pom qm software manual

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POM - QM FOR WINDOWS

Version 3

Software for Decision Sciences:

Quantitative Methods,

Production and Operations Manae!ent

"oward #$ Weisswww$prenha%%$co!&weiss

dsSoftware'prenha%%$co!

#une (), *))+

Copyright (c) 2006 by Pearson Education, Inc., Upper Saddle i!er, "e# $ersey, 0%&'.Pearson Prentice all. *ll rights reser!ed.

http://var/www/apps/conversion/tmp/scratch_3/www.prenhall.com/weiss

mailto:[email protected]

http://var/www/apps/conversion/tmp/scratch_3/www.prenhall.com/weiss

mailto:[email protected]



a%e of .ontents

.hapter /: IntroductionOverview...........................................................................1Hardware and Software Requirements..............................3Installing the Software.......................................................4The Program Grou...........................................................!Starting the Program.........................................................."

The #ain S$reen...............................................................%

.hapter *: 0 Sa!p%e Pro%e!Introdu$tion.....................................................................1&'reating a (ew Pro)lem.................................................13The *ata S$reen..............................................................1!+ntering and +diting *ata...............................................1!The Solution S$reen........................................................1%

.hapter (: he Main Menu

,ile..................................................................................1-+dit..................................................................................&4iew................................................................................&!#odule............................................................................&",ormat.............................................................................&%Tools...............................................................................3/0indow...........................................................................3/Hel.................................................................................3&

.hapter 1: PrintinThe Print Setu S$reen....................................................3"

Information to Print.........................................................3%Page Header Information.................................................3aout.....................................................................3-Printer Otions................................................................4/

v



PO#5# for 0indows

.hapter +: 2raphsIntrodu$tion.....................................................................4&,ile Saving......................................................................43Print.................................................................................43'olors and ,onts.............................................................43

.hapter 3: Modu%esOverview.........................................................................446ggregate 7Produ$tion8 Planning....................................4!6ssem)l 2ine 9alan$ing................................................!%The 6ssignment #odel..................................................."!9rea:even;'ostolume 6nalsis..................................."%'aital Investment...........................................................%1*e$ision 6nalsis............................................................%3,ore$asting......................................................................4

Game Theor...................................................................--Goal Programming........................................................1/&Integer < #i=ed Integer Programming..........................1/"Inventor.......................................................................1/->o) Sho S$heduling 7Sequen$ing8................................11%2aout...........................................................................1&"2earning 7+=erien$e8 'urves.......................................13/2inear Programming......................................................1332o$ation........................................................................13-2ot Si?ing......................................................................14!#ar:ov 6nalsis...........................................................1!/#aterial Requirements Planning...................................1!4 (etwor:s.......................................................................1"1Produ$tivit...................................................................1"!Pro@e$t S$heduling.........................................................1""5ualit 'ontrol..............................................................1%!Relia)ilit......................................................................1&Simulation.....................................................................1!Statisti$s........................................................................1The Transortation #odel.............................................1-30aiting 2ines................................................................1-%

0or: #easurement.......................................................&/"

0ppendices6. 'ustomi?ation due to te=t)oo:.........................................&1/9. Aseful hints for modules....................................................&11

vi



Preface

It is hard to )elieve that POM-QM for Windows 7formerl DS for Windows8 has )een in e=isten$eB first as a *OS rogram and then as a 0indows rogramB forover 1! ears. It seems li:e we have )een using )oth mini$omuters and 0indowsforever )ut in fa$t larges$ale 0indows usage has )een for less than a de$ade. 6tthe time that I finished the original *OS ersionB few students had ersonal$omuters or :new what an ISP was. TodaB sin$e a large ma@orit of students

have their own $omuters this software is even more valua)le than it has ever )een.

The original goal in develoing software was to rovide students with the mostuserfriendl a$:age availa)le for rodu$tion;oerations managementBquantitative methodsB management s$ien$eB and oerations resear$h. 0e aregratified ) the resonse to the revious versions of POM-QM for Windowsindi$ating that we have $learl met our goal.

QM for Windows 7ersion 1./8 was first distri)uted in the summer of 1--" while asearate )ut similar rogramB POM for Windows 7ersion 1.18 was first distri)uted

in the fall of 1--". DS for WindowsB whi$h $ontained all of the modules in )oth POM and QM and also $ame with a rinted manual was first distri)uted in 1--%.ersion & of all three rograms was $reated for 0indows -! and distri)uted in thefall of 1---.

,or this new versionB version 3B we have $ollased the three former rodu$ts intoone rodu$t named POM-QM for Windows. ,or $onsisten$ with ast versionsBwhen using Prenti$eHall te=ts it is ossi)le to install the rogram as POM forWindows or QM for Windows and to disla the POM for Windows or QM forWindows module menu. Regardless of the name of the des:to i$on all of themodules are availa)le to all users. 0e will refer to the rodu$t as POM-QM forWindows throughout this manual.

This is a a$:age that $an )e used to sulement an te=t)oo: in the )road area:nown as *e$ision S$ien$es. This in$ludes Produ$tion and Oerations#anagementB 5uantitative #ethodsB #anagement S$ien$eB or OerationsResear$h.

vii



6ser Friend%iness

6s mentioned a)oveB we have $om)ined all three a$:ages into one a$:age inorder that all models will )e availa)le to the students ese$iall students who

ta:e )oth an Oerations #anagement 'ourse and a 5uantitative #ethods $ourse.0e still allow for student $hoi$e of menu 7PO#B 5# or )oth8 to minimi?e$onfusion. In additionB in order to imrove the understanding of the models wehave added searators )etween models in the model su)menu sele$tion menu. 0ehave $om)ined integer and mi=ed integer rogramming into one module. 0e haveadded an overview ta) to the ro)lem $reation s$reen to hel des$ri)e the otionsthat are availa)le. #anuals in )oth P*, format 7this do$ument8 and 0ord formathave )een added so that users ma easil a$$ess the manual while running the rogram or rint ages from the manual. Tutorials whi$h wal: ou through $ertainoerations ste)ste are in$luded in the "e%p menu. The e=amles used in thismanual are in$luded in the installation. #ore user $ustomi?ation otions are

availa)le in the Aser Information se$tion under the Hel menu.

To the students who use this softwareB I hoe ou find that this software$omlements our te=t well. To the instru$tors who use this softwareB than: oufor $hoosing POM-QM for Windows. I wel$ome our $ommentsB ese$iall ) email at dsSoftwareDrenhall.$om.

i=



There are several individuals at Prenti$e Hall to whom I must give se$ial than:s.Ri$h 0ohl and Tom Tu$:er are the editors with whom I had wor:ed on this ro@e$t for the first " versions. (ot all editors have their :een understanding of$omutersB softwareB te=tsB studentsB and rofessors. 0ithout this understandingand visionB POM-QM for Windows would still )e a vision rather than a realit.# $urrent editorsB #ar: Pfalt?graf and 6lana 9radle have )een instrumental ingetting this version to mar:et. ,ellow Prenti$e Hall authors in$luding 9arrRenderB Ralh StairB 'hu$: TalorB and >a Hei?er have heled me to ma:e thetransition from the original *OS rodu$t to the $urrent 0indows rodu$ts and toimrove the 0indows rodu$t. I am grateful for their man suggestions and thefa$t that the $hose m software as the software to a$$oman their te=ts. ThesuortB en$ouragementB and hel from all of these eole are ver mu$hare$iated. (an$ 0el$her rovides the suort of the Prenti$e Hall we) agesthat are maintained for these rodu$ts. ,inallB I would li:e to e=ress mare$iation to *e))ie 'lare who has )een the mar:eting manager for m

software.

6s alwasB I must e=ress m are$iation and love to m wifeB 2u$iaB for herunderstanding and suort during the man hours that $ontinue to send in front ofm P'. In additionB I am grateful for the valua)le $omments and suggestionsregarding the loo: and feel of the software from 2u$ia and m $hildrenB +rnie and2isa.



.hapter /

IntroductionOverview

0el$ome to Prenti$eHallEs *e$ision S$ien$e software a$:ageC POM-QM forWindows ( a.:.a. POM for Windows and QM for Windows). This a$:age is themost userfriendl software a$:age availa)le in the fields of rodu$tion andoerations managementB quantitative methodsB management s$ien$eB or oerationsresear$h. POM-QM for Windows has )een designed to hel ou to )etter learn andunderstand these fields. The software $an )e used either to solve ro)lems or to$he$: answers that have )een derived ) hand. POM-QM for Windows $ontains alarge num)er of modelsB and most of the homewor: ro)lems in PO# te=t)oo:sor 5# te=t)oo:s $an )e solved or aroa$hed using POM-QM for Windows.

In this introdu$tion and the ne=t four $hatersB we des$ri)e the general features ofthe software. 0e en$ourage ou to read them while running the software on our$omuter. 'hater " $ontains the des$rition of the se$ifi$ models andali$ations availa)le in POM-QM for Windows.

ou will find that the software is ver userfriendl due to the following featuresC

Standardi8ation

The grahi$al user interfa$e for the software is a standard 0indowsinterfa$e. 6none familiar with an standard sreadsheetB word ro$essorBor resentation a$:age in 0indows will easil )e a)le to use the software.This standard interfa$e in$ludes the $ustomar menuB tool)arB status )arBand hel files of 0indows rograms.

+ven though the software $ontains &- modules and more than "/su)modelsB the s$reens for ever module are $onsistentB so that after ou )e$ome a$$ustomed to using one module ou will have an eas time withthe other modules.,ile storage and retrieval is simle. ,iles are oened and saved in the usual0indows fashion andB in additionB files are named ) moduleB whi$h ma:esit eas to find reviousl saved files.

1



PO#5# for 0indows

*ata and resultsB in$luding grahsB $an )e easil $oied and asted )etweenthis ali$ation and other 0indows ali$ations.

F%e9ii%it5

The s$reen $omonents and the $olors $an )e $ustomi?ed ) the user. This$an )e arti$ularl effe$tive on overhead data shows.

