Numerical performance

Numerical tests with different data sets derived from real data collected at the industrial partner in the course of the pilot application reported in Chapter 5 were performed to establish the applicability of the proposed model to problem instances of realistic size. While the numerical performance of mixed-integer programs depends to a large extent on the data set used, some results are provided below. All tests were performed using ILOG OPL 4.2 and CPLEX 10 on a computer with an AMD Athlon XP 2600+ processor and 1 GB memory using a ten-year planning horizon. 

In a first step data sets with increasing complexity with respect to the number of products, markets and potential production facilities were generated. The results of the numerical tests are shown in Table 19. Product and market complexity was increased by further disaggregating the de- 

Run Product/market combinations Data set characteristics Plants Existing Potential MIP-GAP CPU time (hh mm ss) 

In a second step the effect of using different MIP gaps was tested. Since the data underlying the calculations is based on highly uncertain forecasts a MIP gap of 0% is not required. As can be seen with a MIP gap of around 1% calculation times of less than 3 minutes can be achieved. This is clearly sufficient to interactively evaluate a broad number of scenarios. To 

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In the past two decades, a variety of semiclassical initial-value representations have been developed [105-111], which are equivalent within the semiclassical approximation (i.e., they solve the Schrodinger equation to first order in H), but differ in their accuracy and numerical performance. Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator [105, 108, 187, 245, 252-255], which for a general n-dimensional system can be written as... 

At the allocation level product-plant and plant-market allocation decisions can be evaluated. In many cases these decisions take the form of regular single-sourcing restrictions. Alternatively, it might be desired to asses the impact of producing certain products only at a single site that is to be selected by the model or pre-determined by the user to reduce the complexity of the network. As discussed in the context of the numerical performance analysis, the latter type of restriction has to be handled with care because of the strong increase in calculation time that was observed. 

Chapter 3.5 provides the results of numerical performance tests conducted which demonstrate that, despite its complexity, the resulting MILP model can be solved in reasonable time using the standard optimization software ILOG CPLEX 10. 

Commercial and public transportation heavily relies on human operators. There is growing evidence that sleep loss may play a large role in transportation accidents. A committee formed at the 1986 meeting of the Association of Professional Sleep Societies found that numerous performance failures leading to catastrophic events occur most often at times of day coincident with the temporal patterns of brain processes associated with sleep (3). In addition, an investigation in the Netherlands showed that the highest accident rate in public transit accidents occurred in bus drivers who began an early work shift (4). An assessment of the impact of sleep loss in commercial and public transportation is therefore needed. 

The first two properties are algebraic in nature and are used in the formal derivation of the various ASC equations. The third property is concerned with functional analysis. As it is of no use for the formal derivation of ASC methods, it is rarely reported in the chemistry literature. However, it has direct consequences on the comparative numerical performances of the various ASC methods (see Section 1.2.5). 

Some efforts have been done to improve the numerical performances of the PBE exchange functional without modifying its theoretical background. 

We have recently shown that the numerical performances of some of these models are comparable to those of current 3-parameter hybrids like B3LYP [47,48]. In particular, we have obtained the PBEO model, casting the PBE functional in equation (15) [49]. This model provides very good results, both for the termochemistry of molecules belonging to the G2 set (see table 1) and for the corresponding geometric parameters (see table 2). 

This procedure has been theoretically justifled and it appears to work numerically reasonably well if the basis sets arc not too badly chosen. At the Cl level similar, though somewhat more (XMnplicated, procedures have been proposed 106. m) but there is less experience available yet as to their numerical performance. 

We would like to stress that this chapter is a review of coupled cluster theory. It is not primarily intended to provide an analysis of the numerical performance of the coupled cluster model, and we direct readers in search of such information to several recent publications. " Instead, we offer a detailed explanation of the most important aspects of coupled cluster theory at a level appropriate for the general computational chemistry community. Although many of the topics described here have been discussed by other au-thors, ° this chapter is unique in that it attempts to provide a concise, practical introduction to the mathematical techniques of coupled cluster theory (both algebraic and diagrammatic), as well as a discussion of the efficient... 

The force vector / , f,x) should represent the discontinuous influence of the vegetation in a form like (1.7). The model closed by boundary conditions and hypotheses as for turbulent viscosity allows the numerical performance of a practical importance, [453, 658],... 

Problem (3.29) with the formulated boundary conditions possesses a unique solution over the length of the entrance flow region 0 < x < Lx, but the value Lx, the unknown length of the entrance region, is to be chosen at a distance, where no further transformation of the flow field takes place. Lx can be easily adjusted in the course of the numerical performance. 

No simplification can be used for the problem of the backward facing penetrable step but the full Navier—Stokes equations. Therefore, no solution is available to validate the numerical algorithm. To be aware of it, the numerical algorithm shortly described in the previous section was tested over the whole range of the above-mentioned problems. In this case, the outlet boundary condition (3 = which is associated with the steady flow in an infinite duct, was used. The results of two numerical performances for the flow regime Re = 100 and EPR dimensions h = 0.3 and L x = 1, are shown in Fig. 3.16 the halves of flows in each case are symmetric. Let us analyze them. 

The analytical solutions for particular regions and the full numerical performance complement each other. The former can be used in the elaboration of an approximate calculation method. 

FIGURE 5.72 Schematic presentation of the Zimm plot (method of double extrapolation). The data from measurements at several concentrations (C[ to c ) and scattering angles (0, to G ) are presented by empty circles. Then, extrapolation to c = 0 for each angle and to 0 = 0 for each concentration is numerically performed (see the black dots). Both hues, c = 0 and 0 = 0, should meet the ordinate at the point Af, where M is the... 

There are several different aspects of the numerical performance of the approximants to SS-MRCC, which we want to illustrate with our example applications ... 

The numerical performance of DK Hamiltonians of higher than second order was examined for one-electron atoms as well as for the atoms Ag, Au [89], El 12 [94], and El 18 [17] in the SR approximation. For El 12, also its diatomic hydride E112H as well as the corresponding cation E112H and anion E112H were studied at several correlated levels [94]. 

The behavior of some of the most common functionals with respect to these three constraints is reported in table II. The B functional does not obey neither the Levy condition nor the Lieb-Oxford bound, but its numerical performances are better than those provided by the PW functional, which respects all the above mentioned constraints. 

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