It is quite clear that getting all these data might well be a very difficult job. Hence, once the situation is thoroughly studied, relevant models built; these models will be first tested on small samples of modeled data. Once models prove to be working well on modeled data, real world data should be addressed; in case the samples are not large enough statistical data analysis and/or simulation may be addressed to generate additional inputs. In case some data is impossible to get either models should be adjusted or modeled data should be used to make relevant stubs.
3. Methodology and literature review 3.1 Solutions techniques
A number of different operation research based techniques are applied in this master thesis. In this section we mention some of them and describe the way they are applied. The more detailed description of them is provided in the dedicated to them chapters of this thesis.
Time series modeling
According to Box et al. (1976), time series analysis comprises methods for analyzing time series data so as to extract meaningful statistics and other characteristics of the data.
Forecasting models are used to predict future values based on previously observed values, whilst regression analysis is often employed in such a way as to test theories that current values of one or more independent time series affect a current value of another time series.
Significant wave heights as well as wave directions are auto correlated time series (which are moreover correlated with one another between different geographical points) thus for weather modeling different sorts of stochastic processes should be addressed (ARIMA, VAR, N-Markov chains, etc.) with respect to geographical and/or seasonal weather clusters.
This means, we have to find the most relevant approach for modeling of weather at the Norwegian continental shelf and the area around installations of interest.
Discrete event based simulation
According to Robinson (2004) discrete event based simulation models are used to model the operation of a system as a discrete sequence of events in time. Each event occurs at a particular instant in time and marks a change of state in the system. Between consecutive events, no change in the system is assumed to occur; thus the simulation can directly jump in time from one event to the next.
This approach is used for simulation of vessel schedules (including sailing, waiting and servicing of the installations) with respect to modeled weather.
Agent based simulation
Agent based simulation is a class of computational models for simulating actions and interactions of autonomous and their influence on the system as a whole (Niazi et al. 2011).
This methodology occurs when deciding about changes in the sequence of visits within a voyage during sailing or when swapping voyages between the vessels (when for instance one of them is late from the previous route, whereas another one (with the same parameters) is waiting for its departure at the supply base area) or when doing any kinds of dynamic speed adjustments.
Combinatorial optimization
Combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects (Schrijver 2006).
Combinatorial optimization problems could be addressed during the simulation itself for implementing agent based behavior and attempting to do a posteriori optimization of the schedules (rerouting and rescheduling in particular).
Expert assessment theory
Expert evaluation (expert assessment) is the procedure of obtaining system estimates based on the opinions of experts (experts) in order to make the subsequent decision (choice).
This approach should be addressed when estimating distributions of service times at the installations and the supply base, since there are hardly likely any sources of secondary data for that sort of statistics. This approach might as well be addressed when scaling multicriteria decisions.
Threshold aggregation and/or multicriteria ranking
Threshold aggregation is an aggregation procedure based on some threshold rule for construction of an output ranking from individual m-graded rankings with an arbitrary integer
Another approach is to use multicriteria ranking algorithms described by Zopoundis and Doumpos (2002), like TOPSIS, ELECTRE, UTADIS, etc. These approaches use some distance and/or preference measure to rank alternatives with respect to a number of criteria.
Both of these techniques might be addressed at the stage of building an aggregation criterion for key parameters so as to have an aggregated measure of schedules in terms of robustness versus fuel consumption.
3.2 Literature review
In this section we carry out additional review of the literature on the chosen methodology applied to similar problems so as to get a better idea of what to pay especial attention to during our research.
3.2.1 Weather modelling
As it has already been mentioned a very important part of the research is dedicated to weather modeling and forecasting and in particular to analysis of wave heights and wave directions components of weather in the Norwegian offshore zone, where the installations of interest are located. Two groups of studies concerning weather modelling are addressed: those based on statistical distribution analysis and those based on stochastic processes analysis.
Caires and Sterl (2004) present in their article global estimates of long term return values of wind speed and significant wave heights. These estimates are based on data; they also are linearly corrected using estimates based on buoy data. Calculation of return values in their research is based on the peak solver-threshold method. Large amounts of data used in this study provide evidence that the distributions of significant wave height and wind speed data could belong to the family of exponential distributions. Further, the effect of space and time variability of SWH and WS (wind speed) on the prediction of their edge values is addressed. Thus research in this article might well help us model statistical distributions of significant wave heights in the Norwegian continental shelf as well as carry out clustering with respect to both seasons and geographical locations of the installations and routes among them with respect to these distributions. Another detailed example of statistical analysis of waves is presented by Bauer and Staabs (1998). Comparison of different models for wave heights is carried out in this paper. Forristall (2012) presents the paper, studying how well the
Rayleigh distribution matches the observed distribution of wave heights. It is claimed that most of the controversy stems from comparisons are based on different definitions of the significant wave height. Once consistent definitions are used, all available data support the conclusion that the Rayleigh distribution over-predicts the heights of the higher waves in a record. Analysis of 116 hours of hurricane-generated waves in the Gulf of Mexico permitted the empirical fitting of the data to a Weibull distribution. Another paper by Nerzic and Prevosto (1998) describes a model for estimation of maximal wave heights in a given sea state. Authors modify standard Weibull and Rayleigh distributions using a third order Stokes expansion of the so called wave envelope. What is especially important, authors conclude that the suggested approach have been tested on real data in the North Sea and provided much better predictions than standard models. Moreover the proposed model is relatively easy to apply and, thus, could be an effective tool in determining extreme wave and crest heights for offshore structure design purposes.
