Reno Training

Training Guide

Version 1

RENO 1 Training GuidePart Identification: RRE1S-TG-01

ReliaSoft CorporationWorldwide Headquarters1450 South Eastside LoopTucson, Arizona 85710-6703, USASales and Information: [email protected]://www.ReliaSoft.com

© 2005-2009 ReliaSoft Corporation, ALL RIGHTS RESERVED.

Notice of RightsNo part of this document may be reproduced or transmitted, in any form or by any means, for anypurpose, without the express written permission of ReliaSoft Corporation, Tucson, AZ, USA.

DisclaimerInformation in this document is subject to change without notice and does not represent acommitment on the part of ReliaSoft Corporation.

Companies, names and data used herein are fictitious unless otherwise noted.

Use of the software and this document are subject to the terms and conditions set forth in theaccompanying License Agreement.

This software and documentation were developed at private expense; no portion was developedwith government funds.

TrademarksReliaSoft and RENO are trademarks of ReliaSoft Corporation.

Product names and services identified in this document are trademarks of their respectivetrademark holders, and are used for illustration purposes. Their use in no way conveysendorsement or other affiliation with ReliaSoft Corporation.

10 9 8 7 6 5 4

IMPORTANT NOTEREGARDING REGIONAL SETTINGS

Examples in this document assume that you are using the USA regional settings where thedecimal point is a period (.) and the list separator is a comma (,).

RENO is regionally aware (i.e. it will work with your regional settings). Therefore, if yourregional settings use a different decimal definition and list separator, the way that you entervalues and functions will differ from the way presented in this manual. To reproduce theexamples in this document, you must adjust the inputs accordingly.

For example, under USA regional settings, the function f(10.3,2.7) is some function of f withtwo function arguments of 10.3 and 2.7. If you are using German regional settings, this functionshould be entered based on the German regional decimal definition (,) and list separator (;). Thatis, the function would be f(10,3;2,7).

RENO Training Guide 1

1.1 About this Training Guide

This training guide is intended to provide you with many examples to demonstrate the use of RENO. Itbegins with step-by-step examples and then proceeds into more advanced examples. At any time during thetraining, please feel free to ask the instructor(s) any questions you might have.

The examples in this training guide are presented in files that have been shipped with the RENO application.These files are located in subfolders in the Examples folder in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples). The Examples folder is also accessible by clicking the Open ExamplesFolder... link in the What do you want to do? window or by selecting Open Examples Folder... from theHelp menu. Each file contains the complete solution to the problem presented in the example.

Note that the examples included in this guide are not necessarily the most efficient way of accomplishing theunderlying task using RENO. Rather, they are designed to demonstrate how to use various features ofRENO.

1.2 RENO Documentation

Like all of ReliaSoft's standard software products, RENO is shipped with detailedprinted documentation on the product (RENO User's Guide). This training guide isintended to be a supplement to that reference.

1.3 Contacting ReliaSoft

ReliaSoft can be reached at:ReliaSoft CorporationWorldwide Headquarters

1450 South Eastside LoopTucson, AZ 85710-6703 USA

Phone: +1.520.886.0366Fax: +1.520.886.0399

E-mail: [email protected]

For up-to-date product information, visit our Web site at:http://RENO.ReliaSoft.com

Introduction 1


The examples in this training guide have been designed to introduce you to the features available in RENO.This section presents a brief summary of these features. If you are already familiar with RENO’scapabilities, you can proceed to Chapter 4, First Steps.

You can think of RENO as a graphical “stochastic event simulation tool.” It is a powerful and flexibleplatform for visualizing and dynamically simulating nearly any kind of physical, financial or organizationalsystem. Using the familiar flowcharting concept, you can build systems and/or scenarios and then have thesoftware execute the model via simulation. You may also think of RENO as a “visual spreadsheet” thatallows you to graphically create and manipulate data and equations, but with greater flexibility due to itsability to evaluate systems and scenarios that evolve over time.

2.1 Easy to Build Flowchart Models

With RENO, you will use the familiar and intuitive flowcharting concept to build models for the scenariosthat you are trying to understand and/or quantify via simulation. The software provides a full array ofbuilding blocks for these Flowcharts (called “Constructs”) to serve different functions within the model. Forexample, Conditional Blocks allow you to define “If/Then” statements that trigger one response if thecondition is met (TRUE) and another if it is not (FALSE). You can also define various types of globalvariables (called “Definitions”) that are available throughout the project. For example, Random Variablesprovide values based on a specified statistical distribution. You will employ simple point-and-click anddrag-and-drop techniques to use these components to build simple or complex Flowchart models.

2.2 Flexible Interface to Organize the Analysis

RENO’s flexible user interface makes it easy to enter and manage the information required to performanalyses. Using the intuitive, hierarchical "Project Explorer" approach that was introduced in ReliaSoft'sBlockSim software, RENO makes it easy for you to manage one or many Flowchart models together withthe Definitions and Constructs used to build them. The optional Template feature allows you to pre-defineConstructs that can be used in multiple Flowcharts to save time and ensure consistency. Your projects can also include Excel-compatible spreadsheets for your own custom analyses. Thesespreadsheets provide complete in-cell formula support, over 140 built-in functions and a wizard to help youbuild your own custom graphical charts. In addition, the Attachments feature makes it easy to keepsupporting documentation and other related files together with the analysis.

2.3 Item Properties Table to View/Edit Project Information in a Spreadsheet

The Item Properties Table allows you to view and edit the data in your project in an Excel-compatiblespreadsheet format. This makes it easy to quickly update analysis properties without having to open andsave each record individually. It also allows you to export or copy/paste this information to use in reports oryour own custom analyses.

Features Summary 2

2 Features Summary

4 http://RENO.ReliaSoft.com

2.4 Definitions Report to Manage Analysis Components

The Definitions Report provides a summary of the global variables that have been defined in your project.This report identifies the Flowcharts and other Definitions that reference each defined variable. The utilitycan also automatically remove unused Definitions from your project with the click of a button.

2.5 “Intelligent” Features to Help You Build Equations

All equation-enabled windows in RENO provide “intelligent” features to help you save time and avoiderrors when building equations. These include:

Auto-Complete/Intelli-Sense: When you press Ctrl+Space while building an equation, RENO provides alist of recognized variables and functions.

Color-Coding: As you type, RENO automatically color-codes recognized variables, functions andoperands to make it easy for you to spot typing errors, undefined variables, unmatched parentheses, etc.

Variable Preview: RENO provides a summary of the properties of each Definition that you use in yourequations and allows you to view/edit the Definition’s properties with the click of a button.

2.6 Integration with Weibull++ and ALTA

If ReliaSoft’s Weibull++ and/or ALTA software are installed on your computer, you can use them to definethe distribution and parameters for Probabilities and Random Variables based on the distributional analysisfor existing data sets. The Parameter Experimenter is also available.

2.7 More Than 100 Reserved Keywords and Built-In Functions

The integrated Function Wizard provides instant access to the reserved keywords, Definitions and built-infunctions that you will use in the equations for your analyses. This includes:

Definitions: A list of the Constants, Tables, Storage Variables, Equation Variables, Functions,Probabilities and Random Variables defined in the project.

Reserved Keywords: A list of keywords that instruct the software to insert specific information into theequation, such as the value passed from the previous Flowchart Construct (IN), the total number ofsimulations performed (SIMS_TOTAL), the maximum value in a table (MAX_TABLE), etc.

Pre-Defined Functions: A list of built-in functions supported by the software. This includes:

Engineering functions such as Bessel, Error functions, etc.

Math and Trigonometry functions such as Sine, Derivative, Integral, Logarithm, etc.

Financial functions such as Future Value, Cumulative Interest Paid, Periodic Payment for an Annuity,etc.

Statistical functions such as the Weibull distribution, the Fisher Transformation, the Chi-Squareddistribution, etc.

Logical functions such as IF statements and TRUE/FALSE statements.

2.8 Debugger Utility to Step Through Flowchart Simulations

RENO includes an integrated utility to help you validate and “debug” your Flowchart models. This featureallows you to move through each step in the simulation of a Flowchart and watch the values as eachConstruct is executed. You can also watch the progress as the software automatically performs a specifiednumber of simulations in “debug” mode.

2.9 Flexible Array of Simulation Results with Multiple Display Options


2.9 Flexible Array of Simulation Results with Multiple Display Options

RENO’s simulation engine can generate a wide variety of results for your analyses, including averages,sums, arrays, minimum values, maximum values, etc. Simply construct the Flowchart model to generate themetrics of interest to you and then perform the simulation. This information is presented by the software in avariety of ways, including:

During the simulation: You can configure the Simulation Console to display a specific variable ofinterest while the simulation is running and watch the variation as the simulation progresses.

In the Simulation Results Explorer: The Simulation Results Explorer displays the full results of thesimulation in Excel-compatible spreadsheets that present the data in an organized fashion and make iteasy to copy/paste or export results of interest to your reports or customized analyses.

In the Flowchart: After a Flowchart has been simulated, the final values for all Result StorageConstructs will be displayed directly within the chart. You also have the option to show the last valuesfor other Constructs, if desired.

In the Plot Sheet: RENO’s integrated Plot Sheet provides a graphical display of your simulation resultswith the click of a button. You can choose to display the data for any Result Storage Construct in theFlowchart and control other display options, such as scaling, fonts, colors, etc. The utility provides both2-dimensional and 3-dimensional plots to present analysis results in the most effective way to supportdecision-making. You can export plots as Windows metafile graphics that can be inserted into otherdocuments and/or use the RS Draw metafile editor to annotate and customize the plot graphics.

2.10 Sensitivity Analysis to Vary Constants Across Simulation Runs

RENO’s Sensitivity Analysis feature allows you to vary one or two constants across simulation runs. Forexample, if you are analyzing potential investment strategies, you may wish to vary the number of years thatyou will be investing and/or the percentage of your income that will be invested to see how different inputswill affect the final results. Likewise, you may wish to generate reliability or availability results for a rangeof times in order to examine the equipment’s behavior over time. Potential applications are limited only byyour imagination!

Simply enter the starting, ending and increment values for the variable constant(s) and then RENOautomatically performs the specified number of simulations under each set of conditions.

2.11 Perform Multiple Runs to Automatically Estimate Optimum Values

The Multiple Runs feature takes the sensitivity analysis capability a step further by allowing you toconfigure the software to automatically vary the specified constant(s) in order to determine the value thatminimizes or maximizes a specified result. For example, you could use this feature to automaticallydetermine the optimum amount of inventory to maximize profits or the optimum preventive maintenance(PM) interval to minimize costs. The results of each analysis can be displayed in the Plot Sheet, with theoptimum value marked on the plot.


This chapter briefly covers the Definitions and Constructs used to build Flowcharts in RENO and theterminology used for simulation. For complete information on the properties and use of Definitions andConstructs and on simulation, please consult the RENO User’s Guide.

3.1 Definitions

Definitions are global objects that can be called upon by any Flowchart within the project. They cannot,however, be placed directly into Flowcharts; instead, they are called by Constructs, which are placed intoFlowcharts as blocks. Constructs are presented in detail in Section 3.2. Definitions serve to store or generatevalues used in Flowcharts. While the values of some Definitions may change during the simulation process,once simulation has been completed, each Definition contains only the original values that you entered in itsproperties window. If you wish to keep the value(s) of the Definition used during simulation, you must use aResult Storage Block to store the value(s).

There are seven types of Definitions. A brief description of each is given next. The properties of eachDefinition can be specified in the properties window that appears when a new Definition is created, or youcan access the properties window at any time by double-clicking the Definition. In addition, the properties ofa Definition are displayed at the bottom of the Project Explorer when the Definition is selected.

Each type of Definition has an identifier, or name. All identifiers must meet the following requirements.

Must be unique within the project.

Must be text; cannot be a number or anything that evaluates to a number, such as Pi or reservedkeywords.

Cannot contain spaces or any of the following characters:

\ / : * ? “ < > | ‘ ~ ! @ # $ % ^ & ( ) ;

3.1.1 Constants

A Constant stores a fixed numerical value that can be reused. Constants cannot be changed during eachsimulation, but can be varied across runs based on your selections on the Sensitivity Analysis page of theSimulation Console. After simulation, the Constant returns to its original value. The values of the Constantused during simulation are available in the Simulation Results Explorer.

Basics 3

3 Basics


3.1.2 Tables

A Table stores an array of values in rows and columns. Tables can have up to 64,000 rows and 256 columns.The data in a Table are available for both read and write during simulation by using reserved keywords.After simulation, the Table will contain its original data. If you wish to keep the values stored in the Tableduring simulation, you must use a Result Storage Block to store them.