The user $an sele$t the desired outut to rint rather than having to rinteverthing. In additionB several rint formatting otions are availa)le.

There are several referen$es that the user $an sele$t from the HelB AserInformation menu. ,or e=amleB the software $an )e set to automati$allsave a file after data has )een entered and;or to automati$all solve a

ro)lem after data has )een entered.

6ser-oriented desin

The sreadsheette data editor ma:es data entr and editing e=tremeleas. In additionB whenever data is to )e enteredB there is a $lear instru$tiongiven on the s$reen des$ri)ing what is to )e enteredB and when data isentered in$orre$tl a $lear error message is dislaed.

It is eas to $hange from one solution method to another in order to$omare methods and answers. In several $asesB this is siml a one$li$:

oeration. In additionB intermediate stes are generall availa)le for disla.

The disla has )een $olor $oded so that answers will aear in a different$olor from data.

e9too7 custo!i8ation

The software $an )e $ustomi?ed to Prenti$eHall te=t)oo:s in order that themodelsB notation and dislas will mat$h the arti$ular te=t)oo:.

&



6ser support

Adates are availa)le on the Internet through the Prenti$e Hall 0e) site forthis )oo:B httC;;www.renhall.$om;weissB andB hel is availa)le )$onta$ting dsSoftwareDrenhall.$om.

0hat all of this means to ou is that with a minimal investment of time in learningthe )asi$s of POM-QM for Windows, ou will have a ver eastouse means forsolving ro)lems or for $he$:ing our homewor:. Rather than )eing limited toloo:ing at the answers in the )a$: of our te=t)oo:B ou will )e a)le to see thesolutions for most ro)lems. In man $asesB the intermediate stes are dislaed inorder to hel ou $he$: our wor:. In additionB ou will have the $aa)ilit to erform sensitivit analsis on these ro)lems or to solve )iggerB more interesting ro)lems.

"ardware and Software Reuire!ents

.o!puter

The software has minimal sstem requirements. It will run on an I9# P'$omati)le Pentium ma$hine with at least #9 R6# and oerating 0indows&///B 0indows (TB 0indows #+ or 0indows P.

Dis7 Drives&.D-ROM

The software is rovided on a '*. This requires a '*RO# drive.

Monitor

The software has no se$ial monitor requirements. *ifferent $olors are used to ortra different items. 6ll messagesB oututB dataB et$. will show u on anmonitor. Regardless of the te of monitor that ou are usingB the software has the$aa)ilit that allows ou to $ustomi?e $olors and;or fonts and font si?es in thedisla to our li:ing. This is e=tremel useful when using an overhead ro@e$tionsstem. These otions are e=lained in 'hater 3 in the se$tion entitled ,ormat.

Printer

6 rinter is not required to run the software )utB of $ourseB if ou want a hard $o7rintout8 then it is ne$essar to have a rinter atta$hed. The rinting is standard sothat no se$ial featuresB $hara$tersB or rinters are required. It also is ossi)le to rint to a file in order to imort the rintout into a word ro$essor for furtherediting.5poraphic .onventions in this Manua%

3



PO#5# for 0indows

0hen we use )oldfa$eB we are indi$ating something that ou te or ress.

0hen we use a )ra$:etB ; <B we are naming a :e on the :e)oard or a $ommand )utton on the s$reen. ,or e=amle ;F/< means ,un$tion :e ,1B while ;O=< means

the JO:aE )utton on the s$reen.

0e will use ;Return<B ;4nter<B or ;Return&4nter< to mean the :e on our:e)oard that has one of those names. The name of the :e varies on different:e)oards and some even have )oth :es.

0e will use )oldfa$e and $aitali?e onl the first letter to refer to a 0indowsmenu $ommand. ,or e=amleB Fi%e refers to the menu $ommand.

!. 0e will use all $aitals to refer to a tool)ar $ommand su$h as SO2+.

Insta%%in the Software

0e assume in the dire$tions that follow that the hard drive is named 'C and thatthe '*RO# is drive *C. The software is installed in the manner that most rograms designed for 0indows are installed. ,or all 0indows installationsBin$luding this oneB it is )est to )e $ertain that no rograms are running while ouare installing a new one.

Insert the '* with POM-QM for Windows in drive *C. 6fter a little while theinstallation rogram should )egin automati$all. If it does not thenC

,rom the 0indows Start 9utton sele$tB Run. 9rowse the '* for D:setup$po!!v($e9e 7$ase does not matter8. Press ;4nter< or $li$: on ;O=<$

,ollow the setu instru$tions on the s$reen. Generall sea:ingB it is simlne$essar to $li$: ;N4>< ea$h time that the installation as:s a question.

*efault values have )een assigned in the setu rogramB )ut ou ma $hange themif ou li:e. The default folder is 'CKProgram ,ilesKPO#5#3.

The setu rogram will as: ou for registration information su$h as our nameBuniversitB rofessorB and $ourse. 6ll items are otional e=$et for the student;username that must )e given. This name cannot be changed later To $hange the otherinformation from within the rogramB use "e%p, 6ser Infor!ation.

If ou have the '* from the O!erations Management, "e te=t)oo: ) Hei?er andRender the software will automati$all )e installed as POM for Windows and$ustomi?ed to the te=t)oo:. In the futureB if ou have other Prenti$eHall *e$isionS$ien$e te=t)oo:s then the software will )e on the '* in the )a$: of the te=t)oo:

4



PO#5# for 0indows

,inallB the software $omes with a (ormal *istri)ution 'al$ulator. The $al$ulatoris on the Tools menu of the rogram )ut also $an )e used as a stand alone rogramwithout having to oen POM-QM for Windows.

To uninstall the rogram use the usual 0indows uninstall ro$edure 7Start,Settins, .ontro% Pane%, 0dd&Re!ove Prora!s8. The rograms will )eremoved )ut the data files will notF the will have to )e deleted using #'omuter or ,ile +=lorer if ou wish to do so.

In addition to the Start #enuB the installation will la$e a short$ut to the rogramon the des:to. The i$on will )e one of the three i$ons dislaed )elow deendingon the e=a$t '* )eing used. 0hi$hever des:to i$on has )een installed is the i$onthat $an )e used to easil )egin the rogram.

Startin the Prora!

The easiest wa to start the rogram is ) dou)le$li$:ing the rogram i$on that ison the des:to. 6lternativelB ou ma use the standard 0indows means forstarting the rogram. 'li$: on StartB Prora!sB POM-QM for Windows (BPOM-QM for Windows ( in order to use the software. 6fter starting the rogramB

a slash s$reen will aear as dislaed on the ne=t age.

Na!eThe name of the li$ensee will aear in the disla. This should )e our name ifou are running on a standalone $omuter or the networ: name if ou are runningon a networ:.

"



?ersion Nu!erOne imortant ie$e of information is the version num)er of the software. In thee=amleB the ersion is 3./ and this manual has )een designed around that

num)er. 0hile this is ersion 3./ there is also more detailed information a)outthe rogram version that $an )e found using "e%p, 0out 7dislaed at the verend of 'hater 38. In arti$ularB there is a )uild num)er. If ou send email as:ingfor te$hni$al suortB ou should in$lude the )uild num)er with the email.

NO4: If the rogram has )een registered as )eing in a u)li$ la) or on a networ:then at this oint the oening s$reen will $hange and give ou the oortunit toenter our name. This is useful when ou rint our results.

The rogram will start in a $oule of se$onds after the oening disla aears.

he Main Screen

The se$ond s$reen that aears is an emt main menu s$reen. The first time thatthis s$reen aearsB a Ti of the *a form will aear as dislaed )elow. If oudonEt want the Ti of the *a to show u ea$h timeB then un$he$: the )o= at thelower left of the form. If ou $hange our mind later and want to see the Ti of the*aB then go to the "e%p menu.

%



PO#5# for 0indows

Please noti$e the )a$:ground in the middle of the s$reen. This is referred to as agradient. This gradient aears whenever the main s$reen is emt and it aearson other s$reens in the software. ou ma $ustomi?e the disla of the gradient )using For!at, .o%ors as e=lained in 'hater 3.

6fter $losing the Ti of the *aB or if ou have $hosen not to see the tisB the ne=ts$reen is the module sele$tion s$reen 7shown in 'hater &8. In order to disla allof the s$reen $omonentsB we have sele$ted a module and loaded a data file.



The to of the s$reen is the standard 0indows title )ar for the window. 6t the )eginning the title is POM-QM for Windows 7or POM for Windows or QM forWindows8. If ou are using a Prenti$e Hall te=t then the names of the authors ofthe te=ts should aear in this title )ar at the )eginning of the rogram as shown inthe figure on the revious age. 7If notB go to "e%p, 6ser Infor!ation.8 The title )ar will $hange to in$lude the name of the file when a file is loaded or saved asshown a)ove. On the left of the title )ar is the standard 0indows $ontrol )o= and

on the right are the standard minimi?eB ma=imi?eB and $lose )uttons for thewindowsi?ing otions.

9elow the title )ar is a )ar that $ontains the #ain #enu. The menu )ar is ver$onventional and should )e eas to use. The details of the menu otions of Fi%e,4dit, ?iew, Modu%e, For!at, oo%s, Window, and "e%p are e=lained in 'hater3. 6t the )eginning of the rogramB the 4dit otion is not ena)ledB as there is nodata to edit. The Window otion is also disa)ledB sin$e this refers to resultswindows and we have no results. 0hile the menu aears in the standard

0indows osition at the to of the s$reenB it $an )e moved if ou li:e ) $li$:ingon the handle on the left and dragging the mouse.

9elow the menu is a standard tool)ar 7also $alled a )utton )ar or ri))on8. Thistool)ar $ontains short$uts for several of the most $ommonl used menu$ommands. If ou move the mouse over the tool for a)out two se$ondsB an

-



PO#5# for 0indows

e=lanation of the tool 7)alloon hel8 will aear on the s$reen. 6s with mostsoftware a$:agesB the tool)ar $an )e hidden if ou so $hoose 7right $li$: on anof the tool)ars or use ?iew, oo%ars, .usto!i8e8. Hiding the tool)arB allows formore room on the s$reen for the ro)lems. 6s is the $ase with most tool)arsB weallow the tool)ar to float. In order to reosition an of the tool)arsB siml $li$:

on the handle on the left and drag.

