Handout simulasi computer

Department of Industrial Engineering Faculty of Science & Technology The Sunan Kalijaga State Islamic University

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Program Studi Teknik IndustriFakultas Sains dan Teknologi Universitas Islam Negeri Sunan Kalijaga

Tahun 2009

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Chapter 1Chapter 1Chapter 1Chapter 1

Introduction to SimulationIntroduction to SimulationBy : Arya WirabhuanaBy : Arya Wirabhuana

The Opportunity GameThe Opportunity Game

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2

333

3

4

45 400

400

500

500

500500

600

600

600700 200

200

300

300300400

400

400

400500

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Cost to Play: $1000Payoff ($): (A Spinner) x (B Spinner) – (C Spinner)Return ($): Payoff – Cost-to-Play

Spinner A Spinner B Spinner C

Arya Wirabhuana - Industrial Computer Simulation

Department of Industrial Engineering UIN Sunan Kalijaga Yogyakarta

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ProblemsProblems

If il bl ti (f l i th ) i blIf available time (for playing the game) is no problem,and if there is no constraint on available workingcapital, would a prudent person choose to play thisgame (repeatedly)?

(In other words, what is the expected (that is, the

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long-run average) Return?

Alternative SolutionAlternative SolutionApproachesApproaches

S l h bl h i llSolve the problem mathematicallyPerform experiments with real systemPerform experiments with a model (representation) ofthe real system

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Mathematical ModelMathematical Model

E t d lt f i di id l Expected B Spinner ResultExpected results of individualSpinners(long-run individual spinner results)

Expected – A-Spinner Result

Outcome Probability Outcome x Probability

1 1 / 10 0 1

Expected – B-Spinner Result


400 2 / 10 80

500 4 / 10 200

600 3 / 10 180

700 1 / 10 70

Sum (Expected Outcome) : 530

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1 1 / 10 0.1

2 2 / 10 0.4

3 4 / 10 1.2

4 2 / 10 0.8

5 1 / 10 0.5

Sum (Expected Outcome) : 3.0

Expected – C-Spinner Result


200 2 / 10 40

300 3 / 10 90

400 4 / 10 160

500 1 / 10 50

Sum (Expected Outcome) : 340

Mathematical Model Mathematical Model (cont’)(cont’)

What is the Expected Return in the OpportunityWhat is the Expected Return in the OpportunityGame?

Payoff = (A-Spinner) x (B-Spinner) – (C-Spinner)Return = Payoff – (Cost-to-Play)

substitute expected spinner results to get expected Payoff and Return

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substitute expected spinner results to get expected Payoff and Return

Expected Payoff = (3.0) x (530) – (340) = $1,250

Expected Return = $1,250 - $1,000 = $250



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Complicated QuestionComplicated Question

Wh t th h th t ill lWhat are the chances that a person will lose moneyin a single play of the game?

The answer to this question can be developedmathematically, but doing so requires:

computing the relative frequency with which each of the possible

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returns occurs,then using these relative frequencies to determine the cumulativefrequencies for the returns ordered from lowest to highest,and finally looking up the cumulative frequency for negativereturns

Complicated Question Complicated Question (cont’)(cont’)

Thi i d b ti ll f th iThis is done by enumerating all of the variouspossible spinner combinations and using the law ofmultiplication to compute the probability associatedwith each combination(and then using the law of addition to addprobabilities for identical outcomes to determine the

ll b bilit f th t t )

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overall probability of that outcome)



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Solution bySolution byEnumerationEnumeration

Spinner A

1

2

3

Spinner B

400

500

600

Spinner C

200

300

400

Probability

0.032

0.048

0.064

Start Return

$300

$200

$1004 / 10

4 / 10

2 / 10

3 / 10

4 / 10

1 / 10

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4

5

700 500 0.016 $0

1 / 10

and so on, for all othercombinations of Spinner A,

Spinner B and Spinner C

Opportunity GameOpportunity GameOutcomesOutcomes

Return Relative Frequency Cumulative Frequencyq y q y-$1100 0.002 0.002

-$1000 0.012 0.014

-$900 0.025 0.039

-$800 0.029 0.068

… … …

-$200 0.072 0.290

-$100 0.044 0.33434 distinct returns,ranging from-$1100 to $2300

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$0 0.052 0.386

$100 0.074 0.460

$200 0.069 0.528

… … …

$2100 0.004 0.995

$2200 0.003 0.998

$2300 0.002 1.000

-$1100 to $2300



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Opportunity GameOpportunity GameHistogramHistogram

A hi t h i l ti f i fA histogram showing relative frequencies ofvarious “Return” ranges in the Opportunity Game

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25

30

35

40

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0

5

10

15

-1000 -500 0 500 1000 1500 2000 2500

5.2%

31%27.2%

13.2%15%

6.8% 0.2%1.4%

More ComplicatedMore ComplicatedQuestionsQuestions

If h d $2000 i ki it l dIf a person had $2000 in working capital, andenough time to play the game up to 25 times, whatare chances that the person would:

go bankrupt?lose money, but not go bankrupt?break even?make money?

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make money?exceed the expected gain of $6250?

($6250 = 25 x $250)



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More ComplicatedMore ComplicatedQuestions (cont’)Questions (cont’)

A th ti l h l b t k tA mathematical approach can also be taken toanswer each of these questions, but the calculations,although straightforward, are quite tedious!

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S l th bl th ti llSolve the problem mathematicallyPerform experiments with real systemPerform experiments with a model (representation) ofthe real system

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Real SystemReal System

It ld b t t t th “ t it ”It would be easy to construct the “opportunity game”spinners and play the game repeatedly (without dollarconsequences), say 1,000 times, then use theaverage result as an estimate of the expected resultMore generally, experimentation on “the real system”can be done in concept, but often cannot be done in

ti

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practiceExperimenting on the real system requires of coursethat the system exists, and it might not (the goalmight in fact be to design a system)

Real System (cont’)Real System (cont’)

If th t d i t it i ht t b f ibl tIf the system does exists, it might not be feasible toexperiment with it, for reasons such as these:

Economic reasons(it might be prohibitively expensive to interrupt the ongoing use

of the real system)Political reasons

(it might be difficult to get permission from the system’s

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(it might be difficult to get permission from the system s“owners” to experiment with the system)

Real-system experiments might take too long(days, weeks, or months of experimentation might be required,

and so the findings might not be available in time to do any good)



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S l th bl th ti llSolve the problem mathematicallyPerform experiments with real systemPerform experiments with a model (representation) of the real system

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Model of theModel of theReal SystemReal System

F i l ti i i l t h iFor our purposes, simulation is a numerical techniquefor conducting experiments with a model thatdescribes or mimics the behaviour of a systemA model is a representation of a system that behaveslike the system itself behaves

(the model may not behave like the system in all respects, but thed l t b h lik th t t l t i th t th t

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model must behave like the system at least in those respects thatare important for the purpose at hand)



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Model of theModel of theReal System (cont’)Real System (cont’)

I l d l ti h i lIn general, models sometimes are physical, e.g.,blueprints of a housea three dimensional model of a shopping malla mock-up of the control panels in a jetliner

Models sometimes are logical abstractions based onthe rules that govern the operation of a system, for

l

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example,a computer program that plays the “opportunity game” bydetermining spinner results at random and combining the resultsto determine the payoff and return.

Spreadsheet OutputSpreadsheet Output

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Model Classification andModel Classification andSteps in a Simulation StudySteps in a Simulation Study

By : Arya WirabhuanaBy : Arya Wirabhuana

Definition of SimulationDefinition of Simulation

Simulation is the imitation of an operation of a realSimulation is the imitation of an operation of a real-world process or system over time.Simulation is a method of understanding,representing and solving complex interdependentsystem.Simulation is the process of designing a model of areal system and conducting experiments with this

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y g pmodel for the purpose either of understanding thebehavior of the system or of evaluating variousstrategies (with the limits imposed by a criterion or aset of criteria) for the operation of the system.



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Definition of SimulationDefinition of Simulation(cont’)(cont’)

Si l ti i l i t t d th t d l ithSimulation in general is to pretend that one deals witha real thing while really working with an imitation.A flight simulator on a PC is computer model of someaspects of the flight: it shows on the screen thecontrols and what the “pilot” (the youngster whooperates it) is supposed to see from the “cockpit” (his

h i )

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armchair).

When to use ModelWhen to use Model

T fl i l t i f d h th th lTo fly a simulator is safer and cheaper than the realairplane.For precisely this reason, models are used inindustry, commerce and military: it is very costly,dangerous and often impossible to make experimentswith real systems.

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Provided that models are adequate descriptions ofreality (they are valid), experimenting with them cansave money, suffering and even time.



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When to use When to use SimulationsSimulations

Systems which change with time such as a gasSystems which change with time such as a gasstation where cars come and go (called dynamicsystems) and involve randomness (nobody canguess at exactly which time and next cars shouldarrive at the station) are good candidates forsimulation.Modeling complex dynamic systems theoretically

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need too many simplifications and the emergingmodels may not be therefore valid.Simulation does not require that many simplifyingassumptions, making it the only tool even in absenceof randomness.

How to simulate?How to simulate?

Suppose we are interested in a gas station We maySuppose we are interested in a gas station. We maydescribe the behaviour of this system graphically byplotting the number of cars in the station; the state ofthe system.Every time a car arrives the graph increases by oneunit while a departing car causes the graph to dropone unit.

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This graph (called sample path), could be obtainedfrom observation of a real station, but could also beartificially constructed.Such artificial construction and the analysis of theresulting sample path consists of the simulation.



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Types of ModelsTypes of Models

M d l b l ifi d b i th ti lModels can be classified as being mathematical orphysical.A mathematical model uses symbolic notation andmathematical equations to represent a system.A simulation model is particular type of mathematicalmodel of a system.

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Type of SimulationType of Simulation

Si l ti d l b f th l ifi d b iSimulation models may be further classified as being:Static model or Dynamic modelDeterministic model or Stochastic modelDiscrete model or Continuous model

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Static vs DynamicStatic vs Dynamic

St ti d l d d i d l l ifi tiStatic models and dynamic models are classificationby the dependency on timeA static simulation model, sometimes called a MonteCarlo simulation, represents a system at a particularpoint in time.

For example, Mark Six, inventory level

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Dynamic simulation models represent systems inwhich state of the variables change over time. Thesimulation of a bank from 9:00am to 4:00pm is anexample of a dynamic simulation.

For example, service time, waiting time.

Deterministic vsDeterministic vsStochasticStochastic

Cl ifi ti b th t f th i blClassification by the nature of the variablesSimulation models that contain no random variablesare classified as deterministic.

For example, deterministic arrivals would occur at a dentist’soffice if all arrived at the scheduled appointment time.

A stochastic simulation model has one or mored i bl i t

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random variables as input.Random inputs lead to random outputs.

For example, random arrival, random product demand, randomincoming calls.



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Deterministic vsDeterministic vsStochastic (cont’)Stochastic (cont’)

Si th t t d th bSince the outputs are random, they can beconsidered only as estimates of the truecharacteristics of a model.

For example, the simulation of a bank would usually involverandom interarrival times and random service times.

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Discrete vs ContinuousDiscrete vs Continuous

Di t d ti d l d fi d iDiscrete and continuous models are defined in ananalogous manner, classification by system nature.A discrete model is one in which the state variable(s)change only at a discrete set of points in time.The bank is an example of a discrete system, sincethe state variable, the number of customers in the

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bank, changes only when a customer arrives or whenthe service provided a customer is complete.Other examples, busy/idle counter, occupied/freemachine.



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Discrete vs ContinuousDiscrete vs Continuous(cont’)(cont’)

A continuous model is one in which the stateA continuous model is one in which the statevariable(s) change continuously over time.An example is the head of water behind a dam.During and for some time after a rain storm, waterflows into the lake behind the dam.Water is drawn from the dam for flood control and tomake electricity.

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yEvaporation also decreases the water level.But, continuous system can be approximated by adiscrete-event system, depending on the expectedpreciseness and the objective of the study.

ApplicationsApplications-- Service ApplicationsService Applications

St ffiStaffingA bank manager might determine that three tellers onduty results in a tolerable wait for service during mostof the day, but that her customers’ “time in queue” istoo long during the busy lunch hour and in the lateafternoon.

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She could then assess the impacts of addingadditional part-time help during the peak hours.



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ApplicationsApplications-- Service Applications Service Applications (cont’)(cont’)

P d I tProcedure ImprovementMany organizations have learned that internalconsumers are customers.In an effort to improve the responsiveness of theiradministrative and support functions many of thesecompanies are using simulation to model revised

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procedures designed to streamline processing ofpaperwork, telephone calls and other dailytransactions.

Advantages ofAdvantages ofSimulationSimulation

New policies operating procedures decision rulesNew policies, operating procedures, decision rules,information flows, organizational procedures, and soon can be explored without disrupting ongoingoperations of the real system.New hardware designs, physical layouts,transportation systems, and so on, can be testedwithout committing resources for their acquisition.

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Hypotheses about how or why certain phenomenaoccur can be tested for feasibility.Time can be compressed or expanded allowing for aspeedup or slowdown of the phenomena underinvestigation.



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Advantages ofAdvantages ofSimulation (cont’)Simulation (cont’)

Insight can be obtained about the interaction ofInsight can be obtained about the interaction ofvariables.Insight can be obtained about the importance ofvariables to the performance of the system.Bottleneck analysis can be performed indicatingwhere work-in-process, information, materials, and soon are being excessively delayed.

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g y yA simulation study can help in understanding how thesystem operates rather than how individuals think thesystem operates.“What-if” questions can be answered.

Disadvantages ofDisadvantages ofSimulationSimulation

Model building requires special trainingModel building requires special training.Simulation results may be difficult to interpret.Simulation modeling and analysis can be timeconsuming and expensive. Skimping on resourcesfor modeling and analysis may result in a simulationmodel or analysis that is not sufficient for the task.Simulation is used in some cases when an analytical

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Simulation is used in some cases when an analyticalsolution is possible, or even preferable. This mightbe particularly true in the simulation of some waitinglines where closed-form queueing models areavailable.



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Defense of SimulationDefense of Simulation

V d f i l ti ft h b ti lVendors of simulation software have been activelydeveloping packages that contain all or part ofmodels that need only input data for their operation.Many simulation software vendors have developedoutput analysis capabilities within their packages forperforming very thorough analysis.

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Simulation can be performed faster today thanyesterday, and even faster tomorrow. This isattributable to the advances in hardware that permitrapid running of scenarios.

Defense of Simulation Defense of Simulation (cont’)(cont’)

Cl d f d l t bl t l t fClosed-form models are not able to analyze most ofthe complex systems that are encountered inpractice.

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Steps in aSteps in aSimulation StudySimulation Study

Problem formulationProblem formulation

Setting of objectives and overall project plan

Model Conceptualization Data Collection

Model translation

Experimental design

Production runs and analysis

More runs?

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Model translation

Verified?

Validated?

Documentationand reporting

Implementation

No

Yes

NoNo

Yes

Steps in aSteps in aSimulation Study Simulation Study (cont’)(cont’)

P bl f l tiProblem formulationIf the statement is provided by the policy makers, or those thathave the problem, the analyst must ensure that the problem beingdescribed is clearly understood. If a problem statement is beingdeveloped by the analyst, it is important that the policy makersunderstand and agree with the formulation.

Setting of objectives and overall project plan

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The objectives indicate the questions to be answered by simulation.The overall project plan should include a statement of thealternative systems to be considered, and a method for evaluatingthe effectiveness of these alternatives.



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Steps in aSteps in aSimulation StudySimulation Study(cont’)(cont’)

Model conceptualizationModel conceptualizationThis is another important and difficult subject. The basic steps areto consider all the related factors first, then evaluate each one(keep or ignore) and reach the final model.

Data collectionThe more data you have the more complete information youhave the more precise model you can build the bettersolution you would get

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solution you would get.Model translation

Program the model into a computer language. Simulationlanguages are powerful and flexible. In most cases, somecomputer software packages are involved. The model developmenttime is greatly reduce. Furthermore, software packages haveadded features that enhance their flexibility.


Verified?Verified?Verification pertains to the computer program prepared for thesimulation model. Is the computer program performing properly?If the input parameters and logical structure or the model arecorrectly represented in the computer, verification has beencomplete.

Validated?Validation is the determination that a model is an accurate

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Validation is the determination that a model is an accuraterepresentation of the real system. Validation is usually achievedthrough the calibration of the model, an iterative process ofcomparing the model to actual system behaviour and using thediscrepancies between the two, and the insights gained, to improvethe model.



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Experimental designExperimental designThe alternatives that are to be simulated must be determined. Foreach system design that is simulated, decisions need to be madeconcerning the length of the initialization period, the length ofsimulation runs, and the number of replications to be made of eachrun.

Production runs and analysisProduction runs and their subsequent analysis are used to

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Production runs, and their subsequent analysis, are used toestimate measures of performance for the system designs that arebeing simulated.

More runs?The analyst determines of additional runs are needed and whatdesign those additional experiments should follow.


Documentation and reportingDocumentation and reportingProgram documentation:

If the program is going to be used again by the same ordifferent analysts, it may be necessary to understandhow the program operates.The model users can change parameters at will in aneffort to determine the relationships between inputparameters and output measures of performance, or to

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p p p ,determine the input parameters that “optimize” someoutput measure of performance.

Progress report:It provides the important written history of a simulationproject.



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I l t tiImplementationThe success of the implementation phase depends on how well theprevious eleven steps have been performed.

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Concepts of DiscreteConcepts of Discrete--Event Event Si l tiSi l tiSimulationSimulation


Discrete Event ModelDiscrete Event Model

In the discrete approach to system simulation stateIn the discrete approach to system simulation, statechanges in the physical system are represented by aseries of discrete changes or events at specificinstants of time and such models are known asdiscrete event models.The time and state are the two important coordinatesused in describing simulation models.

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Between events, the states of the entities remainconstant.The change in state is brought about by events whichfrom the driving force behind every discrete eventsimulation model.



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System TerminologySystem Terminology

S tSystem:A collection of entities (e.g., people and machines) that interacttogether over time to accomplish one or more goals.

Model:An abstract representation of a system, usually containingstructural logical, or mathematical relationships which describe asystem in terms of state, entities and their attributes, sets,

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system in te ms of state, entities and thei att ibutes, sets,processes, events, activities, and delays.

System state:A collection of variables that contain all the information necessaryto describe the system at any time.

System TerminologySystem Terminology(cont’)(cont’)

E titiEntities:Any object or component in the system which requires explicitrepresentation in the model (e.g., a server, a customer, a machine).

Attributes:The properties of a given entity (e.g., the priority of a waitingcustomer, the routing of a job through a job shop).

List (Set Queue):

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List (Set, Queue):A collection of (permanently or temporarily) associated entities,ordered in some logical fashion (such as all customers currently ina waiting line, ordered by first come, first served, or by priority).



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E tEvent:An instantaneous occurrence that changes the state of a system(such as an arrival of a new customer).

Event notice:A record of an event to occur at the current or some future time,along with any associated data necessary to execute the event; at aminimum, the record includes the event type and the event time.

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minimum, the eco d includes the event type and the event time.

Event list:A list of event notices for future events, ordered by time ofoccurrence; also known as the future event list (FEL).


A ti itActivity:A duration of time of specified length (e.g., a service time or inter-arrival time), which is known when it begins (although it may bedefined in terms of a statistical distribution).

Delay:A duration of time of unspecified indefinite length, which is notknown until it ends (e.g., a customer’s delay in a last-in, first-out

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known until it ends (e.g., a custome s delay in a last in, fi st outwaiting line which, when it begins, depends on future arrivals).

Clock:A variable representing simulated time.



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System TerminologySystem Terminology(Example 2.1)(Example 2.1)

S t t tSystem state:LQ(t), the number of customer waiting to be served at time tLC(t), 0 or 1 indicate counter being idle or busy at time t

EntitiesNeither the customers nor the servers need to be explicitlyrepresented, unless certain customer averages are desired

E t

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Events:Arrival eventService completion

System TerminologySystem Terminology(Example 2.1)(Example 2.1)

A ti itiActivities:Interarrival timeService timeUnconditional wait

Delay:A customer’s wait in queue until counter becomes free

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Conditional wait



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Main ApproachesMain Approaches

Event scheduling approachEvent-scheduling approachconcentrate on the events and how they affect the system state.The simulation evolves over time by executing events in increasingorder of their times of occurrence.Examples: FORTRAN, GASP IV, C++

Process-interaction approachconcentrate on a single entity (e.g. a customer) and the sequenceof events and activities it undergoes as it PASSES THROUGH

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of events and activities it undergoes as it PASSES THROUGHTHE SYSTEM. At any given time, the system may contain manyprocesses (e.g. customers) interacting with each other whilecompeting for a set of resources.Example: GPSS

Main ApproachesMain Approaches(cont’)(cont’)

I t ti d i i t d hInteractive, menu-driven, animated approachRecently available on PCs.Examples: PROMODEL, SIGMA

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Future Event ListFuture Event List(FEL)(FEL)

B th th t h d li d thBoth the event-scheduling and the process-interaction approaches use a variable time advance;that is, when all events and system state changeshave occurred at one instant of simulated time, thesimulation clock is advanced to the time of the nextimminent event on the FEL.

