Table Of Content

Mridula Sarkar Tapan Kumar Roy Florentin Smarandache Neutrosophic Optimization and its Application on Structural Designs Mridula Sarkar Tapan Kumar Roy Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O- Botanic Garden, Howrah-711103, West Bengal, India. [email protected] [email protected] Florentin Smarandache Department of Mathematics, University of New Mexico, Gallup Campus, USA. [email protected] Mridula Sarkar Tapan Kumar Roy Florentin Smarandache Neutrosophic Optimization and its Application on Structural Designs Brussels, 2018 CONTENTS Chapter-1 Basic Notions and Neutrosophic Optimization…………....……….1 1.1 Overview……………………………………………………………………….…..1 1.2 Neutrosophic Set…………………………………...………………………….........2 1.3 Single Valued Neutrosophic Set…………………………………….………...........2 1.4 Complement of Neutrosophic Set……………………………….………………….3 1.5 Containment………………………………………………………………...………3 1.6 Equality of Two Neutrosophic Sets…………………………………………..…….4 1.7 Union of Neutrosophic Sets…………………………………………………...…..4 1.8 Intersection of Neutrosophic Sets………………………………..….................…4 1.9 Difference of two single valued Neutrosophic Set…………………………………5 1.10 Normal Neutrosophic Set………………………………………………………....6 1.11 Convex Neutrosophic Set……………………...…………………………….…...6 1.12 Single Valued Neutrosophic Number(SVNN)…………………………..........…6 1.13 Generalized Triangular Neutrosophic Number(GTNN)…………………………..7 1.14 ,, Cut of Single Valued Triangular Neutrosophic Number(SVTNN)…....8 1.15 Ranking of Triangular Neutrosophic Number……………………………..…….....8 1.16 Nearest Interval Approximation for Neutrosophic Number……………...............11 1.17 Decision Making in Imprecise Environment……………………………….…..…12 1.18 Single-Objective Neutrosophic Geometric Programming…………………….….13 1.19 Numerical Example of Neutrosophic Geometric Programming…………………..15 1.20 Application of Neutrosophic Geometric Programming in Gravel Box Design Problem…………………………………………….………………………………17 1.21 Multi-Objective Neutrosophic Geometric Programming Problem………………..18 1.22 Definition: Neutrosophic Pareto(or NS Pareto) Optimal Solution……….………..22 1.23 Theorem 1…………………………………………………………...……………22 1.24 Theorem 2…………………………………………………….…………………..23 1.25 Illustrated Numerical Example……………………………………………………24 1.26 Application of Neutrosophic Optimization in Gravel Box Design Problem……..26 1.27 Multi-Objective Neutrosophic Linear Programming Problem (MOLPP)…….......27 1.28 Production Planning Problem……..………………………………………………39 1.29 Neutrosophic Optimization (NSO)Technique to two type Single-Objective Minimization Type Nonlinear Programming (SONLP)Problem………………...42 1.30 Neutrosophic Optimization Technique to solve Minimization Type Multi Objective Non-linear Programming Problem for Linear Membership Function...55 1.31 Illustrated Numerical Example…………………………………….……………..63 1.32 Application of Neutrosophic Optimization in Riser Design Problem……………63 1.33 Neutrosophic Optimization(NSO) Technique to Solve Minimization Type Multi Objective Non-linear Programming Problem(MONLP)……………………...….65 1.34 Neutrosophic Goal Programming(NGP)…………………………………………74 1.35 Theorem on Generalized Goal Programming………………………………….…76 I 1.36 Generalized Neutrosophic Goal Programming(GNGP)………………………….78 1.37 Application of Neutrosophic Goal Programming to Bank Three Investment Problem……………………………………………………………………………80 1.38 Numerical Example………………………………………………………………82 1.39 Neutrosophic Non-linear Programming (NNLP) Optimization to solve Parameterized Multi-objective Non-linear Programming Problem (PMONLP)…83 1.40 Neutrosophic Optimization Technique(NSO) to solve Parametric Single-Objectiv Non-linear Programming Problem (PSONLP)…………………………………..88 Chapter-2 Structural Design Optimization……………………………………………94 2.1 S.I Unit Prefixes …………………………………...………………………...….96 2.2 Conversion of U.S Customary Units to S.I Units……………………………….96 2.3 Design Studies…………………………………………………………………..97 2.3.1 Two-Bar Truss(Model-I)………………………………………….….97 2.3.2 Three-Bar Truss(Model-II)……………………………………...