Abstract
Multiple optimum solutions of a multistage allocation problem, well-known to chemical engineers, are analyzed. The number of local optima becomes greater with a decrease in the initial-condition value of the first stage or with an increase in the total stage number. The fact that this behavior is closely related to the flat portion of the profile of a curvef(x), which determines the objective function, is revealed. A construction method by Aris is used to give an excellent insight into this behavior. Moreover, the construction curves ensure that all stationary points are found. Finally, a theorem to discriminate local optima from stationary points, without evaluating second-order derivatives, is presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Takamatsu, T., Sayama, H., andOi, K.,Optimal Design of Extraction Processes by Second-Order Convergence Gradient Methods, Seigyo Kogaku (Journal of the Japan Association of Automatic Control Engineers), Vol. 13, pp. 313–321, 1969.
Caillaud, J. B., andPadmanabhan, L.,An Optimization Technique for a Class of Discrete Allocation Problems, Chemical Engineering Journal, Vol. 3, pp. 14–21, 1972.
Aris, R., Rudd, D. F., andAmundson, N. R.,On Optimum Cross-Current Extraction, Chemical Engineering Science, Vol. 12, pp. 88–97, 1960.
Zahradnik, R.L., andArcher, D. H.,Application of the Discrete Maximum Principle to Cross-Current Extraction, Industrial and Engineering Chemistry Fundamentals, Vol. 2, pp. 238–240, 1963.
Wang, C. S., andFan, L. T.,Optimization of Some Multistage Chemical Processes, Industrial and Engineering Chemistry Fundamentals, Vol. 3, pp. 38–42, 1964.
Lee, E. S.,Optimization by a Gradient Technique, Industrial and Engineering Chemistry Fundamentals, Vol. 3, pp. 373–380, 1964.
Takamatsu, T., Nakanishi, E., andHashimoto, I.,Optimal Control and Its Sensitivity for a Multi-Stage Process, 3rd IFAC Congress, Paper No. 19-F, London, England, 1966.
Oi, K., Sayama, H., andTakamatsu, T.,An Extension of the Davidon-Fletcher-Powell Method to Sensitivity Analaysis, Journal of Chemical Engineering of Japan, Vol. 5, pp. 413–417, 1972.
Aris, R.,The Optimal Design of Chemical Reactors: A Study in Dynamic Programming, Academic Press, New York, New York, 1961.
Aris, R.,Elementary Chemical Reactor Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.
Nishitani, H., Hashimoto, I., andTakamatsu, T.,A Consideration of Multimodality of a Multistage Allocation Problem, 7th Autumn Meeting of the Society of Chemical Engineers, Paper No. B-311, Nagoya, Japan, 1973.
Kitagawa, H., Watanabe, N., Nishimura, Y., andMatsubara, M.,Noninferior Solution of a Bicriterion Programming Problem of a Multistage Extraction Process, 10th Autumn Meeting of the Society of Chemical Engineers, Paper No. B-210, Nagoya, Japan, 1976.
Oi, K., Sayama, H., andTakamatsu, T.,Some Considerations of a Class of Multistage Allocation Problems, Journal of Chemical Engineering of Japan, Vol. 7, pp. 304–309, 1974.
Author information
Authors and Affiliations
Additional information
Communicated by R. Jackson
The authors would like to thank Dr. I. Hashimoto and Dr. H. Nishitani for valuable discussions. Computations were carried out with the assistance of Messrs. H. Unno, H. Nakano, Y. Era, and Y. Ueno. The authors are indebted to the computing centers of Osaka University, Kyoto University, and Nagoya University for the use of their facilities.
Rights and permissions
About this article
Cite this article
Oi, K., Sayama, H. & Takamatsu, T. Multimodality analysis of a class of multistage allocation problems. J Optim Theory Appl 25, 33–48 (1978). https://doi.org/10.1007/BF00933253
Issue Date:
DOI: https://doi.org/10.1007/BF00933253