1. © 2006 Prentice Hall, Inc.D – 1 Operations Management Module D – Waiting-Line Models © 2006 Prentice Hall, Inc. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e

2. © 2006 Prentice Hall, Inc.D – 2 Outline  Characteristics Of A Waiting-Line System  Arrival Characteristics  Waiting-Line Characteristics  Service Facility Characteristics  Measuring the Queue’s Performance  Queuing Costs

3. © 2006 Prentice Hall, Inc.D – 3 Outline – Continued  The Variety Of Queuing Models  Model A(M/M/1): Single-Channel Queuing Model With Poisson Arrivals and Exponential Service Times  Model B(M/M/S): Multiple-Channel Queuing Model  Model C(M/D/1): Constant Service Time Model  Model D: Limited Population Model  Other Queuing Approaches

4. © 2006 Prentice Hall, Inc.D – 4 Learning Objectives When you complete this module, you should be able to: Identify or Define:  The assumptions of the four basic waiting-line models  How to apply waiting-line models  How to conduct an economic analysis of queues Describe or Explain:

5. © 2006 Prentice Hall, Inc.D – 5 Common Queuing Situations Situation Arrivals in Queue Service Process Supermarket Grocery shoppers Checkout clerks at cash register Highway toll booth Automobiles Collection of tolls at booth Doctor’s office Patients Treatment by doctors and nurses Computer system Programs to be run Computer processes jobs Telephone company Callers Switching equipment to forward calls BankCustomer Transactions handled by teller Machine maintenance Broken machines Repair people fix machines Harbor Ships and barges Dock workers load and unload Table D.1

6. © 2006 Prentice Hall, Inc.D – 6 Characteristics of Waiting- Line Systems 1.Arrivals or inputs to the system  Population size, behavior, statistical distribution 2.Queue discipline, or the waiting line itself  Limited or unlimited in length, discipline of people or items in it 3.The service facility  Design, statistical distribution of service times

7. © 2006 Prentice Hall, Inc.D – 7 Arrival Characteristics 1.Size of the population  Unlimited (infinite) or limited (finite) 2.Behavior of arrivals  Scheduled or random, often a Poisson distribution 3.Behavior of arrivals  Wait in the queue and do not switch lines  Balking or reneging

8. © 2006 Prentice Hall, Inc.D – 8 Parts of a Waiting Line Figure D.1 Dave’s Car Wash enterexit Population of dirty cars Arrivals from the general population … Queue (waiting line) Servicefacility Exit the system Arrivals to the system Exit the system In the system Arrival Characteristics  Size of the population  Behavior of arrivals  Statistical distribution of arrivals Waiting Line Characteristics  Limited vs. unlimited  Queue discipline Service Characteristics  Service design  Statistical distribution of service

9. © 2006 Prentice Hall, Inc.D – 9 Poisson Distribution P(x) = for x = 0, 1, 2, 3, 4, … e - x x! whereP(x)=probability of x arrivals x=number of arrivals per unit of time =average arrival rate =average arrival rate e=2.7183 (which is the base of the natural logarithms)

10. © 2006 Prentice Hall, Inc.D – 10 Poisson Distribution Probability = P(x) = e - x x! 0.25 0.25 – 0.02 0.02 – 0.15 0.15 – 0.10 0.10 – 0.05 0.05 – – Probability 0123456789 Distribution for = 2 x 0.25 0.25 – 0.02 0.02 – 0.15 0.15 – 0.10 0.10 – 0.05 0.05 – – Probability 0123456789 Distribution for = 4 x 1011 Figure D.2

11. © 2006 Prentice Hall, Inc.D – 11 Waiting-Line Characteristics  Limited or unlimited queue length  Queue discipline - first-in, first-out is most common  Other priority rules may be used in special circumstances

12. © 2006 Prentice Hall, Inc.D – 12 Service Characteristics  Queuing system designs  Single-channel system, multiple- channel system  Single-phase system, multiphase system  Service time distribution  Constant service time  Random service times, usually a negative exponential distribution

