Path Method Steps

Identify interdependencies among tasks Step 3 requires an understanding of how projects unfold, that is, how tasks are related. It is also a reality check on Step 1. For example, if a task identified in Step 1 does not feed into another task, then the earlier task is not needed. Or if we find that a task could not be started because we would be missing some important information, then we must have omitted a necessary upstream task. The first three steps encourage us to carefully think through our project. 

Construct the network This step is mostly mechanical assuming the first three tasks are completed. Software can do Step 4, as explained later in this chapter. 

Determine the project s minimum completion time and the critical path The CP will be defined shortly when the process used to determine it and the minimum completion time are described. Step 5 can also be performed by software. 

The CPM was developed in 1956 by the Engineering Control Group, a design and construction unit of E.I. du Pont de Nemours and Company. The method was programmed by the Remington Rand Corporation to run on the UNTVAC computer. The first application of CPM was construction of a 10,000,000 chemical plant in Louisville, KY in 1957. There were 800 tasks in this project 

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The above analyses show that it is fairly easy to deal with temperature variation for unidirectional elementary reaction kinetics containing only one reaction rate coefficient. Analyses similar to the above will be encountered often and are very useful. However, if readers get the impression that it is easy to treat temperature variation in kinetics in geology, they would be wrong. Most reactions in geology are complicated, either because they go both directions to approach equilibrium, or because there are two or more paths or steps. Therefore, there are two or more reaction rate coefficients involved. Because the coefficients almost never have the same activation energy, the above method would not simplify the reaction kinetic equations enough to obtain simple analytical solutions. 

If the goal of the experimentation has been to optimize something, the next step after analysing the results of a 2 or a fractional 2 design is to try to make improvement using knowledge provided by the analysis. The most common technique is the method of steepest ascent, also called the gradient (path) method. 

If the function g(x) is twice differentiable, then the above sample path method produces estimators that converge to an optimal solution of the true problem at the same asymptotic rate as the stochastic approximation method, provided that the stochastic approximation method is applied with the asymptotically optimal step sizes (Shapiro 1996). On the other hand, if the underlying probability distribution is discrete and g(x) is piecewise linear and convex, then w.p.l the sample path method provides an exact optimal solution of the true problem for N large enough, and moreover the probability of that event approaches one exponentially fast as A — (Shapiro and Homem-de-MeUo 1999). 

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

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