1. MISSISSIPPI RIVER BARGE ARRIVALS AND UNLOADINGS A Queuing Simulation Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD Port of New Orleans

2. Two Courses of Action Considered COA Number 1 COA Number 2 DOCK CREW OF 6 DOCK CREW OF 12 DOCK CREW OF 6 DOCK CREW OF 12 COURSE OF ACTION NUMBER ONE COURSE OF ACTION NUMBER TWO

3. Two Relevant Variables Daily Evening Barge Arrivals Unloadings VARIABLENUMBERONEVARIABLENUMBERTWO Port of New Orleans

4. Evaluation Criteria 1.Average Number of Barges Unloaded Each evening. 2.Average Number of Barges Delayed Each evening.

5. Simulation Execution Simulate a daily barge arrival.  Simulate a daily barge arrival.

6. Simulation Execution Simulate a daily barge arrival.  Simulate a daily barge arrival.  Simulate a daily barge unloading.

7. Simulation Execution Simulate a daily barge arrival.  Simulate a daily barge arrival.  Simulate a daily barge unloading.  Determine how many, if any, barges remain unloaded at the end of the evening. remain unloaded at the end of the evening.

8. Simulation Execution Simulate a daily barge arrival.  Simulate a daily barge arrival.  Simulate a daily barge unloading.  Determine how many, if any, barges remain unloaded at the end of the evening. remain unloaded at the end of the evening.  Unloaded barges become the beginning balance for the following evening. balance for the following evening.

9. Overnight Barge Arrivals SPREADSHEET NUMBER OF ARRIVALS PROBABILITYCUMULATIVE PROBABILITY RANDOM NUMBER INTERVAL0.13.13 01 - 13 1.17.30 14 - 30 2.15.45 31 - 45 3.25.70 46 - 70 4.20.90 71 - 90 5.101.00 91 - 00

10. Crew of 6 Unloading Rates SPREADSHEET DAILY UNLOADING RATE PROBABILITY CUMULATIVE PROBABILITY RANDOM NUMBER INTERVAL1.05.05 01 - 05 2.15.20 06 - 20 3.50.70 21 - 70 4.20.90 71 - 90 5.101.00 91 - 00

11. Random Number Strings TO GENERATE DAILY ARRIVALS 52 06 50 88 53 30 10 47 99 37 66 91 35 32 00

12. Random Number Strings TO GENERATE DAILY ARRIVALS 52 06 50 88 53 30 10 47 99 37 66 91 35 32 00 TO GENERATE DAILY UNLOADINGS 37 63 28 02 74 35 24 03 29 60 74 85 90 73 59

15. Random Number Strings Day 1 st 2 nd 3 rd Random Number for Daily Arrival 520650 Random Number for Daily Unloading 376328

16. Random Number Strings Day 4 th 5 th 6 th Random Number for Daily Arrival 885330 Random Number for Daily Unloading 027435

17. Random Number Strings Day 4 th 5 th 6 th Random Number for Daily Arrival 885330 Random Number for Daily Unloading 027435

18. Random Number Strings Day 4 th 5 th 6 th Random Number for Daily Arrival 885330 Random Number for Daily Unloading 027435

19. Random Number Strings Day 7 th 8 th 9 th Random Number for Daily Arrival 104799 Random Number for Daily Unloading 240329

20. Random Number Strings Day 7 th 8 th 9 th Random Number for Daily Arrival 104799 Random Number for Daily Unloading 240329

21. Random Number Strings Day 7 th 8 th 9 th Random Number for Daily Arrival 104799 Random Number for Daily Unloading 240329

22. Random Number Strings Day 10 th 11 th 12 th Random Number for Daily Arrival 376691 Random Number for Daily Unloading 607485

23. Random Number Strings Day 10 th 11 th 12 th Random Number for Daily Arrival 376691 Random Number for Daily Unloading 607485

24. Random Number Strings Day 10 th 11 th 12 th Random Number for Daily Arrival 376691 Random Number for Daily Unloading 607485