13. © 2006 Prentice Hall, Inc.D – 13 Queuing System Designs Figure D.3 Departures after service Single-channel, single-phase system Queue Arrivals Single-channel, multiphase system Arrivals Departures after service Phase 1 service facility Phase 2 service facility Service facility Queue Your family dentist’s office McDonald’s dual window drive-through

14. © 2006 Prentice Hall, Inc.D – 14 Queuing System Designs Figure D.3 Multi-channel, single-phase system Arrivals Queue Most bank and post office service windows Departures after service Service facility Channel 1 Service facility Channel 2 Service facility Channel 3

15. © 2006 Prentice Hall, Inc.D – 15 Queuing System Designs Figure D.3 Multi-channel, multiphase system Arrivals Queue Some college registrations Departures after service Phase 2 service facility Channel 1 Phase 2 service facility Channel 2 Phase 1 service facility Channel 1 Phase 1 service facility Channel 2

16. © 2006 Prentice Hall, Inc.D – 16 Negative Exponential Distribution Figure D.4 1.0 1.0 – 0.9 0.9 – 0.8 0.8 – 0.7 0.7 – 0.6 0.6 – 0.5 0.5 – 0.4 0.4 – 0.3 0.3 – 0.2 0.2 – 0.1 0.1 – 0.0 0.0 – Probability that service time ≥ 1 ||||||||||||| 0.000.250.500.751.001.251.501.752.002.252.502.753.00 Time t in hours Probability that service time is greater than t = e -µt for t ≥ 1 µ = Average service rate e = 2.7183 Average service rate (µ) = 1 customer per hour Average service rate (µ) = 3 customers per hour  Average service time = 20 minutes per customer

17. © 2006 Prentice Hall, Inc.D – 17 Measuring Queue Performance 1.Average time that each customer or object spends in the queue 2.Average queue length 3.Average time in the system 4.Average number of customers in the system 5.Probability the service facility will be idle 6.Utilization factor for the system 7.Probability of a specified number of customers in the system

18. © 2006 Prentice Hall, Inc.D – 18 Queuing Costs Figure D.5 Total expected cost Cost of providing service Cost Low level of service High level of service Cost of waiting time MinimumTotalcost Optimal service level

19. © 2006 Prentice Hall, Inc.D – 19 Queuing Models Table D.2 ModelNameExample ASingle channel Information counter system at department store system at department store(M/M/1) NumberNumberArrivalService ofofRateTimePopulationQueue ChannelsPhasesPatternPatternSizeDiscipline SingleSinglePoissonExponentialUnlimitedFIFO

20. © 2006 Prentice Hall, Inc.D – 20 Queuing Models Table D.2 ModelNameExample BMultichannel Airline ticket (M/M/S) counter (M/M/S) counter NumberNumberArrivalService ofofRateTimePopulationQueue ChannelsPhasesPatternPatternSizeDiscipline Multi-SinglePoissonExponentialUnlimitedFIFO channel channel

21. © 2006 Prentice Hall, Inc.D – 21 Queuing Models Table D.2 ModelNameExample CConstant Automated car service wash service wash(M/D/1) NumberNumberArrivalService ofofRateTimePopulationQueue ChannelsPhasesPatternPatternSizeDiscipline SingleSinglePoissonConstantUnlimitedFIFO

22. © 2006 Prentice Hall, Inc.D – 22 Queuing Models Table D.2 ModelNameExample DLimited Shop with only a population dozen machines population dozen machines (finite) that might break NumberNumberArrivalService ofofRateTimePopulationQueue ChannelsPhasesPatternPatternSizeDiscipline SingleSinglePoissonExponentialLimitedFIFO

23. © 2006 Prentice Hall, Inc.D – 23 Model A - Single Channel 1.Arrivals are FIFO and every arrival waits to be served regardless of the length of the queue 2.Arrivals are independent of preceding arrivals but the average number of arrivals does not change over time 3.Arrivals are described by a Poisson probability distribution and come from an infinite population

