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SIMULATION An attempt to duplicate the features, appearance, and characteristics of a real system Applied Management Science for Decision Making, 1e ©

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Presentation on theme: "SIMULATION An attempt to duplicate the features, appearance, and characteristics of a real system Applied Management Science for Decision Making, 1e ©"— Presentation transcript:

1 SIMULATION An attempt to duplicate the features, appearance, and characteristics of a real system Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

2 HISTORY Developed by John von Neumann to solve nuclear physics problems that were too complex and costly to solve manually during the Manhattan Project in 1944 Specialized computer languages were developed in the 1960s such were developed in the 1960s such as GASP, Dynamo,and Simscript as GASP, Dynamo,and Simscript

3 Simulation Applications  Train scheduling  Manufacturing  Sales forecasting  Inventory planning and control  Distribution system design  Plant layout  Aircraft scheduling   War games  Waiting line analysis  Economic forecasting  Ambulance location and dispatching  Traffic light timing  Investment portfolios  Highway planning  Environmental planning

4 The Idea Behind Simulation  To imitate a real-world situation mathematically.  To then study its properties and operating characteristics.  To draw conclusions and make action decisions based on the simulated results.

5 To Use Simulation   Define the problem  Introduce the critical variables associated with the problem  Construct a numerical model  Set up possible courses of action for testing   Run the experiment  Consider the results  Perhaps modify the model or change the data inputs  Decide what course of action to take

6 Relevant Variable Any variable or factor that affects the behavior of the real-world system. The goal is to identify and integrate as many of these variables as possible into the simulation, so as to account for approximately 85% of a system’s actual behavior.

7 Relevant Variables U.S. Econometric Model Relevant Variables U.S. Econometric Model McGraw-Hill – DRI, Lexington, MA. PLUS 1,000 ADDITIONAL VARIABLES NOT SHOWN, WITH MORE BEING IDENTIFIED AND ADDED EVERY YEAR  Trade deficit / surplus for goods  Trade deficit / surplus for services  Prime interest rate  Money supply  Average worker productivity  Unemployment rate  Capital investment by private industry  Government spending

8 Types of System Behavior Simulated Growth or contraction of an economy. Actions of rival firms. Ecosystems such as marshlands & lakes. Product demand. Financial markets. Customer arrivals. Land, air, sea traffic. Spread of epidemics. Enemy tactics in combat. Equipment breakdowns.

9 simulating the “unthinkable”

10 The Iconic Simulation  Uses a physical model to replicate a real-world system.  From the Greek word ίκων ( icon ) or “image”.  A flight simulator is used to train pilots without jeopardizing real people or aircraft.  Wind tunnels are employed to evaluate aircraft designs for fuel efficiency and fuselage integrity.

11 MonteCarloSimulation When a system contains elements that exhibit chance behavior, the Monte Carlo method of simulation is applied. THE MOST USED SIMULATION IN BUSINESS

12 Television Sales Example Suppose a store wanted to estimate a salesperson’s average daily commissions on televisions sold. Assume s/he receives a $10.00 commission on each television sold, and that only one model is offered. Assume also, that only six ( 6 ) levels of sales are possible on a given business day: 0,1,2,3,4, or 5. The store wants to run the simulation over ten ( 10 ) business days.

13 Monte Carlo Simulation Steps 1. Set up a probability distribution for each relevant variable.

14 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 1.20 2.30 3.20 4.10 5.05 ONE RELEVANT VARIABLE WITH 6 POSSIBLE VALUES

15 Monte Carlo Simulation Steps 2. Set up a cumulative probability distribution for the relevant variable.

16 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 1.20.35 2.30.65 3.20.85 4.10.95 5.051.00 HOW TO COMPUTE THE CUMULATIVE PROBABILITIES

17 Monte Carlo Simulation Steps 3. Set up random number intervals for the relevant variable.

18 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 HOW TO SET UP THE RANDOM NUMBER INTERVALS

19 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 HOW TO SET UP THE RANDOM NUMBER INTERVALS THE CUMULATIVE PROBABILITY BECOMES THE UPPER BOUND OF THE RN INTERVAL

20 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 THE COMPLETED SIMULATION SPREADSHEET

21 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 00 - 14 1.20.3515 - 34 2.30.6535 - 64 3.20.8565 - 84 4.10.9585 - 94 5.051.0095 - 99 ALTERNATE SIMULATION SPREADSHEET

22 Random Number Intervals A block of consecutive numbers used to represent each possible value or outcome of a relevant variable in the simulation. SOME RELEVANT VARIABLES HAVE ONLY A FEW POSSIBLE VALUES OR OUTCOMES, BUT OTHERS MAY HAVE HUNDREDS OR THOUSANDS !