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EventEvent--SchedulingSchedulingApproachApproach

Thi li t t i ll t ti f t th tThis list contains all event notices for events thathave been scheduled to occur at a future time.Scheduling a future event means that at the instantan activity begins, its duration is computed or drawnas a sample from a statistical distribution and theend-activity event, together with its event time, isl d th f t t li t

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placed on the future event list.In the real world, most future events are notscheduled but merely happen – such as randombreakdowns or random arrivals.



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EventEvent--SchedulingSchedulingApproachApproach(Example)(Example)

List of all activities’ SCHEDULED TIME OFList of all activities’ SCHEDULED TIME OFCOMPLETION (EVENTS)

The FEL is a SET ordered in completion timest1 < t2 < … < tn

ExampleConsider a single server queue with the following arrival times for thefirst 10 customers:

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f0 4 8 10 13 14 17 20 27 29

and service times for these customers5 5 1 3 2 1 4 7 3 1

Assume that completions are given priority over arrivals


Time = 0FEL4 ARRIVAL5 COMPLETION..

QUEUE0 φ

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.

.



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Time = 4FEL5 COMPLETION8 ARRIVAL10 COMPLETION13 ARRIVAL

QUEUE4 #1

arrival time customer

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14 ARRIVAL17 ARRIVAL20 ARRIVAL27 ARRIVAL29 ARRIVAL

ProcessProcess--InteractionInteractionApproachApproach

A i th lif l f titA process is the life cycle of one entity.This life cycle consists of various events andactivities.Some activities may require the use of one or moreresources whose capacities are limited.These and other constraints cause processes to

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pinteract, the simplest example being an entity forcedto wait in a queue (on a list) because the resource itneeds is busy with another entity.



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ProcessProcess--InteractionInteractionApproach (cont’)Approach (cont’)

I i t i tiIn more precise terms, a process is a time-sequenced list of events, activities, and delays,including demands for resources, that define the lifecycle of one entity as it moves through a system.We see the interaction between two customerprocesses as customer n + 1 is delayed until the

i t ’ “ d i t”

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previous customer’s “end-service event” occurs.

ProcessProcess--InteractionInteractionApproachApproach(Example)(Example)

Customer n

TimeTime

Arrivalevent

Beginservice

End-service-event

Interaction

ActivityDelay

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TimeTime

Arrivalevent

Beginservice

End-service-eventActivityDelay

Customer n + 1



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Mathematical and Statistical Mathematical and Statistical M d l i Si l tiM d l i Si l tiModels in SimulationModels in Simulation


Queueing ModelsQueueing Models

Si l ti i ft d i th l i f iSimulation is often used in the analysis of queueingmodels.Typical measures of system performance includeserver utilization (percentage of time server is busy),length of waiting lines, and delays of customers.Decision maker is involved in trade-offs between

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server utilization and customer satisfaction in termsof line lengths and delays.

Calling population Waiting line Server



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I i l h l i t th l

Flow DiagramFlow Diagram

In a single-channel queueing system there are onlytwo possible events that can affect the state of thesystem.

The entry of a unit into the system or the completion of service on aunit

The server has only two possible states:it i ith b idl

3

it is either busy or idleDeparture event

Being serveridle time

Remove the waiting unitfrom the queue

Being servicing the unit

Another unitwaiting ?

NO YES

Example 2.1Example 2.1

Si l h l t fi t iSingle-channel queue serves customers on a first-in,first-out (FIFO) basis

Customer Number

Arrival Time (Clock)

Time Service Begins (Clock)

Service Time (Duration)

Time Service Ends (Clock)

1 0 0 2 2

2 2 2 1 3

Table 2.4. Simulation Table Emphasizing Clock Times

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2 2 2 1 3

3 6 6 3 9

4 7 9 2 11

5 9 11 1 12

6 15 15 4 19



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Chronological OrderChronological Order

Th f th t t f tThe occurrence of the two types of events

Event Type Customer Number Clock Time

Arrival 1 0

Departure 1 2

Arrival 2 2

Departure 2 3

Arrival 3 6

Table 2.5. Chronological Ordering of Events

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Arrival 4 7

Departure 3 9

Arrival 5 9

Departure 4 11

Departure 5 12

Arrival 6 15

Departure 6 19

Chronological Ordering Chronological Ordering (cont’)(cont’)

N b i t t ti tNumber in system at time t

of c

usto

mer

s in

the

syst

em

1

2

4 5

6

Num

ber o

0 4 8 12 16 20

1 2 3 4 5 6



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TerminologyTerminology

λ i l t ( b f lli it it f ti )λ mean arrival rate (number of calling units per unit of time)μ mean service rate of one server (number of calling units served

per unit of time)1/ μ mean service time for a calling units number of parallel service facilities in the systemLq mean length of the queueL mean number in the system (those in queue + being served)

7

L mean number in the system (those in queue + being served)Wq mean time spent waiting in the queueW mean time spent in the system (Wq + 1/ μ)ρ server utilization factor

Statistical ModelsStatistical Modelsin Simulationin Simulation

Di t Di t ib ti P i (λ)Discrete Distribution – Poisson (λ)estimate “number of arrivals per unit time”

whereP(x) = the probability of X successes given a knowledge of λ

!)(

xexP

xλλ−=

8

λ = expected number of successese = mathematical constant approximated by 2.71828x = number of successes per unit

λ=)(xE λ=)(xV



May 02, 2009 - Page (37)

5

Poisson DistributionPoisson Distribution

D f N(t) i P i ifDef: N(t) is a Possion process ifArrivals occurs individually (at rate λ)N(t) has stationary increments: The distribution of the numbers ofarrivals between t and t+s depends on the length of the interval sand not on the starting point t.N(t) has independent increments: The numbers of arrivals duringnonoverlapping time intervals (t, t+s) and (t’, t’+s’) are

9

independent random variables.

Uniform DistributionUniform Distribution

C ti Di t ib ti U if di t ib tiContinuous Distribution – Uniform distributionA random variable x is uniformly distributed on the interval (a, b):

abxf

−=

1)( bxa ≤≤

)( bU

10

),(~ baUx

2)( baxE +=

12)()(

2abxV −=



May 02, 2009 - Page (38)

6

Uniform Distribution Uniform Distribution (cont’)(cont’)

Th if di t ib ti l it l l iThe uniform distribution plays a vital role insimulation. Random numbers, uniformly distributionbetween zero to 1, provide the means to generaterandom events.

11

Exponential Exponential DistributionDistribution

C ti Di t ib ti E ti l di t ib ti hContinuous Distribution – Exponential distribution hasbeen used to model interarrival times when arrivalsare completely random and to model service timeswhich are highly variable.A random variable x is exponentially distributed with parameter λ>0:

12

xexf λλ −=)( 0≥x

λ1)( =xE 2

1)(λ

=xV



May 02, 2009 - Page (39)

7

MemorylessMemoryless

M lMemoryless

)()()|(

sXPsXandtsXPsXtsXP

>>+>

=>+>

)()(

sXPtsXP

>+>

=

13

ts

ts

ee

e λλ

λ−

−

+−

==)(

)( tXP >=

Example of Example of MemorylessMemoryless

S th t th lif f i d t i l l iSuppose that the life of an industrial lamp, inthousands of hours, is exponentially distributed withfailure rate λ=1/3 (one failure every 3000 hours, onthe average). Find the probability that the industriallamp will last for another 1000 hours, given that it isoperating after 2500 hours.

14

)1()5.2|5.3( >=>> XPXXP3/1−= e

717.0=



May 02, 2009 - Page (40)

1


Properties of Random NumbersProperties of Random NumbersBy : Arya WirabhuanaBy : Arya Wirabhuana

Random NumberRandom NumberGenerationGeneration

A i l ti f t i hi h thA simulation of any system or process in which thereare inherently random components requires amethod of generating or obtaining numbers that arerandom, in some sense.The earliest methods were carried out by hands suchas throwing dice.

2

As computers (and simulation) became more widelyused, increasing attention was paid to methods ofrandom number generation compatible with thecomputers work.



May 02, 2009 - Page (41)

2

Random NumberRandom NumberGeneration (cont’)Generation (cont’)

Th f th h i th 1940’ d 1950’Therefore, the research in the 1940’s and 1950’sturned to numerical or arithmetic ways to generate“random” numbers.These method are sequential, with each new numberbeing determined by one or several of itspredecessors according to a fixed mathematicalf l

3

formula.The first such arithmetic generate generator,proposed by von Neumann and Metropolic in the1940’s is the famous midsquare method.

Midsquare MethodMidsquare Method

A l f id th d f tiAn example of midsquare method for generating auniform [0, 1] random numbers:

i Zi Ui Zi2

0 7182 - 51581124

1 5811 0.5811 33767721

2 7677 0.7677 58936329

3 9363 0.9363 87665769

4

4 6657 0.6657 44315649

5 3156 0.3156 09960336

. . . .

. . . .

. . . .



May 02, 2009 - Page (42)

3

Midsquare MethodMidsquare Method(cont’)(cont’)

Drawbacks for midsquare methodDrawbacks for midsquare methodIf Zi = 0 for some i, then Uj = Zj = 0 for all j > i.Ui+1 is determined by Ui, i.e. Ui+1 is a function of Ui. Therefore,Ui and Ui+1 are not independent.

Nowadays, the random numbers generated fromcomputers are more complicated and appear to beindependent, in that they pass a series of statisticalt t

5

test.But after all, the random numbers generated fromcomputers are still not purely random(pseudorandom, but too awkward to use this term).

Good Random NumberGood Random NumberGenerationGeneration

A “good” arithmetic random number generator shouldA “good” arithmetic random number generator shouldposses several properties:

Above all, the numbers produced should appear to be distributeduniformly on [0, 1] and should not exhibit any correlation witheach other; otherwise, the simulation’s results may be completelyinvalid.From a practical standpoint, we would naturally like the generatorto be fast and avoid the need for a lot of storage.

6

f f f gWe would like to be able to reproduce a given stream of randomnumbers exactly, for at least two reasons. First, this cansometimes make debugging or verification f the computer programeasier. More important, we might want to use identical randomnumbers in simulating different systems in order to obtain a moreprecise comparison.



May 02, 2009 - Page (43)

4

Good Random NumberGood Random NumberGeneration (cont’)Generation (cont’)

Th h ld b i i i th t f d i lThere should be provision in the generator for producing severalseparate “stream” of random numbers. As we shall see, a streamis simply a subsegment of the numbers produced by the generator,with one stream beginning where the previous stream ends.

7

Methods for Methods for GenerationGeneration

I T f T h i (ITT)Inverse Transform Technique (ITT)Exponential distributionUniform distributionTriangular distributionEmpirical discrete distributionEmpirical continuous distribution

A t R j ti T h i

8

Acceptance-Rejection Technique



May 02, 2009 - Page (44)

5

Inverse TransformInverse TransformTechniqueTechnique

Th i t f t h i b d tThe inverse transform technique can be used tosample from the exponential, the Weibull and theuniform distributions, and empirical distribution.Additionally, it is the underlying principle for samplingfrom a wide variety of discrete distributions.A step by step procedure for the inverse transform

9

techniques, illustrated by the exponential distribution,is as follows:Step 1 Compute the cdf of the desired random variable.

For the exponential distribution, the cdf isxexF λ−−=1)( 0≥x

Inverse TransformInverse TransformTechnique (cont’)Technique (cont’)

St 2 S t F( ) R th f XStep 2 Set F(x) = R on the range of X.For the exponential distribution, it becomes

on the range

Since X is a random variable (with the exponentialdistribution in this case), it follows thatis also a random variable, here called R. As will be shown

Re x =− −λ1 0≥x

Re x =− −λ1

10

,later, R has a uniform distribution over the interval (0, 1).



May 02, 2009 - Page (45)

6


St 3 S l th ti F(X) R f X i t f R F thStep 3 Solve the equation F(X) = R for X in terms of R. For theexponential distribution, the solution proceeds as follows:

1)1ln(

1

1

RXRe

ReX

X

−−=−

−=

=−−

−

λ

λ

λ

11

)1ln(1 RX −=λ


Step 4 Generate (as needed) uniform random number R1 R2 R3Step 4 Generate (as needed) uniform random number R1, R2, R3,…,and compute the desired random variable by

where

One simplification that is usually employed is to replaceby to yield

)(1ii RFX −=

)1ln(1)(1ii RRF −

−=−

λ

R1 R

12

by to yield

which is justified since both and are uniformlydistributed on (0, 1).

iR−1 iR

ii RX ln1λ−

=

iR−1iR



May 02, 2009 - Page (46)

7

Uniform DistributionUniform Distribution

C id d i bl X th t i if l di t ib t d thConsider a random variable X that is uniformly distributed on theinterval [a, b].The pdf of X is given by

Step 1 The cdf is given by

⎪⎩

⎪⎨⎧ ≤≤

−=otherwise,0

,1)( bxa

abxf

0 ba

)(1

ab −

13

p f g y

Step 2 Set F(X) = (X – a) / (b – a) = RStep 3 Solving for X in terms of R yields X = a + (b – a) R

⎪⎩

⎪⎨

⎧

>

≤≤−−

<

=

bx

bxaabax

ax

xF

,1

,

,0

)(

Triangular DistributionTriangular Distribution

C id d i bl X hi h h dfConsider a random variable X which has pdf

This distribution is called a triangular distribution withendpoints (0, 2) and mode at 1.Step 1 The cdf is given by

⎪⎩

⎪⎨

⎧≤<−≤≤

=otherwise,0

21,210,

)( xxxx

xf

14

p f g y

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>

≤<−

−

≤<

≤

=

2,1

21,2

)2(1

10,2

0,0

)( 2

2

x

xx

xxx

xF

1 20

1

f(x)



May 02, 2009 - Page (47)

8

Empirical DiscreteEmpirical DiscreteDistributionsDistributions

Service Time Probability Cumulative Probability Random Number

1 0.10 0.10 0.01 – 0.10

2 0.20 0.30 0.11 – 0.30

3 0.30 0.60 0.31 – 0.60

4 0.25 0.85 0.61 – 0.85

5 0.10 0.95 0.86- 0.95

6 0.05 1.00 0.96 – 000

p(x)

15

0 1

1

12

Draw a samplefrom a 0-1 uniform

distribution

Convert the 0-1 sample to an equivalent sample from the

target population

0 1 2 3 4

0.3

0.2

0.1

5 6

Empirical ContinuousEmpirical ContinuousDistributionDistribution

If th d l h b bl t fi d th ti lIf the modeler has been unable to find a theoreticaldistribution that provides a good model for the inputdata, then it may be necessary to use the empiricaldistribution of the data.

Suppose that 100 broken-widget repair times have been collected.The data are summarized in the following table in terms of thenumber of observations in various interval. For example, there

16

f p ,were 31 observations between 0 and 0.5 hour, 10 between 0.5 and1 hour, and so on.

Interval (Hours) Frequency Relative Frequency Cumulative Frequency

0.0 ≤ x ≤ 0.5 31 0.31 0.31

0.5 < x ≤ 1.0 10 0.10 0.41

1.0 < x ≤ 1.5 25 0.25 0.66

1.5 < x ≤ 2.0 24 0.24 1.00



May 02, 2009 - Page (48)

9

Empirical ContinuousEmpirical ContinuousDistribution (cont’)Distribution (cont’)

F(x)

(1.5, 0.66)

(2.0, 1.0)1.0

0.8

0.6

R1=0.83

F(x)

e pr

obab

ility

)( 11

1 =− RFX

17

(1.0, 0.41)

(0.5, 0.31)

0 0.5 1.0 1.5 2.0X1=1.7

5

0.4

0.2

xRepair times

Cum

ulat

ive

( )

75.1

5.10.266.000.1

66.05.1

)(

11

11

=

−⎥⎦

⎤⎢⎣

⎡−−

+=

=

RX

RFX

AcceptanceAcceptance--Rejection Rejection TechniqueTechnique

S th t d t d i th d fSuppose that we need to devise a method forgenerating random variates, X, uniformly distributedbetween ¼ and 1.

Step 1: Generate a random number u ~ U (0, 1)Step 2a: If , accept X = u, then go to step 3.Step 2b: If , reject u, and return to step 1.Step 3: If another uniform random variate on [1/4 1] is needed

25.0≥u25.0<u

18

Step 3: If another uniform random variate on [1/4, 1] is needed,repeat the procedure beginning at step 1. If not, stop.



May 02, 2009 - Page (49)

10

Tests for Random Tests for Random NumbersNumbers

Th d i bl ti f d bThe desirable properties of random numbersuniformity and independence

To insure that these desirable properties areachieved, a number of tests can be performed.The tests can be placed in two categories accordingto the properties of interest.

19

Test for uniformityTest for independence

Tests for Random Tests for Random Numbers (cont’)Numbers (cont’)

F t t U th hi t t tFrequency test: Uses the chi-square test tocompare the distribution of the set of numbersgenerated to a uniform distribution.Runs test: Tests the runs up and down or the runsabove and below the mean by comparing the actualvalues to expected values. The statistic for

i i th hi

20

comparison is the chi-square.



May 02, 2009 - Page (50)

11

Frequency TestFrequency Test

FREQUENCY TESTFREQUENCY TESTRandom numbers about from the uniform distributionand several tests have been developed to test for thiscondition. We will consider the χ2 goodness-of-fittest.The goodness-of-fit test requires that:

21

50 observations in totalExpected frequency of at least five in each classThe following table shows the results of placing a total of 100observations in 10 evenly spaced classes

Frequency Test (cont’)Frequency Test (cont’)

F T tFrequency TestClasses Observed Frequency Expected Frequency (fo – fe)2 / fe

0.00 – 0.10 9 10 0.10

0.10 – 0.20 12 10 0.40

0.20 – 0.30 10 10 0.00

0.30 – 0.40 11 10 0.10

0.40 – 0.50 8 10 0.40

0.50 – 0.60 10 10 0.00

22

0.60 – 0.70 10 10 0.00

0.70 – 0.80 7 10 0.90

0.80 – 0.90 12 10 0.40

0.90 – 1.00 11 10 0.10

100 100 2.40



May 02, 2009 - Page (51)

12

Frequency Test (cont’)Frequency Test (cont’)

The question isThe question is,Do these numbers come from the uniform distribution?Calculating the χ2 statistic from the data using the equation

Gives a value of χ2 = 2.40. In testing the null hypothesis that therandom numbers come from the uniform distribution,

H R U [0 1]

∑ −=

e

eo

fff 2

2 )(χ

23

H0 : Ri ~ U [0, 1]one compares the calculated χ2 to the value obtained from thetable based on (10-1) = 9 degree of freedom and a α = 0.05.This χ2 value is found to be 16.919, which is larger than thecalculated χ2 value.Therefore, we accepted the null hypothesis, and find our randomnumber generation acceptable.

Run Up and Down TestRun Up and Down Test

RUNS UP AND DOWN TESTRUNS UP AND DOWN TESTNumbers can pass a uniformity test and still not berandom.

For example, the numbers 0.00, 0.10, 0.20, 0.30, 0.40, …obviouslyare not random.

The numbers also must be sequentially random to bej d d t l d

24

judged truly random.A variety of runs test can be used for this purpose.We will consider a run up and down test.



May 02, 2009 - Page (52)

13

Run Up and Down Test Run Up and Down Test (cont’)(cont’)

I f b if b i f ll d bIn a sequence of numbers, if a number is followed bya larger number, this is an upward run.Likewise, a number followed by a smaller number is adownstream run.If the numbers are truly random, one would expect tofind a certain numbers of runs up and down.

25

In a sequence of N numbers, one should expect tofind runs equal to the following equation:

902916

312 2 −

=−

=NN δμ


A l th t th f ll i 40As an example, assume that the following 40numbers have been generated.

0.43, 0.32, 0.48, 0.23, 0.90, 0.72, 0.94, 0.11, 0.14, 0.67,

0.61, 0.25, 0.45, 0.56, 0.87, 0.54, 0.01, 0.64, 0.65, 0.32, 0.03,– + – + – + – +

– – –+ +

26

0.93, 0.08, 0.58, 0.41, 0.32, 0.03, 0.18, 0.90, 0.74, 0.32,

0.75, 0.42, 0.71, 0.66, 0.03, 0.44, 0.99, 0.40, 0.51

– – –

– – –

+ + +

+ + + +



May 02, 2009 - Page (53)

14


One should expect to find 26 33 runsOne should expect to find 26.33 runsThere were 26 runs in the sequence of numbers.

