Scaling Criteria Values

Consider a supplier selection problem with m suppliers and n criteria, where/iy denotes the value of criterion for supplier i. Let F denote the supplier criteria matrix. 

Hj will be the ideal value if criterion is maximizing and Ly is its ideal value if it is minimizing. 

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Note that the scaled criteria values will always be < 1, and all the criteria have been changed to maximization. The best value of each criterion is 1,... 

Here all the scaled criteria value will be between 0 and 1 and all the criteria are to be maximized after scaling. 

Illustrative Example of Scaling Criteria Values Example 6.5 Scaling Criteria Values... 

Scaled Criteria Values by the Ideal Value Method (Example 6.5)... 

The scaled criteria values, using the Ideal Value method are computed using Equation 6.5 for TCO and Equation 6.4 for Service and Experience. They are given in Table 6.20. 

Weighted methods for ranking alternatives, such as the rating method and Borda method, require that the criteria values (/jy) are properly scaled so that they are of similar magnitudes. Otherwise, the rankings will be incorrect. There are several methods available for scaling criteria values, such as ideal value method, linear normalization, and vector normalization using Lp norms. 

Note that the scaled criteria values (r j) will always be 1, and all the criteria have been changed to maximizahon. The best value of each criterion is 1, but the worst value need not necessarily be zero. In the next approach, all criteria values will be scaled between 0 and 1, with 1 for the best value and 0 for the worst. 

The ideal values for the three criteria are 65,000, 10, and 9, respectively. Of course, the ideal solution is not achievable. Using the ideal value method, the scaled criteria values are computed using Equation 2.6 for TCO and Equation 2.5 for service and experience. They are given in Table 2.4. 

Scaled Criteria Values by Simple Linearization (Example 2.3)... 

Combined scaled criterion To improve the performance of the above-mentioned scaled procedures, a novel approach has been proposed, consisting of a combined criterion for evaluating BE (40). Scaled BE limits containing an effective constraint have been developed. The proposed BE limits scale with intrasubject variability but incorporate also a GMR-dependent criterion, which makes them less permissive as GMR values depart from unity (40,57). 

The PROMETHEE is an outranking method. Preference values determined from the pairwise comparisons are then analyzed to develop an overall rating value for each variant. These overall rating values are on a scale of -f 1 to — 1. Basic for this method is pair comparing of variants, step by step from viewpoints of all criterions. Result is formidation intensity of preferential between pairs of variants. Values of coefficients Pt ar, as) will depend on difference of criterion values (Formula 19) ... 

The standard state chosen for the calculation of controls its magnitude and even its sign. The standard state is established when the concentration scale is selected. For most solution kinetic work the molar concentration scale is used, so A values reported by different workers are usually comparable. Nevertheless, an important chemical question is implied Because the sign of AS may depend upon the concentration scale used for the evaluation of the rate constant, which concentration scale should be used when A is to serve as a mechanistic criterion The same question appears in studies of equilibria. The answer (if there is a single answer) is not known, though some analyses of the problem have been made. Further discussion of this issue is given in Section 6.1. 

One of the most important characteristics of the emulsifier is its CMC, which is defined as the critical concentration value below which no micelle formation occurs. The critical micelle concentration of an emulsifier is determined by the structure and the number of hydrophilic and hydrophobic groups included in the emulsifier molecule. The hydrophile-lipophile balance (HLB) number is a good criterion for the selection of proper emulsifier. The HLB scale was developed by W. C. Griffin [46,47]. Based on his approach, the HLB number of an emulsifier can be calculated by dividing... 

The design methods de.scribed above rely on correlations of the overall reactor average quantities obtained from experimental tanks of different scales. The most important deficiency of these methods is that local effects are not taken into consideration, while these might be responsible for the overall reactor performance. Accordingly, if none of the above scale-up criteria is found satisfactory (see e.g. data of Middleton et ai, 1986) a more fundamental approach must be applied, although not necessarily as complex as the one presented in Section 5.4.S.2. Such an approach was presented by Paul et al. (1971) who found that the yield of the desired intermediate in a system of consecutive reactions (iodination of L-tjrosine) correlates reasonably with fluctuations of the velocity, So, these fluctuations could be chosen as a criterion for scale-up of the reactor. The average value for u in the upper part of the tank was evaluated from ... 

Agglomerates in a sheared fluid rupture when the hydrodynamic stress exceeds a critical value in dimensionless form the criterion for rupture is Fa > Facrjt. Rupture occurs within a short time of application of the critical stress, and thus can be distinguished from erosion, which occurs over much longer time scales. 

