B – 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Simulation B For...

B – 1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

SimulationSimulationB

For For Operations Management, 9eOperations Management, 9e by by Krajewski/Ritzman/Malhotra Krajewski/Ritzman/Malhotra © 2010 Pearson Education© 2010 Pearson Education

PowerPoint Slides PowerPoint Slides by Jeff Heylby Jeff Heyl


Simulation models can be used 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

Used to conduct experiments without disrupting real systems

Used to obtain operating characteristic estimates in much less time (time compression)

Simulation is useful in sharpening managerial decision-making skills through gaming

SimulationSimulation


The Simulation ProcessThe Simulation Process

Monte Carlo simulation uses random numbers to generate the simulation events Data collection Random-number assignment Model formulation


Data Collection for a SimulationData Collection for a Simulation

EXAMPLE B.1

The 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.



SOLUTION

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

Weekly Production Requirements (hr) Relative Frequency

200 0.05

250 0.06

300 0.17

350 0.05

400 0.30

450 0.15

500 0.06

550 0.14

600 0.02

Total 1.00



To gather these data, all weeks with requirements of 175.00–224.99 hours were grouped in the 200-hour category, all weeks with 225.00–274.99 hours were grouped in the 250-hour category, and so on. The average weekly production requirements for the lathe department are

Weekly Production Requirements (hr)

Relative Frequency

200 0.05

250 0.06

300 0.17

350 0.05

400 0.30

450 0.15

500 0.06

550 0.14

600 0.02

Total 1.00

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



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 is

320(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 (8 machines) 0.30

400 (9 machines) 0.40

440 (10 machines) 0.30

Resources and Costs

Maximum Overtime 100 hrs

Lathe Operators $10/hr

Overtime Cost $25/hr

Subcontracting 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 AssignmentRandom-Number Assignment

A random number is a number that has the same probability of being selected as any other number

The 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

Table B.1 shows the allocation of 100 random numbers to demand events


Random-Number AssignmentRandom-Number Assignment

Event

Weekly Demand

(hr)Probability Random

Number

Existing Weekly

Capacity (hr)

Probability Random Numbers

200 0.05 00-04 320 0.30 00-29

250 0.06 05-10 360 0.40 30-69

300 0.17 11-27 400 0.30 70-99

350 0.05 28-32

400 0.30 33-62

450 0.15 63-77

500 0.06 78-83

550 0.14 84-97

600 0.02 98-99

Table B.1 – Random-Number Assignments to Simulation Events


Model FormulationModel Formulation

Formulating a simulation model entails specifying the relationships among the variables

Decision variables are controlled by the decision maker and will change from one run to the next as different events are simulated

Uncontrollable variables are random events that the decision maker cannot control

Dependent variables reflect the values of the decision variables and the uncontrollable variables

The relationships among the variables are expressed in mathematical terms


Formulating a Simulation ModelFormulating a Simulation Model

EXAMPLE B.2

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.



SOLUTION

Let us 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, we use only 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



ANALYSIS

Table B.2 contains the simulations for the two capacity alternatives at Specialty Steel Products. 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 gray) to be reduced by 18 – 10 = 8 per week.



The average weekly savings would be

Overtime: (12 hours)($25/hours) = $300

Subcontracting: (8 hours)($35/hour) = 280

Total savings per week = $580

This amount falls short of the minimum required savings of $650 per week. Does this outcome mean that we should not add the machine and worker? Before answering, let us look at Table B.3, which shows the results of a 1,000-week simulation for each alternative. The costs (highlighted in lavender) are quite different from those of the 20-week simulations. Now the savings are estimated to be $1,851.50 – $1,159.50 = $692 and exceed the minimum required savings for the additional investment. This result emphasizes the importance of selecting the proper run length for a simulation analysis. We can use statistical tests to check for the proper run length.



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

Table B.2 – 20-Week Simulation of Alternatives




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

Table B.2 – 20-Week Simulation of Alternatives



Table B.3 – Comparison of 1,000-Week Simulations


Idle hours 26.0 42.2

Overtime hours 48.3 34.2

Subcontract hours 18.4 8.7

Cost $1,851.50 $1,159.50


Application B.1Application B.1

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 Probability

1 0.01 8 0.12

2 0.03 9 0.10

3 0.06 10 0.07

4 0.09 11 0.05

5 0.12 12 0.04

6 0.14 13 0.03

7 0.14 1.00



Assign a range of random numbers to each event so that the demand pattern can be simulated.

