Stage values

The radical concentration has the stationary-stage value given by Eq. (6.23). 

The evaluation of the implicit second-stage value junction Q, (x) for a given x requires the solution of ST independent MILP subproblems ... 

At this stage, values may be assumed for AT) and AT2 and heat balances made until equal areas are obtained. There are sufficient data, however, to enable an exact solution to be obtained as follows. 

Finally, the value chain network can have a single value-creation step or multiple value-creation steps in production and distribution where raw materials are processed through several production and distribution steps before being sold to the market. In the work, a multi-stage value chain network is considered. 

In order to reflect these lead times, the concept of a timestamp is introduced. Timestamp is used in computer science documenting the system time when a certain event or transaction occurs e.g. for logging events (N.N. 2007). In the context of future inventory value planning, the time-stamp marks the period, when the first raw material has reached a certain stage in the value chain network included into a specific product. In the example illustrated in fig. 57, the raw material is processed in the same period to be converted into product 1. Therefore, all four value chain steps indexed from one to four occur in the same period and have the same time-stamp one. Conversion into product 2, however, requires additional time caused by production lead times, safety inventory and/or transportation time, that the steps indexed with five and six have a time stamp of two. The timestamp reflects that the inventory value of product 2 is not based on the raw material costs from the same period but based on the raw material costs from the previous period in order to reflect the lead time. Consequently, value chain indices and timestamps are defined for all steps and can cover multiple periods reflecting that raw materials in a global complex multi-stage value chain network can take several months, until they are sold as part of a finished product to the market. 

Table 1. Variation of fractional shielding diamagnetism as a function of composition x during different stages of preparation. The relaxation stage values were reproducible to 5% (determined for two or more samples at several compositions).

Table 1 Applied thermodynamics system delivers value at every stage of life cycle modeling, and throughout the enterprise Stage Value...

Hot stage value where completely molten broad exotherm instead of Tm endotherm on DSC thermograms of first and second heating scans. 

An initial set bf tear variables 7 and V is assumed to initiate the calculations. For most problems it is sufficient to assume a set of V) values based on the assumption of constant molal interstage flows, working up from the bottom of the absorber using specified vapor feeds and any vapor side-stream flows. Generally, an adequate initial set of 7) values can be derived from assumed top-stage and bottom-stage values and a linear variation with stages in-between. 

Equation (12-11) permits a rapid estimation of minimum equilibrium stages. A more convenient form of (12-11) is obtained by replacing the product of the mole fraction ratios by the equivalent product of mole distribution ratios in terms of component distillate and bottoms flow rates d and b, respectively, and by replacing the relative volatility by a geometric mean of the top-stage and bottom-stage values. Thus... 

For other stages, values of these quantities are computed in a similar manner with the following results. 

A distinctive feature of the relaxation method is the large freedom left to the judgment of the computer. This is particularly true in deciding how small to make the e s in (9) at any given stage (value of X). For this reason, the speed of convergence of the method depends very largely on the experience of the computer. 

Who is not familiar with the whiplash effect in logistics It occurs primarily in multi-stage value-added chains. Minor fluctuations on the market (3 to 5%) are reproduced throughout the chain and are intensified. The quantity variation often reaches 50 to 70% (a factor of 20 )."... 

By inserting the stage values into the final equation we get... 

To start the Newton iteration the collocation polynomial used in the last step is extrapolated to obtain guesses for the stage values ... 

Solving this linear system for the stage values X and inserting these values into the last equation results in... 

This kind of methods allows the use of the same stage values ki for all Xn 9) G [xn,Xn- -i]- If the order of the continuous representation is required to be the order of the global error of the method, then there is such a formulation only for very few Runge-Kutta methods. For the DOPRJ45 method a continuous representation of order 4 exists ... 

In general only the stage values lie on the manifold ( /, A) /c(tn + Cih y X) = 0. For methods with 6 = asi for i = 1,..., s also the solution values are on the manifold. Methods with this property are called stiffly accurate Runge-Kutta methods. Note, collocation methods with the interval end point being a collocation point, c = 1, are stiffly accurate by construction. 

OS98] Olsson H. and Soderlind G. (1998) Stage value predictors and efficient Newton iterations in implicit Runge-Kutta methods. SIAM J, Sci, Stat, Comput, to appear. 

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