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Multiple variable problems

In a design situation there will be constraints on the possible values of the objective function, arising from constraints on the variables such as, minimum flow-rates, maximum allowable concentrations, and preferred sizes and standards. [Pg.27]

Some may be equality constraints, expressed by equations of the form  [Pg.27]

The problem is to find values for the variables v to vn that optimise the objective function that give the maximum or minimum value, within the constraints. [Pg.27]

The method of Lagrange s undetermined multipliers is a useful analytical technique for dealing with problems that have equality constraints (fixed design values). Examples of the use of this technique for simple design problems are given by Stoecker (1989), Peters and Timmerhaus (1991) and Boas (1963a). [Pg.27]

The nature of the relationships and constraints in most design problems is such that the use of analytical methods is not feasible. In these circumstances search methods, that require only that the objective function can be computed from arbitrary values of the independent variables, are used. For single variable problems, where the objective function is unimodal, the simplest approach is to calculate the value of the objective function at uniformly spaced values of the variable until a maximum (or minimum) value is obtained. Though this method is not the most efficient, it will not require excessive computing time for simple problems. Several more efficient search techniques have been developed, such as the method of the golden section see Boas (1963b) and Edgar and Himmelblau (2001). [Pg.28]


When the radial variation of temperature must be taken into account, the problem assumes an entirely different character. Each of the equations is now a partial differential equation, and both radial and axial profiles must be calculated a mesh or network of radial and axial lines is set up, and the temperature and composition are calculated for each intersection. A great deal of work has been done on the formulation of difference equations for solving the related diffusion or heat-conduction equations most of this has been directed towards the case in which there is only one dependent variable and in which the source is a linear function of that variable. Although the results obtained for one dependent variable are only partially applicable to the multiple-variable problem,... [Pg.236]

In the development of a SE-HPLC method the variables that may be manipulated and optimized are the column (matrix type, particle and pore size, and physical dimension), buffer system (type and ionic strength), pH, and solubility additives (e.g., organic solvents, detergents). Once a column and mobile phase system have been selected the system parameters of protein load (amount of material and volume) and flow rate should also be optimized. A beneficial approach to the development of a SE-HPLC method is to optimize the multiple variables by the use of statistical experimental design. Also, information about the physical and chemical properties such as pH or ionic strength, solubility, and especially conditions that promote aggregation can be applied to the development of a SE-HPLC assay. Typical problems encountered during the development of a SE-HPLC assay are protein insolubility and column stationary phase... [Pg.534]

A simpler and general discrete time scheduling formulation can also be derived by means of the Resource Task Network concept proposed by Pantelides [10], The major advantage of the RTN formulation over the STN counterpart arises in some problems involving many identical pieces of equipment. In these cases, the RTN formulation introduces a single binary variable instead of the multiple variables used by the STN model. The RTN-based model also covers all the features at the column on discrete time in Table 8.1. In order to deal with different types of resources in a uniform way, this approach requires only three different classes of constraints in terms ofthree types of variables defining the task allocation, the batch size, and the resource availability. Briefly, this model reduces the batch scheduling problem to a simple resource balance problem carried out in each predefined time period. [Pg.173]

The multiple-minimum problem is a severe handicap of many large-scale optimization applications. The state of the art today is such that for reasonable small problems (30 variables or less) suitable algorithms exist for finding all local minima for linear and nonlinear functions. For larger problems, however, many trials are generally required to find local minima, and finding the global minimum cannot be ensured. These features have prompted research in conformational-search techniques independent of, or in combination with, minimization.26... [Pg.16]

It is clear that Eq. (5.70) results from the general relation (5.3). In this case, when k= 2, we have a regression surface whereas, when k>2, a hypersurface is obtained. For surface or hypersurface constructions, we have to represent the corresponding values of the process parameters (factors and one dependent variable) for each axis of the phase s space. The theoretical starting statistical material for a multiple regression problem is given in Table 5.11. [Pg.363]

Another problem encoimtered with pEVA involves microscopy. The agarose surface on which the spores sit is often rmeven due to multiple variables in its preparation, meaning that not all... [Pg.36]

FIGURE 15.3. A systematic approach to troubleshooting a problem with multiple variables... [Pg.319]

The first steps of the algorithm are almost identical to the modified Bums and Carter (1985) algorithm of Section 4.1. Since the demand pattern has been restricted to being the same for each weekday and the same, but of a different value, for each weekend day, rather than totally variable, the extra condition of having pairs of adjacent days off for the type 74 workers can be incorporated. Surprisingly, no change in the size of the workforce is required for the multiple shift problem, and the same bounds can be used. [Pg.1758]

