1. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter 16 Markov Analysis

2. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives Students will be able to Determine future states or conditions using Markov analysis. Compute long-term or steady- state conditions using only the matrix of transition. Understand the use of absorbing state analysis in predicting future conditions.

3. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter Outline 16.1 Introduction 16.2 States and State Probabilities 16.3 Matrix of Transition Probabilities 16.4 Predicting Future Market Share 16.5 Markov Analysis of Machine Operations 16.6 Equilibrium Conditions 16.7 Absorbing States and the Fundamental Matrix: Accounts Receivable Applications

4. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Assumptions of Markov Analysis 1. A finite number of possible states. 2. Probability of change remains the same over time. 3. Future state predictable from current state. 4. Size of system remains the same. 5. States collectively exhaustive. 6. States mutually exclusive.

5. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Markov Process Matrix of Transition New State Current State   P  

6. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Markov Process Equations P 11 P 12 P 13...P 1n P 21 P 22 P 23...P 2n P m1... P mn Matrix of transition probabilities = P =  (i) = State probabilities = [  1  2  3 …  n ]  (i+1) =  (i)P

7. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Predicting Future States

8. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Machine Example: Periods to Reach Equilibrium Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 State 1 1.0.8.66.562.4934.44538.411766.388236.371765.360235.352165.346515.342560.339792.337854 0.0.2.34.438.5066.55462.588234.611763.628234.639754.647834.653484.657439.660207.662145 State 2

9. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Equilibrium Equations

10. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 16-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Markov Process Fundamental Matrix Where I = Identify matrix, and 0 = Null matrix Then And FA indicates the probability that an amount in one of the non-absorbing states will end up in one of the absorbing states.