E X A M I

IOE 474 Review

Lecture 1

Components of a System Entity - Customers Attribute – Buying Habits (groceries) Activity - Strolling through aisle State of a system - # customers in each

aisle Event – (started/finished aisle) Endogenous/exogenenous (activity,

event) – Utilization of aisles by organization of merchandise

Dynamic Model Represents a system as it changes over

time (performance of a checkout counter depends on customer arrival rate)

No random variables – inputs result in know outputs

Static Model Represents a system at a particular point

in time, time independent (closing price of stock, outcome of dice)

Random variables – represent probabilities and uncertainties of system

Discrete Model’s states change only at discrete set of

points in time – number of cars in parking lot

Continuous Model’s states change continuously in time –

Water level in reservoir, weight of person

12 Steps in a Simulation Study Formulation (What is the problem?) Define Project Goal and Plan (Is simulation

appropriate?) Model Conceptualization Or Data Collection Model Translation Verified? (Is code ok?) Validated? (Represents system well?) Experimental Design (What runs to make to

answer question efficiently?) Production Runs & Analysis (Estimate the

performance measures) More Runs? Documentation & Reporting (Scope of this class) Implementation

Lecture 2

Simulation Software Work? System State – a collection of variables

that contain all the information necessary to describe the system

FEL – Future Event List, contains all event notices scheduled to occur in the future

Basic Concepts for Simulation Clock – variable representing

simulated time List – logically ordered collection of

associated entities (queue) Event List – List of event notices

(current events, future events, blocked list)

Activity – specified duration of time, IAT and ST

Delay – unspecified duration of time (wait)

Variables – values for calculations

Set An ordered list of associated entities or

events (queue, chain) Clock

A global variable representing time in simulation

Steps in Discrete Event Simulation Determine what event will occur next Set simulation clock equal to the time

of next event Update statistical variables Perform actions (and computations)

associated with event Schedule events caused by this event

/event list Future Event List

FEL is ordered by time chronologically

Lecture 3

Random Number Generation For value to follow a certain

distribution Use value obtained by Rand() function

as the Cumulative Prob and use an inverse function to obtain the corresponding percentile

Framework Determine the characteristics of each

of the outputs Determine the activities, events and

system states Determine model responses or

performance measures Construct a simulation table

Step, Xn, For each step i, generate a value for

each activities, compute system states and outputs

When simulation ends, use the outputs to compute measures of performance

Lecture 4

Modules in Build Menu Locations – physical locations for machines,

queues, conveyors, resources Entities – people, products, which move through

or are processed by system Processes – that which happens to an entity at a

location Arrivals – the way entities enter a system Resources – required to perform operations, but

without physical location Path Networks – a map for resource motion Attributes – variables for entity or location Variables/arrays/tables/subroutines/macro/file

– just like in other programming languages Streams – control over mechanism for getting

random numbers Entities

Items that are processed by the system, such as products, pallets, or even paperwork

Entities move between locations according to processing logic

Dummy entities can be used to make simulation function correctly

Arrivals Arrival defines the entry of entities into the

system Frequency – IAT

Locations Capacity – refers to the maximum number of

entities at a location Units – refers to the number of similar locations

in a grouping Stats – define what level of detail to collect

(None, basic, time series) Rules – Select Incoming (which entry gets

admittance when selecting all entries that wish to enter), Queuing for output (determines who can compete to go into next location), Selecting a unit (only for more than one unit)

Downtimes – clock downtime, entry downtime, usage downtime, setup downtime

Processes Define the routing of entities through the system

and what operations take place Processes, Operations, Routings

Routings Entities get routed to a location according to a

rule, user condition Move

No routing blocks in operation logic, then all routing will be executed in sequence upon completion of operation logic

Lecture 5

Global Variables Can be accessed from

anywhere in the model Created using variables edit

table Local Variables Created within a logic Type of a local variable is

defined by using “INT” or “REAL” statements

Private to each entity logic processing. Each entity going through this logic creates its own local variable

Statements Wait <time>: simulates the

time it takes to process an entity

Wait Until <Boolean Expression>: Delays the further processing until the Boolean expression is true

Move for <time>: simulates the move time

Combine <number>: accumulates and combines number of entities into a single entry

INC <variable> / DEC <variable>

Lecture 6

Four Steps in Developing an Input Model Collect data from the real system Identify a probability

Develop a frequency distribution or histogram of the data

Choose a family of distributions Choose parameters Evaluate distribution

Discrete Bounded [a,b] distributions Discrete uniform – models complete uncertainty

(equal outcome) Binomial – Number of successes in n trials

Discrete Unbounded [0,infinity] Poisson – number of independent events that

occur in a fixed amount of time or space Negative Binomial – Number of trials required to

achieve k successes Continuous on (-inf,inf)

