Optimization Techniques and their Applications to Mine Systems

Description

This book describes the fundamental and theoretical concepts of optimization algorithms in a systematic manner, along with their potential applications and implementation strategies in mining engineering. It explains basics of systems engineering, linear programming, and integer linear programming, transportation and assignment algorithms, network analysis, dynamic programming, queuing theory and their applications to mine systems. Reliability analysis of mine systems, inventory management in mines, and applications of non-linear optimization in mines are discussed as well. All the optimization algorithms are explained with suitable examples and numerical problems in each of the chapters. Features include: • Integrates operations research, reliability, and novel computerized technologies in single volume, with a modern vision of continuous improvement of mining systems. • Systematically reviews optimization methods and algorithms applied to mining systems including reliability analysis. • Gives out software-based solutions such as MATLAB®, AMPL, LINDO for the optimization problems. • All discussed algorithms are supported by examples in each chapter. • Includes case studies for performance improvement of the mine systems. This book is aimed primarily at professionals, graduate students, and researchers in mining engineering. Cover Half Title Title Page Copyright Page Table of Contents Preface Author Biographies 1 Introduction to Mine Systems 1.1 Definition of a System 1.2 Types of System 1.3 System Approach 1.4 System Analysis 1.5 Elements of a Mining System 1.6 Definition and Classification of Optimization Problem 1.6.1 Based On the Existence of Constraints 1.6.2 Based On the Nature of the Equations Involved 1.6.3 Based On the Permissible Values of the Decision Variables 1.6.4 Based On the Number of the Objective Function 1.7 Solving Optimization Problems 1.7.1 Classical Optimization Techniques 1.7.1.1 Direct Methods 1.7.1.2 Gradient Methods 1.7.1.3 Linear Programming Methods 1.7.1.4 Interior Point Method 1.7.2 Advanced Optimization Techniques 2 Basics of Probability and Statistics 2.1 Definition of Probability 2.2 Additional Theory of Probability 2.3 Probability Distributions 2.4 Common Probability Distribution Functions 2.4.1 Uniform Distribution 2.4.2 Normal Distribution Case 1 Case 2 Case 3 Case 4 2.4.3 Poisson Distribution 2.4.4 Exponential Distribution 2.5 Conditional Probability 2.6 Memoryless Property of the Probability Distribution 2.7 Theorem of Total Probability for Compound Events 2.8 Bayes’ Rule 2.9 Definition of Statistics 2.10 Statistical Analyses of Data 2.10.1 Common Tools of Descriptive Statistics 2.10.1.1 Arithmetic Mean 2.10.1.2 Median 2.10.1.3 Mode 2.10.1.4 Standard Deviation 2.10.1.5 Mean Absolute Deviation 2.10.1.6 Skewness 2.10.1.7 Coefficient of Variation 2.10.1.8 Expectation Or Expected Value 2.10.1.9 Variance and Covariance 2.10.1.10 Correlation Coefficient 2.10.2 Standard Analysis Tools of Inferential Statistics 2.10.2.1 Hypothesis Tests 3 Linear Programming for Mining Systems 3.1 Introduction 3.2 Definition of Linear Programming Problem (LPP) 3.3 Solution Algorithms of LPP 3.3.1 Graphical Method 3.3.1.1 Multiple Solutions 3.3.1.2 Unbounded Solution 3.3.2 Simplex Method 3.3.3 Big-M Method 3.4 Sensitivity Analysis 3.4.1 Graphical Method of Sensitivity Analysis 3.4.2 Sensitivity Analysis of the Model Using Simplex Method 3.5 The Dual Problem 3.5.1 Formulation of Dual Problem for a Given Primal LPP 3.5.2 Dual Simplex Algorithm 3.6 Case Study of the Application of LPP in Optimization of Coal Transportation From Mine to Power Plants 4 Transportation and Assignment Problems in Mines 4.1 Definition of a Transportation Problem 4.2 Types of Transportation Problem 4.3 Solution Algorithms of a Transportation Model 4.3.1 Initial Basic Feasible Solution 4.3.1.1 The North-West Corner Method 4.3.1.2 Matrix Minimum Method 4.3.1.3 Vogel Approximation Method (VAM) 4.3.2 Determination of Optimal Solution 4.3.2.1 The Modified Distribution Method 4.3.2.2 Stepping Stone Method 4.3.3 Solution Algorithm of an Unbalanced Transportation Model 4.3.4 Solution Algorithm of a Transportation Model With Prohibited Routes 4.3.5 Solution Algorithm for Degeneracy Problem 4.4 Assignment Problem 4.4.1 The Hungarian Assignment Method (HAM) 4.5 Case Study On the Application of Transportation