24. © 2006 Prentice Hall, Inc.D – 24 Model A - Single Channel 4.Service times vary from one customer to the next and are independent of one another, but their average rate is known 5.Service times occur according to the negative exponential distribution 6.The service rate is faster than the arrival rate

25. © 2006 Prentice Hall, Inc.D – 25 Model A - Single Channel =Mean number of arrivals per time period =Mean number of arrivals per time period µ=Mean number of units served per time period L s =Average number of units (customers) in the system (waiting and being served) = W s =Average time a unit spends in the system (waiting time plus service time) = µ – µ – 1 Table D.3

26. © 2006 Prentice Hall, Inc.D – 26 Model A - Single Channel L q =Average number of units waiting in the queue = W q =Average time a unit spends waiting in the queue = p=Utilization factor for the system = 2 µ(µ – ) µ Table D.3

27. © 2006 Prentice Hall, Inc.D – 27 Model A - Single Channel P 0 =Probability of 0 units in the system (that is, the service unit is idle) =1 – P n > k =Probability of more than k units in the system, where n is the number of units in the system =µ µ k + 1 Table D.3

28. © 2006 Prentice Hall, Inc.D – 28 Single Channel Example =2 cars arriving/hourµ= 3 cars serviced/hour =2 cars arriving/hourµ= 3 cars serviced/hour L s = = = 2 cars in the system on average W s = = = 1 hour average waiting time in the system L q = = = 1.33 cars waiting in line 2 µ(µ – ) µ – µ – 1 2 3 - 2 1 2 2 3(3 - 2)

29. © 2006 Prentice Hall, Inc.D – 29 Single Channel Example W q = = = 40 minute average waiting time p= /µ = 2/3 = 66.6% of time mechanic is busy µ(µ – ) 2 3(3 - 2) µ P 0 = 1 - =.33 probability there are 0 cars in the system =2 cars arriving/hourµ= 3 cars serviced/hour =2 cars arriving/hourµ= 3 cars serviced/hour

30. © 2006 Prentice Hall, Inc.D – 30 Single Channel Example Probability of More Than k Cars in the System kP n > k = (2/3) k + 1 0.667  Note that this is equal to 1 - P 0 = 1 -.33 1.444 2.296 3.198  Implies that there is a 19.8% chance that more than 3 cars are in the system 4.132 5.088 6.058 7.039

31. © 2006 Prentice Hall, Inc.D – 31 Single Channel Economics Customer dissatisfaction and lost goodwill= $10 per hour W q = 2/3 hour Total arrivals= 16 per day Mechanic’s salary= $56 per day Total hours customers spend waiting per day = (16) = 10 hours 2323 Customer waiting-time cost = $10 10 = $106.67 23 Total expected costs = $106.67 + $56 = $162.67

32. © 2006 Prentice Hall, Inc.D – 32 Multi-Channel Model M=number of channels open =average arrival rate =average arrival rate µ=average service rate at each channel P 0 = for Mµ > P 0 = for Mµ > 11 11M!M!11M!M!1 11n!n!11n!n! Mµ Mµ - Mµ - M – 1 n = 0 µ n µ M + ∑ L s = P 0 + µ( /µ) µ( /µ)M (M - 1)!(Mµ - ) 2 µ Table D.4

33. © 2006 Prentice Hall, Inc.D – 33 Multi-Channel Example = 2 µ = 3 M = 2 = 2 µ = 3 M = 2 P 0 = = 11 112!2!112!2! 1 11n!n!11n!n! 2(3) 2(3) - 2 1 n = 0 23 n 23 2 + ∑ 12 L s = + = (2)(3(2/3) 2 23 1! 2(3) - 2 2 1234 W q = =.0415.0832 W s = = 3/4238 L q = – = 23 34112

