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Dynamic programming has long been applied to numerous areas in mathematics, science, engineering, business, medicine, information systems, biomathematics, artificial intelligence, among others. Applications of dynamic programming have increased as recent advances have been made in areas such as neural networks, data mining, soft computing, and other areas of computational intelligence. The value of dynamic programming formulations and means to obtain their computational solutions has never been greater. This book describes the use of dynamic programming as a computational tool to solve discrete optimization problems.
(1) We first formulate large classes of discrete optimization problems in dynamic programming terms, specifically by deriving the dynamic programming functional equations (DPFEs) that solve these problems. A text-based language, gDPS, for expressing these DPFEs is introduced. gDPS may be regarded as a high-level specification language, not a conventional procedural computer programming language, but which can be used to obtain numerical solutions.
(2) We then define and examine properties of Bellman nets, a class of Petri nets that serves both as a formal theoretical model of dynamic programming problems, and as an internal computer data structure representation of the DPFEs that solve these problems.
(3)We also describe the design, implementation, and use of a software tool, called DP2PN2Solver, for solving DPFEs. DP2PN2Solver may be regarded as a program generator, whose input is a DPFE, expressed in the input specifi- cation language gDPS and internally represented as a Bellman net, and whose output is its numerical solution that is produced indirectly by the generation of solver code, which when executed yields the desired solution. This book should be of value to different classes of readers: students, instructors, practitioners, and researchers. We first provide a tutorial introduction to dynamic programming and to Petri nets. For those interested in dynamic programming, we provide a useful software tool that allows them to obtain numerical solutions. For researchers having an interest in the fields of dynamic programming and Petri nets, unlike most past work which applies dynamic programming to solve Petri net problems, we suggest ways to apply Petri nets to solve dynamic programming problems.
For students and instructors of courses in which dynamic programming is taught, usually as one of many other problem-solving methods, this book provides a wealth of examples that show how discrete optimization problems can be formulated in dynamic programming terms. Dynamic programming has been and continues to be taught as an art, where how to use it must be learned by example, there being no mechanical way to apply knowledge of the general principles (e.g., the principle of optimality) to new unfamiliar problems. Experience has shown that the greater the number and variety of problems presented, the easier it is for students to apply general concepts. Thus, one objective of this book is to include many and more diverse examples. A further distinguishing feature of this book is that, for all of these examples, we not only formulate the DP equations but also show their computational solutions, exhibiting computer programs (in our specification language) as well as providing as output numerical answers (as produced by the automatically generated solver code).
Dynamic Programming.
Introduction to Dynamic Programming.
Applications of Dynamic Programming.
Modeling of DP Problems.
The DP Specification Language gDPS.
DP Problem Specifications in gDPS.
Bellman Nets: A Class of Petri Nets.
Bellman Net Representations of DP Problems.
Design and Implementation of DP Tool.
DP2PN2Solver Tool.
The PN2Solver Modules.
Computational Results.
Java Solver Results of DP Problems.
Other Solver Results.
Conclusions.
A Supplementary Material.
B User Guide for DP2PN2Solver