Variable: decisive random

This problem contains 31 variables and 29 equality constraints (or governing equations) including the objective function. This gives rise to 2 variables as independent (or decision) variables. For a practical reason, the saturation pressure for steam, P, and the fraction of steam generated in the evaporator, which is reused for heating, a.., are selected as the independent variables. A random search technique (26) is adopted to locate the optimal point for each given e. The results are tabulated in Table I, and the trade-off curve is plotted in Figure 3. The relationship between these two objectives is obtained by the least square method as... 

Check violation of decision variable bounds randomly re-initialize the violated decision variables within the bounds. 

Before implementing an infeasible path optimization, it is very helpfiil to carry out preliminary searches by varying the key decision variables, somewhat randomly, to gain insights into the key trade-offs. For these searches, it is probably best not to use optimization algorithms that require derivatives, or approximations to them, such as SQP. A common approach is to use the sensitivity analysis facilities of the process simulators referred to earlier. 

By considering such a language to express solutions, one ignores the fact that in the type of problems we want to study decision variables behave as random variables, and there is always some variability associated with them. No matter how good control systems happen to be, in... 

The principle of sequential analysis consists of the fact that, when comparing two different populations A and B with pre-set probabilities of risks of error, a and /3, just as many items (individual samples) are examined as necessary for decision making. Thus the sample size n itself becomes a random variable. 

Table 10.10 shows the performance of the evolutionary solver on this problem in eight runs, starting from an initial point of zero. The first seven runs used the iteration limits shown, but the eighth stopped when the default time limit of 100 seconds was reached. For the same number of iterations, different final objective function values are obtained in each run because of the random mechanisms used in the mutation and crossover operations and the randomly chosen initial population. The best value of 811.21 is not obtained in the run that uses the most iterations or computing time, but in the run that was stopped after 10,000 iterations. This final value differs from the true optimal value of 839.11 by 3.32%, a significant difference, and the final values of the decision variables are quite different from the optimal values shown in Table 10.9. 

Gupta/Maranas (2003) as one example for a demand uncertainty model present a demand and supply network planning model to minimize costs. Production decisions are made here and now and demand uncertainty is balanced with inventories independently incorporating penalties for safety stock and demand violations. Uncertain demand quantity is modeled as normally distributed random variables with mean and standard deviation. The philosophy to have one production plan separated from demand uncertainty can be transferred to the considered problem. Penalty costs for unsatisfied demand and normally distributed demand based on historical data... 

The other quantity of interest is the VSS. In order to quantify it, we first need to solve the mean value problem, also referred to as the expected value (EV) problem. This can be defined as Min z(x, [) ]) where [ ] = f (Birge, 1982). The solution of the EV problem provides the first stage decisions variables evaluated at expectation of the random realizations. The expectation of the EV problem, evaluated at different realization of the random parameters, is then defined as (Birge, 1982) ... 

For distributions that represent variability, initial decisions may relate to the selection of data on which to base distributions. The problem formulation must identify meaningful populations. Ideally, the data are a random sample from the populations of interest in practice, one may be happy to establish that the data are representative. In addition, data should represent a spatiotemporal scale appropriate for the model. 

The major benefit of 2nd-order Monte Carlo analysis is that it allows analysts to propagate their uncertainty about distribution parameters in a probabilistic analysis. An analyst need not specify a precise estimate for an uncertain parameter value simply because one is needed to conduct the simulation. The relative importance of our inability to precisely specify values for constants or distributions for random variables can be determined by examining the spread of distributions in the output. If the spread is too wide to promote effective decision making, then additional research is required. 

The decision-rule approach tried by Frieden (1974, 1975) will serve to introduce the concept of an object built up from grains. In this approach, both 6 and x are taken to be discrete. That is, at a particular value of independent variable xm, we permit 6 to be only an integral multiple of Ad, the grain size. We may consider the use of random numbers to select locations for grain placement. A decision rule provides the basis for acceptance or rejection... 

Decisions Made by the NIOSH Action Level Criteria. Figure 3 shows the decision contours for the NIOSH Action Level Decision Criteria, and Table I summarizes the decision probabilities for each of the nine sample workplaces. Recall from Equation A-19 that the AL is computed from GSD to provide 95% confidence that no more than 5% of the daily exposures exceed the standard if one randomly collected sample is less than the AL. In terms of the variables used in this paper, (e > 0.05 with p > 0.05 if X > AL). 

Decisions Made with the OSHA Compliance Criteria. The problem of the OSHA compliance officer is significantly different than the problem addressed by the NI0SH Action Level. The compliance officer is not nearly so interested in how many exposures exceed the standard as in whether it can be shown with 95% confidence that the standard was exceeded on the basis of the sample which was collected. Thus, the OSHA Compliance Criteria is not a function of workplace variability, GSD, but only of the random sampling and analytical errors, CV (5). As explained earlier, CV = 0.1 for all illustrations in this paper. In this case, the OSHA Decision Criteria is characterized by AL = 0.8355 and UAL =... 

ELEMENTARY DECISION THEORY, Herman Chemoff and Lincoln E. Moses. Clear introduction to statistics and statistical theory covers data processing, probability and random variables, testing hypotheses, much more. Exercises. 364pp. 5X x 8H. 65218-1 Pa. 38.95... 

For some variables, for example, the relative collision velocity, the cumulative distribution function does not have closed form, and then a third Monte Carlo method must be adopted. Here, another random number R is used to provide a value of v, but a decision on whether to accept this value is made on the outcome of a game of chance against a second random number. The probability that a value is accepted is proportional to the probability density in the statistical distribution at that value. The procedure is repeated until the game of chance is won, and the successful value of v is then incorporated into the set of starting parameters. 

The decisions made by the computer concerning the pressure of the pump-flow rate dependence and of the flow rate of the fresh suspension, are controlled by the micro-device of the execution system (ES). It is important to observe that the majority of the input process variables are not easily and directly observable. As a consequence, a good technological knowledge is needed for this purpose. If we look attentively at the xj — X5 input process variables, we can see that their values present a random deviation from the mean values. Other variables such as pump exit pressure and flow rate (xg,X7) can be changed with time in accordance with technological considerations. 

The EA is embedded into PPSiM to perform an optimization of the decision variables. It interacts with the simulator in two ways. First, the simulator computes the ASAP solution. The initial population of the EA consists of multiple copies of this solution and other random solutions. Secondly, the simulator maps the sequence of recipe steps proposed by the EA into a feasible schedule and evaluates its fitness. The framework is shown in Fig. 3 where it can be seen which tasks are accomplished by the graphical interface, the simulator and the optimizer. 

Mitra et al. (1998) employed NSGA (Srinivas and Deb, 1994) to optimize the operation of an industrial nylon 6 semibatch reactor. The two objectives considered in this study were the minimization of the total reaction time and the concentration of the undesirable cyclic dimer in the polymer produced. The problem involves two equality constraints one to ensure a desired degree of polymerization in the product and the other, to ensure a desired value of the monomer conversion. The former was handled using a penalty function approach whereas the latter was used as a stopping criterion for the integration of the model equations. The decision variables were the vapor release rate history from the semibatch reactor and the jacket fluid temperature. It is important to note that the former variable is a function of time. Therefore, to encode it properly as a sequence of variables, the continuous rate history was discretized into several equally-spaced time points, with the first of these selected randomly between the two (original) bounds, and the rest selected randomly over smaller bounds around the previous generated value (so as... 

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