One ver imortant tool on the standard tool)ar is the SO2+ tool on the farright of the tool)ar. This is what ou ress after ou have entered the data and ouare read to solve the ro)lem. 6lternativelB ou ma use Fi%e, So%ve or ress the;F@< :e. Please note that after ressing the SO2+ toolB this tool will $hange toan +*IT tool. This is how ou go )a$: and forth from entering data to viewingthe solution. ,or two modulesB linear rogramming and transortationB there is onemore $ommand that will aear on the standard tool)ar. This is the ST+P tool7not dislaed in the figure8B and it ena)les ou to ste through the iterationsB

dislaing one iteration at a time.

9elow the standard tool)ar is a format tool)ar. This tool)ar is ver similar to thetool)ars found in +=$elB 0ordB and 0ordPerfe$t. It too $an )e $ustomi?edB movedBhiddenB or floated.

There is one more tool)arB and its default lo$ation is at the )ottom of the s$reen.This )ar is a utilit )ar and it $ontains si= tools. The tool on the left is named#O*A2+. 6 module list $an aear in two was either ) using this tool or theModu%e otion on the main menu. The ne=t tool is named PRI(T S'R++(B and itis there to emulate the old rint s$reen fun$tion in *OS. The ne=t two tools willload files in alha)eti$al order either forward or )a$:wards. This is ver usefulwhen reviewing a num)er of ro)lems in one $hater su$h as the samle files thata$$oman this manual. The two remaining tools allow files to )e saved as +=$elor HT#2 files.

In the $enter are two areasB one of whi$h is the main data ta)le. The ta)le $ontainsa heading or title and then siml rows and $olumns. The num)er of rows and$olumns deends on the moduleB ro)lem teB and se$ifi$ ro)lem. The large

1/



area with no grid is the ta)le )a$:ground. The $ation $olorsB ta)le $olors and )a$:ground $olor $an )e $hanged ) using For!at, .o%ors, as e=lained in'hater 3.

6)ove the data ta)le is an area named the e=tra data )ar for la$ing e=tra ro)leminformation. Sometimes it is ne$essar to indi$ate whether to minimi?e orma=imi?eB sometimes it is ne$essar to sele$t a methodB and sometimes some valuemust )e given. These generall aear a)ove the data. On the right of the e=tradata anel is an instru$tion anel. There is alwas an instru$tion here to hel outo figure out what to do or what to enter. 0hen data is to )e entered into the datata)leB this instru$tion will e=lain what te of data 7integerB realB ositiveB et$.8 isto )e entered. The instru$tion lo$ation $an )e $hanged ) using the ?iew otion.

There also is a form for annotating ro)lems. 6 $omment ma )e la$ed here.0hen the file is savedB the information will )e savedF when the file is loadedB theinformation will aear and the annotation ma )e rinted if so desired.

Towards the )ottom of the s$reen is the status anel. The leftmost anel willdisla the module and su)model name as ou sele$t different modulesB ase=emlified in this illustration where the module is ,ore$asting and the su)modelis Time Series 6nalsisL. The $enter anel $ontains the te of s$reen 7dataBresultsB menuB grahB et$.8 and the rightmost anel has the te=t)oo: name 7if ate=t)oo: has )een sele$ted8. The status )ar $an )e hidden ) using the ?iewotion. This anel $an not )e moved.

11



PO#5# for 0indows

.hapter *

0 Sa!p%e Pro%e!

Introduction

In this $haterB we will run through a samle ro)lem from )eginning to end inorder to demonstrate how to use the a$:age. 0hile not all ro)lems or modulesare identi$alB there is enough similarit that seeing one e=amle will ma:e it ver

eas to use an module in this software. 6s we mentioned in the introdu$tionB thefirst instru$tion is to sele$t a module to )egin the wor:.

In the a)ove figureB we disla the modules as the are listed when ou use the#O*A2+ tool on the utilit )ar 7as oosed to the Modu%e otion in the mainmenu at the to8. 6s ou $an seeB there are &- modules availa)le. The are divided

into three grous. The models in the first grou ti$all are in$luded in all PO#and 5# )oo:sB while the models in the se$ond grou ti$all aear onl inPO# )oo:s and the models in the third grou aears onl in 5# te=ts. Themodels are divided in this fashion so that ou will understand it is $omletel fineto ignore PO# onl modules if ou have a 5# $ourse and vi$eversa.If ou $hoose the Modu%e otion from the #ain #enuB ou get the same moduleslisted in a single list in alha)eti$al order. 7This is dislaed in 'hater 3.8 ou



'hater &C 6 Samle Pro)lem

have the otion on this menu to disla onl the PO# modules or onl the 5#modules.

.reatin a New Pro%e!

GenerallB the first menu otion that will )e $hosen is Fi%eB followed ) eitherNew to $reate a new data set or Open to load a reviousl saved data set. In thefigure that followsB we show the $reation s$reen that is used when a new ro)lemis started. O)viouslB this is an otion that will )e $hosen ver often. The $reations$reens are similar for all modulesB )ut there are slight differen$es that ou willsee from module to module.

The to line $ontains a te=t )o= into whi$h the title of the ro)lem $an )e entered.The default title for ro)lems is initiall M7untitled8L. The default title $an )e$hanged ) ressing the )utton NModif5 Defau%t it%e. ,or e=amleB if ou$hange the default title to MHomewor: Pro)lemL then ever time ou start a new ro)lem the title will aear as Homewor: Pro)lemB and ou would siml needto add the ro)lem num)er to $omlete the title. If ou want to $hange the titleafter $reating the ro)lemB this $an easil )e done ) using the For!at, it%eotion from the main menu )ar or from the tool)ar.

,or man modulesB it is ne$essar to enter the num)er of rows in the ro)lem.Rows will have different names deending on the module. ,or e=amleB in linear rogrammingB rows are M$onstraintsLB while in fore$astingB rows are Mast eriodsL. 6t an rateB the num)er of rows $an )e $hosen with either the s$roll )aror the te=t )o=. 6s is usuall the $ase in 0indowsB the are $onne$ted. 6s oumove the s$roll )arB the num)er in the te=t )o= $hangesF as ou $hange the te=tBthe s$roll )ar moves. In generalB the ma=imum num)er of rows in an module is



PO#5# for 0indows

-/. There are three was to add or delete rows or $olumns after the ro)lem has )een $reated. ou ma use the otions in the 4dit menuB ou ma right $li$: onthe data ta)le whi$h will )ring u )oth $o and insert;delete otions orB to inserta row or insert a $olumn B ou ma use the tools on the tool)ar.

This rogram has the $aa)ilit to allow ou different otions for the default rownames. Sele$t one of the si= otion )uttons in order to indi$ate whi$h stle ofdefault naming should )e used. In most modulesB the row names are not used for$omutationsB )ut ou should )e $areful )e$ause in some modules 7most nota)lPro@e$t #anagement and #RP8 the names might )e relevant to the $omutations.In most modulesB the row names $an )e $hanged ) editing the data ta)le.

#an modules require a num)er of $olumns. This is given in the same wa as thenum)er of rows. The rogram gives ou a $hoi$e of default values for $olumnnames in the same fashion as row names )ut on the ta) named 'olumn (ames.

0e have added an overview ta) to the $reation s$reen in this version of thesoftware. The overview ta) gives a )rief des$rition of the models that areavaila)le and also gives an imortant information regarding the $reation or dataentr for that module.

Some modulesB su$h as the linear rogramming e=amle dislaed on the revious ageB will have an e=tra otion )o=B su$h as for $hoosing minimi?e or ma=imi?e orsele$ting whether distan$es are smmetri$ or not. Sele$t one of these otions. Inmost $asesB this otion $an )e $hanged later on the data s$reen.

0hen ou are satisfied with our $hoi$esB $li$: on the ;O=< )utton. 6t this ointBa )lan: data s$reen will aear as given in the following figure. S$reens will differmodule ) module )ut the will all resem)le the s$reen on the following age.



'hater &C 6 Samle Pro)lem

he Data Screen

The data s$reen was des$ri)ed )riefl in 'hater 1. It has a data ta)le andB forman modelsB there is e=tra information that aears a)ove the data ta)le as shownin the figure )elow.

4nterin and 4ditin Data

6fter a new data set has )een $reated or an e=isting one has )een loadedB the data$an )e entered or edited. +ver entr is in a row and $olumn osition. ounavigate through the sreadsheet using the $ursor movement :es 7or the mouse8.These :es fun$tion in a regular wa with one ver useful e=$etion the ;4nter:e.

The ;4nter< :e ta:es ou to the ne=t $ell in the ta)leB first moving to the right andthen moving down. 0hen a row is finishedB the ;4nter< :e goes to the first $ell inthe ne=t row that $ontains data rather than a row name. ,or e=amleB in the s$reena)oveB if ou are at the end of the row named MSour$e 1 and ou ress ;4nter<Bthe $ursor will move to the $ell with a M/ in the ne=t row. It is ossi)le to set the$ursor to go to the first $ellB the one with the name in itB ) using "e%p, 6serInfor!ation$



PO#5# for 0indows

In additionB if ou use the ;4nter< :e to enter the dataB after ou are done withthe last $ell the rogram will automati$all solve the ro)lem 7saving ou thetrou)le of $li$:ing on the SO2+ tool8. This )ehavior $an )e ad@usted ) using"e%p, 6ser Infor!ation andB in additionB if ou want the rogram toautomati$all romt ou to save the file when ou are done entering dataB this

too $an )e a$$omlished through "e%p, 6ser Infor!ation$

The instru$tion frame on the s$reen will $ontain a )rief instru$tion des$ri)ing whatis to )e done. There are essentiall three tes of $ells in the data ta)le.