25. Random Number Strings Day 13 th 14 th 15 th Random Number for Daily Arrival 353200 Random Number for Daily Unloading 907359

26. Random Number Strings Day 13 th 14 th 15 th Random Number for Daily Arrival 353200 Random Number for Daily Unloading 907359

27. Random Number Strings Day 13 th 14 th 15 th Random Number for Daily Arrival 353200 Random Number for Daily Unloading 907359

28. Simulation Execution EVENING NUMBERDELAYEDPREVIOUSEVENING ARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED 1 st - a - a5233373 2 nd 0060063 0 b 0 b 3 rd 05033283 a – a – WE CAN BEGIN WITH NO DELAYS OR SOME DELAYS FROM THE PREVIOUS EVENING. OVER THE LENGTH OF THE SIMULATION, THE INITIAL BALANCE AVERAGES OUT. b - b - THREE BARGES COULD HAVE BEEN UNLOADED BUT SINCE THERE WERE NO ARRIVALS AND NO BACKLOG, ZERO UNLOADINGS RESULTED. c- THE PROGRAM WOULD HAVE UNLOADED ANY NUMBER OF BARGES UP TO, AND INCLUDING THREE (3), HAD THERE BEEN A POSITIVE BALANCE FOR TOTAL TO BE UNLOADED !,c

29. Simulation Execution EVENING NUMBERDELAYEDPREVIOUSEVENING ARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADED UNLOADINGRANDOMNUMBER NUMBERUNLOADED 4 th 08844021 5 th 35336744 6 th 23013353

30. Simulation Execution EVENING NUMBERDELAYEDPREVIOUSEVENING ARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED 7 th 01000240 8 th 04733031 9 th 29957293 c,d C – THREE BARGES COULD HAVE BEEN UNLOADED BUT SINCE THERE WERE NO ARRIVALS AND NO BACKLOGS, ZERO UNLOADINGS RESULTED. D - THE PROGRAM WOULD HAVE UNLOADED UP TO, AND INCLUDING THREE (3) BARGES, HAD THERE BEEN A POSITIVE BALANCE FOR UNLOADINGS !

31. Simulation Execution EVENING NUMBERDELAYEDPREVIOUSEVENING ARRIVAL RANDOM NUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED 13 th 33525904 14 th 1322373 3d 3d 3d 3d 15 th 00055593 d – FOUR BARGES COULD HAVE BEEN UNLOADED BUT SINCE ONLY THREE WERE IN THE QUEUE, THE NUMBER UNLOADED IS RECORDED AS “3”.

32. Simulation Summary EVENING NUMBERDELAYEDPREVIOUSEVENING ARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED 13 th 33525904 14 th 13223733 15 th 00055593 TOTAL DELAYS = 20 AVERAGE = 1.33 TOTAL ARRIVALS = 41 AVERAGE = 2.73 TOTAL UNLOADINGS = 39 AVERAGE = 2.60

33. Overnight Barge Arrivals SPREADSHEET FOR CREW OF 12 NUMBER OF ARRIVALS PROBABILITYCUMULATIVE PROBABILITY RANDOM NUMBER INTERVAL0.13.13 01 - 13 1.17.30 14 - 30 2.15.45 31 - 45 3.25.70 46 - 70 4.20.90 71 - 90 5.101.00 91 - 00

34. Crew of 12 Unloading Rates SPREADSHEET DAILY UNLOADING RATE PROBABILITY CUMULATIVE PROBABILITY RANDOM NUMBER INTERVAL1.03.03 01 - 03 2.12.15 04 - 15 3.40.55 16 - 55 4.28.83 56 - 83 5.12.95 84 - 95 6.051.00 96 - 00

35. Random Number Strings CREW OF 12 SIMULATION TO GENERATE DAILY ARRIVALS 37 77 13 10 02 18 31 19 32 85 31 94 81 43 31

36. Random Number Strings CREW OF 12 SIMULATION TO GENERATE DAILY ARRIVALS TO GENERATE DAILY UNLOADINGS 37 77 13 10 02 18 31 19 32 85 31 94 81 43 31 69 84 12 94 51 36 17 02 15 29 16 52 56 43 26

