Simulation Model for an Automotive Maintenance and Repair Shop Operations

1. Outline • Model 05-01 • problem statement • detailed ARENA model • model technique • Output Analysis

2. Model 5-1: An Automotive Maintenance and Repair Shop • additional maintenance and repair facility in the suburban area • customer orders (calls) • by appointments, from one to three days in advance • calls arrivals ~ Poisson process, mean 25 calls/day • distribution of calls: 55% for the next day; 30% for the days after tomorrow; 15% for two days after tomorrow • response missing a desirable day: 90% choose the following day; 10% leave

3. An Automotive Repair and Maintenance Shop • service • Book Time, (i.e., estimated service time) ~ 44 + 90*BETA(2, 3) min • Book Time also for costing • promised wait time to customers • wait time = Book Time + one hour allowance • actual service time ~ GAMM(book time/1.05, 1.05) min • first priority to wait customers • customer behavior • 20% wait, 80% pick up cars later • about 60% to 70% of customers arrive on time • 30% to 40% arrive within 3 hours of appointment time

4. Costs and Revenues • schedule rules • at most five wait customers per day • no more than 24 book hours scheduled per day (three bays, eight hours each) • normal cost: $45/hour/bay, 40-hour per week • overtime costs $120/hour/bay, at most 3 hours • revenue from customers: $78/ book hour • penalty cost • each incomplete on-going car at the end of a day: $35 • no penalty for a car whose service not yet started

5. System Performance • simulate the system 20 days to get • average daily profit • average daily book time • average daily actual service time • average daily overtime • average daily number of wait appointments not completed on time

6. Relationship Between Models • Model 5-1: An Automotive Maintenance and Repair Shop • a fairly complicated model • non-queueing type • Model 5-2: Enhancing the Automotive Shop Model • two types of repair bays for different types of cars • customer not on time

7. The Structure of the ARENA Model • Five parts • Control Logic to initialize variables and count days • Generate appointment calls, including a representative initial condition • Make appointments, considering priority of jobs • Service activities • Update performance variables

8. Details of Model 05-01 • logic control and submodels • for each day • first simulate the calls for appointments (of future days) • then simulate the work of the day • vectors • variables and expressions

9. Steps to Prepare a Simulation Program • assumption: already formulated the problem, i.e., fully understood how the system works • for a simple problem: use the crude to detailed pseudo code approach to build the flow of the model • for a complicated problem • first play around with a simplified problem • use paper and pencil to simulate

10. An Illustration for Model 05-01 • a simplified version of Model 05-01 • a week of three days • reservations made two days in advance • Book Time = 1 w.p. 3/4 and = 2 w.p. 1/4 • actual Service time • = 1.2 Book Time w.p. 1/3 • = 0.8 Book Time w.p. 2/3

11. An Illustration for Model 05-01 • each customer equally likely to be leave or wait • every day 4 hours, with at most 1 hour OT • at most 1 customer to leave his car • number of customers in each day • = 2 w.p. 1/3 and = 3 w.p. 2/3 • simulation duration: 4 days

12. Before Simulation • terminating or non-terminating process? • non-terminating • typically simulated for a long time and the initial condition being unimportant • how to set the initial condition if a non-terminating system is simulated for a short time? • empty: is it representative? • not empty: how to make it representative?

13. To Generate a Representative Initial Condition • representative initial condition • day 1: appointments made in previous two days, i.e., day -1 and day 0 • day 2: appointments made in day 0 • idea • generate calls for day -1 and then for day 0 • whenever applicable, schedule appointments on days 1 and 2 • implicitly drop appointments for days -1 and 0

14. Paper and Pencil Simulation of the Simplified System 4 1 2 3 Day -1 0

15. Very Crude Pseudo-Code • 1 Generate a representative initial condition • 2 Simulate the system for 4 days • assumption for the model: ignore the time of calls, assuming that all happen in the morning

16. Refinement of the First Step of the Pseudo-Code Start with an empty 6-day schedule Generate number of calls for Day -1 Generate a representative initial condition Generate Book Timesand schedule them for calls in Day -1 Generate number of calls for Day 0 Generate Book Timesand schedule them for calls in Day 0

17. Refinement of the Second Step of the Pseudo-Code

18. To Implement in ARENA • need further refinement of the pseudo-code • need modifying the pseudo-code to suit the structure of ARENA, e.g., • what are the entities in the ARENA model? • what are the correspondence between the steps in paper and pencil simulation and ARENA? • ….. lots of details

19. Output Analysis • simulation: estimate  = E(X) by observing sample values from the distribution of X • output analysis • point estimator of ? • unbiased estimator of ? • variance of estimator? • efficient estimator of ? • confidence on the range estimator? • # of simulation runs (replications) required?

20. Desirable Functions of Software • interval estimation • comparing alternatives • automatic statistical tests • handy housekeeping for scenarios • automatic searching for optimal • all features available in ARENA

21. Output Analysis • two types of estimates, point and interval • theoretical basis • point estimates: SLLN • interval estimate: CLT

22. define Strong Law of Large Numbers • i.i.d. random variables X1, X2, … • finite mean  and variance 2 •  = E(X)  (X1 + … + Xn)/n

23. Additional Facts • X1, X2, ..., Xn be i.i.d.; finite mean  and variance 2 unbiased estimator of unbiased estimator of2

24. Central Limit Theorem • i.i.d. random variables X1, X2, … • finite mean  and variance 2

25. Central Limit Theorem - Basis to Analyze Terminating Systems • t, 2, and F: useful distributions for range estimation and hypothesis testing of normal random variables Xi’s • CLT: statsitics approximately normal for “large enough”n • can use t, 2, and Ffor (approximate) range estimation and hypothesis testing

26. Differences Between Terminating and Non-Terminating Processes • termination condition and run length • terminating: well-defined, i.i.d. replications • non-terminating: no well-defined length • initial condition • terminating: clear, defined by the problem • non-terminating: unclear, biased by any fixed initial value • random variables for estimation • i.i.d. random variables • stationary version of random variables

27. time Non-Terminating Processes        

28. Terminating Processes • standard outputs • interval estimate of mean Model 05-02 • hypothesis testing of mean Model 05-02 • number of runs Model 06-01 • saving results in an output file for further processing • export from Output Analyzer to a text file Model 06-02 • processed by first by Excel and then Input Analyzer to analyze the output data Model 06-02 • confidence intervals by Output Analyzer Model 06-02 • comparison by Output Analyzer Model 06-03 • sequential determination of number of runs comparison by Output Analyzer Model 12-03

29. Non-Terminating Processes • non-terminating process Model 07-02 • Output Analyzer • replication/deletionModel 07-03 • batch means • sequential batch means • auto-correlation • regenerative simulation