612

79.690

294016

33.263

1402

2

=

=−×

=

=−×

=

δ

δ

μ

96131033.262696.133.26:33.26:

025.0

1

0

−>−=−

=−

=

±=≠=

μ

μμ

XZ

ZHH

27

We consider to accept the generated numbers arerandom.

61.2=δ96.131.061.2

−>−===δ

Z

Table of Random DigitsTable of Random Digits

10097 32533 76520 13586 34673 54876 80959 09117 39292 7494510097 32533 76520 13586 34673 54876 80959 09117 39292 74945

37542 04805 64894 74296 24805 24037 20636 10402 00822 91655

08422 68953 19645 09303 23209 02560 15953 34764 35080 33606

99019 02529 09376 70715 38311 31165 88676 74397 04436 27659

12807 99970 80157 36147 64032 36653 98951 16877 12171 76833

66065 74717 34072 76850 36697 36170 65813 39885 11190 29170

31060 10805 45571 82406 35303 42614 86799 07439 23403 09732

85269 77602 02051 65692 68665 74818 73053 85247 18623 88579

28

63573 32135 05325 47048 90553 57548 28468 28709 83491 25624

73796 45753 03529 64778 35808 34282 60935 20344 35273 88435

98520 17767 14905 68607 22109 40558 60970 93433 50500 73998

11805 05431 39808 27732 50725 68248 29405 24201 52775 67851

…



May 02, 2009 - Page (54)

1


Data Collection and Parameter Data Collection and Parameter E ti tiE ti tiEstimationEstimation


Input ModelingInput Modeling

I l ld i l ti li ti d t i iIn real-world simulation applications, determiningappropriate distributions for input data is a major taskfrom the standpoint of time and resourcerequirements.Faulty models of the inputs will lead to outputs whoseinterpretation may give rise to misleading

d ti

2

recommendations.Steps to develop a useful model for input data

Collect data from the real system of interestIdentify a probability distribution to represent the input process



May 02, 2009 - Page (55)

2

Input ModelingInput Modeling(cont’)(cont’)

Ch t th t d t i ifi i t f thChoose parameters that determine a specific instance of thedistribution familyEvaluate the chosen distribution and the associated parameters forgoodness-of-fit

3

Data CollectionData Collection

Pl d t ll tiPlan your data collection processAlways try to find ways that can help you collect dataefficiently and accurately (equipment, barcoding,receipts, personnel, video, etc)Collect only data that is useful for your project

4



May 02, 2009 - Page (56)

3

IdentifyingIdentifyingthe Distributionthe Distribution

HISTOGRAMSHISTOGRAMSDivide the range of the data into intervalsLabel the horizontal axis to conform to the intervalsselectedDetermine the frequency of occurrences within eachinterval

5

Label the vertical axis so that the total occurrencescan be plotted for each intervalPlot the frequencies on the vertical axis

IdentifyingIdentifyingthe Distributionthe Distribution(cont’)(cont’)

SELECTING THE FAMILY OF DISTRIBUTOINSSELECTING THE FAMILY OF DISTRIBUTOINSRecall if the histogram drawn from your resemblesany kind of statistical distributionUse physical basis (e.g. usage, discrete orcontinuous) of the distribution as a guideUse software

6

The exponential, normal, and Poisson distributionsare frequently encountered and are not difficult toanalyze from a computational standpoint



May 02, 2009 - Page (57)

4


QUANTILE QUANTILE PLOTSQUANTILE-QUANTILE PLOTSEvaluate the fit of the chosen distribution(s)Compare the actual values with the values derivedfrom the chosen distributionThe nearer to become a straight line, the better theaccuracy

7

y99.79 99.56 100.17 100.33

100.26 100.41 99.98 99.83100.23 100.27 100.02 100.4799.55 99.62 99.65 99.8299.96 99.90 100.06 99.85


Observed Valueq-q plot

99 40

99.60

99.80

100.00

100.20

100.40

100.60

100.80

Estim

ated

Observed Value

j Value j Value j Value j Value

1 99.55 6 99.82 11 99.98 16 100.26

2 99.56 7 99.83 12 100.02 17 100.27

3 99.62 8 99.85 13 100.06 18 100.33

4 99.65 9 99.90 14 100.17 19 100.41

5 99.79 10 99.96 15 100.23 20 100.47

8

99.20

99.40

99.40 99.60 99.80 100.00 100.20 100.40 100.60

Observed

⎟⎠⎞

⎜⎝⎛ −−

20211 jF

Estimated Value

j Value j Value j Value j Value

1 99.43 6 99.82 11 100.01 16 100.20

2 99.58 7 99.86 12 100.04 17 100.25

3 99.66 8 99.90 13 100.08 18 100.32

4 99.73 9 99.94 14 100.12 19 100.40

5 99.78 10 99.97 15 100.16 20 100.55



May 02, 2009 - Page (58)

5

Parameter EstimationParameter Estimation

S l M d S l V iSample Mean and Sample VarianceCalculate sample mean ( ) and variance ( ) fromthe collected dataBased on the distribution chosen, convert theparameters from the sample mean and variancewhich is (are) used for the distribution

X 2S

9

Distribution Parameter(s) Suggested Estimator(s)Poisson αExponential λNormal μ, σ 2

X=αX/1ˆ =λ

22ˆ

ˆ

S

X

=

=

σ

μ

GoodnessGoodness--ofof--Fit TestsFit Tests

P id h l f l ( tit ti ) id fProvides helpful (quantitative) guidance forevaluating the suitability of a potential input modelUsed in large samples size dataUse tables to determine accept or reject

10



May 02, 2009 - Page (59)

6

GoodnessGoodness--ofof--Fit TestsFit Tests(cont’)(cont’)

Chi S T tChi-Square TestThis test is applied to for testing the hypothesis that a randomsample of size n of the random variable X follows a specificdistributional formThe test is valid for large sample sizes, for both discrete andcontinuous distributional assumptions

∑ −k EO 22 )(

11

Oi is the observed frequency in the ith class intervalEi is the expected frequency in that class interval

∑=

=i i

ii

EEO

1

20

)(χ


Example 9 13 (Poisson Assumption)Example 9.13 (Poisson Assumption)H0 : the random variable is Poisson distributedH1 : the random variable is not Poisson distributed

For α = 3.64, the probabilities associated with various values of x:

⎪⎩

⎪⎨⎧

==−

otherwise,0

,...2,1,!)( x

xe

xpxαα

12

It is significantly to reject H0 at the 0.05 level of significance.

P(0) = 0.026 P(4) = 0.192 P(8) = 0.020

P(1) = 0.096 P(5) = 0.140 P(9) = 0.008

P(2) = 0.174 P(6) = 0.085 P(10) = 0.003

P(3) = 0.211 P(7) = 0.044 P(11) = 0.001

2.6 19.2 2.0

9.6 14.0 0.8

17.4 8.5 0.3

21.1 4.4 0.1

E(x)=np

68.271.112117,05.0 <=−−χ



May 02, 2009 - Page (60)

7


xi Observed frequency, Oi Expected Frequency, Ei

0 12 2.6

1 10 9.6

2 19 17.4 0.15

3 17 21.1 0.80

4 10 19.2 4.41

5 8 14.0 2.57

i

ii

EEO 2)( −

22 12.2 7.87

13

6 7 8.5 0.26

7 5 4.4

8 5 2.0

9 3 0.8

10 3 0.3

11 1 0.1

100 100.0 27.68

17 7.6 11.62


E l 9 14 (E ti l A ti )Example 9.14 (Exponential Assumption)H0 : the random variable is Exponential distributedH1 : the random variable is not Exponential distributed

Let k = 8, then each interval will have probability p = 0.125

( ) iai eaF λ−−=1

084.0/1ˆ == Xλ

1

14

iaeip λ−−=1

590.1)125.01ln(084.01

1 =−−=a

)1ln(1 ipai −−=⇒λ

,677.11,252.8,595.5,425.3,1590.0 54321 ===== aaaaa755.24,503.16 76 == aa



May 02, 2009 - Page (61)

8


Class Interval Observed frequency, Oi

Percentage Factor Expected Frequency, Ei

[0, 1.590) 19 P(X≤0.159) – P(X ≤0) = 0.125 6.25 26.01

[1.590, 3.425) 10 P(X≤3.425) – P(X ≤1.590) = 0.125 6.25 2.25

[3.425, 5.595) 3 P(X≤5.595) – P(X ≤3.425) = 0.125 6.25 0.81

[5.595, 8.252) 6 P(X≤8.252) – P(X ≤5.595) = 0.125 6.25 0.01

[8.252, 11.677) 1 P(X≤11.677) – P(X ≤8.252) = 0.125 6.25 4.41

[11.677, 16.503) 1 P(X≤16.503) – P(X ≤11.677) = 0.125 6.25 4.41

i

ii

EEO 2)( −

xexXP λ−−=≤ 1)(

15

It is significantly to reject H0 at the 0.05 level of significance.

[11.677, 16.503) 1 P(X≤16.503) P(X ≤11.677) 0.125 6.25 4.41

[16.503, 24.755) 4 P(X≤24.755) – P(X ≤16.503) = 0.125 6.25 0.81

[24.755, ∞) 6 P(X≤ ∞) – P(X ≤24.755) = 0.125 6.25 0.01

50 1.000 50 39.6

6.126.39 2118,05.0

20 =>= −−χχ

Selecting Input ModelsSelecting Input Modelswithout Datawithout Data

Engineering dataEngineering dataA product or process has performance ratings provided by themanufacturer (for example, a laser printer fan produce 4pages/minute)

Expert optionTalk to people who are experienced with the process or similarprocesses.

Ph i l ti l li it ti

16

Physical or conventional limitationsMost real processes have physical limits on performance (forexample, computer data entry cannot be faster than a person cantype)

The nature of the processSelect the family of distribution



May 02, 2009 - Page (62)

1


Model Development and Model Model Development and Model V ifi tiV ifi tiVerificationVerification


Model BuildingModel Building

One of the most important and difficult tasks facing aOne of the most important and difficult tasks facing amodel developer is the verification and validation ofthe simulation model.

To reduce the degree of skeptic about model’s validityTo increase the model’s credibility

Verification is concerned with building the modelright. It is utilized in the comparison of the

2

conceptual model to computer representation thatimplements that conception.

Is the model implemented correctly in the computer?Are the input parameters and logical structure of the modelcorrectly represented?



May 02, 2009 - Page (63)

2

Model BuildingModel Building(cont’)(cont’)

V lid ti i d ith b ildi th i ht d lValidation is concerned with building the right model.It is utilized to determine that a model is an accuraterepresentation of the real system.Validation is usually achieved through the calibrationof the model, an iterative process of comparing themodel to actual system behavior and using thedi i b t th t d th i i ht

3

discrepancies between the two, and the insightsgained, to improve the model.This process is repeated until model accuracy isjudged to be acceptable.


Th fi t t i d l b ildi i t f b iThe first step in model building consists of observingthe real system and the interactions among itsvarious components and collecting data on itsbehavior.

Ask person who are familiar with the system.New questions may arise.Model developers will return to this step of learning true system

4

Model developers will return to this step of learning true systemstructure and behavior.



May 02, 2009 - Page (64)

3


Th d t i d l b ildi i th t tiThe second step in model building is the constructionof a conceptual model.

A collection of assumptions on the components and the structure ofthe system, plus hypotheses on the values of model inputparameters.

The third step is the translation of the operationalmodel into a computer-recognizable form – the

5

model into a computer recognizable form thecomputerized model.

ModelModel--building building ProcessProcess

Real system

Conceptual model1. Assumptions on system components2. Structural assumptions, which define the interactions

between system components

Conceptual validation

CalibrationAndvalidation

6

between system components3. Input parameters and data assumptions

Operational model(Computerizedrepresentation)

Model verification



May 02, 2009 - Page (65)

4

VerificationVerification

Th f d l ifi ti i t th tThe purpose of model verification is to assure thatthe conceptual model is reflected accurately in thecomputerized representation.The conceptual model quite often involves somedegree of abstraction about system operations, orsome amount of simplification of actual operations.

7

Verification asks the question:Is the conceptual model (assumptions on system components andsystem structure, parameter values, abstractions andsimplifications) accurately represented by the operational model?

Three Classes of Three Classes of TechniqueTechnique

C t h iCommon-sense techniquesThorough documentationTraces

8



May 02, 2009 - Page (66)

5

CommonCommon--sense sense TechniquesTechniques

Ch k d b th th it d lChecked by someone other than its developer.Make a flow diagram and follow each event type.Examine the output for reasonableness under avariety of settings of the input parameters.Print the input parameters at the end of simulation toensure that these parameters values have not been

9

pchanged inadvertently.Make the model as self-documenting as possible.

CommonCommon--sense sense Techniques (cont’)Techniques (cont’)

If the operational model is animated verify that whatIf the operational model is animated, verify that whatis seen in the animation imitates the actual system.Use the debugger provided by the simulationsoftware.Use a variety of graphics to represent different modelstates.For example (reasonableness)

10

For example, (reasonableness)Current contents and total count

Current content refers to the number of items ineach component of the system at a given time.Total count refers to the total number of items thathave entered each component of the system by a givetime.



May 02, 2009 - Page (67)

6

OftOft--neglected neglected Documentation Documentation TechniqueTechnique

D t ti i l i t t fDocumentation is also important as a means ofclarifying the logic of a model and verifying itscompleteness.If a model builder writes brief comments in thecomputerized model, plus definitions of all variablesand parameters, and descriptions of each major

ti f th t i d d l it b h

11

section of the computerized model, it becomes muchsimpler for someone else, or the model builder at alater date, to verify the model logic.

Trace TechniqueTrace Technique

A more sophisticated techniqueA more sophisticated technique.A trace is a detailed computer printout which givesthe value of every variable in a computer program,every time that one of these variables changes invalue.The purpose of the trace is to verify the correctnessof the computer program by making detailed paper-

12

p p g y g p pand-paper calculations.Some software allows a selective trace.

Whenever the queue before a certain resource reaches five ormore, turn on the trace.



May 02, 2009 - Page (68)

7

RecommendationsRecommendations

It i d d th t th fi t t l b i dIt is recommended that the first two always be carriedout.Close examination of model output forreasonableness is especially valuable andinformative.A trace can also provide information if it is selective.

13

The generalized trace can be extremely timeconsuming.



May 02, 2009 - Page (69)

1


Model Calibration and Model Model Calibration and Model V lid tiV lid tiValidationValidation


Calibration of ModelsCalibration of Models

Verification and validation although conceptuallyVerification and validation, although conceptuallydistinct, usually are conducted simultaneously by themodeler.Validation is the overall process of comparing themodel and its behavior to the real system and itsbehavior.Calibration is the iterative process of comparing the

2

p p gmodel to the real system, making adjustments (oreven major changes) to the model, comparing therevised model to reality, making additionaladjustments, comparing again, and so on.



May 02, 2009 - Page (70)

2

Comparison of the Comparison of the ModelsModels

The comparison of the model to reality is carried outThe comparison of the model to reality is carried outby a variety of tests

Subjective and ObjectivesSubjective tests usually involve people, who areknowledgeable about one or more aspects of thesystem, making judgments about the model and itsoutput.

3

Objective tests always require data on the system’sbehavior plus the corresponding data produced bythe model. Then one or more statistical tests areperformed to compare some aspect of the systemdata set to the same aspect of the model data set.

Iterative Process of Iterative Process of Calibration a ModelCalibration a Model

Compare model

RealRealsystemsystem

Initial modelInitial model

First revisionFirst revisionof modelof model

Revise

R i

Compare modelto reality

Compare revisedmodel to reality

4

Second revisionSecond revisionof modelof model

Revise

Revise

Compare secondRevision to reality



May 02, 2009 - Page (71)

3

ThreeThree--step Validation step Validation ApproachApproach

B ild d l th t h hi h f liditBuild a model that has high face validityValidate model assumptionsCompare the model input-output transformations tocorresponding input-output transformations for thereal system

5

Face ValidityFace Validity

The first goal of the simulation modelers is toThe first goal of the simulation modelers is toconstruct a model that appears reasonable on itsface to model users and others who areknowledgeable about the real system beingsimulated.The potential users should be involved in modelconstruction. (Conceptual Implementation)

6

Sensitive analysis can be used to check a model’sface validity.

E.g., if the arrival rate of customer were to increase, it would beexpected that …

utilization of servers, lengths of lines, and delays would tend toincrease.



May 02, 2009 - Page (72)

4

Validation of Model Validation of Model AssumptionsAssumptions

Model assumptions fall into two general classes:Model assumptions fall into two general classes:Structural assumptions and data assumptions

Structural assumptions involve questions of how thesystem operates and usually involve simplificationsand abstractions of reality.

E.g., number of tellers may be fixed or variableVerified by actual observation, discussion with managers, etc.

7

Data assumptions should be based on the collectionof reliable data and correct statistical analysis of thedata.

E.g., Interarrival time, service times, etc.Input data analysis

Validating InputValidating Input--Output Output TransformationsTransformations

DecisionDecision

RandomRandomvariablesvariables MODELMODEL

“Black box”“Black box”

OutputOutputvariablesvariables

8

DecisionDecisionvariablesvariables

Input variables Model Output variables



May 02, 2009 - Page (73)

5

InputInput--Output Validation Output Validation –– Artificial Input DataArtificial Input Data

ARTIFICIAL INPUT DATAARTIFICIAL INPUT DATAWhen the model is run using generated randomvariates, it is expected that observed values shouldbe close to collected values.Hypothesis test – average customer delay: (p.380)

( ) minutes34: 20 =YEH 5121== ∑YY

n

9

( )( ) minutes 3.4:

minutes 3.4:

21

20

≠=

YEHYEH

( )( )

82.01

51.2

1

222

2

122

=−

−=

==

∑

∑

=

=

n

YYYS

Yn

Y

n

i ii

ii

nSY

t 020

μ−=

InputInput--Output Validation Output Validation –– Artificial Input DataArtificial Input Data(Example)(Example)

T bl 10 2 ( 383)Table 10.2 (p.383)Replication Y4, Observed

arrival rateY5, Average service

timeY2, Average

Delay1 51 1.07 2.792 40 1.12 1.123 45.5 1.06 2.244 50 5 1 10 3 45

10

4 50.5 1.10 3.455 53 1.09 3.136 49 1.07 2.38

Sample mean 2.51Standard deviation 0.82



May 02, 2009 - Page (74)

6

InputInput--Output Validation Output Validation –– Artificial Input Data Artificial Input Data (cont’)(cont’)

Hypothesis test (cont’):Hypothesis test (cont’):Degree of freedom = n – 1 = 5

Since ,

( )

34.5682.03.451.2

3.4

0

20

−=−

=

==

t

YEμ

571.234.5 5,025.00 =>= tt

11

reject H0 and conclude that the model is inadequatein its prediction of average customer delay at α=0.05.AreAre thesethese assumptionsassumptions metmet inin thethe presentpresent case?case?(p(p..385385))

InputInput--Output Validation Output Validation –– Historical Input DataHistorical Input Data

HISTORICAL INPUT DATAHISTORICAL INPUT DATAAn alternative to generating input data is to use theactual historical record to drive the simulation modeland then to compare model output to system data.

12



May 02, 2009 - Page (75)

7

InputInput--Output Validation Output Validation –– Historical Input DataHistorical Input Data(cont’)(cont’)

Input Data Set

System Output, Zij

Model Output, Wij

Observed Difference, dj

Squared Deviation from Mean,

1 Zi1 Wi1

2 Zi2 Wi2

3 Zi3 Wi3

… … …

( )2dd j −

( )22 dd −

( )23 dd −

111 ii WZd −=

222 ii WZd −=

333 ii WZd −=

( )21 dd −

13

K ZiK WiK ( )2ddK −

∑=

=K

jjd

Kd

1

1 ( )∑=

−−

=K

jjd dd

KS

1

22

11

444 ii WZd −=

InputInput--Output Validation Output Validation –– Historical Input DataHistorical Input Data(Example)(Example)

E l 10 4 ( 392)Example 10.4 (p.392)

0:0:

1

0

≠=

d

d

HH

μμ

37.1585.8705

2.53430 ==

−=

KSd

td

dμ72 10580.7

2.5343

5

×=

=

=

dS

d

K

14

Since ,the null hypothesis cannot be rejected at α = 0.05.

278.237.1 4,025.00 =<= tt



May 02, 2009 - Page (76)

8

I dditi t t ti ti l t t h t ti ti l

InputInput--Output ValidationOutput Validation–– Turing TestTuring Test

In addition to statistical test, or when no statisticaltest is readily applicable, persons knowledgeableabout system behavior can be used to comparemodel output to system output.

SystemSystemff

15

performanceperformance

SimulationSimulationOutputOutput

ReportsReports ?

ConclusionConclusion

Th l f th lid ti i t f ldThe goal of the validation process is twofold:to produce a model that represents true system behavior closelyenough for the model to be used as a substitute for the actualsystem for the purpose of experimenting with the system;to increase to an acceptable level the credibility of the model, sothat the model will be used by managers and other decisionmakers.

