1. © 2007 Pearson Education Simulation Supplement B

2. © 2007 Pearson Education Simulation Simulation: The act of reproducing the behavior of a system using a model that describes the processes of the system. Time Compression: The feature of simulations that allows them to obtain operating characteristic estimates in much less time than is required to gather the same operating data from a real system. Monte Carlo simulation: A simulation process that uses random numbers to generate simulation events.

3. © 2007 Pearson Education Specialty Steel Products Company produces items such as machine tools, gears, automobile parts, and other specialty items in small quantities to customer order. Demand is measured in machine hours. –Orders are translated into required machine-hours. Management is concerned about capacity in the lathe department. Assemble the data necessary to analyze the addition of one more lathe machine and operator. Specialty Steel Products Co. Example B.1

4. © 2007 Pearson Education Weekly ProductionRelative Requirements (hr)Frequency 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02 Total1.00 Total1.00 Specialty Steel Products Co. Example B.1 Historical records indicate that lathe department demand varies from week to week as follows:

5. © 2007 Pearson Education Weekly ProductionRelative Requirements (hr)Frequency 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02 Total1.00 Specialty Steel Products Co. Example B.1 Average weekly production is determined by multiplying each production requirement by its frequency of occurrence. Average weekly production requirements = 200(0.05) + 250(0.06) + 300(0.17) + … + 600(0.02) = 400 hours

6. © 2007 Pearson Education Specialty Steel Products Co. Example B.1 Weekly ProductionRelative Requirements (hr)Frequency 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02 Total1.00 Average weekly production requirements = 400 hours RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 The average number of operating machine-hours in a week is: 320(0.30) + 360(0.40) + 400(0.30) = 360 hours

7. © 2007 Pearson Education Specialty Steel Products Co. Weekly ProductionRelative Requirements (hr)Frequency 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02 Total1.00 Total1.00 Average weekly production requirements = 400 hours RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 400 (10 machines)0.30 The average number of operating machine-hours in a week = 360 Hrs. RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 400 (10 machines)0.40 440 (11 machines)0.30 440 (11 machines)0.30 Experience shows that with 11 machines, the distribution would be: Example B.1

8. © 2007 Pearson Education Specialty Steel Products Co. Assigning Random Numbers Random numbers must now be assigned to represent the probability of each demand event. Random Number: A number that has the same probability of being selected as any other number. Since the probabilities for all demand events add up to 100 percent, we use random numbers between (and including) 00 and 99. Within this range, a random number in the range of 0 to 4 has a 5% chance of selection. We can use this to represent our first weekly demand of 200 which has a 5% probability.

9. © 2007 Pearson Education Event Weekly Demand (hr)Probability 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02 Random numbers in the range of 0-4 have a 5% chance of occurrence. Random numbers in the range of 5-10 have a 6% chance of occurrence. Random numbers in the range of 11-27 have a 17% chance of occurrence. Random numbers in the range of 28-32 have a 5% chance of occurrence. Specialty Steel Products Co. Assigning Random Numbers

10. © 2007 Pearson Education Event Existing Weekly RandomWeeklyRandom Demand (hr)ProbabilityNumbersCapacity (hr)ProbabilityNumbers 2000.0500–043200.3000–29 2500.0605–103600.4030–69 3000.1711–274000.3070–99 3500.0528–32 4000.3033–62 4500.1563–77 5000.0678–83 5500.1484–97 6000.0298–99 If we randomly choose numbers in the range of 00-99 enough times, 5 percent of the time they will fall in the range of 00-04, 6% of the time they will fall in the range of 05-10, and so forth. Specialty Steel Products Co. Assigning Random Numbers

11. © 2007 Pearson Education Specialty Steel Products Co. Model Formulation Formulating a simulation model entails specifying the relationship among the variables. Simulation models consist of decision variables, uncontrollable variables and dependent variables. Decision variables: Variables that are controlled by the decision maker and will change from one run to the next as different events are simulated. Uncontrollable variables are random events that the decision maker cannot control.

