PART IV MODELS FOR ANALYSING SECURITY RISKS AND POLICY IMPLICATIONS
4.1 Stochastic Stability
In the preceding, we assumed that the arrival headway at the first port is a constant h. We now assume that the actual headway varies randomly around a mean value of h. Thus for the first stop
(7) (d(n)− d ) = 1 1−s(h
(n)− h) − ( s 1−s)(d
(n−1) − d ).
Define x(n)= d(n)− d and y(n)= h(n)− h. Then
(8) x(n)= 1
nx(0)vanishes as n tends to infinity, we obtain for large n
(9) vd= vh
provided the series is convergent (provided s< 0.5). We see that the variance of the departure headway is always greater than the variance of the arrival headway (since s> 0) and that the variance of the departure headway tends to infinity as stends to 0.5 from below. The system is therefore stochastically stable provided s< 0.5.
This result may unfortunately not be applied recursively along the route of a container liner, as the series of arrival headways at the second port is autocorrelated. For the second port (where the subscript denotes the port)
x2(n)= 1
(15) E{(y(n))(x1(n−m))} = 0, ∀m >0
Unfortunately, (13) cannot be applied recursively for the third (or sub-sequent) ports as
(16) E{(x1(n))(x2(n−m))} ≠ 0, ∀m >0
Table 3 compares simulation with analytic results. It is assumed that con-tainer ships follow a daily routine with one arrival at each port every 24 hours.
It is assumed that the arrival headway at the first port can vary by up to +/− 1 hour, with a uniform distribution, giving a variance of 0.3387 h2. There are no other sources of random variation in this model. At each port, it is assumed that the rate of arrivals of containers is 1/3 the loading rate, so s1= s2= s3= 1/3. The simulated arrival headway variance at the second port magnifies to 0.8333 h2. The variance calculated by (9) is in good agreement with the simulated value. At the third port, the arrival headway variance magnifies again to 2.9950 h2, in good agreement with the value calculated by (13). By the time the fourth port is reached, what was initially a small headway perturbation has become huge, having a variance of 13.7513 h2.
Headways Mean (h) Simulated variance
(h2)
Calculated variance (h2)
Arrival at 1st port 24.0151 0.3387
Arrival at 2nd port 24.0147 0.8333 0.8467 (from equ.
9)
Arrival at 3rd port 24.0141 2.9950 3.0693 (from equ.
13) Arrival at 4th port 24.0139 13.7513
Table 3: Arrival Headway Variance for Three Ports in Sequence (s1= s2= s3= 1/3)
If the loading rate can be speeded up to four times the arrival rate of containers, the effect on stability is significant. When s1= s2= s3= 1/4, we obtain the results in Table 4.
Headways Mean (h) Simulated variance (h2)
Calculated variance (h2)
Arrival at 1st port 24.0151 0.3387
Arrival at 2nd port 24.0148 0.6689 0.6774 (from equ.
9)
Arrival at 3rd port 24.0143 1.6537 1.6934 (from equ.
13) Arrival at 4th port 24.0138 4.8555
Table 4: Arrival Headway Variance for Three Ports in Sequence (s1= s2= s3= 1/4)
The simulated arrival headway variance at the fourth port is reduced from 13.7513 h2 to 4.8555 h2, with calculated variances at the second and third ports being in good agreement with the simulated values.
5 C O N C L U S I O N S
Taking the unrealistic case of a shipping line calling at equidistant ports, schedule stability requires that loading be twice as fast as the arrival of containers at the terminal. In reality, travel times between ports will vary.
Nonetheless, it is evident the schedule stability depends critically on the speed of loading and unloading.
Even where schedules are stable over time, the variance of arrival headways can grow explosively along the route. The rate of growth of arrival headway variance may, however, be reduced by speeding the rate at which containers are loaded and unloaded.
Hence we see that the performance of the quay crane is essential to the stability of ship schedules. This confirms the importance of quay crane per-formance in the context of ship and port operations.