16



May 02, 2009 - Page (77)

1


Output Analysis for a Single ModelOutput Analysis for a Single ModelBy : Arya WirabhuanaBy : Arya Wirabhuana

Output Analysis for a Output Analysis for a Single ModelSingle Model

O t t l i i th i ti f d t t dOutput analysis is the examination of data generatedby a simulation.Its purpose is to predict the performance of a systemor to compare the performance of two or morealternative system designs.This lecture deals with the analysis of a single

2

system, while next lecture deals with the comparisonof two or more systems.



May 02, 2009 - Page (78)

2

Type of Simulation with Type of Simulation with respect to Output respect to Output AnalysisAnalysis

Wh l i i l ti t t d t di ti tiWhen analyzing simulation output data, a distinctionis made between terminating or transient simulationand steady-state simulation.A terminating simulation is one that runs for someduration of time TE, where E is a specified event (orset of events) which stops the simulation.

3

Example 11.1: Shady Grove Bank operates 8:30 – 16:30, thenTE = 480min.Example 11.3: A communication system consists of severalcomponents. Consider the system over a period of time, TE , untilthe system fails. E = {A fails, or D fails, or (B and C both fail)}

Terminating SimulationTerminating Simulation

When simulating a terminating system the initialWhen simulating a terminating system, the initialconditions of the system at time 0 must be specified,and the stopping time TE, or alternatively, thestopping event E, must be well defined.Whether a simulation is considered to be terminatingor not depends on both the objectives of thesimulation study and the nature of the system.

4

Examples 11.1 and 11.3 are considered the terminating systemsbecause:

Ex. 11.1: the objective of interest is one day’s operation;Ex. 11.3: short-run behavior, from time 0 until the first systemfailure.



May 02, 2009 - Page (79)

3

SteadySteady--state state SimulationSimulation

A t i ti t i t th tA nonterminating system is a system that runscontinuously, or at least over a very long period of time.

For example, assembly lines which shut down infrequently,continuous production systems of many different types, telephonesystems and other communications systems such as the Internet,hospital emergency rooms, fire departments, etc.

A steady-state simulation is a simulation whose

5

objective is to study long-run, or steady-state,behavior of a nonterminating system.The stopping time, TE, is determined not by the nature ofthe problem but rather by the simulation analyst, eitherarbitrarily or with a certain statistical precision in mind.

Stochastic Nature of Stochastic Nature of Output DataOutput Data

Consider one run of a simulation model over a periodConsider one run of a simulation model over a periodof time [ 0, T ]. Since the model is an input-outputtransformation, and since some of the model inputvariables are random variable, it follows that themodel output variables are random variables.The stochastic (or probability) nature of outputvariables will be observed.

6

Example 2.2 (Able-Baker carhop problem)Input: randomness of arrival time and service timeOutput: randomness of utilization and time spent in the system percustomer.



May 02, 2009 - Page (80)

4

Output Analysis for Output Analysis for Terminating Terminating SimulationsSimulations

C id th ti ti f f tConsider the estimation of a performance parameter,θ (or φ), of a simulated system.The simulation output data is of the form {Y1, Y2, … ,Yn} (discrete-time data) for estimating θ.

E.g. the delay of customer i, total cost in week i.

The simulation output data is of the form {Y(t), 0 ≤ t ≤

7

TE} (continuous-time data) for estimating φ.E.g. the queue length at time t, the number of backlogged orders attime t.

Point Estimation: ∑=

=n

iiY

n 1

1θ dttYT

ET

E∫=

0)(1φ

Output Analysis for Output Analysis for Terminating Terminating Simulations (cont’)Simulations (cont’)

B th C t l Li it d Th (CLT) f 30By the Central Limited Theorem (CLT), for n ≥ 30,

where

Interval Estimation:An approximate 100(1 α)% confidence interval for θ

nSt

/)ˆ(

ˆ

)ˆ(ˆ

ˆ

θθθ

θσθθ −=

−=

1

)ˆ()ˆ( 1

2

2

−

−

=∑=

n

YS

n

ii θ

θ

8

An approximate 100(1 - α)% confidence interval for θis given by:

nSt

nSt nn

)ˆ(ˆ)ˆ(ˆ1,2/1,2/

θθθθθ αα −− +≤≤−



May 02, 2009 - Page (81)

5

Output Analysis for Output Analysis for Terminating Terminating Simulations (Example)Simulations (Example)

E l 11 10 (Abl B k C h P bl )Example 11.10 (Able Baker Carhop Problem)Run, r Utilization, Average System Time,

1 0.808 3.74

2 0.875 4.53

3 0.708 3.84

4 0.842 3.98

rρ rw

8420708087508080 +++

9922.0694.0

)036.0)(18.3(808.0)ˆ(ˆˆ

)036.0()4(3

)808.0842.0()808.0808.0()ˆ(ˆ

808.04

842.0708.0875.0808.0ˆ

3,025.0

222

2

≤≤±

±

=−++−

=

=+++

=

ρ

ρσρ

ρσ

ρ

t

L

Number of ReplicationsNumber of Replications

PRECISION LEVELPRECISION LEVELSuppose that an error criterion ε is specified; in otherwords, it is desired to estimate θ by to withinwith high probability, say at least 1 – α.

θ ε±

1,2/)ˆ(≤− εθ

αR

St R

10

2

2/

2

1,2/)ˆ()ˆ(⎟⎟⎠

⎞⎜⎜⎝

⎛≥⎟

⎟⎠

⎞⎜⎜⎝

⎛≥ − ε

θεθ

ααSzStR

R

R



May 02, 2009 - Page (82)

6

Number of ReplicationsNumber of Replications(Example)(Example)

E l 11 12 (Abl B k C h P bl )Example 11.12 (Able Baker Carhop Problem)Suppose that it is desired to estimate Able’sutilization in Example 11.7 to within withprobability 0.95. An initial sample size R0=4 is taken.

Step 1:

04.0±

1344.12)040(

)00518.0()96.1(2

22

2

20

2025.0 ≈==

Sz

11

Step 2:

)04.0( 22ε

R 13 14 15

t0.025, R-1 2.18 2.16 2.14

15.39 15.10 14.832

20

21,025.0

ε

St R−

R = 15 Additional replications:R – R0 = 15 – 4 = 11

Output Analysis for Output Analysis for SteadySteady--State State SimulationsSimulations

Prior to beginning analysis of output data thePrior to beginning analysis of output data, themodeler must take every effort to ensure that theoutput represents an accurate estimate of the truesystem values.One useful technique for improving the reliability ofoutput results from steady-state simulation is toprovide an initialization period for which statistics are

12

not kept.A steady-state condition implies that a simulation hasreached a point in time where the state of the modelis independent of the initial start-up conditions.



May 02, 2009 - Page (83)

7

Output Analysis for Output Analysis for SteadySteady--State State Simulations (cont’)Simulations (cont’)

Th t f ti i d t hi t d t tThe amount of time required to achieve steady-stateconditions is referred to as a warm-up period.Data collection begins after a warm-up period iscompleted.Determining the length of this period can beaccomplished by utilizing moving averages calculated

13

from the output produced by multiple modelreplications.

=)(wYi

∑−=

+ +w

wssi wY )12/(

∑−

−−=+ −

1

)1(

)12/(i

issi iY

wnwifor −+= ,,1K

wifor ,,1K=

WarmWarm--up Periodup Period

Determine A Warm-up Period in a Steady-state Simulation

Period Average Cost w = 5 w = 10 w = 19

1 422.00 422.00 422.00 422.00

2 468.16 522.20 522.20 522.20

3 676.45 502.72 502.72 502.72

4 572.88 568.92 568.92 568.92

5 374.10 571.26 571.26 571.26

6 842.90 560.94 560.94 560.94

7 625.92 587.72 563.86 563.86

8 4 3 08 8 46 4 3 4 3

14

8 473.08 585.46 574.53 574.53

9 685.88 568.25 569.68 569.68

10 528.79 588.95 578.06 578.06

11 500.22 611.93 565.67 565.67

12 716.52 575.28 568.46 558.81

13 443.33 546.57 569.65 561.05

14 487.13 593.43 564.58 563.82



May 02, 2009 - Page (84)

8

Moving AverageMoving AverageAverage Monthly Cost

1200

Moving Average for w = 5

1200

0200

400600800

1000

1 5 9 13 17 21 25 29 33 370

200

400600800

1000

1 5 9 13 17 21 25 29 33 37

Moving Average for w = 10 Moving Average for w = 19

15

0200

400600800

10001200

1 5 9 13 17 21 25 29 33 370

200

400600800

10001200

1 5 9 13 17 21 25 29 33 37

d = 12

Output Analysis for Output Analysis for SteadySteady--State State Simulations (Example)Simulations (Example)

Observed cost during i-th period and j-th replicationPeriod Rep 1 Rep 2 Rep 3 Rep 4 Rep 5

13 376.81 500.97 192.96 509.00 636.92

14 352.05 329.30 587.45 336.11 530.74

15 518.96 634.81 716.81 533.05 1899.13

16 673.88 853.97 563.86 179.72 864.17

17 376.99 1098.67 290.92 205.43 276.93

18 139.26 339.08 563.49 319.10 189.20

19 199.54 4032.35 355.94 138.88 215.99

20 542 79 908 48 633 90 349 55 727 93

Period Rep 1 Rep 2 Rep 3 Rep 4 Rep 5

27 589.21 649.52 544.64 296.36 289.96

28 103.42 936.04 393.21 771.45 151.18

29 219.14 1338.29 163.15 169.59 938.19

30 169.36 841.89 651.41 492.09 232.72

31 791.25 137.11 734.38 807.81 401.16

32 1360.99 274.57 457.19 148.87 231.46

33 530.04 1259.50 497.51 1300.90 990.27

34 198 98 275 20 177 60 723 29 414 02

16

20 542.79 908.48 633.90 349.55 727.93

21 383.47 317.29 165.11 345.11 106.20

22 276.26 387.05 366.19 789.89 613.58

23 336.39 388.63 605.87 315.20 818.90

24 562.06 323.83 1311.94 339.76 312.95

25 931.46 236.80 706.43 484.30 658.11

26 182.82 352.79 991.44 271.73 1815.32

34 198.98 275.20 177.60 723.29 414.02

35 523.28 1012.45 904.77 212.75 523.06

36 633.30 723.07 431.72 245.69 158.22

37 631.47 455.12 1256.28 287.57 351.78

38 807.24 1627.84 994.93 215.50 603.36

39 271.41 138.54 352.43 441.73 352.38

Avg. 469.70 752.74 565.96 427.05 567.14



May 02, 2009 - Page (85)

9

Output Analysis for Output Analysis for SteadySteady--State State Simulations (Example)Simulations (Example)

C fid t I t l f St d St t Si l tiConfident Interval for a Steady-State Simulation

dn

Y

dnY

n

djrj

r −=∑

+= 1),(

514.56705.42796.56574.75270.469),(1)(

1

++++== ∑

=

R

rr dnY

RRY

17

49.15752)(1

1)( 2

1

2 =−−

= ∑=

R

rr YY

RRS

5/)51.125)(78.2(57.556/)(4,025.0 ±=± RRStY04.15657.556 ±=



May 02, 2009 - Page (86)

1


Comparison and Evaluation of Comparison and Evaluation of Alt ti S t D iAlt ti S t D iAlternative System DesignsAlternative System Designs


Basic Concept of Basic Concept of Confidence IntervalConfidence Interval

C fid I t l id f l b dConfidence Interval provides range of values basedon observations from 1 sample, .A probability that the population parameter fallssomewhere within the interval.

Confidence IntervalSample Statistic(Point Estimate)

( )SX ,

2

Confidence Limit(Lower)

Confidence Limit(Upper)

( )XestX n ..1,2/ −± α

X

C.I. for a mean:

nStX

nStX nn ⋅+≤≤⋅− −− 1,2/1,2/ αα μ



May 02, 2009 - Page (87)

2

Output Analysis for Output Analysis for Two SystemsTwo Systems

O f th t i t t f i l ti i thOne of the most important uses of simulation is thecomparison of alternative system designs.

A two-sided 100(1 – α)% C.I. for θ1 – θ2 will always be

Parameter EstimatorSystem 1 θ1

System 2 θ2

1Y

2Y

3

A two sided 100(1 α)% C.I. for θ1 θ2 will always beof the form:( ) ( )21,2/21 .. YYestYY v −±− α

( ) ( ) ( ) ( )21,2/212121,2/21 .... YYestYYYYestYY vv −+−≤−≤−−− αα θθ

Comparison of Comparison of AlternativesAlternatives

C I b d th t diff θ θ ithi thC.I. bounds the true difference θ1 – θ2 within the rangewith probability 1 – α.

( x )0

21 YY −

0( x )

2121 0 θθθθ <⇔<−

0 θθθθ >⇔>−

4

0

21 YY −

0( x )

21 YY −

2121 0 θθθθ >⇔>

2121 0 θθθθ =⇔=−



May 02, 2009 - Page (88)

3

Independent Sampling Independent Sampling with Equal Varianceswith Equal Variances

I d d t li th t diff t dIndependent sampling means that different andindependent random-number streams will be used tosimulate the two systems.

where( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅

−−−=−+

21

2

21212,2

1121

RRS

YYt

p

RRμμ

α( ) ( )

211

21

222

2112

−+−−−

=RR

SRSRS p

5

⎠⎝ 21 21

( )21

2111..RR

SYYes p +=−

( ) ( )212,2/21 ..21

YYestYY RR −±− −+α

Independent Sampling Independent Sampling with Unequal Varianceswith Unequal Variances

If th ti f l i t f l bIf the assumption of equal variances cannot safely bemade, an approximate 100(1 – α)% C.I. for θ1 – θ2 canbe computed as follows.

where( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−−=

2

22

1

21

2121,2

RS

RS

YYt vμμ

α

( )( ) ( )2222

22

221

21 +

=RSRS

RSRSv

6

⎠⎝ 21 ( ) ( )11 2

22

1

11

−+

− RRS

RRS

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=−

2

22

1

21

21..RS

RSYYes

( ) ( )21,2/21 .. YYestYY v −±− α



May 02, 2009 - Page (89)

4

Correlated SamplingCorrelated Sampling

C l t d li th t f h li tiCorrelated sampling means that, for each replication,the same random numbers are used to simulate bothsystem. Therefore, R1 and R2 must be equal, sayR1 = R2 = R.

where21 rrr YYD −= ∑

=

=R

rrD

RD

1

1

R1D μ−

7

( )∑=

−−

=R

rrD DD

RS

1

22

11

RS

DtD

DR

μα =−1,2

( ) ( )R

SYYesDes D=−= 21.... ( )DestD R ..1,2 −± α

Experimental DesignExperimental Design

E i t l d i id f d idiExperimental design provides a way of decidingbefore the runs are made which particularconfigurations to simulate so that the desiredinformation can be obtained with the least amount ofsimulating.The input parameters and structural assumptions

i d l ll d f t d th t t

8

composing a model are called factors, and the outputperformance measures are called responses.



May 02, 2009 - Page (90)

5

Experimental Design Experimental Design (cont’)(cont’)

E h ibl l f f t i ll d l l f thEach possible value of a factor is called a level of thefactor.A combination of factors all at a specified level iscalled a treatment.Factors can be either quantitative or qualitative.These factors are collectively called decision

9

yvariables, or policy variables.

E.g., queue discipline (policy variable), number of physicians(decision variable).

Factorial DesignFactorial Design

S ti i l ti l d tSometimes simulation analyses are used todetermine the effects that various factors exert onselected performance criteria.Factorial designed experiments are one means ofproviding this type of information.The results produced from these experiments can be

10

statistically analyzed to measure the 1) main effects,and 2) interactive effects that selected factors exerton performance indices (system responses).



May 02, 2009 - Page (91)

6

Factorial Design (cont’)Factorial Design (cont’)

A i ff t (d t E ) i th h iA main effect (denote Ei) is the average change in aresponse resulting from raising the ith factor from aspecified “low level” to a specified “high level”.Suppose we perform a simulation to investigate threefactors (lot size, quantity of machines, and set-uptime) regarding their individual effects on a product’s

k

11

makespan.Factor Low Level (–) High Level (+)

Lot Size, E1 5 10Machine Quantity, E2 1 2

Rework Rate, E3 6% 12%

Main Effect FactorMain Effect Factor

A design matrix with 2x2x2 design points ist t dconstructed.

Design Point Factor 1 Level(Lot Size)

Factor 2 Level(Machine Qty)

Factor 3 Level(Rework Rate)

Response(Makespan)

1 + + + R1 = 5.7

2 – + + R2 = 5.0

3 + – + R3 = 12.1

4 – – + R4 = 11.1

5 + + – R5 = 5.7

12

5 R5 5.7

6 – + – R6 = 5.0

7 + – – R7 = 12.1

8 – – – R8 = 11.1

+ R1 – R2 + R3 – R4 + R5 – R6 + R7 – R8

2 k-1Main effect factor #1 E1 =



May 02, 2009 - Page (92)

7

Main Effect Factor and Main Effect Factor and Interactive Effect Interactive Effect FactorsFactors

+ R1 + R2 – R3 – R4 + R5 + R6 – R7 – R8


+ R1 + R2 + R3 + R4 – R5 – R6 – R7 – R8


Interactive effect factor #1 and #2 E12 = + R1 – R2 – R3 + R4 + R5 – R6 – R7 + R8

13

12

Interactive effect factor #1 and #3 E13 =

Interactive effect factor #2 and #3 E23 =

2 k-1

+ R1 – R2 + R3 – R4 – R5 + R6 – R7 + R8

2 k-1

+ R1 + R2 – R3 – R4 – R5 – R6 + R7 + R8

2 k-1

Analyzing Factorial Analyzing Factorial Designed ExperimentsDesigned Experiments

A i t ti ff t t ll if th ff t f iAn interactive effect tells us if the effect of a givenfactor is influenced by the level of another factor.If there is a significant interactive effect, then wecannot be certain that a main effect is due solely tothe raising or lowering of a factor level.Main Effects Raising a lot size from 5 to 10 is

t i d t k

14

Lot Size, E1 0.8Machine Quantity, E2 -6.3Rework Rate, E3 0.0

Interactive EffectsLot Size & Machine Qty, E12 -0.2Lot Size & Rework Rate, E13 0.0Machine Qty & Rework Rate, E23 0.0

to increase product makespan an average of 0.8 days per part.

Adding an additional machine decreased the makespan by an average of 6.3 days.



May 02, 2009 - Page (93)

8

Documentation and Documentation and ConclusionsConclusions

D t ti b di id d i t fiDocumentation can be divided into five areas:Objectives and AssumptionsModel Input ParametersExperimental DesignResultsConclusions

Obj ti d A ti

15

Objectives and AssumptionsAll objectives and assumptions should be recorded at the onset ofany simulation project. Any changes or modifications madeduring the course of building a model need to be included in thefinal report.

Documentation and Documentation and Conclusions (cont’)Conclusions (cont’)

M d l I t P tModel Input ParametersThis section contains a recap of the data used with a simulation.System flow charts, mathematical calculations, performancecriteria, solution constraints, solution restrictions, and any costrelated information should be included.

Experimental DesignThe information summarized in this category is comprised of

16

he info mation summa i ed in this catego y is comp ised ofdescriptions regarding the alternatives investigated, theexperiments designed for comparing alternatives, startingconditions, stopping conditions, a history of the random numberstreams employed with each experiment, and an account for thenumber of model replications performed for each alternative.



May 02, 2009 - Page (94)

9

Documentation and Documentation and Conclusions (cont’)Conclusions (cont’)

R ltResultsThis section is composed of the output data produced by asimulation. It also provides an overview of the statistical analysesperformed on the data. Tables and graphical charts whichillustrate the findings are very beneficial.

ConclusionsOne of the final steps in any decision-making process is to make

17

One of the final steps in any decision making p ocess is to makeconclusions and recommendations. This demands that benefit-to-cost ratios be investigated for each alternative. What are the totalcosts (tangible and intangible) needed to implement an alternative,and what are the total benefits anticipated from doing it?

Risk and UncertaintiesRisk and Uncertainties

Si d i i ki i b d th t fSince decision-making is based on the precept ofprediction, risk and uncertainties are almost alwaysinvolved. The potential labor requirementsforecasted with a given alternative may fall within arange. This can be classified as an uncertainty. Thepotential outcome for an alternative may also vary.This can be designated as a risk Any uncertainties

18

This can be designated as a risk. Any uncertaintiesand risks associated with an alternative should bediscussed in the final documentation.



May 02, 2009 - Page (95)

1


Queueing ModelsQueueing ModelsBy : Arya WirabhuanaBy : Arya Wirabhuana

Queueing SystemQueueing System

Q i S tQueueing System:A system in which items (or customers) arrive at a station, wait ina line (or queue), obtain some kind of service, an then leave thesystem.

