Reference point method

In reference point based methods, the DM first specifies a reference point z S consisting of desirable aspiration levels for each objective and then this reference point is projected onto the Pareto optimal set. That is, a Pareto optimal solution closest to the reference point is found. The distance can be measured in different ways. Specifying a reference points is an intuitive way for the DM to direct the search of the most preferred solution. It is straightforward to compare the point specified and the solution obtained without artificial concepts. Examples of methods of this t rpe are the reference point method and the light beam search . 

The reference point method is based on using a so-called achievement (scalarizing) function (Wierzbicki, 1982). The achievement function measures the distance between the reference point and Pareto optimal solutions and produces a new Pareto optimal solution closest to the reference point. The beauty here is that Pareto optimal solutions are generated no matter how the reference point is specified, that is, they can be attainable or not. We have an example of an achievement function in the problem... 

In the reference point method, the DM specifies a reference point and the corresponding solution of (6.4) is shown to him/her. In addition, the DM is shown k other solutions obtained by slightly shifting the reference point in each coordinate direction. Thus, the DM can compare k + Pareto optimal solutions close to the reference point. Then the DM can set a new reference point (i.e., adjust the reference point according to his/her preferences) and the solution process continues as long as the DM wants to. When the Pareto optimal solutions are generated, the DM learns more about the possibilities and limitations of the problem and, therefore, can use more appropriate reference points. 

Another reference point based method is the light beam search (Jaszkiewicz and Slowinski, 1999). It uses a similar achievement function as the reference point method but combined with tools of multiattribute decision analysis (designed for comparing a discrete set of solutions). Besides a reference point, the DM must supply thresholds for objective functions describing indifference and preference in objective values. This information is used to derive outranking relations between solutions. As a result, incomparable or indifferent solutions are not displayed to the DM. 

In this section, the URP (updated reference-point) method proposed originally for stochastic input is explained (Fujita and Takewaki 2011a). This method can be used as an efficient uncertainty analysis to obtain the robustness function a explained in the previous section. Since the URP method takes full advantage of an approximation of first- and second-order Taylor series expansion in the interval analysis, the formulation of Taylor series expansion in the interval analysis and the achievement of second-order Taylor series expansion proposed by Chen et al. (2009) are explained briefly. 

This is the same as Eq. 12. The evaluation of first- and second-order sensitivities/ x- and/ x-Xi is made at the nominal model (reference point). It should be noted that the correlation among interval parameters is not taken into account. Here a method called the lIRP (updated reference-point) method (Fujita and Takewaki 2011a) is used. This method changes the reference point step by step. The flowchart for finding the upper bound of the objective function is shown in Fig. 17a. 

H.H. Rachford (1966) Numerical calculation of immiscible displacement by a moving reference point method. Society of Petroleum Engineers Journal, June 1966, 87-101. 

With the reference block method the distance law of a model reflector is established experimentally prior to each ultrasonic test. The reference reflectors, mostly bore holes, are drilled into the reference block at different distances, e.g. ASME block. Prior to the test, the reference reflectors are scanned, and their maximised echo amplitudes are marked on the screen of the flaw detector. Finally all amplitude points are connected by a curve. This Distance Amplitude Curve (DAC) serves as the registration level and exactly shows the amplitude-over-distance behaviour" of the reference reflector for the probe in use. Also the individual characteristics of the material are automatically considered. However, this curve may only be applied for defect evaluation, in case the reference block and the test object are made of the same material and have undergone the same heat treatment. As with the DGS-Method, the value of any defect evaluation does not consider the shape and orientation of the defect. The reference block method is safe and easy to apply, and the operator need not to have a deep understanding about the theory of distance laws. 

Table II, 9, A.—Reference Substances for the Calibration of Thermometers BY THE Melting Point Method...

A modified definition of resonance energy has been introduced by Dewar (66T(S8)75, 69JA6321) in which the reference point is the corresponding open-chain polyene. In principle this overcomes the difficulties inherent in comparing observed stability with that of an idealized molecule with pure single and double bonds, as thermochemical data for the reference acyclic polyenes are capable of direct experimental determination. In practice, as the required data were not available, recourse was made to theoretical calculations using a semiempirical SCF-MO method. The pertinent Dewar Resonance Energies are listed in Table 30. 