One te is a regular data $ell into whi$h ou enter either a name or a num)er.0hen entering names and num)ersB siml te the name or num)erF then ressthe ;4nter< :e or one of the dire$tion :es or $li$: on another $ell. If ou te anillegal $hara$terB a message )o= will )e dislaed indi$ating so.

6 se$ond te is a $ell that $annot )e edited. ,or e=amleB the emt $ell in theuer left hand $orner of the ta)le $an not )e edited. 7ou a$tuall $ould asteinto the $ell.8

6 third te is a $ell that $ontains a drodown )o=. ,or e=amleB the signs in alinear rogramming $onstraint are $hosen from this te of )o=B as shown in thefollowing illustration. To see all of the otionsB ress the arrow on the drodown )o=.

0hen ou are finished entering the dataB ress the SO2+ tool on the tool)ar oruse ;F@< or Fi%e, So%ve and a solution s$reen will aear as given in the followingillustration. The original data is in )la$: and the solution is in a $olor. Of $ourseBthese are onl the default valuesB as all $olors ma )e set ) using For!at,

.o%ors.



he So%ution Screen

6n imortant thing to noti$e is that there is more solution information availa)lethan the one ta)le dislaed. This $an )e seen ) the i$ons given at the )ottom.'li$: on these to view the information.

6lternativelB when ou solve the ro)lemB the form )elow $an )e set to aear onto of the solution through "e%p, 6ser Infor!ation.

If ou $li$: on the OPTIO(S )utton then ou $an set u the )ehavior of thesoftware when a ro)lem is solved. The otions are as followsC



The first otion will siml disla the solution. The ne=t three otions are thereto remind ou that more results ma e=ist than the one window )eing dislaed.The se$ond otion will disla the Solutions 0indow whi$h $ontains a )riefdes$rition of ea$h solution 0indow. The third otion will automati$all drodown the 0indow menu. These otions $an )e rest using "e%p, 6serInfor!ation$

It is generall at this oint thatB after reviewing the solutionB ou would $hoose to rint )oth the ro)lem and solution.

(ow that we have e=amined the $reation and solution of a ro)lemB we e=lain allof the otions that are availa)le in the #ain #enu.



.hapter (

he Main Menu

Fi%e

Fi%e $ontains the usual otions that one finds in most 0indows rogramsB as seenin the figure that follows.

These otions are now des$ri)ed.

New

6s demonstrated in the samle ro)lemB this otion is $hosen to )egin a new ro)lem;file. In some $asesB ou will go dire$tl to the ro)lem $reation s$reenB

while in other $ases a ou menu will aear indi$ating the su)models that areavaila)le. 6fter sele$ting a su)modelB ou will go to the $reation s$reen.



Open

This is used to oen;load a reviousl saved file. ,ile sele$tion is the standard0indows $ommon dialog te. 6n e=amle of the s$reen for oening a file isshown )elow. (oti$e that the e=tension for files in the software sstem is given )the first three letters of the module name 7The e=$etions are assem)l line )alan$ing 7Q.)al8 and laout 7Q.oe8 due to $onventions in revious versions and rodu$tivit7Q.rd8 to avoid a $onfli$t with ro@e$t management.8. ,or e=amleB alltransortation files have the e=tension Q.tra. 0hen ou go to the Oen ,ile dialogBthe default value is for the rogram to loo: for files of the te in this module.This $an )e $hanged at the )ottom leftB where it sas M,iles of te.L OtherwiseBfile oening and saving is quite normal. The drive or folder $an )e $hanged withthe drive;folder drodown )o=B a new dire$tor ma )e $reated using the new )utton at the toB and details a)out the files ma )e seen ) using the details )utton at the to right.

It is ossi)le to use "e%p, 6ser Infor!ation to set the rogram to automati$allsolve an ro)lem when it gets loaded. This waB if ou li:eB ou $an )e loo:ingat the solution s$reen whenever ou load a ro)lem rather than at the data s$reen.

Save

Save will rela$e the file without as:ing ou if ou $are a)out overwriting the

revious version of this file. If ou tr to save and have not reviousl named thefileB ou will )e as:ed to name this file.

The names that are legal are standard 0indows file names. In addition to the filenameB ou ma refa$e the name with a drive letter 7with its $olon8 or athdesignation. The software will automati$all aend an e=tension to the name thatou use. 6s mentioned a)oveB the e=tension is the first three letters of the module



name. ou ma te file names in as uer$aseB lower$aseB or mi=ed. +=amles oflegal file names are

samleB samle.traB $Cm,ileB $CKm'ourseKtestB and mro)lem.e=amle.

If ou enter samle.tra and the module is not transortationB then an e=tension will )e added. ,or e=amleB if the module is linear rogrammingB then the name underwhi$h the file will )e saved will )e samle.tra.lin.

Save as

Save as will romt ou for a file name )efore saving. This otion is ver similarto the otion to load a data file. 0hen ou $hoose this otionB the 0indows'ommon *ialog 9o= for ,iles will aear. It is essentiall identi$al to the one reviousl shown for oening files.

Save as 49ce% Fi%e

The software has an otion that allows ou to save most 7)ut not all8 of the ro)lems as +=$el files. The data is transorted to +=$el and the sreadsheet isfilled with formulas for the solutions. In some $asesB +=$elEs Solver ma )erequired in order to get the solution.

,or e=amleB given )elow is the outut from a waiting line model. The lefthandside has the data while the righthand side has the solution. (oti$e the $olor$oding of answer visvis data.

6fter saving as an +=$el fileB the +=$el file aears )elow. (oti$e from the

formula for $ell +% 7shown at the to of the sreadsheet8 that a sreadsheet withformulas was $reated. That isB we did not M$ut and asteL the a)ove s$reen into+=$el 7whi$h is ossi)le8 )ut instead $reated an +=$el sreadsheet witharoriate formulas.



Save as "MA Fi%e

6n ta)leB either data or solutionB ma )e saved as an HT#2 fileB as shown )elow.

If more than one ta)le is on the s$reen at the time that this otion is sele$tedB thenthe a$tive ta)le is the one that is saved.Print



Print will disla a Print Setu s$reen. Printing otions are des$ri)ed in 'hater4. 9oth Save and Print will a$t slightl differentl if a grah is )eing dislaed atthe time that ou use Print or Save.

Print Screen

This will rint the s$reen as it aears. *ifferent s$reen resolutions ma affe$t the rinting. Printing the s$reen is more time $onsuming than a regular rint. Ase thisotion if ou need to demonstrate to our instru$tor e=a$tl what was on thes$reen at the time.

So%ve

There are several was to solve a ro)lem. 'li$:ing on Fi%e, So%ve is ro)a)l theleast effi$ient wa to solve the ro)lem. The tool)ar i$on ma )e usedB as well asthe ;F@< :e. 6lsoB if the data is entered in order 7to to )ottomB left to rightB using

;4nter<8B the rogram will solve the ro)lem automati$all after the last $ell.

6fter solvingB the So%ve otion will $hange to an 4dit otion on )oth the menuand the tool)ar. This is the wa to go )a$: and forth )etween data and solutions. (ote that "e%p, 6ser Infor!ation ma )e used to set the rogram toautomati$all ma=imi?e the solution windows if so desired.

49it The ne=t otion on the Fi%e menu is 49it. This will e=it the rogram. ou will )eas:ed if ou want to e=it the rogram. ou $an eliminate this question ) using"e%p, 6ser Infor!ation$

Aast Four Fi%es

The Fi%e menu $ontains a list of the last four files that ou have used. 'li$:ing onone of these will load the file.



4dit

The $ommands under 4dit $an )e seen in the following illustration. Their uroses are threefold. The first si= $ommands are used to insert or delete rows or$olumns. The se$ond te of $ommand is useful for reeating entries in a $olumnB

and the third te is for $utting and asting )etween 0indows ali$ations. It isalso ossi)le to ena)le the insert;delete and $o otions ) right $li$:ing on thedata or solution ta)le.

1Insert Row will insert a row after the $urrent rowB and Insert .o%u!n willinsert a $olumn after the $urrent $olumn. Insert RowsBsC and Insert .o%u!nsBsC ena)le ou to insert multile rows or $olumns. De%ete Row will delete the $urrentrowB and De%ete .o%u!n will delete the $urrent $olumn.

.op5 4ntr5 Down .o%u!n

This $ommand is used to $o an entr from one $ell to all $ells )elow it in the$olumn. This is not often usefulB )ut it $an save a great deal of wor: when it is.

.op5

'o has five otions availa)le. It is ossi)le to $o the entire ta)leB the $urrentrowB or the $urrent $olumn to the $li)oard. It is ossi)le to $o from the datata)le or an of the solution ta)les. 0hatever is $oied $an then )e asted into this rogram or some other 0indows rogram. 7The $o tool in the tool)ar $oiesthe entire ta)le.8 If ou are at the solution stageB the $oing will )e for the ta)lethat is a$tive.

'o Se$ial will $o the entire ta)le )ut ena)le ou to limit the num)er ofde$imals that are $oied. Save as HT#2 will erform the HT#2 oeration thatwas reviousl des$ri)ed.



Paste

Paste is used to aste in the $urrent $ontents of the $li)oard. 0hen asting into POM-QM for Windows, the asting )egins at the $urrent $ursor osition. ThusB it

is ossi)le to $o a $olumn to a different $olumn )eginning in a different row.This $ould )e done to $reate a diagonal. It is not ossi)le to aste into a solutionta)leB althoughB as indi$ated reviouslB it is ossi)le to $o from a solution ta)le.

/NO4: Right$li$:ing on an ta)le will )ring u 'o otions and if the ta)le isthe data ta)le it will also )ring u the insert and delete otions.

?iew

?iew has several otions that ena)le ou to $ustomi?e the aearan$e of thes$reen.

The oo%ars menu $ontains two otions. The tool)ar $an )e $ustomi?ed 7as $anmost 0indows tool)ars8 or the tool)ar $an )e reset to its original loo:.

The Instruction )ar $an )e dislaed at its default lo$ation in the e=tra data anelor a)ove the dataB or )elow the dataB or as a floating windowB or not at all. TheStatus ar disla $an )e toggled on or off.

