Simulation. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall

Transcription

1 Simulation Chapter

2 Chapter Topics The Monte Carlo Process Computer Simulation with Excel Spreadsheets Simulation of a Queuing System Continuous Probability Distributions Statistical Analysis of Simulation Results Crystal Ball Verification of the Simulation Model Areas of Simulation Application 14-2

3 Overview Analogue simulation replaces a physical system with an analogous physical system that is easier to manipulate. In computer mathematical simulation a system is replaced with a mathematical model that is analyzed with the computer. Simulation offers a means of analyzing very complex systems that cannot be analyzed using the other management science techniques in the text. 14-3

4 Monte Carlo Process A large proportion of the applications of simulations are for probabilistic models. The Monte Carlo technique is defined as a technique for selecting numbers randomly from a probability distribution for use in a trial (computer run) of a simulation model. The basic principle behind the process is the same as in the operation of gambling devices in casinos (such as those in Monte Carlo, Monaco). 14-4

5 Monte Carlo Process Use of Random Numbers (1 of 10) In the Monte Carlo process, values for a random variable are generated by sampling from a probability distribution. Example: ComputerWorld demand data for laptops selling for $4,300 over a period of 100 weeks. Table 14.1 Probability Distribution of Demand for Laptop PC s 14-5

6 Monte Carlo Process Use of Random Numbers (2 of 10) The purpose of the Monte Carlo process is to generate the random variable, demand, by sampling from the probability distribution P(x). The partitioned roulette wheel replicates the probability distribution for demand if the values of demand occur in a random manner. The segment at which the wheel stops indicates demand for one week. 14-6

7 Monte Carlo Process Use of Random Numbers (3 of 10) Figure 14.1 A Roulette Wheel for Demand 14-7

8 Monte Carlo Process Use of Random Numbers (4 of 10) When the wheel is spun, the actual demand for PCs is determined by a number at rim of the wheel. Figure 14.2 Numbered Roulette Wheel 14-8

9 Monte Carlo Process Use of Random Numbers (5 of 10) Table 14.2 Generating Demand from Random Numbers 14-9

10 Monte Carlo Process Use of Random Numbers (6 of 10) Select number from a random number table: Table 14.3 Delightfully Random Numbers 14-10

11 Monte Carlo Process Use of Random Numbers (7 of 10) Repeating selection of random numbers simulates demand for a period of time. Estimated average demand = 31/15 = 2.07 laptop PCs per week. Estimated average revenue = $133,300/15 = $8,

12 Monte Carlo Process Use of Random Numbers (8 of 10) Table

13 Monte Carlo Process Use of Random Numbers (9 of 10) Average demand could have been calculated analytically: n E( x) = P( xi) x i = 1 where: i xi = P( xi) = n= demand value i probability of demand the number of different demand values therefore: E( x) = (.20)(0) + (.40)(1) + (.20)(2) + (.10)(3) + (.10)(4) = 1.5 PC's per week 14-13

14 Monte Carlo Process Use of Random Numbers (10 of 10) The more period simulated, the more accurate the results. Simulation results will not equal analytical results unless enough trials have been conducted to reach steady state. Often difficult to validate results of simulation - that true steady state has been reached and that simulation model truly replicates reality. When analytical analysis is not possible, there is no analytical standard of comparison thus making validation even more difficult

15 Computer Simulation with Excel Spreadsheets Generating Random Numbers (1 of 2) As simulation models get more complex they become impossible to perform manually. In simulation modeling, random numbers are generated by a mathematical process instead of a physical process (such as wheel spinning). Random numbers are typically generated on the computer using a numerical technique and thus are not true random numbers but pseudorandom numbers

16 Computer Simulation with Excel Spreadsheets Generating Random Numbers (2 of 2) Artificially created random numbers must have the following characteristics: 1. The random numbers must be uniformly distributed. 2. The numerical technique for generating the numbers must be efficient. 3. The sequence of random numbers should reflect no pattern

17 Simulation with Excel Spreadsheets (1 of 3) Exhibit

18 Simulation with Excel Spreadsheets (2 of 3) Exhibit

19 Simulation with Excel Spreadsheets (3 of 3) Exhibit

20 Computer Simulation with Excel Spreadsheets Decision Making with Simulation (1 of 2) Revised ComputerWorld example; order size of one laptop each week. Exhibit

