CORPORACIÓN MEXICANA DE
INVESTIGACIÓN EN
MATERIALES, S.A. DE C.V.
PROGRAMA INTERINSTITUCIONAL DE CIENCIA Y TECNOLOGÍA
DOCTORADO EN INGENIERÍA INDUSTRIAL
SIMULACIÓN AVANZADA
Impartida por:
Dr. Ernesto Gutiérrez Miravete
TAREA 2
EJERCICIOS L5.5, L6.11 y L7.4
Sergio Manuel Ramírez Campos
16 de Noviembre de 2003
L5.5 Exercises
1. Consider the operation of fast food restaurant where customers arrive for ordering lunch. The following is
a log of the time (minutes) between arrivals of 40 successive customers. Use Stat::Fit to analyze the data
and fit an appropriate continuous distribution to the data. What are the parameters of this distribution?
11
9
12
8
11
11
13
12
8
10
12
14
7
17
16
8
9
13
15
10
15
14
12
10
11
14
9
16
7
12
15
13
7
16
14
13
7
10
11
15
We observe in the above figure the best distribution adjusted by PROMODEL: Weibull with shape
parameter 4.71 and scale parameter 12.8 forcing a lower bond of 0. This is accord with the feasible arrive
values.
2.
The servers at the above restaurant took the following time (minutes) to serve food to these customers. Use
Stat::Fit to analyze the data and fit an appropriate continuous distribution to the data. What are the
parameters of this distribution?
11
9
12
10
11
11
13
12
8
10
12
14
11
17
13
8
10
13
12
10
15
14
12
10
11
Next graph shows the best distribution: IG with parameters
and Erlang represent a good estimation too.
14
9
16
7
12
15
13
11
13
14
13
12
10
11
15
 = 313 and  = 11.8. Distributions Gamma
3.
The following is the number of incoming calls (each hour for 80 successive hours) to call center set up for
serving customers of a certain Internet service provider. Use Stat::Fit to analyze the data and fit an
appropriate discrete distribution to the data. What are the parameters of this distribution?
12
9
12
I
10
t
11
9t
12
a
k
10
e
11
s
10
12
13
12
8
11
13
12
8
10
8
11
14
11
17
12
14
11
17
13
17
13
10
13
12
8
10
13
12
10
12
12
14
12
10
15
14
12
10
11
10
16
9
16
7
14
9
16
7
12
7
11
13
11
13
15
13
11
13
14
13
10
12
10
11
13
12
10
11
15
11
In the above graph shows the best discrete distribution: Binomial with n = 22 and p = 0.533
4.
Observations were taken on the times to serve online customers at a stockbroker's site (STOCK.com) on
Internet. The times (in seconds) are shown below, sorted in ascending order. Use Stat::Fit and fit an
appropriate distribution to the data. What are the parameters of this distribution?
1.39
3.59
7.11
8.34
11.14
11.97
13.53
16.87
17.63
19.44
21.47
22.55
28.04
28.97
29.05
35.26
37.65
38.21
38.32
39.17
39.49
39.99
41.42
42.53
47.08
51.53
55.11
55.75
55.85
56.96
58.78
60.61
63.38
65.99
66.00
73.55
73.81
74.14
79.79
81.66
82.10
83.52
85.90
88.04
88.40
88.47
92.63
93.11
93.74
98.82
Next graph shows the best distribution: Uniform con parameters  = 0 y  = 98.8. In this case, is the same
distribution if we consider lower bond () the minimum value 1.39 or 0.
L6.11 Exercises
1. Visitors arrive at kid's entertainment park according to an exponential interarrival time distribution with
mean 2.5 minutes. The travel time from the entrance to the ticket windows is normally distributed with
mean of three minutes and standard deviation of 0.5 minute. At the ticket window, visitors wait in a single
line until one of six cashiers is available to serve them. The time for the purchase of tickets is normally
distributed with mean of five minutes and standard deviation of one minute. After purchasing tickets, the
visitors got to the respective gates to enter the park. Create a simulation model, with animation, of this
system. Run simulation model for 200 hours to determine
a. The average and maximum length of the ticketing queue.
