Variational methods in the solution of certain dynamic optimization problems in operations research

Abstract

The objective of this research is to apply the powerful mathematical techniques of the classical calculus of variations and Pontryagin's maximum principle to the modeling and subsequent solution of dynamic problems in operations research. At present, dynamic problems are either approximated by static models, which are then solved by various mathematical programming techniques, or they are represented as staged, discrete-time models for solution by dynamic programming. While many dynamic problems are tractable through one or both of these approaches, others do not respond satisfactorily to either. The present study is presented in two distinct parts. The first part exhibits a compilation and condensation of the major theoretical results from the classical calculus of variations and from modern control theory. The terminology and notation are consistent with current usage in operations research and management science, and many worked example problems are included to illustrate the theory. Many of the proofs and derivations which usually accompany theoretical treatises in variational mathematics are omitted in the interests of clarity and utility. The second part consists of specific applications of variational mathematics in the solution of problems in production control. Two versions of a production phase-out problem and a production modernization problem are modeled and solved. Extensive parametric analysis is performed on the resulting solutions, and numerical examples are given in each case. Not only the optimal production schedule, but also the optimal cost functional for each problem appears as a closed-form function with all parameters intact. Graphs depicting the optimal production-rate functions for the numerical examples are included. The final chapter contains a resume of the study and recommendations for further research topics in this area.