Fuzzy Geometric Programming Techniques and Applications

Description

Nonlinear analysis plays an ever-increasing role in theoretical and applied mathematics, as well as in many other areas of science such as engineering, statistics, computer science, economics, finance, and medicine. Most of the problems in our real life are nonlinear. There are many techniques used for solving nonlinear problems. Geometric programming is one of the best techniques to solve this nonlinear optimization problem. Geometric programming was introduced in 1967 by Duffin, Peterson, and Zener. It is very useful in the applications of a variety of optimization problems and falls under the general class of signomial problems. Geometric programming is a special method used to solve a class of nonlinear programming problems, mainly to solve optimal design problems where we minimize cost and/or weight, maximize volume and/or efficiency, etc. It is an important technique to solve the special type of nonlinear optimization problems. The global optimum of a convex problem can be achieved more quickly than the result of a nonlinear problem. Since its inception, geometric programming has been closely associated with applications in engineering analysis and design problem. Uncertainty in the problem data often cannot be avoided when dealing with practical problems. Errors occur in real-world data for a host of reasons. However, over the last 30 years, the fuzzy set approach has been proved to be useful in such situations. Thousands of research papers on geometric programming problem are published under the fuzzy environment, but only one book, Fuzzy Geometric Programming, by B. Y. Cao of geometric programming under fuzzy environment is published, which mainly focuses on theoretical aspects. Organized into 13 chapters, this book discusses an overall concept of geometric programming, goal geometric programming, and multi-objective geometric programming problem under a crisp and fuzzy environment. The main aim of this book is to develop the concepts of some optimization techniques: geometric programming, modified geometric programming, fuzzy geometric programming, fuzzy modified geometric programming, signomial geometric programming, goal programming, and fuzzy multi-objective geometric programming. In Chap. 1, we have presented geometric programming, modified geometric programming problem, constrained geometric programming problem, and modified geometric programming problem. In Chap. 2, we have presented convexity of signomial functions, unconstrained nonlinear programming problem, primal modified signomial geometric programming problem, unconstrained modified signomial function, constrained nonlinear programming problem and application of problem, and signomial geometric programming technique. In Chap. 3, we have described uncertainty and imprecision, basic fuzzy set theory, operations on fuzzy sets, geometrical interpretation of fuzzy sets, fuzzification, extension principle and its application, definition (extension principle), Cartesian product, and definition (extension principle on the n-dimensional universe). In Chap. 4, we have discussed fuzzy number, generalized fuzzy number, generalized trapezoidal fuzzy number, integral value, fuzzy number and its nearest interval approximation, fuzzy equation, and fuzzy optimization. In Chap. 5, we have presented unconstrained geometric programming problem with fuzzy parametric interval-valued function, geometric programming problem with simple fuzzy parametric coefficients, and geometric programming problem with Zimmermann max-min operators. In Chap. 6, we have studied the unconstrained modified geometric programming problem with fuzzy parametric interval-valued function, unconstrained MGP problem with simple fuzzy parametric coefficients, and unconstrained MGP problem with Zimmermann max-min operator. In Chap. 7, we have discussed constrained geometric programming problem with a fuzzy coefficient, fuzzy parametric geometric programming, and constrained geometric programming under max-min operator. In Chap. 8, we have analyzed constrained modified geometric programming problem with a fuzzy coefficient, fuzzy parametric modified geometric programming, and fuzzy modified geometric programming. In Chap. 9, we have developed an introduction to fuzzy signomial geometric programming problem, unconstrained problem, and constrained problem. In Chap. 10, we have discussed an introduction to goal programming, strength and weakness of goal programming, the importance of weighted goal programming, Chebyshev goal programming model, multi-objective problem, goal programming with logarithmic deviational variable, and fuzzy goal programming problem. In Chap. 11, we have presented an introduction to nonlinear programming function, fuzzy nonlinear programming technique, multi-objective optimization, and inventory models through fuzzy multi-objective geometric programming approach. In Chap. 12, we have presented an uncertain chance-constrained geometric programming model, geometric programming approach under zigzag uncertainty distribution, and multi-objective geometric programming problem under uncertainty. Lastly, in Chap. 13, we have studied intuitionistic fuzzy posynomial geometric programming problem, intuitionistic fuzzy goal programming model, and neutrosophic goal geometric programming problem. Geometric programming has applications across a variety of fields from engineering to economics and will continue to be useful in the future. Geometric programming is an essential course at the postgraduate level in mathematics, applied mathematics, engineering, as well as at the research level in many universities. The topic also forms a course to postgraduate students in engineering in reliability, operations research, circuit design, mathematics optimizations, and engineering problems. Front Matter ....Pages i-xxi Preliminary Concepts of Geometric Programming (GP) Model (Sahidul Islam, Wasim Akram Mandal)....Pages 1-25 Signomial Geometric Programming (GP) Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 27-45 Introduction to Fuzzy Set Theory (Sahidul Islam, Wasim Akram Mandal)....Pages 47-73 Fuzzy Numbers and Fuzzy Optimization (Sahidul Islam, Wasim Akram Mandal)....Pages 75-131 Fuzzy Unconstrained Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 133-153 Fuzzy Unconstrained Modified Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 155-174 Fuzzy Constrained Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 175-189 Constrained Fuzzy Modified Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 191-207 Fuzzy Signomial Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 209-231 Goal Geometric Programming (Sahidul Islam, Wasim Akram Mandal)....Pages 233-258 Fuzzy Multi-objective Geometric Programming (FMOGP) Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 259-286 Geometric Programming Problem Under Uncertainty (Sahidul Islam, Wasim Akram Mandal)....Pages 287-330 Intuitionistic and Neutrosophic Geometric Programming Problem (Sahidul Islam, Wasim Akram Mandal)....Pages 331-355 Back Matter ....Pages 357-359