Design problem formulation

The minimal cost design problem formulated above was solved by Bickel et al. (B7) using the generalized reduced gradient (GRG) method of Abadie and Guigou (H4). If x and u are vectors of state and decision (independent) variables and u) is the objective function in a minimization subject to constraints [Eq. (90)], then the reduced gradient d/du is given by... 

L. Constantine, K. Bagherpour, R. Gani, J.A. Klein D.T. Wu, 1996, Computer Aided product design Problem formulations, methodology and applications, Computers Chemical Engineering, 20 (6/7), 685-702. 

Constantinou, L., Gani, R., Fredenslund, Aa., Klein, J. A., and Wu, D. J., Computer-aided product design, problem formulation and application. Proc. PSE 94, Kyongju, Korea (1994). 

This is the one serious limitation in plastic design problems. Even if the designer did wait for data on one material, chances are the final design might be switched to another plastic or formulation. Thus, as a compromise, data from relatively short-term tests are extrapolated by means of theory to long-term problems. However, when this is done, the limitations inherent in the procedure should be kept in mind. 

Example 4.2 used the method of false transients to solve a steady-state reactor design problem. The method can also be used to find the equilibrium concentrations resulting from a set of batch chemical reactions. To do this, formulate the ODEs for a batch reactor and integrate until the concentrations stop changing. This is illustrated in Problem 4.6(b). Section 11.1.1 shows how the method of false transients can be used to determine physical or chemical equilibria in multiphase systems. 

In order to define and compare in a concise way the different problem formulations, let s designate as... 

As will be seen later, these techniques will prove to be useful when solving design problems in general-purpose software, such as spreadsheets. Many of the numerical problems associated with optimization can be avoided by appropriate formulation of the model. Further details of model building can be found elsewhere12. 

Publications on optimal design of tree networks are further divided into single-branch trees or pipelines (C6, F4, L3, L6, S8) and many-branch trees (B7, C7, F4, Kl, K2, M3, M9, Nl, R5, W10, Y1, Zl). For our purposes, since the pipeline problems can always be solved using the optimization methods developed for the many-branch tree networks, we need to dwell no further on this special case. On the other hand, it is important to note that the form of the objective function could influence the applicability of a given optimization method. For the sake of concreteness, problem formulations and optimization techniques will be discussed in the context of applications. 

In Section 1.5 we briefly discussed the relationships of equality and inequality constraints in the context of independent and dependent variables. Normally in design and control calculations, it is important to eliminate redundant information and equations before any calculations are performed. Modem multivariable optimization software, however, does not require that the user clearly identify independent, dependent, or superfluous variables, or active or redundant constraints. If the number of independent equations is larger than the number of decision variables, the software informs you that no solution exists because the problem is overspecified. Current codes have incorporated diagnostic tools that permit the user to include all possible variables and constraints in the original problem formulation so that you do not necessarily have to eliminate constraints and variables prior to using the software. Keep in mind, however, that the smaller the dimensionality of the problem introduced into the software, the less time it takes to solve the problem. 

You can now see why for line A a branch-and-bound technique is not required to solve the design problem. Because of the way the objective function is formulated, if the ratio (pd ps) = 1, the term involving compressor i vanishes from the first summation in the objective function. This outcome is equivalent to the deletion of compressor i in the execution of a branch-and-bound strategy. (Of course the pipeline segments joined at node i may be of different diameters.) But when... 

Cheng L, Subrahmanian E, Westerberg AW (2003) Design and planning under uncertainty issues on problem formulation and solution. Computers Chemical Engineering 27 (6) 781-801... 

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