3.1.3 Storage Variables

A Storage Variable stores a numerical value that is available for both read and write during simulation. Aftersimulation, the Storage Variable returns to its original value. If you wish to keep the value(s) stored in theStorage Variable during simulation, you must use a Result Storage Block to store the value(s).

3.1.4 Equation Variables

An Equation Variable stores a string expression to be evaluated. The expression can include numericalvalues, standard operands (+, -, *, / ), intrinsic functions (exp, log, sin, etc.), certain reserved keywords,

3.1 Definitions


references to any other Definition(s) except for Functions and references to any Spreadsheets that are namedin accordance with the naming conventions for Definitions.

Note that you must enter all required inputs for each part of the equation. For instance, a reference to a Tablerequires inputs for the column name/number and the row number. Table1(1,2) is a reference to the cell in thefirst column and second row of the Definition called Table1. References to Spreadsheets require inputs forthe sheet number, column name/number and row number. Some reserved keywords also require inputs.

3.1.5 Functions

A Function stores a string expression to be evaluated. Functions are identical to Equation Variables exceptthat a Function can contain arguments that are passed to it during simulation. For instance, the Functionshown here finds the sum of the arguments X1, X2 and X3. An equation of the form Function1(1,2,3) wouldreturn 6.

3 Basics


3.1.6 Probabilities

A Probability returns the probability that a random variable is observed by x. In other words, given x, theProbability returns F(x), where F(x) is the cumulative density function of the specified distribution. Thereturned probability will always be between 0 and 1.

3.1.7 Random Variables

A Random Variable is essentially the opposite of a Probability. Given F(x) (the probability), the RandomVariable returns x, based on a probability distribution.

3.2 Constructs

RENO’s Constructs are the building blocks with which you will create Flowcharts. Each Construct isrepresented by a block in the Flowchart and performs a step in the model. You can customize the appearanceof each block to enhance the presentation of your Flowcharts. Note that the terms “Construct” and “block”are used interchangeably in this guide.

In order to provide the flexibility required to create Flowcharts that model scenarios as realistically aspossible, there are twelve different types of Constructs. A brief description of each is given next. Theproperties of each Construct can be specified in the properties window that appears when a new block iscreated or you can access the properties window at any time by double-clicking the block. In addition, theproperties of a Construct are displayed in the Flowchart Information area when the block is selected.

3.2 Constructs


3.2.1 Standard Blocks

A Standard Block evaluates to a numerical value, based on the equation used in its Evaluates to input box.The expression can include numerical values, standard operands (+, -, *, / ), intrinsic functions (exp, log, sin,etc.), certain reserved keywords, references to Definitions and references to any Spreadsheets that are namedin accordance with the naming conventions for Definitions. A Standard Block may accept up to oneincoming path, and may have multiple outgoing paths.

The properties window for a Standard Block has three tabs or pages. Note that these pages also apply toSubchart Blocks.

3.2.1.1 Item Page

The Item page gives general characteristics of the block, including the equation used to evaluate the block.

Note that you must enter all required inputs for each part of the equation. For instance, a reference to a Tablerequires inputs for the column name/number and the row number. Table1(1,2) is a reference to the cell in thefirst column and second row of the Definition called Table1. References to Spreadsheets require inputs forthe sheet number, column name/number and row number. Some reserved keywords also require inputs.

3 Basics


3.2.1.2 Subchart Page

The Subchart page is used to specify whether the block is a Standard Block or a Subchart Block. Subchartblocks are presented in detail in Section 3.2.12.

3.2.1.3 Attachments Page

The Attachment page displays the attachments, if any, associated with the block. You can add, open or deleteattachments here.

3.2 Constructs


3.2.2 Result Storage Blocks

A Result Storage Block stores numerical values passed to it during simulation. It will accept any number ofincoming paths, and can have any number of outgoing paths. It can hold a single value or an array of values.If called during simulation, it will return the last value it holds.

The Required Number of Executed Paths field allows the simulation to continue when not all theincoming paths should/will be executed. The Required Number of Executed Paths field is presented in detailin Section 3.2.13.

3.2.3 Conditional Blocks

A Conditional Block functions as an "If" statement. That is, it checks the incoming value against a specifiedvalue or a Definition (except for a Function or Table) that evaluates to a numerical value. Possible outcomesare TRUE and FALSE. It accepts one incoming path and has up to two outgoing paths. The value(s) passeddown the outgoing path(s) may consist of a numerical value, a Definition or a reserved keyword. AConditional Block cannot be the final block in a Flowchart.

3 Basics


If Continue execution on both paths is checked, execution will continue on both paths regardless of theconditional outcome. If this option is not checked, execution will continue only on the path corresponding tothe conditional outcome.

3.2.4 Logic Gates

A Logic Gate accepts multiple incoming paths and has up to two outgoing paths. It checks incoming valuesagainst a specified value or a Definition (except for a Function or Table) that evaluates to a numerical value.Possible outcomes are TRUE and FALSE. The value(s) passed down the outgoing path(s) may consist of anumerical value, a Definition or a reserved keyword. A Logic Gate cannot be the final block in a Flowchart.

Gate Type dictates the conditions under which a TRUE or FALSE outcome will be returned. Possible typesare:

AND: the condition is TRUE for all incoming paths.

OR: the condition is TRUE for at least one incoming path.

XOR: the condition is TRUE for exactly one incoming path.

K-out-of-N: the condition is TRUE for at least K out of N incoming paths; K must be specified in theinput box below.

If Continue execution on both paths is checked, execution will continue on both paths regardless of theconditional outcome (i.e. whether the conditions of the gate type are met). If this option is not checked,execution will continue only on the path corresponding to the conditional outcome.

3.2 Constructs



3.2.5 Branch Gates

A Branch Gate accepts one incoming path and has multiple outgoing paths. It checks an incoming valueagainst a different condition for each outgoing path. The value(s) passed down the outgoing path(s) mayconsist of a numerical value, a Definition or a reserved keyword. A Branch Gate cannot be the final block ina Flowchart.

Each path is identified by the name of the next block in the path.

If Check condition on every branch is selected, the incoming value will be checked against every outgoingbranch. Execution continues down every path that evaluates to TRUE. If Continue execution on FALSEbranch is checked below, execution continues down every path, with the value that is passed down the pathdetermined by the conditional outcome.

If Check condition until first branch is TRUE is selected, the incoming value will be checked againsteach branch sequentially, based on the order displayed, and continues only through the first TRUE branch.Execution will also continue on any branch for which Continue execution on FALSE branch is checkedbelow, passing the Condition is FALSE value. You can use the arrows that appear when Check conditionuntil first branch is TRUE is selected to change the order in which the paths are evaluated.

3 Basics


3.2.6 Summing Gates

A Summing Gate can have multiple incoming paths and multiple outgoing paths. It performs a specifiedoperation (addition, subtraction, multiplication or division) on all incoming values and returns a single valueon all outgoing paths.

The order in which the incoming paths will be handled is managed in the Precedence field.

The Required Number of Executed Paths field allows the simulation to continue when not all theincoming paths have been executed. The Required Number of Executed Paths field is presented in detail inSection 3.2.13.

3.2.7 Flag Markers

A Flag Marker marks a spot in the path of the Flowchart. It may have multiple paths in and out. ThisConstruct is mainly intended to be used in conjunction with the Go to Flag Construct. A Go to Flag calls, orsets the simulation point to, a Flag Marker within the same Flowchart.

3.2 Constructs


3.2.8 Go to Flags

A Go to Flag sets the current simulation point to a named Flag Marker within the same Flowchart, thusallowing for loops inside the Flowchart. Note that it is up to you to avoid configurations that would createinfinite loops and other unresolved issues, as RENO cannot check this for you. Since Construct names donot have to be unique, it is also up to you to avoid ambiguity in which Flag Marker the Go to Flag points to.

3.2.9 Counter Blocks

A Counter Block records the number of times the simulation has passed through it.

Three types of Counter Blocks are available:

Global Counter if selected, the counter is never reset.

Simulation Counter if selected, the counter is reset after each simulation (i.e. after each single passthrough the Flowchart).

Run Counter if selected, the counter is reset after each run (i.e. after the specified number ofsimulations have been completed).


3 Basics


3.2.10 Reset Blocks

Each time the Flowchart is simulated, RENO generates new values for any Random Variables and/orProbabilities in the project. While simulation is in progress, these values are constant. A Reset Block forcesRENO to generate new values for Random Variables and Probabilities while simulation is in progress. Anyother Definitions that are dependent on these values are also recalculated.

3.2.11 Stop Flags

A Stop Flag halts the simulation process when it is encountered. No further simulations, runs or analyseswill be performed until you next simulate the Flowchart.

3.2.12 Subchart Blocks

A Subchart Block represents, in one block, the contents of a Flowchart. When inserting a Subchart Block,you must specify the Flowchart to which it refers. Circular references are not permitted (i.e. a SubchartBlock cannot refer to the Flowchart in which it is placed). Subchart blocks are identical to Standard Blocksexcept for the fact that they represent the contents of a Flowchart rather than an expression to be evaluated.

3.2 Constructs


The first page of the properties window for a Subchart Block differs from that of a Standard Block, as shownnext.

Note that any Flowchart used as a subchart must have a single starting point and a single ending point.

3.2.13 Required Number of Executed Paths

During a simulation, some paths may not be executed (e.g. a FALSE path following a Conditional Blockwhen the condition has evaluated to TRUE). Entering the required number of executed paths allows thesimulation to continue when not all of the incoming paths have been executed. As soon as the requirednumber of paths have been executed, the outgoing paths will be executed and the “memory” of the Constructwill be erased. An event will not be triggered again until the required number of executed paths is met again.

In the example shown next, the Summing Gate will send the result of A+B+C+D to all its outgoing pathsonce A, B, C and D have been executed.

If the user had specified 2 as the required number of executed paths and A, B, C and D were executed in thatorder, the Summing Gate would send results to its outgoing paths twice: once with a value of A+B and oncewith a value of C+D.

If there were only 3 incoming paths executed (A, B and C in that order) and the user had specified 2 as therequired number of executed paths, the Summing Gate would send the result of A+B to its outgoing pathsand would ignore C.

3 Basics


This field is used by Result Storage Blocks, Logic Gates, Summing Gates and Counter Blocks.

3.3 Simulation Terminology

RENO’s simulation process can involve several levels. The terminology for each level is as follows.

A simulation is a single pass through the Flowchart. Note that this may pass through blocks multipletimes if your Flowchart includes one or more loops.

A run is a set of simulations. The number of simulations in each run is set on the General page of theSimulation Console.

An analysis is a set of runs. The number of runs in each analysis is determined by the settings on theSensitivity Analysis page of the Simulation Console.

You can also choose to perform multiple analyses, using the settings on the Multiple Analyses page of theSimulation Console.


In addition to information on starting RENO and a brief overview of the software's main interface, thissection guides you through all of the steps involved in creating Flowchart models and obtaining resultsthrough simulation. Working through this example will help you to become familiar with the softwarequickly and easily.

4.1 Starting RENO

RENO is a 32-bit application that has been designed to work with Windows NT, 2000 and XP. The RENOinternal screens and commands are identical regardless of which operating system you are using, and thistraining guide is equally applicable. To start RENO, from Start select Programs, ReliaSoft Office and thenRENO.

4.2 Multiple Document Interface

Every time you begin a new session in RENO, the first window you will encounter is the MultipleDocument Interface (MDI). The MDI is the manager for RENO. It contains the windows you use to createyour project in RENO. The next figure displays the MDI of RENO and its components, which are explained

First Steps 4

4 First Steps


in detail next. The appearance of the MDI will vary depending on the window(s) that are currently open andthe configuration of the Project Explorer and Template Panel.

The status bar in the MDI may not be visible the first time the application is opened. To make the status barvisible, go to the View menu and select Status Bar. The status bar will be located at the bottom of the MDIand displays the number of Flowcharts within the project, the name of the currently active window, the pathand filename of the project and the amount of memory available on your computer.

The MDI serves as the container for all Flowcharts, Spreadsheets, Attachments, the Template Panel and theProject Explorer and manages the different active windows. The MDI remains open until you close theprogram. Closing the MDI has the same effect as terminating the program.

4.3 Getting Help in the RENO Environment

ReliaSoft's RENO includes complete on-line help documentation. This help can be obtained at any time bypressing F1 or by selecting Contents from the Help menu.

