throughput, port revenues, and service times through simulation. Functional relationships are also developed so that ship draft and cargo capacity can be calculated based only on ship length in order to condense the amount of information necessary to create a realistic traffic model. Moreover, based on the actual data studied, an algorithm is presented to replicate the existing crane assignment practice, a problem on which very little research has been published Daganzo 1 989a,b; Peterkofsky and Daganzo 1990 Following the proposed methodology, researchÂers and consultants can formulate a detailed representation of the ship traffic, including all the information that is important for simulating the cargo flow in ship-berth operations. They can then create a traffic generation algorithm to integrate with existing or new port simulation systems. For the case when not so much data are available, some simplifying recommendations are given.
In the following sections, first the methodology is presented in detail and the development steps are explained; then the impleÂmentation of the methodology is illustrated in a U.S. port. The development and calibration of the functional relationships and probability distributions required for modeling ship arrivals and length-draft and length-capacity relationships for containerships, as well as the number of containers handled, are described in detail, supported by actual data wherever possible. Next, the trafÂfic generation algorithm created by the methodology is presented. Finally, validation results show that the algorithm can success fully create ship traffic that replicates not only the container throughput but also the recorded port revenues. The latter are a useful quantity to measure for loss estimation studies, which have become increasingly more important as port managers have to make prudent decisions on port retrofit, expansions, and insurance premium negotiations, which necessarily have to include revÂenues.
Specific introduction about this topic, i.e. why carry
This section describes the procedure with which the existing data should be condensed and analyzed so that a robust and accurate traffic-generating algorithm can be designed. The approach conÂsists of eight steps and is summarized in Fig. 1.
Out a research like the aforementioned one
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Ship shape: The 3D FEA model of a cruise ship created with ANSYS software shows how a hogging wave condition causes a full deflection in the 3D model
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1.3) Structure of this report
Step 1: Decide Duration of Simulation
This step is important in determining the scope and resolution required in modeling. For example, if only a few weeks of simuÂlation are required, then the daily and weekly variations of ship traffic should be studied. On the other hand, if a year of simulated operations is sought, the monthly fluctuations have to be ac-counted for and the seasonality of the traffic should be included. The duration of simulation also will determine the units of the measured parameters.
Step 2: Determine Monthly Ship Arrival Distributions
If long periods of simulated time are required and port revenues are not very sensitive to service time, mean interarrival times can
be measured in hours and total simulated time in months. The interarrival time distribution has to be assumed or inferred from observed data. The most common assumption for interarrival times is that they are exponentially distributed. Although this might not be theoretically correct in terminals with planned arrivÂals, many authors suggest that it is a suitable assumption ~Plumlee 1966; Nicolaou 1967; Jones and Blunden 1968; Miller 1971; Van and Wijbrands 1988; Noritake and Kimura 1990; Radmilovic 1992; Kozan 1997; Shabayek and Yeung 2001!. The exponential interarrival time distribution can be easily calibrated by the total number of ship arrivals every month, and its parameters can be allowed to vary to account for seasonal variations.
Step 3. Determine Ship Types and Monthly Distribution of Ship Arrivals at Terminals
If the simulation is of a multiple-terminal port, it is necessary to determine the percentage of total traffic that goes to each terminal every month and its fluctuations around the year. For multiple-purpose terminals, it is also important to estimate the percentage that each ship type ~containership, bulk cargo, tanker, roll on-roll off!, accounts for at each terminal because each type requires different cargo-handling procedures and has different dimensional characteristics.
Step 4. Develop Functional Relationships of Required Ship Characteristics with Ship Length and Ship Type
If ship particulars, such as draft, cargo capacity and so on, are required for the simulation model and are not readily available, they can be inferred from functional relationships with other feaÂtures such as ship overall length and ship type. For this reason, regression relationships can be developed to condense existing data so that only a few ship characteristics have to be analyzed and modeled. Data points for the development of these functional relationships can be retrieved in either the port's ship inventory or in ship specification databases. In the implementation described in this paper, regression relationships were developed for container-ships between their length and capacity, measured in twenty-foot equivalent units ~TEUs!, and between their draft and length.
