Supplement E Simulation Copyright ©2013 Pearson Education, Inc. publishing as Prentice HallE- 01.

Supplement ESimulation

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall E- 01

What is Simulation?

Simulation

The act of reproducing the behavior of a system using a model that describes the processes of the system.

E- 02Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall

• To analyze a problem when the relationship between variables is nonlinear, or when the situation involves too many variables or constraints to handle with optimizing approaches.

• To conduct experiments without disrupting real systems.

• To obtain operating characteristic estimates in much less time (time compression).

• To sharpen managerial decision-making skills through gaming.

Reasons for Using Simulation

Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall E -03

The Simulation Process

• Monte Carlo simulation

– A simulation process that uses random numbers to generate simulation events

• Data collection

• Random-number assignment

– A random number is a number that has the same

probability of being selected as any other number (see

Appendix 2)

• Model formulationCopyright ©2013 Pearson Education, Inc. publishing as Prentice Hall E - 04

Example E.1: Specialty Steel ProductsThe Specialty Steel Products Company produces items, such as machine tools, gears, automobile parts, and other specialty items, in small quantities to customer order.

Because the products are so diverse, demand is measured in machine-hours.

Orders for products are translated into required machine-hours, based on time standards for each operation.

Management is concerned about capacity in the lathe department.

Assemble the data necessary to analyze the addition of one more lathe machine and operator.

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Example E.1 : Specialty Steel Products

Historical records indicate that lathe department demand varies from week to week as follows:

Weekly Production Requirements (hour) Relative Frequency

200 0.05250 0.06300 0.17350 0.05400 0.30450 0.15500 0.06550 0.14600 0.02

Total 1.00



To gather these data:• Weeks with requirements of 175.00–224.99

hours were grouped in the 200-hour category. • Weeks with 225.00–274.99 hours were grouped

in the 250-hour category, and so on.

200(0.05) + 250(0.06) + 300(0.17) + ... + 600(0.02) = 400 hours


The average weekly production requirements for the lathe department are:


Employees in the lathe department work 40 hours per week on 10 machines. However, the number of machines actually operating during any week may be less than 10. Machines may need repair, or a worker may not show up for work. Historical records indicate that actual machine-hours were distributed as follows:

Regular Capacity (hr) Relative Frequency 320 (8 machines) 0.30 360 (9 machines) 0.40 400 (10 machines) 0.30

The average number of operating machine-hours in a week is320(0.30) + 360(0.40) + 400(0.30) = 360 hours



The company has a policy of completing each week’s workload on schedule, using overtime and subcontracting if necessary.

Regular Capacity (hr) Relative Frequency 360 (9 machines) 0.30 400 (10 machines) 0.40 440 (11 machines) 0.30

Resources and CostsMaximum Overtime 100 hrsLathe Operators $10/hrOvertime Cost $25/hrSubcontracting Cost $35/hr

To justify adding another machine and worker to the lathe department, weekly savings in overtime and subcontracting costs should be at least $650. Management estimates from prior experience that with 11 machines the distribution of weekly capacity machine-hours would be


Random-Number Assignment

• Random number –A number that has the same probability of

being selected as any other number

• Events in a simulation can be generated in an unbiased way if random numbers are assigned to the events in the same proportion as their probability of occurrence.


Random-Number AssignmentEVENT

Weekly Demand

(hour)Probability Random

Number

Existing Weekly

Capacity (hr)

Probability Random Numbers

200 0.05 00-04 320 0.30 00-29250 0.06 05-10 360 0.40 30-69300 0.17 11-27 400 0.30 70-99350 0.05 28-32400 0.30 33-62450 0.15 63-77500 0.06 78-83550 0.14 84-97600 0.02 98-99


Model Formulation

• Decision variables

• Uncontrollable variables (random)

• Dependent variables



Formulate a simulation model for Specialty Steel Products that will estimate idle-time hours, overtime hours, and subcontracting hours for a specified number of lathes.