Managerial Problem Related to Queueing Systems:Analysis Problems

Need to know if a given system is performing

2

Need to know if a given system is performingsatisfactorily.

Design ProblemsWant to design the features of a system to accomplishan overall objective.



May 02, 2009 - Page (96)

2

Characteristics of Characteristics of Queueing SystemQueueing System

A t l ti hi h i th ll ti f llA customer population, which is the collection of allpossible customers.An arrival process, which is how the customers fromthis population arrive.A queueing process, which consists of

the way in which the customers wait for service and

3

the queueing discipline, which is how they are then selected forservice.

Characteristics of Characteristics of Queueing System Queueing System (cont’)(cont’)

A i hi h i th d t t hi hA service process, which is the way and rate at whichcustomers are served.A departure process, which is one of the following:

Items leave the system completely after being served, resulting in aone-step queueing system.Items, once finished at a station, proceed to some other workstation to receive further service resulting in a network of queues

4

station to receive further service, resulting in a network of queues.



May 02, 2009 - Page (97)

3

Queueing SystemQueueing System

Components of a Queueing System:Servers

CustomerPopulation

ArrivalProcess

System

.

.

.

WaitingCustomers

QueueingProcess

Depart

Departure Processes:

Servers

ServiceProcess

5

Network of queues

Arrival Depart

One-stepqueueing system

ArrivalDepart

p

Arrival ProcessArrival Process

Cl f I t i l TiClasses of Interarrival Times:Deterministic, in which each successive customer arrives after thesame fixed and known amount of time. A classic example is anassembly line, where the jobs arrive at a station at unvarying timeintervals (called the cycle time).Probabilistic, in which the time between successive arrivals isuncertain and varies. Probabilistic interarrival times ared b d b b b l d b

6

described by a probability distribution.



May 02, 2009 - Page (98)

4

Arrival Process Arrival Process (Example)(Example)

Exponential DistributionExponential Distribution

where λ is the average number of arrivals per unit of time.

Probability of next customer arriving within T units of previousarrival:P (interarrival time ≤ T) = 1- e-λT

tetf λλ −=)(

7

For example:P (interarrival time ≤ 1/6 hour) = 1- e-20(1/6)

= 0.964

Queueing ProcessQueueing Process

Si l li Q i S tSingle-line Queueing System:A queueing system in which the customers waiting in a single linefor the next available server.

Multiple-line Queueing System:A i t i hi h i i t l t f

8

A queueing system in which arriving customers may select one ofseveral lines in which to wait for service.



May 02, 2009 - Page (99)

5

Classifications of Classifications of Queueing ModelsQueueing Models

Q i Di i liQueueing DisciplinesFirst-In-First-Out (FIFO)Last-In-First-Out (LIFO)Priority Selection

Notations (Kendall)A / B / c / K / L

9

where A: interarrival time distributionB: service time distributionc: no. of servicesK: the system capacityL: the size of the calling population

Classifications of Classifications of Queueing Models Queueing Models (cont’)(cont’)

Th i l S b l A t d ib i t i lThe arrival process: Symbol A to describe interarrivaldistribution:

D = deterministic interarrival timeM = probabilistic interarrival time with exponential distributionG = probabilistic interarrival time with general distribution other

than exponential.The service process: Symbol B to describe the service timedi t ib ti

10

distribution:D = deterministic service timeM = probabilistic service time with exponential distributionG = probabilistic service time with general distribution other than

exponential.



May 02, 2009 - Page (100)

6

Classifications of Classifications of Queueing Models Queueing Models (cont’)(cont’)

Th i thi b d t hThe queueing process: this number c denotes how manyparallel stations or channels are in the system (Servers areassumed to be identical in rate of service).When the waiting space and/or customer population size isfinite:

K = Maximum number of customers that can be in the system at any onetime

b f ll l t ti l t t l b f t th t

11

= number of parallel stations plus total number of customers that canwait for service

L = Total number of customers in the populationExample:

M/M/3//10 = a system that has room for an infinite number of customer (Khas been left off) but only 10 possible customers exists.

Performance MeasuresPerformance Measures

P f M f E l ti Q iPerformance Measures for Evaluating a QueueingSystem

Customer number

12

Transient Phase Steady State Phasenumber

(in order of arrival)



May 02, 2009 - Page (101)

7

Common performance Common performance MeasuresMeasures

A i l t λArrival rate – λService rate of one server – μServer utilization– ρMean number in the system – LMean queue length – Lq

Average time in the system W

13

Average time in the system – WAverage waiting time – Wq

Steady-state probability distribution – Pn, n = 0,1,2,…

Relationships among Relationships among Performance MeasuresPerformance Measures

W = Wq + (1 / μ)

Average timein the system = Average

waiting time + Averageservice time

Average no. of customersin the system = Average no. of arrivals

per unit of time× Average time

in the system

14

Lq = λ × W q

L = λ × W

Average no. of customersin the queue = Average no. of arrivals

per unit of time× Average time

in the queue



May 02, 2009 - Page (102)

8

Relationships among Relationships among Performance Measures Performance Measures (Example)(Example)

S λ 12 4 L 3Suppose λ = 12, μ = 4, Lq = 3

41

123===

λq

q

LW

21

41

411

=+=+=μqWW

15

62112 =×=×= WL λ

MM / / MM / 1 Queueing / 1 Queueing SystemSystem

A l i Si l Li Si l Ch l Q iAnalyzing a Single-Line Single-Channel QueueingSystem with Exponential Arrival and ServiceProcesses (M / M / 1)Queueing System for Weigh Station

λ = the average number of trucksarriving per hour

16

= 60μ = the average number of trucks

that can be weighed per hour= 66 Weigh Station



May 02, 2009 - Page (103)

9

Formulas (Formulas (MM / / MM / 1)/ 1)

P f M G l F lPerformance Measure General FormulaUtilization

Average number in line

Average waiting time in queue

ρρ−

=1

2

qL

ρ== q

q

LW

μλρ =

17

Average waiting time in queue

Average waiting time in system

)1( ρμλ −qW

)1(11ρμμ −

=+= qWW

Formulas (Formulas (MM / / MM / 1) / 1) (cont’)(cont’)

P f M G l F lPerformance Measure General FormulaAverage number in system

Probability that no customers are in system

Probability that an arriving customer has to wait 01 PPw −=

ρ−=10P

ρρλ−

=×=1

WL

18

Probability that an arriving customer has to wait

Probability of n customers in the system

0w

)1(0 ρρρ −=×= nnn PP



May 02, 2009 - Page (104)

10

ComputationComputation

Utili ti ( ) λ / 60 / 66 0 9091Utilization (ρ): ρ = λ / μ = 60 / 66 = 0.9091Prob.(no customers are in system) (P0): P0 = 1 - ρ = 1 – 0.9091 = 0.0909Average no. in line (Lq): Lq = ρ2 / (1 - ρ) = 9.0909Average waiting time in the queue (Wq): Wq = Lq / λ = 9.0909 / 60 = 0.152Average waiting time in the system (W): W = Wq + 1 / μ = 0.1667Average no. in the system (L): L = λ ⋅ W = 60 ⋅ 0.1667 = 10Prob (arriving customer having to wait) (P ): P = 1 - P0 = ρ = 0 9091

19

Prob.(arriving customer having to wait) (Pw): Pw 1 - P0 ρ 0.9091Prob.(n customers in the system) (Pn): Pn = ρn × P0

n 0 1 2 3 …

pn 0.0909 0.0826 0.0751 0.0683 …

M M / / M M / / cc Queueing Queueing SystemSystem

A l i Si l Li M lti l Ch l Q iAnalyzing a Single-Line Multiple-Channel QueueingSystem with Exponential Arrival and ServiceProcesses (M / M / c)Queueing System forWeigh Stationwith Two Scales

20

c = 2 serversλ = 70 trucks per hourμ = 40 trucks per hour on each scale

Weigh Station



May 02, 2009 - Page (105)

11

Formulas (Formulas (MM / / MM / / cc))

P f M G l F lPerformance Measure General FormulaServer utilization

Probability that no customers are in system

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

∑−

ρρ1

1!

1!)(

110

ccP

cc n

μλρc

=

21

Average number in line

( ) ⎥⎦

⎢⎣

⎟⎠

⎜⎝ −⎟⎠

⎜⎝

⎟⎠

⎜⎝∑= ρ1!!0 cnn

( ) 20

1

)1(!)(

ρρ

−=

+

ccPcL

c

q

Formulas (Formulas (MM / / MM / / cc) ) (cont’)(cont’)

P f M G l F lPerformance Measure General FormulaAverage waiting time in queue

Average waiting time in system

Average number in system

μ1

+= qWW

λq

q

LW =

01)( ρρ ++ PccL

c

22

Average number in system( ) 2

0

)1(!)(

ρρρ

−+=

cccL

W⋅= λ



May 02, 2009 - Page (106)

12

ComputationComputation

Utili ti ( ) λ / 70 / (2 40) 0 875Utilization (ρ): ρ = λ / c μ = 70 / (2 ⋅ 40) = 0.875Prob.(no customers are in system) (P0):

75.275.11!1

75.1!0

75.1!)1(

)(!1)(

!0)(

!)( 101101

0

=+=+=−

+++=−−

=∑ c

cccn

c cc

n

n ρρρρL

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

∑−

= ρρρ

11

!1

!)(

11

0

0

cc

nc

Pc

c

n

n

23

25.12853125.1875.01

1!2

75.11

1!)( 2

=⋅=−

⋅=−

⋅ρ

ρc

c c

06667.0151

25.1275.21

0 ==+

=P

Computation (cont’)Computation (cont’)

A i li (L )Average no. in line (Lq):

( )

7168.506667.0643398.1

06667.0875.01

1!22

75.12

3

=⋅⋅=

⋅−

⋅⋅

=

( ) 20

1

)1(!)(

ρρ

−=

+

ccPcL

c

q

24

Average waiting time in the queue (Wq):Average waiting time in the system (W):

Average no. in the system (L):

08167.070/7168.5/ === λqq LW

10667.040/108167.0)/1( =+=+= μqWW

4669.710667.070 =⋅=⋅= WL λ



May 02, 2009 - Page (107)

13

Economic Analysis of Economic Analysis of Queueing SystemQueueing System

A i W IAmerican Weavers, Inc.The plant has a large number of machines that jam frequently.Machines are repaired on a first-come-first-serve basis by one ofseven available repair persons.An average of 10 to 12 machines are out of operation at any onetime due to jams.Hiring more repair people will reduce the number of jammed

25

g p p p f jmachines. How many should be hired?

Economic Analysis of Economic Analysis of Queueing System Queueing System (cont’)(cont’)

M d li C t S tModeling Current System:Current number of repair persons (c = 7).The occurrence of jammed machines can be approximated by aPoisson arrival process with an average rate of 25 per hour(λ = 25).Each jammed machine requires a random amount of time forrepair that can be approximated by an exponential distribution

26

p pp y pwith an average service time of 15 minutes, which is an averagerate per server of four machines per hour (μ = 4).



May 02, 2009 - Page (108)

14

Analyzing the Current Analyzing the Current SystemSystem

M / M / Q S i iM / M / c Queue StatisticsNumber of identical servers 7Mean arrival rate 25.0000Mean service rate per server 4.0000Mean server utilization (%) 89.2857Expected number of customers in queue 5.8473Expected number of customers in system 12 0973

27

Expected number of customers in system 12.0973Probability that a customer must wait 0.7017Expected time in the queue 0.2339Expected time in the system 0.4839

Cost AnalysisCost Analysis

t it f ti $50/hcs = cost per server per unit of time = $50/hourcw = cost per unit of time for a customer waiting in the system

= $100/hour for lost production, etc.L = average number of customers in the systemTotal Cost = cost of crew + cost of waiting

= cs ⋅ c + cw ⋅ L= 50 ⋅ 7 + 100 ⋅ 12 0973

28

50 7 + 100 12.0973= $1559.73 per hour



May 02, 2009 - Page (109)

15

Cost Analysis (cont’)Cost Analysis (cont’)

Crew size Expected no in the system Hourly cost ($)Crew size Expected no. in the system Hourly cost ($)

7 12.0973 50 × 7 + 100 × 12.0973 = 1559.73

8 7.7436 50 × 8 + 100 × 7.7436 = 1174.36

9 6.7863 50 × 9 + 100 × 6.7863 = 1128.63

10 6.7594 50 × 10 + 100 × 6.4594 = 1145.94

11 6.3330 50 × 11 + 100 × 6.3330 = 1183.30

16001800

29

0200400600800

1000120014001600

7 8 9 10 11

Service CostWaiting CostTotal Cost



May 02, 2009 - Page (110)

INDUSTRIAL COMPUTER SIMULATION

CHAPTER 12

By : Arya Wirabhuana

1

Model Verification and Validation

Outline

• Important of Model Verification and Validation

• Model Verification

• Model Validation

• Model Building, Verification and Validation

• Validation Examples

2



May 02, 2009 - Page (111)

Important of Model Verification and Validation

• Model building is, by nature, very error prone• The modeler must translate the real world system• The modeler must translate the real‐world system into a conceptual model

• Then the conceptual model must be translated into a simulation models.

• This translation is iterative• In this, there is plenty of room for making errorsIn this, there is plenty of room for making errors • Verification and validation processes can reduce or eliminate these errors.

Important of Model Verification and Validation

• Model Verification is the process of determining whether the simulation modeldetermining whether the simulation modelcorrectly reflects the conceptual model.

• Model Validation is the process of determining whether the conceptual model correctly reflects the real systemM d l V ifi ti d V lid ti iti l• Model Verification and Validation are critical to the success of simulation project.



May 02, 2009 - Page (112)

Reason for Neglect

The primary reasons fro neglecting this i t t ti itimportant activity are:

• Time and budget pressures

• Laziness

• Overconfidence

• Ignorance



May 02, 2009 - Page (113)

Practices That Facilitate Verification and Validation

• Poor modeling practicesB ild d l ith littl th ht b t• Build models with little or no thought about being able to verify or validate the model

• Models contain spaghetti code that is difficult for anyone, including the modeler, to follow.

• Becomes more acute as models grows in complexity

Practices That Facilitate Verification and Validation

To create models that ease the difficulty of V ifi ti d V lid tiVerification and Validation

• Reduce the amount of complexity of the model

• The code is readable and understandable

• Finally model data and logic code should be• Finally, model data and logic code should be thoroughly and clearly documented



May 02, 2009 - Page (114)

Objectives of Verification and Validation

• To produce a representative model of the t d t dsystem under study

• To increase the model credibility

• To gradually refine the model during the development process

3

Model Verification

• Model Verification is the process of determining whether the model operates asdetermining whether the model operates as intended (it runs correctly).

• Building the model right• Tries to detect unintended errors in the model data and logic and removes them.

• Verification is the process of debugging the model



May 02, 2009 - Page (115)

Model Verification

• Errors or bugs in a simulation model are of t ttwo types:

• Syntax errors –grammatical errors

• Semantic errors – Associated with the meaning of the modeler and are therefore difficult to detectare therefore difficult to detect.

– Often they are logical errors

Model Verification

Preventive MeasuresG t it i ht th fi t ti ( d l ith b• Get it right the first time (models with no bugs the first time)

• In practice, this isn’t always possible as bugs are often sneaky and difficult to prevent

Practices minimizing bugs are:• BE CAREFUL



May 02, 2009 - Page (116)

Preventive Measures

• Use structured programming

• Five principles of structured programming– Top‐down design

– Modularity

– Compact modules

– Stepwise refinement– Stepwise refinement

– Structured control

Establishing a Standard for Comparison

• One simple standard is a common sense– If your simulation is giving totally bizarre results

• To construct an analytic model of the problem (with simplified assumption) if at all possible, e.g.,– To run simulation without downtimeTo run simulation without downtime

– A queuing system without balking



May 02, 2009 - Page (117)

Verification Techniques

• Conduct model code reviews

• Check the output for reasonableness

• Watch simulation for correct behavior

• Use the trace and debug facilities provided with the software

Model Validation

• Model Validation is the process of determining whether the model is meaningful andwhether the model is meaningful and accurate representation of the real system (Hoover and Perry, 1990)

• Building the right model• Stakeholders and customers should become h il i l d i th lid tiheavily involved in the validation process

• Can be very time consuming



May 02, 2009 - Page (118)

Model Validation

• Not interested in achieving absolute validity b t l f ti l liditbut only functional validity.

• Functional validity: The process of establishing that the model’s output behavior has sufficient accuracy for the model’s intended purpose over the domain of the model’s p pintended applicability (Sargent, 1998)

Model Validation

• Through simulation we convey the user or consumer that the simulation results can be consumer that the simulation results can betrusted and used to make a real world decisions.

• For existing systems, the model behavior should correspond to that of the actual system.

• In case of a new system, the input data should accurately reflect the design specification of the systemsystem.



May 02, 2009 - Page (119)

Model Validation

• Determining Model Validity

• There is no simple test to determine the validity of a model.

• Validation is an inductive process

• As with model verification, it is common to use a combination of techniques whenuse a combination of techniques when validating a model.

Model Validation

Techniques for validating a model• Watching the animationWatching the animation• Comparing with other models• Conducting degeneracy and extreme condition test• Checking for face validity• Testing against historical data• Performing sensitivity analysis• Running traces• Conducting Turing test



May 02, 2009 - Page (120)

Steps in Validation

1.‐ Build a model with high face validity.

2.‐ Validate model assumptions.

3.‐ Compare the model’s input‐output transformations against those in the real system.

9

Face Validity

• Face validity is concerned with the bl f th d l treasonableness of the model to

knowledgeable peers.

• Sensitivity analysis can help checking for face validity.

10



May 02, 2009 - Page (121)

Validating Model Assumptions

• Types of Assumptions– Structural

– Data

• Structural assumptions must be checked against the real system.

• Data assumptions must be checked by

11

• Data assumptions must be checked by statistical testing.

Validating Transformations

• The range of outputs of the model for a given f i t t bl thrange of inputs must resemble the one

observed in the real system.

• Use historical data.

• Validate on the main response variables.

• What to do if the model represents a non

12

• What to do if the model represents a non‐existing system?



May 02, 2009 - Page (122)

Maintaining Validation

• We need to maintain validation as system ifi ti t d t l i ht t dspecifications tend to evolve right up to, and

often even after system implementation

• We need to maintain validation (keep updating the model to continually reflect current system design specification)y g p )

Optimum Level of Validation



May 02, 2009 - Page (123)

Model Building, Verification and Validation

Steps in Model Building1.‐ Observe the real system

2.‐ Construct conceptual model and perform conceptual validation

3.‐ Translate conceptual model into a computer model and perform verification

5

4.‐ Calibrate, verify and validate at every step

Verification and Validation

• Verification– Building the model right

• Validation– Building the right model

• Verification and Validation must be conducted simultaneously throughout the model

4

simultaneously throughout the model development process



May 02, 2009 - Page (124)

Validation and Calibration

• Validation compares the model to the real tsystem.

• Calibration adjusts the model to make it more representative of the real system.

• Validation and Calibration must be performed all the time and until the very last minute

8

all the time and until the very last minute.

Validation Examples

• Discuss two cases, i.e. (See p.216‐219)– St. Hospital and Medical Centre

– HP Surface Mount Assembly Line



May 02, 2009 - Page (125)


CHAPTER 13CHAPTER 13


Simulation Output Analysis1

Statistical Analysis of Simulation Output (1)

• Because random input variables are used to drive the model the output measures of the simulation aremodel, the output measures of the simulation are also going to be random.

• The performance metrics based on a model’s output are only estimates and not precise measures.

• Running the simulation once provides a single random sample of the model’s fluctuating modelrandom sample of the model s fluctuating model output that may or may not be representative of the expected or average model output.

2



May 02, 2009 - Page (126)


• Statistical analysis of the simulation output is based on inferential or descriptive statisticson inferential or descriptive statistics.

• In descriptive statistics, it is dealt with a population, samples from the population, and a sample size.

• The idea behind inferential statistics is to gather a large enough sample size to draw valid inferences about the population while keeping the sampleabout the population while keeping the sample‐gathering time and cost at a minimum.

3


• The same statistical concepts apply to conducting an experiment with a simulation model of a system thatexperiment with a simulation model of a system that apply to sampling from a population.– Samples are generated by running the experiment.– The population size often very large or infinite (there are infinite number of possible outcomes from the experiment).

– A random variable is used to express each possibleA random variable is used to express each possible outcome of an experiment as a continuous or discrete number.

– Experiment sample, called replications, are independent.

4



May 02, 2009 - Page (127)


• Multiple independent runs or replications of a i l ti id lti l b ti th tsimulation provide multiple observations that can be used to estimate the expected value of the model’s output response.

5

Simulation Replications

• One run of the simulation constitutes one replication of the experiment.p

• The outcome of a replication represents a single sample.

• To obtain a sample size n, it is needed to run n independent replications.

