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.
Author(s): Neha Gupta, Irfan Ali
Publisher: CRC Press
Year: 2021
Language: English
Pages: 252
City: Boca Raton
Cover
Title Page
Copyright Page
Preface
Table of Contents
Authors’ Biographies
Section I: Essential Mathematics and Software
1. Basic Calculus
1.1 Distance Formula for Points in the Plane
1.2 Circle Centered at the Origin
1.3 Straight Lines
1.4 Angle of Inclination
1.5 Parallel and Perpendicular Lines
1.6 Function
1.7 Domain
1.8 Functions Graphs
1.9 Composite Functions
1.10 Even and Odd Functions-Symmetry
1.11 Piecewise Defined Functions
1.12 Shifting Graphs
1.13 Average Rate of Change and Secant Lines
1.14 Continuity
1.14.1 Lipschitz Continuity
1.15 Slope of a Curvilinear Function
1.16 Vertical Tangent
1.17 Derivative of the Function
1.18 Differentiable in Interval
1.19 Extreme Values of Functions
1.20 Local and Absolute (Global) Extrema
1.21 Increasing Functions and Decreasing Functions
1.22 Tests Based on First Derivative
1.23 Concavity
1.24 Inflection Point
1.25 Linearization Approximation
2. Matrix and Determinant Algebra
2.1 Matrix
2.2 Types of Matrix
2.3 Algebra of Matrix
2.3.1 Addition and Subtraction of Matrix
2.3.2 Multiplication by a Scalar
2.3.3 Matrix Multiplication
2.3.4 Laws of Algebra
2.4 Transpose of Matrix
2.5 Trace of Matrix
2.6 Non-Singular Matrix
2.7 Inverse of a Matrix
2.8 Orthogonal Matrix
2.9 Rank of Matrix
2.10 Determinants and Non-Singularity
2.11 Three Order Determinants
2.12 Linear Equations
2.13 Vector Algebra
2.13.1 Metric Spaces
2.13.2 Normed Linear Spaces
2.13.3 Norms
2.13.4 Forbenlus Norm
2.13.5 Eigenvalues Vectors
3. LINGO-18: Optimization Software
3.1 Introduction
3.1.1 Key Benefits of LINGO
3.1.2 Installing LINGO
3.1.3 Window Types in LINGO
3.1.4 Model Creation and Solution in LINGO
3.1.5 Logical Operators and Set Looping Functions
3.1.6 Variable Domain Functions
3.1.7 Shortcut Keys
Section II: Optimization
4. Introduction to Optimization
4.1 Introduction
4.1.1 Unconstrained vs Constrained
4.1.2 Linear vs Non-Linear
4.2 Basic Definitions
4.2.1 Affine Sets
4.2.2 Convex Set
4.2.3 Convex Function
4.2.4 Concave Function
4.3 Convex and Concave Properties
4.4 Convex Optimization Problems
4.5 Concave Maximization Problem
4.5.1 Quasi Concave & Quasi Convex Function
4.5.1.1 Properties of Quasi Concave Functions
4.6 Convexity of Smooth Function
4.6.1 Quasi Concavity of Smooth Function
4.6.2 Pseudo Concave Function
4.6.3 Pseudo Convex Function
4.6.4 Cone
4.6.5 Hyperplanes and Half Spaces
4.7 Univariate Optimization Problems
4.7.1 Optimality Condition
4.8 Multivariate Optimization Problems
4.8.1 Optimality Condition
4.9 Gradient Vector of f (x)
4.9.1 Hessian Matrix of f (x)
5. Linear Optimization Problems
5.1 Introduction
5.2 General Form of LP Model
5.2.1 Components of LP Model
5.2.2 Assumptions
5.2.3 General Mathematical Model
5.2.4 Properties of Solution to an LPP
5.3 Formulation of an LPP
5.3.1 Product Mix Problem
5.3.2 Resource Allocation Problem
5.3.3 Diet Problem
5.3.4 The Capital Budgeting Problem
5.3.5 Assignment Problem
5.3.6 Transportation Problem
5.4 Integer Programming
5.5 Graphical Solution of LPP
5.6 Theory of Simplex Method
5.6.1 Standard Form
5.6.2 Computational Procedure of Simplex Method
5.6.3 Maximization Problem
5.6.4 Degeneracy and Cycling in LPP
5.6.5 Artificial Basis Technique
5.6.6 Minimization Problem
5.7 Solving LPP using LINGO-18
5.7.1 Product Mix Profit Maximization Problem
5.7.2 Product Sales through Advertisement Problem
5.7.3 Profit Maximization Problem
5.7.4 Total Production Cost Minimization Problem