Step 5. Model Cargo Loaded and Unloaded at Each Terminal
Accurate representation of the cargo that is loaded and unloaded from each ship is one of the most important tasks of the traffic modeling procedure because port revenues are proportional to cargo throughput. In this step, the various ships that arrive at each terminal have to be divided into homogeneous classes with reÂspect to loaded and discharged cargo, and the distributions correÂsponding to those classes have to be determined. Since dockage fees depend not only on service time but also on ship length, the arriving ships can be classified according to the length classes determined by port tariff. The contribution of each length class to each terminal's traffic is estimated, and distributions for the loaded and discharged cargo are developed. When no data are available to justify the use of a particular distribution, based on the observations made in the analysis of the data in this study, the loaded and discharged cargo can be approximated reasonably well by normal random variables. In the case of ships that only either load or discharge, their percentage in the total population of that length class has to be estimated, and the cargo distributions will be application of the methodology, the percentage of the ships of zero load or discharge was around 5%. To reduce the number of parameters that need to be estimated and increase the sample size whenever possible, samples from different classes can be merged, provided that there are enough indications that they come from a homogeneous population.
Another significant issue that has to be addressed is the numÂber of handling units used to service the ship. A handling unit is defined as a set of equipment and personnel that transfers the cargo on and off the ship, for example, cranes for containerships, pumps for tankers, and so on. The total ship service time depends heavily not only on the quantity of cargo but also on the number of handling units, so a predictive relationship should be develÂoped to relate the two. In the case of a new terminal, this can come from an appropriate crane-scheduling algorithm. For existÂing terminals, such a relationship can be inferred from observed data.
Step 6: Tabulate Results and Create Algorithm
In this step, all the findings coming from the analysis of the preÂvious steps are summarized and condensed so that the traffic properties that need to be modeled emerge. Then the flowchart of a traffic generator is laid out, showing the steps required to simuÂlate the necessary ship arrival times, specifications, and cargo. The simulating algorithm has to ensure that the statistical strucÂtures will be preserved and that the output will be compatible with the needs of the simulation environment.
Step 7. Calibrate, Verify, and Validate Algorithm
Once the algorithm for ship traffic generation is in place, the form and parameters of the distributions used have to be determined. When adequate existing data are available, they are used for disÂtribution fitting. Otherwise, projections of traffic flow, empirical distributions, and simplifying assumptions can compensate for the lack of detailed information.
Verification and validation are also very important stages in the development of a credible traffic simulation model. First, it is important to check that the random number generators produce random variables with the desired properties ~verification!. Next, the assumptions made for the port traffic have to describe the observed traffic adequately to a desired accuracy ~validation!. InÂtegrating the simulated traffic with the port operations model and validating the results of the port simulation also provide helpful feedback with regard to the validity of the traffic-generating alÂgorithm.
Step 8: Modify Algorithm If Necessary
In this stage, the conclusions of the validation procedure are used to improve, whenever necessary, the structure and assumptions of the traffic-generation algorithm. After modifications take place, calibration and validation procedures are repeated until the out-come has satisfactory properties.
Introduction (general) about this chapter
The above-described methodology was implemented as part of a port simulation environment developed to study the performance of ports after catastrophic events, such as earthquakes and hurriÂcanes. Data were obtained from a port in the United States whose actual name and location could not be included for proprietary reasons. For that port, at least one year of simulated ship traffic was required for representing incoming ships at its container terÂminals. Because the calculation of revenue losses after a major earthquake event was among the objectives of the simulation, emphasis was placed on the adequate representation of the deterÂminants of port revenues. The following sections describe impleÂmentation of the methodology in this particular framework, toÂgether with the challenges encountered and insight gained from analysis of the port traffic data.
With respect to revenues, the port collects the following fees reÂgarding the movement of containers from the ship to the port:
Dockage: a minimum fee is charged for the first 24 h and then
charged in 6-h increments; determined by the length of the
ship and the duration of its stay at the berth;
Wharfage: fee that depends on the number of TEUs loaded and discharged to and from the ship;
Craneage: collected from the tenants each time a crane loads or unloads a TEU;
Demurrage: fee collected per TEU per day if containers disÂcharged from the ship or waiting to be loaded remain in the port premises more than 5 days; and
Wharf storage: daily fee charged per TEU if the owner wants to store containers at the storage facilities of the port for a longer time.
