Unit 2 monte carlo simulation

solved problems on monte carlo simulation

1. MONTE CARLO SIMULATION

2. System: The physical process of interest Model: Mathematical representation of the system Models are a fundamental tool of science, engineering, business, etc. Models always have limits of credibility Simulation: A type of model where the computer is used to imitate the behavior of the system Monte Carlo simulation: Simulation that makes use of internally generated (pseudo) random numbers 2 Basics

3. 3 Ways to Study System Focus of course System Experiment w/ actual system Experiment w/ model of system Physical Model Mathematical Model Analytical Model Simulation Model

4. “The Monte Carlo method is a numerical solution to a problem that models objects interacting with other objects . A Monte Carlo simulation is a model used to predict the probability of different outcomes when the intervention of random variables is present. Monte Carlo simulations help to explain the impact of risk and uncertainty in prediction and forecasting models. A variety of fields utilize Monte Carlo simulations, including finance, engineering, supply chain, and science. The basis of a Monte Carlo simulation involves assigning multiple values to an uncertain variable to achieve multiple results and then to average the results to obtain an estimate. It represents an attempt to model nature through direct simulation of the essential dynamics of the system in question. In this sense the Monte Carlo method is essentially simple in its approach. 4

5. B usiness and finance are plagued by random variables, Monte Carlo simulations have a vast array of potential applications in these fields. Monte Carlo Method: A Monte Carlo simulation takes the variable that has uncertainty and assigns it a random value. The model is then run and a result is provided. This process is repeated again and again while assigning the variable in question with many different values. Once the simulation is complete, the results are averaged together to provide an estimate. 5

6. let’s consider a simple system with simple inputs: 6 As A, B, C and D are always the same, the output will always be the same and it can be easily calculated I magine that input A has a range of possible values – the output will also be variable. And when there are many more possible inputs and all of them have a range of possible values, the output is not that simple to calculate. That’s where you need to use Monte Carlo simulation.

7. Steps in monte carlo simulation: Step 1:Clearly define the problem. Step 2:Construct the appropriate model. Step 3:Prepare the model for experimentation. Step 4:Using step 1 to 3,experiment with the model. Step 5:Summarise and examine the results obtained in step 4. Step 5:Evaluate the results of the simulation. 7

8. A manufacturing company keeps stock of a special product. Previous experience indicates the daily demand as given below 8 Daily demand 5 10 15 20 25 30 probability 0.01 0.20 0.15 0.50 0.12 0.02 Simulate the demand for the next 10 days. Also find the daily average demand for that product on the basis of simulated data. Consider the following random numbers: 82,96,18,96,20,84,56,11,52,03

9. Solution : Step 1:Generate tag values 9 Daily demands Probability Cumulative probability Tag values(Random num range) 5 0.01 0.01 00-00 10 0.20 0.21 01-20 15 0.15 0.36 21-35 20 0.50 0.86 36-85 25 0.12 0.98 86-97 30 0.02 1.00 98-99 Step 2: Simulate for 10 days Days Random num Daily demand 1 82 20 2 96 25 3 18 10 4 96 25 5 20 10 6 84 20 7 56 20 8 11 10 9 52 20 10 03 10 Average demand=(20+25+10+25+10+20+20+10+20+10)/ 10 = 170/10=17 units/day

10. 2)A tourist car operator finds that during the past few months the cars use has varied so much that the cost of maintaining the car varied considerably. During the past 200 days the demand for the car fluctuated as below 10 Trips per week Frequency 16 1 24 2 30 3 60 4 40 5 30 Using random numbers 82,96,18,96,20,84,56,11,52,03, simulate the demand for 10 week period

11. Solution: Step 1:Generate tag values 11 Trips/week frequency Probability Cumulative probability Tag values 16 16/200=0.08 0.08 00-07 1 24 24/200=0.12 0.20 08-19 2 30 30/200=0.15 0.35 20-34 3 60 60/200=0.30 0.65 35-64 4 40 40/200=0.20 0.85 65-84 5 30 30/200=0.15 1.00 85-99 Frequency-Number of occurrences, Total num of occurrences(16+24+30+60+40+30 )=200 Step 2: Simulation for next 10 week Weeks Random Num Trips/week 1 82 4 2 96 5 3 18 1 4 96 5 5 20 2 6 84 4 7 56 3 8 11 1 9 52 3 10 03 Avg trips/week=28/10=2.8≈3 trips/week

