Norwegian University of Science and Technology TPK4161 - Supply chain analytics Author: Jørn Vatn September 10, 2020 Contents 1 Introduction 2 1.1 Course content ....................................... 2 1.2 Learning outcome ...................................... 2 2 Probability theory 3 2.1 Basic probability notation ................................. 3 2.1.1 Event ........................................ 3 2.1.2 Probability ..................................... 3 2.1.3 Probability and Kolmogorov’s axioms ...................... 4 2.1.4 The law of total probability ............................ 5 2.1.5 Bayes theorem ................................... 6 2.1.6 Stochastic variables ................................ 7 2.2 Common probability distributions ............................ 10 2.2.1 The normal distribution .............................. 11 2.2.2 The exponential distribution ........................... 12 2.2.3 The Weibull distribution .............................. 12 2.2.4 The gamma distribution .............................. 13 2.2.5 The inverted gamma distribution ......................... 13 2.2.6 The lognormal distribution ............................ 14 2.2.7 The binomial distribution ............................. 14 2.2.8 The Poisson distribution .............................. 14 2.2.9 The inverse-Gauss distribution .......................... 15 2.2.10 The triangular distribution ............................ 16 2.2.11 The PERT distribution .............................. 16 2.3 Assessment of parameters in parametric distributions .................. 17 2.4 Distribution of sums, products and maximum values .................. 18 2.4.1 Distribution of sums ................................ 18 2.4.2 Distribution of a product ............................. 19 2.4.3 Distribution of maximum values ......................... 19 3 Introduction to modelling 22 3.1 Deterministic and probabilistic models .......................... 22 3.2 Problem formulation - Modelling ............................. 22 4 Discrete event simulation 25 4.1 Introduction ......................................... 25 4.2 Components of a Discrete-Event Simulation ....................... 26 1 4.3 Simulation Engine Logic .................................. 27 4.4 Implementing the pending event set (PES) ........................ 28 4.5 Library of functions for generating pseudorandom numbers .............. 31 4.6 A simple failure and repair model ............................. 31 5 Linear, dynamic, non-linear and stochastic programming 34 5.1 Introduction to programming problems .......................... 34 5.2 Linear programming .................................... 34 5.2.1 Motivating example ................................ 35 5.2.2 Linear programming problem on standard form ................. 35 5.2.3 Solving the linear programming problem by the SIMPLEX method ...... 36 5.2.4 Demonstration of the SIMPLEX method .................... 37 5.2.5 Summing up the SIMPLEX method ....................... 39 5.2.6 Unique optimal, multiple optimal and unbounded solutions .......... 40 5.2.7 Shadow prices .................................... 40 5.2.8 Solving the LP problem by a computer ..................... 41 5.2.9 Mixed integer programming ............................ 45 5.3 Dynamic programming ................................... 49 5.3.1 Worked example .................................. 51 5.4 Nonlinear programming .................................. 56 5.5 Stochastic programming .................................. 58 5.5.1 Introduction .................................... 58 5.5.2 Discretization .................................... 59 5.5.3 The Value of the Stochastic Solution ....................... 61 5.5.4 Expected value of perfect information ...................... 62 5.5.5 Scenario building .................................. 63 5.5.6 How to perform discretization? .......................... 64 6 Flow and network modelling 66 6.1 Transportation problems .................................. 66 6.1.1 Worked example in MS Excel ........................... 67 6.2 Job assignment problems ................................. 68 6.2.1 Maximal-flow problems .............................. 69 6.2.2 A worked maximal-flow problem in MS Excel .................. 70 6.3 Project management .................................... 71 6.4 Critical Path Method (CPM) ............................... 73 6.5 Linear programming (LP) ................................. 74 6.5.1 Slack ......................................... 75 6.6 Program Evaluation and Review Technique (PERT) .................. 76 6.7 Successive schedule planning (SSP) ............................ 77 6.8 Monte Carlo simulation (MCS) .............................. 80 6.9 Penalty for default ..................................... 