23 Random Number Interval Assumption ( Television Daily Sales Simulation ) We arbitrarily select a block of consecutive numbers from “1” to “100” and assign those numbers to each television set “sale event” on a proportional basis. 0 Sales 1 Sale 2 Sales 3 Sales 4 Sales 5 Sales WE JUST AS EASILY COULD HAVE USED A CONSECUTIVE BLOCK OF NUMBERS FROM “1” to “1,000” or “1” to “10,000”

24 In Other Words…… 15% 01 – 15 Since P ( 0 Sales ) = 15%, the numbers 01 – 15 will be assigned to, and represent that particular demand event. 20% 16 – 35 Since P ( 1 Sale ) = 20%, the numbers 16 – 35 will be assigned to, and represent that particular demand event. 30% 36 – 65 Since P ( 2 Sales ) = 30%, the numbers 36 – 65 will be assigned to, and represent that particular demand event. 15% First 15% of NUMBERS 20% Next 20% of NUMBERS 30% Next 30% of NUMBERS

25 Monte Carlo Simulation Steps 4. Generate random numbers.

26 Generating Random Numbers  If the simulation is large and involves many trials, computer programs are available to generate the needed random numbers.  If the simulation is small and involves few trials, random number tables are available from which to select the random numbers.

27 Table of Random Numbers 52 06 50 88 53 30 10 47 99 37 66 91 35 32 00 84 57 07 37 63 28 02 74 35 24 03 29 60 74 85 90 73 59 55 17 60 82 57 68 28 05 94 03 11 27 79 90 87 92 41 09 25 36 77 69 02 36 49 71 99 32 10 75 21 95 89 38 97 73 37 77 13 10 01 18 32 85 94 81 43 58 33 26 50 48 61 18 85 23 08 54 17 12 80 69 24 84 92 16 49 59 27 88 21 62 69 48 31 12 73 05 15 30 34 87 01 14 FROM A STATISTICS OR MATHEMATICS TEXT

28 If the random number intervals used 2-digit numbers, what would we need to do to a random number table? Answer We would need to break the numbers in the random number table into 2-digit numbers as well. Question

29 Table of Random Numbers 52 06 50 88 53 30 10 47 99 37 66 91 35 32 00 84 57 07 37 63 28 02 74 35 24 03 29 60 74 85 90 73 59 55 17 60 82 57 68 28 05 94 03 11 27 79 90 87 92 41 09 25 36 77 69 02 36 49 71 99 32 10 75 21 95 89 38 97 73 37 77 13 10 01 18 32 85 94 81 43 58 33 26 50 48 61 18 85 23 08 54 17 12 80 69 24 84 92 16 49 59 27 88 21 62 69 48 31 12 73 05 15 30 34 87 01 14 FROM A STATISTICS OR MATHEMATICS TEXT

30 If the random number intervals used 3-digit numbers, what would we need to do to a random number table? Answer We would need to break the numbers in the random number table into 3-digit numbers as well. Question

31 Table of Random Numbers 52 06 50 88 53 30 10 47 99 37 66 91 35 32 00 84 57 07 37 63 28 02 74 35 24 03 29 60 74 85 90 73 59 55 17 60 82 57 68 28 05 94 03 11 27 79 90 87 92 41 09 25 36 77 69 02 36 49 71 99 32 10 75 21 95 89 38 97 73 37 77 13 10 01 18 32 85 94 81 43 58 33 26 50 48 61 18 85 23 08 54 17 12 80 69 24 84 92 16 49 59 27 88 21 62 69 48 31 12 73 05 15 30 34 87 01 14

32 Monte Carlo Simulation Steps 5. Simulate a series of trials.

33 Simulating The Experiment We simulate outcomes by simply selecting random numbers from the random number table and noting the random number interval into which each random number falls.

34 Execution If the random number generated is “51”, it falls within the random number interval “36 – 65” which represents a daily demand of two television sets. WE HAVE JUST SIMULATED A DAILY DEMAND OF TWO TELEVISION SETS FOR THE 1 st DAY!

35 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 IF RANDOM NUMBER SELECTED IS “51” THEN SIMULATED DAILY DEMAND IS TWO TELEVISIONS

36 Execution If the random number generated is “08”, it falls within the random number interval “01 – 15” which represents a daily demand of zero television sets. WE HAVE JUST SIMULATED A DAILY DEMAND OF ZERO TELEVISION SETS FOR THE 2 nd DAY!