12. For a particular shop the daily demand of an item is given as follows, Use random numbers 25,39,65,76,12,05,73,89,19,49.Find the average daily demand. Daily demand 5 10 15 20 25 30 Probability 0.01 0.20 0.15 0.50 0.12 0.02 Solution: Generate tag values 12 Daily demand Probability Cumulative probability Tag values 0.01 0.01 00 10 0.20 0.21 01-20 20 0.15 0.36 21-35 30 0.50 0.86 36-85 40 0.12 0.98 86-97 50 0.02 1.00 98-99

13. Step 2: Simulation for 10 days 13 Days Random num Daily demand 1 25 20 2 39 30 3 65 30 4 76 30 5 12 10 6 05 10 7 73 30 8 89 40 9 19 10 10 49 30 Avg daily demand= 240/10=24

14. An automobile company manufactures around 150 scooters.Daily production varies from 146 to 154,the probability distribution is given below. Step 1:Generate tag values for production/day Step 1:Generate tag values for production/day 14 Production/day 146 147 148 149 150 151 152 153 154 probability 0.04 0.09 0.12 0.14 0.11 0.10 0.20 0.12 0.08 The finished scooters are transported in a lorry accomodading150 scooters. using the following random numbers 80,81,76,75,64,43,18,26,10,12,65,68,69,61 , 57 simulate 1)Average number of scooters waiting in the factory 2)Average number of empty space in the lorry

15. Step 1:Generate tag values for production/day 15 Production/day probability Cumulative probability Tag values 146 0.04 0.04 00-03 147 0.09 0.13 04-12 148 0.12 0.25 13-24 149 0.14 0.39 25-38 150 0.11 0.50 39-49 151 0.10 0.60 50-59 152 0.20 0.80 60-79 153 0.12 0.92 80-91 154 0.08 1.00 92-99 Step 2: Simulate for 15 days to get avg no of waiting scooters and empty space, lorry can accommodate 150 scooters Days Random num Production/day No of scooters waiting No of empty space in lorry 1 80 153 3 - 2 81 153 3 - 3 76 152 2 - 4 75 152 2 - 5 64 152 2 - 6 43 150 - - 7 18 148 - 2 8 26 149 - 1 9 10 147 - 3 10 12 147 - 3 11 65 152 2 - 12 68 152 2 - 13 69 152 2 - 14 61 152 2 - 15 57 151 1 -       Total=21 Total=9 Avg no of scooters waiting = 21/15=1.4 Avg No of space in the lorry = 9/15=0.6

16. An automobile production line turns out about 100 cars/day, but deviation occur owing to many causes.Production of cars are described by the probability distribution given below. 16 Production/day 95 96 97 98 99 100 101 102 103 104 105 106 probability 0.03 0.05 0.07 0.10 0.15 0.20 0.15 0.10 0.07 0.05 0.03 0.06 Finished cars are transported across the bay at the end of each day by ferry. If ferry has space for only 101 cars, what will be the average number of cars waiting to be shipped and what will be the average number of empty space on ship? Simulate the production of cars for next 15 days, consider the random numbers 97,02,80,66,96,55,50,29,58,51,04,86,24,39,47.

17. Step 1:generate tag values 17 Production/day Probabillity Cumulative probability Tag values 95 0.03 0.03 00-02 96 0.05 0.08 03-07 97 0.07 0.15 08-14 98 0.10 0.25 15-24 99 0.15 0.40 25-39 100 0.20 0.60 40-59 101 0.15 0.75 60-74 102 0.10 0.85 75-84 103 0.07 0.92 85-91 104 0.05 0.97 92-96 105 0.03 1.00 97-99

18. Step 2:Simulate for 15 days, ferry can transport 101 cars 18 Days Random numbers Productions/day No of cars waiting Empty space in the ship 1 97 105 |105-101|=4 - 2 02 95 - (101-95) =6 3 80 102 1   4 66 101 - - 5 96 104 3 - 6 55 100 - 1 7 50 100 - 1 8 29 99 - 2 9 58 100 - 1 10 51 100 - 1 11 04 96 - 5 12 86 103 2 - 13 24 98 - 3 14 39 99 - 2 15 47 100 - 1       Total=10 Total=23 Avg num of cars waiting=10/15 Avg empty space in the ship=23/15

