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E-raamat: Optimization with LINGO-18: Problems and Applications [Taylor & Francis e-raamat]

(Amity University, Uttar Pradesh, IN), (Aligarh Muslim University, India)
  • Formaat: 252 pages, 35 Tables, black and white; 8 Line drawings, color; 36 Line drawings, black and white; 8 Illustrations, color; 36 Illustrations, black and white
  • Ilmumisaeg: 18-Oct-2021
  • Kirjastus: CRC Press
  • ISBN-13: 9781003048893
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  • Taylor & Francis e-raamat
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Optimization with LINGO-18: Problems and ApplicationsSuurem pilt
  • Formaat: 252 pages, 35 Tables, black and white; 8 Line drawings, color; 36 Line drawings, black and white; 8 Illustrations, color; 36 Illustrations, black and white
  • Ilmumisaeg: 18-Oct-2021
  • Kirjastus: CRC Press
  • ISBN-13: 9781003048893
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9781003048893

This book presents fundamental concepts of optimization problems and its real-world applications in various fields. The core concepts of optimization, formulations and solution procedures of various real-world problems are provided in an easy-to-read manner. The unique feature of this book is that it presents unified knowledge of the modelling of real-world decision-making problems and provides the solution procedure using the appropriate optimization techniques.

The book will help students, researchers, and faculty members to understand the need for optimization techniques for obtaining optimal solution for the decision-making problems. It provides a sound knowledge of modelling of real-world problems using optimization techniques. It is a valuable compendium of several optimization techniques for solving real-world application problems using optimization software LINGO. The book is useful for academicians, practitioners, students and researchers in the field of OR. It is written in simple language with a detailed explanation of the core concepts of optimization techniques. Readers of this book will understand the formulation of real-world problems and their solution procedures obtained using the appropriate optimization techniques.



The book is a valuable compendium comprising several optimization applications-based case studies. Chapters are written in a reader-friendly manner; theoretical concepts with applications have also been discussed.