Average length = 0.003579 and maximum length = 3
b.
The average number of customers completing ticketing per hour.
Average minutes/customer in system: 7.9837, then
(60 min/hour)/(7.9837 min/customer) = 7.51531 customers/hour
c.
The average utilization of the cashiers.
Cashier 1
67.19%
d.
2.
Cashier 2
53.76%
Cashier 3
38.09%
Cashier 4
23.95%
Cashier 5
12.56%
Cashier 6
4.94%
Do you recommend that management add more cashiers?
The system has enough capacity yet, so I do not recommend more cashiers.
A consultant recommended that six individual queues be formed at the ticket window (one for each cashier)
instead of one common queue. Create a simulation model, with animation, of this system. Run the
simulation model for 200 hours to determine
a. The average and maximum length of the ticketing queues.
Cashier 1
Cashier 2
Cashier 3
Cashier 4
Cashier 5
Cashier 6
Average length
0.002022
0.0007345
0.001544
0.001202
0.001682
0.001758
Maximum length
1
1
1
1
1
1
b.
The average number of customers completing ticketing per hour.
Average minutes/customer in system: 7.9964, then
(60 min/hour)/(7.9964 min/customer) = 7.5033 customers/hour
c.
The average utilization of the cashiers.
Cashier 1
Cashier 2
Cashier 3
33.43%
33.30%
33.83%
Cashier 4
33.48%
Cashier 5
33.32%
Cashier 6
33.41%
d.
3.
Do you agree with the consultant's decision? Would you recommend a raise for the consultant?
There is not a significant difference between both proposals. I would not recommend a raise for the
consultant.
At the Southern California Airline's traveler check-in facility, three types of customers arrive: passengers
with e-ticket (Type E), passengers with paper ticket (Type T), and passengers that need to purchase ticket
(Type P). The interarrival distribution and the service times for these passengers are given in the table.
Create s simulation model, with animation, of this system. Run the simulation model for 2000 hours. If each
type of passenger is served by separate gate agents, determine the following:
Types of Traveler
Type E
Type T
Type P
a.
Type E
0.371559
7
Type P
1.40508
15
Type E
21613
10.806
Type T
11386
5.693
Type P
7728
3.864
The average utilization of the gate agents.
Average utilization
d.
Type T
1.45677
14
The average number of customers of each type completing check-in procedures per hour.
Total entries
Average customer/hour
c.
Service Time Distribution
Normal(3, 1)
Normal(8, 3)
Normal(12, 3)
The average and maximum length of the three queues.
Average length
Maximum length
b.
Interarrival Distribution
Exponential(mean 5.5 min.)
Exponential(mean 10.5 min.)
Exponential(mean 15.5 min.)
Gate agent type E
54.07%
Gate agent type T
75.74%
Gate agente type P
77.21%
Would you recommend one single line for check-in for all three types of travelers? Discuss the pros
and cons for such a change.
One single line
One line for each type of customer
Advantage
Disadvantage
Advantage
Disadvantage
Average
Increase
Decrease
utilization of the
gate agents
Average attended
Decrease
Increase
customers
Average length
Increase
Decrease
of queue
I would recommend one line for each type of customer because additionally the gate agent's
specialization would permits to give a better service.
4.
Raja & Rani, a fancy restaurant in Santa Clara, holds a maximum of 100 diners. Customers arrive
according to an exponential distribution with a mean of 35 minutes. Customers stay in the restaurant
according a triangular distribution with a minimum of 30 minutes, a maximum of 60 minutes, and a mode
of 45 minutes. Create a simulation model, with animation, of this system.
Explanation: With 100 diners, the restaurant takes much time to fill. Then I considered 20 diners and the
results are following.
a.
Beginning empty, how long is it before the restaurant fill?
3.6521 hours before the restaurant fill.
b.
c.
What is the total number of diners entering the restaurant before it fills?
50 diners.
What is the utilization of the restaurant?
60.65%
5.