MinutesRandom Numbers

MinutesRandom Numbers

1 00–00 8 59-70

2 01–03 9 71-80

3 04–09 10 81-87

4 10–18 11 88-92

5 19–30 12 93-96

6 31–44 13 97-99

7 45–58



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

Random Number

Time to Arrival

Arrival 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



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

Random Number

Time to Arrival

Arrival 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 0 0:27 0:33 6

49 7 0:34 0 0:34 0:40 6

68 8 0:42 0 0:42 0:48 6

11 4 0:46 1 0:48 0:54 8

40 6 0:52 1 0:54 1:00 hr. 8

93 12 1:04 0 1:04 1:10 6

61 8 1:12 0 1:12 1:18 6

48 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


Computer SimulationComputer Simulation

It is important to simulate a process long enough to achieve steady state, so that the simulation is repeated over enough time that the average results for performance measures remain constant

Manual simulations can be excessively time-consuming

Simulating these real-world situations manually can become too time-consuming

Simple simulation models can be developed using Excel



Using Excel spreadsheets for 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



Figure B.1 – A Spreadsheet with 100 Random Numbers Generated with RAND( )


Excel Simulation for BestCarExcel Simulation for BestCar

EXAMPLE B.3

The 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 0.05

1 0.15

2 0.20

3 0.30

4 0.20

5 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.



SOLUTION

Figure B.2 simulates 50 weeks of sales at BestCar.

The bottom right of the spreadsheet shows that the average weekly sales is 2.88 cars, for $57,600 per week.

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.

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



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. 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.



Figure B.2 – BestCar Simulation Model



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 SoftwareSimQuick 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


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

Figure B.3 – Flowchart of Passenger Security Process



Figure B.4 – Simulation Results of Passenger Security Process

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 1Solved 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.



Estimate how many units will be produced in an hour. Then simulate 60 minutes of operation for Mel and Danny. The random numbers have already been selected in Table B.4 for each of the four uncontrolled variables. For example, the third column provides the random numbers for determining Mel’s setup time for each batch, and the fifth column provides the random numbers for determining Mel’s processing times.

Setup Time (min) Probability Run Time per Unit (sec) Probability

1 0.10 5 0.10

2 0.20 6 0.20

3 0.40 7 0.30

4 0.20 8 0.25

5 0.10 9 0.15



SOLUTION

Except 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



Mel 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

Table B.4 – Simulation Results for Mel and Danny



Thus, the total of average setup and run times per batch is 251.5 seconds. However, this estimate is probably too high. Keep in mind that Mel and Danny operate in sequence and that Danny cannot begin to do work until it has been completed by Mel (see batch 2 of Table B.4). Nor can Mel start anew batch until Danny is ready to accept the previous one. 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.



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.



SOLUTION

A first step in using SimQuick is to draw a flowchart of the process using SimQuick’s building blocks. Figure B.5(a) shows that the one-teller bank (both the original and the variation with a check-reading machine) can be modeled with four building blocks: an entrance (modeling the arrival of customers at the bank), a buffer (modeling the waiting line), a workstation (modeling the teller), and a final buffer (modeling served customers). The two-teller variation can be modeled with five building blocks, as shown in Figure B.5(b).



BufferServed Customers

BufferServed Customers

WorkstationTeller

WorkstationTeller 1

WorkstationTeller 2

BufferLine

BufferLine

EntranceDoor

EntranceDoor

FIGURE B.5A – Flowchart of Bank Flowchart for a One-Teller Bank

Figure B.5B – Flowchart for a Two-Teller Bank



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. Figure B.6 shows the key results for the model of the original one-teller process as output by SimQuick.

ElementTypes

ElementNames Statistics Overall

Means

Entrance(s) Door Service Level 0.90

Buffer(s) Line Mean Inventory

Mean cycle time

4.47

11.04Figure B.6 – Simulation Results of Bank



The numbers shown in Figure B.6 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 customers, and the mean cycle time drops to 0.71 minutes. 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. These statistics, together with cost information, should help management select the best process.

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