• To understand how independent replications are produced in simulation, it is essential to understand p ,the concepts:– Random number generation– Initial seed values

6



May 02, 2009 - Page (128)

Random Number Generation (1)

• At the heart of a stochastic simulation is the random number generator.Th d b t d li• The random number generator produces a cycling sequence of pseudo‐random numbers that are used to generate observations from various distributions that drive the stochastic processes represented in the simulation model.

• As pseudo‐random numbers, the random number sequence or stream can be repeated if desiredsequence, or stream, can be repeated if desired.

• After the random number generator sequences through all random numbers in its cycle, the cycle starts over again through the same sequence.

• The length of the cycle before it repeat is called the cycle period, which extremely long.

7

Random Number Generation (2)

Example of a cycling pseudo-random stream produced by a random number generator with a very short cycle length

8



May 02, 2009 - Page (129)

Initial Seed Values (1)

• The random number generator requires the initial seed values.

• Identical random number sequences produce identical results.

• If a different seed value is appropriately selected to initialize the random number generator, the simulation will produce different results because it will be driven segment of numbers from a the

d brandom number stream.• This is how the simulation experiment is replicated to collect statistically independent observations of the simulation model’s output response.

9

Initial Seed Values (2)

• To replicate a simulation experiment, the simulation model is initialized to its starting conditions, allmodel is initialized to its starting conditions, all statistical variables for the output measures are reset, and a new seed value is appropriately selected to starts the random number generator.

• Each time an appropriate seed value is used to start the random number generator, the simulation produces a unique output responseproduces a unique output response.

• Repeating the process with several appropriate seed values produces a set of statistically independent observations of a model’s output response.

10



May 02, 2009 - Page (130)

Performance Estimation

• Assuming it is possessed an adequate ll ti i d d t b ticollection independent observations,

statistical methods can be applied to estimate the expected or mean value of the model’s output response.

• Two types of estimates:yp– Point estimates

– Interval estimates

11

Point Estimates (1)

• A point estimate is a single value estimate of a parameter of interestparameter of interest.

• Point estimates are calculated for:– Mean– Standard deviation

• The sample mean estimates the population xmean μ.

• The sample standard deviation s estimates the population standard deviation σ.

12



May 02, 2009 - Page (131)

Point Estimates (2)

• Sample mean:x

n

∑

• Sample standard deviation

n

xx i

i∑== 1

( )2−∑ xxn

13

( )

11

−=∑=

n

xxs i

i

Point Estimates (3)

Number of haircuts given 12 Saturdays

14



May 02, 2009 - Page (132)

Interval Estimates (1)

• A point estimate gives a little information about how accurately it estimates the trueabout how accurately it estimates the true value of the unknown parameter.

• Interval estimates constructed using x and s provide information about how far off the point estimate x might be from the true mean μ.

• The method used to determine the interval estimate is referred to as confidence interval estimation.

15


• A confidence interval is a range within it can be had a certain level of confidence that thebe had a certain level of confidence that the true mean falls.

• The interval is symmetric about x, and the distance that endpoint is from x is called half‐width (hw).A fid i t l i d th• A confidence interval is expressed as the probability P that unknown true mean μ lies within the interval x ± hw.

• The probability is called the probability level.16



May 02, 2009 - Page (133)


• If the sample observations used to compute xd i d d t d lland s are independent and normally

distributed, the half‐width of a confidence interval for a given level of confidence is:

( )n

sthw n 2/,1α−=

• The confidence intervals :

17

n

( ) αμ −=+≤≤− 1hwxhwxP


• The width of the interval indicates the accuracy the point of estimateaccuracy the point of estimate.

• The width of the confidence interval is affected by the variability in the output of the system and the number of observations collected (sample size).Th h lf idth ill h i k if (1) th l i• The half‐width will shrink if (1) the sample size n is increased or (2) the standard deviation s is reduced.

18



May 02, 2009 - Page (134)

Number of Replications (Sample Size) (1)

• Often it is needed to know the sample size or b f li ti d d t t bli hnumber of replication needed to establish a

particular confidence interval for a specified amount of absolute error (denote by e) between the point estimate of the sample mean x and the unknown true mean μ.

19


• Number of replications:

• Number of replications (rough approximation)

( ) 22/,1

⎥⎦

⎤⎢⎣

⎡= −

est

n n α

( ) 2⎤⎡ Z

hwe =

20

( )2/' ⎥⎦⎤

⎢⎣⎡=

esZn α



May 02, 2009 - Page (135)


• Number of replications (rough approximation) b d th l ti ( )based on the relative error (re)

( )

( )

2

2/

1

'

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

=xresZn α

21

( )1 ⎥⎦⎢⎣ + re


Number of haircuts given 18 Saturdays

22



May 02, 2009 - Page (136)

Statistical Issues with Simulation Output (1)

• A single simulation run (replication) represents a single sample of possible outcomes (values of asingle sample of possible outcomes (values of a random number) from the simulation.

• Averaging the observation from a large number of runs comes closer to the true expected performance of the model, but it is still only a statistical estimate.

• Constructing a confidence interval is a good idea b i id f h f ff hbecause it provides a measure of how far off the estimate is from the model’s true expected performance.

23


• The following three statistical assumptions must be met regarding the sample ofmust be met regarding the sample of observations used to construct the confidence interval:– Observations are independent so that no correlation exists between consecutive observations.

– Observations are identically distributed the entire duration of the process (that is, they are time invariant).

– Observations are normally distributed. 24



May 02, 2009 - Page (137)


Simulation Output from a Series of Replications

25


m n

m

yx j

ij

i

∑== 1

n

∑

( )

11

2

−

−=∑=

n

xxs

n

ii

( )

26

n

xx i

i∑== 1

( )n

stx n 2/,1α−±



May 02, 2009 - Page (138)

Replication technique

27

Terminating and Nonterminating Simulations

• Determining experimental procedures, such as how many replications and how to gatherhow many replications and how to gather statistics, depends on whether we should conduct a terminating or nonterminating simulation.

• A terminating simulation is conducted when it is interested in the behavior of the systemis interested in the behavior of the system over a particular period.

• A nonterminating simulation is conducted when it is concerned with the steady‐state behavior of a system. 28



May 02, 2009 - Page (139)

Terminating Simulations

• A terminating simulation is one in which the simulation starts at a defined state or time and endssimulation starts at a defined state or time and ends when it reaches some other defined state or time.

• A common type of terminating system is one that starts empty, runs for a while, and then empties out again before shutting down.

• Terminating simulations are not intended to measure• Terminating simulations are not intended to measure the steady‐state behavior of a system.

• Average measures of performance based on the entire duration of the simulation are of little meaning.

29

Nonterminating Simulations (1)

• Simulation in which the steady‐state (long‐term average) behavior of the system is beingterm average) behavior of the system is being analyzed

• The simulation could theoretically go on indefinitely with no statistical change in behaviorA it bl l th f ti t th d l i• A suitable length of time to run the model in order to collect statistics on the steady‐state behavior of the system must be determined.

30



May 02, 2009 - Page (140)

Nonterminating Simulations (2)

• A steady‐state condition is not one in which the observations are all the same, or even one for which ,the variation in observations is any less than during a transient condition.

• It means only that all observations throughout the steady‐state period will have approximately the same distribution.

• Nonterminating systems begins with a warm‐upp (or ) d d ll dtransient) state and gradually move to a steady‐state.

• Once the initial transient phase has diminished to the point where the impact of the initial condition on the system’s response is negligible, it is considered to reach steady‐state.

31

Experimenting with Terminating Simulations (1)

• Experiments are usually conducted by making l i l ti li ti f thseveral simulation runs or replications of the

period of interest using a different seed value for each run.

• This procedure enables statistically independent and unbiased observations to be pmade on the response variables of interest in the system over the period simulated.

32



May 02, 2009 - Page (141)

Experimenting with Terminating Simulations (2)

• Three important questions to answer in i th i trunning the experiment:

– What should be the initial state of the model?

– What is the terminating event or time?

– How many replications should be made?

33

Selecting the Initial Model State

• The initial state of the model represents how th t l k ( i i iti li d) t ththe system looks (or is initialized) at the beginning of the simulation.

• All terminating simulations do not necessarily begin with the system being empty and idle.

34



May 02, 2009 - Page (142)

Selecting a Terminating Event

• A terminating event is an event that occurs during the simulation that causes the simulation run to endthe simulation that causes the simulation run to end.

• The terminating event might occur when a certain time of day is reached or when certain conditions are met.

• If the termination event is based on the satisfaction of conditions other than time then the time that theof conditions other than time, then the time that the simulation will end is not known in advance.

35

Determining the Number of Replications

• Experiments are conducted by making several simulation runs or replications of the period ofsimulation runs or replications of the period of interest using a different stream of random numbers for each run.

• The procedure enables statistically independent and unbiased observations to be made on the response variables of interest in the system over the periodvariables of interest in the system over the period simulated.

• The answer to the question of how many replications are necessary is usually based on the analyst’s desired half‐width of a confidence interval.

36



May 02, 2009 - Page (143)

Experimenting with Nonterminating Simulations

• Nonterminating simulations are conducted h i t t d i th t d t twhen we are interested in the steady‐state

behavior of the simulation model.

• The following topics must be addressed:– Determining and eliminating the initial warm‐up bias

– Obtaining sample observations

– Determining run length

37

Determining the Warm‐up Period (1)

• In a steady‐state simulation, it is interested in th t d t t b h i f th d lthe steady‐state behavior of the model.

• If a model starts out empty of entities, it is usually takes some time before it reaches steady‐state.

• In a steady‐state condition the responseIn a steady‐state condition, the response variables in the system exhibit statistical regularity.

38



May 02, 2009 - Page (144)


• The time that it takes to reach steady state is a function of the activity times and the amount of yactivity taking place.

• In modeling steady‐state behavior, it is concerned with the problem of determining when a model reaches steady state.

• The start‐up period is usually referred to as the warm‐up period (transient period).

• Gathering any statistics starts when the warm‐up period is passed.

• This way is done to eliminate any bias due to observations during the transient state of the model.

39


Behavior of model’s output response as it reaches steady state

40



May 02, 2009 - Page (145)


• The easiest and most straightforward h f ti ti th ti (ifapproach for estimating the warm‐up time (if

the averaged output response is flat and shows a repeating pattern)– Running a preliminary simulation of the system (5 ~ 10 replications)

– Averaging the output values at each time step across simulation

– Observing at what time the system reaches statistical stability.

41


• When the model’s output response is erratic, it i f l t th it ith iit is useful to smooth it with a moving average.

• A moving average is constructed by calculating the arithmetic average of the w most recent data points (averaged output responses) in the data set.

• The w is the moving‐average window).

42



May 02, 2009 - Page (146)


• The formulation for computing moving averages:

( )

y

wmwiw

ywy

i

si

w

wssi

i ,,1,12

1

L −+=+

=

∑

∑

−

+

−=+

43

wii

yis

si

,,1,12

)1( L=−

=∑

−−=+


Welch Moving Average based on five replications

44



May 02, 2009 - Page (147)

Obtaining Sample Observations

• For computing confidence intervals, i d d t l b ti t bindependent sample observations must be created.

• For steady‐state simulations, two approaches for obtaining independent sample observations:– Running multiple replication

– Interval batching

45

Independent Observations via Replications (1)

• Running multiple replication for nonterminating simulation is similar to thenonterminating simulation is similar to the way it is done for terminating simulations.

• The only differences are:– The initial warm‐up bias must be determined and eliminated.An appropriate length must be determined– An appropriate length must be determined.

• Once the replications are made, confidence intervals can be computed.

46



May 02, 2009 - Page (148)

Independent Observations via Replications (2)

• Advantages:– Ensuring independent samples

• Disadvantages– Long running time

– Causing biased results from each replication

47

Independent Observations via Batch Mean (1)

• Interval batching (batch mean technique) is a th d i hi h i l l i dmethod in which a single, long run is made

and statistics are collected during separate periods.

• The simulation is run only once to collect observations for statistical analysis purposes. y p p

• Placing the output values from the simulation run into groups or batches forms the set of observations.

48



May 02, 2009 - Page (149)


49


50



May 02, 2009 - Page (150)


m n

m

yx j

ij

i

∑== 1

n

∑

( )

11

2

−

−=∑=

n

xxs

n

ii

( )

51

n

xx i

i∑== 1

( )n

stx n 2/,1α−±


52



May 02, 2009 - Page (151)

Determining Run Length (1)

• Determining the run length for a steady‐state simulation is more difficult because the simulationsimulation is more difficult because the simulation could be run indefinitely.

• Running extremely long simulations can be very time consuming, so the objective is to determine an appropriate run length to ensure a representative sample of the steady‐state response of the system.Th l h f h i l i f d• The length of the simulation run for a steady‐state simulation depends on the interval between the least frequency occurring event and the type of sampling method (replication or interval batching) used.

53


• If running independent replications, it is ll d id t th i l ti lusually a good idea to run the simulation long

enough past the warm‐up point to let every type of event (including rare ones) happen many times.

54



May 02, 2009 - Page (152)


• The choice between running simulation i d d t li ti b t h i t lindependent replications or batch intervals can be made based on how long it takes the simulation to reach steady state.

• If the warm‐up period is short, then the running independent replications is preferred.g p p p

• If the warm‐up period is long, then batch interval method is preferred.

55

• Periodic

56



May 02, 2009 - Page (153)

ProModel’s Simulation Options (1)

57

ProModel’s Simulation Options (2)

• Standard– ProModel collects output statistics for one or more preplications.

– No interval length can be specified.• Batch Mean

– The method of batch means, or interval batching, is a way to collect independent samples when simulating steady‐state systems.

• Periodic• Periodic– Useful primarily in terminating or non‐steady state simulations where it is interested in the system behavior during different periods (e.g. peak or full periods) of activity

58



May 02, 2009 - Page (154)


CHAPTER 14


Comparing Systems

Introduction

• In many cases, simulations are conducted to compare two or more alternative designs of acompare two or more alternative designs of a system with the goal of identifying the superior system relative to some performance measure.

• Comparing alternative system design requires careful analysis to ensure that differencescareful analysis to ensure that differences being observed are attributable to actual differences in performance and ni .



May 02, 2009 - Page (155)

Hypothesis Testing (1)

• A null hypothesis, denoted H0, is drafted to state the value of μ is not significantly different than the valuevalue of μ1 is not significantly different than the value of μ2 at the α level of significance.

• An alternative hypothesis, denoted H1,is drafted to oppose the null hypothesis H0. For example, H1 could state that μ1 and μ1 are different.

• Formally• Formally,

H0: μ1 = μ2 or equivalently H0: μ1 − μ2 = 0

H1: μ1 ≠ μ2 or equivalently H1: μ1 − μ2 ≠ 0


• The α level of significance refers to the probability of making a Type I errorprobability of making a Type I error.

• A Type I error occurs when H0 is rejected, but in fact H0 is true.

• A Type II error occurs when it is failed to reject H0 but in fact H1 is true.

• Hypothesis testing methods are designed such that the probability of making a type II error, β, is as small as possible for a given value of α.



May 02, 2009 - Page (156)


Comparing Two Alternative Designs

• Methods used to compare two alternative t d isystem designs:

– Welch confidence interval for comparing two systems

– Paired‐t confidence interval for comparing two systems



May 02, 2009 - Page (157)

Comparing Two Alternative Designs (1)

Comparing Two Alternative Designs (2)

• Hypothesis:

H0: μ1 − μ2 = 0

H1: μ1 − μ2 ≠ 0



May 02, 2009 - Page (158)

Welch Confidence Interval for Comparing Two Systems (1)

• The Welch confidence interval method requires that the observations drawn from each populationthe observations drawn from each population (simulated system) be normally distributed and independent within a population and between populations.

• The Welch confidence interval method does not require that the number of samples drawn from one population (n ) equal the number of samples frompopulation (n1) equal the number of samples from the other population (n2).

• This approach does not require that two population have equal variances (σ1

2 = σ22 = σ2).

Welch Confidence Interval for Comparing Two Systems (2)

• The Welch confidence interval for an α level of i ifisignificance:

( ) ( )[ ] αμμ −=+−≤−≤−− 1212121 hwxxhwxxP

22

21

2/ssthw df +=

212/, nn

thw df +α

[ ][ ] ( ) [ ] ( )11 2

22

221

21

21

22

221

21

−+−

+≈

nnsnnsnsnsdf



May 02, 2009 - Page (159)

Paired‐t Confidence Interval for Comparing Two Systems (1)

• Paired‐t confidence interval method requires that the observations drawn from each population (simulatedobservations drawn from each population (simulated systems) be normally distributed and independent within a population.

• The paired‐t confidence interval method does not require that the observations between populations be independent.Th i d fid i l h d i• The paired‐t confidence interval method requires that the populations have equal variances (σ1 = σ2 = σ).

Paired‐t Confidence Interval for Comparing Two Systems (2)

n

∑n

xx j

j∑=

−

− = 1)21(

)21(

[ ]1

2)21()21( −∑

=−− xx

n

jj

11

−=

ns j



May 02, 2009 - Page (160)

Comparing More Than Two Alternative Designs

• Methods:– Bonferroni approach

– Advanced statistical models (ANOVA)



May 02, 2009 - Page (161)

The Bonferroni Approach for Comparing More Than Two Alternative Systems (1)

• The Bonferroni approach is useful when there are more than two alternative system designsare more than two alternative system designs to compare with respect to some performance measure.

• Given K alternative system designs to compare, the null hypothesis H0 and alternative hypothesis H becomehypothesis H1 become

H0 : μ1 = μ1 = ... = μK = μ for K alternative system

H1 : μi ≠ μi’ for at least one pair i ≠ i’

The Bonferroni Approach for Comparing More Than Two Alternative Systems (2)

• The null hypothesis H0 states that the means from the K populations (mean output of the K differentthe K populations (mean output of the K different simulation models) are not different.

• The alternative hypothesis H1 states that at least one pair of the means are different.

• The Bonferroni approach is very similar to the two confidence interval methods that it is based on

i fid i l d i if hcomputing confidence intervals to determine if the true mean performance of one system (μi) is significantly different than the true mean performance of another system (μi’).



May 02, 2009 - Page (162)

• Number of pairwise comparison for K did t d icandidate designs

• Probability of m confidence intervals being simultaneously correct

( )2

1−=

KK

simultaneously correct

( ) ( ) ⎟⎠

⎞⎜⎝

⎛−=−≥ ∑

=

m

iimP

111correct are statements interval confidence all αα

( ) ( ) 21,,1,21

−=−

= KKiKKi Lαα

( )n

stx qpn

qpi )(2/,1

)(−−

− ± α

n



May 02, 2009 - Page (163)

Advanced Statistical Models for Comparing More Than Two Alternative Systems (1)

• Analysis of variance (ANOVA) in conjunction with a multiple comparison test provides awith a multiple comparison test provides a means for comparing a much larger number of alternative designs.

• Given K alternative system designs to compare, the null hypothesis H0 and alternative hypothesis H becomehypothesis H1 become

H0 : μ1 = μ1 = ... = μK = μ for K alternative system

H1 : μi ≠ μi’ for at least one pair i ≠ i’



May 02, 2009 - Page (164)



May 02, 2009 - Page (165)

Analysis of Variance (1)

• Analysis of variance (ANOVA) allow us to partition the total variation in the output response from thethe total variation in the output response from the simulated system into two component – Variation due to the effect of the treatments – Variation due to experimental error (the inherent variability in the simulated system)

• For this problem, we are interested is knowing if the variation due to the treatment is sufficient tovariation due to the treatment is sufficient to conclude that the performance of one strategy is significantly different than the other with respect to mean throughput of the system.


• Assumptions:– The observations are drawn from normally distributed populations.

– The observations are independent within a strategy and between strategies.

• The variance reduction technique based on common random number (CRN) cannot be used with this method.



May 02, 2009 - Page (166)


• The fixed‐effects model:

τi: the effect of the ith treatment (ith strategy) as a deviation from the overall (common to all treatments) population mean μ.

njKix ijiij ,,1;,,1, LL ==++= ετμ

εij: the associated error with this observation (in the context of simulation, the εij : the random variation of the response xij that occurred during the jth replication of the ith treatment).


• Assumptions for the fixed‐effects model:– the sum of all τi equals to zero.– the error terms εij are independent and normally distributed with a mean of zero and common variance.



May 02, 2009 - Page (167)

Analysis of Variance (5)• Sum of squares i:

xn

ijn

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎞⎛ ∑

• Grand total:

• Overall mean:

Kin

x jjn

jiji ,,1,SS 1

1

2 L=⎟⎠

⎜⎝−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∑∑ =

=

∑ ∑∑= ==

==K

i

K

ii

n

jij xxx

1 1.

1..

nKNNx

N

xx

K

i

n

jij

===∑∑= = ;..1 1

..


