Optimization Techniques and their Applications to Mine Systems

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