21 Computer Simulation with Excel Spreadsheets Decision Making with Simulation (2 of 2) Order size of two laptops each week. Exhibit

22 Simulation of a Queuing System Burlingham Mills Example (1 of 3) Table 14.5 Distribution of Arrival Intervals Table 14.6 Distribution of Service Times 14-22

23 Simulation of a Queuing System Burlingham Mills Example (2 of 3) Average waiting time = 12.5days/10 batches = 1.25 days per batch Average time in the system = 24.5 days/10 batches = 2.45 days per batch 14-23

24 Simulation of a Queuing System Burlingham Mills Example (3 of 3) Caveats: Results may be viewed with skepticism. Ten trials do not ensure steady-state results. Starting conditions can affect simulation results. If no batches are in the system at start, simulation must run until it replicates normal operating system. If system starts with items already in the system, simulation must begin with items in the system

25 Computer Simulation with Excel Burlingham Mills Example Exhibit

26 Continuous Probability Distributions A continuous function must beused for continuous distributions. Example : f(x) = x, 0 x 4 where x = time (minutes) 8 Cumulativeprobability of x: x x x 1x 1 1 x 2 F(x) = dx x dx 0 8 = 8 = F(x) = x 2 16 Let F(x) = the random number r r = x2 16 x = 4 r By generating a random number,r, a value x for "time" is determined. Example : if r =.25, x = 4.25 = 2 minutes 14-26

27 Machine Breakdown and Maintenance System Simulation (1 of 6) Bigelow Manufacturing Company must decide if it should implement a machine maintenance program at a cost of $20,000 per year that would reduce the frequency of breakdowns and thus time for repair which is $2,000 per day in lost production. A continuous probability distribution of the time between machine breakdowns: f(x) = x/8, 0 x 4 weeks, where x = weeks between machine breakdowns x = 4*sqrt(r i ), value of x for a given value of r i

28 Machine Breakdown and Maintenance System Simulation (2 of 6) Table 14.8 Probability Distribution of Machine Repair Time 14-28

29 Machine Breakdown and Maintenance System Simulation (3 of 6) Revised probability of time between machine breakdowns: f(x) = x/18, 0 x 6 weeks where x = weeks between machine breakdowns x = 6*sqrt(r i ) Table

30 Machine Breakdown and Maintenance System Simulation (4 of 6) Simulation of system without maintenance program (total annual repair cost of $84,000): Table

31 Machine Breakdown and Maintenance System Simulation (5 of 6) Simulation of system with maintenance program (total annual repair cost of $42,000): Table

32 Machine Breakdown and Maintenance System Simulation (6 of 6) Results and caveats: Implement maintenance program since cost savings appear to be $42,000 per year and maintenance program will cost $20,000 per year. However, there are potential problems caused by simulating both systems only once. Simulation results could exhibit significant variation since time between breakdowns and repair times are probabilistic. To be sure of accuracy of results, simulations of each system must be run many times and average results computed. Efficient computer simulation required to do this

33 Machine Breakdown and Maintenance System Simulation with Excel (1 of 2) Original machine breakdown example: Exhibit

34 Machine Breakdown and Maintenance System Simulation with Excel (2 of 2) Simulation with maintenance program. Exhibit

35 Statistical Analysis of Simulation Results (1 of 2) Outcomes of simulation modeling are statistical measure such as averages. Statistical results are typically subjected to additional statistical analysis to determine their degree of accuracy. Confidence limits are developed for the analysis of the statistical validity of simulation results

36 Statistical Analysis of Simulation Results (2 of 2) Formulas for 95% confidence limits: upper confidence limit lower confidence limit = x+ ( 1. 96)( s/ n) = x ( 1. 96)( s/ n) where x is the mean and s the standard deviation from a sample of size n from any population. We can be 95% confident that the true population mean will be between the upper confidence limit and lower confidence limit

37 Simulation Results Statistical Analysis with Excel (1 of 3) Simulation with maintenance program. Exhibit

38 Simulation Results Statistical Analysis with Excel (2 of 3) Exhibit

39 Simulation Results Statistical Analysis with Excel (3 of 3) Exhibit

40 Crystal Ball Overview Many realistic simulation problems contain more complex probability distributions than those used in the examples. However there are several simulation add-ins for Excel that provide a capability to perform simulation analysis with a variety of probability distributions in a spreadsheet format. Crystal Ball, published by Decisioneering, is one of these. Crystal Ball is a risk analysis and forecasting program that uses Monte Carlo simulation to provide a statistical range of results