• Degree of freedom total (corrected)

• Degree of freedom treatment

• Degree of freedom error

1corrected)df(total −= N

1nt)df(treatme −= K

KN −=df(error)



May 02, 2009 - Page (168)


• Sum of squares error:K

• Sum of squares treatment:

• Sum of squares total (corrected):

∑=

=K

iii

1SSSSE

⎥⎦

⎤⎢⎣

⎡−⎟

⎠

⎞⎜⎝

⎛= ∑

= Kxx

n

K

iii

2..

1

2.

1SST

SSESSTSSTC +=


• Mean square treatment:SST

• Mean square error:

• Calculated F statistics:

nt)df(treatmeSSTMST =

MST

df(error)SSEMSE =

MSEMST

CALC =F



May 02, 2009 - Page (169)


Source of V i ti

Degree of F d

Sum of S

Mean S

Calculated F St ti tiVariation Freedom Squares Square F Statistics

Treatment K – 1 SST MST MST/MSE

Error N – K SSE MSE

Total N – 1 SSTC

Multiple Comparison Test (1)

• The hypothesis test suggests that not all designs are the same with respect to adesigns are the same with respect to a particular response, but it does not identify which designs perform differently.

• The Fisher’s least significant difference (LSD) test is used to identify which designs perform differentlydifferently.

• It is generally recommended to conduct a hypothesis test prior to the LSD test to determine if one or more pairs of treatments are significantly different.



May 02, 2009 - Page (170)


• The LSD test requires the calculation of a test t ti ti d t l t ll i istatistics used to evaluate all pairwise comparisons of the sample mean from each population.

• Number of pairwise comparisons for Kcandidate designs = K(K – 1)/2g ( )/

• The LSD test statistics:

( ) ( )n

t MSE2LSD )2/df(error),( αα =


• The decision rule: If the difference in the l l d th LSDsample mean response values exceeds the LSD

test statistics, the population mean response values are significantly different at a given level of significance.

( ) differentlysignificanareandthenLSDIf μμαxx >− ( )ce.significan of level at the

differently significanare andthen ,LSD If '

αμμα i'iii xx >



May 02, 2009 - Page (171)


• Several



May 02, 2009 - Page (172)

One‐way ANOVA with SPSS (1)

Factor (candidate design)

Observation


Multiple comparison test



May 02, 2009 - Page (173)





May 02, 2009 - Page (174)


10 56,30200 1,37371 ,43441 55,31931 57,28469 54,480 58,33010 54,63300 1,16549 ,36856 53,79926 55,46674 52,140 56,01010 57,39200 ,65794 ,20806 56,92134 57,86266 56,110 58,30030 56,10900 1,57266 ,28713 55,52176 56,69624 52,140 58,330

1,0002,0003,000Total

STRAOBSN Mean

Std.Deviation Std. Error

LowerBound

UpperBound

95% ConfidenceInterval for Mean

Minimum Maximum

Descriptives


3,195 2 27 ,057OBS

LeveneStatistic df1 df2 Sig.

Test of Homogeneity of Variances



May 02, 2009 - Page (175)


38,619 2 19,310 15,749 ,000

33,105 27 1,226

71,724 29

BetweenGroupsWithinGroupsTotal

OBS

Sum ofSquares df

MeanSquare F Sig.

ANOVA


Multiple Comparisons

Dependent Variable: OBSLSD

1,66900* ,495 ,002 ,65294 2,68506-1,09000* ,495 ,036 -2,10606 -7,4E-021 66900* 495 002 2 68506 65294

(J) STRA2,0003,0001 000

(I) STRA1,000

2 000

MeanDifference

(I-J) Std. Error Sig.LowerBound

UpperBound

95% ConfidenceInterval

Multiple Comparisons

-1,66900* ,495 ,002 -2,68506 -,65294-2,75900* ,495 ,000 -3,77506 -1,742941,09000* ,495 ,036 7,39E-02 2,106062,75900* ,495 ,000 1,74294 3,77506

1,0003,0001,0002,000

2,000

3,000

The mean difference is significant at the .05 level.*.



May 02, 2009 - Page (176)

Factorial Designs and Optimization (1)

• In simulation experiments, it is sometimes intended in finding out how different decision variable settingsin finding out how different decision variable settings impact the response of the system rather than simply comparing one candidate system to another.

• There are often many decision variables of interests for complex systems.

• Rather than run hundreds of experiments for every ibl i bl i i l d ipossible variable settings, experimental design

techniques can be used as a shortcut for finding those decision variables of greatest significance.


• Using experimental design terminology, decision variables are referred to as factors and the outputvariables are referred to as factors and the output measures are referred to as responses.

• Once the response of interest has been identified and the factors that are suspected of having an influence in this response defined, a factorial design method that prescribes how many runs to make andmethod that prescribes how many runs to make and what level or value to use for each factor is used.



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Relationship between factors (decision variables) and o tp t responsesand output responses


• One type of experiment that looks at the combined effect of multiple factors on system response iseffect of multiple factors on system response is referred to as a two‐level, full factorial design.

• For experiments in which a large number of factors are considered, a fractional‐factorial design is used to strategically select a subset of combinations to test in order to “screen‐out” factors with little or no impact on system performanceimpact on system performance.

• In case of there are many combinations, optimization techniques are used to search for the combination that produces the most desirable response.



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Variance Reduction Techniques

• The variance of a performance measure computed from the output of simulations can be reduced.p

• Reducing the variance allows to estimate the mean value of a random variable within a desired level of precision and confidence with fewer replications (independent observations).

• The reduction in the required number of replications is achieved by controlling how random numbers are

d “d ” h h l d lused to “drive” the events in the simulation model.• The use of common random number (CRN) is perhaps one of the most popular variance reduction techniques.

Common Random Number (1)

• The common random number (CRN) technique was invented for comparing alternative system designs.p g y g

• The CRN technique provides a means for comparing alternative system designs under more equal experimental conditions,

• This is helpful in ensuring that the observed differences in the performance of two system designs are due to the differences in the designs and

d ff l dnot to differences in experimental conditions.• The goal is to evaluate each system under the exact same circumstances to ensure a fair comparison.



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• The common random number (CRN) technique was invented for comparing alternative system designs.p g y g

• The CRN technique provides a means for comparing alternative system designs under more equal experimental conditions,

• This is helpful in ensuring that the observed differences in the performance of two system designs are due to the differences in the designs and

d ff l dnot to differences in experimental conditions.• The goal is to evaluate each system under the exact same circumstances to ensure a fair comparison.


Unique seed value assigned for each replicationsfor each replications



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Unique random numberstream assigned to eachstream assigned to eachstochastic element insystem



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CHAPTER 15


Modeling Manufacturing Systems

Characteristics of Manufacturing Systems (1)• Manufacturing systems: A class of processing systems in which entities (raw materials, y ( ,components, subassemblies, pallets or container loads, etc.) are routed through a series of workstations, queues, and storage areas.

• Entities in manufacturing systems: Inanimate objects that have a more controlled entry and routing sequence.

d• Entity production:– frequently performed according to schedule to fill predefined quotas,

– triggered by low finished goods inventories (make‐to‐stock production),

– triggered by orders (make/assemble‐to‐order production).



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Characteristics of Manufacturing Systems (2)

• Manufacturing system– Utilizing varying degrees of mechanization and automation for entity processing and material movement.

• In the most highly automated facilities– There may be very little, if any, human involvement to be considered in the simulation study.

Characteristics of Manufacturing Systems (3)

• Implications for modeling manufacturing systems:O ti ti ft h littl if i bilit– Operation times often have little, if any, variability

– Entity arrivals frequently occur at set times or conditions

– Routing sequences are usually fixed from the start

– Entities are often processed in batches

– Equipment reliability is frequently a key factor

– Material handling can significantly impact entity flowMaterial handling can significantly impact entity flow

– We are usually interested in the steady‐state behavior of the system



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Manufacturing Terminology (1)

• Operation: An activity performed on a product t k t ti (t f ti l blat a workstation (transformational, assembly, nontransformational).

• Workstation (work center): The place or location where an operation is performed consisting of one or more machines and/or g /personnel.


• NC machine: A machine tool whose spindle d t bl ti t ti ll t ll dand table action are automatically controlled

by a computer.

• Machining center: An NC machine with a tool exchanger that provides greater capability than a stand‐alone NC machine.



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• Master production schedule: The schedule defining what end products are to bedefining what end products are to be produced for a given period.

• Production plan: A detailed schedule of production for each individual component comprising each end product.

• Bottleneck: The bottleneck is traditionally throught of as the workstation that has the highest utilization or largest ratio of required processing time to time available.


• Setup: The activity required to prepare a k t ti t d diff t t tworkstation to produce different part type.

• Job: – The activity or task being performed (general sense)

– Each individual part (mass production)p ( p )

– A customer order to be produced (job shop)



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• Machine cycle time: The time required to perform a single operationperform a single operation.– Floor‐to‐floor time (load/unload)

• Capacity: – Holding capacity (tank or storage bin), – Production capacity (machine).

• Theoretical capacity• Effective capacity• Expected capacity• Actual capacity


• Scrap rate: The percentage of defective parts that are removed from the system followingthat are removed from the system following an operation.

• Reliability: The average time a machine or piece of equipment whenever if fails. – Mean time between failure (MTTF)

l b l h f l h d l d• Availability: The percentage of total scheduled time that a resource is actually available for production – Function of reliability and maintainability.



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• Preventive or scheduled maintenance: P i di i t (l b i ti l i )Periodic maintenance (lubrication, cleaning) performed on equipment to keep it in running condition.

• Unit load: A consolidated group of parts that is containerized or palletized for movement pthrough the system.– Minimizing handling through consolidation

– Providing a standardized pallet or container as the movement item.

Use of Simulation in Manufacturing (1)

• Simulation has proved effective in helping to sort the complex issues surrounding manufacturing decisions.complex issues surrounding manufacturing decisions.

• Kochan (1986) notes that, in manufacturing systems, “the possible permutations and combinations of workpieces, tools, pallets, transport vehicles, transport routes, operations, etc., and their resulting performance, are almost endless. Computer simulation has become an absolute necessity in thesimulation has become an absolute necessity in the design of practical systems, and the trend toward broadening its capabilities in continuing as systems move to encompass more and more of the factory”.



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• Simulation in manufacturing covers the range from real‐time cell control to long‐range technologyreal time cell control to long range technology assessment, where it is used to assess the feasibility of new technologies prior to committing capital funds and corporate resources.

• Simulations used to make short‐term decisions usually require more detailed models with a closer resemblance to current operations that what wouldresemblance to current operations that what would be found in long‐term decision‐making models.

• Simulation helps evaluate the performance of alternative designs and the effectiveness of the alternative operating policies.


Decision range in manufacturing simulation



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• Design decisions:– What type and quantity of machines, equipment, and tooling should

b d?be used?– How many operating personnel are needed?– What is the production capability (throughput time) for a given

configuration?– What type and size of material handling systems should be used?– How large should buffer and storage areas be?– What is the best layout of the factory?– What automation controls work the best?– How balanced is a production line?– Where are the bottlenecks (bottleneck analysis)?– What is the impact of machine downtime on production (realibility

analysis)?– What is the effect of setup time on production?– Should storage be centralized or localized?– What is the effect of vehicle or conveyor speed on part flow?


• Operational decisions:– What is the best way to schedule preventive maintenance?y p– How many shifts are needed to meet production requirements?

– What is the optimum production batch size?– What is the optimum sequence for producing a set of jobs?

– What is the best way to allocate resources for a particular set of tasks?set of tasks?

– What is the effect of a preventive maintenance policy as opposed to a corrective maintenance policy?

– How much time does a particular job spend in a system (makespan or throughput time analysis)?

– What is the best production control method (kanban, MRP, or other) to use?



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Applications of Simulation in Manufacturing

• Method analysis• Plant layout• Plant layout• Batch sizing• Production control• Inventory control• Supply chain planningP d ti h d li• Production scheduling

• Real‐time control• Emulation

Method Analysis (1)

• Method analysis looks at alternative ways of processing and handling material.processing and handling material.

• One of the decisions that addresses manufacturing methods pertains to the type and level of automation to use in the factory.

• There are three types of automation to consider in manufacturing: operation automation, material h dli i d i f i ihandling automation, and information automation.

• The degree of automation ranges from manual to fully automatic.



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Method Analysis (2)Areas of simulation

19

Plant Layout (1)

• With the equipment‐intensive nature of manufacturing systems, plant layout is the importantmanufacturing systems, plant layout is the important to the smooth flow of product and movement of resources through the facility.

• A good layout results in a streamlined flow with minimal movement and thus minimizes material handling and storage costs.Si l i h l id if i ffi i fl• Simulation helps to identify inefficient flow patterns and to create better layouts to economize the flow of material.



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Plant Layout (2)

Material flow system

Plant Layout (3)

• In laying out equipment in a facility, the designer must choose between a process layout, a product p y , playout, and a part family layout of machines.

• In a process layout, machines are arranged together based on the process they perform.

• In a product layout, machines are arranged according to the sequence of operations that are performed on a product.

• In a part family layout, machines are arranged to process parts of the same family. Part family layouts are also called as group technology cells.



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Plant Layout (4)

Comparing between(a)process layout( )p y(b)product layout(c)cell layout

Plant Layout (5)

• Deciding which layout to use will depend largely on the variety and lot size of parts that are produced. y p p

• If a wide variety of parts are produced in small lot sizes, a process layout may be the most appropriate.

• If the variety is similar enough in processing requirements to allow grouping into part families, a cell layout is best.

• If the variety is small enough and the volume is y gsufficiently large, a product layout is best.

• Often a combination of topologies is used in the same facility.



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Batch Sizing (1)

• Batching decisions play a major role in meeting the flow and efficiency goals of ameeting the flow and efficiency goals of a production facility.

• A batch or lot of parts refers to quantity of parts grouped together for some particular purpose. T i ll th diff t b t h t• Typically three different batch types are spoken of in manufacturing: production batch, move batch, and process batch.

Batch Sizing (2)

• The production batch (production lot) consists of all of the part of one type that begin production beforeof the part of one type that begin production before a new part type is introduced to the system.

• The move or transfer batch is the collection of parts that are grouped and moved together from one workstation to another.

• The process batch is the quantity of parts that are• The process batch is the quantity of parts that are processed simultaneously at a particular operation and usually consists of a single part.



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Batch Sizing (3)

• The deciding which size to use for each ti l b t h t i ll b dparticular batch type is usually based on

economic trade‐offs between in‐process inventory costs and economies of scale associated with larger batch sizes.

• Larger batch sizes usually result in lower setup g y pcosts, handling costs, and processing costs.



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Production Control (1)

• Production control governs the flow of material between individual workstationsmaterial between individual workstations.

• Simulation helps plan the most effective and efficient method for controlling material flow.

• Common method for controlling material flow are:– Push control– Pull control– Drum‐buffer‐rope (DBR) control


• Push controlTh d ti i d i b k t ti it d– The production is driven by workstation capacity and material availability.

– Each workstation seeks to produce as much as it can, pushing finished work forward to the next workstation.

– In make‐to‐stock production, the triggering mechanism is a drop in finished goods inventory.

– In make‐to‐order or assemble‐to‐order production, a master production schedule drives production by scheduling through material requirement planning (MRP) or other backward or forward scheduling system.



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• Pull controlTh d t d d t i di t ti t– The downstream demand triggers preceding station to produce a part with no more than one or two parts at a station at any given time.

– Corresponding to Just‐in‐time (JIT) or lean manufacturing philosophy, which advocates the reduction of inventories to a minimum.

h b l f l d l– The basic principle of JIT: To continuously reduce scrap, lot sizes, inventory, and throughput time as well as eliminate the waste associated with non‐value added activities such as material handling, machine setup, and rework.


• Pull control (cont.)– Be implemented using a method the Japanese call kanban

– Implemention of the kanban system:• Withdrawal kanban, authorizing the retrieval of work items.

P d ti k b th i i th d ti f• Production kanban, authorizing the production of more parts.



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• Drum‐buffer‐rope (DBR) control– The production is driven by workstation capacity andThe production is driven by workstation capacity and material availability.

– Based on the theory of constraints and seek to improve production by managing the bottleneck operation.

– The DBR approach basically lets the bottleneck operation pace the production rate by acting as a drum.

– The buffer is inventory needed in front of the bottleneck yoperation to ensure that it is never starved for parts.

– The rope represents the tying or synchronizing of the production release rate to the rate of bottleneck/

Inventory Control

• Inventory controlTh Pl i h d li d di t hi f– The Planning, scheduling, and dispatching of inventory to support the production activity.

• Goals:– Responding promptly to customer demandsMi i i i d d i d i h– Minimizing order and setup costs associated with supplying inventory

– Minimizing the amount inventory



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Inventory Control (2)

• Decisions affecting inventory levels and costs:– How much inventory should be ordered when inventory is replenished (the economic order quantity)?

– When should more inventory be ordered (the reorder point)?

– What should the optimum inventory levels to maintain?


• Two common approaches to deciding when to order more inventory and how much:more inventory and how much:– the reorder point system– the MRP system

• The reorder system: An inventory control system where inventory for a particular items is replenished to a certain level wherever if falls below a defined l llevel.

• The MRP system: Inventory is ordered to match the timing of the demand for products over the next planning period



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• Benefits of simulation models over traditional l ti d l f i t l ianalytic models for inventory planning:

– Greater accuracy

– Greater flexibility

– Easier to model

– Easier to understandEasier to understand

– More informative output

– More suitable for management


• Two common approaches to deciding when to order more inventory and how much:more inventory and how much:– the reorder point system– the MRP system

• The reorder system: An inventory control system where inventory for a particular items is replenished to a certain level wherever if falls below a defined l llevel.

• The MRP system: Inventory is ordered to match the timing of the demand for products over the next planning period



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• Centralized storage: – The storage is placed in a centralized location.

– Creating excessive handling of material and increased response times.

• Decentralized storage: – The point‐of‐use storage with parts kept whereThe point of use storage with parts kept where they are needed.

– Eliminating needless handling and reducing response time.

Supply Chain Management (1)

• Supply chain management is a key area for improving efficiencies in a corporationimproving efficiencies in a corporation.

• A supply chain is made up of suppliers, manufacturers, distributors, and customers.

• Two main goals with respect to supply chain management: – to reduce costs – to improve customer service, which is achieved by reducing delivery time.



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Supply Chain Management (2)

• Supply chain network analysis tools:– looking at the problem in the aggregatelooking at the problem in the aggregate.– helping identify optimum site locations and inventory levels given a general demand.

• Simulation approaches:– looking at the system on a more detailed level– analyzing the impact of interdependencies and variation over timeover time

– “accurately” tracing each transaction through the network at a detailed level, incorporating cost and service impacts of changes in inventories policies.

Production Scheduling (1)

• Production schedules determines the start d fi i h ti f j b t b d dand finish times for jobs to be produced.

• The use of simulation for production scheduling is called simulation‐based scheduling.

• Types of scheduling problems:Types of scheduling problems:– Static

– Dynamic



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• Four principal goals or objectives in h d lischeduling:

– Minimize job lateness or tardiness

– Minimize the flow time or time jobs spend in production

– Maximize resource utilization

– Minimize production cost


Rules for making decision of which job to process next:



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• Traditional scheduling methods fail to account f fl t ti i titi dfor fluctuations in quantities and resource capacity.

• Simulation for dynamic scheduling deals with short time periods and less concerns about long‐term statistical fluctuations in the system.g y


• Characteristics for building simulation‐based h d lischeduling

– The model captures an initial state.

– Operation times are usually based on expected times.

– Anomaly conditions (machine failures and the like) are ignored.

– The simulation is run only until the required production has been met.



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• Simulation can work with material requirement planning (MRP) advancedrequirement planning (MRP), advanced planning and scheduling (APS), and manufacturing execution system (MES) to simulate the plans generated by these systems in order to test their viability.

• It can also produce a detailed schedule• It can also produce a detailed schedule showing which resources were used and at what times.

Real‐Time Control

• During actual production, simulation has been integrated with manufacturing cells to perform real‐g g ptime analysis such as next task selection and dynamic routing decisions.

• Benefits of using the simulation for cell management:– Logic used in the simulator does not need to be recorded for the controller.A i ti biliti ll t b it d– Animation capabilities allow processes to be monitored.

– Statistical‐gathering capabilities of simulation automatically provide statistics on selected measures of performance.



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Emulation

• A special use of simulation in manufacturing, particularly in automated systems has been in theparticularly in automated systems, has been in the area of hardware emulation.

• As an emulator, simulation takes inputs from the actual control system, mimics the behavior that would take place in the actual system, and then provides feedback signals to the control systemprovides feedback signals to the control system.

• The feedback signals are synthetically created rather than coming from the actual hardware devices.