Demurrage and wharf storage revenues are small compared to craneage, wharfage, and dockage because the cargo-handling companies have their own storage facilities where they dispatch the cargo, and they properly arrange for the containers to stay less than 5 days in the port premises. For the purposes of the simulaÂtion model, craneage, wharfage, and dockage are tracked, which generates the main portion of the cargo-related revenues. From the description of the problem parameters given above, it be-comes evident that the important features to be captured are the frequency of ship traffic, the ship lengths, the number of TEUs that each ship has to load and discharge, and the total time the ship stays on dock within 6 h resolution. Revenue-generating opÂerations related to tugboat and yard operations ~hostler moveÂments, and so on! may add complications but are not considered significant enough to be included in the model. Thus ship service time was inferred indirectly as a function of the total number of TEUs handled, the service rate of the handling systems, and the number of cranes used. Statistical analysis of the available data led to the creation and calibration, first, of a ship traffic generator program, and then of an algorithm for crane assignment, as de-scribed in the following sections.
Presentation of the model
The data that became available to the writers took the following three forms:
Ship's log report for all port terminals. The log report included the ship arrival and departure times with 1 min resolution, the number of cranes used for cargo handling, and the berths where the ships were docking;
Dockage records in U.S. dollars for each terminal; and Loading and discharge records in revenue tons for all container terminals. To convert revenue tons to TEUs, the equivalence used is 12 revenue tons to 1 TEU, based on communications with the port authority.
Only one month's data ~December 1999! were available in the ship's log report; thus the detailed information about exact inter arrival times, service times, and number of cranes used was limÂited. On the other hand, one year ~1999! of dockage, loading, and discharge data was used. It should also be noted that dockage, craneage, and wharfage tariffs changed that year ~in October 1999! so that two different tariffs had to be used. Through the dockage records and the tariffs, the ship lengths and approximate service times were inferred, and the TEUs processed were inÂferred from loaded and discharged revenue tons. Since the data were available in hard copies only, a significant amount of pre-processing was required to convert the data to digital form. This included scanning, sorting all dockage and revenue-ton records for each terminal, and cross validating with log report data when-ever possible. The data were sufficient to capture the traffic of eight of the port's container terminals.
Presentations of the functions of the variables
As mentioned previously, only one month of data had a resolution high enough to fit a distribution to the interarrival times. More-over, when the arriving ships were assigned to terminals, the sample size was not big enough for a distribution to emerge clearly. However, even though the arrivals of the ships at each terminal are scheduled and not random, when the ship traffic going to the whole port was considered, the distribution of the interarrival times fitted very well the exponential distribution shown in Fig. 2. This phenomenon has also been observed in telecommunications where the superposition of many indepenÂdent non-Poisson streams with uniformly sparse arrival rates yields approximately a Poisson arrival pattern ~Cinlar 1974. The exponential interarrival time assumption also agrees with the findings of other researchers Noritake and Kimura 1990; Kozan 1995; Shabayek and Yeung 2002.
In the absence of more data with high resolution, the exponenÂtial distribution was adopted to describe the monthly ship arrivals at the port. The rate parameter was calibrated from the total numÂber of monthly ship arrivals. The assignment of ships to the variÂous terminals was assumed random with probabilities equal to the percentages of ships that went to the individual terminals for that particular month. A plot of the monthly arrivals at the port in Fig. 3 shows that the arrival process is nonhomogeneous but with little variation except for the months of July and December, the latter usually a month with increased traffic because of the Christmas market. The distribution of the ships to the terminals also varied from month to month. Fig. 4 shows that even though some termi nals have almost constant traffic, others exhibit large variation throughout the year.
Based on the above observations, the ship arrivals appear to be best modeled as a filtered nonhomogeneous Poisson process with constant monthly rates. The filtering probabilities are constant over a single month but allowed to change every month. The thinning algorithm Lewis and Shedler 1979 was applied to genÂerate the nonhomogeneous Poisson process. The special case of the thinning algorithm used here generates a stationary Poisson process with a constant rate l* 5maxt$l(t)% and arrival times ti* , and then rejects each ti* as an arrival with probability 1 2l(ti*)/l*. Simply changing the rate parameter in the Poisson arrivals-generating algorithm would create inconsistencies in the arrivals if the differences in rates are large Law and Kelton 2000. Using thinning, the transitions from heavy to less heavy traffic and vice versa are smooth.
