mississippi river barge arrivals and unloadings a queuing simulation applied management science for...

MISSISSIPPI RIVER BARGE ARRIVALS AND UNLOADINGS

A Queuing SimulationA Queuing Simulation

Applied Management Science for Decision Making, 1e Applied Management Science for Decision Making, 1e © 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro , PhD© 2012 Pearson Prentice-Hall, Inc. Philip A. Vaccaro , PhD

Port of New OrleansPort of New Orleans

Two Courses of Action ConsideredTwo Courses of Action Considered

COA Number 1 COA Number 2COA Number 1 COA Number 2

DOCK CREW OF 6 DOCK CREW OF 12DOCK CREW OF 6 DOCK CREW OF 12

COURSE OF ACTIONCOURSE OF ACTIONNUMBER ONENUMBER ONE

COURSE OF ACTIONCOURSE OF ACTIONNUMBER TWONUMBER TWO

Two Relevant VariablesTwo Relevant Variables

Daily Evening BargeDaily Evening Barge

ArrivalsArrivalsDaily Evening BargeDaily Evening Barge

UnloadingsUnloadings

VARIABLEVARIABLENUMBERNUMBER

ONEONE

VARIABLEVARIABLENUMBERNUMBER

TWOTWO

Port of New OrleansPort of New Orleans

Evaluation CriteriaEvaluation Criteria

1. Average Number of Barges Unloaded Each evening.

2. Average Number of Barges Delayed Each evening.

Simulation ExecutionSimulation Execution

Simulate a daily barge arrival.Simulate a daily barge arrival.



Simulate a daily barge unloading.Simulate a daily barge unloading.




Determine how many, if any, barges Determine how many, if any, barges remain unloaded at the end of the evening.remain unloaded at the end of the evening.




Determine how many, if any, barges 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 beginningUnloaded barges become the beginning balance for the following evening.balance for the following evening.

Overnight Barge ArrivalsOvernight Barge Arrivals

SPREADSHEET

NUMBER OF ARRIVALS

PROBABILITY CUMULATIVE

PROBABILITY

RANDOM NUMBER

INTERVAL

00 .13.13 .13.13 01 - 1301 - 13

11 .17.17 .30.30 14 - 3014 - 30

22 .15.15 .45.45 31 - 4531 - 45

33 .25.25 .70.70 46 - 7046 - 70

44 .20.20 .90.90 71 - 9071 - 90

55 .10.10 1.001.00 91 - 0091 - 00

Crew of 6 Unloading RatesCrew of 6 Unloading Rates

SPREADSHEET

DAILY UNLOADING

RATEPROBABILITY

CUMULATIVE

PROBABILITY

RANDOM NUMBER

INTERVAL

11 .05.05 .05.05 01 - 0501 - 05

22 .15.15 .20.20 06 - 2006 - 20

33 .50.50 .70.70 21 - 7021 - 70

44 .20.20 .90.90 71 - 9071 - 90

55 .10.10 1.001.00 91 - 0091 - 00

Random Number StringsRandom Number Strings

TO GENERATE DAILY ARRIVALS

52 06 50 88 53 30 10 47 99 37 66 91 35 32 00



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


Day1st 2nd 3rd

Random Number

for

Daily Arrival52 06 50

Random Number

for Daily

Unloading37 63 28


Day4th 5th 6th

Random Number

for


Random Number

for Daily

Unloading02 74 35


Day7th 8th 9th

Random Number

for


Random Number

for Daily

Unloading24 03 29


Day10th 11th 12th

Random Number

for


Random Number

for Daily

Unloading60 74 85


Day13th 14th 15th

Random Number

for


Random Number

for Daily

Unloading90 73 59


EVENINGEVENING

NUMBERNUMBERDELAYEDDELAYEDPREVIOUSPREVIOUSEVENINGEVENING

ARRIVALARRIVALRANDOMRANDOMNUMBERNUMBER

BARGEBARGEARRIVALARRIVALNUMBERNUMBER

TOTAL TOTAL TO BETO BE

UNLOADEDUNLOADED

UNLOADINGUNLOADINGRANDOMRANDOMNUMBERNUMBER

NUMBERNUMBERUNLOADEDUNLOADED

11stst - - aa 5252 33 33 3737 33

22ndnd 00 0606 00 00 6363 00bb

33rdrd 00 5050 33 33 2828 33

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,c


EVENINGEVENING





UNLOADEDUNLOADED



44thth 00 8888 44 44 0202 11

55thth 33 5353 33 66 7474 44

66thth 22 3030 11 33 3535 33


EVENINGEVENING





UNLOADEDUNLOADED



77thth 00 1010 00 00 2424 00

88thth 00 4747 33 33 0303 11

99thth 22 9999 55 77 2929 33

c,dc,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 !