……99 2.3.3 Design Criteria for Thickness Optimization……………………......104 2.3.4 Welded Beam Design Formulation……………………………….109 Chapter-3 Truss Design Optimization using Neutrosophic Optimization Technique: A Comparative Study…………………………………………………….115 3.1 General Formulation of Single-objective Structural Model……………..………...116 3.2 Neutrosophic Optimization Technique to Solve Single-objective Structural Optimization Problem (SOSOP)………………………………………..….117 3.3 Numerical Solution of Two Bar Truss Design using Single Objective NSO Technique…………………………………………………………………...128 3.4 Conclusion…………………………………………………………………………136 Chapter-4 Multi-Objective Neutrosophic Optimization Technique and its Application to Structural……………………………………………………………138 4.1 General form of Multi-Objective Truss Design Model………………………....…139 4.2 Solution of Multi-objective Structural Optimization Problem (MOSOP) by Neutrosophic Optimization Technique……………………………………...140 4.3 Numerical Solution of Solution of Multi-objective Structural Optimization Problem (MOSOP) by Neutrosophic Optimization Technique……………..147 4.4 Conclusion ………………………………………………………………….....….155 Chapter-5 Optimization of Welded Beam Structure using Neutrosophic Optimization Technique: A Comparative Study……………………………………………..156 5.1 Welded Beam Design (WBD)and its Optimization in Neutrosophic Environment…………………………………………………...…………….....158 II 5.2.1 Crisp Formulation of WBD……………………………………..…..159 5.2.2 WBD Formulation in Neutrosophic Environment……………….....161 5.2.3 Optimization of WBD in Neutrosophic Environment……………....162 5.2 Numerical Solution of WBD by Single Objective Neutrosophic Optimization Technique…………………………………………………………………..…167 5.3 Conclusion…………………………………………………………...…...……....173 Chapter-6 Multi-Objective Welded Beam Optimization using Neutrosophic Optimization Technique: A Comparative Study………………………………………………175 6.1 General Form of Multi-Objective Welded Beam Design(MOWBD)…………......176 6.2 Solution of Multi-Objective Welded Beam Design (MOWBD) Problem by Neutrosophic Optimization(NSO) Technique………………………………177 6.3 Numerical Solution of Welded Beam Design using Multi-Objective Neutrosophic Optimization Technique………………………………...……185 6.4 Conclusion..………………………………………………………………………..197 Chapter-7 Multi-Objective Welded Beam Optimization using Neutrosophic Goal Programming Technique………………………………………..…………...…..198 7.1 General Formulation of Multi-objective Welded Beam Design……….............…200 7.2 Generalized Neutrosophic Goal Optimization Technique to Solve Multi- objective Welded Beam Optimization Problem (MOWBP)……………..…201 7.3 Numerical Solution of Welded Beam Design by GNGP, based on Different Operator……………………………………………………………………..204 7.4 Conclusion………………………………………………………………………....211 Chapter-8 Truss Design Optimization with Imprecise Load and Stress in Neutrosophic Environment…………………………………………………………………….212 8.1 Multi-Objective Structural Design Formulation…………………………..213 8.2 Parametric Neutrosophic Optimization Technique to Solve Multi-Objective Structural Optimization Problem………………………………………...…214 8.3 Numerical Solution of Three Bar Truss Design using Parametric Neutrosophic Optimization Technique……………………………………218 8.4 Conclusion…………………………………………………………….....…227 Chapter-9 Optimization of Welded Beam with Imprecise Load and Stress by Parameterized Neutrosophic Optimization Technique…………….…..229 9.1 General Formulation of Single-Objective Welded Beam Design……………....…230 9.2 NSO Technique to Optimize Parametric Single-Objective Welded Beam Design(SOWBD)……………………………………………………………231 9.3 Numerical Solution of Parametric Welded Beam Design Problem by NSO Technique………………………………………………………………..….234 9.4 Conclusion……………………………………………………………...………….242 Chapter-10 Optimization of Thickness of Jointed Plain Concrete Pavement Using Neutrosophic Optimization Technique…………….…………………...…243 10.1 Formulation for Optimum JPCP Design…………………………………....244 III 10.1.1 Design input parameters…………………...………………………..245 10.1.2 Design method………………………………..…………………….245 10.2 Neutrosophic Optimization……………………………………………..…..248 10.3 Numerical Illustration of Optimum JPCP Design based on IRC:58-2002…251 10.4 Conclusion………………………..