Fuzzy geometric programming techniques and applications.

*(English)*Zbl 1420.90001
Forum for Interdisciplinary Mathematics. Singapore: Springer (ISBN 978-981-13-5822-7/hbk; 978-981-13-5823-4/ebook). xxi, 359 p. (2019).

This book is concerned with a timely and well-established area of geometric programming (GP) augmented by mechanisms of fuzzy sets. Fuzzy sets are involved in the formulation of the generic programming problem \(\text{Min}\,f(x)\) under the constraints described as monomials or posynomials, \(x>0\). Their introduction reflects the aspect of uncertainty present in real-world environment and thus makes the formulated optimization task better aligned with the essence of the problem.

The book is organized in a coherent way. It exposes the reader to some introductory material covered mostly in Chapters 1–4. Chapters 1 and 2 form a prerequisite to concepts of geometric programming and signomial geometric programming. Chapter 3 offers a brief introduction to fuzzy sets. Chapter 4 is predominantly focused on fuzzy numbers and their arithmetic. Various optimization problems are discussed in the consecutive chapters: unconstrained GP programming (Chapter 5 and 6), constrained GP programming (Chapters 7 and 8), fuzzy signomial GP, and goal programming (Chapter 10). Furthermore discussed are problems of GP under uncertainty (Chapter 12) and intuitionistic and neutrosophic GP (Chapter 13).

The material is well illustrated by a suite of carefully structured and convincing examples, where the role of fuzzy sets is apparent and aligned with the modeling purposes as well as the specificity of the problem at hand. Perhaps some discussion on constructing fuzzy sets could have offered a more rounded exposure of the application-oriented side of the material. One can also be aware that some formulas presented in the book are only approximations – a fact that has not been noted by the authors (e.g,. the result of multiplication and division of two triangular fuzzy numbers is not a triangular fuzzy number, see Chapter 4, page 78). The references are sufficient albeit in some cases a bit outdated.

As the whole, the book offers a good introduction to the subject and reports on the progress in the area.

The book is organized in a coherent way. It exposes the reader to some introductory material covered mostly in Chapters 1–4. Chapters 1 and 2 form a prerequisite to concepts of geometric programming and signomial geometric programming. Chapter 3 offers a brief introduction to fuzzy sets. Chapter 4 is predominantly focused on fuzzy numbers and their arithmetic. Various optimization problems are discussed in the consecutive chapters: unconstrained GP programming (Chapter 5 and 6), constrained GP programming (Chapters 7 and 8), fuzzy signomial GP, and goal programming (Chapter 10). Furthermore discussed are problems of GP under uncertainty (Chapter 12) and intuitionistic and neutrosophic GP (Chapter 13).

The material is well illustrated by a suite of carefully structured and convincing examples, where the role of fuzzy sets is apparent and aligned with the modeling purposes as well as the specificity of the problem at hand. Perhaps some discussion on constructing fuzzy sets could have offered a more rounded exposure of the application-oriented side of the material. One can also be aware that some formulas presented in the book are only approximations – a fact that has not been noted by the authors (e.g,. the result of multiplication and division of two triangular fuzzy numbers is not a triangular fuzzy number, see Chapter 4, page 78). The references are sufficient albeit in some cases a bit outdated.

As the whole, the book offers a good introduction to the subject and reports on the progress in the area.

Reviewer: Witold Pedrycz (Edmonton)

##### MSC:

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90C30 | Nonlinear programming |

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

90C29 | Multi-objective and goal programming |

03E72 | Theory of fuzzy sets, etc. |

90C10 | Integer programming |

65K05 | Numerical mathematical programming methods |

90B05 | Inventory, storage, reservoirs |