EVENINGEVENING





UNLOADEDUNLOADED



1313thth 33 3535 22 55 9090 44

1414thth 11 3232 22 33 7373 33dd

1515thth 00 0000 55 55 5959 33

d – FOUR BARGES COULD HAVE BEEN UNLOADED BUT SINCE ONLY THREE WERE IN THE QUEUE, THE NUMBER UNLOADED IS RECORDED AS “3”.

Simulation SummarySimulation Summary

EVENINGEVENING





UNLOADEDUNLOADED



1313thth 33 3535 22 55 9090 44

1414thth 11 3232 22 33 7373 3 3

1515thth 00 0000 55 55 5959 33

TOTAL DELAYS = 20AVERAGE = 1.33

TOTAL ARRIVALS = 41AVERAGE = 2.73

TOTAL UNLOADINGS = 39AVERAGE = 2.60

Overnight Barge ArrivalsOvernight Barge Arrivals

SPREADSHEET FOR CREW OF 12

NUMBER OF ARRIVALS

PROBABILITY CUMULATIVE

PROBABILITY

RANDOM NUMBER INTERVAL

00 .13.13 .13.13 01 - 1301 - 13

11 .17.17 .30.30 14 - 3014 - 30

22 .15.15 .45.45 31 - 4531 - 45

33 .25.25 .70.70 46 - 7046 - 70

44 .20.20 .90.90 71 - 9071 - 90

55 .10.10 1.001.00 91 - 0091 - 00

Crew of 12 Unloading RatesCrew of 12 Unloading RatesSPREADSHEET

DAILY UNLOADING RATE PROBABILITY

CUMULATIVE

PROBABILITY

RANDOM NUMBER INTERVAL

11 .03.03 .03.03 01 - 0301 - 03

22 .12.12 .15.15 04 - 1504 - 15

33 .40.40 .55.55 16 - 5516 - 55

44 .28.28 .83.83 56 - 8356 - 83

55 .12.12 .95.95 84 - 9584 - 95

66 .05.05 1.001.00 96 - 0096 - 00

Random Number StringsRandom Number StringsCREW OF 12 SIMULATIONCREW OF 12 SIMULATION


37 77 13 10 02 18 31 19 32 85 31 94 81 43 31

Random Number StringsRandom Number StringsCREW OF 12 SIMULATIONCREW OF 12 SIMULATION


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

Simulation ExecutionSimulation ExecutionCREW OF TWELVECREW OF TWELVE

11stst -- 3737 22 22 6969 22

22ndnd 00 7777 44 44 8484 44

33rdrd 00 1313 00 00 1212 00

44thth 00 1010 00 00 9494 00

EVENINGEVENINGDELAYEDDELAYEDPREVIOUSPREVIOUSEVENINGEVENING



TOTALTOTALTO BETO BE

UNLOADEDUNLOADED




55thth 00 0202 00 00 5151 00

66thth 00 1818 11 11 3636 11

77thth 00 3131 22 22 1717 22

88thth 00 1919 11 11 0202 11





UNLOADEDUNLOADED




1212thth 00 9494 55 55 5252 33

1313thth 22 8181 44 66 5656 44

1414thth 22 4343 22 44 4343 33

1515thth 11 3131 22 33 2626 33





UNLOADEDUNLOADED



TOTAL DELAYS = 6AVERAGE = 0.4

TOTAL ARRIVALS = 31AVERAGE = 2.07

TOTAL UNLOADINGS = 31AVERAGE = 2.07

ScoreboardScoreboard

BargesBarges Crew of 6Crew of 6 Crew of 12Crew of 12AVERAGE DAILY AVERAGE DAILY

DELAYSDELAYS 1.331.33 .40.40AVERAGE DAILYAVERAGE DAILY

UNLOADINGSUNLOADINGS2.602.60 2.072.07

AVERAGE DAILYAVERAGE DAILY

ARRIVALSARRIVALS2.732.73 2.072.07

Possible Relevant VariablesPossible Relevant Variables

WindsWinds CurrentsCurrents FogFog TemperatureTemperature River IceRiver Ice Seasonal Barge TrafficSeasonal Barge Traffic Competing DocksCompeting Docks PrecipitationPrecipitation