Fu%% Screen will turn all of the )ars 7tool)arB $ommand )arB instru$tionB andstatus )ar8 on or off.

Eoo! will generate a small form allowing ou to redu$e or in$rease the si?e of the

$olumns. It is easier to use the ?oom tool on the standard tool)ar.

'olors $an )e set to Monochro!e 7)la$: and white8 or from this state to theirOriina% .o%ors. This formerl was ver useful when overhead devi$es dislaedmu$h )etter in mono$hrome than in $olor. TodaB the ro@e$tors are so owerfulthat mono$hrome is generall not required.

Modu%e



6 drodown list with all of the modules in alha)eti$al order will aear. The#O*A2+ tool on the utilit tool)ar )elow the data area is a se$ond wa to get alist of modules. 6t the )ottom of the list are otions for indi$ating whether ouwant to disla onl the PO# modules 7as dislaed8B onl the 5# modules or allof the modules.



For!at

For!at has several otions for the disla of data and solution ta)lesB as $an )eseen in the following illustration. In additionB there are some additional formatotions availa)le in the format tool)ar.

.o%ors

The $olors for all of the dislas $an )e set. There are five ta)s as shown )elow.These otions will $reate ermanent $hanges whereas the foreground and )a$:ground tools on the format )ar will $hange onl the $urrent ta)le.,urthermoreB the $olor settings are for the entire ta)leB while the format tools ma )e used for either the entire ta)le or for sele$ted $olumns.

The first ta) is for setting the $olors in the data ta)leB and the se$ond ta) is forsetting the $olors in the solution ta)les. That isB it is ossi)le to have the dislas

of the data and the disla of the results aear differentlB whi$h $an )e helful.,or either the data or the resultsB ou ma set the )a$:ground and foreground$olors for rows to alternate ) using the odd and even otions. This ma:es readinglong ta)les easier.



In order to set the $olorsB first sele$t the ta)le roert that ou want to setB thensele$t foreground or )a$:ground if ali$a)leB then sele$t rows if ali$a)le andthen $li$: on the $olor. ,or e=amleB $li$: on od5, Foreround, Odd and then$li$: on the red $olor )o= and the foreground for ever other row will )e$ome red.

The $hanges here will )e maintained throughout until ou return to this s$reen andreset the $olors. If ou want to ma:e $hanges in onl one ta)le for one ro)lemBthen it ma )e easier to use the tool)ar otions for foreground and )a$:ground. 6lsoB the foreground and )a$:ground $olor sele$tion toolsB as well as the )oldBitali$B and underline toolsB ma )e used on individual $olumns if ou sele$t these$olumns )efore ressing on the tool.

The third ta) allows ou to $ustomi?e the $olors in the anels 7statusB instru$tion8.The fourth ta) $an )e used to set the gradient that aears on several of the s$reens7ro)lem $reationB emt data s$reen8B and the fifth ta) allows ou to reset the$olors to their original 7fa$tor8 settings.

Other For!at Options

The font teB stleB and si?e for all ta)les $an )e set. eros $an )e set to dislaas )lan:s rather than ?eros in the data ta)le. The grid line disla $an )e setto hori?ontalB verti$alB )othB or none. The ro)lem title that aears in the datata)leB and whi$h was $reated at the $reation s$reenB $an )e $hanged .



In order to give some idea of the e=tensive formatting $aa)ilities availa)leB wedisla a samle of an overl formatted s$reen.

In order to $reate this s$reen we used For!at, .o%ors and $hanged the )a$:ground and foreground $olors of odd rows to give us the alternating rows. 0ealso went to the anels ta) and $hanged the status anel )a$:ground to a dar:$olor and the foreground to a light $olor. 6fter this we sele$ted the $olumn named

M'iL and used the )a$:ground tools on the tool)ar to reset the $olors for this$olumn onlB then we sele$ted M9ru$eL and M2aurenL and used the 9old and Itali$tools on the tool)ar rese$tivelB andB finallB we sele$ted M9rianL and used theforeground tool.

Returning to the For!at menuB o)serve that the ta)le $an )e squee?ed ore=anded . That isB the $olumn widths $an )e de$reased or in$reased. +a$h ressof the tool $hanges the $olumn widths ) 1/ er$ent. This is ver useful if 7results8ta)les are wider than the s$reen. The tool)ar has the ?oom otion whi$h ma also )e used for resi?ing the $olumn widths.NO4C 6ll ta)les $an have their $olumn widths $hanged ) $li$:ing on the line

searating the $olumns and dragging the $olumn divider left or right *ou)le$li$:ing on this line will not automati$all ad@ust the $olumn width as it does in+=$el.

The inut $an )e $he$:ed or not. It is a good idea to alwas $he$: the inutB )utnot $he$:ing allows ou to ut entries into $ells that otherwise $ould not )e utthere.

1Nu!er of Deci!a%sB .o!!a and Fi9ed are used to format the dislaed or rintedoutut. The .o!!a otion will disla num)ers greater than --- with a $omma. TheNu!er of deci!a%s dro down )o= $ontrols the ma=imum num)er of de$imalsdislaed. If ou have it set to M// then .333 would )e dislaed as .33 while 1.! would )e dislaed as 1.!. If ou turn on the Fi9ed 7de$imal8 otion then all num)ers wouldhave & de$imals. Thus 1.! would )e dislaed as 1.!/ and line u with 1.33.



oo%s

1The software should find the 0indows $al$ulator if ou sele$t the .a%cu%atorotion. If notB a $al$ulator is availa)le for simle $al$ulationsB in$luding squareroot. (um)ers ma )e $oied from the $al$ulator and asted into an individual

$ell in the data ta)le. 6 Nor!a% Distriution .a%cu%ator is availa)le for erforming $al$ulations related to the normal distri)ution. This is arti$ularluseful for fore$asting and ro@e$t management. See the "e%p s$reen forinformation on how to use the (ormal *istri)ution 'al$ulatorB or use the hafa$e on the $al$ulator to get ste)ste instru$tions. 6n e=amle of the (ormal'al$ulator aears in 'hater " in the se$tion on ro@e$t management. The same$omutations $an )e done in the Statisti$s module )ut the $al$ulator is a little moreintuitive to use.

There is an area availa)le to 0nnotate ro)lems. If ou want to write a note toourself a)out the ro)lemB sele$t annotation. The note will )e saved with the fileif ou save the file. 6n e=amle of annotation aears in 'hater 1. In order toeliminate the annotation $omletelB the )o= must )e )lan: 7) deleting8 and thenthe file must )e resaved. 0hen ou rintB ou have an otion to rint the note ornot.

Window6 samle of the 0indow otions aears in the ne=t illustration. This menu otionis ena)led onl at the solution s$reen. (oti$e that in this e=amle there are si=different outut s$reens that $an )e viewed. The num)er of windows deends onthe se$ifi$ module and ro)lem.

9elow we are dislaing the s$reen after using the i%e otion from the Windowmenu when the s$reen resolution was set to 1&/ ) 1/&4. 0ith this resolution it



ma )e ver useful to tile in order to see all of the availa)le solution windows. Infa$tB using "e%p, 6ser Infor!ation ou $ould set all solution windows to oen ufor ever ro)lem. O)viouslB the value of this otion deends on our s$reenresolution.



"e%p

The "e%p otions are dislaed ne=t. The third otion is the toi$ and will give ades$rition of the moduleB the data required for inutB the outut resultsB and theotions availa)le in the module. It is worthwhile to loo: at this s$reen at least one

time in order to )e $ertain that there are no unsuse$ted differen$es )etween ourassumtions and the assumtions of the rogram. If there is anthing to )e warneda)out regarding the otionB it will aear on the hel s$reen as well as in 'hater "of this manual.

ip of the Da5

The ip of the Da5 will )e dislaed. ,rom thisB it will )e ossi)le to set the ti todisla all of the time or not to disla.

4-!ai% support

This will use our email to set u a message to )e sent to Prenti$e Hall. The firstste is to $li$: on the main )od of the message and then to aste 7'TR2 orSHI,TI(S8 the information that the rogram has $reated into the )od of themail.

Prora! 6pdate

This otion oints ou to www.renhall.$om;weiss. Adates are on the download age.

Manua%

The rogram $omes with this manual in )oth P*, form and as a 0ord do$ument.The P*, manual requires 6do)e a$ro)at reader whi$h is availa)le free throughhttC;;www.ado)e.$om;.

utoria%s



The rogram $omes with tutorials that are #a$romedia ,lash dislas 7e=e files8whi$h show ou how to erform $ertain oerations.

6ser Infor!ation

The user information form is shown )elow. The first ta) $an )e used to $hange thename of the $ourseB instru$tor or s$hool. The student name is set at the time ofinstallation of the software and $annot )e $hanged.



The se$ond ta) is used to set several of the otions that have )een dis$ussed to this oint.

The third ta) is used to set the te=t)oo:. There are some differen$es )etweendislasB models availa)le and $omutations for different te=t)oo:s.



0out POM-QM for Windows

The last hel otion is a standard 6)out disla. (oti$e the )uild num)er 79uild&/8 after the version num)er. If ou send email requesting helB lease )e sure toin$lude this )uild num)er. 6lsoB noti$e the 0e) site lo$ation www.renhall.$om;weiss. This site $ontains udates



'hater 4C Printing

.hapter 1

Printin

he Print Setup Screen

6fter sele$ting Print from the menu 7or tool)ar8B the Print Setu s$reen will )edislaed as shown in the figure )elow. There are several otions on this s$reen whi$hare divided over five ta)s. The first ta) is shown )elow.

9efore e=amining the ta)s lease noti$e that the )ottom of the form $ontains three frameswhi$hB if $li$:edB will $hange the format )etween )la$:;white and $olorB ortrait andlands$aeB and 6S'II and grid stle rinting.

