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

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

Operations Operations ManagementManagement

SimulationSimulationModule FModule F


Two ways to address waiting linesTwo 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


SimulationSimulation 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


Real World Variables Which Are Real World Variables Which Are Probabilistic in NatureProbabilistic 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


New Orleans Barge Unloading SimulationNew 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.


Table of Random NumbersTable of Random Numbers52 06 50 88 53 30 10 47 99 37 66 91 3537 63 28 02 74 35 24 03 29 60 74 85 9082 57 68 28 05 94 03 11 27 79 90 87 9269 02 36 49 71 99 32 10 75 21 95 90 9498 94 90 36 06 78 23 67 89 85 29 21 2596 52 62 87 49 56 49 23 78 71 72 90 5733 69 27 21 11 60 95 89 68 48 17 89 3450 33 50 95 13 44 34 62 63 39 55 29 3088 32 18 50 62 57 34 56 62 31 15 40 9090 30 36 24 60 82 51 74 30 35 36 85 0150 48 61 18 85 23 08 54 17 12 80 69 2427 88 21 62 69 64 48 31 12 73 02 68 0045 14 46 32 13 49 66 62 74 41 86 98 92


Distribution 1 – Inter-Arrival TimesDistribution 1 – Inter-Arrival TimesTime Between

Arrivals (Hours)Probability Cumulative

ProbabilityRandom Number Interval

36 0.13 0.13 01 – 1324 0.17 0.30 14 – 3012 0.15 0.45 31 – 458 0.25 0.70 46 – 706 0.20 0.90 71 - 904 0.10 1.00 91 - 00


Distribution 2 – Unloading TimesDistribution 2 – Unloading TimesUnloading

Times (Hours)Probability Cumulative

ProbabilityRandom-Number Interval

24 0.05 0.05 01 – 05

12 0.15 0.20 06 – 20

8 0.50 0.70 21 – 70

6 0.20 0.90 71 – 90

4 0.10 1.00 91 - 00


New Orleans Barge Unloading SimulationNew Orleans Barge Unloading Simulation

From Dist 1 From Dist 2From Random Number Table

20182829174154499

01742403150150847

01508241421423610

01148351061062430

1810667410082853

210024027674688

07682868682450

05286344443606

01683788852

Waiting Time

Unloading Ends

Unloading Time

Unloading Starts

Arrival Time

Int Arr Time

Rnd#

Rnd#


New Orleans Barge Unloading SimulationNew Orleans Barge Unloading SimulationAverage

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


Simulation Using ExcelSimulation Using Excel

RAND ( ) =

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

37 63 28 02 74 35 24 03 29 60 74 85 90

82 57 68 28 05 94 03 11 27 79 90 87 92

69 02 36 49 71 99 32 10 75 21 95 90 94

98 94 90 36 06 78 23 67 89 85 29 21 25

96 52 62 87 49 56 49 23 78 71 72 90 57

33 69 27 21 11 60 95 89 68 48 17 89 34

50 33 50 95 13 44 34 62 63 39 55 29 30

88 32 18 50 62 57 34 56 62 31 15 40 90

90 30 36 24 60 82 51 74 30 35 36 85 01

50 48 61 18 85 23 08 54 17 12 80 69 24

27 88 21 62 69 64 48 31 12 73 02 68 00

45 14 46 32 13 49 66 62 74 41 86 98 92

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)


Metric ConstructionMetric ConstructionMetric 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.


Metric ConstructionMetric 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


Metric Construction – The StrikeMetric 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 ))))


Metric Construction – The WeatherMetric 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


Metric Construction – ContinuingMetric 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


Metric Construction – The ResultsMetric 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


Billy-Bob’s Bait ShackBilly-Bob’s Bait ShackBilly-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.


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’sBait Shack


ArrivalsArrivals 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 ShackBilly-Bob’s Bait Shack


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)


WaitingWaiting


The restThe 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)


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