Absentee RatesAbsentee Rates Barge SizesBarge Sizes Additional Crew Staffing Additional Crew Staffing

OptionsOptions Local Economy Effect Local Economy Effect

on Barge Trafficon Barge Traffic Crew TrainingCrew Training

BARGE SIMULATIONBARGE SIMULATION

Repeating Random Number StringsRepeating Random Number Strings

Used for generating arrival and unloading ratesUsed for generating arrival and unloading ratesfor for bothboth crew staffing options if you want to crew staffing options if you want to

isolate and observe the impact of each staffingisolate and observe the impact of each staffingoption on the dock-river system.option on the dock-river system.

ANY DIFFERENCES FOUND IN THE UNLOADINGRATES WOULD BE DIRECTLY ATTRIBUTABLE TO

THE CREW SIZE ITSELF, SINCE ALL OTHER ELEMENTS OF THE SIMULATION HAD BEEN

HELD CONSTANT!

Non-Repeating Random Number StringsNon-Repeating Random Number Strings

Used for generating arrival and unloading ratesUsed for generating arrival and unloading ratesfor for bothboth crew staffing options if you want to crew staffing options if you want to

test for consistent results of the impact of each test for consistent results of the impact of each staffing option on the dock-river system.staffing option on the dock-river system.

TO YIELD VALID CONCLUSIONS HOWEVER, YOU MUSTINSURE THAT THE SIMULATION HAS RUN OVER A

SUFFICIENTLY LONG PERIOD OF TIME IN ORDER TOALLOW THE NUMBERS TO “SETTLE DOWN” TO

THEIR LONG-TERM AVERAGES.

Barge Simulation PostscriptBarge 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 potential, 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,THIS DATE IS AVAILABLE FROM HUMAN RESOURCES,MARKETING, ACCOUNTING, AND FINANCE.MARKETING, ACCOUNTING, AND FINANCE.

QM for WINDOWSQM for WINDOWSCOMMENTSCOMMENTS

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 variable separately, insert the simulated outcomes on a spreadsheet, and then manually calculate the outcomes of the variables’ interactions.

THIS APPROACH IS FEASIBLE FOR ONLY THE MOST ELEMENTAL SIMULATIONSTHIS APPROACH IS FEASIBLE FOR ONLY THE MOST ELEMENTAL SIMULATIONS

ExampleExample

EVENINGEVENING





UNLOADEDUNLOADED



11stst - - 5252 3 33 3737 3

22ndnd 00 0606 0 00 6363 0

33rdrd 00 5050 3 33 2828 3

SIMULATEDVIA

QM for WINDOWSor QM EXCEL

SIMULATEDVIA

QM for WINDOWSor QM EXCEL

NOTREQUIRED

NOT REQUIRED

MANUALLYMANUALLYENTEREDENTERED

These simulatedbarge arrivals

would be insertedon our manualspreadsheet

These simulated bargeunloadings wouldbe inserted on our

manual spreadsheet

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

Templateand

Sample Data

TemplateAnd

Sample Data

Templateand

Sample Data

Inventory Policy SimulationInventory Policy Simulation

Establishing an inventory control doctrinefor 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 realisticbusiness application

Electric Drill DemandElectric Drill Demand

Daily Demand

Frequency

( days )

Probability Cumulative

Probability

Random

No. Interval

0 15 .05 .05 01 - 05

1 30 .10 .15 06 - 15

2 60 .20 .35 16 - 35

3 120 .40 .75 36 - 75

4 45 .15 .90 76 - 90

5 30 .10 1.00 91 - 00

∑∑= = 300 300 ∑= ∑= 1.001.00

11stst relevant variable relevant variable

Electric Drill Electric Drill Reorder Lead TimeReorder Lead Time

LEAD TIME

(DAYS)

Frequency

(ORDERS) Probability

Cumulative

Probability

RN

Interval

1 1010 .20 .20 01 - 20

2 2525 .50 .70 21 - 70

3 1515 .30 1.00 71 - 00

∑∑ = = 50 50 ∑ = ∑ = 1.001.00

22ndnd relevant variable relevant variable

The SimulationThe Simulation

The 1The 1stst inventory policy to be simulated: 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 dayOrder 10 drills at a time when the shelf stock falls to five drills or less at the end of the business day