34. © 2006 Prentice Hall, Inc.D – 34 Multi-Channel Example Single Channel Two Channels P0P0P0P0.33.5 LsLsLsLs 2 cars.75 cars WsWsWsWs 60 minutes 22.5 minutes LqLqLqLq 1.33 cars.083 cars WqWqWqWq 40 minutes 2.5 minutes

35. © 2006 Prentice Hall, Inc.D – 35 Constant Service Model Table D.5 L q = 2 2µ(µ – ) Average length of queue W q = 2µ(µ – ) Average waiting time in queue µ L s = L q + Average number of customers in system W s = W q + 1µ Average waiting time in system

36. © 2006 Prentice Hall, Inc.D – 36 Net savings= $ 7 /trip Constant Service Example Trucks currently wait 15 minutes on average Truck and driver cost $60 per hour Automated compactor service rate (µ) = 12 trucks per hour Arrival rate ( ) = 8 per hour Compactor costs $3 per truck Current waiting cost per trip = (1/4 hr)($60) = $15 /trip W q = = hour 8 2(12)(12 - 8) 112 Waiting cost/trip with compactor = (1/12 hr wait)($60/hr cost)= $ 5 /trip Savings with new equipment = $15 (current) - $ 5 (new)= $10 /trip Cost of new equipment amortized= $ 3 /trip

37. © 2006 Prentice Hall, Inc.D – 37 Limited Population Model Service factor: X = Average number running: J = NF(1 - X) Average number waiting: L = N(1 - F) Average number being serviced: H = FNX Average waiting time: W = Number of population: N = J + L + H T T + U T(1 - F) XF

38. © 2006 Prentice Hall, Inc.D – 38 Limited Population Model Service factor: X = Average number running: J = NF(1 - X) Average number waiting: L = N(1 - F) Average number being serviced: H = FNX Average waiting time: W = Number of population: N = J + L + H T T + U T(1 - F) XF D =Probability that a unit will have to wait in queue N =Number of potential customers F =Efficiency factorT =Average service time H =Average number of units being served U =Average time between unit service requirements J =Average number of units not in queue or in service bay W =Average time a unit waits in line L =Average number of units waiting for service X =Service factor M =Number of service channels

39. © 2006 Prentice Hall, Inc.D – 39 Finite Queuing Table Table D.7 XMDF.0121.048.999.0251.100.997.0501.198.989.0602.020.999 1.237.983.0702.027.999 1.275.977.0802.035.998 1.313.969.0902.044.998 1.350.960.1002.054.997 1.386.950

40. © 2006 Prentice Hall, Inc.D – 40 Limited Population Example Service factor: X = =.091 (close to.090) For M = 1, D =.350 and F =.960 For M = 2, D =.044 and F =.998 Average number of printers working: For M = 1, J = (5)(.960)(1 -.091) = 4.36 For M = 2, J = (5)(.998)(1 -.091) = 4.54 2 2 + 20 Each of 5 printers requires repair after 20 hours (U) of use One technician can service a printer in 2 hours (T) Printer downtime costs $120/hour Technician costs $25/hour

41. © 2006 Prentice Hall, Inc.D – 41 Limited Population Example Service factor: X = =.091 (close to.090) For M = 1, D =.350 and F =.960 For M = 2, D =.044 and F =.998 Average number of printers working: For M = 1, J = (5)(.960)(1 -.091) = 4.36 For M = 2, J = (5)(.998)(1 -.091) = 4.54 2 2 + 20 Each of 5 printers require repair after 20 hours (U) of use One technician can service a printer in 2 hours (T) Printer downtime costs $120/hour Technician costs $25/hour Number of Technicians Average Number Printers Down (N - J) Average Cost/Hr for Downtime (N - J)$120 Cost/Hr for Technicians ($25/hr) Total Cost/Hr 1.64$76.80$25.00$101.80 2.46$55.20$50.00$105.20

42. © 2006 Prentice Hall, Inc.D – 42 Operations Management Module E – Learning Curves © 2006 Prentice Hall, Inc. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e

