1. Copyright 2009 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Operations Management - 6 th Edition Chapter 13 Supplement Roberta Russell & Bernard W. Taylor, III Simulation

2. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-2 Lecture Outline   Monte Carlo Simulation   Computer Simulation with Excel   Areas of Simulation Application

3. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-3 Simulation   Mathematical and computer modeling technique for replicating real-world problem situations   Modeling approach primarily used to analyze probabilistic problems It does not normally provide a solution; instead it provides information that is used to make a decision   Physical simulation Space flights, wind tunnels, tread-mills for tires   Mathematical-computerized simulation Computer-based replicated models

4. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-4 What Discrete Stochastic Simulation is  Probabilistic occurrences  Activity completions  Processes  Precedence relationships  Probabilistic routing  Events, Entities and Attributes

5. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-5 Some definitions  Activity—a task that requires time to complete, has time duration  Event – an instant in time at which the system undergoes state change  Entity—an object that engages the system  Attributes—information elements carried along with entities and attributes

6. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-6 Activities and Events  All activities begin and end with an event—nothing interesting happens in- between

7. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-7 An example—Southwest Airlines airline gate turn-- ACTIVITIES  Disembark passengers  Cabin cleanup  Embark passengers  Unload baggage  Load Baggage  Refuel  Remove waste  Refurbish snacks and drinks

8. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-8 EVENTS for the airline gate turn  Arrival at gate  Beginning of unloading  Completion of passenger unloading  Beginning of cleanup  Ending of cleanup  Beginning of passenger loading  Ending of passenger loading  Beginning of baggage unloading  Ending of baggage unloading

9. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-9 ACTIVITIES & EVENTS  Activities always have time duration. That time duration is in general random That time duration is in general random  Events are instants in time  Activities are sometimes engaged in by entities  Activities, events and entities have attributes

10. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-10 Events, Entities and Attributes  Entities may be permanent or temporary Customers, students, piece parts, messages, boxes, items,--TEMPORARY Customers, students, piece parts, messages, boxes, items,--TEMPORARY Universities, cities, companies, facilities, servers, professors, service areas -- PERMANENT Universities, cities, companies, facilities, servers, professors, service areas -- PERMANENT

11. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-11 Both entities and events possess ATTRIBUTES Attributes of a server entity—mean service time, std. dev. of service time, probability distribution type Attributes of a server entity—mean service time, std. dev. of service time, probability distribution type attributes of an event--event type, time the event is to occur, no of entities assoc. with it, links to the specific entities attached to the event attributes of an event--event type, time the event is to occur, no of entities assoc. with it, links to the specific entities attached to the event

12. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-12 A typical service system scenario  In the early morning hours between 7 and 9 a.m., arrivals to convenience stores are larger than normal. If there are more than 6 people waiting in line, new arrivals will balk and go somewhere else. People arrive at the rate of 1 every second, but the time is exponentially distributed. Patrons shop for a time period that is uniformly distributed between 3 and 5 minutes.

13. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-13 Convenience Store Scenario, continued  It takes the checkout clerk an average of 43 sec to collect money from a customer and provide them with a receipt, but this time is normally distributed with a std dev. of 30 sec.. People will automatically enqueue themselves in front of the checkout stand.  The manager can hire a second clerk, who is less well paid but also much slower

14. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-14 Convenience Store Scenario, continued  The store manager is interested in the average waiting time of his patrons in the queue the average waiting time of his patrons in the queue the average number of customers that balked the average number of customers that balked

15. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-15 With one store clerk:  average waiting time is 128 seconds  number of balked customers is 76 out of 1000 customers  Check out clerk is busy 86% of the time checking out customers

16. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-16 With two store clerks:  average waiting time is 42 secs for the first server 69 secs for the second 69 secs for the second  There are no balked customers  Servers are busy 70% and 40% of the time, respectively

17. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-17 How does randomness come into play?  Probabilistic activity durations  Probabilistic routing “decisions”  Probabilistic arrivals

18. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-18 How is randomness created within a digital computer?  Monte Carlo--the computer generation of random numbers Sample the clock?--no--not replicable Sample the clock?--no--not replicable maintain a huge file of random numbers--no maintain a huge file of random numbers--no Takes up too much space in primary memory Takes up too much space in primary memory On secondary storage, its too slow On secondary storage, its too slow when deciding to fetch from disk as opposed to primary memory, the time required. is 500,000 times longer when deciding to fetch from disk as opposed to primary memory, the time required. is 500,000 times longer Use an ALGORITHM? --YES, YES Use an ALGORITHM? --YES, YES

19. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-19 Why use an algorithm?  The sequence it generates will be deterministic  Doesn’t take up much space in primary storage  Takes up no space on secondary storage  Far faster than getting data off of the hard drive

20. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-20 An Algorithm for Generating Random Numbers  Must be fast (short and sweet)  Must be capable of generating numbers that have all of the characteristics of randomness, but in fact are deterministic  Multiplicative Congruence is one method

21. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-21 About Random Numbers  Uniform on the interval zero to one  They are completely independent and therefore un-correlated  We represent them this way: U(0,1)

22. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-22 Multiplicative Congruence Algorithm  C I+1 = K*C i function random(float u, int I) function random(float u, int I) I = I * 1220703125; if I<0 then I = I + 2147483647 + 1; else U = I * 0.4656613E-9; return “and” end;

23. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-23 Notes  Generates a sequence on the entire interval of 32-bit integers--0 to 2147483647  Maps these onto the real interval of 0 to 1  If the first multiplication causes integer overflow, the resultant number I will be negative--it is made positive by adding the largest 32-bit integer representable +1 The last multiplication is like dividing the number by the largest integer possible 1/2147483647 =.4656613x10 to the minus 9

24. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-24 You can easily generate random numbers in an EXCEL spreadsheet using the function RAND()

25. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-25 What about non-uniform random numbers? ExponentialNormalGammaPoissonLognormalRectangularTriangular

26. Supplement 12-26 ONE ANSWER: Use the inverse transformation method Every non-uniform random variate has an associated cumulative distribution function F(x) whose values are contained within the interval 0 to 1 and whose values are uniformly distributed over this interval If x is a non uniform random variate, y = F(x) is uniformly distributed over the interval 0 to 1.

27. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-27 Strategy If the inverse of the cumulative distribution function F(x) exists so that x = F -1 (y) can be determined, then 1) simply generate a random number uniformly distributed on the interval 0 to 1 2) call this number y and apply the inverse transformation F -1 (y) to obtain a random number x with the appropriate distribution.

28. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-28 Exponential Random variates 1) generate a random number U using the program given above 2)then apply EXPRND = -XMEAN * ALOG(U) (This is simply using the inverse distribution method)

29. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-29 When Analytic inverses of the cumulative distribution function are unavailable You can use a table function (this is what we will do in the subsequent problems) You can use specialized algorithms that have been developed by academics over thirty-five years of cumulative research

30. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-30 Outputs Outputs  The animation reveals bottlenecks, idleness  We are also interested in productivity, cycle time (time in the system), wait time,blocked time, productivity, cycle time (time in the system), wait time,blocked time, number of trips made in a given period of time, number of trips made in a given period of time, system throughput within a given period of time system throughput within a given period of time  We can get this from the statistical reports provided after the simulation is finished

31. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-31 Monte Carlo Simulation  Select numbers randomly from a probability distribution  Use these values to observe how a model performs over time  Random numbers each have an equal likelihood of being selected at random

32. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-32 Distribution of Demand LAPTOPS DEMANDEDFREQUENCY OFPROBABILITY OF PER WEEK, xDEMANDDEMAND, P(x) 0200.20 1400.40 2200.20 3100.10 4100.10 1001.00

33. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-33 Roulette Wheel of Demand 90 80 60 200 x = 2 x = 0 x = 4 x = 3 x = 1

34. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-34 Generating Demand from Random Numbers DEMAND,RANGES OF RANDOM NUMBERS, xr 00-19 120-59r = 39 260-79 380-89 490-99

35. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-35 Random Number Table

36. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-36 Random Numbers in Excel

37. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-37 15 Weeks of Demand Average demand = 31/15 = 2.07 laptops/week WEEKrDEMAND (x)REVENUE (S) 13914,300 27328,600 37228,600 47528,600 53714,300 60200 787312,900 898417,200 91000 104714,300 1193417,200 122114,300 1395417,200 1497417,200 156928,600  = 31$133,300

38. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-38 Computing Expected Demand E(x) = (0.20)(0) + (0.40)(1) + (0.20)(2) + (0.10)(3) + (0.10)(4) = 1.5 laptops per week Difference between 1.5 and 2.07 is due to small number of periods analyzed (only 15 weeks)Difference between 1.5 and 2.07 is due to small number of periods analyzed (only 15 weeks) Steady-state result an average result that remains constant after enough trials

39. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-39 Simulation in Excel Enter this formula in G6 and copy to G7:G20 Enter “=4300*G6” in H6 and copy to H7:H20 Generate random numbers for cells F6:F20 with the formula “=RAND()” in F6 and copying to F7:F20 = AVERAGE (G6:G20)

40. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-40 Simulation in Excel (cont.)

41. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-41 Decision Making with Simulation This formula entered in G7 and copied to G8:G105 =VLOOKUP (F6,LOOKUP,2) in H6 and copied to H7:H105 =G6*50 entered into cell L6 and copied to L7:L105 Shortages computed by entering =MIN(G6-H6,0) in I6 and copying to I7:I105

42. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-42 Decision Making with Simulation (cont.) New formula for two laptops ordered per week

43. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-43 Areas of Simulation Application  Waiting Lines/Service Complex systems for which it is difficult to develop analytical formulas Determine how many registers and servers are needed to meet customer demand  Inventory Management Traditional models make the assumption that customer demand is certain Simulation is widely used to analyze JIT without having to implement it physically Simulation is widely used to analyze JIT without having to implement it physically

44. Copyright 2006 John Wiley & Sons, Inc.Supplement 12-44 Areas of Simulation Application (cont.)  Production and Manufacturing Systems Examples: production scheduling, production sequencing, assembly line balancing, plant layout, and plant location analysis Machine breakdowns typically occur according to some probability distributions  Capital Investment and Budgeting Capital budgeting problems require estimates of cash flows, often resulting from many random variables Simulation has been used to generate values of cash flows, market size, selling price, growth rate, and market share

45. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-45 Areas of Simulation Application (cont.)  Logistics Typically include numerous random variables, such as distance, different modes of transport, shipping rates, and schedules to analyze different distribution channels  Service Operations Examples: police departments, fire departments, post offices, hospitals, court systems, airports Complex operations that no technique except simulation can be employed  Environmental and Resource Analysis Examples: impact of manufacturing plants, waste-disposal facilities, nuclear power plants, waste and population conditions, feasibility of alternative energy sources

46. Copyright 2009 John Wiley & Sons, Inc.Supplement 12-46 Copyright 2009 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without express permission of the copyright owner is unlawful. Request for further information should be addressed to the Permission Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information herein.