Design the simulation model to terminate after 20 weeks of simulated lathe department operations.



• Use the first two rows of random numbers in the random number table for the demand events and the third and fourth rows for the capacity events. Because they are five-digit numbers, only use the first two digits of each number for our random numbers.

• The choice of the rows in the random-number table was arbitrary.

• The important point is that we must be consistent in drawing random numbers and should not repeat the use of numbers in any one simulation.



To simulate a particular capacity level, we proceed as follows:

Step 1:Draw a random number from the first two rows of the table. Start with the first number in the first row, then go to the second number in the first row, and so on.

Step 2:Find the random-number interval for production requirements associated with the random number.

Step 3:Record the production hours (PROD) required for the current week.

Step 4:Draw another random number from row 3 or 4 of the table. Start with the first number in row 3, then go to the second number in row 3, and so on.



Step 5:Find the random-number interval for capacity (CAP) associated with the random number.

Step 6:Record the capacity hours available for the current week.

Step 7: If CAP ≥ PROD, then IDLE HR = CAP – PROD.Step 8: If CAP < PROD, then SHORT = PROD – CAP.

If SHORT ≤ 100, then OVERTIME HR = SHORT and SUBCONTRACT HR = 0.If SHORT > 100, then OVERTIME HR = 100 and SUBCONTRACT HR = SHORT – 100.

Step 9:Repeat steps 1–8 until you have simulated 20 weeks.



• We used a unique random-number sequence for weekly production requirements for each capacity alternative and another sequence for the existing weekly capacity to make a direct comparison between the capacity alternatives.

• Based on the 20-week simulations, we would expect average weekly overtime hours (highlighted in orange) to be reduced by 41.5 – 29.5 = 12 hours and subcontracting hours (highlighted in blue) to be reduced by 18 – 10 = 8 per week. (See Slide E-20)



The average weekly savings would be:

Overtime: (12 hours)($25/hours) = $300Subcontracting: (8 hours)($35/hour) = 280

Total savings per week = $580

• This amount falls short of the minimum required savings of $650 per week.

• The savings are estimated to be $1,851.50 – $1,159.50 = $692 and exceed the minimum required savings for the additional investment from a 1000 week simulation.

• This result emphasizes the importance of selecting the proper run length for a simulation analysis.



10 Machines 11 Machines

WeekDemand Random Number

Weekly Production (hr)

Capacity Random Number

Existing Weekly Capacity (hr)

Idle Hours

Overtime Hours

Subcontract Hours


Idle Hours

Overtime Hours

Subcontract Hours

1 71 450 50 360 90 400 50

2 68 450 54 360 90 400 50

3 48 400 11 320 80 360 40

4 99 600 36 360 100 140 400 100 100

5 64 450 82 400 50 440 10

6 13 300 87 400 100 440 140

7 36 400 41 360 40 400

8 58 400 71 400 440 40

9 13 300 00 320 20 360 60

10 93 550 60 360 100 90 400 100 50

11 21 300 47 360 60 400 100

12 30 350 76 400 50 440 90


Example E.2: Specialty Steel Products


WeekDemand Random Number

Weekly Production (hr)

Capacity Random Number


Idle Hours

Overtime Hours

Subcontract Hours


Idle Hours

Overtime Hours

Subcontract Hours

13 23 300 09 320 20 360 60

14 89 550 54 360 100 90 400 100 50

15 58 400 87 400 440 40

16 46 400 82 400 440 40

17 00 200 17 320 120 360 160

18 82 500 52 360 100 40 400 100

19 02 200 17 320 120 360 160

20 37 400 19 320 80 360

Total 490 830 360 890 590 200

Weekly average 24.5 41.5 18.0 44.5 29.5 10.0

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Results from 20-week simulation

E- 20



Idle hours 26.0 42.2

Overtime hours 48.3 34.2Subcontract hours 18.4 8.7

Cost $1,851.50 $1,159.50


Results from 1,000-week simulation.