Connections between the variables
In general, it would be desirable to include the ship's draft and TEU capacity in the ship characteristics. However, no data were available from the port to determine the draft or TEU capacity of the various ships. To overcome this problem, ship specifications were collected from a containership database in Delft, the Nethâ€‘ erlands SHIPDES 2000. Using these data, the following regressionrelationship was established between ship length overall LOA and draft:Draft5
Fig. 4. Monthly distribution of ships at terminals
3.4) Presentation of the Model
Table 1. Statistics and Parameters of Length-Draft Regression
Ship length overall
Standard error s
The values of the regression parameters, sample sizes, and regression statistics are shown in Table 1. A plot of the linear fit in Fig. 5 shows that this regression can describe adequately the reÂlationship between the ship's LOA and the ship's draft, although for the LOA.200 m branch the coefficient of determination is somewhat low (R250.33) due to dispersion in the data and relaÂtively small sample size. It is noted that the ship draft cannot increase linearly with ship length due to constraints in the ship design coming from limited port depths. Thus, for greater ship lengths a nonlinear regression relationship would be more approÂpriate. However, for the scope of this study and the lengths conÂsidered ~less than 330 m!, the linear approximation is deemed sufficient.
Using the same data, a linear relationship was fitted between the logarithms of the TEU capacities and the logarithms of the ship lengths. The regression equation hence is of the form
with parameters A 526.208, B 52.586, coefficient of determinaÂtion R250.912, and standard error s50.232. The high value of R2 and Fig. 6 show that there is indeed a strong correlation beÂtween the logarithms of these quantities.
Since most of the tracked revenues are directly proportional to the number of containers handled by the port, it is important to cap ture their distribution accurately. For the calculation of dockage, 26 classes of ships are used according to the ship LOA, ranging from 30 to 405 m with an increment of every 15 m. For this reason, the ships for each terminal were grouped into the same classes. Empirical distributions were used to describe the percent-age of each ship length class for each terminal. For each length class, two separate sample populations were formed: one for the discharged TEUs and one for the loaded TEUs.
It was observed in the data that some ships only loaded ~zero discharge! or only discharged ~zero load! cargo. These samples ~zero load and zero discharge! were separated from the rest of the populations and modeled separately. Essentially, each of the load and discharge probability distribution functions ~PDFs! is a mixÂture of a delta function that represents the zeros and a regular PDF that models the nonzero discharged or loaded containers. Distributions were fit to each population, and the percentage of zero load and zero discharge was computed. Whenever the sample seemed homogeneous, different length classes were merged. For all the merged length classes, the Kruskal-Wallis test ~Law and Kelton 2000! was performed to verify homogeneity.
For some length classes, no distribution could be fitted sucÂcessfully; thus the empirical cumulative distribution function ~CDF! was used in the traffic generator. Goodness of fit was deÂtermined in each case by both chi-square and Kolmogorov-Smirnov tests at a 5% significance level. The fitted parameters were also tested visually by plotting empirical and sample PDFs, CDFs, and quantile-quantile plots. The distribution-fitting proceÂdure was performed in the S-Plus environment ~Chambers 1998!, which readily provided most of the statistics and visualization tools.
For some cases, however, mixtures of distributions had to be used. The maximum likelihood estimation of the parameters for these distributions is rather complex and requires numerical miniÂmization of the negative log-likelihood function. The approach followed is described in Venables and Ripley ~1999!. It was found that normalizing the data using the scaling and center proposals by Huber ~1981! facilitated the computations significantly. The mixtures fitted were mostly bimodal normal distributions; their PDFs can be expressed in the form
Presentation of the Results
Several different distributions were fitted successfully to the data. These included normal, lognormal, exponential, Weibull, beta, and mixtures of normals. However, the most frequent type of distribution for the loaded and unloaded TEUs was the normal distribution. Based on this observation, it would be reasonable to model the handled TEU distributions as normal in the absence of more detailed information.