Preface iii
Authors' Biographies xi
Section I: Essential Mathematics and Software
1 Basic Calculus
2(20)
1.1 Distance Formula for Points in the Plane
2(1)
1.2 Circle Centered at the Origin
2(1)
1.3 Straight Lines
2(1)
1.4 Angle of Inclination
3(1)
1.5 Parallel and Perpendicular Lines
3(1)
1.6 Function
3(1)
1.7 Domain
4(1)
1.8 Functions Graphs
5(1)
1.9 Composite Functions
5(1)
1.10 Even and Odd Functions-Symmetry
5(1)
1.11 Piecewise Defined Functions
6(1)
1.12 Shifting Graphs
6(2)
1.13 Average Rate of Change and Secant Lines
8(1)
1.14 Continuity
8(2)
1.14.1 Lipschitz Continuity
9(1)
1.15 Slope of a Curvilinear Function
10(1)
1.16 Vertical Tangent
11(1)
1.17 Derivative of the Function
11(1)
1.18 Differentiable in Interval
12(1)
1.19 Extreme Values of Functions
13(1)
1.20 Local and Absolute (Global) Extrema
13(2)
1.21 Increasing Functions and Decreasing Functions
15(1)
1.22 Tests Based on First Derivative
16(1)
1.23 Concavity
17(1)
1.24 Inflection Point
18(2)
1.25 Linearization Approximation
20(2)
2 Matrix and Determinant Algebra
22(14)
2.1 Matrix
22(1)
2.2 Types of Matrix
23(1)
2.3 Algebra of Matrix
23(1)
2.3.1 Addition and Subtraction of Matrix
23(1)
2.3.2 Multiplication by a Scalar
23(1)
2.3.3 Matrix Multiplication
23(1)
2.3.4 Laws of Algebra
24(1)
2.4 Transpose of Matrix
24(1)
2.5 Trace of Matrix
25(1)
2.6 Non-Singular Matrix
25(1)
2.7 Inverse of a Matrix
25(1)
2.8 Orthogonal Matrix
26(1)
2.9 Rank of Matrix
26(1)
2.10 Determinants and Non-Singularity
26(2)
2.11 Three Order Determinants
28(2)
2.12 Linear Equations
30(1)
2.13 Vector Algebra
31(5)
2.13.1 Metric Spaces
31(1)
2.13.2 Normed Linear Spaces
32(1)
2.13.3 Norms
33(1)
2.13.4 Forbenlus Norm
34(1)
2.13.5 Eigenvalues Vectors
34(2)
3 LINGO-18: Optimization Software
36(10)
3.1 Introduction
36(10)
3.1.1 Key Benefits of LINGO
36(1)
3.1.2 Installing LINGO
37(1)
3.1.3 Window Types in LINGO
37(1)
3.1.4 Model Creation and Solution in LINGO
38(4)
3.1.5 Logical Operators and Set Looping Functions
42(1)
3.1.6 Variable Domain Functions
43(1)
3.1.7 Shortcut Keys
44(2)
Section II: Optimization
4 Introduction to Optimization
46(22)
4.1 Introduction
46(1)
4.1.1 Unconstrained vs Constrained
46(1)
4.1.2 Linear vs Non-Linear
47(1)
4.2 Basic Definitions
47(4)
4.2.1 Affine Sets
47(1)
4.2.2 Convex Set
48(1)
4.2.3 Convex Function
48(1)
4.2.4 Concave Function
49(2)
4.3 Convex and Concave Properties
51(1)
4.4 Convex Optimization Problems
52(1)
4.5 Concave Maximization Problem
52(4)
4.5.1 Quasi Concave & Quasi Convex Function
53(3)
4.5.1.1 Properties of Quasi Concave Functions
54(2)
4.6 Convexity of Smooth Function
56(3)
4.6.1 Quasi Concavity of Smooth Function
57(1)
4.6.2 Pseudo Concave Function
57(1)
4.6.3 Pseudo Convex Function
57(1)
4.6.4 Cone
58(1)
4.6.5 Hyperplanes and Half Spaces
58(1)
4.7 Univariate Optimization Problems
59(2)
4.7.1 Optimality Condition
59(2)
4.8 Multivariate Optimization Problems
61(4)
4.8.1 Optimality Condition
61(4)
4.9 Gradient Vector off(x)
65(3)
4.9.1 Hessian Matrix off(x)
65(3)
5 Linear Optimization Problems
68(47)
5.1 Introduction
68(1)
5.2 General Form of LP Model
69(3)
5.2.1 Components of LP Model
69(1)
5.2.2 Assumptions
69(1)
5.2.3 General Mathematical Model
70(2)
5.2.4 Properties of Solution to an LPP
72(1)
5.3 Formulation of an LPP
72(5)
5.3.1 Product Mix Problem
72(2)
5.3.2 Resource Allocation Problem
74(1)
5.3.3 Diet Problem
74(1)
5.3.4 The Capital Budgeting Problem
75(1)
5.3.5 Assignment Problem
76(1)
5.3.6 Transportation Problem
77(1)
5.4 Integer Programming
77(1)
5.5 Graphical Solution of LPP
78(5)
5.6 Theory of Simplex Method
83(8)
5.6.1 Standard Form
83(1)
5.6.2 Computational Procedure of Simplex Method
84(2)
5.6.3 Maximization Problem
86(2)
5.6.4 Degeneracy and Cycling in LPP
88(1)
5.6.5 Artificial Basis Technique
88(1)
5.6.6 Minimization Problem
89(2)
5.7 Solving LPP using LINGO-18
91(11)
5.7.1 Product Mix Profit Maximization Problem
92(1)
5.7.2 Product Sales through Advertisement Problem
93(1)
5.7.3 Profit Maximization Problem
94(1)
5.7.4 Total Production Cost Minimization Problem
95(2)
5.7.5 Product Manufacturing Problem
97(1)
5.7.6 Product Manufacturing Problem
97(1)
5.7.7 Production Cost Minimization Problem
98(2)
5.7.8 Profit Maximization Problem
100(1)
5.7.9 Cost Minimization Problem
101(1)
5.8 Concept of Duality in LPP
102(13)
5.8.1 Conversion of Primal to Dual
103(3)
5.8.2 Importance of Duality Concepts
106(3)
5.8.3 Properties of Primal-dual LPPs
109(1)
5.8.4 Economic Interpretation of Duality
109(2)
5.8.5 Dual Simplex
111(4)
6 Non-Linear Optimization Problems
115(26)
6.1 Introduction
115(2)
6.1.1 Basic Definitions
115(2)
6.1.2 Some Properties
117(1)
6.2 Lagrange Multiplier
117(1)
6.3 Kuhn-Tucker Conditions
118(1)
6.4 Solution of Non-Linear Optimization Problems using LINGO-18
119(3)
6.5 Quadratic Programming
122(7)
6.5.1 Wolf's Method
123(3)
6.5.2 Beale's Method
126(1)
6.5.3 Algorithm
126(3)
6.6 Solution of Quadratic Programming Problems using LINGO-18
129(2)
6.7 Convex Programming
131(7)
6.7.1 A Feasible Direction
131(1)
6.7.2 A Usable Feasible Direction
131(1)
6.7.3 An Outline of the Methods of Feasible Directions
132(1)
6.7.4 Rosen's Gradient Projection Method
132(3)
6.7.5 Kelly's Method
135(3)