37. Simulation Execution CREW OF TWELVE 1 st -3722692 2 nd 07744844 3 rd 01300120 4 th 01000940 EVENING DELAYEDPREVIOUSEVENINGARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED

38. Simulation Execution CREW OF TWELVE 5 th 00200510 6 th 01811361 7 th 03122172 8 th 01911021 EVENING DELAYEDPREVIOUSEVENINGARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED

39. Simulation Execution CREW OF TWELVE 12 th 09455523 13 th 28146564 14 th 24324433 15 th 13123263 EVENING DELAYEDPREVIOUSEVENINGARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED TOTAL DELAYS = 6 AVERAGE = 0.4 TOTAL ARRIVALS = 31 AVERAGE = 2.07 TOTAL UNLOADINGS = 31 AVERAGE = 2.07

40. ScoreboardBarges Crew of 6 Crew of 12 AVERAGE DAILY DELAYS 1.33.40 AVERAGE DAILY UNLOADINGS2.602.07 ARRIVALS2.732.07

41. Possible Relevant Variables  Winds  Currents  Fog  Temperature  River Ice  Seasonal Barge Traffic  Competing Docks  Precipitation  Absentee Rates  Barge Sizes  Additional Crew Staffing Options  Local Economy Effect on Barge Traffic on Barge Traffic  Crew Training BARGE SIMULATION

42. Repeating Random Number Strings Used for generating arrival and unloading rates for both crew staffing options if you want to isolate and observe the impact of each staffing option on the dock-river system. ANY DIFFERENCES FOUND IN THE UNLOADING RATES WOULD BE DIRECTLY ATTRIBUTABLE TO THE CREW SIZE ITSELF, SINCE ALL OTHER ELEMENTS OF THE SIMULATION HAD BEEN HELD CONSTANT!

43. Non-Repeating Random Number Strings Used for generating arrival and unloading rates for both crew staffing options if you want to test for consistent results of the impact of each staffing option on the dock-river system. TO YIELD VALID CONCLUSIONS HOWEVER, YOU MUST INSURE THAT THE SIMULATION HAS RUN OVER A SUFFICIENTLY LONG PERIOD OF TIME IN ORDER TO ALLOW THE NUMBERS TO “SETTLE DOWN” TO THEIR LONG-TERM AVERAGES.

44. Barge Simulation Postscript  If the data were also analyzed in terms of barge delay opportunity costs, extra crew hiring costs, idle time costs, insurance, and barge traffic po- tential, a better quality staffing decision might have been attained.  The simulation should also have been executed under other crew size options. THIS DATE IS AVAILABLE FROM HUMAN RESOURCES, MARKETING, ACCOUNTING, AND FINANCE.

45. QM for WINDOWS COMMENTS   This program cannot simultaneously accommodate two or more relevant variables.  Every simulation is custom-built, and therefore presents too many design options for assimilation into a general-purpose software program.  An alternative would be to run each relevant varia- ble separately, insert the simulated outcomes on a spreadsheet, and then manually calculate the out- comes of the variables’ interactions. THIS APPROACH IS FEASIBLE FOR ONLY THE MOST ELEMENTAL SIMULATIONS

46. Example EVENING NUMBERDELAYEDPREVIOUSEVENING ARRIVALRANDOMNUMBERBARGEARRIVALNUMBERTOTAL TO BE UNLOADEDUNLOADINGRANDOMNUMBER NUMBERUNLOADED 1 st -5233373 2 nd 00600630 3 rd 05033283 SIMULATED VIA QM for WINDOWS or QM EXCEL SIMULATED VIA QM for WINDOWS or QM EXCEL NOT REQUIRED NOT REQUIRED MANUALLYENTERED

49. These simulated barge arrivals would be inserted on our manual spreadsheet

53. These simulated barge unloadings would be inserted on our manual spreadsheet

55. Average Daily Delays ( 20/15 days ) = 1.33 Barges Average Daily Arrivals ( 41/15 days ) = 2.73 Barges Average Daily Unloadings ( 39 / 15 days ) = 2.60 Barges

