PREDICTABLE UNCERTAINTY ABOUT TERMINAL OPERATIONS IN THE SEA

%is paper considers a problem of planning short term operations in a bulk terminal faced by port management when making tactical decisions. Ship loading and discharging, cargo stocking, the maintenance and service of facility equipment are regular operations of a bulk terminal which is a subsystem of a sea port the conduct of which is subject to di+cult-to-predict or unforeseen in!uences. %e problem that port management encounters in day to day operations looks into making the best possible plan with the scheduled duration of operations/states and transition instants, considering various internal and external factors in!uencing terminal performance. A state and transition model is used for deriving e3ective solutions to obtaining the state order and state transition time of a bulk terminal with an objective of minimizing operational costs. %e behaviour of the terminal is tested applying the stochastic and deterministic method.


Introduction
Cargo transhipment in sea ports consisting of numerous operations together with various in uences results in speci c port behaviour and cargo turnover. In this context, port behaviour means the manner in which port operations take place. In such an operational environment, decisions must be made not only concerning ship scheduling and cargo distribution but also on how to employ port facilities to full capacity, when and how to upgrade facilities, a schedule of repair and maintenance works and plan work shi s. Such decisions are o en inuenced by the events that cannot be predicted with certainty. e major objective of planning port operations is to diminish port vacancy, thus minimizing operational costs while assuring that service rendered to ships is in line with widely accepted standards (Česnauskis 2007;Hsu and Hsieh 2007;Jaržemskis and Vasilis Vasiliauskas 2007;Afandizadeh and Moayedfar 2008;Paulauskas and Bentzen 2008;Vasilis Vasiliauskas and Barysienė 2008;Imai et al. 2009;Liu et al. 2009;Su and Wang 2009;Chen and Zeng 2010).
During the last decade, a signi cant amount of attention has been directed towards port/terminal management problems. Veeke and Ottjes (1999) describe the way of detailed modelling and simulation of a new container-handling concept (Improved Port/Ship Interface) of a container terminal that has a direct in uence on planning port operations. Cullinane et al. (2005) applied the mathematical programming approach to estimate the e ciency of container port production. Cullinane (2002) also investigated possible methods and their applications for productivity and e ciency modelling ports and terminals.
Even though a wide range of planning problems within shipping industry has received signi cant attention from researchers so far, there are still problems that have to be addressed, i.e. uncertainty about planning port operations. When dealing with port and ship operations, there is a lot of uncertainty due to weather conditions, mechanical problems and strikes. us, optimization under uncertainty is an important eld within Operation Research, see the survey by Gendreau et al. (1996). e problem of optimization under uncertainty also exists in bulk terminals, but is of a somewhat di erent character. Limited storage capacity and facility output necessitate planning terminal operations to prevent storage over ow and unoccupied terminal capacities. In contrast to the vast body of literature dedicated to transportation planning problems, relatively little attention has been directed to the problem of planning port operations. Radić and Bošnjak (1997) formulate the concept of the generalized tra c model using the methodology of a general system theory. e model focuses on the level that is not technically speci c and describes general trafc behaviour with applicability to di erent transportation subsystems.
is paper also concentrates on understanding terminal behaviour and addresses the question whether terminal operations show deterministic or stochastic behaviour. e major contribution of this work is made to determining the model of states and transitions (ST model) for bulk terminal behaviour observation which is based on a comparison of deterministic states and transitions (DST method) and stochastic states and transitions (SST method). Besides, we developed a method for a stochastic interpretation of terminal behaviour. e ST model can assist port management in making short term tactical operational decisions, such as planning human resources (shi s planning), maintenance and repair work, facility/machinery engagement in daily/weekly cargo operations, etc. We analyzed the worksheets of a bulk terminal at Bakar port for the period of two years (2007)(2008). To de ne various operations of the terminal for the model, we took data for the year 2007 only. Upon obtaining model results, a comparison considering data on terminal operations in 2008 and conclusions on which the method better ts a real example has been made as well as a proposal for the measures minimizing operational costs has been put forward. e next section provides a description of the problem and is followed by the mathematical model used to solve it presenting the results of the experiment on the problem of bulk terminal operations. Finally, practical extensions are outlined.