43. © 2006 Prentice Hall, Inc.D – 43 Outline  Learning Curves In Services And Manufacturing  Applying The Learning Curve  Arithmetic Approach  Logarithmic Approach  Learning-Curve Coefficient Approach  Strategic Implications of Learning Curves  Limitations of Learning Curves

44. © 2006 Prentice Hall, Inc.D – 44 Learning Objectives When you complete this module, you should be able to: Identify or Define:  What a learning curve is  Examples of learning curves  The doubling concept

45. © 2006 Prentice Hall, Inc.D – 45 Learning Objectives When you complete this module, you should be able to: Describe or Explain:  How to compute learning curve effects  Why learning curves are important  The strategic implication of learning curves

46. © 2006 Prentice Hall, Inc.D – 46 Learning Curves  Based on the premise that people and organizations become better at their tasks as the tasks are repeated  Time to produce a unit decreases as more units are produced  Learning curves typically follow a negative exponential distribution  The rate of improvement decreases over time

47. © 2006 Prentice Hall, Inc.D – 47 Learning Curve Effect Figure E.1 Cost/time per repetition Number of repetitions (volume) 0

48. © 2006 Prentice Hall, Inc.D – 48 Learning Curves T x L n = Time required for the n th unit whereT=unit cost or unit time of the first unit L=learning curve rate n=number of times T is doubled First unit takes 10 labor-hours 70% learning curve is present Fourth unit will require doubling twice — 1 to 2 to 4 Hours required for unit 4 = 10 x (.7) 2 = 4.9 hours

49. © 2006 Prentice Hall, Inc.D – 49 Learning Curve Examples Table E.1 Example Improving Parameters Cumulative Parameter Learning- Curve Slope (%) (%) Model -T Ford production Price Units produced 86 Aircraft assembly Direct labor-hours per unit Units produced 80 Equipment maintenance at GE Average time to replace a group of parts Number of replacements 76 Steel production Production worker labor-hours per unit produced Units produced 79

50. © 2006 Prentice Hall, Inc.D – 50 Learning Curve Examples Table E.1 Example Improving Parameters Cumulative Parameter Learning- Curve Slope (%) (%) Integrated circuits Average price per unit Units produced 72 Hand-held calculator Average factory selling price Units produced 74 Disk memory drives Average price per bit Number of bits 76 Heart transplants 1-year death rates Transplants completed 79

51. © 2006 Prentice Hall, Inc.D – 51 Uses of Learning Curves Internal:labor forecasting, scheduling, establishing costs and budgets External:supply chain negotiations Strategic:evaluation of company and industry performance, including costs and pricing

52. © 2006 Prentice Hall, Inc.D – 52 Arithmetic Approach  Simplest approach  Labor cost declines at a constant rate, the learning rate, as production doubles Nth Unit Produced Hours for Nth Unit 1100.0 2 80.0= (.8 x 100) 4 64.0= (.8 x 80) 8 51.2= (.8 x 64) 16 41.0= (.8 x 51.2)

53. © 2006 Prentice Hall, Inc.D – 53 Logarithmic Approach Determine labor for any unit, T N, by T N = T 1 (N b ) whereT N =time for the N th unit T 1 =hours to produce the first unit b=(log of the learning rate)/(log 2) = slope of the learning curve

54. © 2006 Prentice Hall, Inc.D – 54 Logarithmic Approach Determine labor for any unit, T N, by T N = T 1 (N b ) whereT N =time for the N th unit T 1 =hours to produce the first unit b=(log of the learning rate)/(log 2) =slope of the learning curve Learning Rate (%) b 70–.515 75–.415 80–.322 85–.234 90–.152 Table E.2

55. © 2006 Prentice Hall, Inc.D – 55 Logarithmic Example Learning rate = 80% First unit took 100 hours T N =T 1 (N b ) T 3 =(100 hours)(3 b ) =(100)(3 log.8/log 2 ) =(100)(3 –.322 ) =70.2 labor hours