60. Template and Sample Data

61. Template And Sample Data

66. Template and Sample Data

67. Template and Sample Data

70. Inventory Policy Simulation Establishing an inventory control doctrine for an item having variable daily demand and variable reorder lead time. The goal is to minimize the ordering, holding, and stockout costs involved. a more realistic business application

71. Electric Drill Demand Daily Demand Frequency ( days ) ProbabilityCumulative Probability Random No. Interval 015.05 01 - 05 130.10.1506 - 15 260.20.3516 - 35 3120.40.7536 - 75 445.15.9076 - 90 530.101.0091 - 00 ∑= 300 ∑= 1.00 1 st relevant variable

72. Electric Drill Reorder Lead Time LEAD TIME ( DAYS ) Frequency ( ORDERS )Probability Cumulative Probability RN Interval 110.20 01 - 20 225.50.7021 - 70 315.301.0071 - 00 ∑ = 50 ∑ = 1.00 2 nd relevant variable

73. The Simulation The 1 st inventory policy to be simulated: Q = 10 units R = 5 units Regardless of the simulated lead time period, an order will not arrive the next morning but at the beginning of the following working day Order 10 drills at a time when the shelf stock falls to five drills or less at the end of the business day

74. 1-1006190NO-- 20963360NO-- 30657330YES021 40394502NO-- 5101052370NO-- DAY UNITS RECEIVED BEGINNING INVENTORYRANDOMNUMBER DEMAND ENDINGINVENTORYLOSTSALES ORDER?RANDOM NUMBER LEAD TIME a – 1 st order is placed b – generates 1 st lead time c – next random number in series d – no order placed because of outstanding order from previous day a bc d The Simulation

75. 60769340YES332 70432220NO-- 80230200NO-- 9101048370NO-- 100788430YES141 DAY UNITSRECEIVEDBEGINNINGINVENTORYRANDOMNUMBER DEMAND ENDINGINVENTORYLOSTSALES ORDER ? RANDOM NUMBER LEAD TIME f – order placed at end of 6 th day arrives f The Simulation ∑ = 41 2 3 units ending inventory number of lost sales number of orders placed SUMMARY STATISTICS

76. Simulation Results AVERAGE ENDING INVENTORY 41 units / 10 days = 4.1 units per day AVERAGE LOST SALES 2 sales lost / 10 days =.2 unit per day AVERAGE NUMBER OF ORDERS PLACED 3 orders / 10 days =.3 order per day

77. Simulation Costs Daily Order Cost $10.00 per order x.3 daily orders = $3.00 Daily Holding Cost $.03 per unit per day x 4.1 units per day = $.12 Daily Stockout Cost $8.00 per lost sale x.2 daily lost sales = $1.60 Total Daily Cost = $4.72 ( TOTAL ANNUAL COSTS = $944.00 )

78. Simulation Postscript  We must now compare this potential inventory control doctrine to others.  Perhaps we might evaluate every pair of values for Q ( 6 to 20 units ) and R ( 3 to 10 units ) : After simulating all reasonable combinations of Q and R, we select the pair yielding the lowest total inventory cost

79. Fast Food Drive-Through Simulation ARRIVAL RN for TIME between ARRIVALS TIME BETWEEN ARRIVALS TIME RN for SERVICE TIME SERVICE TIME WaitingTime CUSTOMER LEAVES 1 st 14 1 min. 11:0188 3 min. 011:04 2 nd 74 3 min. 11:0432 2 min. 011:06 3 rd 27 2 min. 11:0636 011:08 4 th 03 1 min. 11:0724 111:09 ( ASSUME THE DRIVE-THROUGH OPENS AT 11:00 AM )

80. Generator Breakdown Simulation

81. Generator Breakdown Simulation TIME BETWEEN RECORDED MACHINE FAILURES (hours) PROBABILITYCUMULATIVEPROBABILITYRANDOM NUMBER INTERVAL ½.05 01 - 05 1.06.1106 - 11 1 ½.16.2712 - 27 2.33.6028 - 60 2 ½.21.8161 - 81 3.191.0082 - 00 ∑1.00