e Problem
Regular operations of a bulk terminal are ship loading and discharging, cargo stocking, the maintenance and service of facility equipment and distribution of cargo to/from hinterland. One of the major problems and endeavour of port management is to create a plan of the most e cient operations in order to achieve optimal cargo turnover in a manner that will minimize total operating costs and maximize facility capacities. Although the operations are coordinated similarly to those carried out in all real systems (Hess et al. 2007), their conduct is subject to di cult-to-predict or unforeseen in uences that may be of internal (machinery breakdown, strike of longshoremen, etc.) or external nature (bad weather, port-hinterland transportation bottleneck, etc.). e approach taken here in contribution to solving the problem is to consider terminal operations either they are planned or in uenced by impact factors as states in which the terminal can be in a given instant. We also take that transitions between states are either planned or subjected to impact factors. We assume that impact factors are stochastic variables since they cannot be predicted with certainty. e objective of the paper is to identify a particular state in which the bulk terminal will be at a given mo-ment in the future starting from an assumption that in the beginning, it was in idle state and that state switching occurred with designated transition probabilities. Besides, we will try to answer the question whether the observed bulk terminal behaves as a deterministic system, i.e. according to the logical terminal operation ow, or as a stochastic system, meaning that in uences causing state transition disorder are not negligible. In the rst case, the duration of each state and time of transits at a given period are known with certainty, so there is no need for advanced mathematical procedures to estimate terminal behaviour (Hess et al. 2008). However, in the second case, the impact of odd factors is considerable, and thus the use of an appropriate probabilistic method in order to follow terminal behaviour appears to be necessary.
We set up the ST model of the terminal taking into account its activity, behaviour, states and transitions. A lack of uniformity in the case of cargo arrival at the terminal and the impossibility of predicting the exact time and quantity of cargo arriving on the terminal are the main reasons for the stochastic property of its operations. Since transitions in the DST method are exactly known, at the second stage, we take the e ort of quantifying state transitions with the probability distribution of the SST method only. We examine two approaches to quantifying state transitions. e rst one consists of setting up a system of di erential equations for terminal operations with an assumption that the terminal has discrete states expressed with probabilities. e second approach de ning bulk terminal operations as Markov processes and setting up the matrix of transition probabilities yield state probabilities and lead quickly to an accepTable solution to the ST model essential for any practical application.

e Model
In this section, we set up the ST model of terminal operations. First, we de ne the DST method for observing deterministic terminal behaviour and develop the SST method for stochastic behaviour.
To simplify a procedure of de ning terminal behaviour, we take into account that the terminal exists in one of ve states at a given instant. ese states include: -S 1 -idle state (no operations on the terminal except data processing, i. e. the collection and analysis of weather reports, cargo/ships related information); -S 2 -preparatory state (operations carried on the terminal just before ship arrival, i. e. the preparation of facility/cargo/longshoremen for cargo operations); -S 3 -transhipment state (cargo loading and/or discharging; from economical perspective, the most desirable state of the terminal); -S 4 -closing state (operations performed immediately a er nishing ship loading/discharging, i. e. paper-work, ship departure operation); -S 5 -repair and maintenance state (regular main-tenance of equipment, repair in case of machinery breakdown). A terminal has deterministic behaviour if the order of states and their durations are exactly known in advance. Having de ned only ST-structure with operation ow, one can easily deduce in which state the terminal will be in the future instant. In the DST method, a set of states and transitions between these states for a bulk terminal is formed around ST-structure shown in Fig. 1.
Transitions between terminal states de ned through the DST method are: -I 12 -from the idle to preparatory state at ship arrival; -I 23 -a er ending preparation, transition to cargo transhipment state; -I 34 -back to the closing state a er transhipment ends; -I 45 -from the closing state to maintenance; -I 51 -the idle state follows maintenance state. On a terminal that behaves in a deterministic manner, transitions from state to state follow logical terminal work ow. e occurrence of these transitions is certain. erefore, in an instant, the transition between adjacent states has probability that equals one while other transitions are not possible.
A terminal has stochastic behaviour if the order of states and transitions do not follow logical work ow due to various internal and external unforeseen in uences on regular operations. For researching such a system, the SST method will be developed. In this case, ST-structure may be de ned from Fig 2. e SST method, in addition to transitions de ned through the DST method, comprises the following transitions: -I 15 -from the idle to repair and maintenance state; -I 21 -transition from the preparatory to idle state due to bad weather or the strike of longshoremen, etc.; -I 25 -breakdown of facility causes transition to repair state; -I 31 -back to the idle state if adverse events occur during transhipment; -I 35 -breakdown of facility causes transition to repair state; -I 41 -to the idle state upon a ship leaves the terminal; -I 52 -from maintenance state to the preparatory state due to early ship arrival; -I 53 -switch back to transhipment state a er failure is removed. e stochastic ow of operations yields additional transitions between nonadjacent states with various transition probabilities. Since states and transitions are subject to stochastic changes and therefore can be expressed with probabilities that should be quanti ed for a real example, we set a system of di erential equations for the bulk terminal. We derive a system of Kolmogorov equations using the graph of terminal states (Fig. 2 dp p p p p p dt dp p p p dt dp p p p dt dp p p dt dp where: p i is the probability of state i, i = 1, …, 5; O ij is transition probability from state i to state j; i, j = 1, …, 5; and t is time.
Since condition each probability can be expressed in terms of other probabilities and thus diminish the number of equations by one. To solve the system of di erential equations for probabilities of states p 1 (t), p 2 (t),…, p N (t), the initial probability distribution p 1 (0), p 2 (0),…, p i (0),…, p N (0) the sum of which is equal to unity has to be speci ed. If in a special case the state of terminal S at the initial moment t = 0 is exactly known, S(0) = s i , then p i (0) = 1 and other initial probabilities are zeros. In our case, if the system of differential equations is set on the basis of ST structure, then a solution to the system presents probabilities of nding the terminal in one of the ve possible states depending where: transition probabilities O ij represent transitions from state i to state j between two consecutive state changes. In the matrix, each probability should be quanti ed separately for the DST and SST method. e initial state is given by a vector of states P 0 of the bulk cargo terminal: , 5 are the probabilities of states at the initial moment of terminal observation. If the initial state vector P 0 and the matrix of transition probabilities P are given, then the probabilities of all states of terminal P (n) can be found from the formula: ( ) 0 n n P P P where n denotes the ordinal number of steps, n≥1.

