1. Simulation

2. Introduction What is Simulation? –Try to duplicate features, appearance, and characteristics of real system. Idea behind Simulation –Imitate real-world situation mathematically. –Study its properties and operating characteristics. –Draw conclusions and make action decisions based on results of simulation.

3. Process of a Simulation

4. Advantages And Disadvantages Of Simulation Advantages Relatively straightforward and flexible. Used to analyze large and complex real-world situations. Allows “what-if ? ” types of questions. Does not interfere with real-world system. Allows study of interactive effects of individual components or variables to determine which ones are important. Time compression. Allows for inclusion of real-world complications.

5. Advantages And Disadvantages Of Simulation Disadvantages Good models can be very expensive. Often it is a long, complicated process to develop model. Does not generate optimal solutions to problems. Managers must generate all of conditions and constraints for solutions to be examined. Each simulation model is unique and not easily transferable.

6. Monte Carlo Simulation Applicable when system contains elements that exhibit chance behavior. Experimentation based on chance elements through random sampling. Steps of Monte Carlo Simulation - –Set up probability distribution for each variable in model subject to chance. –Use random numbers to simulate values from probability distribution for each variable in Step 1. –Repeat process for series of replications or trials.

7. Auto Tire Shop Example Monthly demand for radial tires over past 60 months. Assume past demand rates will hold in future. Convert data to probability distribution. Divide each demand frequency by total number of months 60. Distributions can either be empirical or known such as normal, binomial, Poisson, or exponential patterns.

8. Step 2 - Simulate Values From the Probability Distributions Simulate demand for a specific month? Actual demand value is 300, 320, 340, 360, 380, or 400. There is 5% chance monthly demand is 300, –10% chance that it is 320. –20% chance that it is 340. –30% chance that it is 360. –25% chance that it is 380. –10% chance that it is 400. Harry’s Auto Tire Shop

9. Step 2 - Simulate Values From the Probability Distributions For long run - Expected monthly demand=  (demand D i ) x (probability of D i ) = (300)(0.05) + (320)(0.10) + (340)(0.20) + + (360)(0.30) + (380)(0.25) + (400)(0.10) = 358 tires In short term, occurrence of demand may be quite different from these probability values. Auto Tire Shop

10. Random Numbers In simulation, use random numbers to achieve preceding objectives. Random number is number that has been selected by totally random process. Assume generate an integer valued random number from set 0, 1, 2, …, 97, 98, 99. One way to do this would be: 1.Take 100 identical balls and mark each one with unique number between 00 and 99. 2. Put all balls in large bowl and mix thoroughly. 3. Select one ball from bowl and write down number. 4. Replace ball in bowl and mix again. Go to step 2.

11. Random Numbers Instead of balls in bowl, one could have used spin of roulette wheel that has 100 slots to accomplish this task. Another commonly used means is to choose numbers from table of random digits such as table of random numbers. Table of random numbers appears on next slide.

12. Table of Random Numbers

13. Using Random Numbers to Simulate Demand Auto Tire Shop

14. Using Simulation To Compute Expected Profit Using this information, simulate and calculate average profit per month from of auto tires. Harry’s Auto Tire Shop

15. Simulation Process 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

16. Simulation Process 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 =

17. Simulation Process 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 = 200(0.05)

18. Simulation Process 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 = 200(0.05) + 250(0.06)

19. Simulation Process 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 = 200(0.05) + 250(0.06) + 300(0.17) + … + 600(0.02) = 400 hours

20. Simulation Process 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

21. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours

22. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours Average weekly operating machine hours =

23. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours Average weekly operating machine hours = 320(0.30)

24. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours Average weekly operating machine hours = 320(0.30) + 360(0.40) +

25. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours Average weekly operating machine hours = 320(0.30) + 360(0.40) + 400(0.30)

26. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours Average weekly operating machine hours = 320(0.30) + 360(0.40) + 400(0.30) = 360 hours

27. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours Average weekly operating machine hours = 360 hours

28. Simulation Process 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 RegularRelative Capacity (hr)Frequency 320 (8 machines)0.30 360 (9 machines)0.40 400 (10 machines)0.30 Average weekly production requirements = 400 hours Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30

29. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event Weekly Demand (hr)Probability

30. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event Weekly Demand (hr)Probability 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02

31. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event Weekly RandomWeeklyRandom Demand (hr)ProbabilityNumbersCapacity (hr)ProbabilityNumbers 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02

32. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event Weekly RandomWeeklyRandom Demand (hr)ProbabilityNumbersCapacity (hr)ProbabilityNumbers 2000.0500–04 2500.0605–10 3000.1711–27 3500.0528–32 4000.3033–62 4500.1563–77 5000.0678–83 5500.1484–97 6000.0298–99

33. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event Weekly RandomWeeklyRandom Demand (hr)ProbabilityNumbersCapacity (hr)ProbabilityNumbers 2000.0500–043200.30 2500.0605–103600.40 3000.1711–274000.30 3500.0528–32 4000.3033–62 4500.1563–77 5000.0678–83 5500.1484–97 6000.0298–99

34. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event 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

35. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event 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 Simulation Process 1.Draw a random number. 2.Find the random number interval for production. 3.Record the production hours. 4.Draw another random number. 5.Find the random number interval for capacity. 6.Record the capacity hours. 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 to simulate 20 weeks. 10 Machines Existing DemandWeeklyCapacityWeeklySub- RandomProductionRandomCapacityIdleOvertimecontract WeekNumber(hr)Number(hr)HoursHoursHours

36. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event 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 Simulation Process 1.Draw a random number. 2.Find the random number interval for production. 3.Record the production hours. 4.Draw another random number. 5.Find the random number interval for capacity. 6.Record the capacity hours. 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 to simulate 20 weeks. 10 Machines Existing DemandWeeklyCapacityWeeklySub- RandomProductionRandomCapacityIdleOvertimecontract WeekNumber(hr)Number(hr)HoursHoursHours 1714505036090 2684505436090 3484001132080 49960036360100140 5644508240050 61330087400100 7364004136040 85840071400 9133000032020 10935506036010090 Total490830360 Weekly average24.541.518.0

37. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event 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 Simulation Process 1.Draw a random number. 2.Find the random number interval for production. 3.Record the production hours. 4.Draw another random number. 5.Find the random number interval for capacity. 6.Record the capacity hours. 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 to simulate 20 weeks. 10 Machines Existing DemandWeeklyCapacityWeeklySub- RandomProductionRandomCapacityIdleOvertimecontract WeekNumber(hr)Number(hr)HoursHoursHours 1714505036090 2684505436090 3484001132080 49960036360100140 5644508240050 61330087400100 7364004136040 85840071400 9133000032020 10935506036010090 Total490830360 Weekly average24.541.518.0 Comparison of 1000-week Simulations 10 Machines11 Machines

38. 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 Average weekly operating machine hours = 360 hours RegularRelative Capacity (hr)Frequency 360 (9 machines)0.30 400 (10 machines)0.40 440 (11 machines)0.30 Simulation Process Event 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 Simulation Process 1.Draw a random number. 2.Find the random number interval for production. 3.Record the production hours. 4.Draw another random number. 5.Find the random number interval for capacity. 6.Record the capacity hours. 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 to simulate 20 weeks. 10 Machines Existing DemandWeeklyCapacityWeeklySub- RandomProductionRandomCapacityIdleOvertimecontract WeekNumber(hr)Number(hr)HoursHoursHours 1714505036090 2684505436090 3484001132080 49960036360100140 5644508240050 61330087400100 7364004136040 85840071400 9133000032020 10935506036010090 Total490830360 Weekly average24.541.518.0 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