For nonKydi ocai bon organics for which normal boiling points are unknown or expected vapor pressures are below 15 kPa, the reference substance method of Othmer and YiF as given by Eq. (2-44) is recommended. 

Transport numbers are intended to measure the fraction of the total ionic current carried by an ion in an electrolyte as it migrates under the influence of an applied electric field. In essence, transport numbers are an indication of the relative ability of an ion to carry charge. The classical way to measure transport numbers is to pass a current between two electrodes contained in separate compartments of a two-compartment cell These two compartments are separated by a barrier that only allows the passage of ions. After a known amount of charge has passed, the composition and/or mass of the electrolytes in the two compartments are analyzed. Erom these data the fraction of the charge transported by the cation and the anion can be calculated. Transport numbers obtained by this method are measured with respect to an external reference point (i.e., the separator), and, therefore, are often referred to as external transport numbers. Two variations of the above method, the Moving Boundary method [66] and the Eiittorff method [66-69], have been used to measure cation (tR+) and anion (tx ) transport numbers in ionic liquids, and these data are listed in Table 3.6-7. 

Although the p.z.c. is difficult to determine experimentally, and although the values obtained vary with the method used, it is of fundamental significance in electrochemistry, since it provides information on adsorption of ions and molecules, i.e. if the potential is negative with respect to the p.z.c. cations will tend to be adsorbed and anions repelled, and vice versa. The p.z.c. appears to be a natural reference point for a rational scale of potentials defined by... 

Preliminary research has shown that Brillouin fiber-optic sensing systems provide a possible method to detect leaks and third-party intrusion on a pipeline over distances of 25 km or more. Their intrinsic response to both temperature and mechanical strain allows for the separation of these parameters and the detection of anomalies in the scan profiles. In addition, the same sensor could be integrated into the pipeline system to detect possible ground movement relative to fixed reference points. Limited test results on surface loads associated with the intrusion of vehicles and people on a pipeline have demonstrated the sensitivity of the system and its ability to discriminate loads at different soil depths. 

Lebel [224] has described an automated chelometric method for the determination of sulfate in seawater. This method utilises the potentiometric end-point method for back titration of excess barium against EDTA following precipitation of sulfate as barium sulfate. An amalgamated silver electrode was used in conjunction with a calomel reference electrode in an automatic titration assembly consisting of a 2.5 ml autoburette and a pH meter coupled to a recorder. Recovery of added sulfate was between 99 and 101%, and standard deviations of successive analyses were less than 0.5 of the mean. 

Methods of experimental design discussed in most basic statistics books can be applied equally well to minimizing fix) (see Chapter 2). You evaluate a series of points about a reference point selected according to some type of design such as the ones shown in Figure 6.1 (for an objective function of two variables). Next you move to the point that improves the objective function the most, and repeat. 

Part—III exclusively treats Electrochemical Methods invariably and extensively used in the analysis of pharmaceutical substances in the Official Compendia. Two important methods, namely potentiometric methods (Chapter 16) deal with various types of reference electrodes and indicator electrodes, automatic titrator besides typical examples of nitrazepam, allopurinol and clonidine hydrochloride. Amperometric methods (Chapter 17) comprise of titrations involving dropping-mercury electrode, rotating—platinum electrode and twin-polarized microelectrodes (i.e., dead-stop-end-point method). 

Calculation, thermodynamic optimization of phase diagrams. The knowledge of phase equilibria, phase stability, phase transformations is an important reference point in the description and understanding of the fundamental properties of the alloys and of their possible technological applications. This interest has promoted a multi-disciplinary and multi-national effort dedicated not only to experimental methods, but also to techniques of optimization, calculation and prediction of... 

Points with a constant Euclidean distance from a reference point (like the center) are located on a hypersphere (in two dimensions on a circle) points with a constant Mahalanobis distance to the center are located on a hyperellipsoid (in two dimensions on an ellipse) that envelops the cluster of object points (Figure 2.11). That means the Mahalanobis distance depends on the direction. Mahalanobis distances are used in classification methods, by measuring the distances of an unknown object to prototypes (centers, centroids) of object classes (Chapter 5). Problematic with the Mahalanobis distance is the need of the inverse of the covariance matrix which cannot be calculated with highly correlating variables. A similar approach without this drawback is the classification method SIMCA based on PC A (Section 5.3.1, Brereton 2006 Eriksson et al. 2006). 

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