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NEUTROSOPHIC OPERATIONAL RESEARCH Volume I Editors: Prof. Florentin Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou Dedication Dedicated with love to our parents for the developments of our cognitive minds, ethical standards, and shared do-good values & to our beloved families for the continuous encouragement, love, and support. Acknowledgment The book would not have been possible without the support of many people: first, the editors would like to express their appreciation to the advisory board; second, we are very grateful to the contributors; and third, the reviewers for their tremendous time, effort, and service to critically review the various chapters. The help of top leaders of public and private organizations inspired, encouraged, and supported the development of this book. Peer Reviewers Prof. M. Caldas Departamento De Matematica, Universidade Federal Fluminense, Rua Mario Santos Braga, s/N, 24020-140, Niteroi, RJ Brasil. Prof. M. Ganster Department of Mathematics, Graz University of Technology Steyrergasse 30, 8010 Graz, Austria. Prof. S. Jafari College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark. Neutrosophic Operational Research Volume I Foreword by John R. Edwards Preface by the editors Pons Brussels, 2017 Editors: Prof. Florentin Smarandache University of New Mexico Mathematics & Science Department Dr. Mohamed Abdel-Basset Faculty of Computers and Informatics, Operations Research Dept. Zagazig University, Egypt Dr. Yongquan Zhou College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China Pons Publishing House / Pons asbl Quai du Batelage, 5 1000 - Bruxelles Belgium DTP: George Lukacs ISBN 978-1-59973-520-7 Neutrosophic Operational Research Volume I Table of Contents Foreword .............................................................................................................. 7 Preface ................................................................................................................. 9 I Neutrosophic Linear Programming Problems ................................................. 15 Abdel-Nasser Hussian, Mai Mohamed, Mohamed Abdel-Baset, Florentin Smarandache II Neutrosophic Linear Fractional Programming Problems ............................... 29 Abdel-Nasser Hussian, Mai Mohamed, Mohamed Abdel-Baset, Yongquan Zhou, Florentin Smarandache III Neutrosophic Integer Programming Problems .............................................. 49 Mohamed Abdel-Baset, Mai Mohamed, Abdel-Nasser Hessian, Florentin Smarandache IV Neutrosophic Goal Programming ................................................................. 63 Ibrahim M. Hezam, Mohamed Abdel-Baset, Florentin Smarandache V Taylor Series Approximation to Solve Neutrosophic Multiobjective Programming Problem ............................................................................. 77 Ibrahim M. Hezam, Mohamed Abdel-Baset, Florentin Smarandache VI Multi-objective Cylindrical Skin Plate Design Optimization based on Neutrosophic Optimization Technique .................................................... 91 Mridula Sarkar, Tapan Kumar Roy VII Neutrosophic MULTIMOORA: A Solution for the Standard Error in Information Sampling ............................................................................ 105 Willem K. M. Brauers, Alvydas Baležentis, Tomas Baležentis VIII Multi-objective Geometric Programming Problem Based on Neutrosophic Geometric Programming Technique ...................................................... 131 Pintu Das, Tapan Kumar Roy 5 Editors: Prof. Florentin Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou IX Using Neutrosophic Sets to Obtain PERT Three-Times Estimates in Project Management .......................................................................................... 143 Mai Mohamed, Mohamed Abdel-Basset, Abdel-Nasser Hussien, Florentin Smarandache X A Critical Path Problem Using Triangular Neutrosophic Number .............. 155 Mai Mohamed, Mohamed Abdel-Baset, Florentin Smarandache, Yongquan Zhou XI A Critical Path Problem in Neutrosophic Environment ............................. 167 Mai Mohamed, Mohamed Abdel-Baset, Florentin Smarandache, Yongquan Zhou XII Integrated Framework of Optimization Technique and Information Theory Measures for Modeling Neutrosophic Variables ................................... 175 Mona Gamal Gafar, Ibrahim El-Henawy XIII New Neutrosophic Sets via Neutrosophic Topological Spaces ............... 189 Wadei Al-Omeri, Florentin Smarandache 6 Neutrosophic Operational Research Volume I Foreword John R. Edwards This book is an excellent exposition of the use of Data Envelopment Analysis (DEA) to generate data analytic insights to make evidence-based decisions, to improve productivity, and to manage cost-risk and benefit- opportunity in public and private sectors. The design and the content of the book make it an up-to-date and timely reference for professionals, academics, students, and employees, in particular those involved in strategic and operational decision- making processes to evaluate and prioritize alternatives to boost productivity growth, to optimize the efficiency of resource utilization, and to maximize the effectiveness of outputs and impacts to stakeholders. It is concerned with the alleviation of world changes, including changing demographics, accelerating globalization, rising environmental concerns, evolving societal relationships, growing ethical and governance concern, expanding the impact of technology; some of these changes have impacted negatively the economic growth of private firms, governments, communities, and the whole society. 7 Neutrosophic Operational Research Volume I Preface Prof. Florentin Smarandache Dr. Mohamed Abdel-Basset Dr. Yongquan Zhou This book treats all kind of data in neutrosophic environment, with real- life applications, approaching topics as linear programming problem, linear fractional programming, integer programming, triangular neutrosophic numbers, single valued triangular neutrosophic number, neutrosophic optimization, goal programming problem, Taylor series, multi-objective programming problem, neutrosophic geometric programming, neutrosophic topology, neutrosophic open set, neutrosophic semi-open set, neutrosophic continuous function, cylindrical skin plate design, neutrosophic MULTIMOORA, alternative solutions, decision matrix, ratio system, reference point method, full multiplicative form, ordinal dominance, standard error, market research, and so on. The first chapter (Neutrosophic Linear Programming Problems) proposes some linear programming problems based on neutrosophic environment. Neutrosophic sets are characterized by three independent parameters, namely truth-membership degree (T), indeterminacy-membership degree (I) and falsity- membership degree (F) which are more capable to handle imprecise parameters. The authors (Abdel-Nasser Hussian, Mai Mohamed, Mohamed Abdel-Baset, and Florentin Smarandache) also transform the neutrosophic linear programming problem into a crisp programming model by using neutrosophic set parameters. To measure the efficiency of the proposed model, several numerical examples are solved. A solution procedure is proposed in the second chapter (Neutrosophic Linear Fractional Programming Problems), to solve neutrosophic linear fractional programming (NLFP) problem where cost of the objective function, the resources and the technological coefficients are triangular neutrosophic numbers. The NLFP problem is transformed into an equivalent crisp multiobjective linear fractional programming (MOLFP) problem. By using the proposed approach, the transformed MOLFP problem is reduced to a single objective linear programming (LP) problem which can be solved easily by suitable LP problem algorithm. The proposed procedure is illustrated through a 9