PO#5# for 0indows



'hater 4C Printing

Infor!ation to Print

The otions in the frame deend on whether Print was sele$ted from the datas$reen or from the solution s$reen. ,rom the data s$reenB the onl otion that willaear is to rint the data. HoweverB from the solution s$reen there will )e one

otion for ea$h s$reen of solution values.

,or e=amleB in the linear rogramming e=amle a)ove there are si= differentoutut dislas as well as a grah availa)le and annotation sin$e this file had anote atta$hed. ou $an sele$t whi$h of these will )e rinted. In generalB the data is rinted when rinting the outut andB thereforeB it is seldom ne$essar to rint thedataB meaning that all rinting $an )e erformed after the ro)lem is solved.

a%es vs$ 4uations

,or the mathemati$al rogramming tes of modulesB there is an otion availa)lea)out the stle of rinting. The ro)lem $an )e rinted in regular ta)ular form orin equation form. 0e show an e=amle of ea$h ne=t.

au%ar For!

Results ----------

x y RHS Dual

Maximize 3 3

labor hours 3 4 <= 14 0.5

material (ou!"s# $ 4 <= 15 0.%5

4uation For!

Results ----------

&')M)*+, 3x 3y

labor hours, 3x 4y <= 14

material (ou!"s#, $x 4y <= 15

Printin 2raphs

If ou sele$t to rint the grahsB the software will allow ou to sele$t whi$h grahsshould )e rinted. ,or e=amleB Pro@e$t #anagement results in$lude three Gantt

$harts and a re$eden$e grah. ou $an sele$t whi$h grahs ou would li:e fromthe list that is resented as shown )elow.



'hater 4C Printing

If all ou want is one of the grahsB it is also ossi)le to do our grah rintingfrom the grah s$reen 7des$ri)ed in the ne=t $hater8 rather than from the results

s$reen. ,urthermoreB if ou want to $ontrol the si?e of the rinted grahB use theotions in the ne=t $hater.

Pae "eader Infor!ation

The ta) for the age header information is dislaed )elow. There are si= ie$es ofinformation that $an )e $hosen to aear on the header. The first three otions willaear on the first header lineB and the se$ond three will aear on the se$ondheader line. If the software is registered as )eing on a networ: or la) then thestudent name will in$lude the name of the la) followed ) the name of the student

that was entered when the rogram was started. To ma:e ermanent $hanges to the$ourse name or instru$tor nameB use "e%p,

6ser Infor!ation

Pae Aa5out

PO#5# for 0indows



'hater 4C Printing

The ta) for age laout is shown )elow.

Print as

There are two stles of rinting that ma )e used. The most $ommon and fastestwa to rint is as 6S'II 7lain te=t8. In additionB ou ma also rint it as a gridsimilar to the one that aears on the s$reen. ThusB ou ma format the grid andthen go to the rint otion and rint a highl formatted grid. The formatted gridsta:e longer to rint than the lain te=t rinting.

Paper Orientation

The aer $an )e rinted in regular fashion 7ortrait8 or it $an )e rinted sidewas7lands$ae8.

0nswers

6nswers $an )e )oldB itali$B $olorB or an $om)ination of the three. *o not loo: for$olor if ou do not have a $olor rinter. In fa$tB if ou set the rinting to use $oloron a )la$:;white rinterB the $olor answers generall aear lighter This is usuallnot the desired $hara$teristi$.

Spacin

The rinting ma )e singlesa$ed 7highl suggested8 or dou)lesa$ed.



'hater 4C Printing

Marins

The leftB rightB toB and )ottom margins $an )e set from ?ero to 1 in$h inin$rements of .1 in$hes. The margin is a)ove or )eond an natural margin that the rinter itself has. #argins of ?ero allow for the most rinting a$ross the age.

Ma9i!u! .o%u!n Widths

The ma=imum widths of the $olumns 7in $hara$ters8 $an )e set. The leftmost$olumn whi$h is usuall namesB $an )e set searatel from the other $olumns. Thisis useful if ou want to $omress ta)les.

Printer Options

The ta) for the rinter otions aears )elow.

Print to and If the Fi%e 0%read5 49ists

It is ossi)le to rint to the rinter or to rint to a file. If ou rint to a file ou will )e as:ed for a file name. 6n name $an )e given. ou also have the otion ofadding the rinting to a file that was alread there 7aending8 or erasing the file )efore rinting 7rela$e file8.

Print i!in

Printing $an o$$ur ea$h time that ou use Print or it $an wait until the endB wheneverthing will )e rinted at on$e. Printing ea$h time is generall refera)leB )utthere are some situations where ou want to wait until the end )e$ause this masave aer or minimi?e the num)er of tris to a networ: rinter.

PO#5# for 0indows



'hater 4C Printing

.hane Defau%t Printer

If ou have more than one rinterB ou ma $hange the rinter using this otion.This $hanges the 0indows default rinter and ma affe$t other rograms If ou rint as a gridB the rinter sele$ted is alwas the 0indows default rinter

regardless of what ou sele$t in this window.



PO#5# for 0indows

.hapter +

2raphs

Introduction

#an of the modules have the $aa)ilit to disla grahs or $hartsas one of the outut otions. Some of the modules have more than

one grah asso$iated with them. ,or e=amleB as shown in the figure )elowB four different ro@e$t management grahs are availa)le. Thegrah to )e dislaed is $hosen using the ta). There are severalotions that ou have when a grah aearsB and we e=lain thoseotions in this $hater.



'hater !C Grahs

0hen a grah is oenedB two things will o$$ur. The first is that the grahwill )e dislaed $overing the entire area )elow the e=tra dataF the se$ondis that some of the menu otions will $hange or e=e$ute differentl.

Fi%e Savin

The file save otion )oth under Fi%e on the main menu and on thetool)ar will save the 7a$tive8 grah rather than saving the file. The file mastill )e saved ) using Fi%e, Save as or ) going to a results window otherthan the grah window.

Print

Print now will rint the grah rather than resenting the general rintsetu s$reen. The rint grah otions are shown )elow. The grah $an )e

rinted in two si?esB and $an )e rinted as either ortrait 7.! ) 118 orlands$ae 711 ) .!8. Small grahs $an )e rinted at the to or )ottom ofthe age. ThusB there is slightl more $ustomi?ation of grah rintingavaila)le through this method than when rinting the grahs as art of theoututB as des$ri)ed in the revious $hater.

.o%ors Font

The foreground $olors and the )a$:ground $olors ma )e $hanged )using the tools on the format tool)ar. 'hanging the font name on the tool)arwill $hange the font for the headings and la)els in the grahs. 'li$:ing on )old on the tool)ar will $hange the font in the grahs to )old.



PO#5# for 0indows



'hater "C #odules

.hapter 3Modu%es

Overview

In this $haterB we detail ea$h of the &- modules. 0e e=lain the inut required for ea$hmoduleB the otions availa)le for modeling and solvingB and the different outut s$reensand reorts that $an )e seen and rinted. Re$all that the menu $an )e set to disla onlPO# modulesB onl 5# modules or all modules. ,or modules that are in )oth the PO#

and 5# menus we disla the POM-QM for Windows i$onB

in this manual. ,or all modules that are in the PO# onl menuB we disla the POM forWindows i$onB

while for all modules that are in the 5# onl menu we disla the QM for Windows i$onB

.

,or e=amleB in the first moduleB aggregate lanningB whi$h aears on the ne=t ageB wesee the PO# i$on sin$e aggregate lanning aears in the POM onl& men' )ut not in theQM onl& men'. ,inallB the e=amles used in this manual have )een installed in the+=amles folder in the PO#5# for 0indows folder 7Program ,ilesKPO#5#38.

0reate BProductionC P%annin

4!



PO#5# for 0indows

Produ$tion lanning is the means ) whi$h we reare our rodu$tion quantitiesfor the medium term 7generall one ear8. 6ggregate lanning refers to the fa$tthat the rodu$tion lanning is usuall $arried out a$ross rodu$t lines. 0e willuse the terms aggregate lanning and rodu$tion lanning inter$hangea)l. Themain lanning diffi$ult is that demands var from month to month. 0e want to

:ee rodu$tion as sta)le as ossi)le et maintain minimum inventor ande=erien$e a minimum of shortages. 0e must )alan$e the $osts of rodu$tionBovertimeB su)$ontra$tingB inventorB shortagesB and $hanges in rodu$tion levels.

In some $asesB aggregate lanning ro)lems might require the use of thetransortation or linear rogramming modules. The se$ond su)model in theaggregate lanning module $reates and solves a transortation model of aggregate lanning for $ases where all of the $osts are identi$al. The transortation model isalso availa)le as one of the methods for the first su)model.

he 0reate P%annin Mode%

Produ$tion lanning ro)lems are $hara$teri?ed ) a demand s$heduleB a set of$aa$itiesB various $osts and a method for handling shortages. 'onsider thefollowing e=amle.

49a!p%e / - S!ooth Production

'onsider a situation where demands in the ne=t four eriods are for 1&//B 1!//B1-//B and 14// units. 'urrent inventor is / units. Suose that regular time$aa$it is &/// units er month and that overtime and su)$ontra$ting are not a

$onsideration. The $osts are U for ea$h unit rodu$ed during regular timeB U3 forea$h unit held er eriodB U4 for ea$h eriod that we are short a unitB U! for ea$hunit ) whi$h we in$rease rodu$tion from the revious eriodB and U" for ea$hunit ) whi$h we redu$e rodu$tion from the revious eriod. The s$reen for thise=amle is dislaed )elow.

6)ove the dataB we have two $onsiderations shortage handling and the method touse for erforming the lanning.

Shortage handling. In rodu$tion lanning there are two models for handlingshortages. In one modelB shortages are )a$:ordered. That isB demands $ana$$umulate and )e met in later eriods. In another modelB the shortages )e$omelost sales. That isB if ou $annot satisf the demand in the eriod in whi$h it isrequestedB the demand disaears. This otion is a)ove the data ta)le.

4"



'hater "C #odules

Methods.

,ive methods are availa)leB whi$h we will demonstrate. Please note that smooth rodu$tion a$$ounts for two methods.