R E F E R E N C E S
Newell, G.F. and Potts, R.B. (1964) ‘‘Maintaining a bus schedule’’. Proc., Second Conference. Australian Road Research Board, Melbourne, Vol. II, pp. 388–393.
Nicholson, A. and Kong, M.H. (2004) ‘‘Assessing the Effect of Congestion on Bus Service Reliability’’. The Second International Symposium on Trans-portation Network Reliability, New Zealand, pp. 21–27.
P R E D I C T I N G T H E P E R F O R M A N C E O F C O N TA I N E R T E R M I N A L O P E R AT I O N S
U S I N G A RT I F I C I A L N E U R A L N E T WO R K S
Richard Linn
Boeing 787 Program, Everett, WA 98204-1710, USA Jiyin Liu
Business School, Loughborough University, Leicestershire LE11 3TU, UK Yat-wah Wan
National Dong Hwa University, Hualien, Taiwan Chuqian Zhang
Information Technology, Columbia University, New York, USA
Abstract
With high average quay crane (QC) rate generally associated with short vessel turna-round times and minimum operations delays, the QC rate is often adopted as the performance indicator by container terminals. It is frustrating for container terminals to realize only in retrospect the decrease of the QC rate but lose the opportunity to reverse the trend. In this study, we identify factors that affect the QC rate. Based on these factors we develop artificial neural network (ANN) models to predict the QC rates of the next planning period, for both the overall rate of a container terminal and the individual rates of specific vessel types. The models are trained and tested using data collected from container terminals in Hong Kong. The results show that the average relative prediction error is small, especially for the models predicting QC rates of specific vessel types. Such predictions also lead to possible remedial actions to increase the QC rates.
1 I N T R O D U C T I O N
Container terminals play a critical role in global transportation. Their effective operation ensures orderly flows of containers between ocean-going vessels and
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landside transportation. Operation in a container terminal includes the gate house, the quayside, and the container storage yard processes and the latter two represent the majority of physical container handling activities. Figure 1 shows schematically the operations of a container terminal (Zhang et al., 2003).
Vessels bring in inbound containers. When a vessel is berthed, inbound containers are discharged from the vessel by the quay cranes (QCs) and transported by the internal trucks (ITs) to their storage blocks in the yard.
When a container arrives at its storage block, a yard crane (YC) lifts up the container from the IT and puts it in the block for temporary storage. Consign-ees then send external trucks (XTs) to the terminal to pick up the inbound containers. When an XT comes to a storage block, the YC there retrieves the required container and places it onto the XT which takes the container out of terminal through the gate house.
Figure 1: A Schematic Diagram of a Container Terminal
Outbound containers flow in the reversed direction. They are sent in by XTs from shippers and stored by YCs in a block. When their vessel is berthed, the outbound containers are retrieved again by the YCs and transported by the ITs to the QCs for vessel loading. Transit containers are brought in by a vessel, stored in the yard, only for transit onto its next vessel. Containers stored in the yard may be relocated, or marshalled, to different locations in the yard to streamline future operations. Marshalling a container requires an IT
b lock s in sto ra g e y a rd
trip between the two locations and two YC moves for retrieving and storing the container at the origin and the destination, respectively.
Terminal operations involve many decisions, such as assigning berths and QCs to arriving vessels, assigning storage space for incoming containers, scheduling YCs and ITs, deciding plans for marshalling moves, etc. The operations and the decisions involved in different areas of a terminal interact with each other. Due to the scale and complexity of the terminal operations, it is impossible to make all decisions altogether in one model to optimize the performance of a terminal. To simplify the operations problem and to make feasible decisions, a hierarchical and distributed approach is often adopted.
Longer-term and global decisions concerning the whole terminal are made first. With known results from these decisions, more short-term and local decisions concerning a specific area or function are then made. Such an approach usually works well to keep smooth operations of the terminal. For detailed description of terminal operations, decision problems and previous research on different specific decision problems, see Vis and de Koster (2003), Steenken et al. (2004) and Murty et al. (2005).