4.4 A Note About Repeatability of Results Under Simulation

As you may be aware, results obtained through repeated simulations of the same model will exhibit somevariability. The use of a seed in simulation forces the software to use the same sequence of random numbersin each simulation, resulting in repeatability. However, you need to be aware that when you use a seed, thesame stream of random numbers is utilized, but the order of the utilizations of the stream may differdepending on the way the model was constructed. Two people constructing the same model in separatesystems may add Constructs or create Definitions in a different sequence. In other words, two people mayhave the same Random Variables, A and B, defined in a project. If in one project A was created before B, Awill take the first random value from the stream and B will take the second. If the other person (in the otherproject) defined the same model but created B before A, B will take the first random value from the streamand A will take the second, thus yielding slightly different answers, even though both persons may haveused the same seed in simulation.

Because of this, the results you obtain for the examples in this training guide may differ slightly from theresults presented in the text. The illustrations and results in this guide are taken from the example files

4.5 First Steps Example


included with the application; a seed is used in these files for internal repeatability. You will not need to usea seed for your simulations.


This example is presented in the RENO file Hinge Assembly.rnp located in the General subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\General).

A hinge is made up of four components, A, B, C and D.

During manufacturing, each part has some variability in its dimensions. Assume that the dimensions(widths) of the parts are distributed as follows:

A - Normal distribution with mean = 2 and std = .02

B - Normal distribution with mean = 2 and std = .02

C - Normal distribution with mean = 30 and std = .2

D - Normal distribution with mean = 34.5 and std = .5

Determine the number of items that will not fit within D, or the expected times that (A+B+C) will be greaterthan D.

The following general flowchart shows one way that this problem could be solved using simulation.

Compute A+B+C

Is A+B+C>D

Count=Count+1

Yes

NO

Draw RandomValues for A , B, C and D

Re

pea

t for N

Sim

ula

tions

4 First Steps


The steps necessary to solve this problem in RENO are presented in the sections that follow.

At this time, we assume that you have started the application.

4.5.1 Create a New Project

To create a project, click Create a New Project in the What do you want to do? window that may appearat startup, select New Project from the File menu or click the New Project icon in the MDI toolbar.

The New Project Wizard will be displayed. On the first page, select Blank Project to create a newproject from scratch, then click Next. On the second page, select A new blank Flowchart. You will notneed a template for this example. Click Next. On the third page, click Finish to create the project. TheMDI will look like the one shown next.

4.5.2 Define the Random Variables

First, you will need to create Random Variables, A, B, C and D, each of which will generate a value in therange specified by the distribution for each part.



To add a new Random Variable, select Add New Definition then Add Random Variable from theProject menu. The properties window for the new Random Variable will appear, as shown next.

Change the Identifier to A, then select Normal from the Distribution drop-down menu. Input theparameters: 2 for Mean and .02 for Std, as given in the background information for the example. Theproperties window will look like the one shown next.

Click Save & New to save the current Random Variable and leave the window open to create a new one.

Repeat this process to define the Random Variables B, C and D. When you have entered the informationfor Random Variable D, click OK to save the variable and close the window.

Defining these four Random Variables is equivalent to the first requirement in the general flowchart.

4.5.3 Create the Flowchart

The next step is to create a model of the problem. The RENO Flowchart is as follows:

It includes three steps executed sequentially from left to right (based on the direction of the arrows).

4 First Steps


4.5.3.1 Use a Standard Block to Compute A+B+C

To add a Standard Block to the Flowchart, click in the Flowchart window to make it active. Select AddBlock from the Flowchart menu or click the Add New Block icon in the Flowchart Constructs toolbar.

Double-click the new block that appears in the Flowchart to open its properties window. Enter ComputeA+B+C for the block’s name and (A+B+C) in the Evaluates to field. The properties window will looklike the one shown next.

This block adds the values of A, B and C and passes the sum along its outgoing path.

Click OK. The block’s new name will appear below it in the Flowchart.

4.5.3.2 Use a Conditional Block to See if A+B+C Is Greater than D

Add a Conditional Block to the Flowchart by selecting Add Block then Add Conditional Block fromthe Flowchart menu or by clicking the Add Conditional Block icon.

Double-click the new Conditional Block. In the properties window that appears, enter >D for the block’sname.

Select > from the drop-down menu in the Condition field, then enter D in the field directly to the right ofthe conditional operator. This is the value against which the input will be checked.



Enter 1 in the Condition is TRUE field. The properties window will look like the one shown next.

The input from the Standard Block (i.e. the sum of A+B+C) is passed to the Conditional, which checks itagainst the value of D. If the condition is met (i.e. the value of A+B+C is greater than the value of D), thenthe Conditional Block evaluates to True and it continues execution by passing 1 to the next Construct (asdefined by the “Condition is TRUE” input). Note that there is no requirement to include a FALSE path. If apath is omitted, RENO will realize that there are no paths to continue down and thus terminate the currentsimulation.

Click OK.

4.5.3.3 Use Result Storage Block to Count the Number of Times that A+B+C > D

Add a Result Storage Block to the Flowchart by selecting Add Block then Add Result Storage from theFlowchart menu or by clicking the Add Result Storage icon.

In the properties window for the Result Storage Block, enter No Fit for the name and select Sum ofResults from the list of storage types. The properties window will look like the one shown next.

4 First Steps


A Result Storage Block stores values passed to it across different simulations. In this case, the block willhold a sum of all the values passed to it; thus, at the end of N simulations, it will hold the number of timesthat the Conditional Block was true (i.e. the number of times A+B+C was greater than D).

Click OK. The Flowchart will now look like the one shown next.

4.5.3.4 Join the Blocks to Complete the Flowchart

The final step in creating the Flowchart is joining the blocks to dictate the order in which information ispassed between them.

To create a relationship line between blocks, you can hold down the Alt key, select Join Blocks from theFlowchart menu or click the Join Blocks icon.

The cursor will change to display a small cross hair. Click the Standard Block, hold down the left mousebutton and drag a line to the Conditional Block. When the cross hair is located above the ConditionalBlock, release the mouse button to create a relationship line.

Repeat this procedure to create a relationship line between the Conditional Block and the Result StorageBlock. Note that this relationship is automatically labeled “T”. By default, RENO designates the firstoutgoing path drawn from a Conditional Block or Logic Gate as the TRUE path and the second outgoingpath as the FALSE path. You can change these designations by double-clicking the relationship line, ifdesired. For this example, keep this path designated as the TRUE path.



To stop adding relationships in the Flowchart and return the cursor to its normal mode, release the Altkey, right-click the Flowchart background or de-select the Join Blocks option. The Flowchart will looklike the one shown next.

Note that it is not necessary to create a loop (i.e. an instruction to repeat the steps for N simulations) asshown in the general Flowchart, since RENO is designed to run multiple simulations of a Flowchart.

4.5.4 Simulate the Flowchart

Once the Flowchart has been created, you then instruct RENO to run a specified number of simulations.After simulation, you can view the results stored in any Result Storage Block.

To open the Simulation Console, select Simulate from the Tools menu or click the Simulate icon.

The Simulation Console will open, as shown next.

4 First Steps


Enter 1000 for the Number of Simulations and click Simulate to run the simulations. The SimulationStatus Display will appear, showing the progress as each simulation is performed. When simulation iscomplete, the Simulation Console will look like the one shown next.

Note that the current value of the Result Storage Block, No Fit, is displayed in the Display VariableStatus area at the top of the window. Click Close to return to the Flowchart and note that the value is alsodisplayed above the Result Storage Block, as shown next.

A+B+C exceeded D in 162 out of 1000 simulations, or 16.2% of the time. In this case, the answer is givennot as a percentage but rather as a count. A minor modification to the Flowchart can create a model thatreturns results as a percentage, as described next.

4.5.5 Normalize the Results

One way to cause results to be returned as a percentage is to add an additional Standard Block to normalizethe results. Rather than adding 1 to the Result Storage Block every time the condition is true, this block willallow us to add (1/(Number of Simulations))*100. This will produce results as a percentage based on all ofthe simulations.



Add another Standard Block to the Flowchart and define it as follows:

This involves the use of two reserved keywords, IN and SIMS_TOTAL. IN represents the value passed to

the block and SIMS_TOTAL evaluates to the total number of simulations that we are running.1 You canenter the equation manually or click the Function Wizard icon or the Build Equation link to use theFunction Wizard to create the equation. You can also press Ctrl+Space to bring up a menu of Definitionsand available keywords for use in the equation. Note that recognized Definitions and keywords areautomatically color coded to assist in reading the equation.

In the Flowchart, select the relationship line between the Conditional Block and the Result Storage Blockby clicking it. You can then delete the line by selecting Delete from the Edit menu or clicking the Deleteicon.

Note that you can click and drag the relationship line to a new destination instead, if you prefer (i.e. click therelationship line and drag it to the Normalize block so that it connects the Conditional Block and theStandard Block).

1. A complete list of reserved keywords is available in Chapter 4 of the RENO User’s Guide.

4 First Steps


Connect the blocks in the Flowchart as shown next, using the Join Blocks function described in Section4.5.3.4.

Simulate the Flowchart again, using the settings specified previously. When simulation is complete, theFlowchart will look like the one shown next.

The Result Storage Block now displays the value as a percentage.

4.5.6 Save and Close the Project

If you are using a fully functional copy of the software (i.e. not a demonstration version), save theanalysis. To do this, click the Save icon



or select Save from the File menu. The Save As window will appear. Type Hinge Assembly as the filename and accept the default file type, RENO Project file (*.rnp). By default, RENO will save the file to

your My Documents folder; you can select another location if you prefer.2

Click Save to save the file.

Close the project by selecting Close from the File menu. You will now be looking at the MDI with noprojects open.

Note that this example is based on the RENO file Hinge Assembly.rnp located in the General subfolder ofthe Examples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\General). If you are working with a demonstration version of the RENOsoftware, you will not be able to save the file you created for this example, but you can view the file shippedwith the application if you wish to refer to this example in the future.

2. For the examples in this training guide, we have created a Files folder in the application directory.


This chapter provides the following step-by-step examples, designed to introduce you to the features of theRENO software:

Example 1 - Estimating Dice Roll Probabilities - page 35

Example 2 - Risk Analysis: Gym Lockers - page 41

Example 3 - Maximizing Profits: The Newspaper Boy Problem - page 49

Example 4 - Maximizing Profits: Airline Seat Allocation - page 55

Example 5 - Minimizing Costs: Preventive Maintenance - page 63

Example 6 - Reliability: Snow Blower - page 70

Example 7 - Simple Reliability and Availability Metrics - page 76

Example 8 - Stress-Strength Calculations - page 88

Example 9 - Failure and Usage Distributions: Washing Machine Usage - page 92

Example 10 - Risk Analysis: Suntan Lotion - page 99

Example 11 - Probability: Rain Prediction - page 102

Example 12 - Probability: Buffon’s Needle - page 108

Example 13 - Financial Analysis: Retirement Strategy - page 113

Each example is based on a RENO project file (*.rnp) that is shipped with the software, and the full solutionis presented in that file. You may wish to view the file that is provided, or to create a new project for eachexample and work through the steps on your own. Keep in mind that if you are using a demonstrationversion of the software, you will not be able to save the files that you create.

5.1 Example 1 - Estimating Dice Roll Probabilities

This example is presented in the RENO file Dice Roll.rnp located in the Casino subfolder of the Examplesfolder located in your application directory (e.g. C:\Program Files\ReliaSoft\RENO\Examples\Casino).

Consider two simple questions regarding the probabilities associated with a dice roll:

When you roll a single 6-sided die, what is the probability of rolling a 4?

When you roll two 6-sided dice, what is the probability of rolling a pair of 4s?

Step-by-Step Examples 5

5 Step-by-Step Examples


5.1.1 Create the Definitions

Define two Random Variables to describe the possible results from rolling each die, as shown next forthe first die.

5.1.2 Build the First Flowchart

To determine the probability of rolling a 4 with a single die, construct the Flowchart as follows:

Use a Standard Block to obtain the result of the dice roll from the “DiceRoll1” Random Variabledescribed above.



Use a Conditional Block to check whether the roll was a 4. If true, then a 1 is passed down the TRUEpath. A FALSE path is not required for this example.

Use a Result Storage Block to keep a sum of the number of times that a 4 is rolled (across allsimulations).



Use a Standard Block to convert the sum to a percentage.

In this equation, the reserved keyword IN represents the sum from the previous construct and thereserved keyword SIMS_TOTAL represents the total number of simulations.

Use another Result Storage Block to store the final percentage (given as a decimal).