19. Strong is a dentist who schedules all her patients for 30 minutes appointment. Some of the patients take more or less than 30min depending on the type of dental works to be done. The following summary shows the various categories of work,their probability and the time actually needed to complete the work 19 Category Filling crown cleaning extracting checkup Time required 45 60 15 45 15 Number of patients 40 15 15 10 20 Simulate the dentist clinic for 4 hrs and find out the avg waiting time for the patients as well as the idleness of doctor.Assume that the ptients show up at the clinic at exactly scheduled time. Arrival time starts at 8AM.Use the following random number for handling the same 40,82,11,34,25,66,19,79

20. category Time required No of patients(Frequency) probability Cumulative probability Tag values Filling 45 40 0.40 0.40 00-39 Crown 60 15 0.15 0.55 40-54 Cleaning 15 15 0.15 0.70 55-69 Extracting 45 10 0.10 0.80 70-79 Checkup 15 20 0.20 1.00 80-99     Total=100       20 Random num Category Time required (min) Arrival time of patients Service time Start time End time waiting time for patients(min) Idleness of doctor 40 crown 60 8.00 8.00 9.00 - 82 checkup 15 8.30 9.00 9.15 30(9-8.30) - 11 Filling 45 9.00 9.15 10.00 15 - 34 Filling 45 9.30 10.00 10.45 30 - 25 Filling 45 10.00 10.45 11.30 45 - 66 Cleaning 15 10.30 11.30 11.45 60 - 19 Filling 45 11.00 11.45 12.30 45 - 79 Extracting 45 11.30 12.30 1.15 60 - Step 1:find the cumulative probability and tag values Step 2:Simulate for 4 hrs Avg waiting time for patients=(30+15+30+45+60+45+60)/8=285/8=35.62 min≈36min Waiting time for patients=(start time of service-arrival time)

21. Bright Bakery keeps stock of a popular brand of cake. Previous experience indicates the daily demand as given below : Consider the following sequence of random numbers; 48, 78, 19, 51, 56, 77, 15, 14, 68,09 . Using this sequence simulate the demand for the next 10 days. Find out the stock situation if the owner of the bakery decides to make 30 cakes every day. Also estimate the daily average demand for the cakes on the basis of simulated data. 21 Daily demand 10 20 30 40 50 Probability 0.01 0.20 0.15 0.50 0.12 0.02

22. Daily demand Probability Cumulative probability Tag values 0.01 0.01 00 10 0.20 0.21 01-20 20 0.15 0.36 21-35 30 0.50 0.86 36-85 40 0.12 0.98 86-97 50 0.02 1.00 98-99 22 Step 1:find the cumulative probability and tag values Step 2:Simulate for 10 days, make 30 cakes every day Days Random num Daily Demand Stock condition 1 48 30 - 2 78 30 - 3 19 10 20 4 51 30 - 5 56 30 - 6 77 30 - 7 15 10 20 8 14 10 20 9 68 30 - 10 09 10 20 Avg daily demand=220/10=22

23. Verification and validation are critical parts of practical implementation Verification pertains to whether software correctly implements specified model Validation pertains to whether the simulation model (perfectly coded) is acceptable representation Are the assumptions reasonable? Accreditation is an official determination (U.S. DoD ) that a simulation is acceptable for particular purpose(s) Project Appraisal We can evaluate the likely profitability of a project using these techniques in the light of many uncertainties using this technique.   23 Verification, Validation, and Accreditation

24. RISK ANALYSIS AND MONTE CARLO SIMULATION Risk analysis is the systematic study of uncertainties and risks we encounter in business, engineering, public policy, and many other areas. Risk analysts seek to identify the risks faced by an institution or business unit, understand how and when they arise, and estimate the impact (financial or otherwise) of adverse outcomes . Uncertainty and risk are issues that virtually every business analyst must deal with, sooner or later . Monte Carlo simulation is a powerful quantitative tool often used in risk analysis . Uncertainty is an intrinsic feature of some parts of nature – it is the same for all observers. But risk is specific to a person or company – it is not the same for all observers . Most business and investment decisions are choices that involve “taking a calculated risk” – and risk analysis can give us better ways to make the calculation . Risk analysis in computers are done using what-if analysis. 24