………………………………………....254 Chapter-11 Multi-Objective Structural Design Optimization Based on Neutrosophic Goal Programming Technique………………………………………..…...………….256 11.1 Multi-Objective Structural Model………………………………………......257 11.2 Solution of Multi-objective Structural Optimization Problem (MOSOP) by Generalized Neutrosophic Goal Optimization Technique…..…………...…258 11.3 Numerical Illustration………………………………………...…………….261 11.4 Conclusion……………………………………………………………......…267 Chapter-12 Multi-objective Cylindrical Skin Plate Design Optimization based on Neutrosophic Optimization Technique…………………………………………...….269 12.1 Multi-Objective Structural Model Formulation…………………………......270 12.2 Solution of Multi-objective Structural Optimization Problem (MOSOP) by Neutrosophic Optimization Technique…………………………………..….270 12.3 Numerical Illustration…………………………………………………...…..271 12.4 Conclusion……………………………………………………………......…274 Chapter-13 APPENDIX-A……………………………………………………..……….275 13.1 Crisp Set…………………………………………...………………………...….275 13.2 Fuzzy Set……………………..……………………………………………..….275 13.3 Height of a Fuzzy Set………………………………..………………………....276 13.4 Normal Fuzzy Set……………………………………………………………....276 13.5  Cut of Fuzzy Set…………………...…………………………………..…..276 13.6 Union of Two Fuzzy Sets………………………………..…………………..…276 13.7 Intersection of Two Fuzzy Sets………………………………...……………....276 13.8 Convex Fuzzy Set…………………………………………………………..…..277 13.9 Interval Number………………………………………………………….....….277 13.10 Fuzzy Number………...……………………………………………..……..….277 13.10.1 Trapezoidal Fuzzy Number(TrFN)……………………………..…..279 13.11  Cut of Fuzzy Number………………………………………………….….279 13.12 Generalized Fuzzy Number (GFN)………………………………………....….280 13.13 Nearest Interval Approximation of Fuzzy Number………………………...…..281 13.14 Intuitionistic Fuzzy Set……………………..………………………………......282 13.15 , Level Or ,Cuts…………………………...……..…………....283 13.16 Convex Intuitionistic Fuzzy Set………………………………………...……..283 13.17 Union of Two Intuitionistic Fuzzy Sets…………………………………..…....284 IV 13.18 Intersection of Two Intuitionistic Fuzzy Sets………………………………..…284 13.19 Generalized Intuitionistic Fuzzy Number(GIFN)…….…………………....…..284 13.20 Generalized Triangular Intuitionistic Fuzzy Number(GTIFN)…...……………284 13.21 Level Set or  Cut of Intuitionistic Fuzzy Number………...……………285 13.22 Arithmetic Operation of Triangular Intuitionistic Fuzzy Number (TIFN)…......285 13.23 Nearest Interval Approximation for Intuitionistic Fuzzy Number…………......286 13.24 Parametric Interval Valued Function…………………………….………..…...287 13.25 Ranking of Triangular Intuitionistic Fuzzy Number…………………………...288 Chapter-14 APPENDIX-B……………………………………………………………...290 14.1 Geometric Programming(GP) Method…………………………………….…...290 14.2 Posynomial Geometric Programming Problem……………………………….291 14.3 Signomial Geometric Programing Problem…………………………..……….292 14.4 Fuzzy Geometric Programming (FGP) ………………………………………..293 14.5 Numerical Example of Fuzzy Geometric Programming……………………….295 14.6 Intuitionistic Fuzzy Geometric Programming………………………………….297 14.7 Fuzzy Decision Making……………………………………………..…….......301 14.8 Additive Fuzzy Decision………………………………………………….……301 14.9 Intuitionistic Fuzzy Optimization(IFO) Technique to solve Minimization Type Single Objective Non-linear Programming (SONLP) Problem ……...…….....302 14.10 Fuzzy Non-linear Programming (FNLP) Technique to Solve Multi-Objective Non Linear Programming (MONLP) problem……………………………….....306 14.11 An Intuitionistic Fuzzy(IF) Approach for Solving Multi-Objective Non-Linear Programming(MONLP) Problem with Non-linear membership and Non-linear Non-membership Function……………………………………………….……..308 14.12 Intuitionistic Fuzzy Non-linear Programming (IFNLP) Optimization to solve Parametric Multi-Objective Non-linear Programming Problem (PMONLP)…………...…………………………………………………………..311 14.13 Fuzzy and Intuitionistic Fuzzy Non-linear Programming (IFNLP) Optimization to solve Parametric Single-Objective Non-linear Programming (PSONLP) Problem…………………………………………………………....…316 Bibiliography…………………………………………………………………….322 V