Model in Mining System Exercise 4 5 Integer Linear Programming for Mining Systems 5.1 Definition 5.2 Formulation of ILP 5.3 Solution Algorithms of an ILP 5.3.1 Cutting Plane Method Or Gomory’s Cut Method 5.3.2 Branch and Bound (B&B) Algorithm 5.4 Case Study of the Application of Mixed Integer Programming (MIP) in Production Scheduling of a Mine Appendix 5 Exercise 5 6 Dynamic Programming for Mining Systems 6.1 Introduction 6.2 Solution Algorithm of Dynamic Programming 6.3 Example 1: Maximising Project NPV 6.3.1 Backward Recursion Algorithm 6.3.2 Forward Recursion Algorithm 6.4 Example 2: Decision On Ultimate Pit Limit (UPL) of Two-Dimensional (2-D) Blocks 6.5 Example 3: Stope Boundary Optimization Using Dynamic Programming 6.6 Case Study of Dynamic Programming Applications to Determine the Ultimate Pit for a Copper Deposit Appendix 6 Loading Data Block Economic Value Calculation Preparing Data for Dynamic Programming Running Dynamic Programming in 2D Running Dynamic Programming in 3D Generating a File for Surpac Or Any Mine Planning Software 7 Network Analysis for Mining Project Planning 7.1 Introduction 7.2 Representation of Network Diagram 7.3 Methods of Determining the Duration of a Project 7.3.1 Critical Path Method (CPM) 7.3.2 Program Evaluation and Review Technique (PERT) 7.3.2.1 PERT Analysis Algorithm 7.4 Network Crashing 8 Reliability Analysis of Mining Systems 8.1 Definition 8.2 Statistical Concepts of Reliability 8.3 Hazard Function 8.4 Cumulative Hazard Rate 8.5 Reliability Functions 8.5.1 Reliability Calculation With an Exponential Distribution Function 8.5.2 Reliability Calculation With a Normal Probability Density Function 8.5.3 Reliability Calculation With a Weibull Distribution Probability Density Function 8.5.4 Reliability Calculation With a Poisson Distribution Probability Mass Function 8.5.5 Reliability Calculation for a Binomial Distribution 8.6 Mean Time Between Failure (MTBF) and Mean Time to Failure (MTTF) 8.7 Maintainability and Mean Time to Repair (MTTR) 8.8 Reliability of a System 8.8.1 System Reliability On a Series Configuration 8.8.2 System Reliability On Parallel Configuration 8.8.3 System Reliability of a Combination of Series and Parallel System 8.8.4 System Reliability of K-Out-Of-N Configuration 8.8.5 System Reliability of Bridge Configuration 8.8.6 System Reliability of Standby Redundancy 8.9 Availability 8.10 Improvement of System Reliability 8.10.1 Redundancy Optimization 8.11 Reliability Analysis to a Mine System: A Case Study 8.11.1 Introduction 8.11.2 Data 8.11.3 Exploratory Data Analysis 8.11.4 Estimating the Best Fit Probability Density Function (PDF) for TBF and TTR 8.11.5 Reliability Analysis for Estimation of Maintenance Schedule 9 Inventory Management in Mines 9.2 Costs Involved in Inventory Models 9.3 Inventory Models 9.3.1 Deterministic Model 9.3.1.1 Basic Economic Order Quantity (EOQ) Model 9.3.1.2 EOQ Model With Planned Shortages 9.3.1.3 EOQ Model With Price Discounts 9.3.1.4 Multi-Item EOQ Model With No Storage Limitation 9.3.1.5 Multi-Item EOQ Model With Storage Limitation 9.3.2 Fixed Time-Period Model 9.3.3 Probabilistic EOQ Model 10 Queuing Theory and Its Application in Mines 10.1 Introduction 10.2 Kendall Notation 10.3 Probability Distributions Commonly Used in Queuing Models 10.3.1 Geometric Distribution 10.3.2 Poisson Distribution 10.3.3 Exponential Distribution 10.3.4 Erlang Distribution 10.4 Relation Between the Exponential and Poisson Distributions 10.5 Little’s Law 10.6 Queuing Model 10.6.1 M/M/1 Model 10.6.1.1 Time-Dependent Behaviour of the Flows of Dump Trucks 10.6.2 M/M/s Queuing System 10.6.3 Infinite Server Queue Model (M/M/8) 10.6.4 (M/M/s): (FCFS)/K/K Queuing System Determining the Model Performance Measures Results and Discussion 10.7 Cost Models 10.8 Case Study for the Application of Queuing Theory for Shovel-Truck Optimization in an Open-Pit Mine 11 Non-Linear Algorithms for Mining Systems 11.1 Introduction 11.2 Stationary Points 11.3 Classifications of Non-Linear Programming 11.3.1 Unconstrained Optimization Algorithm for Solving Non-Linear Problems 11.3.2 Constrained Optimization Algorithm for Solving Non-Linear Problems KKT Conditions for Minimization of the Objective Function at the Stationary Point Indices Parameters Variables Objective Function Constraints 11.4 Case Study On the Application of Non-Linear Optimization for Open-Pit Production Scheduling Bibliography Index