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- 1. Chapter 10 Monte Carlo Simulation and the Evaluation of Risk Chemical Engineering Department West Virginia University Copyright - R.Turton and J. Shaeiwitz 2012 1
- 2. Outline Copyright - R.Turton and J. Shaeiwitz 2012 2 Causes of uncertainty in profitability calculations Forecasting Quantification of risk Best-case - worst-case Monte-Carlo method and probability distributions Using CAPCOST
- 3. Factors Affecting Profitability From Table 10.1 Cost of Fixed Capital Investment1 -10 to +25 Construction Time -5 to +50 Start-up Costs and Time -10 to +100 Sales Volume -50 to +150 Price of Product -50 to +20 Plant Replacement and Maintenance Costs -10 to +100 Income Tax Rate -5 to +15 Inflation Rates -10 to +100 Interest Rates -50 to + 50 Working Capital -20 to +50 Raw Material Availability and Price -25 to +50 Salvage Value -100 to +10 Profit -100 to +10 Copyright - R.Turton and J. Shaeiwitz 2012 3
- 4. Forecasting – Prediction of Future Trends Copyright - R.Turton and J. Shaeiwitz 2012 4 demand supply Quantity of X demanded, Q (per year) Demand: As P demand increases Supply: As P more supply will become available Market will reach equilibrium when Supply = Demand New plant comes on line – so supply curve shifts down and Pequilib
- 5. Historical Data Copyright - R.Turton and J. Shaeiwitz 2012 5 • Variation around trend line = ± 35c/gal • Build Plant in 1998 or 2005!
- 6. Difficulty in Forecasting Copyright - R.Turton and J. Shaeiwitz 2012 6 According to Yogi Berra “It’s tough to make predictions, especially about the future”
- 7. Quantifying Risk Copyright - R.Turton and J. Shaeiwitz 2012 7 Example 10.1 and 10.2 R= $75 million per year COMd = $30 million per year FCIL = $150 million NPV = $17.12 million What if variation of 3 parameters is R – 20% to +5%, COMd –10% to +10%, FCIL +30% to –20%?
- 8. Quantifying Risk Copyright - R.Turton and J. Shaeiwitz 2012 8 Best Case – Worst Case Scenario Worst Case (all figures in $million or $million/yr) R = (75)(0.8) = 60 COMd = (30)(1.1) = 33 FCIL = (150)(1.3) = 195 Best Case R = (75)(1.05) = 78.75 COMd = (30)(0.9) = 27 FCIL = (150)(0.8) = 120 NPV = -59.64 NPV = 53.62 What does this tell us? - not much!
- 9. Quantifying Risk Copyright - R.Turton and J. Shaeiwitz 2012 9 The problem with the best case –worst case scenario is that neither case is very likely! If each variation were equally likely, i.e., the high, average, and low values could each occur with the same probability then we would have 33 = 27 equally possible outcomes
- 10. Quantifying Risk Copyright - R.Turton and J. Shaeiwitz 2012 10 Scenario R1 COMd 1 FCIL 1 Probability of Occurrence 1 -20% -10% -20% (1/3)(1/3)(1/3) = 1/27 2 -20% -10% 0% 3 -20% -10% +30% 4 -20% 0% -20% 5 -20% 0% 0% 6 -20% 0% +30% 7 -20% +10% -20% 8 -20% +10% 0% 9 (worst) -20% +10% +30% 10 0% -10% -20%
- 11. Quantifying Risk Copyright - R.Turton and J. Shaeiwitz 2012 11 Assign Probabilities to values using probability distributions leads to the Monte Carlo Method (MC) We use an 8-step method to describe MC
- 12. Quantifying Risk Copyright - R.Turton and J. Shaeiwitz 2012 12 1. All the parameters for which uncertainty is to be quantified are identified. 2. Probability distributions are assigned for all parameters in step 1 above. 3. A random number is assigned for each parameter in step 1 above. 4. Using the random number from step 3, the value of the parameter is assigned using the probability distribution (from step 2) for that parameter. 5. Once values have been assigned to all parameters, these values are used to calculate the profitability (NPV or other criterion) of the project. 6. Steps 3, 4, and 5 are repeated many times (say 1000). 7. A histogram and cumulative probability curve for the profitability criteria calculated from step 6 are created. 8. The results of step 7 are used to analyze the profitability of the project.
- 13. Probability Distributions Copyright - R.Turton and J. Shaeiwitz 2012 13 Uniform Distribution Probability density function p(x) a b 1 b - a a b 1 0 Cumulative probability function P(x) x p(x) x P(x)
- 14. Probability Distributions Copyright - R.Turton and J. Shaeiwitz 2012 14 Triangular Distribution Probability density function, p(x) a b c 2 c - a a b c 1 0 Cumulative probability function, P(x) P(x) x x p(x)
- 15. Probability Distributions Copyright - R.Turton and J. Shaeiwitz 2012 15 Triangular Distribution – used in CAPCOST Triangular probability density function: (10.9) Triangular cumulative probability function (10.10) 2( ) ( ) for ( )( ) 2( ) ( ) for ( )( ) x a p x x b c a b a c x p x x b c a c b 2 ( ) ( ) for ( )( ) ( ) ( )(2 ) ( ) for ( ) ( )( ) x a P x x b c a b a b a x b c x b P x x b c a c a c b
- 16. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 16 Monte Carlo Method 1. Identify parameters = R, COMd, FCIL 2. Probability distributions assigned – use low, medium and high values for a, b, c in triangular distribution 3. and 4. As an example – look at R a = 60, b = 75, c = 78.75 (-20% - +5%, BC = 75) P(x = b) = (b-a)2/(c-a)(b-a) =15/18.75= 0.8 Generate a random number (RN) (0,1) = 0.3501
- 17. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 17 Monte Carlo Method Since RN < 0.8 use first part of Eqn (10.10) 2 2 ( ) ( ) for ( )( ) ( 60) 0.3501 69.92 (78.75 60)(75 60) x a P x x b c a b a x x
- 18. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 18 b = 75 First part of curve – Eqn (10.10) x<b 0.3501 x = 69.92 0.80
- 19. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 19 • Using R = x = 69.92 • Choose RNs for COMd and FCIL and repeat procedure to get values for these parameters • Calculate NPV • Repeat many times (1000) and plot frequency (distribution) of NPV Figure 10.15 shows NPV distribution for this problem
- 20. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 20
- 21. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 21 Project B Project A 1.00 0.50 0.00 -40 -20 0 20 40 60 0.17 0.02 Net Present Value ($ Millions) Cumulative Probability Figure 8.16: A Comparison of the Profitability of Two Projects Showing the NPV with Respect to the Estimated Cumulative Probability from a Monte Carlo Analysis
- 22. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 22
- 23. Monte Carlo Method Copyright - R.Turton and J. Shaeiwitz 2012 23 Results using Capcost for Monte Carlo Simulations
- 24. Summary Copyright - R.Turton and J. Shaeiwitz 2012 24 • The quantification of risk allows a more complete interpretation of the economic potential of a new project • The Monte-Carlo method is a convenient tool for quantifying the risk associated with factors affecting a project’s profitability • Capcost may be used to run Monte-Carlo simulations on a process

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