هزینه های کیفیت و معیارهای مستحکم در بهینه سازی طراحی فرایند شیمیایی

The identification and incorporation of quality costs and robustness criteria is becoming a critical issue while addressing chemical process design problems under uncertainty. This article presents a systematic design framework that includes Taguchi loss functions and other robustness criteria within a single-level stochastic optimization formulation, with expected values in the presence of uncertainty being estimated by an efficient cubature technique. The solution obtained defines an optimal design, together with a robust operating policy that maximizes average process performance. Two process engineering examples (synthesis and design of a separation system and design of a reactor and heat exchanger plant) illustrate the potential of the proposed design framework. Different quality cost models and robustness criteria are considered, and their influence in the nature and location of best designs systematically studied. This analysis reinforces the need for carefully considering/addressing process quality and robustness related criteria while performing chemical process plant design

At the design stage of a process system, decisions have to be made in the presence of high uncertainty level. For instance, equipment configuration and dimensions, and their operating conditions have to be decided on the basis of an available process model, whose parameters may be uncertain, and on external information, which commonly exhibits a random behavior. Taguchi (1986) approach to quality engineering provides a robust design strategy aimed at determining nominal settings for the design variables (parameter design) and their associated tolerance limits (tolerance design), in order to reduce process sensitivity to uncertainty. The traditional Taguchi methodology, which is based on running statistically designed experiments on a process prototype, is not, however, directly applicable to early process system design. On the other hand, process design and optimization under uncertainty (Pistikopoulos, 1995) offers a systematic optimization-based vehicle to address process system design issues in the presence of uncertainty. However, in most such optimization studies, robustness issues are not explicitly considered, although attempts to link robustness/quality engineering aspects to stochastic process design optimization have begun to appear in the literature (Straub & Grossmann, 1993, Diwekar & Rubin, 1994, Bernardo & Saraiva, 1998, Samsatli, Papageorgiou & Shah, 1998 and Georgiadis & Pistikopoulos, 1999). In this article, we introduce a systematic design framework for process quality that embeds Taguchi's method and other robustness criteria within a stochastic optimization formulation. Quality related constraints are relaxed and process robustness is guaranteed through the explicit incorporation of robustness criteria in the optimization formulation, such as penalty terms in the objective function and/or limits on the variance of quality variables. With the relaxation strategy mentioned above, feasibility tests are not required, and thus the objective function expected value is obtained through integration over the entire uncertainty space. As a consequence, the original two-stage optimization problem was transformed into a single-level stochastic optimization formulation. The computation of multiple integrals over the uncertainty space is a critical numerical issue in stochastic process design. Integration techniques applied so far to this kind of problems include Gaussian quadrature and stratified sampling techniques. In the first case, the number of points where the integrand function need to be evaluated increases exponentially with the integral dimension (the number of uncertain parameters), making the problem untreatable for a reasonably large number of uncertain parameters (Pistikopoulos & Ierapetritou, 1995). On the other hand, sampling techniques may be computationally more attractive, since the number of points required does not necessarily increase with the number of uncertain parameters. However, even the most efficient sampling techniques, such as the Hammersley sequence sampling (HSS) introduced by Diwekar & Kalagnanam, 1997a and Diwekar & Kalagnanam, 1997b require some hundreds of points to achieve a reasonable accuracy. At the numerical level, the present work employs a cubature technique (Stroud, 1971) to compute the multiple integrals involved in the stochastic problem formulation. When all uncertain parameters are normally distributed, a specialized cubature formula is applied, reducing significantly the number of points needed when compared with other integration strategies, such as product Gauss rules or efficient sampling techniques (Bernardo & Saraiva, 1998). The remaining parts of this paper are structured as follows. First, the proposed mathematical problem formulation, addressing process quality, is developed, based upon a two-stage stochastic optimization framework. Next, robustness criteria and their implementation are described in more detail. Finally, the proposed formulation is illustrated through two chemical process design examples (synthesis and design of a separation system and design of a reactor and heat exchanger system).

We have developed a stochastic optimization framework for conducting process design under uncertainty that takes explicitly into account process robustness and product quality issues. Although the formulation is based in a two-stage approach, under the assumption of perfect information and control during process operation, the design problem has been formulated as a single-level stochastic optimization problem, where a Taguchi's perspective of continuous quality loss is adopted. The incorporation of different robustness criteria and their applicability has also been discussed in the context of process engineering applications. An efficient cubature technique, suitable to integrate normally distributed uncertainties, has been applied for the estimation of multiple integrals involved in the formulation, reducing significantly the computational effort required, when compared with other integration methodologies, such as product Gauss rules or stratified sampling techniques. The potential of the proposed design framework is illustrated with two process engineering application examples, where different robustness criteria are studied — nominal-the-best Taguchi loss functions, one-sided Taguchi loss functions, maximum variance and quantile constraints for a quality related variable. From such case studies, one can see quite clearly what implications and consequences are derived from considering different models to express quality related issues while addressing chemical process design problems.