37 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 IF RANDOM NUMBER SELECTED IS “08” THEN SIMULATED DAILY DEMAND IS ZERO TELEVISIONS

38 Execution If the random number generated is “62”, it falls within the random number interval “36 – 65” which represents a daily demand of two television sets. WE HAVE JUST SIMULATED A DAILY DEMAND OF TWO TELEVISION SETS FOR THE 3rd DAY!

39 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 IF RANDOM NUMBER SELECTED IS “62” THEN SIMULATED DAILY DEMAND IS TWO TELEVISIONS

40 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 IF RANDOM NUMBER SELECTED IS “01” THEN SIMULATED DAILY DEMAND IS ZERO TELEVISIONS

41 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 IF RANDOM NUMBER SELECTED IS “29” THEN SIMULATED DAILY DEMAND IS ONE TELEVISION

42 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 IF RANDOM NUMBER SELECTED IS “69” THEN SIMULATED DAILY DEMAND IS THREE TELEVISIONS

43 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15 01 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 IF RANDOM NUMBER SELECTED IS “94” THEN SIMULATED DAILY DEMAND IS FOUR TELEVISIONS

44 Simulated Sales for Ten Business Days Day 1 ( 2 TVs )Day 1 ( 2 TVs ) Day 2 ( 0 TVs )Day 2 ( 0 TVs ) Day 3 ( 2 TVs )Day 3 ( 2 TVs ) Day 4 ( 0 TVs )Day 4 ( 0 TVs ) Day 5 ( 1 TV )Day 5 ( 1 TV ) Day 6 ( 3 TVs )Day 6 ( 3 TVs ) Day 7 ( 2 TVs )Day 7 ( 2 TVs ) Day 8 ( 2 TVs )Day 8 ( 2 TVs ) Day 9 ( 4 TVs )Day 9 ( 4 TVs ) Day 10 ( 2 TVs )Day 10 ( 2 TVs ) Simulated Average Daily Sales ( 18/10 = 1.8 TVs ) Σ = 18 TVs

45 TELEVISION SALES SIMULATION NUMBER OF TELEVISIONS SOLD Sales Probability CumulativeProbability RANDOM NUMBER INTERVAL 0.15.1501 - 15 1.20.3516 - 35 2.30.6536 - 65 3.20.8566 - 85 4.10.9586 - 95 5.051.0096 - 00 THE LONG-TERM EVENT PROBABILITIES

46 To Test if the Simulation Was Run Over Sufficient Time.15(0) +.20(1) +.30(2) +.20(3) +.10(4) +.05(5) = 2.05 televisions Long-term probabilities Daily demand events Compute the expected value of average daily television set sales ( in units )

47 Conclusion  Simulated average daily TV sales over 10 days = 1.80 units  Expected Value of average daily TV sales = 2.05 units  This simulation has not run a sufficient number of days! THE DIFFERENCE BETWEEN THE TWO NUMBERS IS JUST TOO SIGNIFICANT

48 How Simulation Duplicates “Reality”   Every possible outcome that a relevant variable can assume, together with its respective long- term probability, is entered into the model.

49 How Simulation Duplicates “Reality”   Every possible outcome that a relevant variable can assume, together with its respective long- term probability, is entered into the model.  Randomly occuring outcomes for each relevant variable are generated by random number inter- vals and random number strings.

50 How Simulation Duplicates “Reality”   Every possible outcome that a relevant variable can assume, together with its respective long- term probability, is entered into the model.  Randomly occuring outcomes for each relevant variable are generated by random number inter- vals and random number strings.  Since “reality” equals long-term probabilities + randomness, simulation achieves both!

51

52

53

54

55 The number of possible values for the relevant variable The number of days we wish to run the simulation

56 The program will choose the random numbers for you, if you select this option. We select RNs that are 2 places to the right of the decimal

57

58 The expected value of sales over time

59

60 Use this option to insert your own random numbers

61 We get the same simulated daily sales and the same average daily sales under this method

62

63 We need to run this simulation more than just 10 days. The observed frequencies do not come close to the long-term frequencies.

64

65

66

67

68

69 Template and Sample Data

70 Template and Sample Data

71 Here, we allow the program to select its own random numbers

72

73 Here, we specify the random numbers we want to use to simulate daily demand

74

75 SIMULATION Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

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