82. Generator Breakdown Simulation REPAIR TIME REQUIRED ( HOURS ) PROBABILITYCUMULATIVEPROBABILITYRANDOMNUMBERINTERVAL 1.28.28 01 - 28 2.52.80 29 - 80 3.201.00 81 - 00 Total1.00 a – MAINTENANCE TIME IS ROUNDED TO HOURLY TIME BLOCKS a

83. 1572 02:00 071 03:00 1 217 1.5 03:30 602 05:30 2 3362 05:30 772 07:30 2 472 2.5 08:00 492 10:00 2 5853 11:00 762 13:00 2 6312 13:00 953 16:00 3 BREAKDOWNNUMBER TIME BETWEEN BREAKDOWNS RANDOM NO. TIMEBETWEENBREAKDOWNS TIME OF BREAKDOWN TIMEMECHANIC FREE TO BEGIN THIS REPAIR REPAIR TIME RANDOM NO. REPAIR TIME REQUIRED TIME REPAIR ENDS NO. HRS. MACHINEDOWN

84. 13332 01:0004:00 402 06:00 5 1489 3 04:0006:00 422 08:00 4 1513 1.5 05:3008:00 522 10:00 4.5 BREAKDOWNNUMBER TIME BETWEEN BREAKDOWNS RANDOM NO. TIMEBETWEENBREAKDOWNS TIME OF BREAKDOWN TIMEMECHANIC FREE TO BEGIN THIS REPAIR REPAIR TIME RANDOM NO. REPAIR TIME REQUIRED TIME REPAIR ENDS TOTAL NO. HRS. MACHINESDOWN 44 Generator Breakdown Simulation

85. Simulation Results Simulation of fifteen (15) generator breakdowns spanned 34 hours of operation. The clock began at 00:00 hours of day 1 and ran until the final repair at 10:00 hours of day 2. THE TOTAL NUMBER OF HOURS THAT GENERATORS WERE OUT OF SERVICE IS COMPUTED TO BE 44 HOURS

86. Simulation Costs Service Maintenance Cost 34 hours x $30.00 per hour = $1,020.00 Simulated Machine Breakdown Cost 44 hours x $75.00 lost per down hour = $3,300.00 Total Simulated Maintenance Cost $4,320.00

87. Simulation Applications Applied Management Science for Decision Making, 1e © 2011 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

88. Solved Problems Simulation Modeling Computer-Based Manual Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD

89. Simulation Modeling Lundberg’s Car Wash The number of cars arriving per hour at Lundberg’s Car Wash during the past 200 hours of operation is observed to be as follows: CarsArrivingFrequencyCarsArrivingFrequency=<30760 420840 530=>90 650∑200

90. Simulation Modeling Lundberg’s Car Wash REQUIREMENT: 1.Set up a probability and cumulative probability distribution for the variable of car arrivals. 2.Establish random number intervals for the above variable. 3.Simulate fifteen (15) hours of car arrivals and compute the average number of arrivals per hour. 4. Compute the expected number of cars arriving using the expected value formula. Compare this with the results ob- tained in the simulation. Note: Select the random numbers needed from the 1 st column of Table 15.5, beginning with the digits “52”.

91. Simulation Modeling Lundberg’s Car Wash Number of Cars ProbabilityCumulativeProbability Random Number Interval 3 or less 0.000.00--- 40.100.1001-10 50.150.2511-25 60.250.5026-50 70.300.8051-80 80.201.0081-00 9 or more 0.001.00---

92. Simulation Modeling Lundberg’s Car Wash HourRN Simulated Arrivals HourRNSimulatedArrivals 15279888 237610908 382811506 469712276 598813456 696814818 733615667 8506∑=105 105/15 = 7.00 cars Average hourly arrivals

93. Simulation Modeling Lundberg’s Car Wash (.10 x 4) + (.15 x 5) + (.25 x 6) + (.30 x 7) + (.20 x 8) = 6.35 Expected Value Arrival Events Probabilities The average number of arrivals in the simulation was “ 7.00 “. If enough simulations were performed, the average number computed would approach the expected value.