A total of 30 and 32 different distributions were used to model the discharged and loaded TEUs, respectively, for all eight conÂtainer terminals and all length classes. A typical mixture fit with sample parameters is shown in Fig. 7, together with a nonparaÂmetric fit for comparison. It can be seen that the mixture of norÂmals with parameters yielded by the above procedure can give a very good representation of the data. The observation that the TEU distributions sometimes follow mixture distributions may indicate that modeling the container throughput with a single value-for example, the average number of TEUs per ship as done in Shabayek and Yeung ~2002!, or a fixed percentage of capacity as done in Turner ~2000!-may introduce errors in the revenue and service time calculations.
The data available for the use of cranes in ship-berth operations were limited to one month. Examination of the data in Fig. 8 shows a relation between the average total number of TEUs handled ~loaded and discharged! and the number of cranes used for each ship. This observation is not surprising because the more containers are to be processed, the more cranes will be used in order to minimize the docking time, during which the dockage fee is collected. A rule of thumb that can be used in this case is that one crane is used for every 400 TEUs to be handled. The mini-mum and maximum number of TEUs that correspond to each number of cranes utilized ~here from 1 through 4! can be used in developing an algorithm for crane assignment roughly based on the number of available cranes and the total number of TEUs to be handled, as shown in Table 2.
Depending on the range of TEUs the ship has to load and discharge, the ship will be assigned a number of cranes according to how many are currently available but within the specified mini-mum and maximum bounds. Although we do not attempt to solve the complex problem of crane scheduling, the above observations can provide a useful insight regarding how the problem is handled in current practice.
Discussion of the Results
Traffic Generation Algorithm
Based on the analysis of the existing data, a computational algoÂrithm was developed to simulate the monthly ship traffic at the port terminals for this study. Although the current application of this algorithm is for containerships, it could be easily modified to accommodate multiple types of ships. This algorithm produces the input that is then used by a port operations simulation module. Communication between the two modules is achieved through ASCII files. The input file that the traffic generator uses contains the following information:
â€¢ The mean ship interarrival times for each of the 12 months of a year. These times correspond to the frequency with which ships arrive at the port;
Table 2. Minimum and Maximum Required Number of Cranes Given Total Number of TEUs to Be Handled
Upper bounds of the length classes for all the classes of ship
lengths that come to the port. For this port, 14 length classes
were used from 120 to 135 m through 315 to 330 m;
Parameters of the draft-LOA regression;
Parameters of the cargo capacity ~in TEUs!-LOA regression;
Two tables that contain the information of the percentage of
zero-load and zero-discharge ships for each length class and
each terminal; and
Two tables containing for each terminal the type and paramÂeters of the loaded and discharged cargo.
The traffic generator and the port operation simulation algoÂrithms were implemented through GPSS/H ~Schriber 1991!, an environment specifically designed for simulation of manufacturÂing and queuing systems. The algorithm also uses the empirical distributions of the length classes, the percentages of total ship arrivals for each terminal, and the parameters of the function that calculates the minimum and maximum number of cranes re-quested. These parameters were not included in the input file, but were hard-coded in this implementation of the algorithm because of the limitations of GPSS/H.
The output file generated contains the following: a counter for the generated ship, its draft, length, cargo capacity, number of discharged and loaded TEUs, minimum and maximum number of cranes requested, and arrival time, counted in hours from the beginning of the simulation. The arrival time is converted to a date, and the interarrival time from the previous ship is computed for easier verification of the outputs.
The flowchart for generating ship arrivals and their properties is shown in Figs. 9, 10, and 11. After the input file is read, ship arrivals are generated as a Poisson process with the maximum of the monthly rates, and then arrivals are thinned out according to the current monthly arrival rate. Next, terminal assignment is deÂcided using the empirical distribution of the ship arrivals to terÂminals. After the terminal is assigned, the ship length and type
Table 3. Dockage Statistics Comparison for One Year's Traffic-Terminals 1-4
Coefficient of variation
here always containership! are generated from the empirical disÂtributions and are conditional on the terminal. Then Eqs. ~1! and ~2! are used to determine the ship's draft and cargo capacity as functions of its length.
