Non-linear and dynamic programming: useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Delhi ; Bengaluru ; Chennai ; Kochi
CBS Publishers & Distributors Pvt Ltd
2020
|
| Edition: | First edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | 334 Seiten Illustrationen |
| ISBN: | 9789389688887 |
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| adam_text | Contents 1. INTRODUCTION І Miilil ӀЖйЯвЖ ІІВіІтОйі. «ա» BİBİMİ ®elÉliÄ «■и |»M|| Οββίο 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 2. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3. 3.1 3.2 3.3 3.4 3.5 3.6 1-34 Introduction Matrix Type of Matrices Operations on Matrices Properties of Matrix Addition Properties of Multiplication of Matrix by a Scalar Multiplication of Matrices Determinant of a Square Matrix Properties of Determinants Evaluation of a Determinant by Sarrus Diagram Minors and Cofactors . Singuİar and Non-Singuİar Matrix Transpose of a Matrix Properties of Transpose of a Matrix Symmetric Matrix Skew-Symmetric Matrix Rank of a Matrix Echelon Form of a Matrix Inverse of a Matrix Convex Set Some Related Definitions Convex Hull Convex Function and Convex Polyhedron Feasible and Basic Feasible Solutions______________________________ SEQUENCING Introduction General· Sequencing Problem Sequencing Decision Problem for n Jobs and Two Machines Johnson s Method (For n-Jobs 2-Machines) Sequencing Decision Problem of n-Job and Three Machines Sequencing Decision Problems for n-Jobs and m-Machines Sequencing Problem involving Two Jobs and m Machines Graphical Method_______________________________________________ CLASSICAL OPTIMIZATION TECHNIQUES Introduction Maxima and Minima Necessary Condition Necessary Condition Sufficient Condition Maxima and Minima of a Function of Several Independent Variables for the Existence of Maxima or Minima for Maxima and Minima for Maxima and Minima: The Lagrange s Condition of the Function of
Three Independent Variables 1 1 1 3 4 5 7 7 8 8 9 9 10 10 11 11 11 12 13 15 16 18 18 28 35-50 35 35 36 37 37 38 45 45 51-102 51 59 59 60 61 70
а ч 44 * V X ՝ Հհ ֊χ լ ֊1 ř· Հ s Հ ? ЛЛ Հ ՀՀ : 3.7 Maxima and Minima for a Function of Three Independent VariaWes : The Lagrange s Condition 70 3.8 Lagrange s Method of Undetermined Multipliers 74 82 3.9 Lagrangian Multipliers Method in Non-Linear Programming 3.10 Sufficient Conditions for Maximum or Minimum of the Objective Function 85 3.11 Solution of Non-Linear Programming Probİems when Constraints are not Equality NON-LINEAR PROGRAMMING : FORMULATION AND GRAPHICAL SOLUTIONS 103-118 4.1 4.2 4.3 4.4 4.5 ИМ Ік , 7 4 ж » К в і Ь? 5. 103 103 103 104 107 Introduction General Non-Linear Programming Problems (GNLPP) Canonical Form of Non-Linear Programming Problem Mathematical Formulation of Non-Linear Programming Problems Graphical Solution of a Non-Linear Programming Problem 119-148 QUADRATIC AND SEPARABLE PROGRAMMING Introduction Quadratic Programming^ Kuhn-Tucker Conditions for Quadratic Programming Problems Wolfe s Modified Simplex Method Beale s Method Separable Programming Some General Definitions Related to Separable Programming Piecewise Linear Approximation of Non-Linear Functions Mixed Integer Approximation of Separable Problems Validity of Mixed Integer Approximation 5.11 Method of Solution of Separable Programming Probİems 119 119 120 120 132 143 143 143 144 144 14Я 5.1 5.2 5.3 5.4 5.5 5.6 5.7 с о 5.8 5.9 5.10 б. ал 6.2 б.з б·4 6.5 6.6 6.7 6.8 6.9 FRACTIONAL PROGRAMMING 7.2 7.3 7.4 7.5 7.6 8. 149-164 GEOMETRIC PROGRAMMING Introduction General Form of Geometric Programming Formulation of Geometric Programming Problem Geometric-Arithmetic Mean Inequality More
General Formulation of Geometric Programming Problem Necessary Condition for Optimality Primal Geometric Programming Problem withEquality Constraints Constrained Optimization Problem Degree of Difficulty Fractional Programming Mathematical Formulation of Linear Fractional Programming Linear Fractional Programming Algorithm Graphical Method for Linear Fractional Programming Charmes and Cooper Method for Linear Fractional Programming STOCHASTIC PROGRAMMING 8.1 Introduction 8.2 Approaches for solving a Stochastic Programming Problem ОііііІІіІІМІІмІИ „ 149 149 149 150 151 151 157 160 162 165-172 165 165 166 169 172 174-178 173 173 і §