82 7 Markov processes and queueing theory 85 7.1 Markov processes ...................................... 85 7.1.1 Markov state equations .............................. 87 7.1.2 Time dependent solution for the Markov process ................ 87 7.1.3 Steady state solution for the Markov process .................. 88 2 7.1.4 Mean time to first passage to a given state ................... 89 7.2 Birth-death processes .................................... 90 7.3 Queue theory models .................................... 92 7.3.1 The M=M=1 queue ................................. 93 7.3.2 The M=M=1=N queue ............................... 97 7.3.3 The M=M=C=N queue ............................... 98 7.3.4 The M=Ek=1=N queue ............................... 99 7.3.5 Final remarks .................................... 101 8 Inventory models 102 8.1 Introduction ......................................... 102 8.2 The classical economic order quantity .......................... 102 8.3 Probabilistic models .................................... 103 8.3.1 The newsboy problem ............................... 103 8.3.2 A lot size, reorder point policy; (r; Q) ...................... 105 9 Reliability and maintenance 110 9.1 Definitions .......................................... 110 9.2 Reliability terminology ................................... 110 9.2.1 System structure analysis ............................. 113 9.2.2 Systems of independent components ....................... 115 9.3 Maintenance model - Single component considerations ................. 117 9.3.1 Preventive maintenance .............................. 117 9.3.2 Single activity - Calendar based maintenance .................. 118 9.3.3 Synchronization of maintenance and production ................. 119 9.3.4 Predictive maintenance .............................. 120 9.3.5 Predictive maintenance and cyber physical systems ............... 122 10 Decision under uncertainties 127 10.1 Introduction ......................................... 127 10.1.1 Overview of the method .............................. 127 10.2 Basic concepts ....................................... 128 10.2.1 Discrete end consequences vs attribute vector .................. 129 10.2.2 Maximising expected utility ............................ 133 10.2.3 Examples with one decision node ......................... 135 10.2.4 Decision trees .................................... 136 11 Parameter estimation 142 11.1 Introduction ......................................... 142 11.2 The MLE principle ..................................... 142 11.3 Method of moments – PERT distribution ........................ 143 11.4 The LS principle ...................................... 144 11.5 Bayesian meothds ...................................... 147 12 Forecasting 149 12.1 Introdcution ......................................... 149 12.2 Naïve approach ....................................... 149 12.3 Average approach ...................................... 150 3 12.4 Moving average method .................................. 150 12.5 Simple exponential smoothing method .......................... 150 12.6 Holt’s method ........................................ 151 12.7 Holt-Winters additive method ............................... 152 12.8 Holt-Winters multiplicative method ............................ 153 Bibliography 153 Index 158 4 Chapter 1 Introduction This course compendium is covering the main aspects of the topics in the course TPK4161 - Supply Chain Analytics. TPK4161 was first lectured autumn 2017. Excel files used will be uploaded on Blackboard. 1.1 Course content Introduction to mathematical modelling as a tool to address challenges in production logistics and supply chains. Problem formulation and choice of modelling. Linear, dynamic and non-linear pro- gramming. Flow and network modelling. Queueing models and Markov chains. Some analytical results and use of discrete event simulation. Monte Carlo simulation Systems dynamics modelling. Forecasting. Statistical process control. Reliability and maintenance of the production line. Syn- chronization of maintenance and production activities. Multivariate regression analysis for analysis of performance data. Statistical techniques for estimation of model parameters. Machine learning and Big Data. Quantitative methods for artificial intelligence. Methods for economical analysis and especially activity based calculation. Models and visualization of cyber physical systems in real time. Decision trees. Expected Utility theory. 1.2 Learning outcome Knowledge: Basic insight in mathematical formulation of operations and supply chain manage- ment problems. Ability to analyse and understand real problems in order to develop realistic models where text book examples are not sufficient. Understand