Flexibility analysis is of prime importance in chemical process systems, since it permits the creation of chemical processes, which can satisfy all design specifications in spite of changes in internal and external factors during the operation stage. The modern foundations of flexibility analysis were de®eloped in the 1980s by Grossman and coworkers. They formulated solution approaches for the main problems of flexibility ( ) analysis feasibility test , flexibility index, and the two-stage optimization problem. All the formulations are based on the supposition that during the operation stage there are enough experimental data to accurately determine uncertain parameter ®alues, but in practice, these assumptions are not likely to be met. This article discusses extensions of ( ) the feasibility test and the two-stage optimization problem TSOP , which take into account the possibility of accurately estimating some of the uncertain parameters while estimating with less accuracy the remaining uncertain parameters. To sol®e the TSOP, the split and bound approach was de®eloped based on a partitioning of the uncertainty region and estimation of bounds on the performance objecti®e function. Three computa-( tional experiments show the importance of taking into account the possibility or lack) thereof of obtaining ®alues of greater accuracy for uncertain parameters at the operation stage. IntroductionDesign specifications usually have to be met during the Ž . design and synthesis of chemical processes CP . However, the satisfaction of such design specifications is complicated by the presence of uncertainty in the process model parameters. The issue then is how could one guarantee satisfaction of all design specifications at the operation stage? Flexibility analysis addresses this problem. The first articles in flexibility Ž analysis appeared in the 1970s Takamatsu et al., 1973; . Grossmann and Sargent, 1978;Johns et al., 1978 . However, foundations of the modern theory of flexibility analysis were Ž laid in the 1980s Halemane and Grossmann, 1983; Swaney . and Grossmann; 1985;Grossmann and Floudas, 1987 . For-Ž mulations of the feasibility test and the flexibility index all . subproblems of flexibility analysis , and the two-step optimization problem were given in these articles. In addition, some solution approaches were suggested.Correspondence concerning this article should be addressed to L. E. K. Achenie. where x is a vector of state variables, z is a vector of control variables, and is a vector of uncertain variables. Equation 5 Ž describes the state of the CP i.e., material and energy bal-. ances , while the inequalities in Eq. 6 are design specifications. Here, dim hsdim x and x can be obtained from Eq. 5, either analytically or numerically using fixed values of the variables d, z, . Thus With the preceding definitions, the two-stage optimization Ž . Ž problem designated here as TSOP1 is given as Halemane . and Grossmann, 1983Ž . Ž . 1Ä 4 where E . . . is the mathematical expectation over the re-T Ž .U Ž . gion ...
Process uncertainty is almost always an issue during the design of chemical processes (CP). In the open literature it has been shown that consideration of process uncertainties in optimal design necessitates the incorporation of process flexibility. Such an optimal design can presumably operate reliably in the presence of process and modeling uncertainty. Halemane and Grossmann (1983) introduced a feasibility function for evaluating CP flexibility. They also formulated a two-stage optimization problem for estimating the optimal design margins. These formulations, however, are based implicitly on the assumption that during the operation stage, uncertain parameters can be determined with enough precision. This assumption is rather restrictive and is often not met in practice. When available experimental information at the operation stage does not allow a more precise estimate of some of the uncertain parameters, new formulations of the flexibility condition and the optimization problem under uncertainty are needed. In this article, we propose such formulations, followed by some computational experiments.
Optical fibers are used in diverse applications ranging from ocean and terrestrial cables to remote sensors and to light guides in medical applications. The information and telecommunications industries are other fields where optical fibers are actively used. However, the full potential of optical fibers has not been realized because of performance and production limitations. In the manufacturing process, high draw speeds result in large temperature gradients, which often lead to inferior quality fiber with poor optical and mechanical properties. On the other hand, low draw speeds lead to superior quality fiber with better optical and mechanical properties. Here, quality is achieved at the expense of a reduced rate of production. Therefore, the realization of good quality fiber without unduly compromising the production throughput can be assessed through the solution of an optimization model. The objective is to maximize the fiber draw speed subject to constraints on fiber mechanical and transmission properties. Additional complications arise in the optimization problem when uncertainties in process and model parameters are accounted for.
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