Join the blocks by either holding down the Alt key, selecting Join Blocks from the Flowchart menu orclicking the Join Blocks icon. Click each source block and drag a relationship line to the destinationblock. Return the cursor to its normal mode by releasing the Alt key, right-clicking the Flowchartbackground or de-selecting the Join Blocks option.



5.1.3 Simulate the First Flowchart

In the Simulation Console, enter 5000 for the number of simulations.

When simulation is complete, the results are displayed in the Simulation Results Explorer and in theFlowchart, as shown next.

The estimated probability of rolling a 4 is 16.26%.

5.1.4 Build the Second Flowchart

Add a new Flowchart to the project by selecting Add Flowchart from the Project menu.

You can rename the new Flowchart by right-clicking the new Flowchart in the Project Explorer andselecting Rename Item from the shortcut menu that appears. In the text box, type Roll Pair of 4s andpress Enter.

To determine the probability of rolling a pair of 4s with two dice, construct the Flowchart as follows:



In this case, two Standard Blocks are required to obtain the result from each die and a Logic Gate is usedinstead of a Conditional Block. With AND selected for Gate Type, this construct requires that both pathsmeet the conditional requirement in order for the condition to be TRUE.

It is important to note that the Required Number of Executed Paths field in the Logic Gate must be set to2. This ensures that both incoming paths are executed before the Logic Gate is executed.

The rest of the Flowchart is identical to the one described above.

5.1.5 Simulate the Second Flowchart

In the Simulation Console, enter 10,000 for the number of simulations.

5.2 Example 2 - Risk Analysis: Gym Lockers



The probability of rolling a pair of 4s is 2.83%.


This example is presented in the RENO file Lockers.rnp located in the Risk Analysis subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Risk Analysis).

Consider a gym’s lockers that are used by members to store clothes and other valuables. The lockersthemselves cannot be locked; however, locks can be purchased for the lockers.

Suppose that:

The value of the items in the lockers is normally distributed with a mean of $500 and a standarddeviation of 150.

The possibility of a thief stealing the items from a given locker in a year is 5% if unlocked and 1% iflocked.

A lock costs $5. The lock can fail (according to a Weibull distribution with beta = 2 and eta = 900)and the member has to replace it.

Members lose their keys (average time to key loss being 600 days), and it costs another $10 to breakthe lock whenever a key is lost.

Determine the expected cost in one year of buying vs. not buying a lock.




For this example, three Random Variables are needed:

The value of the locker’s contents:

The time to lock failure:

The time to key loss (using an exponential distribution with Lambda = 1/Mean):



5.2.2 Build the Flowchart

There are two costs involved in this problem: the cost of the lost belongings and the cost of acquiring andmaintaining a lock. We will begin with estimating the cost of lost belongings. Because this cost is present in both cases (with orwithout a lock), we will use a subchart to define this cost so that the effort is not duplicated.

5.2.2.1 Subchart: Cost of Lost Belongings

The subchart is:

Note that there is a single starting point and a single ending point. This is a requirement for all subcharts.When you create the Flowchart, you can rename it to StolenCosts by right-clicking it in the Project Explorerand selecting Rename Item from the shortcut menu. Type the new name in the text box that appears, thenpress Enter.

In building this subchart, it is important to keep in mind that there is a different probability of occurrence foreach case (with lock or without lock). Therefore, the probability should not be defined in the subchart, butrather passed into the subchart from the main Flowchart. This in turn would then be accessible with thekeyword IN.

A Flowchart cannot have a Conditional Block (or a Summing Gate, Logic Gate or Branch Gate) as astarting block. Therefore, begin this Flowchart with a Standard Block named “Start,” which evaluates toIN.



Define the Conditional Block as follows:

This Conditional Block draws a random number uniformly distributed from 0 to 100 and tests to see if it isless than or equal to the fixed probability of items being stolen from the locker (i.e. 5% if unlocked and 1%if locked). If true, this indicates that the items were stolen and the value of the loss is passed to the TRUEpath. If false, then "0" is passed to the FALSE path. In order to exit the subchart without terminating the simulation, you will need to define paths for both theTrue and False situations that lead eventually to a single ending point. To do that, use two Standard Blocks,“Stolen” and “Not Stolen,” each evaluating to IN. These continue simulation by passing the values passed tothem to the next Construct.

To join the two paths you will need to utilize a Construct that is designed to accept multiple inputs. Oneoption is to use a Summing Gate, since a Summing Gate outputs the sum of active inputs. In this case, onlyone input is possible, so the output will be either the value passed along the TRUE path (Value) or the valuepassed along the FALSE path (0).

Define the Summing Gate as follows:

Name: S

Operation: +


When you join blocks to the Summing Gate, their names appear in the Precedence field in the SummingGate properties window. Since this Summing Gate is performing addition, the order of precedence isunimportant.

Note that the order in which you create the relationship lines from the Conditional Block to the “Stolen”and “Not Stolen” blocks determines which is identified as the TRUE or FALSE path. By default, RENOdesignates the first outgoing path drawn from a Conditional Block or Logic Gate as the TRUE path andthe second outgoing path as the FALSE path. You can change these designations by double-clicking therelationship line, if needed.



5.2.2.2 Main Flowchart

You can now create the main Flowchart, “Lock,” containing models for both cases.

Add a new Flowchart to the project by selecting Add Flowchart from the Project menu, then rename theFlowchart to Lock.

When the Flowchart is complete, it will look like the one shown next. The steps that follow guide you increating this Flowchart.

Note that the models for both cases are in the same Flowchart. RENO will simulate each part of a Flowchart,so you can have multiple cases or subroutines within one Flowchart.

For the case where no lock is used, the Flowchart is:

Add a Standard Block that evaluates to the desired probability for this case (5%).



This will be passed to the subchart, represented in the Flowchart by the Subchart Block called “StolenCosts.” The subchart will return either the value of the loss or 0 if there was no loss.

Add a Subchart Block to the Flowchart. In the window that appears, select the StolenCosts flowchart asthe subchart that the block will represent. Name the Subchart Block Stolen Costs.

Add a Result Storage Block named Expected Costs to store an average value of all the outcomes(including the zeroes).


For the case where a lock is used, the Flowchart is:

Because the two cases are so similar, you may wish to copy the model you have created in the previous stepand then modify it, rather than recreating each of the Constructs. You can do this by clicking inside theFlowchart, holding down the left mouse button and dragging a box around the blocks to be selected, thenselecting Copy from the Edit menu. Click in the Flowchart to return focus to it, then select Paste from theEdit menu. The copy will be placed in the Flowchart and will be selected. Click one of the blocks in thecopy and drag the copy to the desired location.

For the “Probability” block, make sure it evaluates to 1, as the probability of theft in cases where a lockis used is 1%.



In this case, an additional block is added to include the additional costs of purchasing and maintaining alock. A simple way to account for the additional costs is to create an Equation Variable to compute theadditional costs.

The equation is derived as follows:

The following number of lock failures are expected over a year:

(INT(365/LockFail))

where INT is a function intrinsic to RENO that rounds a function down to the nearest integer, since youcannot have a partial failure.

The following number of key losses are expected over a year:

(INT(365/KeyLoss))

These numbers of events, multiplied by the cost of each respective type of event, will result in the additionalexpected costs over the year.

Add a Standard Block as follows:



Note that if you did not use the Equation Variable and instead defined the equation directly in the StandardBlock as follows, you would obtain identical results.

Join the blocks for this model. To include the new block in the copy of the original model, you can deletethe relationship line between the Subchart Block “Stolen Costs” and the Result Storage Block “ExpectedCosts” by selecting it and selecting Delete from the Edit menu or clicking the Delete icon, then creatingnew relationship lines. If you prefer, you can click and drag the relationship line to a new destinationinstead (i.e. click the relationship line and drag it to the “Total Costs” block so that it connects theSubchart Block and the Standard Block), then create a new relationship line between “Total Costs” and“Expected Costs.”


The next step is to simulate the Flowchart. In the Simulation Console, we enter 1000 for the number ofsimulations then click Simulate. Once simulated, the Flowchart looks like the one shown next.

Over 1000 simulations, the expected yearly cost of using a lock averaged $42.85, while the expected yearlycost of not using a lock averaged $22.72.

5.3 Example 3 - Maximizing Profits: The Newspaper Boy Problem



This example is presented in the RENO file Newspaper Boy.rnp located in the Optimization subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Optimization).

A newsboy can purchase newspapers at 15 cents each and sell them for 25 cents. The demand is normallydistributed with a mean of 100 and standard deviation of 25. The problem is to determine how manynewspapers to buy, given that if he buys too many he will lose money on the ones he does not sell, while ifhe buys too few he will lose money by not having enough newspapers to sell.

The exact algebraic solution to this problem is given by:

1-P(Q) = Cost/Sell

= 0.6

Q = 93.67

The RENO solution is described next.


Define a Random Variable to describe the demand for newspapers.

Define a Constant that represents the number of papers that the newspaper boy buys. This will be variedduring the simulation to determine the optimum quantity.



Define an Equation Variable to calculate the newspaper boy’s supply cost, which is the quantity ofpapers purchased (represented by the “Quantity” Constant) multiplied by 15 cents.


Construct the Flowchart as follows:

Use a Standard Block to determine the demand for the newspapers based on the Random Variable called“Demand.”

Use a Conditional Block to check whether the amount of papers on hand (represented by the Constantcalled “Quantity”) is equal to or greater than the demand determined in the previous step. If true, then thequantity of papers sold is equal to the Random Variable “Demand” and this number is passed to the



TRUE path. If false, then the quantity of papers sold is equal to the Constant called “Quantity” and thisnumber is passed to the FALSE path.

Use two Standard Blocks called “Profit 1” and “Profit 2” to calculate the amount of revenue from theTRUE and FALSE paths. When supply meets or exceeds demand, this is the demand quantity passedfrom the Conditional (represented in the equation by the reserved keyword IN) multiplied by the price(25 cents) less the supply cost (represented by the Equation Variable called “Cost”), as shown next. Asimilar equation is defined for the FALSE path, where the number of papers sold (IN) is equal to theConstant called “Quantity.”



Use a Result Storage Block to calculate and store the average profit from all simulations.

Join the blocks as shown in the Flowchart above, making sure that the TRUE and FALSE paths from theConditional Block are correct.


For this example, it is important to note that a single simulation run provides a single answer; thus, multipleruns will be needed to arrive at an optimum solution. This can be done using the Sensitivity Analysis page inthe Simulation Console, which allows you to run the simulations multiple times and vary a constant witheach run.

To first explore the area from 50 to 150, specify 500 simulations on the General page of the SimulationConsole then set the Sensitivity Analysis page as shown next.

Click Simulate then, after the simulation completes, close the Simulation Console.

RENO will perform 500 simulations per run, with the Quantity set to 50 for the first run, 60 for the secondrun, and so on up to 150.



Click the Plot icon.

The plot indicates that the optimum value lies somewhere between 80 and 100.

The next plot shows the results from repeating the simulation with the Quantity varied from 80 to 100 andincremented by 1.

Given the small number of simulations (500), noise (variation due to simulation) is present. The next step isto increase the number of simulations to 50,000 and focus on the region between 90 and 96, incrementing by1, as shown next. Note that the plot shown here has had its scaling changed. To make your plot look like this

ReliaSoft RENO - www.ReliaSoft.com

Average Profit vs. Quantity

Flowchart: Flowchart1 - Simulations per run: 500 - Sensitivity Analysis (One Way): Quantity - Multiple Analyses Disabled

Quantity

Aver

age

Prof

it

10.000 200.00048.000 86.000 124.000 162.0000.000

10.000

2.000

4.000

6.000

8.000

Flowchart1

Average Profit

User's NameCompany10/17/200512:19:01 PM




Quantity

Aver

age

Prof

it

80.000 100.00084.000 88.000 92.000 96.0007.000

8.000

7.200

7.400

7.600

7.800

Flowchart1

Average Profit




one, de-select Automatic Scaling on the Control Panel and enter 7.4 for the Y-axis minimum and 7.7 for theY-axis maximum. You can leave the values of 90 for the X-axis minimum and 96 for the X-axis maximum.

In addition to the manual technique described above, RENO can automatically perform multiple analysesdesigned to determine the quantity that will maximize revenue. To do this, specify 2000 simulations on theGeneral page, specify 60 to 120 with an increment of 12 on the Sensitivity Analysis page and configure theMultiple Analyses page as shown next.