56. © 2006 Prentice Hall, Inc.D – 56 Coefficient Approach TN=T1CTN=T1CTN=T1CTN=T1C whereT N =number of labor-hours required to produce the N th unit T 1 =number of labor-hours required to produce the first unit C=learning-curve coefficient found in Table E.3

57. © 2006 Prentice Hall, Inc.D – 57 Learning-Curve Coefficients Table E.3 70%85% UnitNumber (N) TimeUnit TimeTotal TimeUnit TimeTotal Time 11.0001.0001.0001.000 2.7001.700.8501.850 3.5682.268.7732.623 4.4902.758.7233.345 5.4373.195.6864.031 10.3064.932.5837.116 15.2486.274.5309.861 20.2147.407.49512.402

58. © 2006 Prentice Hall, Inc.D – 58 Industry and Company Learning Curves Figure E.2 Price per unit (log scale) Accumulated volume (log scale) Gross profit margin Loss (a)(a)(a)(a) (c)(c)(c)(c) (b)(b)(b)(b) Company cost Industry price

59. © 2006 Prentice Hall, Inc.D – 59 Coefficient Example First boat required 125,000 hours Labor cost = $40/hour Learning factor = 85% T N =T 1 C T 4 =(125,000 hours)(.723) =90,375 hours for the 4 th boat 90,375 hours x $40/hour = $3,615,000 T N =T 1 C T 4 =(125,000 hours)(3.345) =418,125 hours for all four boats

60. © 2006 Prentice Hall, Inc.D – 60 Coefficient Example Third boat required 100,000 hours Learning factor = 85% 100,000.773 = 129,366 hours New estimate for the first boat

61. © 2006 Prentice Hall, Inc.D – 61 Strategic Implications To pursue a strategy of a steeper curve than the rest of the industry, a firm can: 1.Follow an aggressive pricing policy 2.Focus on continuing cost reduction and productivity improvement 3.Build on shared experience 4.Keep capacity ahead of demand

62. © 2006 Prentice Hall, Inc.D – 62 Limitations of Learning Curves  Learning curves differ from company to company as well as industry to industry so estimates should be developed for each organization  Learning curves are often based on time estimates which must be accurate and should be reevaluated when appropriate

63. © 2006 Prentice Hall, Inc.D – 63 Limitations of Learning Curves  Any changes in personnel, design, or procedure can be expected to alter the learning curve  Learning curves do not always apply to indirect labor or material  The culture of the workplace, resource availability, and changes in the process may alter the learning curve

64. © 2006 Prentice Hall, Inc.D – 64 Operations Management Module F – Simulation © 2006 Prentice Hall, Inc. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e

65. © 2006 Prentice Hall, Inc.D – 65 Outline  What Is Simulation?  Advantages and Disadvantages of Simulation  Monte Carlo Simulation  Simulation of a Queuing Problem  Simulation and Inventory Analysis

66. © 2006 Prentice Hall, Inc.D – 66 Learning Objectives When you complete this module, you should be able to: Identify or Define:  Monte Carlo simulation  Random numbers  Random number interval  Simulation software

67. © 2006 Prentice Hall, Inc.D – 67 Learning Objectives When you complete this module, you should be able to: Describe or Explain:  The advantages and disadvantages of modeling with simulation  The use of Excel spreadsheets in simulation

68. © 2006 Prentice Hall, Inc.D – 68 What is Simulation?  An attempt to duplicate the features, appearance, and characteristics of a real system 1.To imitate a real-world situation mathematically 2.To study its properties and operating characteristics 3.To draw conclusions and make action decisions based on the results of the simulation

69. © 2006 Prentice Hall, Inc.D – 69 Computer Analysis

70. © 2006 Prentice Hall, Inc.D – 70 Simulation Applications Ambulance location and dispatching Assembly-line balancing Parking lot and harbor design Distribution system design Scheduling aircraft Labor-hiring decisions Personnel scheduling Traffic-light timing Voting pattern prediction Bus scheduling Design of library operations Taxi, truck, and railroad dispatching Production facility scheduling Plant layout Capital investments Production scheduling Sales forecasting Inventory planning and control Table F.1