DST Terminal Operations
On the basis of ST structure for deterministic terminal behaviour (Fig. 1) and bearing in mind that the transition from the state to the consecutive state is certain, the matrix of transition probabilities for the DST method is given by: Considering the simplicity of the matrix, the probability of states a er n steps can be easily obtained. For example, following 12 steps, the terminal will be in S 3 (transhipment state) with probability equal to one. Limiting distribution for Markov process ^ǹ X is de ned by:^( erefore, limiting probability in the long run (n→∞) of nding the Markov chain in state j is approximately 1/5, no matter in which state the chain began at time 0. Furthermore, in our case, Z j gives the long run mean fraction of time that terminal is in state j.

SST Terminal Operations
Data derived from terminal work were used to assemble a problem of stochastic terminal operations. To de ne various operations in the terminal applying the SST method, we analyzed the worksheets of the bulk terminal in Bakar port for the year 2007. e performed operations include the transportation of bulk cargo from/to terminal, loading/discharging cargo to/from ships, the inspection of ship and cargo, the distribution of cargo to shore stock, the maintenance and repair of facility equipment and customs procedures. We also took data on the frequency of machinery failure, bad weather and strike caused stoppages of operations and congestions on the terminal. ese data served as a basis for the population of the stochastic matrix of transition probabilities for bulk terminal behaviour (see matrix (6)). Worksheet data on the terminal show that upon the receipt of ship arrival notice the terminal switches from the idle to preparatory state in 98% cases resulting in the probability of 0.98. Further probabilities are obtained analogous in respect to their own meaning.
We can summarize the transition probabilities matrix for the states of the bulk terminal in the port of Bakar bearing in mind that the sum of probabilities by rows is one: e solution was obtained employing computer-assisted evaluation program WinQSB (Chang 2003) having an integrated Markov modelling and simulation tool based on discrete space and continuous-time Markov model. Similarity in the procedure of determining port capacity having a di erent approach can be drawn to Kia et al. (2002) and Wang et al. (2002). A er data entry in the transition table, the de ned number of periods (n = 1,…, 12 steps), the initial state vector of the terminal at time t = 0, 0 1,0,0,0,0 P ª º ¬ ¼ and the probabilities of the ve terminal states are obtained and presented in Fig. 3.
Starting from the idle state, simulation shows in which state the terminal will appear most probably a er each transition (step). e probability of the most noTable state decreases with the number of simulation steps and the terminal approaches steady state probabilities. As an illustration, Fig. 3 shows that transitions from state to state do not have to follow a logical order of state transitions (terminal operations).
For example, in the fourth step, the terminal most probably resides in state S 5 representing the logical position order of the terminal in that step. However, in the next step, the terminal is found with the highest probability to reside in state S 2 and not in state S 1 as expected.
e reason of such turn in terminal state order can be explained referring to the previously mentioned numerous probabilistic in uences impacting terminal operation ow. In order to calculate the average duration of each terminal state, we analyzed terminal work sheets for the year 2007 separately for time intervals associated with ship arrivals when terminal operations had DST behaviour and those with SST behaviour. As predicted, the averages of the duration of states S 2 , S 4 and S 5 have low standard deviation, so we took them as representative values. For states S 1 and S 3 , deviation from average is high and cannot be taken into further calculations. e duration of state S 1 depends on the actual time of ship arrival while the duration of state S 3 is in uenced by the size of ships arriving and the quantity of cargo manipulated. e duration of each state in SST terminal operations is generally longer than the equivalent one in DST since, unexpected in uences cause additional waiting time and longer working procedures. For illustration, Table 1 shows a seven days period of terminal operation. Besides, we made analogous simulations for 52 weeks taking the same matrix of transition probabilities but starting each week with a di erent matrix of the initial states formed, considering the real state in which the terminal is found.
Assuming that the terminal switches states from S 1 to S 5 in deterministic order, terminal operation ow as well as state duration and the moments of their transitions can be laid down under DST as presented in Table 1. is is the case if the terminal perfectly follows logical operation ow and can be de ned as deterministic terminal behaviour. In other words, this approach is based on estimating state duration and the moments of transition between states in line with terminal operation plan that does not include disturbances caused by unforeseen or hard to predict in uences. e order of terminal operation ow in Table 1 under SST is derived considering the obtained state probabilities representing stochastic terminal behaviour. e moments of transition to the next state can be derived knowing the duration of each terminal state. Di erences between deterministic and stochastic terminal behaviour in state transition order and state duration result in discrepancy between the moments when the terminal switches among the states.
A er obtaining state probabilities for 12 steps in the SST method and deducing those in the DST method, a comparison of results with real-world operations ow for the year 2008 and the selection of the best-t method for further short term planning follow. We evaluated the order of state transitions on a weekly basis and the -  e proposed procedures for reducing the e ect of stochastic events include timely data collection on ship scheduling and expected time of arrival, hinterland connection congestions, cargo distribution and quantity and weather forecast. Besides, it is vital to keep regular maintenance, testing and control of equipment, education and training of employees as well as carrying out appropriate drills. e measures that are particularly e ective comprise proper contingency planning and adhering to contingency procedures in case of the occurrence of an event that disturbs planned terminal operations. is way, if the terminal operates in line with a working plan, even if it is a contingency plan, the operations will be departing back from SST to DST mode. In order to make the contingency plans as appropriate as possible for real situations, it is important to having collected and analyzed data on past terminal operations under the in uence of adverse events applying an appropriate method for creating practical contingency procedures.

