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

2. B – 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. 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

3. B – 3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. The Simulation Process Monte Carlo simulation uses random numbers to generate the simulation events  Data collection  Random-number assignment/Model formulation  Experimentation

4. B – 4 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Data 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.

5. B – 5 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Data Collection for a Simulation SOLUTION Historical records indicate that lathe department demand varies from week to week as follows: Weekly Production Requirements (hr)Relative Frequency 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02 Total1.00

6. B – 6 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Data Collection for a Simulation 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 2000.05 2500.06 3000.17 3500.05 4000.30 4500.15 5000.06 5500.14 6000.02 Total1.00 200(0.05) + 250(0.06) + 300(0.17) +... + 600(0.02) = 400 hours

7. B – 7 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Data Collection for a Simulation 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

8. B – 8 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Data Collection for a Simulation 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 Overtime100 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

9. B – 9 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Random-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

10. B – 10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Random-Number Assignment Event Weekly Demand (hr) Probability Random Number Existing Weekly Capacity (hr) Probability Random Numbers 2000.0500-043200.3000-29 2500.0605-103600.4030-69 3000.1711-274000.3070-99 3500.0528-32 4000.3033-62 4500.1563-77 5000.0678-83 5500.1484-97 6000.0298-99 Table B.1 – Random-Number Assignments to Simulation Events

11. B – 11 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Model 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

12. B – 12 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating 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.

13. B – 13 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model 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.

14. B – 14 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model 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.

15. B – 15 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model 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

16. B – 16 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model 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.

17. B – 17 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model The average weekly savings based on 20-week simulation 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? For more reliable results 1000-week simulation is considered.

18. B – 18 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model 10 Machines11 Machines Week Demand Random Number Weekly Production (hr) Capacity Random Number Existing Weekly Capacity (hr) Idle Hours Overtime Hours Subcontract Hours Existing Weekly Capacity (hr) Idle Hours Overtime Hours Subcontract Hours 171450503609040050 268450543609040050 348400113208036040 49960036360100140400100 564450824005044010 61330087400100440140 7364004136040400 8584007140044040 913300003202036060 1093550603601009040010050 11213004736060400100 1230350764005044090 Table B.2 – 20-Week Simulation of Alternatives

19. B – 19 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model 10 Machines11 Machines Week Demand Random Number Weekly Production (hr) Capacity Random Number Existing Weekly Capacity (hr) Idle Hours Overtime Hours Subcontract Hours Existing Weekly Capacity (hr) Idle Hours Overtime Hours Subcontract Hours 1323300093202036060 1489550543601009040010050 15584008740044040 16464008240044040 170020017320120360160 18825005236010040400100 190220017320120360160 20374001932080360 Total490830360890590200 Weekly average 24.541.518.044.529.510.0 Table B.2 – 20-Week Simulation of Alternatives

20. B – 20 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Formulating a Simulation Model Table B.3 – Comparison of 1,000-Week Simulations 10 Machines11 Machines Idle hours26.042.2 Overtime hours48.334.2 Subcontract hours18.48.7 Cost$1,851.50$1,159.50 The estimated savings = $1,851.50 – $1,159.50 = $692 This exceed the minimum required savings for the additional investment.

21. B – 21 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Application 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. MinutesProbabilityMinutesProbability 10.0180.12 20.0390.10 30.06100.07 40.09110.05 50.12120.04 60.14130.03 70.141.00

22. B – 22 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Application B.1 Assign a range of random numbers to each event so that the demand pattern can be simulated. Minutes Random Numbers Minutes Random Numbers 100–00859-70 201–03971-80 304–091081-87 410–181188-92 519–301293-96 631–441397-99 745–58

23. B – 23 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Application B.1 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 5070:070 0:136 6380:150 0:216 95120:27 49 68 11 40 93 61 48

24. B – 24 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Application B.1 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 5070:070 0:136 6380:150 0:216 95120:270 0:336 4970:340 0:406 6880:420 0:486 1140:4610:480:548 4060:5210:54 1:00 hr.8 93121:040 1:106 6181:120 1:186 4871:190 1:256

25. B – 25 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Application B.1 Random Number Time to Arrival Arrival Time Number in Drive Service Begins Departure Time Minutes in System 82101:290 1:356 0931:3211:351:419 0831:3511:411:4712 7291:4411:471:539 98131:570 2:03 hrs.6 4162:030 2:096 3962:090 2:156 6782:170 2:236 1142:2112:232:298 1142:2512:292:3510 0012:2622:352:4115 0732:2922:412:4718 6682:3722:472:5316 0012:3832:532:5921 2952:4332.59 3:05 hrs.22

26. B – 26 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Computer 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

27. B – 27 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Computer Simulation 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

28. B – 28 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Excel 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) 00.05 10.15 20.20 30.30 40.20 50.10 Total1.00 The selling price per car is $20,000. Design a simulation model that determines the probability distribution and mean of the weekly sales.

29. B – 29 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Excel Simulation for BestCar 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.

30. B – 30 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Excel Simulation for 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. 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.

31. B – 31 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Excel Simulation for BestCar Figure B.2 – BestCar Simulation Model

32. B – 32 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Computer Simulation 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

33. B – 33 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Solved Problem 1 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)ProbabilityRun Time per Unit (sec)Probability 10.105 20.206 30.4070.30 40.2080.25 50.1090.15

34. B – 34 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Solved Problem 1 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

35. B – 35 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Solved Problem 1 MelDanny Batch No. Start Time Random No. Setup Random No. Process Cumulative Time Start Time Random No. Setup Random No. Process Cumulative Time 10.007145075:10 2129498:40 25.105036389:30 47383813:50 39.3031373813:50 04117615:50 413.509659920:20 21282823:40 520.2025292923:50 32353728:00 623.5000115625:5028:0066357732:10 728.0000199930:3032:1055311636:10 832.1010261835:3036:1031335740:20 936.1009173838:3040:2024270843:40 1040.2079495945:50 66361850:10 1145.5001141748:0050:1088423655:10 1250.1057345754:2055:1021261858:30 1355.1026246758:2058:3097531764:40 Table B.4 – Simulation Results for Mel and Danny

36. B – 36 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Solved Problem 1 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.