■ і·· 9. 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 10. 10.1 10.2 10.3 10.4 10.5 10.6 10.7 11. 11.1 11.2 11.3 11.4 11.5 12. 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13. 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 179-212 ONE-DIMENSIONAL MINIMIZATION METHODS Introduction Basic Defi nitions Region Elimination Methods Elimination Methods Fibonacci Method Golden Section Method Exhaustive Search Method Dichotomous Search Method Interpolation Methods Direct Root Method ______________________________ 179 179 180 181 184 192 195 195 198 206 213-232 UNCONSTRAINED OPTIMIZATION TECHNIQUES Introduction Unconstrained Minimization Methods Some Direct Method Some More Unconstrained Direct Search Methods Some Descent (or Indirect) Methods Steepest Descent or Cauchy Method Fletcher-Reeves Method ___________________________________ 213 213 214 225 226 227 230 233-252 CONSTRAINED OPTIMIZATION TECHNIQUES Introduction Characteristics of a Constrained Problem Constrained Minimization Methods Direct Methods Indirect Methods _________________________ 233 233 233 233 246 253-278 DYNAMIC PROGRAMMING 253 Introduction 253 Bellman s Principle of Optimality 253 Multistage Decision Problem 254 Characteristic of a Dynamic Programming Problem Solution of a Multi-Stage Problem of Dynamic Programming with Finite Number 254 of Stages 255 Types of Problems 267 Solution of L.P.P. by Dynamic Programming 272 Solution of Inventory Problem by a Dynamic Programming Technique 277 Difference between Dynamic and Linear Programming Problem______ MAXIMUM FLOW AND MINIMUM POTENTIAL IN NETWORKS Introduction Graphs
and their Representation Multigraph Graph Terminology Types of Graphs Graph Isomorphism Homeomorphic Graph Automorphism .Subgraphs Types of Subgraphs 279-330 279 279 281 281 282 287 288 288 290 290 ase··։
13.11 13.12 13.13 13.14 13.15 13.16 13.17 13.18 13.19 13.20 13.21 13.22 13.23 13.24 13.25 13.26 13.27 292 Walks, Paths and Circuits 297 Connected, Disconnected Graphs and Components 299 Euter Graph : Königsberg Bridge Problem 301 Labelled Graph 302 Directed Graph 307 Weighted Graph 308 Shortest Path Problems 315 Trees 316 MinimaUy Connected Graph 316 Arborescences 317 Maximum Flow Problems 318 Network Flows 319 Formulation of Maximum Flow Problem as a Linear Programming Problem 319 Labelling Routine Algorithm 320 Max-Flow Algorithm 326 Extension of Max-Flow Algorithm 328 Potential Difference 331-334 INDEX
r I— I I ———- _ Non-Linear and Dynamic Programming is an introductory text aimed at undergraduate Contents ----------===! and postgraduate students of science and AA Introduction Sequencing engineering. It is also valuable as a resource book A Classical Optirzation Techniques for practicing scientists and engineers who need d Non-linear Programming: Formulation to use linear and non-linear programming. The and Graphical Solutions Quadratic and Separable---relationship to the fundamental mathematical A — Programming -91 A Geometric Programming----A Stochastic Programming Methods--A Unconstrained Optimization-Techniques ======42 A Constrained Optimization---—Techniques-----A Fractional Programming A One-dimensional Minimization A Dynamic Programming d Maximum Flow and Minimum Potential m Networks been leaching undergraduate and postgraduate classes for well over two
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| adam_txt |
Contents 1. INTRODUCTION І Miilil ӀЖйЯвЖ ІІВіІтОйі. «ա» BİBİMİ ®elÉliÄ «■и |»M|| Οββίο 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 2. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3. 3.1 3.2 3.3 3.4 3.5 3.6 1-34 Introduction Matrix Type of Matrices Operations on Matrices Properties of Matrix Addition Properties of Multiplication of Matrix by a Scalar Multiplication of Matrices Determinant of a Square Matrix Properties of Determinants Evaluation of a Determinant by Sarrus Diagram Minors and Cofactors . Singuİar and Non-Singuİar Matrix Transpose of a Matrix Properties of Transpose of a Matrix Symmetric Matrix Skew-Symmetric Matrix Rank of a Matrix Echelon Form of a Matrix Inverse of a Matrix Convex Set Some Related Definitions Convex Hull Convex Function and Convex Polyhedron Feasible and Basic Feasible Solutions_ SEQUENCING Introduction General· Sequencing Problem Sequencing Decision Problem for n Jobs and Two Machines Johnson's Method (For n-Jobs 2-Machines) Sequencing Decision Problem of n-Job and Three Machines Sequencing Decision Problems for n-Jobs and m-Machines Sequencing Problem involving Two Jobs and m Machines Graphical Method_ CLASSICAL OPTIMIZATION TECHNIQUES Introduction Maxima and Minima Necessary Condition Necessary Condition Sufficient Condition Maxima and Minima of a Function of Several Independent Variables for the Existence of Maxima or Minima for Maxima and Minima for Maxima and Minima: The Lagrange's Condition of the Function of