Based on these settings, RENO will perform up to 3 analyses (or fewer if the results converge to within0.0001 of each other before 3 analyses are complete). Each analysis will consist of the number of runsspecified on the Sensitivity Analysis page. In the first analysis, each run will consist of 2000 simulations.Because we have selected to increase the simulations by 2000 in each subsequent analysis, each run in thesecond analysis will consist of 4000 simulations and each run in the third analysis will consist of 6000simulations.




Quantity

Aver

age

Prof

it

90.000 96.00091.200 92.400 93.600 94.8007.400

7.700

7.460

7.520

7.580

7.640

Flowchart1

Average Profit


5.4 Example 4 - Maximizing Profits: Airline Seat Allocation


The resulting plot of Average Profit vs. Quantity will include a marker and a line to identify the optimumvalue.

From this plot, we can see that the optimum number of newspapers for the newsboy to buy is 94.


This example is presented in the RENO file Airline Seat Allocation.rnp located in the Optimization subfolderof the Examples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Optimization).

A given aircraft can have a maximum of 100 coach seats. Two coach seats are removed for each first classseat that is included. For example, if 4 first class seats are included, then the maximum number of coachseats is 92.

Assume the following:

Demand for coach tickets is based on a Weibull distribution with beta = 2 and eta = 45.

Demand for first class tickets is based on a Weibull distribution with beta = 1.2 and eta = 15.

Revenue for each coach ticket is $500.

Revenue for each first class ticket is $1,500.

Operating costs are fixed.

Determine the optimum allocation of first class seats, assuming that only increments of four seats areallowed.



Flowchart: Flowchart1 - Simulations per run: 2000 - Sensitivity Analysis (One Way): Quantity - Multiple Analyses (Maximize): Average Profit

Quantity

Aver

age

Prof

it

60.000 120.00072.000 84.000 108.00094.0005.888

7.689

6.248

6.608

6.969

7.329

7.633Flowchart1

Analyses Variable

MaximumAnalysis 1Analysis 2Analysis 3

User's NameCompany10/17/200510:14:22 AM




Define two Random Variables to describe the demand for coach tickets and first class tickets.

Define a Constant to specify the number of first class seats. This will be varied during the simulation toestimate the optimum number.

Define an Equation Variable to calculate the number of coach seats, which depends on the number offirst class seats.





Use a Standard Block to obtain the demand for coach seats from the Random Variable.

Use a Conditional Block to check whether the demand for coach seats is less than the number of coachseats (which is calculated by the Equation Variable). If it is (i.e. the condition is True), then the numberof coach seats sold will be equal to the demand, so the demand quantity is passed down the TRUE path.If demand is greater than or equal to the number of coach seats (i.e. the condition is False), then the



number of coach seats sold will be equal to the number of available coach seats, so the available quantityis passed down the FALSE path.

Use two Standard Blocks to calculate the revenue from the TRUE and FALSE paths, as shown next forthe situation when supply exceeds demand.

In this equation, the reserved keyword IN represents the quantity of seats sold (passed from theConditional Block), which is then multiplied by the $500 price of a coach ticket.



Join the blocks for this portion of the model, making sure that the TRUE and FALSE paths from theConditional Block are correct.

Create a similar configuration to describe the revenue for first class seats. You will need:

A Standard Block that evaluates to “DemandFirst.”

A Conditional Block like the one shown next.



Two Standard Blocks to calculate the Revenue from the TRUE and FALSE paths, as shown next forthe situation when supply exceeds demand.

Join the blocks for this portion of the model, making sure that the TRUE and FALSE paths from theConditional Block are correct. Your flowchart will now look like the one shown next.

A Summing Gate collects the revenue values from all four incoming paths and performs a summation.Note that the Required Number of Executed Paths field in the Summing Gate must be set to 2. Thisensures that both incoming paths are executed before the Summing Gate is executed.



The result is then passed to a Result Storage Block that stores the average total revenue.

Join the blocks for the remaining part of the model.


On the General page of the Simulation Console, specify 2000 simulations. You may wish to specify Y-Axis limits of -40,000 to 40,000, as shown next, for the most informative display during simulation.



On the Sensitivity Analysis page of the Simulation Console, specify to vary the Constant FCSeats from0 to 48, with an increment of 16, as shown next.

RENO will perform multiple runs of 2000 simulations each, with “FCSeats” equal to 0 in the first run, 20 inthe second run, 40 in the third run, and so on. In this case, this comes to six runs.

The Multiple Analyses feature in the Simulation Console allows you to perform the Sensitivity Analysismultiple times in order to maximize or minimize a Result Storage Block.

On the Multiple Analyses page of the Simulation Console, select to perform multiple analyses todetermine the first class seat quantity that will maximize revenue. Choose Maximize for the Action andaccept Total Revenue as the input for Result Storage. Specify 3 for Max. Analyses and 0.0001 for Std.Deviation. Enter 2000 in the Increase Simulations field and select Use Integers Only with a minimumincrement of 4, as shown next.

Based on these settings, RENO will perform up to 3 analyses (or fewer if the results converge to within0.0001 of each other before 3 analyses are complete). Each analysis will consist of the number of runsspecified on the Sensitivity Analysis page (6 in this case). In the first analysis, each run will consist of 2000simulations. Because we have selected to increase the simulations by 2000 in each subsequent analysis, eachrun in the second analysis will consist of 4000 simulations and each run in the third analysis will consist of

5.5 Example 5 - Minimizing Costs: Preventive Maintenance


6000 simulations. Note that by specifying to use only integers with a minimum increment of 4, we haveensured that only multiples of 4 are used as RENO narrows the value range of FCSeats over multipleanalyses. This meets the specification that only increments of four seats are allowed.

Run the simulation, then close the Simulation Console.

5.4.4 Plot the Results

In the Flowchart Control Panel, click the Plot icon to plot the simulation results.

The plot that is created will look like the one shown next.

The plot identifies 24 as the optimum quantity of first class seats.


This example is presented in the RENO file Preventive Maintenance.rnp located in the Optimizationsubfolder of the Examples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Optimization).

Consider a unit that operates for 10,000 hours and has a Weibull failure distribution (beta = 3, eta = 1000hours) and a Weibull repair distribution (beta = 2, eta = 10 hours). The costs of downtime are based on theamount of time that the unit is down: $20 per hour. The preventive maintenance (PM) cost is fixed at $10 perinstance. Determine the PM interval with the minimum cost.


Total Revenue vs. FCSeats

Flowchart: Seat Allocation - Simulations per run: 2000 - Sensitivity Analysis (One Way): FCSeats - Multiple Analyses (Maximize): Total Revenue

FCSeats

Tota

l Rev

enue

0.000 48.0009.600 19.200 28.800 38.40024.00019839.113

23139.696

26440.279

29740.861

33041.444

36342.027 Seat Allocation

Analyses Variable

MaximumAnalysis 1Analysis 2Analysis 3





Define a Random Variable called “TTF” to describe the failure distribution for the unit and anotherRandom Variable called “TTR” to describe the repair distribution.

Define three Constants to describe the operating time for the equipment, the fixed cost for each PM andthe cost per each hour of downtime.

Define another Constant called “PM Time” to specify the PM interval. This will be set to 1 by defaultand varied during the simulation in order to determine the optimum time.



Define an Equation Variable to calculate the cost of corrective maintenance (CM), which is the cost perCM (Time to Repair x Downtime Cost per Hour) multiplied by the number of CMs (Operating Time /Time to Failure).



Use a Standard Block to obtain the failure time from the “TTF” Random Variable.

Use a Conditional Block to determine whether the failure time is equal to or greater than the specifiedPM interval. If true, then pass 2 to the TRUE path. If false, then pass 3 to the FALSE path. Note that the



numbers passed are arbitrary since the blocks following do not use the values sent by the ConditionalBlock.

In the FALSE path (which indicates that the failure occurred before the PM), use a Standard Block toobtain the corrective maintenance cost from the “CorrectiveCost” Equation Variable.



In the TRUE path (which indicates that the failure did not occur before the PM), use a Standard Block tocalculate the preventive maintenance cost. This is the cost per PM multiplied by the number of PMs(Operating Time / Time to PM).

Use a Result Storage Block to store the average of the PM and CM costs from all simulations.

Join the blocks in the Flowchart. Be sure that the TRUE and FALSE paths from the Conditional Blockare correct; note that in this Flowchart, FALSE is the top path.




On the General page of the Simulation Console, specify 1000 simulations.

On the Sensitivity Analysis page of the Simulation Console, specify to vary the Constant PMTime from100 to 500, with an increment of 100.

On the Multiple Analyses page of the Simulation Console, select to perform multiple analyses todetermine the time that will minimize the costs. Choose Minimize for the Action and accept Cost as the



input for Result Storage. Specify 5 for Max. Analyses and 0.001 for Std. Deviation. Enter 100 in theIncrease Simulations field.

Run the simulation then close the Simulation Console.

5.5.4 Plot the Results

In the Flowchart Control Panel, click the Plot icon to plot the simulation results. The optimum value willbe marked on the Costs vs. PMTime plot, as shown next.

To minimize costs, preventive maintenance should be performed at an interval of 337.5 hours.


Costs vs. PMTime

Flowchart: Flowchart1 - Simulations per run: 1000 - Sensitivity Analysis (One Way): PMTime - Multiple Analyses (Minimize): Costs

PMTime

Cost

s

100.000 500.000180.000 260.000 420.000337.500468.156

1000.000

574.525

680.894

787.263

893.631

518.676

Flowchart1

Analyses Variable

MinimumAnalysis 1Analysis 2Analysis 3Analysis 4Analysis 5




5.6 Example 6 - Reliability: Snow Blower

This example is presented in the RENO file Snow Blower.rnp located in the Reliability subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Reliability).

Reliability analysis on a certain snow blower yielded a Weibull life distribution with a beta of 2 and an eta of400 hrs.

The snow blower will be deployed in a part of the country where the probability that it will snow on anygiven winter day is 1 in 10. If it snows, then the snow depth is normally distributed with a mean of 6 inchesand a standard deviation of 2. Furthermore, the area of the driveways in that region is also normallydistributed with a mean of 300 square feet and a standard deviation of 40.

If the snow blower clears 30 cubic feet an hour, what would the reliability of the blower be assuming 100winter days?


Define a Probability called “SnowProbability” and a Constant called “SnowThreshold” to determinewhether it will snow on a given day. If a random number uniformly distributed from 0 to 1 (generated by“SnowProbability”) is less than or equal to the fixed probability that it will snow on any given winter day(defined by “SnowThreshold”), this indicates that it snowed during the simulation.



Define a Random Variable called “Snowfall_inches” to describe the depth of the snow.

Define a Random Variable called “Area_to_clean_ft2” to describe the area of the driveways.

Define an Equation Variable to describe the number of hours that the snow blower will be used based onthe amount of snowfall and the area of the driveways.

The equation is derived as follows:

The expected depth of the snow, in feet, is:

Snowfall_inches/12



Multiplying this by the area of the driveways (Area_to_clean_ft2) gives the number of cubic feet ofsnow to be cleared.

Divide this number by the number of cubic feet the snow blower can clear in an hour (30) todetermine the number of hours it will take to clear the snow, then multiply that figure by the 100winter days to determine the number of hours of usage for the winter.



Use a Standard Block to generate a random number uniformly distributed from 0 to 1, based on theProbability called “SnowProbability.”

Use a Conditional Block to check whether the random number is less than or equal to the fixedprobability of snow on any given winter day (based on the Constant called “SnowThreshold”). If true,



this indicates that it snowed and the value of the Equation Variable “Usage_Hours” is passed to theTRUE path. If false, this indicates that it did not snow and 0 is passed to the FALSE path.

Use Standard Blocks called “Snow” and “No Snow,” each evaluating to IN, to continue the TRUE andFALSE paths coming from the Conditional Block.



Use a Summing Gate to accept the two incoming paths and perform a summation on the snow blowerusage information that is passed to the construct. The result is one outgoing path.

Use a Result Storage Block called “Usage” to store an average of the snow blower usage informationthat is passed to the construct.

Use a Standard Block called “Reliability” to calculate the snow blower reliability.



This equation uses RENO’s intrinsic function, WEIBULL_REL, which returns the Weibull reliability ata given time. The inputs to this intrinsic function are (Time, Beta, Eta, Gamma). It is important to notehere that RENO is regionally aware (i.e. it will work with your regional settings). Therefore, if yourregional settings use a different decimal definition and list separator, the way that you enter values andfunctions will differ from the way presented here. For example, if you are using German regionalsettings, the inputs to this function should be entered based on the German regional decimal definition (,)and list separator (;). The use of the reserved keyword IN indicates that the daily usage time from theprevious “Usage” construct will be inserted into the equation for Time.