Physical intuition is needed in order to justify the fundamental relationships in step (a). Once the physical problem is converted into a mathematical one (step [b]), physical intuition is no longer needed and the gear must shift to mastering the how. At this point, a good handle of calculus becomes indispensable, in fact, a prerequisite for the successful completion of this material. Especially important is familiarity with functions of multiple variables, partial derivatives and path integrations. [Pg.27]

The purpose of the multiple-variable model is to provide an apprentice technician with a comprehensive view of the large scope of operations he or she will be exposed to in the chemical processing industry. When all of the equipment pieces are combined into a full-scale plant, it is easier to see how each system operates and the potential problems that troubleshooters will encounter. Nine of the troubleshooting models have been combined to make the multivariable model shown in Figure 17-15. [Pg.381]

This feature allows the system to determine a second type of group, reporting details of the accident, which could help point to evidence of origin of the errors. Especially for those accidents that have relation with a cognitive vector. Our study showed different scenarios when the accidents are correlated with multiple variables. This possibility, of course, is due to the ability of Aviation DataBase System (Martins, 2007, 2010), which allows the referred type of analysis. It is necessary to identify accurately the problems or errors that contribute to the pilots making it impossible to act properly. These problems could point, eventually, to an temporary incompetence of the pilot due to limited capacity or lack of training appropriateness of automation in aircraft. We must also consider many other reasons that can alleviate the effective participation or culpability of the pilot. Addressing these problems to a systemic view expands the frontiers of research and prevention of aircraft accidents. [Pg.384]

Another general characteristic of multistage columns is that they usually require controUing several interrelated variables using a number of interrelated manipulated variables. This is a multiple-input, multiple-output (MMO) problem as compared with the basic control action in a single-input, singleoutput (SISO) problem. In a multiple variable process each manipulated variable may affect more than one controlled variable due to process interactions. The multiple variable control problem in multistage columns may be handled either by multiple control loops or by a multivariable controller. [Pg.415]

Unconstrained optimization refers to the situation where there are no inequality constraints and all equality constraints can be eliminated by variable substitution in the objective function. First we consider single-variable optimization, followed by optimization problems with multiple variables. Because optimization techniques are iterative in nature, we focus mainly on efficient methods that can be applied on-line. Most RTO applications are multivariable... [Pg.373]

In engineering terms this is often referred to as a special multiple-level-multiple-variable optimisation problem. Multiple-level means that each of the parameters such as specification, comprises requirements, and is with varying degrees of complexity. Multiple-variable implies that there is more than one variable or factor involved. Optimisation aims... [Pg.250]

Step 1 To solve a Stokes flow problem by this program the inertia term in the elemental stiffness matrix should be eliminated. Multiplication of the density variable by zero enforces this conversion (this variable is identified in the program listing). [Pg.215]

Partial least-squares path modeling with latent variables (PLS), a newer, general method of handling regression problems, is finding wide apphcation in chemometrics. This method allows the relations between many blocks of data ie, data matrices, to be characterized (32—36). Linear and multiple regression techniques can be considered special cases of the PLS method. [Pg.426]

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). [Pg.429]

Selective and Override Control When there are more controlled variables than manipulated variables, a common solution to this problem is to use a selector to choose the appropriate process variable from among a number of available measurements. Selec tors can be based on either multiple measurement points, multiple final control elements, or multiple controllers, as discussed below. Selectors are used to improve the control system performance as well as to protect equipment from unsafe operating conditions. [Pg.733]

One operational problem with analog alarms is that noise in the variable can cause multiple alarms whenever its value approaches a limit. This can be avoided by defining a deadband on the alarm. For example, a high alarm would be processed as follows ... [Pg.769]

Composite materials have many distinctive characteristics reiative to isotropic materials that render application of linear elastic fracture mechanics difficult. The anisotropy and heterogeneity, both from the standpoint of the fibers versus the matrix, and from the standpoint of multiple laminae of different orientations, are the principal problems. The extension to homogeneous anisotropic materials should be straightfor-wrard because none of the basic principles used in fracture mechanics is then changed. Thus, the approximation of composite materials by homogeneous anisotropic materials is often made. Then, stress-intensity factors for anisotropic materials are calculated by use of complex variable mapping techniques. [Pg.343]


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