Conclusions
1. We have answered the question whether the observed port terminal behaves as a deterministic system, i.e. according to the logical terminal operation ow, or as a stochastic system with a degree of variation in the order of terminal states. 2. e method may be used for short term tactical decision making identifying a particular state in which the terminal will be at a given instant. One of the major shortcomings of SST compared to DST terminal operations is represented by time lost and the consumption of more resources on overcoming the effects of unforeseen events resulting in the ine ciency of operational costs. 3. While the results are not as close to optimality as port management would desire, this is a tough reallife problem and an attempt to solve it is deriving high-quality solutions quickly which is essential for any practical application. e presented ST model can serve as a theoretical base for modelling the technological operations of other port terminals and tra c systems.
nal results showed that the data obtained using the SST method matched practice in 38 cases (weeks), whereas applying the DST method embraced 9 cases and in 5 cases state transitions followed some other order. We may conclude that the observed terminal had stochastic behaviour in 73% cases in 2008. A comparison of the DST solution to the corresponding SST solution indicates that the later one better emulates the logic of bulk terminal operations ow. erefore, if under the existing situation and without overtaking speci c measures for improving operational e ectiveness port management follows the SST method in planning short term operations in the terminal, the plan is expected to be more feasible. However, bearing in mind that SST terminal operations draw longer working procedures and therefore time lost on overcoming the e ects of unforeseen events consume more resources, to perform even better, management should strive for adhering to the plan based on the DST method. Terminal operations that follow the DST method will certainly reduce operational costs as shown in the following section. For example, see Machuca et al. (2007).

Cost Comparison
Considering that limiting probability Z j gives the long run mean fraction of time that the terminal is in state j and each visit to state j incurs a cost of c j , then the long run mean cost per unit time C is 0 n j j j C c Z ¦ . e cost distribution of DST terminal operations obtained from Bakar terminal nancial plan for each of the ve states is C DST = (1.00, 2.37, 5.21, 1.93, 3.05). e cost of each particular state represents the sum of operational costs providing that terminal operations follow deterministic states and transitions order without any stochastic in uences, such as machinery failure, bad weather or the strike of longshoremen. e cost of the idle state has been taken as a referential and the cost of other states have been reduced to it. As a result, cost distribution and Z j = 0.2 lead to the long run mean cost per unit time of 2.712.
Limiting distribution for stochastic terminal operations is Z = (0.19, 0.21, 0.23, 0.22, 0.15). e cost distribution of SST terminal operations obtained from Bakar terminal nancial datasheets for the year 2007 and for the cases when operations did not follow deterministic states and transitions order is C SST = (1.00, 2.57, 5.63, 2.09, 3.28). Consequently, the long run mean cost per unit time equals to 2.972. e cost distribution of SST terminal operations di ers from that of DST which can be explained by various stochastic in uences present in SST terminal operations. As expected, each unanticipated and therefore unplanned event draws extra costs in all states except the idle state. Di erence in the long run mean cost per unit time equalling to 9.6% is considerable, although comprehensible SST terminal operations require additional manpower, equipment and overtime work. Also, it generates time wastage of waiting for work related decisions to be made.