94. Simulation Modeling Time Between Arrivals (minutes ) Probability10.20 20.25 30.30 40.15 50.10 Local Bank A local bank A local bank collected one month’s arrival and service rates at its single-teller at its single-tellerdrive-through station. These data are shown here:ServiceTime(minutes)Probability10.10 20.15 30.35 40.15 50.15 60.10

95. Simulation Modeling Local Bank REQUIREMENT : 1.Simulate a one-hour time period from 1:00 P.M. to 2:00 P.M. for the single-teller drive-through station. FOR THE TIME BETWEEN CUSTOMER ARRIVALS, USE THE RN STRING: 52,37,82,69,98,96,33,50,88,90,50,27,45,81,66,74,30,59,67 FOR THE CUSTOMER SERVICE TIME, USE THE RN STRING: 60,60,80,53,69,37,06,63,57,02,94,52,69,33,32,30,48,88

96. Simulation Modeling Local Bank Time Between Arrivals Probability Random Number Interval10.2001-20 20.2521-45 30.3046-75 40.1576-90 50.1091-00

97. Simulation Modeling Local Bank Service Time Probability Random Number Interval10.1001-10 20.1511-25 30.3526-60 40.1561-75 50.1576-90 60.1091-00

98. Simulation Modeling Local Bank RANDOM NUMBER TIME BETWEEN ARRIVALS ACTUAL TIME TIME SERVICE BEGINS RANDOM NUMBER SERVICE TIME SERVICECOMPLETE WAIT TIME (MINUTES) 5231:031:036031:060 3721:051:066031:091 8241:091:098051:140 6931:121:145331:172 9851:171:176941:210

99. Simulation Modeling Local Bank RANDOM NUMBER TIME BETWEEN ARRIVALS ACTUAL TIME TIME SERVICE BEGINS RANDOM NUMBER SERVICE TIME SERVICECOMPLETE WAIT TIME (MINUTES) 9651:221:223731:250 3321:241:250611:261 5031:271:276341:310 8841:311:315731:340 9041:351:350211:360

100. Simulation Modeling Local Bank RANDOM NUMBER TIME BETWEEN ARRIVALS ACTUAL TIME TIME SERVICE BEGINS RANDOM NUMBER SERVICE TIME SERVICECOMPLETE WAIT TIME (MINUTES) 5031:381:389461:440 2721:401:445231:474 4521:421:476941:515 8141:461:513331:545 6631:491:543231:575

101. Simulation Modeling Local Bank RANDOM NUMBER TIME BETWEEN ARRIVALS ACTUAL TIME TIME SERVICE BEGINS RANDOM NUMBER SERVICE TIME SERVICECOMPLETE WAIT TIME (MINUTES) 7431:521:573032:005 3021:542:004832:036 5931:572:038852:086 6732:00---------TOTAL40

102. Simulation Modeling Local Bank Cost of Customer Waiting 40 minutes per hour X 7 hours per day X 200 days per year X $1.00 per minute = $56,000.00

103. Simulation Modeling Local Bank Total Costs Drive-Through Depreciation per year - $12,000.00 + Salary and Benefits for one teller per year - $16,000.00 + Customer Waiting Cost per year - $56,000.00 = $84,000.000

104. Simulation Modeling Local Bank Total Costs for Two Drive-Throughs Drive-Through Depreciation per year - $20,000.00 + Salary and Benefits for two tellers per year - $32,000.00 + Customer Waiting Cost per year - $1,400.00 = $53,400.000

105. Simulation Modeling Local Bank Cost Savings With Two Tellers $84,000.00 ( 1 drive-through ) - $53,400.00 ( 2 drive-throughs ) $30,600.00 The conclusion is to place two teller booths in use. It is critical to replicate this simulation for a much longer time period before drawing any firm conclusions, however.

106. Solved Problems Simulation Modeling Computer-Based Manual Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro, PhD