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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 ...

In the design of a chemical process (CP), certain design specifications (for example, those related to process economics, process performance, safety, and the environment) must be satisfied. During the operation of the plant, because design models have uncertainties associated with them, we need to ensure that, within the region of uncertainty, all design specifications are satisfied. In recent years, research has focused on the investigation of the process flexibility (Biegler et al. Systematic Methods of Chemical Process Design; Prentice Hall: Englewood Cliffs, NJ, 1997) based on the feasibility function (Halemane and Grossmann AIChE J. 1983, 29 (3), 425) which is a measure of the CP's ability to meet design specifications under uncertainty. Several researchers have proposed methods for calculating the process feasibility function, which involves solving a very complex multiextremal and nondifferentiable optimization problem. Current methods for calculation of the flexibility function use an enumeration procedure (explicit or implicit), which in the worst case can require a large number of iterations. To try to address this issue, in this paper, we have introduced an efficient approach, which avoids enumeration. Through examples, we have shown that the new method leads to a small number of iterations and has low CPU requirements.

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Two problems-calculation ojthe Jeasibility test and the two-stage optimization problem (TSOP)-arise in the design oj engineering systems (chemical processes, electrical circuits) under conditions of uncertainty ojoriginal inJormation, and are considered here. The solution oj the first problem allows an estimate oj the ability oj an engineering system to preserve its capacity [or work under changing external and internal factors during operations. Solving the TSO P permits an engineering system to preserve its capacity Jor work under inexact knowledge oj model coefficients. Directly solving both problems requires the use oj multiextremal non-differentiable optimization methods. In this paper, methods oj solving both problems are suggested, which use only chemical methods of nonlinear programming For calculation of the feasibility test we propose an algorithm based on a spatial branch-and-bound method. Also, an efficient procedure for calculation oj an upper estimation oj[easibility test is developed. Two algorithms for solving TSO P are given. A distinctive feature oj the algorithms is that during execution the upper and the lower estimates oj the optimal value oj an objective Junction oj TSO P are calculated. The approach is based on the concept oj a 'branch-and-bound' method.