Use another Result Storage Block called “Snow Blower Reliability” to store the last value.






The reliability is 98.5975%.

5.7 Example 7 - Simple Reliability and Availability Metrics

This example is presented in the RENO file Compute Reliability (Simple).rnp located in the Reliabilitysubfolder of the Examples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Reliability).

Consider a unit with the following failure and repair distributions:

Failure: Weibull with beta = 2 and eta = 100

Repair: Weibull with beta = 2 and eta = 10

Use RENO to estimate the MTTF, Reliability, Availability and other results.


Create two Random Variables to describe the failure and repair distributions.



Create two Constants to describe the time of interest for the reliability and availability calculations.

5.7.2 Compute MTTF

5.7.2.1 Build the Flowchart

To compute the MTTF, construct the Flowchart as follows:

Use a Standard Block to obtain the failure time from the distribution defined in the Random Variablecalled “WeibullTTF.”

Use a Result Storage Block to store the average result.



5.7.2.2 Simulate the Flowchart

On the General Page of the Simulation Console, specify 5000 simulations.


The MTTF is 88.2032.

5.7.3 Compute Reliability Given Time


To compute the Reliability for a given time, construct the Flowchart as follows:

In this case, the time at which the Reliability is calculated is 20, based on the Constant called“TimeForReliability.”



Use a Standard Block to obtain the failure time from the distribution defined in the “WeibullTTF”Random Variable, as described above.

Use a Conditional Block to check whether the failure occurred before the time of interest. If false, then a1 is passed to a Result Storage Block that stores a sum (i.e. the number of times, across all of thesimulations, that the unit did not fail before the specified time). When you join the blocks, RENOautomatically labels the first path as the TRUE path. Double-click the relationship line to change it to theFALSE path. A TRUE path is not required for this Flowchart.

Use a Standard Block to convert the sum to a percentage. In this equation, the reserved keyword INrepresents the sum from the previous construct and the reserved keyword SIMS_CUR represents the



index for the current simulation. This fraction is multiplied by 100 to convert the results to a percentage.Another Result Storage Block is used to store and display the final result.




The reliability at the given time is 96.48%.



5.7.4 Create Reliability vs. Time Plot

You can also use the above Flowchart to create a plot of Reliability vs. Time. To do this, you will need to re-simulate the Flowchart.



On the Sensitivity Analysis page of the Simulation Console, select to vary the TimeForReliabilityConstant from 0 to 200 with increments of 20, as shown next.




5.7.4.2 Plot the Results


Note that this plot displays the Result Storage Block called “Reliability” plotted against the varied Constant,“TimeForReliability.” If desired, you can use the Select Result Storages area in the Control Panel to selectthe Result Storage Block called “Survive” to be plotted either instead of or along with “Reliability.”

5.7.5 Compute Instantaneous Availability Given Time


To compute the Instantaneous Availability for a given time, construct the Flowchart as follows:


Reliability (%) vs. TimeForReliability

Flowchart: Simple Reliability - Simulations per run: 1000 - Sensitivity Analysis (One Way): TimeForReliability - Multiple Analyses Disabled

TimeForReliability

Rel

iabi

lity

(%)

0.000 200.00040.000 80.000 120.000 160.0000.000

100.000

20.000

40.000

60.000

80.000

Simple Reliability

Reliability (%)




Use a Standard Block to obtain the time to failure from the failure distribution defined in the“WeibullTTF” Random Variable, as described above.

Use a Conditional Block to check whether the unit is still operating at the time of interest, which is 100,based on the Constant called “TimeForAvailability.”



If true, then a 1 is passed to a Result Storage Block that stores a sum (i.e. the number of times, across allof the simulations, that the unit did not fail before the specified time).

If false, then the failure time is passed to a Standard Block that adds the repair time (based on the“WeibullTTR” Random Variable).

Your Flowchart so far will be configured as follows (note that the blocks have been joined in this figureto clarify the relationships).

For units that fail and are repaired (i.e. are on the FALSE path), use a Conditional Block to checkwhether repair is complete and the unit is back in operation at the time of interest. If true, pass the



incoming value (the sum of the failure time and the repair time, as calculated by the Standard Block“Time Up”) down the TRUE path. No FALSE path is needed.

Use a Reset Block to cause RENO to generate new values for the Random Variables.

Use a Standard Block to add the new failure time (based on the “WeibullTTF” Random Variable) to theincoming value. This represents the time of the next failure for the unit that has failed and been repaired.

Use a Go to Flag to pass the value to a Flag Marker, which is placed before the first Conditional Block,creating a loop as follows.

Add a Flag Marker between “Get TTF” and “UP.”



Add a Go to Flag after “Get New Failure Time” and select the Flag Marker “Time” in the Go to FlagMarker field.

By checking the next failure time against the time of interest for the availability estimate, this configurationallows you to see whether the repaired unit is up at the time of interest, regardless of how many times it isrepaired before that time.

Your Flowchart so far will be configured as follows (note that the blocks have been joined in this figure toclarify the relationships).

As described above for the Reliability Flowchart, a Standard Block is placed after “UP Count” and usedto convert the results to a percentage and a Result Storage Block is used to store the final result.

When you join the blocks in the Flowchart, note that it is not necessary to draw a relationship linebetween the Go to Flag and the Flag Marker, as the path is specified by the Go to Flag, but you can drawthe line to make the relationship visually clear, if desired.








5.7.6 Create Availability vs. Time Plot

You can use the same Sensitivity Analysis simulation procedure described in Section 5.7.4 to create anAvailability vs. Time plot for this analysis, as shown next for times ranging from 10 to 100.

5.8 Example 8 - Stress-Strength Calculations

This example is presented in the RENO file Stress-Strength.rnp located in the Reliability subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Reliability).

In this simple example, we explore stress-strength calculations. Given a stress distribution (Weibull withbeta = 2 and eta = 200) and a strength distribution (Weibull with beta = 2 and eta = 500), determine theprobability that stress will exceed strength (i.e. the unit will fail).


Define two Random Variables to describe the stress and strength distributions.


Availability vs. TimeForAvailability

Flowchart: Simple Availability Plot - Simulations per run: 5000 - Sensitivity Analysis (One Way): TimeForAvailability - Multiple Analyses Disabled

TimeForAvailability

Avai

labi

lity

10.000 100.00028.000 46.000 64.000 82.00090.520

99.420

92.300

94.080

95.860

97.640

Simple Availability Plot

Availability




Define an Equation Variable to subtract stress from strength.



Use a Standard Block and the Equation Variable to subtract stress from strength.



Use a Conditional Block to check whether (Strength – Stress) is less than or equal to 0. If true, then stressexceeded strength, the unit failed, and a 1 is passed to the TRUE path. If false, then stress did not exceedstrength and the unit did not fail. There is no need to continue on the FALSE path for this example.

Use a Result Storage Block to keep a summation of the number of times that the unit failed.

Use a Standard Block to normalize the results by converting them to a percentage that represents thenumber of failures out of all the simulations. In this equation, the reserved keyword IN represents the



sum from the previous construct and the reserved keyword SIMS_TOTAL represents the number ofsimulations. This fraction is multiplied by 100 to convert the results to a percentage.

Use a Result Storage Block to store the final calculated percentage.






5.9 Example 9 - Failure and Usage Distributions: Washing Machine Usage

This example is presented in the RENO file Failure and Usage Distribution.rnp located in the Reliabilitysubfolder of the Examples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Reliability).

Consider the case of a washing machine that can have varied loads. Assume the times to failure of thewashing machine follow a Weibull distribution. When it is being used under a 10 lb. load, the washingmachine fails according to a Weibull distribution with beta = 3 and eta = 2,500 hr. When it is being usedunder a 5 lb. load, the washing machine fails according to a Weibull distribution with beta = 3 and eta =10,000 hr. Assume that customer loads vary and are described by a normal distribution with mean = 5 andstd = 1.

Determine the Reliability of the machine for 2000 hrs of operation.

Obtain a plot of Reliability vs Time from 1000 hrs to 10,000 hrs (using 1000 hr increments).

Generate a data set of times-to-failure for subsequent life data analysis.


Define an Equation Variable to describe the life-stress relationship, which is used to determine the scaleparameter, eta, for each load. The equation refers to a Random Variable called “Usage Load,” whichdescribes the distribution for customer loads.

In the “LSR” Equation Variable, an inverse power life-stress relationship is assumed to describe the effect ofloading on life.

ReliaSoft’s Accelerated Life Testing Reference provides more information on this model.



The Constants “k” and “n” are the variables of that relationship obtained by solving this model for thegiven conditions, or for L(V=5)=10,000 and L(V=10)=2,500.

Define the Constant “Life,” which will be used to specify the time for which Reliability is estimated andalso can be varied during the simulation to obtain reliability estimates for a range of times.

Define a Random Variable to describe the failure distribution for the washing machine.



5.9.2 Determine the Reliability at 2000 Hours


To estimate the reliability at 2000 hours, construct the Flowchart as follows:

Use a Standard Block to obtain the time-to-failure from the Random Variable called “Failure Time.”

The Conditional Block checks whether the failure time is less than or equal to 2000 hrs (defined with theConstant called “Life”). If true, then 100 is passed to the TRUE path and 0 is passed to the FALSE path.

If false, then 100 is passed to the FALSE path and 0 is passed to the TRUE path1 To achieve this, be sure

1. This example shows another way to normalize results instead of using a Standard Block as shown in other examples.Here, normalization is achieved by passing 100 down each of the Conditional Block’s outgoing paths rather than passing1.



that Continue execution on both paths is checked, and that 0 is entered for both the Condition isTRUE - “F” Path and the Condition is FALSE - “T” Path fields.

It is important to note that this Conditional Block tests for product failure before the time of interest (“Life”).If the condition is true, the product has failed. Thus, results passed down the TRUE path refer to aprobability of failure. When building this Flowchart, make sure that the “T” path and the “F” path go to thecorrect Result Storage Blocks. By default, RENO designates the first outgoing path drawn from aConditional Block as the “T” path and the second outgoing path as the “F” path. You can change thesedesignations by double-clicking the relationship line if necessary.

On the TRUE path, use a Result Storage Block to store an average of the failures (across all simulations).On the FALSE path, use a Result Storage Block to store an average of the successes. The Result StorageBlock called “Reliability,” on the FALSE path, is shown next.






The reliability of the washing machine for 2000 hours of operation is 98.5%.

5.9.3 Create Reliability vs. Time Plot

You can also use the above Flowchart to create a plot of Reliability vs. Time. To do this, you will need to re-simulate the Flowchart.


Keep the same settings on the General page of the Simulation Console.



On the Sensitivity Analysis page of the Simulation Console, select to vary the Life Constant from 1000to 10,000 with increments of 1000, as shown next.


5.9.3.2 Plot the Results


Note that you may need to change the Result Storage Block in the Plot Sheet Control Panel fromProbability of Failure to Reliability.


Reliability vs. Life

Life

Relia

bilit

y

1000.000 10000.0002800.000 4600.000 6400.000 8200.00040.750

99.800

52.560

64.370

76.180

87.990

Reliability Plot

Reliability




5.9.4 Generate Data Set for Life Data Analysis


To generate a set of times-to-failure that can be used for subsequent life data analysis, construct theFlowchart as follows:

The “Time to Failure” Standard Block is defined as described above for the previous Flowchart.

Use a Result Storage Block to store an array of the failure times obtained by the Standard Block.



Click Simulate to run the simulation.

5.10 Example 10 - Risk Analysis: Suntan Lotion


When simulation is complete, click Details to display the table of results in the Simulation ResultsExplorer. When the Simulation Results Explorer opens, click the (+) beside Result Storages in theControl Panel, then select Time to Failure Array to display the Result Storage Block results, as shownnext. Note that the Transpose Results command in the Tools menu may be required to place each resultinto a separate row (rather than a separate column), as shown here. Because the Simulation ResultsExplorer has only 256 columns, you must place the results in rows to view all of them. Only the first fiveresults are displayed in the picture.

Close the Results Explorer when you have reviewed the results.


This example is presented in the RENO file Sun Tan.rnp located in the Risk Analysis subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Risk Analysis).

Assume that chemical X is added to a suntan lotion. The manufacturing process contains some minorvariability. The concentration of chemical X is normally distributed with a mean of 0.005 and a standarddeviation of 0.0001 ml of chemical X per ml of lotion.

The amount of lotion applied by a person also varies and is known to be normally distributed with a mean of100 ml and a standard deviation of 20 ml.

