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Lecture6

MGMT 650Simulation – Chapter 13

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AnnouncementsAnnouncements

HW #4 solutions and grades posted in BB HW #4 average = 111.30 Final exam today Open book, open notes…. Proposed class structure for today

Lecture – 6:00 – 7:50 Class evaluations – 7:50 – 8:00 Break – 8:00 – 8:30 Final – 8:30 – 9:45

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Lecture6

SimulationChapter 13

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Simulation Is …Simulation Is … Simulation – very broad term

methods and applications to imitate or mimic real systems, usually via computer

Applies in many fields and industries

Simulation models complex situations

Models are simple to use and understand

Models can play “what if” experiments

Extensive software packages available

ARENA, ProModel Very popular and powerful method

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ApplicationsApplications Manufacturing facility Bank operation Airport operations (passengers, security, planes,

crews, baggage, overbooking) Hospital facilities (emergency room, operating

room, admissions) Traffic flow in a freeway system Waiting lines - fast-food restaurant, supermarkets Emergency-response system Military

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Example – Simulating Machine BreakdownsExample – Simulating Machine Breakdowns

The manager of a machine shop is concerned about machine breakdowns.

Historical data of breakdowns over the last 100 days is as follows

Simulate breakdowns for the manager for a 10-day period

Number of Breakdowns Frequency

0 10

1 30

2 25

3 20

4 10

5 5

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Simulation ProcedureSimulation ProcedureNumber of Breakdowns Frequency Probability Cum Prob Corresponding Random Numbers

0 10 0.10 0.10 01 to 101 30 0.30 0.40 11 to 402 25 0.25 0.65 41 to 653 20 0.20 0.85 61 to 854 10 0.10 0.95 86 to 955 5 0.05 1.00 96 to 00

100

Day Random Number Simulated # of Breakdowns1 90 42 73 33 82 34 16 15 94 46 92 47 68 38 5 09 84 310 91 4

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Expected number of breakdowns = 1.9 per day

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Statistical AnalysisStatistical AnalysisDay # Replication 1 Replication 2 Replication 3 Replication 4 Replication 5 Replication 6 Replication 7 Replication 8 Replication 9 Replication 10

1 1 4 4 1 0 2 5 1 1 02 3 5 0 2 0 4 3 2 0 33 3 1 1 1 2 1 0 1 1 54 1 2 2 3 2 3 1 1 1 15 2 0 1 1 1 5 2 2 0 26 0 2 3 1 3 1 4 3 2 27 1 1 2 2 1 2 1 1 1 08 3 3 2 2 0 4 2 1 3 29 1 1 1 2 2 1 4 0 1 4

10 5 1 3 2 3 2 1 0 1 1

2.00 2.00 1.90 1.70 1.40 2.50 2.30 1.20 1.10 2.00

95 % confidence interval for mean breakdowns for the 10-day period is given by:

]955.1,665.1[)10

458.0(262.281.1

21;1

n

stx

n

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Monte Carlo SimulationMonte Carlo Simulation

Monte Carlo method: Probabilistic simulation technique used when a process has a random component

Identify a probability distribution

Setup intervals of random numbers to match probability distribution

Obtain the random numbers Interpret the results

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Example 2 – Simulating a Reorder Example 2 – Simulating a Reorder Policy Policy

The manager of a truck dealership wants to acquire some insight into how a proposed policy for reordering trucks might affect order frequency

Under the new policy, 2 trucks will be ordered every time the inventory of trucks is 5 or lower

Due to proximity between the dealership and the local office, orders can be filled overnight

The “historical” probability for daily demand is as follows

Simulate a reorder policy for the dealer for the next 10 days Assume a beginning inventory of 7 trucks

Demand (x) P(x)

0 0.50

1 0.40

2 0.10

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Example 2 SolutionsExample 2 Solutions

x P(x) Cum P(x) RN Day RN Demand Begin Inv End Inv Reorder0 0.5 0.5 01 to 50 1 81 1 7 6 01 0.4 0.9 51 to 90 2 20 0 6 6 02 0.1 1.0 91 to 00 3 82 1 6 5 2

4 34 0 7 7 05 85 1 7 6 06 35 0 6 6 07 10 0 6 6 08 14 0 6 6 09 84 1 6 5 210 92 2 7 5 2

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In-class Example using MS-ExcelIn-class Example using MS-Excel The time between mechanics’ requests for tools in a

AAMCO facility is normally distributed with a mean of 10 minutes and a standard deviation of 1 minute.

The time to fill requests is also normal with a mean of 9 minutes and a standard deviation of 1 minute.

Mechanics’ waiting time represents a cost of $2 per minute.