Care needs to be taken if some components are present in trace quantities. If an estimated concentration is 0.5 ppm and the calculated value is 1 ppm, the scaled error is 100%. This is much too large an error for most variables and yet the absolute error might be acceptable for a trace component. In other situations, it might be necessary to define trace components with a high precision. A trace component threshold can be set, below which the convergence criterion is ignored. 

In distinguishing physicists and chemists on the basis of the identity criterion of values and problems, it is appropriate to emphasize the molecule as the heart of the chemists problem-solving concerns. Historically, the chemist s molecules have differed from the physicist s objects of study not just in linear scale but in the multifunctionality of the chemical molecule s character and behavior and in its capacity for generating wholly new objects, in addition to moving through space. The natural history tradition of chemistry is just as important as its natural philosophy tradition. 

The criterion of mean-unbiasedness seems to be occasionally overemphasized. For example, the bias of an MLE may be mentioned in such a way as to suggest that it is an important drawback, without mention of other statistical performance criteria. Particularly for small samples, precision may be a more important consideration than bias, for purposes of an estimate that is likely to be close to the true value. It can happen that an attempt to correct bias results in lowered precision. An insistence that all estimators be UB would conflict with another valuable criterion, namely parameter invariance (Casella and Berger 1990). Consider the estimation of variance. As remarked in Sokal and Rohlf (1995), the familiar sample variance (usually denoted i ) is UB for the population variance (a ). However, the sample standard deviation (s = l is not UB for the corresponding parameter o. That unbiasedness cannot be eliminated for all transformations of a parameter simply results from the fact that the mean of a nonlinearly transformed variable does not generally equal the result of applying the transformation to the mean of the original variable. It seems that it would rarely be reasonable to argue that bias is important in one scale, and unimportant in any other scale. 

Figure 11 Voltage (in percent of full scale) vs current at 7 Tesla magnetic field for a commercial low temperature superconductor using a voltage criterion of different values to indicate the importance of specifying the criterion. Ref. 30.

The next step is the definition of deep. The choice of quantitative values will again involve some arbitrariness. However, a further complication is that there is at present no universally accepted qualitative criterion. For instance, from the point of view of energy level calculations it is often convenient to define deep states as noneffective-mass-like or as those with a localized potential (see, for example, Bassani and Pastori Parravicini, 1975 Jaros, 1980). However, the disadvantage of this definition of deep is that it includes many isoelectronic states that are very shallow on an energy scale. On the other hand, if one uses an energy criterion, should the states be deep with respect to some fraction of the band gap, with respect to kT, or with respect to some shallow levels In this chapter we shall adopt an energy criterion for deep, and we shall require that our states be deep enough to be important in recombination. The importance of deep levels in recombination under many conditions of practical interest was already realized in the early work of Hall... 

Aerated stirred tanks, bubble columns, and airlifts are usually used for aerobic fermentations. One criterion of scaling-up aerated stirred tank fermentor is k a, approximate values of which can be estimated by Equation 7.36a or b. For the turbulent range, a general correlation for k a in aerated stirred fermentors is of the following type [3] ... 

Satisfactory results were obtained with the pilot-scale fermentor at a rotational impeller speed N of 1.5 s and air rate (30 °C) of 0.5 in min T The density and viscosity of the broth are 1050 kg m and 0.002 Pa s, respectively. Data from the turbine impellers [3] showed that kf a can be correlated by Equation 12.13, with values m = n = 2/3. Using k a as the scale-up criterion, estimate the impeller speed and the air rate for the production-scale fermentor that will give results comparable with the pilot-scale data. 

The Aae criterion was applied to the problem of the piperidine equilibrium (10 11) by examining the H-NMR spectrum of the deutero derivatives 12, 13, and 14 at — 85°C when ring inversion is slow on the NMR time scale.37 The Aae values are recorded in Table I. The preferred conformation for the N-substituted piperidines 13 and 14 was assigned as lone-pair axial because for these compounds Aae values were similar to those in quinolizidine. However, in piperidine (12), where the Aae value is similar to that in cyclohexane, the lone pair was assumed to occupy the equatorial position in the preferred conformation. In support of this contention, protonation of... 

The cumulative probability functions listed in Table 2.1.6 can be used to give some measure of correlation between the various criteria for orbital size. Once a particular value for P is chosen, it can be equated to each of the expresssions in turn and the resulting transcendental equation solved numerically. The p values thus obtained for P = 0.50, 0.90, 0.95, and 0.99 are tabulated in Table 2.1.7. The significant quantites are the size ratios, with the Is p value as standard, which provide a rational scale of relative orbital size based on any adopted probability criterion. As the prescribed P value approaches unity, the size ratios gradually decrease in magnitude and orbitals in the same shell tend to converge to a similar size. 

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