With regard to the number of TEUs to be discharged and loaded, first it is decided if the ship will have zero discharge through a Bernoulli trial. The probability of nonzero discharge depends on ship length and terminal. If the ship has no containers to discharge, it is assumed that the ship will load, and conseÂquently the number of loaded TEUs is generated conditional on length and terminal. If the ship has nonzero discharge, it will be assigned a number of discharged TEUs, again conditional on length and terminal. Decisions on whether or not the ship is going to load any containers are again based on Bernoulli trials. For this case, though, the probability of nonzero load depends not only on the ship length and terminal but is also conditional in the event that the ship had nonzero discharge. If the ship loads, the number of loaded TEUs is generated from the distribution that correÂsponds to its length and terminal. In both discharge and loading, the number of TEUs is checked to be nonnegative and less than the calculated capacity of the ship; otherwise it is generated again. Given the total number of containers in TEUs to be processed, the minimum and maximum number of cranes to be requested is chosen from Table 2. After these properties are assigned, the pro-gram records the ship and its particulars in the output file.
The traffic generator also includes a timer that keeps track of the days and months so that the arrival rates can be adjusted accordingly. It can create multiple replications of ship traffic records for any given number of months ~in the present impleÂmentation, 12!, using different seed for its random number genÂerators. This option is useful when port operations simulation requires multiple replications instead of regenerative cycles, as is the case when statistics of the transient behavior are studied.
For purposes of verification, ship traffic streams were generated by the ship traffic generator algorithm, and then the statistics for
the ship arrival times, lengths, distribution to terminals, number of TEUs, percentage of ships with zero load or zero discharge, and other features were calculated. The parameters for the asÂsumed distributions were calculated and compared with the fitted parameters. Goodness of fit tests were performed again to verify that the ''true'' ~fitted! parameters could fit the simulated data. When empirical distributions were used, the simulated CDFs were compared with the observed ones.
For purposes of validation, two sets of ship traffic streams, one deterministic and the other stochastic, were used as input to the simulation model. For the deterministic input, one month of ship traffic data was fed to the simulation model, where all the ship and operation parameters were the ones observed from the records. Then the simulated dockage fees and the recorded ones were compared and found to match very accurately. For the stoÂchastic input, the ship traffic generator was used to produce one year's ship traffic. The crane assignment algorithm and service rate distributions were used in the calculation of service times.
where Ts is the measured service time and ~TEU/crane! is the total number of handled TEUs divided by the number of cranes used. After analysis of the port data, it was determined that r fits very well a lognormal distribution with different parameters for each type of container handling system. The particular port under study has three different handling systems. When the port operaÂtions are simulated, the total number of TEUs is generated by the traffic simulation algorithm, the number of cranes is determined by the crane assignment algorithm, and the service rate is generÂated from a lognormal distribution. Then service time Ts for a ship, useful for the calculation of dockage fees, is calculated as shown below:
Coefficient of variation
Fig. 12. Histograms of 1 year dockage fees at terminal 7-comparison of model and recorded output
Dockage fees were calculated as functions of the service time, the ship length, and the port tariff. The statistics for 1 year's dockage fees for the actual and simulated records were calculated and compared. Tables 3 and 4 show that in general the two are in good agreement. Fig. 12 compares the histograms of dockage fees taken from one year's simulated and observed data at a particular terminal. The value at the most frequent bin reflects the 24 h dockage fees for ships of the most frequent length classes.
This paper presents a methodology for modeling ship traffic in ports and describes implementation of the proposed procedure in creating a traffic simulation model for a port in the United States. Statistical analysis of port traffic and revenue records was per-formed to formulate and calibrate a ship-traffic generation algoÂrithm to be used in conjunction with a port operations simulation environment. It was shown that regression relationships can be developed between ship length and ship draft as well as between ship length and TEU capacity in order to condense the amount of data required for efficient representation of ship specifications. A relationship was also shown between the average number of TEUs handled and the number of cranes used, and an algorithm was developed to represent the existing crane assignment practice in that port. To simulate well the dockage revenues that port trafÂfic was generating, a breakdown in length classes was proposed and loaded and discharged cargo distributions were developed for each class. Based on analysis of the available data, it can be concluded that the normal distribution can be used as a first apÂproximation of the cargo distributions in the absence of more information. Comparisons between the distributions of dockage fees coming from simulation and actual records showed that the proposed assumptions performed well in the simulation of port revenues.