11 -- 1010 0606 11 99 00 NONO -- --

22 00 99 6363 33 66 00 NONO -- --

33 00 66 5757 33 33 00 YESYES 0202 11

44 00 33 9494 55 00 22 NONO -- --

55 1010 1010 5252 33 77 00 NONO -- --

DAYDAYUNITS

RECEIVEDBEGINNINGINVENTORY

RANDOMRANDOMNUMBERNUMBER DEMAND

ENDINGENDINGINVENTORYINVENTORY

LOSTLOSTSALESSALES

ORDER? RANDOMNUMBER

LEAD TIME

a – 1st order is placedb – generates 1st lead timec – next random number in seriesd – no order placed because of outstanding order from previous day

aa bbcc

dd


66 00 77 6969 33 44 00 YESYES 3333 22

77 00 44 3232 22 22 00 NONO -- --

88 00 22 3030 22 00 00 NONO -- --

99 1010 1010 4848 33 77 00 NONO -- --

1010 00 77 8888 44 33 00 YESYES 1414 11

DAYDAYUNITSUNITS

RECEIVEDRECEIVEDBEGINNINGBEGINNINGINVENTORYINVENTORY

RANDOMRANDOMNUMBERNUMBER DEMAND

ENDINGENDINGINVENTORYINVENTORY

LOSTLOSTSALESSALES

ORDER ?

RANDOMNUMBER

LEAD TIME

f – order placed at end of 6th day arrives

ff


∑ = 41 2 3 units

endinginventory

numberof lostsales

numberof orders

placed

SUMMARYSTATISTICS

SimulationSimulation Results Results

AVERAGE ENDING INVENTORY 41 units / 10 days = 4.1 units per day

AVERAGE LOST SALES2 sales lost / 10 days = .2 unit per day

AVERAGE NUMBER OF ORDERS PLACED3 orders / 10 days = .3 order per day

Simulation CostsSimulation 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 )

Simulation PostscriptSimulation 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

Fast Food Fast Food Drive-Through Drive-Through

SimulationSimulation

ARRIVALARRIVAL

RN for RN for TIME TIME

betweenbetweenARRIVALSARRIVALS

TIME TIME BETWEEN BETWEEN ARRIVALSARRIVALS

TIMETIME

RN for RN for SERVICESERVICE

TIMETIMESERVICESERVICE

TIMETIME

WaitingWaiting

TimeTimeCUSTOMERCUSTOMER

LEAVESLEAVES

11stst 1414 1 min.1 min. 11:0111:01 8888 3 min.3 min. 00 11:0411:04

22ndnd 7474 3 min.3 min. 11:0411:04 3232 2 min.2 min. 00 11:0611:06

33rdrd 2727 2 min.2 min. 11:0611:06 3636 2 min.2 min. 00 11:0811:08

44thth 0303 1 min.1 min. 11:0711:07 2424 1 min.1 min. 11 11:0911:09

( ASSUME THE DRIVE-THROUGH OPENS AT 11:00 AM )( ASSUME THE DRIVE-THROUGH OPENS AT 11:00 AM )

Generator BreakdownGenerator Breakdown

SimulationSimulation

Generator Breakdown SimulationGenerator Breakdown Simulation

TIME BETWEENTIME BETWEEN

RECORDED RECORDED MACHINE MACHINE

FAILURES (hours)FAILURES (hours)PROBABILITYPROBABILITY CUMULATIVECUMULATIVE

PROBABILITYPROBABILITY

RANDOMRANDOM

NUMBERNUMBERINTERVALINTERVAL

½ ½ .05 .05 01 - 05

11 .06 .11 06 - 11

1 ½ 1 ½ .16 .27 12 - 27

22 .33 .60 28 - 60

2 ½ 2 ½ .21 .81 61 - 81

33 .19 1.00 82 - 00

∑ 1.00


REPAIR TIMEREPAIR TIME

REQUIREDREQUIRED

( HOURS )( HOURS )