Three Independent Variables 1 1 1 3 4 5 7 7 8 8 9 9 10 10 11 11 11 12 13 15 16 18 18 28 35-50 35 35 36 37 37 38 45 45 51-102 51 59 59 60 61 70
а ч 44 * V X ՝' Հհ ֊χ լ ֊1 ř· Հ "s'Հ ? ЛЛ Հ ՀՀ : 3.7 Maxima and Minima for a Function of Three Independent VariaWes : The Lagrange's Condition 70 3.8 Lagrange's Method of Undetermined Multipliers 74 82 3.9 Lagrangian Multipliers Method in Non-Linear Programming 3.10 Sufficient Conditions for Maximum or Minimum of the Objective Function 85 3.11 Solution of Non-Linear Programming Probİems when Constraints are not Equality NON-LINEAR PROGRAMMING : FORMULATION AND GRAPHICAL SOLUTIONS 103-118 4.1 4.2 4.3 4.4 4.5 ИМ Ік , 7 4 ж » К в і Ь? 5. 103 103 103 104 107 Introduction General Non-Linear Programming Problems (GNLPP) Canonical Form of Non-Linear Programming Problem Mathematical Formulation of Non-Linear Programming Problems Graphical Solution of a Non-Linear Programming Problem 119-148 QUADRATIC AND SEPARABLE PROGRAMMING Introduction Quadratic Programming^ Kuhn-Tucker Conditions for Quadratic Programming Problems Wolfe's Modified Simplex Method Beale's Method Separable Programming Some General Definitions Related to Separable Programming Piecewise Linear Approximation of Non-Linear Functions Mixed Integer Approximation of Separable Problems Validity of Mixed Integer Approximation 5.11 Method of Solution of Separable Programming Probİems 119 119 120 120 132 143 143 143 144 144 14Я 5.1 5.2 5.3 5.4 5.5 5.6 5.7 с о 5.8 5.9 5.10 б. ал 6.2 б.з б·4 6.5 6.6 6.7 6.8 6.9 FRACTIONAL PROGRAMMING 7.2 7.3 7.4 7.5 7.6 8. 149-164 GEOMETRIC PROGRAMMING Introduction General Form of Geometric Programming Formulation of Geometric Programming Problem Geometric-Arithmetic Mean Inequality More
General Formulation of Geometric Programming Problem Necessary Condition for Optimality Primal Geometric Programming Problem withEquality Constraints Constrained Optimization Problem Degree of Difficulty Fractional Programming Mathematical Formulation of Linear Fractional Programming Linear Fractional Programming Algorithm Graphical Method for Linear Fractional Programming Charmes and Cooper Method for Linear Fractional Programming STOCHASTIC PROGRAMMING 8.1 Introduction 8.2 Approaches for solving a Stochastic Programming Problem ОііііІІіІІМІІмІИ „ 149 149 149 150 151 151 157 160 162 165-172 165 165 166 169 172 174-178 173 173 і §
■ і·· 9. 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 10. 10.1 10.2 10.3 10.4 10.5 10.6 10.7 11. 11.1 11.2 11.3 11.4 11.5 12. 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13. 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 179-212 ONE-DIMENSIONAL MINIMIZATION METHODS Introduction Basic Defi nitions Region Elimination Methods Elimination Methods Fibonacci Method Golden Section Method Exhaustive Search Method Dichotomous Search Method Interpolation Methods Direct Root Method _ 179 179 180 181 184 192 195 195 198 206 213-232 UNCONSTRAINED OPTIMIZATION TECHNIQUES Introduction Unconstrained Minimization Methods Some Direct Method Some More Unconstrained Direct Search Methods Some Descent (or Indirect) Methods Steepest Descent or Cauchy Method Fletcher-Reeves Method _ 213 213 214 225 226 227 230 233-252 CONSTRAINED OPTIMIZATION TECHNIQUES Introduction Characteristics of a Constrained Problem Constrained Minimization Methods Direct Methods Indirect Methods _ 233 233 233 233 246 253-278 DYNAMIC PROGRAMMING 253 Introduction 253 Bellman's Principle of Optimality 253 Multistage Decision Problem 254 Characteristic of a Dynamic Programming Problem Solution of a Multi-Stage Problem of Dynamic Programming with Finite Number 254 of Stages 255 Types of Problems 267 Solution of L.P.P. by Dynamic Programming 272 Solution of Inventory Problem by a Dynamic Programming Technique 277 Difference between Dynamic and Linear Programming Problem_ MAXIMUM FLOW AND MINIMUM POTENTIAL IN NETWORKS Introduction Graphs
and their Representation Multigraph Graph Terminology Types of Graphs Graph Isomorphism Homeomorphic Graph Automorphism .Subgraphs Types of Subgraphs 279-330 279 279 281 281 282 287 288 288 290 290 ase··։