Application E.1: Famous Chamois Famous Chamois is an automated car wash that advertises that your car can be finished in just 15 minutes. The time until the next car arrival is described by the following distribution.

Minutes Probability Minutes Probability1 0.01 8 0.122 0.03 9 0.103 0.06 10 0.074 0.09 11 0.055 0.12 12 0.046 0.14 13 0.037 0.14 1.00


Application E.1 : Famous Chamois Assign a range of random numbers to each event so that the demand pattern can be simulated.

Minutes Random Numbers Minutes Random

Numbers1 00–00 8 59-702 01–03 9 71-803 04–09 10 81-874 10–18 11 88-925 19–30 12 93-966 31–44 13 97-997 45–58


Application E.1Simulate the operation for 3 hours, using the following random numbers, assuming that the service time is constant at 6 minutes (or :06) per car.

Random Number

Time to

ArrivalArrival Time

Number in Drive

Service Begins

Departure Time

Minutes in System

50 7 0:07 0 0:07 0:13 6

63 8 0:15 0 0:15 0:21 6

95 12 0:27

49

68

11

40

93

61

48


Application E.1Simulate the operation for 3 hours, using the following random numbers, assuming that the service time is constant at 6 minutes (or :06) per car.


Random Number

Time to

ArrivalArrival Time

Number in Drive

Service Begins

Departure Time

Minutes in System

50 7 0:07 0 0:07 0:13 663 8 0:15 0 0:15 0:21 695 12 0:27 0 0:27 0:33 649 7 0:34 0 0:34 0:40 668 8 0:42 0 0:42 0:48 6

11 4 0:46 1 0:48 0:54 840 6 0:52 1 0:54 1:00 hr. 893 12 1:04 0 1:04 1:10 661 8 1:12 0 1:12 1:18 648 7 1:19 0 1:19 1:25 6

Random Number

Time to Arrival Arrival Time Number in

Drive Service Begins Departure Time

Minutes in System

82 10 1:29 0 1:29 1:35 6

09 3 1:32 1 1:35 1:41 9

08 3 1:35 1 1:41 1:47 12

72 9 1:44 1 1:47 1:53 9

98 13 1:57 0 1:57 2:03 hrs. 6

41 6 2:03 0 2:03 2:09 6

39 6 2:09 0 2:09 2:15 6

67 8 2:17 0 2:17 2:23 6

11 4 2:21 1 2:23 2:29 8

11 4 2:25 1 2:29 2:35 10

00 1 2:26 2 2:35 2:41 15

07 3 2:29 2 2:41 2:47 18

66 8 2:37 2 2:47 2:53 16

00 1 2:38 3 2:53 2:59 21

29 5 2:43 3 2.59 3:05 hrs. 22


Application E.1

Application E.1

Analysis– The average time a car is in the system:

(234/25) = 9.36 minutes

– The percentage of cars that take more than 15 minutes: (4/25)100 = 16%


Computer Simulation

• Steady state

– The state that occurs when the simulation is repeated over enough time that the average results for performance measures remain constant.

• Manual simulations can be excessively time-consuming.

• Simple simulation models can be developed using Excel.


Computer Simulation

• Random numbers can be generated using the RAND function.

• Excel can translate random numbers into values for the uncontrollable variables using the VLOOKUP function.


Computer Simulation


Example E.3: BestCarThe BestCar automobile dealership sells new automobiles. The BestCar store manager believes that the number of cars sold weekly has the following probability distribution:

Weekly Sales (cars) Relative Frequency (probability)0 0.051 0.152 0.203 0.304 0.205 0.10

Total 1.00

The selling price per car is $20,000. Design a simulation model that determines the probability distribution and mean of the weekly sales.


Example E.3 : BestCar

• The first step in creating this spreadsheet is to input the probability distribution, including the cumulative probabilities associated with it.