Smooth rodu$tion will have equal rodu$tion in ever eriod. This $an )eset a$$ording to the gross demand or the net demand 7gross demand minusinitial inventor8.

Produ$e to demand will $reate a rodu$tion s$hedule that is identi$al to thedemand s$hedule.

'onstant regular time rodu$tionB followed ) overtime and su)$ontra$tingif ne$essar. The lesser $ost method will )e sele$ted first.

6n rodu$tion s$heduleC in whi$h $ase the user must enter the amounts to )e rodu$ed in ea$h eriod.

The transortation model

Quantities

Demand . The demands are the driving for$e of aggregate lanning and these are to )e given in one $olumn.

a!acities - reg'lar time, oertime, and s'bcontracting. The rogram allows forthree tes of rodu$tion regular timeB overtimeB and su)$ontra$tingB and$aa$ities for these are to )e given in the ne=t three $olumns. If the methodsele$ted is the userdefined methodB then these are not viewed as $aa$ities )utrather as rodu$tion quantities. 0hen de$iding whether to use overtime orsu)$ontra$tingB the rogram will alwas first sele$t the one that is less e=ensive.

4%



PO#5# for 0indows

.osts

The $osts for the ro)lem are all la$ed in the far right $olumn of the data s$reen.

Prod'ction costs - reg'lar time, oertime, and s'bcontracting . These are the er

unit rodu$tion $osts deending on when and how the unit is made.

*nentor& (holding) cost. This is the amount $harged for holding one unit for one eriod. The total holding $ost is $harged against the ending inventor. 9e $arefulB )e$ause while most te=t)oo:s $harge against the ending inventorB some te=t)oo:s$harge against average inventor during the eriod.

Shortage cost . This is the amount $harged for ea$h unit that is short in a given eriod. 0hether it is assumed that the shortages are )a$:logged and satisfied assoon as sto$: )e$omes availa)le in a future month or are lost sales is indi$ated )the otion )o= a)ove the data ta)le. Shortage $osts are $harged against endofmonth levels.

ost to increase !rod'ction. This is the $ost due to having $hanges in the rodu$tion s$hedule. It is given on a erunit )asis. The $ost for in$reasing rodu$tion entails hiring $osts. It is $harged against the $hanges in the amount ofregular time rodu$tion )ut not $harged against an overtime or su)$ontra$ting rodu$tion volume $hanges. If the initial rodu$tion level is ?eroB then there will )e no $harge for in$reasing rodu$tion in the first eriod.

ost to decrease !rod'ction. This is similar to the $ost of in$reasing rodu$tion

and is also given on a erunit )asis. HoweverB this is the $ost for redu$ing rodu$tion. It is $harged onl against regular time rodu$tion volume $hanges.

Other .onsiderations

*nitial inentor&. Oftentimes we have a starting inventor from the end of the revious month. The starting inventor is la$ed in the far right $olumn towardsthe )ottom.

+nits last !eriod . Sin$e some of the $osts are for $hanges in rodu$tion quantitiesfrom eriod to eriodB it is ne$essar to in$lude the rodu$tion in the eriod rior

to the start of the ro)lem. These units aear in the far right $olumn at the )ottom.

he So%ution

4



'hater "C #odules

In the first e=amleB shown )elowB we have $hosen the smooth rodu$tion methodand )a$:orders. The demands are 1&//B 1!//B 1-//B and 14//B and the regulartime $aa$it of &/// e=$eeds this demand. There is no initial inventor. Thenum)ers reresent the rodu$tion amounts. The $osts $an )e seen toward the )ottom of the $olumns. The s$reen $ontains information on )oth a eriod)

eriod )asis and summar )asis. (oti$e the $olor $oding of the data 7)la$:8Bintermediate $omutations 7magenta8 and results 7)lue8.

eg'lar time !rod'ction. The amount to )e rodu$ed in regular time is listed inthe MRegular time rodu$tionL $olumn. This amount is determined ) the rogramfor all otions e=$et Aser *efined. In this e=amleB )e$ause the gross 7or net8demand is "///B there are 1!// units rodu$ed in regular time in ea$h of the 4 eriods. If the total demand is not an even multile of the num)er of eriodsB thene=tra units will )e rodu$ed in as man eriods as ne$essar in order to meet thedemand. ,or e=amleB had the total demand )een "//&B the rodu$tion s$hedulewould have )een 1!/1 in the first and se$ond eriods and 1!// in the other two eriods.

The ending inventor is reresented ) one of two $olumns either MInventorL orMShortage.L

*nentor& (holding). The a$$umulated inventor aears in this $olumn if it is

ositive. In the e=amleB there is a ositive inventor of 3// units in eriods 1 and&B no inventor 7a$tuall a shortage8 in eriod 3B and neither an inventor norshortage at the end of eriod 4.

Shortages. If there is a shortageB the amount of the shortage aears in this$olumn. In the e=amleB the 1// in the shortage $olumn for eriod 3 means that

4-



PO#5# for 0indows

1// units of demand have not )een met. 9e$ause we have $hosen the )a$:logotionB the demands are met as soon as ossi)leB whi$h is in the last eriod.

(o in$rease or de$rease from month to month o$$ursB so these $olumns do notaear in this disla.

Total . The total num)ers of units demandedB rodu$edB in inventorB shortB or inin$reased and de$reased rodu$tion are $omuted. In the e=amleB "/// units weredemanded and "/// units were rodu$ed and there was a total of "// unitmonthsof inventorB 1// unitmonths of shortageB and / in$reased or de$reased rodu$tion unitmonths.

osts. The totals of the $olumns are multilied ) the aroriate $ostsB ieldingthe total $ost for ea$h of the $ost $omonents. ,or e=amleB the "// units ininventor have )een multilied ) U3 er unitB ielding a total inventor $ost of

U1//B as dislaed.

Total cost . The overall total $ost is $omuted and dislaed. ,or this strategB thetotal $ost is U!/B&//.

2raph

Two grahs are availa)le in this module.

!/



'hater "C #odules

It is ossi)le to disla a )ar grah of rodu$tion in ea$h eriod 7not shown8 and itis also ossi)le to disla a grah of the $umulative rodu$tion versus the$umulative demand 7shown a)ove8.

49a!p%e *: Startin inventor5 and previous production

0e have made two modifi$ations to the revious e=amle. These modifi$ations$an )e seen in the following s$reen. In the MInitial InventorL lo$ationB we have la$ed 1//. In additionB we have $hanged the method to use the net demand.

+=amining the MRegular time rodu$tionL $olumn in the outut that followsindi$ates that the total rodu$tion is !-// rather than the "/// from the reviouse=amle due to the initial inventor. Thus we need onl rodu$e 14%! er month.

!1



49a!p%e (: 6sin overti!e and sucontractin

In the ne=t e=amle shown in the following s$reenB we ta:e our original e=amle7without starting inventor8 and redu$e the $aa$it to 1/// for regular time. 0ehave in$luded $aa$ities of 1// for overtime and -// for su)$ontra$tingB and wehave in$luded unit $osts for overtime and su)$ontra$ting of U- and U11Brese$tivel. This $an )e seen as follows.

9e$ause there is not enough regular time $aa$itB the rogram loo:s to overtimeand su)$ontra$ting. It first $hooses the one that is less e=ensive. ThereforeB in thise=amleB the rogram first ma:es 1/// units on regular timeB then 1// units onovertime 7U-;unit8B then 4// units 7of the -// availa)le8 on su)$ontra$tingU11;unit8.

49a!p%e 1: When sucontractin is %ess e9pensive than overti!e

In the following s$reenB we show a $ase where su)$ontra$ting is less e=ensivethan overtime. That isB the onl $hange we have made from the revious s$reen isto ma:e the overtime $ost U13 rather than U-. This timeB the rogram first $hoosessu)$ontra$ting andB sin$e there is suffi$ient $aa$itB overtime is not used at all.



49a!p%e +: Aost sa%es - case /

0e have ta:en the revious e=amle and we have $hanged from )a$:orders to lostsalesB as $an )e seen )elow.

The outut shows a shortage of 1// units at the end of eriod 3. In the ne=t eriodBwe rodu$e 1!// units even though we need onl 14// units. These e=tra 1//units are not used to satisf the eriod 3 shortageB sin$e these have )e$ome lostsales. The 1// units go into inventorB as $an )e seen from the inventor $olumnin eriod 4. It does not ma:e sense to use the smooth rodu$tion model and havelost sales. In the endB the total demand is not reall "///B sin$e 1// of the saleswere lost.



49a!p%e 3: he produce to de!and Bno inventor5C strate5

0e have ta:en our first e=amle and toggled the method to disla the following rodu$e to demand or $hase strateg.

(oti$e that the rogram has set the MRegular time rodu$tionL $olumn equal to thedemand $olumn. The inventor is not dislaed )e$ause it is alwas / under thisotion. 0ith rodu$tion equal to demand and no starting inventorB there will )eneither $hanges in inventor nor an shortages. The rodu$tion rates will in$reaseand;or de$rease. In this e=amleB rodu$tion in eriod 1 was 1&// and rodu$tionin eriod & was 1!//. ThereforeB the in$rease $olumn has a 3// in it for eriod &.The rogram will not list an in$rease in eriod 1 if no initial rodu$tion is given.The total in$reases have )een %//F de$reases !//.

*ncrease. The $hange in rodu$tion from the revious eriod to this eriod o$$ursin this $olumn if the $hange reresents an in$rease. (oti$e that the rogramassumes that no $hange ta:es la$e in the first eriod in this e=amle. In thise=amleB there is no $hange in other eriods )e$ause rodu$tion is $onstant underthe smooth rodu$tion otion.

Decrease. If rodu$tion de$reasesB this de$rease aears in this $olumn.

49a!p%e G: Increase and decrease charin

The revious e=amle had in$reases and de$reases in rodu$tion. These in$reasesand de$reases are a$$ounted for ) regular time rodu$tion. In the followings$reenB we have redu$ed the regular time $aa$it in order to for$e rodu$tionthrough regular time and overtime.