It is suspected that 10% of the population may, depending on the amount used, have an allergic reaction tochemical X. From the 10% that are susceptible, the probability that somebody will have an allergic reactionis a function of the amount of the chemical applied. This probability is given by a Weibull distribution withbeta = 2.5 and eta = 1.8.

Determine the percentage of the population that is expected to experience an allergic reaction.




Define a Random Variable to describe the concentration of chemical X in the suntan lotion and aRandom Variable to describe the user’s application of the lotion.

Define an Equation Variable to describe the amount of chemical X applied by the user, which is theamount of lotion applied multiplied by the concentration of chemical X in the lotion.

Define a Probability to describe the likelihood that a susceptible user will experience an allergic reaction.

This probability is based on the Weibull distribution and the amount of exposure calculated by the“Chemical_Applied” variable.





Use a Standard Block to calculate the percentage of the population affected based on the probability of areaction.

This equation multiplies the value of the Probability by the percentage of the population that issusceptible (0.1) and by 100 to convert the figure into a percentage.

Use a Result Storage Block to store an average of the results.






Based on these results, 0.4203% of the population are expected to experience a reaction to the suntan lotion.

5.11 Example 11 - Probability: Rain Prediction

This example is presented in the RENO file Rain Prediction.rnp located in the Misc Fun subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Misc Fun).

Suppose that whether or not it rains today depends on previous weather conditions over the last two days.Specifically, suppose that:

If it has rained for the past two days, then it will rain tomorrow with probability 0.8.

If it rained today but not yesterday, then it will rain tomorrow with probability 0.4.

If it rained yesterday but not today, then it will rain tomorrow with probability 0.3.

If it has not rained in the past two days, then it will rain tomorrow with probability 0.2.

We can transform the above model into a Markov chain by saying that the state at any time is determined bythe weather conditions during both that day and the previous day. In other words, we can say that the processis in:

State 0 if it rained both today and yesterday.

State 1 if it rained today and but not yesterday.

State 2 if it rained yesterday but not today.



State 3 if it did not rain either yesterday or today.

The preceding would then represent a four-state Markov chain having the following transition probabilitymatrix:

The limiting probabilities for these states are found respectively to be 8/30, 4/30, 4/30 and 14/30. Therefore,we have:

Proportion of Rain = 0.8 * 8 / 30 + 0.4 * 4/30 + 0.3 * 4/30 + 0.2 * 14/30 = 0.4

The RENO solution is presented next.


Define three Storage Variables to keep track of the weather conditions for Yesterday, Today andTomorrow, as shown next for Today. At each iteration, Yesterday and Today are reset to reflect the newscenario, i.e. Today becomes Yesterday and Tomorrow becomes Today.

Define four Constants to represent the probability of rain tomorrow: if it rained for the last two days(R_R = .8); if it rained today but not yesterday (NR_R = .4); if it rained yesterday but not today (R_NR =.3); and if it has not rained today or yesterday (NR_NR = .2). The “R_R” Constant is shown next as anexample.



Define a uniform Probability to determine the probability that it will rain on a given day.

Define a Constant called “Loop” to specify the number of loops that will be performed during a singlesimulation run.

In addition, define two Storage Variables called “Probability” and “Rain Counter” to hold the values thatare passed during the simulation for the probability of rain tomorrow and the number of days that it rains,respectively. These should also be reset after each simulation.




Construct the flowchart as follows:

The “Start” Flag that begins each loop of the simulation has ordered relations (identified with numbers overthe lines) leading first to the Standard Block that indicates whether it rained Yesterday and then to theStandard Block that indicates whether it rained Today (based on the previous loop of the simulation). If theexecution were to happen in the opposite order, the information on whether or not it rained today would beoverwritten for the next loop. You can set the ordered relations by selecting the source block (i.e. the FlagMarker), choosing Ordered Relations, then drawing the relationship lines. The Standard Blocks are shownnext.

A Logic Gate construct checks to see whether it has rained for the last two days. If true, then the R_Rprobability (.8) is passed to the Result Storage Block called “Probabilities,” which holds the last valuepassed to it and updates the Storage Variable called “Probability.” If false, then the simulation moves to aConditional Block that checks to see whether it rained yesterday. If true, then the R_NR probability (.3) ispassed. If false, then the simulation moves to another Conditional Block that checks to see whether it rained



today. If true, then the NR_R probability (.4) is passed and if false, then the NR_NR probability (.2) ispassed. (The extra Standard Block, “Didn’t Rain For Two Days,” in the FALSE path is just used to pass theNR_NR probability to the Result Storage Block, and evaluates to IN.)

Another Conditional Block uses the probability that is passed from the Result Storage Block (.8, .4, .3 or .2)and the uniform probability of rain, “ProbabilityRain,” to determine whether it will rain tomorrow.

Two Standard Blocks, “Rain” and “No Rain,” evaluate to IN and update the Storage Variable “Tomorrow.”



A Counter Block called “Rain Counter” keeps track of how many days it rains.

The loop that is required to solve this example is constructed by using a Counter Block called “DayCounter” that is reset after each simulation to keep track of how many days have been simulated. This valueis passed to a Conditional Block called “Done?” that checks the “Loop” Constant to determine whether thespecified number of days, 5000, have been simulated.

If the specified number of days have not yet been simulated, then the FALSE path leads to a Reset Block thatpoints to a Go To Flag that points to the “Start” Flag, so that the simulation can move through the flowchartagain.



Once the specified number of days have been simulated, the Standard Block “Done” calculates the numberof days that it rained divided by the number of days simulated.

The Result Storage “Rain Portion” stores the last value passed to it.


After performing 1 simulation (since the loop is included in the flowchart), the results are displayed in theSimulation Results Explorer and in the flowchart. The proportion of rainy days is found to be 40.34%, whichis similar to the result from the Markov chain analysis.

5.12 Example 12 - Probability: Buffon’s Needle

This example is presented in the RENO file Buffon’s Needle.rnp located in the Misc Fun subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Misc Fun).

More than 200 years before Metropolis coined the name “Monte Carlo” method, George Louis Leclerc,Comte de Buffon, proposed the following problem.

“If a needle of length l is dropped at random on the middle of ahorizontal surface ruled with parallel lines a distance d > l apart,what is the probability that the needle will cross one of thelines?”

Buffon asked what was the probability that the needle would fallacross one of the lines. That will occur only if A < lsin().

In this case, the distance from the end of the needle to thehorizontal line (denoted as “A”) is randomly distributed [0, d).The angle at which the needle falls (denoted as theta, ) is alsorandomly distributed [0, 180).

Even though this probability can be easily obtained, it is interesting to note that the great mathematicianLaplace perceived an amusing approximation for pi, . Laplace argued that if Buffon’s experiment isperformed with the needle being dropped N times, with M hits, then pi could be approximated by (2 * N /M).

This example attempts to approximate this value.




Define two Constants to describe the distance between the two horizontal lines (d) and the length of theneedle (l).

Define two Random Variables to describe the distance from the end of the needle to the horizontal line(A) and the angle at which the needle falls (Theta), as shown next.

Define an Equation Variable to determine the span of the needle based on the length of the needle and theangle at which it falls (lsin()). If this is greater than A, then the needle crossed the horizontal line. SINand RADIANS are pre-defined functions in RENO that can be typed into equations or inserted via theFunction Wizard.





Use a Standard Block to obtain the distance from the end of the needle to the horizontal line (A), basedon the Random Variable.

Use a Conditional Block to determine whether the needle crossed the horizontal line. If the span of theneedle (determined by the Equation Variable called “Hit”) is greater than or equal to the distance thatwas passed from the preceding Construct (A), then the needle crossed the horizontal line and a 1 ispassed to the TRUE path. A FALSE path is not required for this example.

Use a Result Storage Block to store a sum of the number of times that the needle crossed the horizontalline (i.e. M in Laplace’s equation to approximate ).



Use a Standard Block to enter Laplace’s equation to approximate . In this equation, the reservedkeyword IN represents the value of M passed from the previous construct and the reserved keywordSIMS_TOTAL represents the total number of simulations (i.e. N in Laplace’s equation).

Use another Result Storage Block to store the final result.




Specify to perform 1000 simulations and vary the seed from 1 to 10 with increments of 1, as shown next.This will result in 10 runs of 5000 simulations each, using a different set of randomly generated numbersfor each run to give us a feel for the variation in the results.

5.13 Example 13 - Financial Analysis: Retirement Strategy


The results are displayed in the Simulation Results Explorer, as shown next.

The flowchart displays the last value of , corresponding to seed 10.


This example is presented in the RENO file Retirement Strategy.rnp located in the Financial Analysissubfolder of the Examples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Financial Analysis).

In planning for your retirement, you normally invest a percentage of your income in “safe investments”(bank) or in “riskier investments” (stock market). For this example, let’s assume that you are choosing toinvest some of your money in the bank and some in the stock market, let’s say NASDAQ and tied to theNASDAQ Composite Index.

First, you do your research on the two investment vehicles and determine the following:

The interest from the bank investment follows a generalized gamma distribution with mean = 1.7406,standard deviation = 0.1141 and lambda = 4.3106.

The profit from NASDAQ investment follows a normal distribution with mean = 13 and standarddeviation = 28.



Note that you can use the Weibull++ software to analyze data on average bank interest rates and NASDAQannual returns to obtain the distribution and parameters for this analysis.

You will choose to invest X% of your income per year for the next Y years. Assume that your currentincome is $40,000 per year and, based on past history, that your income will increase yearly by a percentagethat is normally distributed with mean = 4 and standard deviation = 1.5.

You will decide to put Z% of your investment money in the bank for the first year and the remainder inNASDAQ. Given the volatility of NASDAQ, the following strategy is applied for subsequent years:

Any profits made in the stock market at the end of each year will be transferred into the savings accountfor safe keeping. Thus, the amount invested in NASDAQ will be restricted to the initial amount put inplus annual contributions and minus any losses.

The amount invested in NASDAQ will not increase as your income increases every year. It will alwaysbe equal to the first year investment.

You can use RENO to model this investment strategy and determine a variety of different results. Forexample, you may wish to:

Estimate your investment income over the next 20 years if you invest 5% of your yearly income with50% going to savings.

Compare your investment income after 20 years, varying the investment portion from 0% to 20% of youryearly income (with 50% going to savings).

Compare your investment income after 20 years, varying the investment portion from 0% to 20% of youryearly income and varying the amount invested in savings from 0% to 100%.


Define two Random Variables to describe the returns from the BANK and NASDAQ investmentvehicles.



Define a third Random Variable to describe the yearly increase in your income.

Define three Constants to set the number of years that will be analyzed, the percentage of your yearlyincome that will be invested and the percentage of the investment that will go to NASDAQ each year.The values of these Constants can be varied during the simulation to determine specific results ofinterest.

(Note that in the example file shipped with the software, three separate constants are defined for Years.This is simply so that each flowchart uses a different constant, enabling the example file to displayconsistent simulated results for each flowchart.)

Define two Equation Variables to calculate the amount of money that will be invested in the bank eachyear. This is determined by calculating the investment amount for the year and then subtracting theamount that will be invested in NASDAQ.



Define a third Equation Variable to calculate the threshold that will be used to determine whether youearned money in the stock market during a given year (and will therefore move those earnings into thebank account).

Note that these Equation Variables refer to defined Storage Variables (i.e. “CurrentSalary,” “StocksPortion”and “CurrentYear”), which hold values obtained from the flowchart during simulation. You will also need todefine three more Storage Variables for use during simulation: “AdditionalEarnings,” “BankIncome” and“StocksIncome.” “CurrentSalary” has a Start Value of 40,000, reflecting the salary you are starting at. Allother Storage Variables have a Start Value of 0. All of the Storage Variables are reset after each simulation.





Use a Standard Block to calculate the amount of money that will be invested in NASDAQ for the firstyear and place this value into the Storage Variable “StocksPortion.”



Use two Standard Blocks to calculate the amount of money in the bank and in NASDAQ at the end of theyear and to store these values in the Storage Variables “BankIncome” and “StocksIncome,” respectively.

For the BANK investment vehicle, this is the amount invested in the bank for the given year(SavingsPortion) plus any NASDAQ earnings from the previous year (AdditionalEarnings) plus the amountof money in the bank account at the end of the previous year (BankIncome) multiplied by the interest earnedduring the year. (1+BankInterest/100). For the NASDAQ investment vehicle, this is the amount invested instocks for the given year (StocksPortion) plus the amount of money in the NASDAQ account at the end ofthe previous year (StocksIncome) multiplied by the interest earned during the year (1+StockProfit/100).