A particularly important for our case research considering wave heights time series is described by Guedes Soares and Cunha (2000) who generalize the application of univariate models of the long-term time series of significant wave height to the case of the bivariate series of significant wave height and mean period. A brief review of the basic features of multivariate autoregressive models is presented, and then applications are made to the wave time series of Figueira da Foz, in Portugal. It is demonstrated that the simulated series from these models exhibit the correlation between the two parameters, a feature that univariate series cannot reproduce. An application to two series of significant wave height from two neighboring stations shows the applicability of this type of models to other type of correlated data sets. This is exactly the case of our research since we have a set of correlated between each other auto correlated time series of significant wave heights and wave directions. A neural networks approach for improving the quality of prediction of significant wave heights is suggested by Makarynskyy (2003); this approach might well be used in our case as well when simulating the prediction of wave heights during the voyage.
However it should be mentioned that none of the papers described above considers the Markov stochastic processes for wave height modeling and/or prediction, which are addressed by Halvorsen-Weare and Fagerholt (2011), which makes it necessary to do additional and probably more advanced research for finding the most appropriate model for modeling stochastic processes of significant wave heights and wave directions at the Norwegian continental shelf.
3.2.2 Event based and agent based simulation
Another subject of interest for us regarding event based simulation might be the paper by Goldsman et al. (2002) discussing the issues concerning the simulation of transportation systems. In particular, a number of implementation tricks that are designed to make the modeling and coding processes more efficient and transparent are demonstrated in that paper.
Authors also present examples involving the simulation of commercial airline and military sealift operations. Even though the article has a different from ours scope, it might still be useful due to the implementation tricks concerning modelling and coding described.
Yet another aspect of simulation that will be applied in our research may be agent based behavior of the entities (in order to carry out a posteriori optimization of the schedules of supply vessels). This approach should also be studied in the appropriate literature. For instance, Arentze and Timmermans (2002) describe the conceptual development, operationalization and empirical testing of a Learning-based Transportation Oriented Simulation System. This activity-based model of activity-travel behavior is derived from theories of choice heuristics that consumers apply when making decisions in complex environments. The model, one of the most comprehensive of its kind, predicts which activities are conducted and decides for such factors as when, where, for how long, with whom, and chooses the transport mode involved. In addition, various situational, temporal, spatial, spatial-temporal and institutional constraints are incorporated in their model. Another paper concerning agent based behavior in transportation was presented by Wahle et al.
(2002). This group of researchers studies the impact of real-time information in a two-route scenario using agent-based simulation. In particular, they address a basic two-route scenario with different types of information and study the impact of it using simulations. The road users are modeled as agents, which is a natural and promising approach to describe them.
Different ways of generating current information are tested.
4. Data analysis of weather parameters
Time series analysis and modeling for significant wave height and wave direction at offshore locations has quite some applications in engineering, scheduling of vessels and organizing of other sorts of offshore operations. It is a useful complement to the models based on statistical distributions of the corresponding parameters which characterize weather in different areas.
Whereas the distribution-based models provide probabilities of occurrence of independent events at random points in time, time series-based models also take into consideration autocorrelation between the consecutive events and provide researchers with an opportunity to build a close to reality model based on the corresponding discrete and/or continuous time-based stochastic processes.
4.1 Methods for constructing point estimates and their properties
By definition a sample of size n X { ... }x1 xn – is a set of n observations over, received from an experiment.
By definition an estimatorAˆ – is some statistics Aˆ A Xˆ ( ) :Rn N RK used to estimate unknown parameters.
Estimator Aˆmight have the following properties:
Aˆ is Consistent if ˆ P ,
n
A A A
;
Aˆ is Strongly Consistent if ˆ a s. . ,
n
A A A
;
Aˆ is Unbiased if E AA{ }ˆ A, n 1;
Aˆ is Asymptotically Unbiased if A{ˆ } 0 E A A n
;
Aˆ is Efficient if it is Unbiased and ˆ arg min
{ } ,
A
A V A
where V A{ }ˆ EA{(A A A Aˆ )(ˆ ) }T is a covariance matrix of Aˆ;
Aˆ is Asymptotically Normal if n A A(ˆ )dNn(0, )V ;
Aˆ is Asymptotically Efficient if its asymptotical covariance matrix is a lower bound of covariance for all consistent asymptotically normal estimators.