6.8 Solution of Convex Programming Problems using LINGO-18
138(3)
7 Optimization Under Uncertainty
141(35)
7.1 Introduction
141(1)
7.2 Fuzzy Optimization
141(10)
7.2.1 Introduction
141(2)
7.2.2 Operations of Fuzzy Sets
143(2)
7.2.3 Cardinality of Fuzzy Set
145(6)
7.2.3.1 Scalar Cardinality
145(1)
7.2.3.2 Relative Cardinality
145(4)
7.2.3.3 a-cut Set
149(1)
7.2.3.4 Strong a-cut
149(1)
7.2.3.5 Level Set
149(2)
7.3 Fuzzy Numbers
151(1)
7.3.1 Triangular Fuzzy Number
151(1)
7.3.2 Trapezoidal Fuzzy Number
151(1)
7.4 Defuzzification Methods
152(4)
7.4.1 Center of Sums Method
152(2)
7.4.2 Center of Gravity Method
154(1)
7.4.3 a-cut Method
155(1)
7.5 Solving Optimization Problem with Fuzzy Numbers using LINGO-18
156(5)
7.5.1 Case I: When Coefficients in Objective Function are Fuzzy Numbers
157(1)
7.5.2 Case II: When Constraint Coefficients are Fuzzy Numbers
158(1)
7.5.3 Case III: When RHS Parameters are Fuzzy Numbers
159(2)
7.6 Stochastic Optimization Problem
161(6)
7.6.1 Situation I: Parameters in Objective Function Cj are Random Variables
161(1)
7.6.2 Situation II: Availability/Requirement Vector bi are Probabilistic
162(2)
7.6.3 Situation III: When aii are Random Variables
164(1)
7.6.4 Situation IV: General Case-All Parameters are Random Variables
165(2)
7.7 Some Numerical Examples using LINGO-18
167(3)
7.8 Interval Optimization Problem
170(6)
7.8.1 Introduction
170(1)
7.8.2 Interval Arithmetic
170(1)
7.8.3 Formulation of Interval Optimization Problem
171(1)
7.8.4 Algorithm
172(1)
7.8.5 Numerical Example using LINGO-18
173(3)
8 Multi-Objective Optimization
176(24)
8.1 Introduction
176(5)
8.1.1 Pareto Optimal Solution
178(1)
8.1.2 Ideal Point
179(1)
8.1.3 Anti-ideal Point
180(1)
8.1.4 Nadir Point
180(1)
8.1.5 Pareto Frontier
181(1)
8.2 Multi-Objective Optimization Techniques
181(7)
8.2.1 Weighted Technique
181(1)
8.2.2 Goal Programming Technique
182(3)
8.2.2.1 Algorithm of Goal Programming Technique
183(1)
8.2.2.2 Another Standard Form of GP
184(1)
8.2.3 Lexicographic Goal Programming Technique
185(1)
8.2.4 s-Constraints Technique
186(1)
8.2.5 Fuzzy Goal Programming Technique
186(2)
8.2.6 Fuzzy Goal Programming with Tolerance
188(1)
8.3 Numerical Example using LINGO-18
188(8)
8.3.1 Solution through Weighted Technique
189(1)
8.3.2 Solution through Goal Programming Technique
190(1)
8.3.3 Solution through Lexicographic Goal Programming Technique
191(1)
8.3.4 Solution through s-Constraints Technique
192(1)
8.3.5 Solution through Goal Programming Technique
193(2)
8.3.6 Solution through Fuzzy Goal Programming Technique
195(1)
8.4 Multi-Objective Optimization Problem with Fuzzy Numbers
196(1)
8.4.1 Algorithm
197(1)
8.5 Numerical Example using LINGO-18
197(3)
9 Applications of Optimization
200(30)
9.1 Optimization Problems in Finance
200(6)
9.2 Optimization Problems in Marketing
206(6)
9.3 Optimization Problems in Human Resource Management
212(2)
9.4 Vendor Selection Optimization Problem
214(3)
9.5 Diet Optimization Problem
217(4)
9.6 Operations Management Optimization Problems
221(4)
9.7 Transportation Optimization Problem
225(2)
9.8 Assignment Optimization Problem
227(3)
References 230(3)
Index 233
Neha Gupta is Assistant Professor at Amity School of Business, Amity University Uttar Pradesh, Noida, India. She holds a PhD in Operations Research from the Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India. She is a Gold Medalist in M.Sc. (Operations Research). Her broad research areas include Optimization Modelling, Decision Science, and Operations Research. She is a life-time member of the Operational Research Society of India (ORSI). She has published more than 25 research papers in journals of national and international repute. She is Editor of the International Journal of Mathematics and Systems Science, International Journal of Data Mining, Modelling and Management, Inderscience Publications, and reviewer of many journals of international repute. She has recently authored a book entitled Mathematical Programming: Techniques & Applications and is currently editing two books to be published by CRC Press, Taylor & Francis and Springer Nature.

Irfan Ali is currently Assistant Professor at the Department of Statistics and Operations Research, Aligarh Muslim University. He obtained B.Sc., M.Sc., M.Phil., and PhD in Statistics from the Aligarh Muslim University. Dr. Ali was a recipient of the Postgraduate Merit Scholarship award during his M.Sc. in Statistics in 2008 and was awarded the UGC-BSR Scholarship during his PhD (Statistics) program in 2013. He performs research in Applied Statistics and Optimization. He has completed a research project, UGC-Start-Up Grant Project, UGC New Delhi, India. He has published more than 75 research papers in reputed journals and serves as a reviewer for several journals. He is currently editing two books to be published by Taylor & Francis and Springer Nature. He is a life-time member of various professional societies: Operational Research Society of India, Indian Society for Probability and Statistics, Indian Mathematical Society and The Indian Science Congress Association. He is the associate editor of Aligarh Journal of Statistics (AJS).
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