Servers represent a cost of $1 per minute. Simulate arrivals for the first 9 mechanic requests and

determine Service time for each request Waiting time for each request Total cost in handling all requests

Assume 1 server only

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AAMCO SolutionsAAMCO SolutionsInter-request time Cum Int-req time Service Time Service Begins Service Ends Wait Time

11.70 11.70 8.80 11.70 20.50 0.007.08 18.78 10.28 20.50 30.78 1.725.38 24.16 8.98 24.16 33.14 0.005.92 30.08 8.26 30.08 38.34 0.006.17 36.25 8.49 36.25 44.74 0.007.10 43.35 8.61 43.35 51.96 0.006.58 49.93 8.76 49.93 58.69 0.007.52 57.45 8.09 57.45 65.54 0.005.71 63.16 8.91 63.16 72.08 0.00

Sum of wait 1.72

Server cost/min 1Waiting cos/min 2

Total cost 75.52

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Discrete Event SimulationDiscrete Event SimulationExample 1 - A Simple Processing Example 1 - A Simple Processing

SystemSystem

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Discrete Event Simulation Discrete Event Simulation Example 2 - Electronic Assembly/Test Example 2 - Electronic Assembly/Test

SystemSystem

Produce two different sealed elect. units (A, B) Arriving parts: cast metal cases machined to accept the electronic parts Part A, Part B – separate prep areas Both go to Sealer for assembly, testing – then to Shipping (out) if OK,

or else to Rework Rework – Salvaged (and Shipped), or Scrapped

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Part APart A Interarrivals: expo (5) minutes From arrival point, proceed immediately to Part A Prep

area Process = (machine + deburr + clean) ~ tria (1,4,8)

minutes Go immediately to Sealer

Process = (assemble + test) ~ tria (1,3,4) min. 91% pass, go to Shipped; Else go to Rework

Rework: (re-process + testing) ~ expo (45) 80% pass, go to Salvaged; Else go to Scrapped

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Part BPart B Interarrivals: batches of 4, expo (30) min. Upon arrival, batch separates into 4 individual parts From arrival point, proceed immediately to Part B Prep

area Process = (machine + deburr +clean) ~ tria (3,5,10)

Go to Sealer Process = (assemble + test) ~ weib (2.5, 5.3) min.,

different from Part A, though at same station 91% pass, go to Shipped; Else go to Rework

Rework: (re-process + test) = expo (45) min. 80% pass, go to Salvaged; Else go to Scrapped

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Run Conditions, OutputRun Conditions, Output

Start empty & idle, run for four 8-hour shifts (1,920 minutes)

Collect statistics for each work area on Resource utilization Number in queue Time in queue

For each exit point (Shipped, Salvaged, Scrapped), collect total time in system (a.k.a. cycle time)

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Simulation Models Are BeneficialSimulation Models Are Beneficial

Systematic approach to problem solving Increase understanding of the problem Enable “what if” questions Specific objectives Power of mathematics and statistics Standardized format Require users to organize

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Simulation ProcessSimulation Process

1. Identify the problem

2. Develop the simulation model

3. Test the model

4. Develop the experiments

5. Run the simulation and evaluate results

6. Repeat 4 and 5 until results are satisfactory

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Different Kinds of SimulationDifferent Kinds of Simulation

Static vs. Dynamic Does time have a role in the model?

Continuous-change vs. Discrete-change Can the “state” change continuously or only at

discrete points in time? Deterministic vs. Stochastic

Is everything for sure or is there uncertainty? Most operational models:

Dynamic, Discrete-change, Stochastic

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Advantages of SimulationAdvantages of Simulation

Solves problems that are difficult or impossible to solve mathematically

Flexibility to model things as they are (even if messy and complicated)

Allows experimentation without risk to actual system

Ability to model long-term effects

Serves as training tool for decision makers

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Limitations of SimulationLimitations of Simulation

Does not produce optimum solution

Model development may be difficult

Computer run time may be substantial

Monte Carlo simulation only applicable to random systems

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Fitting Probability Distributions to Existing Fitting Probability Distributions to Existing DataData

Data Summary

Number of Data Points = 187Min Data Value = 3.2

Max Data Value = 12.6Sample Mean = 6.33Sample Std Dev = 1.51

Histogram Summary

Histogram Range = 3 to 13

Number of Intervals = 13

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ARENA – Input AnalyzerARENA – Input AnalyzerDistribution Summary

Distribution: Gamma Expression: 3 + GAMM(0.775, 4.29)

Square Error: 0.003873

Chi Square Test Number of intervals = 7 Degrees of freedom = 4

Test Statistic = 4.68 Corresponding p-value = 0.337

Kolmogorov-Smirnov Test Test Statistic = 0.0727

Corresponding p-value > 0.15

Data Summary

Number of Data Points = 187Min Data Value = 3.2

Max Data Value = 12.6Sample Mean = 6.33Sample Std Dev = 1.51

Histogram Summary

Histogram Range = 3 to 13Number of Intervals = 13

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Simulation in IndustrySimulation in Industry

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Course ConclusionsCourse Conclusions Recognize that not every tool is the best fit for every problem Pay attention to variability

Forecasting Inventory management - Deliveries from suppliers

Build flexibility into models Pay careful attention to technology

Opportunities Improvement in service and response times

Risks Costs involved Difficult to integrate Need for periodic updates Requires training

Garbage in, garbage out Results and recommendations you present are only as reliable as the model and its

inputs Most decisions involve tradeoffs Not a good idea to make decisions to the exclusion of known information