71. © 2006 Prentice Hall, Inc.D – 71 Select best course Examine results Conduct simulation Specify values of variables Construct model Introduce variables The Process of Simulation Figure F.1 Define problem

72. © 2006 Prentice Hall, Inc.D – 72 Advantages of Simulation 1.Relatively straightforward and flexible 2.Can be used to analyze large and complex real-world situations that cannot be solved by conventional models 3.Real-world complications can be included that most OM models cannot permit 4.“Time compression” is possible

73. © 2006 Prentice Hall, Inc.D – 73 Advantages of Simulation 5.Allows “what-if” types of questions 6.Does not interfere with real-world systems 7.Can study the interactive effects of individual components or variables in order to determine which ones are important

74. © 2006 Prentice Hall, Inc.D – 74 Disadvantages of Simulation 1.Can be very expensive and may take months to develop 2.It is a trial-and-error approach that may produce different solutions in repeated runs 3.Managers must generate all of the conditions and constraints for solutions they want to examine 4.Each simulation model is unique

75. © 2006 Prentice Hall, Inc.D – 75 Monte Carlo Simulation The Monte Carlo method may be used when the model contains elements that exhibit chance in their behavior 1.Set up probability distributions for important variables 2.Build a cumulative probability distribution for each variable 3.Establish an interval of random numbers for each variable 4.Generate random numbers 5.Simulate a series of trials

76. © 2006 Prentice Hall, Inc.D – 76 Probability of Demand (1)(2)(3)(4) Demand for Tires Frequency Probability of Occurrence Cumulative Probability 010 10/200 =.05.05 120 20/200 =.10.15 240 40/200 =.20.35 360 60/200 =.30.65 440 40/200 =.20.85 530 30/ 200 =.15 1.00 200 days 200/200 = 1.00 Table F.2

77. © 2006 Prentice Hall, Inc.D – 77 Assignment of Random Numbers Daily Demand Cumulative Probability Interval of Probability Random Numbers 0.05.05 01 through 05 1.10.15 06 through 15 2.20.35 16 through 35 3.30.65 36 through 65 4.20.85 66 through 85 5.151.00 86 through 00 Table F.3

78. © 2006 Prentice Hall, Inc.D – 78 Table of Random Numbers 5250605205 3727806934 8245533355 6981693209 9866373077 9674064808 3330638845 5059571484 8867020284 9060948377 Table F.4

79. © 2006 Prentice Hall, Inc.D – 79 Simulation Example 1 Select random numbers from Table F.3 DayNumberRandomNumberSimulated Daily Demand 1523 2373 3824 4694 5985 6965 7332 8503 9885 10905 39Total 3.9 Average

80. © 2006 Prentice Hall, Inc.D – 80 Simulation Example 1 DayNumberRandomNumberSimulated Daily Demand 1523 2373 3824 4694 5985 6965 7332 8503 9885 10905 39Total 3.9 Average Expected demand = ∑ (probability of i units) x (demand of i units) =(.05)(0) + (.10)(1) + (.20)(2) + (.30)(3) + (.20)(4) + (.15)(5) =0 +.1 +.4 +.9 +.8 +.75 =2.95 tires 5 i=1

81. © 2006 Prentice Hall, Inc.D – 81 Queuing Simulation Number of Arrivals Probability Cumulative Probability Random-Number Interval 0.13.13 01 through 13 1.17.30 14 through 30 2.15.45 31 through 45 3.25.70 46 through 70 4.20.90 71 through 90 5.101.00 91 through 00 Overnight barge arrival rates Table F.5

