Given a set of scenarios , the two stage stochastic MILP can be applied to solve the first stage decisions to optimality, as discussed in previous sections. It takes into account the ‘risks’ associated with each of the scenarios . In this section we discuss how we construct such a scenario set for the LNG case.
The stochastic processes associated with LNG train breakdowns, shipment delays and spot opportunity arrivals are independent, so we can discuss them separately. Given the scenarios sets for each of these three stochastic processes, we construct the overall scenario set as all combinations that can be constructed from them. To illustrate this procedure, suppose we flip a coin and roll a dice having the usual two and six possible outcomes (scenarios) respectively. These outcomes can be regarded as individual scenario sets:
Scenario set of a coin: { } Scenario set of a dice: { }
If we combine these sets to obtain an overall scenario set, the number of scenarios contained in it is the Cartesian product of the individual sets:
Overall scenario set: { }
For the LNG case, we first define the individual sets (1) for the LNG breakdown stochastic process, (2) for Shipment delay stochastic process and end this section with the construction of the overall scenario set as the Cartesian product of these sets. In this thesis, we do not implement the set for the stochastic process associated with spot opportunity arrivals to avoid lengthy definitions and arithmetic. However, it could be implemented along the same lines as for the other sets and .
LNG train breakdown scenarios
Given the current breakdown state of the LNG trains in period 1, we distinguish between four scenarios per LNG train . If the train is down for remaining days, we face a single scenario that represents the remaining number of days that is needed to fix the train is decreased by one day:
,
where denotes whether or not LNG train is operative in period for breakdown scenario .
On the other hand, if LNG train is operational, we consider the following four different breakdown scenarios, denoted by the index :
1. LNG train is up in period ,
2. LNG train is down in period , and is operational again in period , 3. LNG train is down in periods 2 and 3, and is operational again in period , 4. LNG train is down in period to , and is operational again in period ,
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Deterministic behavior for first period
Stochastic behavior for periods 2 to 5
Scenario Scenario Scenario Scenario
3
3 ̂3 3 3 3 3
̂ ̂ 3
̂ ̂ 3 ̂
Average case behavior for remaining periods
̂ { } .
Otherwise (case 2), if the LNG train is broken in the first period (so ), we consider the scenario in which LNG train is under maintenance for the residual days. When it is fixed, it behaves like it is in its average case behavior for the remainder of the planning horizon. To summarize, if LNG train is broken in period , we have to consider a single scenario that represents the residual number of maintenance days being decreased by one day. On the other hand, if the train is working in the first period, we consider the four different scenarios as stated above.
Note that the number of distinct breakdown scenarios is either 1 (if ) or 4 (if ) per LNG train. However, we always define four scenarios per LNG train for convenience in later definitions. In case of a single scenario, we therefore duplicate it four times.
The total number of breakdown scenarios in the set is thus , as we have the four scenarios for each of the LNG trains. Suppose the model contains two LNG trains, so , we have scenarios in each period . Note that the number of scenarios increases dramatically if additional LNG trains are added to the model.
Shipment delay scenarios
The parameter ̃ denotes the stochastic number of available ships in period . Based on the Annual Delivery Plan (ADP), we know the expected number of ships, defined by as if the model was deterministic. However, uncertainties are not taken into account when constructing this schedule.To model stochastic behavior, we start with the assumption that at most a single ship is delayed in each time period . In each period , we distinguish between four different shipment scenarios :
1. No ships are delayed in any of the periods
2. A single ship is delayed by one day in period , no ships are delayed in other periods 3. A single ship is delayed in period by one day, no ships are delayed in other periods
4. In both periods and a single ship is delayed by one day, no ships are delayed in other periods
Given the initial number of ships that is available in period according to the ADP, we calculate the number of ships in case of the four shipment scenarios denoted by index :
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Deterministic behavior for first period
Stochastic behavior for periods 2 to 4
Scenario Scenario Scenario Scenario
3
3 3 3 3 3 3 3 3 3
3
Average case behavior for remaining periods
{ }
We assume that at least one ship arrives each day in the ADP, so no maximum operator is necessary to ensure non- negativity of the . The probability vector containing the delay probabilities associated with each of the scenarios is
defined as , such that the summation over its elements, denoted by , equals 1.
If we combine the delay scenarios with the breakdown scenarios of the previous section by means of their Cartesian product, we obtain the scenario set that contains scenarios. Its elements are , for
and . The probability of occurrence associated with scenario is the element-wise multiplication (known as Hadamard product) of the breakdown and delay probability vectors and :
The summation of these probabilities of occurrence over all scenarios adds up to , as is necessary for to be a probability distribution.