Normal – distribution of a process that can be thought of as the sum of a number of component processes

Continuous on [0,inf) Exponential – time between independent events,

memory less

Q-Q Plot construction Step 1: Order the data

X(j) – smallest to largest J – n

Step 2: q=(j-0.5)/n q(j)

Step 3: Compute gamma(y) based on the inverse of F(x).=F-1(q(j))=q(j)=gamma(j)

Step 4: Plot gamma(j) vs X(j) Yi’s are known as order statistics If F is from an appropriate distribution, should

create a straight line Continuous on [0,inf)

Gamma – an extremely flexible distribution used to model nonnegative random variables

Erlang – Models processes that can viewed as the sum of exponentially distributed processes

Continuous on [0,inf) Weibull – models the time to failure Lognormal – process that can be thought of as

the multiplication of a number of component processes

Continuous on [a,b] Beta – extremely flexible distribution used to

model bounded random variables Continuous uniform – complete uncertainty Triangular – minimum, most-likely, and

maximum values are known

Lecture 7

MLE – used to estimate the parameters we need for the distribution we have chosen. (choosing the parameters)

Testing the Distribution Evaluate distribution and associated parameters If fit is poor – repick distribution family

Chi-Squared Goodness of fit test X1, X2, … Xn are n i.i.d. samples from RV X H0: X has pdf f(x) Definte pi=p(ai-1<=x<=ai)

Probability that an observation falls within the ith interval

Oi: numbers of observations that fall in the ithinterval Oi~Bin(n,pi) – Binomal-Normal-Chi-Square

Ei=E[Oi] – expected numbers of observations that fall in the ith interval

Guideline – choose n and pi Calculate chi-squared statistic :∑(Oi-Ei)^2/Ei Compare to X^2(k-1-s),alpha

S: number of parameters estimated Alpha: significance level

If x^20>X^20(k-1-s,alpha) Reject H0 Picking ai’s

Equal probability – continuous Equal interval – ai-ai+1=ai+a-ai (discrete)

Lambda = 1/Xbar

K-S Goodness of Fit Test Formalize the idea behind the q-q plot Small sample sizes Designed to test: H0:X1,X2,…Xn CDF of U(0,1), F(x)=x, 0<=x<=1 Sn(x) : the accumulated counts of Xis until x,

normalized by the total sample size, N Sn(x) = Counts of samples X<=x/n

Steps of KS test for Uniform CDF 1 – Rank the Ris from smallest to largest 2- Compute

D+=max{i/N-Ri}, for I from 1 to N D-=max{Ri-(i-1)/N}, I from 1 to N Sample statistic – D=max{D+,D-}

Find critical value, Dalpha If D>Dalpha, reject H0

P-Values Significance level (alpha) – prob of falsely reject

H0 Higher the p-value, the better the fit is

Anderson Darling Weighted average of the squared differences

between observed and hypothesized distributions

Weight function: 1/F(x)*(1-F(x))

Lecture 8 – Logic Elements

General Functions Math functions

+ - * / Mod ** INC, DEC EXP(), LN(), Sqrt() Rand(), Round() FORMAT() Real(0, Trunc

Distribution Functions Discrete:

BI(n,p) GEO(p) P(frequency): poisson

Continuous E(m) – exponential G(a,b): Gamma(shape, scale) U(m, hr): Uniform(mean, half range) N(m,s): Normal(mean, stdev)

Actions and Statements General Actions

Activate, Animate, Debug, Display, DynPlot, Log Pause, report, reset Return, send, sound, trace

General Control Statements If. Then. Else Begin End, Break, Do..Until, Do.. While, Goto

General Options Stop, graphic, prompt

System Functions General

Cap, contents, freecaps, freeunits, entries, units Percentop, percentUtil, timeused Clock, CalMin, CalHour, Calday…. ArrayDims

Entity-specific: Groupqty, resqty, ownedresource Resource-Specific: Last, Next Downtime-Specific: Down, DTLeft, ForLocation Cost: GetCost, GetResRate, IncEntCost

Entity/Resource Related Operation Statements Resource related (Free, Get, Jointly Get, Use) Entity-Related (Accum, Group, create, route,

move, move for, wait, wait until, match) Distinguishing Entities

Method 1: Multiple entities Method 2: Attributes Method 3: Priorities

Lecture 9

Resources General: human labor,

special tools, perform processing on entities, maintain locations and resources, etc

Transporters: human labor, AGV, forklift, etc. To move entities from one location to another

Static: Do not move virtually (stay where they are created during the animation)