13.11 13.12 13.13 13.14 13.15 13.16 13.17 13.18 13.19 13.20 13.21 13.22 13.23 13.24 13.25 13.26 13.27 292 Walks, Paths and Circuits 297 Connected, Disconnected Graphs and Components 299 Euter Graph : Königsberg Bridge Problem 301 Labelled Graph 302 Directed Graph 307 Weighted Graph 308 Shortest Path Problems 315 Trees 316 MinimaUy Connected Graph 316 Arborescences 317 Maximum Flow Problems 318 Network Flows 319 Formulation of Maximum Flow Problem as a Linear Programming Problem 319 Labelling Routine Algorithm 320 Max-Flow Algorithm 326 Extension of Max-Flow Algorithm 328 Potential Difference 331-334 INDEX
r I— I I ———- _ Non-Linear and Dynamic Programming is an introductory text aimed at undergraduate Contents ----------===! and postgraduate students of science and AA Introduction Sequencing engineering. It is also valuable as a resource book A Classical Optirzation Techniques for practicing scientists and engineers who need d Non-linear Programming: Formulation to use linear and non-linear programming. The and Graphical Solutions Quadratic and Separable---relationship to the fundamental mathematical A — Programming -91 A Geometric Programming----A Stochastic Programming Methods--A Unconstrained Optimization-Techniques ======42 A Constrained Optimization---—Techniques-----A Fractional Programming A One-dimensional Minimization A Dynamic Programming d Maximum Flow and Minimum Potential m Networks been leaching undergraduate and postgraduate classes for well over two |
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| spelling | Pundir, Sudhir Kumar Verfasser aut Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations Dr.Sudhir Kumar Pundir, M.Sc., M.Phil, NET (JRF), Ph.D., head Department of Mathematics, S.D. (P.G.) College Muzaffarnagar (U.P.) First edition New Delhi ; Bengaluru ; Chennai ; Kochi CBS Publishers & Distributors Pvt Ltd 2020 334 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Algorithmus (DE-588)4001183-5 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Algorithmus (DE-588)4001183-5 s Nichtlineare Optimierung (DE-588)4128192-5 s DE-604 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032806027&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032806027&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Pundir, Sudhir Kumar Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations Algorithmus (DE-588)4001183-5 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| subject_GND | (DE-588)4001183-5 (DE-588)4128192-5 |
| title | Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations |
| title_auth | Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations |
| title_exact_search | Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations |
| title_exact_search_txtP | Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations |
| title_full | Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations Dr.Sudhir Kumar Pundir, M.Sc., M.Phil, NET (JRF), Ph.D., head Department of Mathematics, S.D. (P.G.) College Muzaffarnagar (U.P.) |
| title_fullStr | Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations Dr.Sudhir Kumar Pundir, M.Sc., M.Phil, NET (JRF), Ph.D., head Department of Mathematics, S.D. (P.G.) College Muzaffarnagar (U.P.) |
| title_full_unstemmed | Non-linear and dynamic programming useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations Dr.Sudhir Kumar Pundir, M.Sc., M.Phil, NET (JRF), Ph.D., head Department of Mathematics, S.D. (P.G.) College Muzaffarnagar (U.P.) |
| title_short | Non-linear and dynamic programming |
| title_sort | non linear and dynamic programming useful for undergraduate and postgraduate students of mathematics statistics computer science physical science management engineering and other professional courses and competitive examinations |
| title_sub | useful for undergraduate and postgraduate students of mathematics, statistics, computer science, physical science, management, engineering and other professional courses and competitive examinations |
| topic | Algorithmus (DE-588)4001183-5 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd |
| topic_facet | Algorithmus Nichtlineare Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032806027&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032806027&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT pundirsudhirkumar nonlinearanddynamicprogrammingusefulforundergraduateandpostgraduatestudentsofmathematicsstatisticscomputersciencephysicalsciencemanagementengineeringandotherprofessionalcoursesandcompetitiveexaminations |