From shipping lines’ perspective, the performance of a terminal is measured by vessel turnaround times. To incorporate the objective of shipping lines into theirs, terminals often use the (overall) quay crane (QC) rate as a unified performance indicator for the terminal efficiency. Suppose that in a given period the total number of containers, N, loaded onto and discharged from the vessels and the total QC working hours spent, T, are known. Then the QC rate is the ratio of N to T. There can be a QC rate defined for a specific vessel type. In that case, the QC rate of type i vessels is the ratio of Nito Ti, where Niis the total number of containers loaded onto and discharged from type i vessels, and Tiis the total QC working hours spent on type i vessels.
The performance of terminal operations, as reflected in QC rates, fluc-tuates. Sometimes the fluctuations, such as that induced by changes of work-load over time, are known ahead of time. However, more often than not once a low performance is observed it is too late to correct, leaving the terminal operating in low efficiency.
In this chapter we propose to tackle the problem by predicting the terminal performance. Driven by the practical needs in Hong Kong container termi-nals, the goal of this study is to provide a satisfactory prediction of terminal performance, so that preventive remedial actions can be taken if the perform-ance is going to be poor. Rule-base procedures and variants of linear regres-sion have been tried in practice to such a prediction-correction problem without leading to concrete results. The main hurdle is the complex relation-ships among the variables in the terminal operations. Such relationrelation-ships are not only non-linear but also difficult to describe by any simple functions.
Before any sensible predication can be made, the modelling and the character-ization of these relationships are already difficult when done using rules or regression models. Instead of relying on rules or regression we adopt an
artificial neural network (ANN) approach to predict the future QC rate from the current available data. This approach captures the non-linear interaction of variables through searching, with historical data fed in, the best parameter values of an ANN. Conceptually an ANN can fit to any amount of data to any degree of accuracy, subject to the allowable computation effort. The approach has been applied to classifications, pattern recognition, optimization and prediction for problem contexts as diversified as business, engineering design, medical diagnoses, agriculture, etc. See, for example, references in Du et al.
(2002), Magdon-Ismail and Atiya (2002), Mena (2003), Ray et al. (2005), Wong et al. (2000), Zhao et al. (2003) and Kominakis et al. (2002) for various applications; see, for example, Murray (1995), Mehrotra et al. (1997) and Haykin (1999) for the systematic development of the theories of ANN.
The remainder of the chapter is organized as follows. First, we identify the input factors affecting the QC rates in the next section. The structure of the ANN models and the training method are then presented. Subsequently we evaluate the performance of the ANN models and suggest ways of using the models to identify remedial actions. Finally conclusions are drawn.
2 I D E N T I F I C AT I O N O F I N P U T FA C TO R S
For prediction purposes, we divide an operation day into a number of basic periods. Our task in each period is to predict the QC rates in the next period, both for the overall rate and for the rates of specific vessel types. To develop such a model, we need to identify factors affecting the QC rates with informa-tion obtainable in the current period. We worked closely with the terminals in Hong Kong port analysing the operations flow and data records. The analysis leads to the following principles for determining the input factors of the prediction model:
u an input factor should have a significant impact on the QC rate;
u the total number of input factors must be manageable;
u the factors should be representative for the status of the whole terminal (even though some factors may be in an aggregate form); and u the factors can be obtained from information available in the current
period.
The execution of the principles appears to be an art rather than a science.
The amount and status of equipment is easy to keep track of. However, the activities and factors in a container terminal interlock so much that one can never quantify the actual effect of such equipment on the performance of a container terminal. Some factors for the container terminal, e.g. the intensities of activities, are intangible in nature such that any tangible indictors are at best approximation for such factors. Moreover, the appropriate number and the
best forms of input factors are just a matter of choice. With rounds of discussion and testing, eventually we settle on a total about 20 input factors.
In the next two subsections, we first describe the development of the ANN model for prediction of the overall QC rate and then proceed to develop ANN models to predict QC rates of specific vessel types.