United Electronics manufactures small custom electronic assemblies. There are four stations through
which the parts must be processed: assembly, soldering, painting, and inspection. Orders arrive with an
exponential interarrival distribution (mean 20 minutes). The process time distributions are shown in the
table.
Assembly
Soldering
Painting
Inspection
Uniform(112,15) minutes
Normal(36,10) minutes
Uniform(40,70) minutes
Exponential(8) minutes
The soldering operation can be performed on three jobs at a time. Painting can be done on four jobs at a
time. Assembly and inspection are performed on one job at a time. Create a simulation model, with
animation, of this system. Simulate this manufacturing system for 100 days, eight hours each day. Collect
and print statistics on the utilization of each station, associated queues, and the total number of jobs
manufactured during each eight-hour shift (average).
Average jobs per shift = 2441/100 = 24.41 jobs
6.
Consider the exercise 5 with the following enhancements. Ten percent of all finished assemblies are sent
back to soldering for rework after inspection, five percent are sent back to assembly for rework after
inspection, and one percent of all assemblies fail to pass and are scrapped. Create a simulation model, with
animation, of this system. Simulate this manufacturing system for 100 days, eight hours each day. Collect
and print statistics on the utilization of each station, associated queues, total number of jobs assembled,
number of assemblies sent for rework to assembly and soldering, and the number of assemblies scrapped
during each eight-hour shift (average).
Assemblies sent for rework to assembly: 376
Assemblies sent for rework to soldering: 2060
Average of assemblies scraped per day: 38/100 = 0.38
Total number of jobs assembled: 1523
7.
Small appliances are assembled in four stages (Centers 1, 2, and 3 and Inspection) at Pomona Assembly
Shop. After each assembly step, the appliance is inspected or tested and if a defect is found, it must be
corrected and then checked again. The assemblies arrive at a constant rate of one assembly per minute.
The times to assembly, test, and correct defect are normally distributed. The mean and standard deviation
of the times to assemble, inspect, and correct defects, as well as the likelihood of an assembly error, are
shown in the following table. If an assembly is found defective, the defect is corrected and it is inspected
again. After a defect is corrected, the likelihood of another defect being found is the same as during the
first inspection. We assume in this model that an assembly defect is eventually corrected and then it is
passed on to the next station.
Center
1
2
3
Assembly time
Standard
Mean
Deviation
.7
.2
.75
.25
.8
.15
Inspect time
Standard
Mean
deviation
.2
.05
.2
.05
.15
.03
P(error)
.1
.05
.03
Correct time
Standard
Mean
deviation
.2
.05
.15
.04
.1
.02
Simulate for one year (2000 hours) and determine the number of good appliances shipped in a year.
Number of good appliances shipped in a year: 3458
8.
Salt Lake City Electronics manufactures small custom communication equipment. Two different job types
are to be processed within the following manufacturing cell. The necessary data are given in the table.
Job
type
1
2
Number
of
batches
15
25
Number
of jobs
per
batch
5
3
Time
between
Assembly
Inspection
batch
time
Soldering time Painting time
time
arrivals
Trial(5,7,10) Normal(36,10) Uniform(40,70) Exponential(8) Exp(14)
Trial(10,15)
Uniform(30,40) Exponential(5) Exp(10)
Simulate the system for 100 days, eight hours each day, to determine the average number of jobs waiting
for different operations, number of jobs of each type finished each day, average cycle time for each type of
job, and the average cycle time for all jobs.
Explanation: I considered that process soldering is necessary for both job types.
The average number of jobs waiting for different operations are shown in column "Average contents" in the
"location name" that starts with Q.
The number of jobs of each type finished each day:
Job 1 = 2840/800 = 3.55
Job 2 = 2304/800 = 2.88
Average cycle time:
Job 1 = 20242.47/(568x5) = 7.127 min/job
Job 2 = 20348.79/(768x3) = 8.832 min/job
Total average cycle time = 7.979 min/job
9.
Six dump trucks at the DumpOnMe facility in Riverside are used to haul coal from the entrance of a small
mine to railroad. Figure L6.39 provides a schematic of the dump truck operation. Each truck is loaded by
one of two loaders. After loading, a truck immediately moves to the scale to be weighed as soon as
possible. Both the loaders and the scale have a first-come, first-served waiting line (or queue) for trucks.