CUMULATIVECUMULATIVE


RANDOMRANDOM

NUMBERNUMBER

INTERVALINTERVAL

11 .28.28 .28.28 01 - 2801 - 28

22 .52.52 .80.80 29 - 8029 - 80

33 .20.20 1.001.00 81 - 0081 - 00

TotalTotal 1.001.00

a – MAINTENANCE TIME IS ROUNDED TO HOURLY TIME BLOCKSa – MAINTENANCE TIME IS ROUNDED TO HOURLY TIME BLOCKS

aa

1 57 2 02:00 02:00 07 1 03:00 1

2 17 1.5 03:30 03:30 60 2 05:30 2

3 36 2 05:30 05:30 77 2 07:30 2

4 72 2.5 08:00 08:00 49 2 10:00 2

5 85 3 11:00 11:00 76 2 13:00 2

6 31 2 13:00 13:00 95 3 16:00 3

BREAKDOWNBREAKDOWNNUMBERNUMBER

TIME BETWEENTIME BETWEENBREAKDOWNSBREAKDOWNSRANDOM NO.RANDOM NO.

TIMETIMEBETWEENBETWEEN

BREAKDOWNSBREAKDOWNS

TIME OFTIME OFBREAKDOWNBREAKDOWN

TIMETIMEMECHANICMECHANICFREE TO FREE TO

BEGIN THISBEGIN THISREPAIRREPAIR

REPAIR TIMEREPAIR TIMERANDOM NO.RANDOM NO.

REPAIR TIMEREPAIR TIMEREQUIREDREQUIRED

TIME TIME REPAIRREPAIRENDSENDS

NO. HRS.NO. HRS.MACHINEMACHINE

DOWNDOWN

13 33 2 01:00 04:00 40 2 06:00 5

14 89 3 04:00 06:00 42 2 08:00 4

15 13 1.5 05:30 08:00 52 2 10:00 4.5

BREAKDOWNBREAKDOWNNUMBERNUMBER

TIME BETWEENTIME BETWEENBREAKDOWNSBREAKDOWNSRANDOM NO.RANDOM NO.

TIMETIMEBETWEENBETWEEN

BREAKDOWNSBREAKDOWNS

TIME OFTIME OFBREAKDOWNBREAKDOWN

TIMETIMEMECHANICMECHANICFREE TO FREE TO

BEGIN THISBEGIN THISREPAIRREPAIR

REPAIR TIMEREPAIR TIMERANDOM NO.RANDOM NO.

REPAIR TIMEREPAIR TIMEREQUIREDREQUIRED

TIME TIME REPAIRREPAIRENDSENDS

TOTALTOTALNO. HRS.NO. HRS.

MACHINESMACHINESDOWNDOWN

44


Simulation ResultsSimulation 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 TOTALNUMBER OF HOURSTHAT GENERATORS

WERE OUT OFSERVICE IS

COMPUTED TO BE44 HOURS

Simulation CostsSimulation Costs

Service Maintenance CostService Maintenance Cost34 hours x $30.00 per hour = $1,020.0034 hours x $30.00 per hour = $1,020.00

Simulated Machine Breakdown CostSimulated Machine Breakdown Cost44 hours x $75.00 lost per down hour = $3,300.0044 hours x $75.00 lost per down hour = $3,300.00

Total Simulated Maintenance Cost$4,320.00

Simulation ApplicationsSimulation Applications


Solved ProblemsSolved Problems

Simulation ModelingSimulation ModelingComputer-BasedComputer-Based

ManualManual


Simulation ModelingSimulation ModelingLundberg’s Car WashLundberg’s Car Wash

The number of cars arriving per hour at The number of cars arriving per hour at Lundberg’s Car Wash during the past 200 Lundberg’s Car Wash during the past 200

hours of operation is observed to be as follows:hours of operation is observed to be as follows:

CarsCars

ArrivingArriving

FrequencyFrequency CarsCars

ArrivingArriving

FrequencyFrequency

=<3=<3 00 77 6060

44 2020 88 4040

55 3030 =>9=>9 00

66 5050 ∑∑ 200200


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 1st column of Table 15.5 , beginning with the digits “52”.