41 Crystal Ball Simulation of Profit Analysis Model (1 of 15) Recap of Western Clothing Company break-even and profit analysis: Price (p) for jeans is $23 variable cost (c v ) is $8 Fixed cost (c f ) is $10,000 Profit Z = vp - c f v c break-even volume v = c f /(p - c v ) = 10,000/(23-8) = pairs

42 Crystal Ball Simulation of Profit Analysis Model (2 of 15) Modifications to demonstrate Crystal Ball Assume volume is now volume demanded and is defined by a normal probability distribution with mean of 1,050 and standard deviation of 410 pairs of jeans. Price is uncertain and defined by a uniform probability distribution from $20 to $26. Variable cost is not constant but defined by a triangular probability distribution. Will determine average profit and profitability with given probabilistic variables

43 Crystal Ball Simulation of Profit Analysis Model (3 of 15) 14-43

44 Crystal Ball Simulation of Profit Analysis Model (4 of 15) Exhibit

45 Crystal Ball Simulation of Profit Analysis Model (5 of 15) Exhibit

46 Crystal Ball Simulation of Profit Analysis Model (6 of 15) Exhibit

47 Crystal Ball Simulation of Profit Analysis Model (7 of 15) Exhibit

48 Crystal Ball Simulation of Profit Analysis Model (8 of 15) Exhibit

49 Crystal Ball Simulation of Profit Analysis Model (9 of 15) Exhibit

50 Crystal Ball Simulation of Profit Analysis Model (10 of 15) Exhibit

51 Crystal Ball Simulation of Profit Analysis Model (11 of 15) Exhibit

52 Crystal Ball Simulation of Profit Analysis Model (12 of 15) Exhibit

53 Crystal Ball Simulation of Profit Analysis Model (13 of 15) Exhibit

54 Crystal Ball Simulation of Profit Analysis Model (14 of 15) Exhibit

55 Crystal Ball Simulation of Profit Analysis Model (15 of 15) Exhibit

56 Verification of the Simulation Model (1 of 2) Analyst wants to be certain that model is internally correct and that all operations are logical and mathematically correct. Testing procedures for validity: Run a small number of trials of the model and compare with manually derived solutions. Divide the model into parts and run parts separately to reduce complexity of checking. Simplify mathematical relationships (if possible) for easier testing. Compare results with actual real-world data

57 Verification of the Simulation Model (2 of 2) Analyst must determine if model starting conditions are correct (system empty, etc). Must determine how long model should run to insure steady-state conditions. A standard, fool-proof procedure for validation is not available. Validity of the model rests ultimately on the expertise and experience of the model developer

58 Some Areas of Simulation Application Queuing Inventory Control Production and Manufacturing Finance Marketing Public Service Operations Environmental and Resource Analysis 14-58

59 Example Problem Solution (1 of 6) Willow Creek Emergency Rescue Squad Minor emergency requires two-person crew Regular emergency requires a three-person crew Major emergency requires a five-person crew 14-59

60 Example Problem Solution (2 of 6) Distribution of number of calls per night and emergency type: Calls Probability Emergency Type Minor Regular Major Probability Manually simulate 10 nights of calls 2. Determine average number of calls each night 3. Determine maximum number of crew members that might be needed on any given night

61 Example Problem Solution (3 of 6) Step 1: Develop random number ranges for the probability distributions. Calls Probability Cumulative Probability Random Number Range, r , 00 Emergency Type Minor Regular Major Probability Cumulative Probability Random Number Range, r ,

62 Example Problem Solution (4 of 6) Step 2: Set Up a Tabular Simulation (use second column of random numbers in Table 14.3)

63 Example Problem Solution (5 of 6) Step 2 continued: 14-63

64 Example Problem Solution (6 of 6) Step 3: Compute Results: average number of minor emergency calls per night = 10/10 =1.0 average number of regular emergency calls per night =14/10 = 1.4 average number of major emergency calls per night = 3/10 = 0.30 If calls of all types occurred on same night, maximum number of squad members required would be

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