• These inputs values are highlighted in yellow in cells B6:B11 of the spreadsheet, with corresponding demands in D6:D11 on Slide E-34.

• The cumulative values provide a basis to associate random numbers to the corresponding demand, using the VLOOKUP() function.


Example E.3: BestCar

• Excel’s logic identifies for each week’s random number (in column H) which demand it corresponds to in the Lookup array defined by $C$6:$D$11 on Slide E-34.

• Once it finds the probability range (defined by column C) in which the random number fits, it posts the car demand (in column D) for this range back into the week’s sales (in column I).

• Finally, the results table is created at the lower left portion of the spreadsheet to summarize the simulation output.

• Percentage and cumulative columns next to the frequency column show the frequencies in percentage and cumulative percentage terms.


Example E.3 : BestCar


Simulation with Two Uncontrollable Variables


Simulation with More Advanced Software

• Even more computer power comes from commercial, prewritten simulation software.

• Simulation programming can be done in general-purpose programming languages such as VISUAL BASIC, FORTRAN, or C++.

• Special simulation languages, such as GPSS, SIMSCRIPT, and SLAM, are also available.

• Simulation is also possible with powerful PC-based packages, such as SimQuick, Extend, SIMPROCESS, ProModel, and Witness.


SimQuick Software

• Easy-to-use package that is simply an Excel spreadsheet with some macros.

• Models can be created for a variety of simple processes.

• A first step with SimQuick is to draw a flowchart of the process using SimQuick’s building blocks.

• Information describing each building block is entered into SimQuick tables.


Buffer Done

SimQuick Software

Workst. Add. Insp. 2

Workst. Add. Insp. 1

Buffer Sec. Line 2

Dec. Pt. DP

Workst. Insp. 1

Workst. Insp. 2

Buffer Sec. Line 1

Entrance Arrivals

Flowchart of Passenger Security Process


SimQuick Software

Simulation Results of Passenger SecurityProcess

ElementTypes

ElementNames Statistics Overall

Means

Entrance(s) Door Objects entering process 237.23

Buffer(s) Line 1 Mean Inventory

Mean cycle time

5.97

3.12

Line 2 Mean Inventory

Mean cycle time

0.10

0.53

Done Final Inventory 224.57


Solved Problem 1• A manager is considering production of several products in an

automated facility. • The manager would purchase a combination of two robots. • The two robots are capable of doing all the required operations. • Every batch of work will contain 10 units. • A waiting line of several batches will be maintained in front of Mel. • When Mel completes its portion of the work, the batch will then be

transferred directly to Danny.

Waiting line Mel Danny

Each robot incurs a setup before it can begin processing a batch. Each unit in the batch has equal run time. The distributions of the setup times and run times for Mel and Danny are identical. But because Mel and Danny will be performing different operations, simulation of each batch requires four random numbers from the table.


Solved Problem 1

• Estimate how many units will be produced in an hour. • Then simulate 60 minutes of operation for Mel and

Danny.

Setup Time (min) Probability Run Time per Unit

(sec) Probability

1 0.10 5 0.102 0.20 6 0.203 0.40 7 0.304 0.20 8 0.255 0.10 9 0.15


Solved Problem 1Except for the time required for Mel to set up and run the first batch, we assume that the two robots run simultaneously. The expected average setup time per batch is:

The expected average run time per batch (of 10 units) is:

[(0.1 1 min) + (0.2 2 min)(0.4 3 min)(0.2 4 min) + (0.1 5 min)]

= 3 minutes or 180 seconds per batch

[(0.1 5 sec) + (0.2 6 sec) + (0.3 7 sec) + (0.25 8 sec) + (0.15 9 sec)]

= 7.15 seconds/units 10 units/batch = 71.5 seconds per batch


Solved Problem 1Mel Danny

Batch No.