82. © 2006 Prentice Hall, Inc.D – 82 Queuing Simulation (1)Day(2)Number Delayed from Previous Day (3) Random Number (4)Number of Nightly Arrivals(5)Total to be Unloaded(6) Random Number (7) Number Unloaded 10523333 200600630 305033283 408844021 535336744 623013353 701000240 804733031 929957293 1043726603 1136636744 1229157854 1333525904 1413223733 1500055593 204139

83. © 2006 Prentice Hall, Inc.D – 83 Queuing Simulation Average number of barges delayed to the next day = = 1.33 barges delayed per day 20 delays 15 days Average number of nightly arrivals = = 2.73 arrivals per night 41 arrivals 15 days Average number of barges unloaded each day = = 2.60 unloadings per day 39 unloadings 15 days

84. © 2006 Prentice Hall, Inc.D – 84 Inventory Simulation (1) Demand for Ace Drill (2)Frequency(3)Probability(4)CumulativeProbability(5) Interval of Random Numbers 015.05.05 01 through 05 130.10.15 06 through 15 260.20.35 16 through 35 3120.40.75 36 through 75 445.15.90 76 through 90 530.101.00 91 through 00 3001.00 Table F.8 Daily demand for Ace Drill

85. © 2006 Prentice Hall, Inc.D – 85 Inventory Simulation (1) Demand for Ace Drill (2)Frequency(3)Probability(4)CumulativeProbability(5) Interval of Random Numbers 110.20.20 01 through 20 225.50.70 21 through 70 315.301.00 71 through 00 501.00 Table F.9 Reorder lead time

86. © 2006 Prentice Hall, Inc.D – 86 Inventory Simulation 1.Begin each simulation day by checking to see if ordered inventory has arrived. If if has, increase current inventory by the quantity ordered. 2.Generate daily demand using probability distribution and random numbers. 3.Compute ending inventory. If on-hand is insufficient to meet demand, satisfy as much as possible and note lost sales. 4.Determine whether the day's ending inventory has reached the reorder point. If it has, and there are no outstanding orders, place an order. Choose lead time using probability distribution and random numbers.

87. © 2006 Prentice Hall, Inc.D – 87 Inventory Simulation (1)Day(2)UnitsReceived(3) Beginning Inventory (4) Random Number (5)Demand(6) Ending Inventory (7)LostSales(8)Order?(9)RandomNumber(10)LeadTime 11006190No 20963360No 30657330Yes021 40394502No 5101052370No 60769340Yes332 70432210No 80230200No 9101048370No 100788430Yes141 412 Table F.10 Order quantity = 10 units Reorder point = 5 units

88. © 2006 Prentice Hall, Inc.D – 88 Inventory Simulation Average ending inventory = = 4.1 units/day 41 total units 10 days Average lost sales = =.2 unit/day 2 sales lost 10 days = =.3 order/day 3 orders 10 days Average number of orders placed

89. © 2006 Prentice Hall, Inc.D – 89 Inventory Simulation Daily order cost=(cost of placing 1 order) x (number of orders placed per day) =$10 per order x.3 order per day = $3 Daily holding cost=(cost of holding 1 unit for 1 day) x (average ending inventory) =50¢ per unit per day x 4.1 units per day =$2.05 Daily stockout cost=(cost per lost sale) x (average number of lost sales per day) =$8 per lost sale x.2 lost sales per day =$1.60 Total daily inventory cost=Daily order cost + Daily holding cost + Daily stockout cost =$6.65

90. © 2006 Prentice Hall, Inc.D – 90 Using Software in Simulation  Computers are critical in simulating complex tasks  General-purpose languages - BASIC, C++  Special-purpose simulation languages - GPSS, SIMSCRIPT 1.Require less programming time for large simulations 2.Usually more efficient and easier to check for errors 3.Random-number generators are built in

91. © 2006 Prentice Hall, Inc.D – 91 Using Software in Simulation  Commercial simulation programs are available for many applications - Extend, Modsim, Witness, MAP/1, Enterprise Dynamic, Simfactory, ProModel, Micro Saint, ARENA  Spreadsheets such as Excel can be used to develop some simulations