Travel time form a loader to the scale is considered negligible. After being weighed, a truck begins travel
time (during which time the truck unloads), and then afterward returns to the loaders queue. The
distributions of loading time, weighing time, and travel time are shown in the table.
Loading time
Weighing time
Travel time
Uniform(5,10) minutes
Uniform(2,5) minutes
Triangular(0,12,15) minutes
Figure L6.39 Schematic of dump truck operation for DumpOnMe.
Traveling
Loading
Weighing
Loading
Scale
Loading
a.
Create a simulation model, with animation, of this system. Simulate for 100 days, eight hours each day.
b.
Collect statistics to estimate the loader and scale utilization (percentage of time busy).
c.
About how many trucks are loaded each day on average?
(11850/100) = 118.5 trucks/day
10. At the Pilot Pen Company, a molding machine produces pen barrels of three different color-red, blue, and
green-in the ratio of 3:2:1. The molding time is triangular(3,4,6) minutes per barrel. The barrels got to a
filling machine where ink of appropriate color is filled at the rate of 20 pens per hour (exponentially
distributed). Another molding machine makes of three different colors-red, blue, and green-in the ratio of
3:2:1. The molding time is triangular(2,3,4) minutes per cap. At the next station, caps and filled barrel of
matching colors are joined together. Simulate for 300 hours. Find the average number of pens produced per
hour. Collect statistics on the utilization of the molding machines and the joining equipment.
From next figures:
Pens/hour = 4043/800 = 5.05
Molding machine for pen barrels % Util. 97.32
Molding machine for caps
% Util. 100.00
Assembly area
% Util 99.85
11. Customers arrive at the NotWeitBurger hamburger stand with an interarrival time that is exponential
distributed with a mean of one minute. Out of 10 customers, five buy hamburger and a drink, three buy a
hamburger, and two buy just a drink. One server handles the hamburger while another handles the drink. A
person buying both items needs to wait in line for both servers. The time it takes to serve a customer is
normally distributed with a mean of 70 seconds for each item. Simulate for 100 days, eight hours each day.
Collect statistics on the number of customers served each day, size of the queues, and utilization of the
servers. What changes would you suggest to make the system more efficient?
Explanation: I considered a standard deviation of 0.3 minutes in the service distribution.
Number of customers/day = 44819/100 = 448.19
Average contents of queue = 1536.5/800 = 1.92 customers/hour
12. Workers who work at the Detroit ToolNDie plant must check out tools from a tool crib. Workers arrive
according to an exponential distribution with a mean time between arrivals of five minutes. At present,
three tool crib clerks staff the tool crib. The time to serve a worker is normally distributed with a mean of
10 minutes and a standard deviation of two minutes. Compare the following servicing methods. Simulate
for 24-hour period and collect data.
a.
b.
c.
Workers form a single queue, choosing the next available tool crib clerk.
Workers enter the shortest queue (each clerk has his/her own queue).
Workers choose one of three queues at random.
% Util. S.1
% Util. S.2
% Util. S.3
% Total Util.
Total Exits
Worker in operation
Single queue
72.35
64.44
53.35
63.38%
269
83.35%
Shortest queue
92.05
67.53
37.45
65.67%
278
64.83%
Queues at random
71.23
64.13
73.68
47.68%
295
49.50%
There is not a significant difference among the three methods.
L7.4 Exercises
1. For the example in Section L7.1, insert a DEBUG statement when a garment is sent back for rework. Verify
that the simulation model is actually sending back garments for rework to the location named Label_Q.
2.
For the example in Section L6.1 (Pomona Electronics), trace the model to verify that the circuit boards of
type B following the routing as given in Table L6.1.
3.
For the example in Section L6.5 (Poly Casting Inc.), run the simulation model and launch the debugger
form the options menu. Turn on the Local Information in the Basic Debugger. Verify the values of the
variables WIP and Prod_Qty.