12. © 2007 Pearson Education Specialty Steel Products Co. Example B.2 1.Using the Appendix 2 random number table, draw a random number from the first two rows of the table. Start with the first number in the first row, then go to the second number in the first row. 2.Find the random-number interval for production requirements associated with the random number. 3.Record the production hours (PROD) required for the current week. 4.Draw another random number from row three or four of the table. 5.Find the random-number interval for capacity (CAP) associated with the random number. 6.Record the capacity hours available for the current week. Simulating a particular capacity level

13. © 2007 Pearson Education 7.If CAP > PROD, then IDLE HR = CAP – PROD 8.If CAP < PROD, then SHORT = PROD – CAP If SHORT < 100 then OVERTIME HR = SHORT and SUBCONTRACT HR = 0 If SHORT > 100 then OVERTIME HR = 100 and SUBCONTRACT HR = SHORT – 100 9.Repeat steps 1 - 8 until you have simulated 20 weeks. Specialty Steel Products Co. Example B.2 Simulating a particular capacity level

14. © 2007 Pearson Education Specialty Steel Products Co. 20-week simulation 10 Machines Existing DemandWeeklyCapacityWeeklySub- RandomProductionRandomCapacityIdleOvertimecontract WeekNumber(hr)Number(hr)HoursHoursHours 1714505036090 2684505436090 3484001132080 49960036360100140 5644508240050 61330087400100 7364004136040........................ 20374001932080 Total490830360 Weekly average24.541.518.0

15. © 2007 Pearson Education Comparison of 1000-week Simulations 10 Machines11 Machines Idle hours26.042.2 Overtime hours48.334.2 Subcontract hours18.48.7 Cost$1,851.50$1,159.50 Specialty Steel Products Co. 1000-week simulation A steady state occurs when the simulation is repeated over enough time that the average results for performance measures remain constant.

16. © 2007 Pearson Education Monte Carlo Simulation Application B.1 Famous Chamois Car Wash Car Arrival Distribution (time between arrivals) Famous Chamois is an automated car wash that advertises that your car can be finished in just 15 minutes. The time until the next car arrival is described by the following distribution. MinutesProbabilityMinutesProbability 10.0180.12 20.0390.10 30.06100.07 40.09110.05 50.12120.04 60.14130.03 70.141.00

17. © 2007 Pearson Education Famous Chamois Car Wash: Random Number Assignment Assign a range of random numbers to each event so that the demand pattern can be simulated. Minutes Random Numbers Minutes Random Numbers 100–00859-70 201–039 71  80 304–091081-87 410–181188-92 519–3012 93  96 631–4413 97  99 745–58 Monte Carlo Simulation Application B.1

18. © 2007 Pearson Education Monte Carlo Simulation Famous Chamois Car Wash: Simulation Simulate the operation for 3 hours, using the following random numbers, assuming that the service time is constant at 6, (:06) minutes per car. Random Number Time to Arrival Arrival Time Number in Drive Service Begins Departure Time Minutes in System 5070:070 0:136 63? 95 49 68

19. © 2007 Pearson Education

20. Computer Simulation  The simulation for Specialty Steel Products demonstrated the basics of simulation.  However it only involved one step in the process, with two uncontrollable variables (weekly production requirements and three actual machine-hours available) and 20 time periods.  Simple simulation models with one or two uncontrollable variables can be developed using Excel, using its random number generator.  More sophisticated simulations can become time consuming and require a computer.

21. © 2007 Pearson Education

22. BestCar Auto Dealer Simulation Model  Below is a probability distribution for the number of cars sold weekly at BestCar (See Example B.3).  The selling price per car is $20,000. Design a simulation model that determines the probability distribution and mean of the weekly sales.

23. © 2007 Pearson Education

24. Simulation with SimQuick  Simquick is an easy-to-use package that is an Excel spreadsheet with some macros.  Let’s simulate the passenger security process at one terminal of a medium-sized airport between the hours of 8 A.M. and 10 A.M.  Passengers arrive in a single line and go through one of two inspection stations consisting of a metal detector and a carry- on baggage scanner.  After this, 10% are randomly selected for an additional inspection handled by one of two stations.  Management wants to examine the effects of increasing the number of random inspections to 15% and 20%.  They also want to consider a 3rd station for the 2nd inspection

25. © 2007 Pearson Education Flowchart of Passenger Security Process Entrance Arrivals Buffer Sec. Line 1 Buffer Buffer Sec. Line 2 Buffer Buffer Done Work St. Add. Insp. 1 Work St. Add. Insp. 1 Work St. Add. Insp. 2 Work St. Add. Insp. 2 Work St. Insp. 1 Work St. Insp. 2 Dec. Pt. DP Information about each block is entered into Simquick. Other information needed in statistical distribution form: (1) when people arrive, (2) inspection times, & (3) % of passengers randomly selected. Arrivals and inspection times are acquired through observation; % randomly selected is a policy decision

26. © 2007 Pearson Education Simulation Results of Passenger Security Process 224.57Final inventoryDone 0.53Mean cycle time 0.10Mean inventoryLine 2 3.12Mean cycle time 5.97Mean inventoryLine 1Buffer(s) 237.23Objects entering processDoorEntrance(s) MeansNames Types OverallStatisticsElement