Number of CarsNumber of Cars ProbabilityProbability CumulativeCumulative

ProbabilityProbability

Random Number Random Number IntervalInterval

3 or less3 or less 0.000.00 0.000.00 ------

44 0.100.10 0.100.10 01-1001-10

55 0.150.15 0.250.25 11-2511-25

66 0.250.25 0.500.50 26-5026-50

77 0.300.30 0.800.80 51-8051-80

88 0.200.20 1.001.00 81-0081-00

9 or more9 or more 0.000.00 1.001.00 ------


HourHour RNRN Simulated Simulated ArrivalsArrivals

HourHour RNRN SimulatedSimulated

ArrivalsArrivals

11 5252 77 99 8888 88

22 3737 66 1010 9090 88

33 8282 88 1111 5050 66

44 6969 77 1212 2727 66

55 9898 88 1313 4545 66

66 9696 88 1414 8181 88

77 3333 66 1515 6666 77

88 5050 66 ∑=105105/15 = 7.00 carsAverage hourly

arrivals


((.10.10 x 4) + x 4) + (.15(.15 x 5) + ( x 5) + (.25.25 x 6) + ( x 6) + (.30.30 x 7) + ( x 7) + (.20.20 x 8) = 6.35 x 8) = 6.35

ExpectedValue

Arrival EventsArrival Events

ProbabilitiesProbabilities

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.

Simulation ModelingSimulation Modeling

Time Time BetweenBetween

Arrivals Arrivals

(minutes(minutes))


11 0.200.20

22 0.250.25

33 0.300.30

44 0.150.15

55 0.100.10

Local BankLocal Bank

A local bank A local bank collected one collected one

month’s arrival month’s arrival and service ratesand service rates

at its single-teller at its single-teller drive-through drive-through station. These station. These

data are shown data are shown here:here:

ServiceService

TimeTime

(minutes)(minutes)


11 0.100.10

22 0.150.15

33 0.350.35

44 0.150.15

55 0.150.15

66 0.100.10

Simulation ModelingSimulation ModelingLocal BankLocal 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


Time Between Time Between ArrivalsArrivals

ProbabilityProbability Random NumberRandom Number

IntervalInterval

11 0.200.20 01-2001-20

22 0.250.25 21-4521-45

33 0.300.30 46-7546-75

44 0.150.15 76-9076-90

55 0.100.10 91-0091-00


Service TimeService Time ProbabilityProbability Random NumberRandom Number

IntervalInterval

11 0.100.10 01-1001-10

22 0.150.15 11-2511-25

33 0.350.35 26-6026-60

44 0.150.15 61-7561-75

55 0.150.15 76-9076-90

66 0.100.10 91-0091-00


RANDOM RANDOM NUMBERNUMBER


ACTUAL ACTUAL TIMETIME

TIME TIME SERVICE SERVICE BEGINSBEGINS


SERVICE SERVICE TIMETIME

SERVICESERVICE

COMPLETECOMPLETEWAIT TIME WAIT TIME (MINUTES)(MINUTES)

5252 33 1:031:03 1:031:03 6060 33 1:061:06 00

3737 22 1:051:05 1:061:06 6060 33 1:091:09 11

8282 44 1:091:09 1:091:09 8080 55 1:141:14 00

6969 33 1:121:12 1:141:14 5353 33 1:171:17 22

9898 55 1:171:17 1:171:17 6969 44 1:211:21 00








SERVICESERVICE


9696 55 1:221:22 1:221:22 3737 33 1:251:25 00

3333 22 1:241:24 1:251:25 0606 11 1:261:26 11

5050 33 1:271:27 1:271:27 6363 44 1:311:31 00

8888 44 1:311:31 1:311:31 5757 33 1:341:34 00

9090 44 1:351:35 1:351:35 0202 11 1:361:36 00








SERVICESERVICE


5050 33 1:381:38 1:381:38 9494 66 1:441:44 00

2727 22 1:401:40 1:441:44 5252 33 1:471:47 44

4545 22 1:421:42 1:471:47 6969 44 1:511:51 55

8181 44 1:461:46 1:511:51 3333 33 1:541:54 55

6666 33 1:491:49 1:541:54 3232 33 1:571:57 55








SERVICESERVICE


7474 33 1:521:52 1:571:57 3030 33 2:002:00 55

3030 22 1:541:54 2:002:00 4848 33 2:032:03 66

5959 33 1:571:57 2:032:03 8888 55 2:082:08 66

6767 33 2:002:00 ------ ------ ------ TOTALTOTAL 4040


Cost of Customer Waiting

40 minutes per hour X

7 hours per dayX

200 days per yearX

$1.00 per minute=

$56,000.00


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


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


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.

Solved ProblemsSolved Problems

Simulation ModelingSimulation ModelingComputer-BasedComputer-Based

ManualManual


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random number stringsto

random number stringsday1st

coa number

daily barge arrival

daily arrivals52

daily unloadings37

unloaded barges

following evening