1. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-1 Operations Management Simulation Module F

2. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-2 Two ways to address waiting lines  Queuing theory  Certain types of lines can be described mathematically  Requires that assumptions are valid for your situation  Systems with multiple lines that feed each other are too complex for queuing theory  Simulation  Building mathematical models that attempt to act like real operating systems  Real-world situations can be studied without imposing on the actual system

3. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-3 Simulation  Imitate a real-world situation mathematically  Study its properties and operating characteristics  Draw conclusions and make action recommendations  Benefits:  Able to handle complex situations with multiple unknowns, non-standard distributions, and different interactions  Fast  Low cost  Low risk

4. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-4 Real World Variables Which Are Probabilistic in Nature  Inventory demand  Lead time for orders to arrive  Time between machine breakdowns  Times between arrivals at a service facility  Service times  Times to complete project activities  Number of employees absent from work each day

5. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-5 New Orleans Barge Unloading Simulation Following long trips down the Mississippi River from industrial mid-western cities, fully loaded barges arrive in New Orleans. Distribution 1 gives the inter-arrival times for the barges. Distribution 2 gives the unloading times.

6. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-6 Table of Random Numbers 52065088533010479937669135 37632802743524032960748590 82576828059403112779908792 69023649719932107521959094 98949036067823678985292125 96526287495649237871729057 33692721116095896848178934 50335095134434626339552930 88321850625734566231154090 303624608251743035368501 50486118852308541712806924 27882162696448311273026800 45144632134966627441869892

7. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-7 Distribution 1 – Inter-Arrival Times Time Between Arrivals (Hours) ProbabilityCumulative Probability Random Number Interval 360.13 01 – 13 240.170.3014 – 30 120.150.4531 – 45 80.250.7046 – 70 60.200.9071 - 90 40.101.0091 - 00

8. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-8 Distribution 2 – Unloading Times Unloading Times (Hours) ProbabilityCumulative Probability Random- Number Interval 240.05 01 – 05 120.150.2006 – 20 80.500.7021 – 70 60.200.9071 – 90 40.101.0091 - 00

9. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-9 New Orleans Barge Unloading Simulation From Dist 1From Dist 2 From Random Number Table 20182829174154499 01742403150 847 0150824142 3610 0114835106 2430 1810667410082853 210024027674688 07682868 2450 05286344 3606 01683788852 Waiting Time Unloading Ends Unloading Time Unloading Starts Arrival Time Int Arr Time Rnd #

10. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-10 New Orleans Barge Unloading Simulation Average Time in Port 11.3 hrs17.1 hrs Average Wait Time (Hours) Total Wait Time (Hours) Average Time to Unload (Hours) Average Time Between Arrivals (Hours) 15.7 hrs 11.3 + 4.4 hrs 4.4 hrs 40 hrs / 940 hrs102 hrs / 9154 hrs / 9

11. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-11 Simulation Using Excel RAND ( ) = 52065088533010479937669135 37632802743524032960748590 82576828059403112779908792 69023649719932107521959094 98949036067823678985292125 96526287495649237871729057 33692721116095896848178934 50335095134434626339552930 88321850625734566231154090 303624608251743035368501 50486118852308541712806924 27882162696448311273026800 45144632134966627441869892 We can create specific distributions, e.g. the normal distribution: NORMINV ( RAND ( ), mean, standard deviation ) We can use if / then statements: = IF ( X < Y, value if true, value if false)

12. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-12 Metric Construction Metric Construction is trying to determine the expected profit for a project. If they finish on time, they will make a $550,000 profit. For every week the project is early, they gain an additional $50,000. For every week late, they lose $60,000. The project is to be completed in 120 weeks. A labor strike is possible with their upcoming contract negotiations. The labor relations staff has estimated the probability of no strike at 45%, the probability of a 3-week strike is 25%, a 4-week strike is 15%, a 5-week strike is 10%, and a 6-week strike is 5%. Metric based their 120 week completion time on the average number of rainy days. Based on weather service input, they estimate a normal distribution for rainy days with a mean of 0 (from their 120-week schedule) and a standard deviation of 20 weeks.

13. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-13 Metric Construction Model Parameters (given information) Profit = 550,000 + (50,000*weeks early) – (60,000*weeks late) Generate Length of Strike, S Generate weather effect, W Expected completion = 120 + S + W Compute Profits repeat

14. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-14 Metric Construction – The Strike The labor relations staff has provided us with the probability distribution for a strike. We can use the random number generator just like we did for the New Orleans barge unloading problem. Use nested if statements:  If random number <.45 S = 0  If random number >.45 and <.70 S = 3  If random number >.70 and <.85 S = 4  If random number >.85 and <.95 S = 5  If random number >.95 S = 6 = IF (C3 < 0.45, 0, IF (C3 < 0.70, 3, IF (C3 < 0.85, 4, IF (C3 < 0.95, 5, 6 ))))

15. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-15 Metric Construction – The Weather We will use a normal distribution for weather delay. Our mean is 0 (from our current schedule) and the standard deviation is 20 weeks: = NORMINV ( RAND ( ), 0, 20 ) We may want to round this to an integer

16. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-16 Metric Construction – Continuing  Completion time for the project will be 120 weeks plus the strike plus the weather change (note when this is negative it decreases project length which is a good thing) L = 120 + S + W  Profits = $550,000 + $50,000 week early – $60,000 week late If L > 120 = 550,000 – 60,000 (L – 120) If L < 120 = 550,000 + 50,000 (120 – L)  Lets run this thing for 50 weeks

17. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-17 Metric Construction – The Results  50 trials (and we could easily do 500) is much more informative than plugging in average, best and worst cases.  Fast to complete and fast to re-run with modified assumptions

18. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-18 Billy-Bob’s Bait Shack Billy-Bob’s Bait Shack is trying to decide between two different ways of serving customers. Presently, Billy-Bob and his willing assistant Bubba work as a team. When a customer enters the bait shack, one person gets the bait while the other rings up the purchase and collects the money. In this system, customers arrive following a uniform distribution with a range from 0 – 15 minutes. The team of Billy-Bob and Bubba service customers following a standard distribution with an average time of 6 minutes and a standard deviation of 2 minutes. In the proposed system, Billy-Bob and Bubba will each do everything for a customer, creating two servers. Customers will wait in a single line. Service will still follow a normal distribution, but now the average time will be 11 minutes, with a standard deviation of 3 minutes. When both servers are available, Billy-Bob will take the next customer.

19. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-19 Initialize model: i = 0, arrival time (0) = 0, completion time (0) = 0 New customer: i = i+1 Arrival time(i) = arrival time (i-1) + IAT Wait? No - AT(i) > CT(i-1)Yes - AT(i) < CT(i-1) Start (i) = AT(i)Start (i) = CT(i-1) Wait (i) = S(i) – AT(i) Generate processing time – p(i) Completion time: CT(i) = S(i) + p(i) System time = CT(i) – AT(i) Generate inter-arrival time (IAT) Billy-Bob’s Bait Shack

20. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-20 Arrivals Inter-arrival time is uniformly distributed between 0 and 15 minutes Value = a + [ random number * ( b – a ) ] a = smallest possible value b = largest possible value In this case: IAT(i) = RAND( ) * 15 AT(i) = AT(i – 1) + IAT(i) Billy-Bob’s Bait Shack

21. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-21 We have to wait when the customer ahead of us is not completed with service when we arrive  If AT(i) < CT(i-1) then AT(i) has to wait start time: S(i) = CT(i-1)  If AT(i) > CT(i-1) then AT(i) does not have to wait start time: S(i) = AT(i)  Start time = if ( AT(i) < CT(i – 1), CT(i – 1), AT(i) )  Wait time: W(i) = S(i) – AT(i) Billy-Bob’s Bait Shack Waiting

22. © 2004 by Prentice Hall, Inc., Upper Saddle River, N.J. 07458 F-22 The rest  Processing time is normally distributed with a mean of 6 minutes and a standard deviation of 2 minutes  p(i) = NORMINV ( RAND( ), 6, 2 )  Completion time = when this transaction is completed  CT(i) = S(i) + p(i)  System time = time from entering the system to leaving the system  CT(i) – AT(i) Billy-Bob’s Bait Shack