Use a Conditional Block to check whether the value of "StocksIncome" exceeds the value of"SavingsThreshold" in order to determine whether you have earned or lost money in NASDAQ duringthe current year and pass the values required to update the storage variables accordingly.



In the False path, a Standard Block called "Save Additional Earnings" calculates the amount of theearnings, if any, and updates the "AdditionalEarnings" storage variable. In the True path, a StandardBlock called "Save Stocks Income" updates the "StocksIncome" storage variable.

Next, use a Standard Block to calculate next year's salary and update the "CurrentSalary" storagevariable.

Once you have defined the model for the first year of investment, the next step is to create a loop in theflowchart to model each subsequent year until the specified number of years have been simulated. To dothis, insert a Start flag and a Reset block before the Standard Blocks that calculate the amount of moneyin each investment vehicle at the end of a given year. The flag identifies the starting point for the loopand the Reset block regenerates the random variables for each year. Also insert a Go To Flag Markerafter the "Save New Salary" block, which points to the Start flag and creates the loop.

Within the loop, insert a Counter Block to track the number of loops that have been executed (i.e.number of years that have been simulated). Note that the Counter Block requires two paths in, which



keeps it from executing until both incoming paths have been executed, and that its value is stored in theStorage Variable “CurrentYear.”

Then insert a Conditional Block to check whether the required number of years have been simulated, asspecified in the Constant "Years." If not, then the simulation continues to the "Earned Money"Conditional Block as described above and the simulation continues to loop through the flowchart.



When the required number of years have been looped through, the simulation continues to a SummingGate, which leads to two Standard Blocks that normalize the "BankIncome" and "StocksIncome"amounts across all simulations, as shown next for “BankIncome.”

A Result Storage block then stores the sum of the bank and stock incomes, which represents the totalinvestment income for the given investment strategy.


To estimate your investment income over the next 20 years if you invest 5% of your yearly income with 50%going to savings, perform 1,000 simulations and vary the Constant “Years” from 1 to 20 with an incrementof 1. The results are displayed in the following plot:


TOTAL vs. Years1

Flowchart: 1. 5% Invested - vary years - Simulations per run: 1000 - Sensitivity Analysis (One Way): Years1 - Multiple Analyses Disabled

Years1

TOTA

L

1.000 20.0004.800 8.600 12.400 16.2002175.747

110243.340

23789.265

45402.784

67016.303

88629.821

1. 5% Invested - vary years

TOTAL




To compare your investment income after 20 years, varying the investment portion from 0% to 20% of youryearly income, configure the Sensitivity Analysis page of the Simulation console as follows and repeat thesimulation.

The results are displayed in the following plot.ReliaSoft RENO - www.ReliaSoft.com

TOTAL vs. InvestmentPercent

Flowchart: 2. After 20 Years - vary investment - Simulations per run: 1000 - Sensitivity Analysis (One Way): InvestmentPercent - Multiple Analyses Disabled

InvestmentPercent

TOTA

L

0.000 20.0004.000 8.000 12.000 16.0000.000

436155.865

87231.173

174462.346

261693.519

348924.692

2. After 20 Years - vary investment

TOTAL




To compare your investment income after 20 years, varying the investment portion and the amount investedin savings, configure the Sensitivity Analysis page as follows and repeat the simulation.

The results are displayed in the following three-dimensional plot.

53642

107284

160926

214568

268210

321852

375494

429136

482778

TOTAL


This chapter provides the following examples, designed to provide further practice in building RENOFlowcharts to model probabilistic scenarios:

Example 1 - Reliability and Availability: Aircraft Readiness - page 125

Example 2 - Risk Analysis: Complex Fuel System Risk Analysis - page 126

Example 3 - Probability: Blackjack - page 127

Example 4 - Probability: Buffy’s Mood - page 127

Example 5 - Probability: Lottery - page 128

Example 6 - Probability: Running Shoes - page 128

Example 7 - Probability: Roulette - page 129

Note that these examples are not covered in detail. Each example is based on a RENO project file (*.rnp)that is shipped with the software, and the full solution is presented in that file.

6.1 Example 1 - Reliability and Availability: Aircraft Readiness

This example is presented in the RENO file Aircraft Readiness.rnp located in the Reliability subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Reliability).

Consider a fleet of aircraft in which each aircraft can be either ready to fly or failed. Assume an 85%probability of the aircraft being ready to fly.

If an aircraft cannot fly, it is sent to the maintenance facility where it is repaired based on a lognormal repairdistribution with mean = 2 hours and std = 0.7.

Each sortie is 1.5 hours and it takes 30 minutes to refuel and check an aircraft before it can undertakeanother sortie. These sorties can take place only during a 6 hour period each day, 5 days a week.

Also assume that an aircraft can fail during flight with a 30% probability. If the aircraft fails during flight,the sortie is considered to be successful; however, the aircraft must undergo repair before its next flight.

Determine the expected number of successful sorties per day per aircraft.

Practice Questions 6

6 Practice Questions


6.2 Example 2 - Risk Analysis: Complex Fuel System Risk Analysis

This example is presented in the RENO file Complex Fuel System.rnp located in the Risk Analysis subfolderof the Examples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Risk Analysis).

DISCLAIMER: This example does not represent any real system, nor are the values or logic used hereinrepresentative of any real world scenario. It uses arbitrary numbers, cases and logic to illustrate the use ofRENO constructs.

A complex fuel system is susceptible to leaks, as follows:

The leak failure mode is governed by a Weibull distribution with a shape parameter of 3 and a scaleparameter of 1,000 days.

Each leak can be classified as one of three types, with a specified fixed probability for each type:

Atomizing Leak – 10%

Minor Leak – 20%

Major Leak – 70%

Atomizing Leak An atomizing leak is undetectable.

It is a risk issue if it fails to ventilate and an ignition source is present at the same time.

Assume a MTTF of 1000 days for the ventilation mechanism and a fixed probability of an ignitionequal to 20%.

If an ignition occurs, a fire or an explosion may result 40% of the time.

If a fire occurs, an explosion will not occur. If a fire does not occur, an explosion will occur. Theprobability that it is a fire rather than an explosion is 80%.

Minor Leak A minor leak is undetectable.



If ignited, it can cause a fire 80% of the time.

Major Leak A major leak is detectable 90% of the time.



If ignited, it can cause a fire 99% of the time.

Regarding Personnel At any time, there is a 30% chance that personnel will be in the compartment.

The number of personnel in the compartment is normally distributed with mean of 5 and standarddeviation of 1.

If personnel are in the compartment, there is a 50% probability that they may be exposed/injured by thefire and a 99% probability that they may be exposed/injured by the explosion.

6.3 Example 3 - Probability: Blackjack


Determine the number of personnel who may be injured if 100 units operate for 3000 days, and what type ofinjury they will sustain.


This example is presented in the RENO file Blackjack.rnp located in the Casino subfolder of the Examplesfolder located in your application directory (e.g. C:\Program Files\ReliaSoft\RENO\Examples\Casino).

Using a single deck that is reshuffled after every hand, play a game of blackjack against the dealer. Thedealer stays at 17 and higher, while you play based on the strategy defined in the MyStrategy table.

In the MyStrategy table, you hit (1) or stay (0) depending on what the dealer is showing (D1-D10) and yourhand.

Note that in this simple example all hands payoff 1-1 (including a blackjack) and no splits or double downsare contemplated.

6.4 Example 4 - Probability: Buffy’s Mood

This example is presented in the RENO file Buffy’s Mood.rnp located in the Misc Fun subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Misc Fun).

On any given day, Buffy is either cheerful (C), so-so (S) or gloomy (G).

If she is cheerful today, then she will be C, S or G tomorrow with respective probabilities of 0.7, 0.2, 0.1.

If she is so-so today, then she will be C, S or G tomorrow with respective probabilities of 0.4, 0.3, 0.3.

6 Practice Questions


If she is gloomy today, then she will be C, S or G tomorrow with respective probabilities 0.2, 0.4, 0.4.

What proportion of time is Buffy cheerful?

An algebraic solution to this can be obtained using Markov chains where the transition matrix is:

with three states: cheerful, so-so and gloomy. The limiting probabilities for these states are found to be 30/59, 16/59 and 13/59, respectively. Therefore, Buffy is cheerful 30/59 or about 51% of the time. Use RENOto obtain a similar answer.

6.5 Example 5 - Probability: Lottery

This example is presented in the RENO file Lottery.rnp located in the Casino subfolder of the Examplesfolder located in your application directory (e.g. C:\Program Files\ReliaSoft\RENO\Examples\Casino).

Consider a simple lottery game with a number of balls (possible outcomes) equal to MaxNumber (e.g. ifMaxNumber is 10 then 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are possible outcomes). Assume that the balls that aredrawn are equal to the NumberOfBalls. Drawn balls are not replaced, so each number can only appear once.

Determine the number of wins and losses that can be expected if there are 10 balls and your ticket has 6numbers.

6.6 Example 6 - Probability: Running Shoes

This example is presented in the RENO file Running Shoes.rnp located in the Misc Fun subfolder of theExamples folder located in your application directory (e.g. C:\ProgramFiles\ReliaSoft\RENO\Examples\Misc Fun).

A man leaves his house each morning and goes for a run. He is equally likely to leave either from his front orback door. Upon leaving the house, he chooses a pair of running shoes from beside the door or goes runningbarefoot if there are no shoes at the door from which he leaves. On his return, he is equally likely to enter,and leave his running shoes, either by the front or back door. If he owns a total of k pairs of running shoes,what proportion of the time does he run barefoot?

To find an algebraic solution, let Xn be the number of shoes at the front door on day n, then Xn is a Markov

Chain with transition matrix :

which is doubly stochastic since row sums and column sums are 1. So for all i if our runnerowns k pairs of shoes. Use RENO to obtain simulated results.

i 1 k 1+ =

6.7 Example 7 - Probability: Roulette



This example is presented in the RENO file Roulette.rnp located in the Casino subfolder of the Examplesfolder located in your application directory (e.g. C:\Program Files\ReliaSoft\RENO\Examples\Casino).

The house edges on all possible bets for American roulette are described in the following table. It detailsexpected gains and losses per 38 games for $1 bet. Each number has a probability of 1/38 of being a winner.

You can use RENO to simulate different bets, including:

Only Straight Up Bet

Only Color Bet

Etc.

Use RENO to analyze a straight up bet.


This chapter provides the answers to the Practice Questions presented in the previous chapter.

Note that these examples are not covered in detail. Each example is based on a RENO project file (*.rnp)that is shipped with the software, and the full solution is presented in that file.

7.1 Example 1 - Reliability and Availability: Aircraft Readiness

After running 10,000 simulations, the results are displayed in the Simulation Results Explorer and in theflowchart. The expected number of sorties per day per aircraft is 2.0375.

7.2 Example 2 - Risk Analysis: Complex Fuel System Risk Analysis

After performing 1000 simulations, the results are displayed in the Simulation Results Explorer and in theflowchart are as follows:

Injury Due to Explosion from Atomizing Leak – 2.7857

Injury Due to Fire from Atomizing Leak – 1.7999

Injury Due to Fire from Minor Leak – 7.1932

Injury Due to Fire from Major Leak – 3.2101

Total Injuries = 14.9889


Assuming that the player started with $100, after 1000 simulations, the player’s final balance is $99.96.

7.4 Example 4- Probability: Buffy’s Mood

After a single simulation (because the Flowchart includes a loop), the results are displayed in the SimulationResults Explorer and in the flowchart. Buffy is shown to be cheerful 51.56% of the time.

Note that in order to approach the limiting probabilities, this scenario needs to be run multiple times. Thedependence of a given scenario on the previous day is achieved by looping over the model while keepingtrack of Buffy’s mood on the previous day. Because the flowchart contains a loop, the example should besolved using a single simulation.

Answers to Practice Questions 7

7 Answers to Practice Questions


7.5 Example 5 - Probability: Lottery

After performing 500 simulations, the results are:

Wins: 4

Losses: 496

This can be repeated for different numbers of balls and winning numbers on the lottery ticket. Note that asthe odds against winning get larger, you need to run the simulation longer in order to realize any winningoutcomes.

7.6 Example 6 - Probability: Running Shoes

After a single simulation (because the Flowchart includes a loop), the results are displayed in the SimulationResults Explorer and in the flowchart. The man runs barefoot 14.89% of the time.


After performing 1000 simulations, the results are displayed in the Simulation Results Explorer and in theflowchart, as shown next.