Scenario tree

The stochastic SC design— planning model is presented next. This section has been divided in two parts (i) the scenario tree representation of uncertainty and (ii) the corresponding deterministic equivalent model. 

7 Capturing Dynamics in Integrated Supply Chain Planning 

Additionally, it will be useful for modeling purposes to define sets Ti, Lt and AHi fii. Ti is the subset of planning periods t that are associated with event 1. For example, T = [ti, ta and T2 = fc+i, fe] in the scenario tree shown in Fig.7.3. Lt is the reciprocal set of T . Lt is the event I which is related to period t. Here, A// is defined to be given by = hi hp c hi), that is AHpn denotes the event 

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All possible evolutions of the demands for three periods are depicted by means of a scenario tree in Figure 9.3. The numbers above each node represent the possible outcomes of the demands and thus the possible observations. Each path from the root node to a leaf of the tree represents a single scenario a>. Each scenario contains one of all possible combinations of the demand outcomes. With two different realizations per period, the scenario tree for three periods consists of 2 = 23 = 8 scenarios. The demands for period i = 4 are not considered in the figure because... 

The sequence of decisions obtained from the scheduler (Figure 9.4) has a tree structure. This structure results from the scenario tree of the uncertain demand parameters (Figure 9.3). Due to the moving horizon scheme, the decisions and the observations alternate at each period and the decisions are functions of the observations. Each point in time where a decision is made is called a stage. The result is a multi-stage tree where each stage corresponds to a period. 

Fig. 9.5 Uncertainties the multi-stage scenario tree of period 1 and its two-stage approximation (both with a horizon of H = 2 periods).

De Vos, C., Saatkamp, H.W., Nielen, M., and Huime, R.B.M. (2003). Sensitivity analysis to evaluate the impact of uncertain factors in a scenario tree model for classical swine fever introduction. Working Paper, Wageningen University, The Netherlands. 

The predictive model within the control algorithm consists in a multistage stochastic MILP. A scenario based approach is applied. Refer to Puigjaner and Lainez for scenario tree description and stochastic formulation indices details (1, hi). 

Nickel et al. (2012) D T EIN MIP, SP Dynamics evaluation of investment decisions and development of scenario tree for representing demand and interest rate uncertainty... 

In this step, the information of abnormal states with the safety measures in IPL (Independent Protection Layer) is added to fault propagation scenarios. The system that we developed creates a scenario tree using fault propagation scenarios. Two or more causes are shown to one hazard in the created scenario tree. 

Using result by HAZOP system, the fault propagation scenario is created. The information of propagation is stored to the data base in the system. The analysis result shows the cause of propagation and identifies the hazards by the database. From this data base, it can remove the necessary information to create scenario tree. The proposed system creates the scenario tree of fault propagation automatically. This scenario tree system is developed to calculate automatically the accident frequency quantitatively. The model of the fault propagation scenario is created from many results of HAZOP system. It is shown in Figure 3. EiO is... 

To search information of hazards in the database is repeated. It is retrieved in the same way about branch condition. A search with some conditions perform on all equipment in the fault propagation scenario. By using the fault propagation scenarios, it is cleared that there are multiple causes for one consequence event. To visualize the position of the equipment that causes the fault propagation in chemical plant is very important. Since effect of fault propagation involved in the equipment becomes clear, it is possible to determine where to add new safety equipment by this technique. The safety equipment that located in a higher place from a branch point in the scenario tree can reduces the probability of hazard to two or more scenarios. 

There are more than one cause event in the scenario tree. The probability of occurrence of consequence event is equal to the sum of the probability of occurrence of each cause event as expressed in (2). 

Figure 4. Scenario tree created from fault propagation...

In the scenario tree, there is one consequence event and one cause event. Safety measures placed more than one in the fault propagation scenario. And they work effectively. The probability of occurrence of consequence event is given by equation (4). 

N = the number of safety equipment placed in the scenario tree... 

The probability of consequence event in the scenario tree is calculated according to equation (5). The probability of consequence events is obtained by summing the probability each scenario including the safety measures. Two deviations propagate in the scenario tree in Figure 7. The deviations are... 

Therefore the probability of occurrence of the top event of this scenario tree is 0.00072 per year. 

Assume that there are ILI events in which uncertainty (0 unfolds over the planning horizon. The values that random factors can take in event I can be identified by hi which belongs to set Hi. The information about the uncertain parameters can be represented by using a scenario tree. For more detailed explanation, the interested reader is directed to Birge and Louveaux (2013). Figure7.3 shows a scenario tree in which two realizations of the uncertain parameters are considered at each of three distinct events ( L = 3 a Hi = 2). 

The information about the scenario tree is introduced into the model by adding two indexes to the decision variables I and hi. As an example, variable associated with a decision that is made when solely the combination of events hi is known, however the decision will be materialized at period t when event I unveils. A subscript hi below a variable also indicates that is a (/ -I- l)-stage variable. For instance, a (/ -I-1)-stage variable such as would be related to a recourse decision... 

In order to follow the structure of the scenario tree, when an equation requires variables associated with previous periods (i.e., / - 1), these variables must be related to a combination of events that is ancestor of the combination of events being evaluated by the equation (/ e L j, hi e AHi fi,) so as to guarantee the non-anticipativity principle. This fact will be recalled in most of the equations presented below. 

Here, SC denotes the set of binary variables of the model, whereas SC represents the set of continuous variables. This model considers demand as uncertain parameter, however, it can be easily extended to include other parameters uncertainty. The only change that would be required is to add the indexes I and hi to the new uncertain parameter. These indexes identify and locate the parameter inside the scenario tree. Problems considering prices and interest rates uncertainty have been dealt with in Chap. 7. 

Non-anticipativity of the decision process is an inherent component of stochastic optimization. The non-anticipativity principle ensures that the solution for stage t does not depend on unavailable information. If two scenarios s and / are indistinguishable at time t on the basis of information available about them at time t, then the decisions associated with scenarios s and s until time t must be the same. Namely, a set of decisions related to two different scenarios, if it is to make sense, cannot require different courses of action at time t if there is no way to distinguish between those two scenarios at time t (Rockafellar and Wets 1991). This principle is shown in Fig. A.6 by using a scenario tree representation for the different scenarios in which uncertain parameter may disclose throughout the time horizon. Such a principle can be mathematically posed as expressed in Eq. (A.23). 

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