5.7.5 Product Manufacturing Problem
5.7.6 Product Manufacturing Problem
5.7.7 Production Cost Minimization Problem
5.7.8 Profit Maximization Problem
5.7.9 Cost Minimization Problem
5.8 Concept of Duality in LPP
5.8.1 Conversion of Primal to Dual
5.8.2 Importance of Duality Concepts
5.8.3 Properties of Primal-dual LPPs
5.8.4 Economic Interpretation of Duality
5.8.5 Dual Simplex
6. Non-Linear Optimization Problems
6.1 Introduction
6.1.1 Basic Definitions
6.1.2 Some Properties
6.2 Lagrange Multiplier
6.3 Kuhn-Tucker Conditions
6.4 Solution of Non-Linear Optimization Problems using LINGO-18
6.5 Quadratic Programming
6.5.1 Wolf’s Method
6.5.2 Beale’s Method
6.5.3 Algorithm
6.6 Solution of Quadratic Programming Problems using LINGO-18
6.7 Convex Programming
6.7.1 A Feasible Direction
6.7.2 A Usable Feasible Direction
6.7.3 An Outline of the Methods of Feasible Directions
6.7.4 Rosen’s Gradient Projection Method
6.7.5 Kelly’s Method
6.8 Solution of Convex Programming Problems using LINGO-18
7. Optimization Under Uncertainty
7.1 Introduction
7.2 Fuzzy Optimization
7.2.1 Introduction
7.2.2 Operations of Fuzzy Sets
7.2.3 Cardinality of Fuzzy Set
7.2.3.1 Scalar Cardinality
7.2.3.2 Relative Cardinality
7.2.3.3 α-cut Set
7.2.3.4 Strong α-cut
7.2.3.5 Level Set
7.3 Fuzzy Numbers
7.3.1 Triangular Fuzzy Number
7.3.2 Trapezoidal Fuzzy Number
7.4 Defuzzification Methods
7.4.1 Center of Sums Method
7.4.2 Center of Gravity Method
7.4.3 α-cut Method
7.5 Solving Optimization Problem with Fuzzy Numbers using LINGO-18
7.5.1 Case I: When Coefficients in Objective Function are Fuzzy Numbers
7.5.2 Case II: When Constraint Coefficients are Fuzzy Numbers
7.5.3 Case III: When RHS Parameters are Fuzzy Numbers
7.6 Stochastic Optimization Problem
7.6.1 Situation I: Parameters in Objective Function Cj are Random Variables
7.6.2 Situation II: Availability/Requirement Vector bi are Probabilistic
7.6.3 Situation III: When aij are Random Variables
7.6.4 Situation IV: General Case—All Parameters are Random Variables
7.7 Some Numerical Examples using LINGO-18
7.8 Interval Optimization Problem
7.8.1 Introduction
7.8.2 Interval Arithmetic
7.8.3 Formulation of Interval Optimization Problem
7.8.4 Algorithm
7.8.5 Numerical Example using LINGO-18
8. Multi-Objective Optimization
8.1 Introduction
8.1.1 Pareto Optimal Solution
8.1.2 Ideal Point
8.1.3 Anti-ideal Point
8.1.4 Nadir Point
8.1.5 Pareto Frontier
8.2 Multi-Objective Optimization Techniques
8.2.1 Weighted Technique
8.2.2 Goal Programming Technique
8.2.2.1 Algorithm of Goal Programming Technique
8.2.2.2 Another Standard Form of GP
8.2.3 Lexicographic Goal Programming Technique
8.2.4 ε-Constraints Technique
8.2.5 Fuzzy Goal Programming Technique
8.2.6 Fuzzy Goal Programming with Tolerance
8.3 Numerical Example using LINGO-18
8.3.1 Solution through Weighted Technique
8.3.2 Solution through Goal Programming Technique
8.3.3 Solution through Lexicographic Goal Programming Technique
8.3.4 Solution through ε-Constraints Technique
8.3.5 Solution through Goal Programming Technique
8.3.6 Solution through Fuzzy Goal Programming Technique
8.4 Multi-Objective Optimization Problem with Fuzzy Numbers
8.4.1 Algorithm
8.5 Numerical Example using LINGO-18
9. Applications of Optimization
9.1 Optimization Problems in Finance
9.2 Optimization Problems in Marketing
9.3 Optimization Problems in Human Resource Management
9.4 Vendor Selection Optimization Problem
9.5 Diet Optimization Problem
9.6 Operations Management Optimization Problems
9.7 Transportation Optimization Problem
9.8 Assignment Optimization Problem
References
Index