Manufacturing Modeling Techniques

• Modeling machine setup

• Modeling machine load and unload time

• Modeling rework and setup

• Modeling transfer machines

• Continuous process systems



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Modeling Machine Setup

• Machine setup: tooling and other machine dj t tadjustments

• Changing part type

• Setup– Time

Resources required to perform the setup– Resources required to perform the setup

Modeling Machine Load and Unload Time

• Approaches for handling load/unload ti itiactivities:

– Ignore them

– Combine them with the operation time

– Model them as a movement or handling activity



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Modeling Rework and Scrap

• DecisionsR j t th t– Reject the part

– Correct the defect at the current station

– Route the part to an offline rework station

– Recycle the part through the station caused the defect

• The first consideration when modeling rework operationsoperations– Determine whether they should be included in the model al all.

Modeling Transfer Machines

• Machines that have multiple serial stations th t ll t f th i k f d ithat all transfer their work forward in a synchronized fashion

• Transfer machines:– Rotary

– In‐lineIn line

• In‐line transfer machines



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• Issues:– Finding the optimum number of pallets in a closed, nonsynchronous pallet system

– Finding the necessary buffer sizes to ensure

Continuous Process Systems (1)

• Production of bulk substances or materials h h i l li id l ti t lsuch as chemicals, liquids, plastics, metals,

textiles, paper



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CHAPTER 16


Modeling Material Handling Systems

Material Handling Principles (1)

• Planning principle: The plan is the prescribed course of action and how to get there At acourse of action and how to get there. At a minimum it should define what, where, and when so that the how, and who can be determined.

• Standardization principle: Material handling methods equipment controls and softwaremethods, equipment, controls, and software should be standardized to minimize variety and customization.



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• Work principle: Material handling work ( l i ht t it ti(volume x weight or count per unit time x distance) should be minimize. Shorten distances and use gravity where possible.

• Ergonomic principle: Human factors (physical and mental) and safety must be considered in ) ythe design of material handling tasks and equipment.


• Unit load principle: Unit loads should be i t l i d t hi th t i lappropriately sized to achieve the material

flow and inventory objectives.

• Space utilization principle: Effective and efficient use should be made of all available (cubic) space. Look at overhead handling ( ) p gsystem.



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• System principle: The movement and storage system should be fully integrated to form a coordinated,should be fully integrated to form a coordinated, operational system that spans receiving, inspection, assembly, packaging, unitizing, order selection, shipping, transportation, and the handling of returns.

• Automation principle: Material handling operations should be mechanized and/or automated where feasible to improve operational efficiency increasefeasible to improve operational efficiency, increase responsiveness, improve consistency, and predictability, decrease operating costs, and eliminate repetitive or potentially unsafe manual labor.


• Environmental principle: Environmental impact and energy consumption should beimpact and energy consumption should be considered as criteria when designing or selecting alternative equipment and material handling systems.

• Life cycle cost principle: A thorough economic analysis should account for the entire life cycleanalysis should account for the entire life cycle of all material handling equipment and resulting systems.



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Material Handling Classification

• Conveyors

• Industrial vehicles

• Automated storage/retrieval systems

• Carousels

• Automated guided vehicle systems

• Cranes and hoists

• Robots

Conveyors (1)

• A conveyor is track, rail, chain, belt. or some other device that provides continuous movement of loadsdevice that provides continuous movement of loads over a fixed path.

• Conveyors are generally used for high volume movement over short to medium distances.

• Some overhead or towline systems move material over longer distances.

• Overhead systems most often move parts individually on carriers, especially if the parts are large.



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Conveyors (2)

• Floor‐mounted conveyors usually move multiple items at a time in a box or containermultiple items at a time in a box or container.

• Conveyor speeds range from 20 to 80 feet per minute, with high‐speed sortation conveyors reaching speeds of up 500 fpm in general merchandising operations. C b ith it d• Conveyors may be either gravity or powered.

• Gravity conveyors are easily modeled as queues.

Conveyor Types

• Function or configuration

• Physical characteristics



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Conveyor Types by Function or Configuration

• Accumulation conveyor: Permitting loads to queue up behind each other.up behind each other.

• Transport conveyor: Transporting loads from one point to another (usually nonaccumulating)

• Sortation conveyor: Providing diverting capability to divert loads off the conveyor.

• Recirculating conveyor: Causing loads to recirculate until they are ready for diverting.

• Branching conveyor: Be characterized by a main line with either merging or diverting branches or spurs.

Conveyor Types by Physical Characteristics

• Belt conveyors

• Roller conveyors

• Chain conveyors

• Trolley conveyor

• Power‐and‐free conveyors

• Tow conveyors

• Monorail conveyors



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Operational Characteristics

• Load transport– A conveyor spans the distance of travel along which loads y p gmove continuously.

• Capacity– The capacity is determined by the speed and load spacing.– It is a function of the minimum allowable interload spacing (which is the length of a queue position n the case of accumulation conveyors) and the length of the conveyor.Intentional control may be imposed on the number of– Intentional control may be imposed on the number of loads that are permitted.

• Entity pickup and delivery– Conveyors usually don’t pick up and drop off loads.

Modeling Conveyor Systems (1)

• Modeling standpointA l ti– Accumulation conveyors

– Nonaccumulation conveyors

• Accumulation conveyor: A class of conveyors that provide queuing capability and permit independent or nonsynchronous part movement.

• Nonaccumulation conveyor Usually powered• Nonaccumulation conveyor: Usually powered conveyors having a single drive that all movement on the conveyor occurs in unison, synchronously.



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• Spacing characteristic:Fi d i– Fixed spacing

– Random load spacing

• Fixed spacing: Conveyors with fixed load spacing (segmented conveyors) that require loads to be introduced to the conveyor at fixed intervals.

• Random load spacing Conveyors permitting parts to• Random load spacing: Conveyors permitting parts to be placed at any distance from another load on the same conveyor.


16



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• Performance measures:– Throughput capacity

– Delivery time

– Queue length (for accumulation conveyors)

• Decision variables:– Conveyor speed– Conveyor speed

– Accumulation capacity

– Number of carriers


• Questions to be answered:Wh t i th i i d th t till t– What is the minimum conveyor speed that still meet throughput requirement?

– What is the throughput capacity of the conveyor?

– What is the load delivery time for different activity levels?

– How much queuing is needed on accumulation conveyors?

– How many carriers are needed on a trolley or power‐and‐y y pfree conveyor?

– What is the optimum number of pallets that maximizes productivity recirculating conveyor?



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Industrial Vehicles (1)

• Industrial vehicles include all push or powered carts and vehicles that generally have free movement.and vehicles that generally have free movement.

• Powered vehicles such as lift trucks are usually utilized for medium‐distance movement of batched parts in container or pallet.

• For short moves, manual or semipowered carts are useful.

• Single‐load transporters are capable of moving only one load at a time from one location to another.

• Multiple‐load transporters can move more than one load at a time. (e.g. picking transporters, towing transporters).

Variety of Industrial Vehicles (1)



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Variety of Industrial Vehicles (2)

Modeling Industrial Vehicles (1)

• Modeling an industrial vehicle involves a resource that moves along a path network.that moves along a path network.

• Paths are typically open aisles in which bidirectional movement is possible and passing is permitted.

• Deployment strategies (work searches, idle vehicle parking, etc.) must be capable of being incorporated into the model.

• Because industrial vehicle are often human‐operated, they may take breaks and be available only during certain shifts.



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Modeling Industrial Vehicles (2)

• Performance measures:– Vehicle utilization– Response time– Move rate capability

• Decision variables:– Number of vehicles– Task prioritization– Empty vehicle positioning

• Typical questions:h h d b f h l h dl h d– What is the required number of vehicles to handle the required

activity?– What is the best deployment of vehicles to maximize utilization?– What is the best deployment of empty vehicles to minimize response

time?

Automated Storage/Retrieval Systems (1)

• An automated storage/retrieval system (AR/RS) consists of one or more automated(AR/RS) consists of one or more automated storage/retrieval machines that store and retrieve material to and from a storage facility such as storage racks.

• The system operates under computer control.Th l i t id d hi h d it• The goal is to provide random, high‐density storage with quick load access, all under computer control.



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• An AR/RS is characterized by one or more aisles of racks storage having one or more storage/retrievalracks storage having one or more storage/retrieval (S/R) machines (sometimes called vertical or stacker cranes) that store and retrieve material into and out of rack storage system.

• Material is pickup up for storage by the S/R machine at an input (pickup) stationat an input (pickup) station.

• Retrieved material is delivered by the S/R machine to an output (deposit) station to be taken away.


• Usually there is one S/R machine per aisle; however, there may also be one S/R machine assigned to two y / gor more aisles or perhaps even two S/R machines assigned to a single aisle.

• For higher storage volume, some AR/RSs utilize double‐deep storage, which may require load shuffling to access loads.

• Where even higher storage density is required and l d d l d h dlonger‐term storage is needed, multidepth or deep lane storage systems are used.

• AS/RSs that use pallet storage racks are usually referred to as unit‐load systems, while bin storage rack systems are known as multiload systems.



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• The throughput capacity of an AR/SR is a function of the rack configuration and speed of the S/R machine. g p /

• Throughput is measured in terms of how many single or dual cycles can be performed per hour.

• A single cycle is measured as the average time required to pick up a load at the pickup station, store the load in a rack location, and return to the pickup station.

• A dual cycle is the time required to pick up a load at the input stations, store the load in a rack location, retrieve a load from another rack location, and deliver the load to the output station.


28



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Modeling AS/RSs (1)

• Typical input for precision modeling of single‐deep AS/RSs with each aisle having a captive S/R machine:– Number of aisles– Number of S/R machines– Rack configuration (bays and tiers)– Bay or column width– Tier or row height– Input point(s)– Output point(s)Output point(s)– Zone boundaries and activity profile is activity zoning is utilized– S/R machine speed and acceleration/deceleration– Pickup and deposit times– Downtime and repair time characteristics

Modeling AS/RSs (2)

• Common performance measures:– S/R machine utilizationS/R machine utilization– Response time– Throughput capability

• Decision variables:– Rack configuration and number of aisles– Storage and retrieval sequence and priorities– First‐in, first‐out or closest item retrieval– Empty S/R machine positioning– Aisle selection (random, round robin, or another method)



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Modeling AS/RSs (3)

• Typical questions:– What is the required number of aisles to handle the qrequired activity?

– Should storage activity be performed at separate time from retrieval activity, or should they be intermixed?

– How can dual cycling (combining a store with a retrieve) be maximized?

– What is the best stationing of empty S/R machines to minimize response time?p

– How can activity zoning (that is, storing fast moves closer than slow movers) improve throughput?

– How is response time affected by peak periods?

Carousels

• A carousel is one class of storage and retrieval tsystem.

• Carousels are essentially moving racks that bring the material to the retriever (operator or robot) rather than sending retriever to the rack location.



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Carousel Configurations

• A carousel storage system consists of a collection of bins that revolve in either acollection of bins that revolve in either a horizontal or vertical direction.

• The typical front‐end activity on a carousel is order picking.

• If a carousels are used for WIP storage, bins i ht t d l th lmight enter and leave the carousel.

• A variation of the carousel is a rotary rack consisting of independently operating tiers.

Modeling Carousels

• Carousels are easily modeled by defining an appropriate response time representing the time forappropriate response time representing the time for the carousel to bring a bin into position for picking.

• In addition to response times, carousels may have capacity considerations.

• The current contents may even affect response time, especially if the carousel is used to store multipleespecially, if the carousel is used to store multiple bins of the same item such as WIP storage.



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Automated Guided Vehicle Systems (1)

• An automated guided vehicle system (AGV) is th t k l hi h ta path network along which computer‐

controlled, driverless vehicle transport loads.

• AGV systems are usually used for medium activity over medium distances.

• If parts are large they are moved individually;If parts are large, they are moved individually; otherwise parts are typically consolidated into a container or onto a pallet.

Automated Guided Vehicle Systems (2)

• Operationally, an AGV is more flexible than conveyors, which provide fixed‐path, fixed‐point pickup andwhich provide fixed path, fixed point pickup and delivery.

• However, they cannot handle the high activity rates of a conveyor.

• On other extreme, an AGV is not as flexible as industrial vehicles, which provide open‐path, any‐

i i k d d lipoint pickup and deliver.• An AGV can handle higher activity rates and eliminate the need for operators.



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Layout of Paths for the AGV System

Modeling an AGV (1)

• Modeling an AGVS is very similar to modeling i d t i l hi l ( hi h it i i )an industrial vehicle (which it is in a sense)

except that the operation is more controlled with less freedom of movement.

• Path are generally unidirectional, and no vehicle passing is allowed. p g



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Modeling an AGV (2)

• Common performance measures:– Resource utilizationResource utilization– Load throughput rate– Response time– Vehicle congestion

• Decision variables:– Number of vehicles– Work search rules– Park search rules– Placement of crossover and bypass paths

Modeling an AGV (3)

• Typical questions:– What is the best path layout to minimize travel time?p y– Where are the potential bottleneck areas?– How many vehicles are needed to meet activity requirement?

– What are the best scheduled maintenance/recharging strategies?

– Which task assignment rule maximize vehicle utilization?What is the best idle vehicle deployment that minimizes– What is the best idle vehicle deployment that minimizes response time?

– Is there any possibility of deadlocks?– Is there any possibility of collisions?



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Cranes and Hoists

• Cranes are floor‐, ceiling‐, or wall‐mounted mechanical devices for short‐ to medium‐distance, ,discrete movement of material.

• Cranes utilize a hoist mechanism for attaching and positioning loads.

• Aside from bridge and gantry cranes, most other types of cranes and hoists can be modeled as an industrial vehicles.

• A bridge crane consists of a beam that bridges a bay (wall to wall) and moves on two tracks mounted on either wall.

Crane Management (1)

• Managing crane movement requires an d t di f th i it f l d t bunderstanding of the priority of loads to be

moved as well as the move characteristics of the crane.

• One must find the optimum balance of providing adequate response time to high‐p g q p gpriority moves while maximizing the utilization of the cranes.



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Crane Management (2)

• Typical rules for managing bridge cranes that tend to minimize the waiting time of loaded cranes andminimize the waiting time of loaded cranes and maximize crane throughput. – Cranes moving to drop points have priority over cranes moving to pickup points.

– If two cranes are performing the same type of function, the crane that was assigned its task earliest is given priority.

– To break on rule 1 and 2, the crane closest to its drop‐offTo break on rule 1 and 2, the crane closest to its drop off point is given priority.

– Idle cranes are moved out of the way of cranes that have been given assignment.

Modeling Bridge Cranes (1)

• For single cranes a simple resource can be defined that moves along a path that is resultant of the x andthat moves along a path that is resultant of the x and y movement of the bridge and hoist.

• ProModel provides a powerful bridge crane modeling capabilities that greatly simplify crane modeling.

• Common performance measures:– Crane utilization– Load movement rate– Response time– Percentage of time blocked by another crane



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• Common performance measures:C tili ti– Crane utilization

– Load movement rate– Response time– Percentage of time blocked by another crane

• Decision variables:– Work search rules– Park search rules– Multiple‐crane priority rule


• Typical questions:– Which task assignment rules maximize crane utilization?

– What idle crane parking strategy minimizes response time?

– How much time are cranes blocked in a multicrane system?



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Robots (1)

• Robots are programmable, multifunctional manipulators used for handling material ormanipulators used for handling material or manipulating a tool such as a welder to process material.

• Classification by the type of coordinate system:– Cylindrical or polar robots (for machine loading)

Cartesian coordinate robots (for assembly work)– Cartesian coordinate robots (for assembly work)

– Revolute or anthropomorphic coordinate robots (for a processing tool)

Robots (2)

• Cartesian or gantry robots can be modeled easily as cranes.easily as cranes.

• When used for handling, cylindrical or revolute robots are generally used – to handle a medium level of movement activity over very short distances

– to perform pick‐and‐place or load/unload f tifunctions.

• Robots generally move parts individually rather than in a consolidated load.



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Robots (3)

• One of the applications of simulation is in designing the cell control logic for a robotic work cell (a machining, assembly, inspection, or a combination cell).

• Robotic cells are characterized by a robot with 3 to 5 degrees of freedom surrounded by workstations.

• The workstation is fed parts by an input conveyor or other accumulation device, and parts exit from the cell on a similar device.

• Each workstation has one or more buffer positions which• Each workstation has one or more buffer positions which parts are brought if the workstation usually is busy.

• A robotic cell usually handles more than one part type, and each part type may have different sequence of workstations.

• In addition to part handling, the robot may be required to handle tooling and fixture.

Sequential Robot Operations Serving Two Machines



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Modeling Robots (1)

• In modeling robots, it is sometimes presumed that the kinematics of the robot need to be simulated.the kinematics of the robot need to be simulated.

• Kinematic simulation is a special type of robot simulation used for cycle time analysis and offline programming.

• For discrete event simulation, it is necessary to know only the move times from every pickup point to

d i ievery deposit point.• The advantage to having a specific robot construct in a simulation product is primarily for providing the graphic animation of the robot.

Modeling Robots (2)

• Common performance measures:– Robot utilization– Response time– Throughput

• Decision variables– Pickup sequence– Task prioritization– Idle robot positioningp g

• Typical questions:– What is priority of tasks results in the highest productivity?– Where is the best position for the robot after completing each particular drop‐off?



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CHAPTER 17


Modeling Service Systems1

Characteristics of Service Systems

• Capricious entities

• Random and fluctuated entity arrivals

• Complex resource decisions

• Fluctuated resource work pace

• Highly variable processing times

• Front‐room and back‐room activities of services



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Performance Measures (1)

• Performance measures: – External performance criteria

– Internal performance criteria

• External performance criteria: – Maximizing profits

– Maximizing customer satisfaction– Maximizing customer satisfaction

• Simulation modeling and analysis: internal performance criteria

Performance Measures (2)

• Typical internal performance measures:– Service time

– Waiting time

– Queue length

– Resource utilization

– Service level (the percentage of customers whoService level (the percentage of customers who can be promptly serviced without any waiting)

– Abandonment rate (the percentage of impatient customers who leave the system)



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Use of Simulation in Service Systems

• Design Decisions

• Management Decisions

5

Design Decisions

• How much capacity should be provided in service and waiting areas?

• What is the maximum throughput capability of the service system?

• What are the equipment requirements to meet service demand?

• How long do customers have to wait before being serviced?• Where should the service and waiting areas be located?• How can work flow and customer flow be streamlined?• What effect would information technology have on reducing

non value‐added time?



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Management Decisions

• What is the best way to schedule personnel?Wh t i th b t t h d l• What is the best way to schedule appointments for customers?

• What is the best way to schedule carriers or vehicles in transportation systems?

• How should specific jobs be prioritized?• What is the best way to deal with emergency situations when a needed resource is unavailable?

Applications of Simulation in Service Industries

• Process design

• Method selection

• System layout

• Staff planning

• Flow control



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Types of Service Systems

• Service factory

• Pure service shop

• Retail service store

• Professional service

• Telephonic service

• Delivery service

• Transportation service

Service Factory

• Systems in which customers are provided services using equipment and facilities requiring low laborusing equipment and facilities requiring low labor involvement

• Low labor costs, high equipment and facility costs• Front‐room and back‐room activities• Primary factors: waiting time, service time, convenience of location

• Examples: banks (branch operations), restaurants, copy centers, barbers, check‐in counters of airlines, hotels, car rental agencies



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Pure Service Shop

• Longer service times than service factories• Greater service customization• Greater service customization• Short front‐room activity time, long back‐room activity time

• Primary consideration: quality of service• Secondary consideration: delivery time, priceE l h it l i h• Examples: hospitals, repair shops (automobiles), equipment rental shops, banking (loan processing), courtroom, prisons

Retail Service Store

• Large facility• Many product options of customersMany product options of customers• High degree of labor intensity, low degree of customization

• Primary factor influencing customers: price• Secondary factors: service quality, delivery time• Interesting factors for customers: convenience of location assistance with finding products quicklocation, assistance with finding products, quick checkout

• Examples: department stores, grocery stores, hardware stores, convenience stores



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Professional Service

• Services provided by a single person or a small f t i ti l fi ldgroup of experts in a particular field

• High customization, expensive resources

• Duration of the service: long, extremely variable, and difficult to predict

• Examples auditing services tax preparation• Examples: auditing services, tax preparation, legal services, architectural services, construction services, tailor services

Telephonic Service

• Services provided over the telephone

• No face‐to‐face contact with the customers

• Issues: overflow calls, reneges, redials

• The most important criterion: delivery time

• Examples: technical support services (hotlines), mail‐order services, airline and hotel reservations



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Delivery Service

• Ordering, shipping, and delivery of goods, raw t i l fi i h d d t t i t fmaterials, or finished products to points of use

or sale

• Time windows constraints for customers

• Consideration factors for customers: convenience fast delivery quality of theconvenience, fast delivery, quality of the products (perishable or fragile goods)

Transportation Systems

• Movement of people from one place to anotheranother

• Fixed routes• Fast transportation• Two types of systems: multiple pickup and drop‐off points, single pickup and drop‐off points

• Examples: airlines, railroads, cruise lines, mass transit systems, limousine services



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