Dynamic: Do not have a fixed place to stay, move on a network path

Path Networks Paths of motion for resources

Using Resources USE: get and use resources

for specified period of time GET: Get resources as mad

available JOINTLY GET: Get none

until all requested resources are available at once

FREE: Release resource

Lecture 10 – Path Networks and Dynamic Resources

Path Networks Used to having dynamic

resources move on them Entities move themselves

– can be assigned to move on a path network as well as in move logic

Types: Passing, non-passing, and crane

Using Dynamic Resources USE, GET, JOINTLY GET

are used the same way as in the static resources

MOVE WITH: gets the available resource, use it to move the entity from one location to another

Lecture 11 – Random Number Generators

ProModel Distribution Functions Continuous including

Uniform: RAND(X) is uniform [0,x) Uniform: U(a,b), a=mean, b=half-range Normal: N(a,b), a=mean, b=standard deviation

Discrete Binomial: BI(a,b) Geometric: Geo(a) Poisson: P(a), a=mean

Stream A long circular list of ‘random’ numbers Numbers picked from list in order Usually numbers are uniform (0,1) How streams affect Simulations?

Outcome 1: The same system seems different Outcome 2: The better system seems worse

Random Number RN streams created by RNG RNG ideas

Random time Fractional part of square roots of prime numbers Pseudo-random number generators

A good RNG produces RN fast, long cycle, replicable, repeatable

X0 is called the seed – starting point

Linear Congruential Method: Iterate: Xi+1 = (A*Xi+C) mod M

A- multiplier Xo – seed (starting point) C – Increment M – Modulus (large prime number)

Tests for Pseudo Random Numbers Uniformity (order doesn’t matter) Independence (order does matter)

Uniformity Test Null: H0: Xi’s ~ U(0,1) Alternative: H1: Xis not U(0,1)

Chi-squared goodness of fit Kolmogorov Smirov goodness of fit (preferred)

Independence Test Ho: Xis are independent, H1: Xis not

independent Alpha=P(type I error) = P(reject H0|H0 is

true) Alpha is commonly chosen to be 0.1 or 0.05 Two-tallied, alpha is halved

Chi-squared Goodness of fit H0: Ris ~ U(0,1), H1: Ris not U(0,1)

Runs test for independence H0: Ris are independentt/H1: are not

independent

Lecture 12 – Routing/debugging

Advanced Routing Locations – Rules Processing – Routing – Rule ROUTE Statement

Incoming Entities Oldest by Priority – waiting longest Random Least available capacity Last selected location – Oldest by

priority Highest Attribute value/Lowest

Queuing for Output No Queuing – entities that have

completed their operations are free to process any routing logic independent from other entities that have finished their operations

FIFO LIFO By Type Highest/lowest attribute

Trace Lists events happening in the

simulation Can check “trace” option under

simulation/options to start trace automatically when the simulation starts

Debug DEBUG statement brings up

ProModel’s debugger Step through logic one statement at a

time and examine the values of variables and attributes

WAIT UNTIL statement WAIT UNTIL <condition>

Delays processing for the current block of statements until condition is true

Can use variables, arrays, location attributes

Cannot use system functions, entity attributes

Homework 1

Spaghetti dinner simulation Activities

Fill pan with water Heat water Cook Noodles Drain Noodles Wait 5 minutes for

noodles to cool

Events Turn on water Turn off water Turn on stove Water boils Add noodles to water Turn off stove Start draining noodles Start eating

State variables Temperature of water,

location of noodles, time

Homework 2

Q1 IAT: Lookup(Rand(), CDF, Time

between Arrivals) Service Time: Lookup(Rand(),

CDF, Service Time) Drive in Departure Time:

IF(Drive-in Served = “YES”, MAX(Arrival Time, Sum of Previous Departure times) + Service Time, -1)

Drive in served: IF(Arrival Time > LARGE(Previous Departure times), “Yes”, “No”)

Q2 – service station with 3 available spots, FIFO, no spots, leave service Entities: Customers Attributes: Arrival time, car type,

how much money they spend Events: Arrival of customer (A),

departure of a customer (D) Activities: Customer being waited,

being served Event Notice

Event notice (Ai,t): the ithcustomer arrived at time t.

Event notice (Di,t): the ith

customer departed at time t. System States:

LQ(t): customers in queue at the service station

LS(t): 1 or 0 indicating worker being busy or idle.

Homework 3 – ProModel

Good.

Homework 4

Homework 5 – ProModel

If pizza is bad more than twice – throw away Attribute: Fail_Pizza Variable: GoodPizza

Move Logic: INC GoodPizza, INC Fail_Pizza

RULE: TRM – If Fail_Times < 2, 1

Trashcan - if Fail_Times = 2

Homework 6 – ProModel

Homework 6 Continued