Start Time

Random No. Setup Random

No. Process Cumulative Time

Start Time

Random No. Setup Random

No. Process Cumulative Time

1 0.00 71 4 50 7 5:10 5:10 21 2 94 9 8:40

2 5.10 50 3 63 8 9:30 9:30 47 3 83 8 13:50

3 9.30 31 3 73 8 13:50 13:50 04 1 17 6 15:50

4 13.50 96 5 9 9 20:20 20:20 21 2 82 8 23:40

5 20.20 25 2 92 9 23:50 23:50 32 3 53 7 28:00

6 23.50 00 1 15 6 25:50 28:00 66 3 57 7 32:10

7 28.00 00 1 99 9 30:30 32:10 55 3 11 6 36:10

8 32.10 10 2 61 8 35:30 36:10 31 3 35 7 40:20

9 36.10 09 1 73 8 38:30 40:20 24 2 70 8 43:40

10 40.20 79 4 95 9 45:50 45:50 66 3 61 8 50:10

11 45.50 01 1 41 7 48:00 50:10 88 4 23 6 55:10

12 50.10 57 3 45 7 54:20 55:10 21 2 61 8 58:30

13 55.10 26 2 46 7 58:20 58:30 97 5 31 7 64:40


Solved Problem 1• The total of average setup and run times per batch is 251.5

seconds. • Even though the robots used the same probability

distributions and therefore have perfectly balanced production capacities, Mel and Danny did not produce the expected capacity of 14 batches because Danny was sometimes idle while waiting for Mel (see batch 2) and Mel was sometimes idle while waiting for Danny (see batch 6).

• The simulation shows the need to place between the two robots sufficient space to store several batches to absorb the variations in process times.

• Subsequent simulations could be run to show how many batches are needed.


Solved Problem 2

• Customers enter a small bank, get into a single line, are served by a teller, and finally leave the bank.

• Currently, this bank has one teller working from 9 A.M. to 11 A.M.

• Management is concerned that the wait in line seems to be too long.

• Therefore, it is considering two process improvement ideas: adding an additional teller during these hours or installing a new automated check-reading machine that can help the single teller serve customers more quickly.

• Use SimQuick to model these two processes.


Solved Problem 2

BufferServed Customers


WorkstationTeller

WorkstationTeller 1

WorkstationTeller 2

BufferLine

BufferLine

EntranceDoor

EntranceDoor

Flowchart for a One-Teller Bank

Flowchart for a Two-Teller BankCopyright ©2013 Pearson Education, Inc. publishing as Prentice Hall E- 46



WorkstationTeller

WorkstationTeller 1

WorkstationTeller 2

BufferLine

BufferLine

EntranceDoor

EntranceDoor


Solved Problem 2• Three key pieces of information need to be entered: when

people arrive at the door, how long the teller takes to serve a customer, and the maximum length of the line.

• Each of the three models is run 30 times, simulating the hours from 9 A.M. to 11 A.M.

ElementTypes

ElementNames

Statistics OverallMeans

Entrance(s) Door Service Level 0.90

Buffer(s) Line Mean Inventory

Mean cycle time4.47

11.04

Simulation Results of BankCopyright ©2013 Pearson Education, Inc. publishing as Prentice Hall E- 47

Solved Problem 2• The numbers shown are averages across the 30 simulations. • The service level for Door tells us that 90 percent of the simulated

customers who arrived at the bank were able to get into Line. • The mean inventory for Line tells us that 4.47 simulated customers

were standing in line. • The mean cycle time tells us that simulated customers waited an

average of 11.04 minutes in line.• When we run the model with two tellers, we find that the service

level increases to 100 percent, the mean inventory in Line decreases to 0.37 customer, and the mean cycle time drops to 0.71 minute.

• When we run the one-teller model with the faster check-reading machine we find that the service level is 97 percent, the mean inventory in Line is 2.89 customers, and the mean cycle time is 6.21 minutes.



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