(oti$e that the in$rease $olumn onl has a value in it in the se$ond eriod whenregular time rodu$tion went from 1&// to 1!// units. The regular time rodu$tion remains at 1!//F even though overtime in$reasesB this does not showu in the in$rease $olumns. 0e do not $harge against su$h in$reases.

he ransportation Mode% of 0reate P%annin

The transortation model of aggregate lanning $ontains data whi$h is nearlidenti$al to the models @ust e=amined. The onl differen$e is that thetransortation model does not $onsider $hanges in rodu$tion levels so there is nodata entr allowed for in$rease and de$rease $osts or for units last eriod. The$reation s$reen will as: for the num)er of eriods and whether shortages areallowed or not. The similarit to the revious s$reens $an )e seen )elow. (oti$ethat there is onl one entr for ea$h of the $osts. ThusB this model $an not )e used

for situations where the $osts $hange from eriod to eriod. ou must formulatethese ro)lems ourself using the transortation model from the #odule menurather than this transortation su)model of aggregate lanning.

1NO4: The transortation model that is the se$ond su)model in the New menu$an also )e a$$essed as the last method in the first su)modelB



The solution s$reen is dislaed )elow. The window on the left $ontains the rodu$tion quantities as e=ressed in transortation form. The window on the rightsummari?es the rodu$tion quantitiesB unitmonths of holding 7and shortage ifali$a)le8 and the $osts.

It is even more o)vious that this is a transortation ro)lem if we e=amine these$ond window of outut whi$h is the transortation model itself.

The large num)ers 7----8 have )een entered in order to re$lude the rogram from

)a$:ordering. If ou li:eB this ta)le $ould )e $oiedF ou $ould then oen theTransortation modelB $reate a newB emt ta)le that is 13 ) 4 and aste this datain to that ta)le.



0sse!%5 Aine a%ancin

This model is used to )alan$e wor:loads on an assem)l line. ,ive heuristi$ rules$an )e used for erforming the )alan$e. The $$le time $an )e given e=li$itl or

the rodu$tion rate $an )e given and the rogram will $omute the $$le time.This model will not slit tas:s. Tas: slitting is dis$ussed in more detail in a laterse$tion.

he Mode%

The general framewor: for assem)l line )alan$ing is di$tated ) the num)er oftas:s that are to )e )alan$ed. These tas:s are artiall orderedB as shownB fore=amle in the re$eden$e diagram that follows.

Method. The five heuristi$ rules that $an )e $hosen areC

1. 2ongest oeration time&. #ost following tas:s

3. Ran:ed ositional weight4. Shortest oeration time!. 2east num)er of following tas:s



NO4: Ties are )ro:en in an ar)itrar fashion if two tas:s have the same riorit )ased on the rule given. (ote that tie)rea:ing $an affe$t the final results.

The remaining arameters areC

&cle time. The $$le time $an )e given in one of two was. One wa is when the$$le time is given dire$tl as shown a)ove. 0hile this is the easiest methodB it ismore $ommon to determine the $$le time from the demand rate. The $$le time is$onverted into the same units as the times for the tas:s. 7See +=amle &8.

Tas time 'nit . The time unit for the tas:s is given ) this drodown )o=. oumust $hoose se$ondsB hoursB or minutes. (oti$e that the $olumn heading for thetas: times will $hange as ou sele$t different time units.

Tas names. The tas: names are essential for assem)l line )alan$ing sin$e thedetermine the re$eden$es. 'ase does not matter.

Tas times. The tas: times are given.

Precedences. +nter the re$eden$esB one er $ell. If there are two re$eden$esthe must )e entered in two $ells. *o not enter Ma, b. In fa$tB a $omma will not )ea$$eted. (oti$e that in the re$eden$e list in the s$reen a)ove we have ted )otha and /. 6s mentioned reviouslB the $ase of the letters is irrelevant.

49a!p%e /

In this e=amle we have si= tas:s named a through f . The re$eden$e diagram forthis ro)lem aears a)ove. The time to erform ea$h tas: is a)ove the tas:. 6lsoBnote that the tas:s that are read at the )eginning of the )alan$e are tas:s a and b.,inallB in this first e=amleB we use a $$le time of 1/.

So%ution

1The following s$reen $ontains the solution to our first e=amle. The solutions$reen $onsists of two windows as shown )elow. The window on the left gives the$omlete results for the method $hosen while the window on the right gives thenum)er of stations required 7not the theoreti$al num)er8 when using ea$h

)alan$ing rule. The solution s$reen will alwas have the same aearan$e and$ontain the same information regardless of the rule that is $hosen for the )alan$e.6lsoB as shown in the summar window on the rightB in this $ase ea$h rule leads to3 stations. This is not alwas the $ase as will )e demonstrated later in this se$tion.Station n'mbers. The station num)ers aear in the far left $olumn. The aredislaed onl for the first tas: that is loaded into ea$h station. In this e=amleBthree stations are required.



ran:ed ositional weight than c. The results disla indi$ates the ran:ed ositionalweight in the $olumn named Mread tas:s.L

Shortest Operation i!e

6nother rule that is used sometimes is to give riorit to the tas: that ta:es theleast amount of time.

Aeast Nu!er of Fo%%owers

The last rule that is availa)le is the least num)er of followers.

49a!p%e (: What to do if %onest operation ti!e wi%% not fit

Some )oo:s and some software do not al the longest oeration time rule roerl. If the tas: with the longest time will not fit into the stationB the tas: with

the se$ond longest time should )e la$ed in the station if it will fit.

In the following s$reen we resent data for eight tas:s. (oti$e that tas:s b, c, e,and f immediatel follow tas: a.

The )alan$e aears )elow for a $$le time of ! se$onds. 6fter tas: a is$omletedB tas:s b, c, d, and e are read. Tas: b is longest )ut will not fit in the 4se$onds that remain at station 1. ThereforeB tas: c is inserted into the )alan$e. 0e$aution ou that if the answer in our )oo: differs from the rogramB ou should$he$: if the )oo: has negle$ted to ut in the tas: with the longest oeration timethat will fit.



49a!p%e 1: Sp%ittin tas7s

If the $$le time is less than the amount of time to erform a se$ifi$ tas:B there isa ro)lem. 0e erform what is termed tas: slitting )ut whi$h in realit isa$tuall duli$ation. ,or e=amleB suose that the $$le time is & minutes and atas: ta:es ! minutes. Then we have the tas: erformed 3 times 7) three eole atthree ma$hines indeendent of one another8. The effe$t is that 3 units will )e done

ever ! minutesB whi$h is equivalent to one unit ever 1.33 minutesB whi$h fitsinto the & minute $$le.

(owB the a$tual wa that the three eole wor: ma var. 0hile other rogramswill slit tas:sB the assumtions var from rogram to rogram. Rather thanma:ing assumtionsB we leave it to ou to slit the tas:s ) dividing the tas: timearoriatel.

Suose that in +=amle 1 we wanted to use a $$le time of ! se$onds. Then it isne$essar to reli$ate )oth tas:s d and f sin$e the will not fit in the $$le time.

The aroa$h to use is to solve the ro)lem ) dividing the tas: times ) &B sin$ethis reli$ation is needed. 0e resent the results in the following s$reen. (oti$ethat different rules lead to different minimum num)er of stations



2raphs

Two different grahs are availa)le. The first is a re$eden$e grahB as shown inthe figure )elow. Please note that there ma )e several different was to draw a re$eden$e grah.

The se$ond grah 7not dislaed here8 is of time used at ea$h station. In a erfe$t

world these would all )e the same 7a erfe$t )alan$e8.



he 0ssin!ent Mode%

The assignment model is used to solve the traditional onetoone assignment ro)lem of assigning emloees to @o)sB emloees to ma$hinesB ma$hines to @o)sB

et$. The model is a se$ial $ase of the transortation method. In order to generatean assignment ro)lemB it is ne$essar to rovide the num)er of @o)s and ma$hinesand to indi$ate whether the ro)lem is a minimi?ation or ma=imi?ation ro)lem.The num)er of @o)s and num)er of ma$hines do not have to )e equal )ut usuallthe are.

Ob2ectie f'nction. The o)@e$tive $an )e to minimi?e or to ma=imi?e. This is set atthe $reation s$reen )ut $an )e $hanged in the data s$reen.

49a!p%e

The ta)le )elow shows data for a %)% assignment ro)lem. Our goal is to assignea$h saleserson to a territor at minimum total $ost. There must )e e=a$tl onesaleserson er territor and e=a$tl one territor er saleserson.

#ort 'i 9ru$e 9eth 2auren +ddie 9rian

Pennslvania (ew >erse (ew or: ,lorida'anada

#e=i$o+uroe

1&331&1!4&

4/1&

!44!!43%3&

%134

Q%%"3%1

%"!

%&%&3"!%%

%"&3

!434%&"&3

&44

-%"44-"!!

-/&3

-"!"&&3%

441&

Q 9ru$e is not allowed to wor: in the state of Pennslvania.

The data stru$ture is nearl identi$al to the stru$ture for the transortation model.The )asi$ differen$e is that the assignment model does not disla sulies anddemands sin$e the are all equal to one.

NO4: To tr to re$lude an assignment from )eing madeB su$h as 9ru$e toPennslvania in this e=amleB enter a ver large $ost. If ou te H9 , the rogramwill la$e a $ost of -B--- for minimi?ation ro)lems or a rofit of -B--- for

ma=imi?ation ro)lems in that $ell.



he So%ution

/ssignments. The M6ssignsL in the main )od of the ta)le are the assignments thatare to )e made. In the e=amleB #ort is to )e assigned to PennslvaniaB 'i to,loridaB 9ru$e to 'anadaB 9eth to (ew or:B 2auren to (ew >erseB +ddie to

+uroeB and 9rian to #e=i$o.

Total cost . The total $ost aears in the uer left $ellB U1-1 in this e=amle.

The assignments $an also )e given in list formB as shown in the s$reen )elow.

The marginal $osts $an )e dislaed also. ,or e=amleB if we want to assign #ortto (ew >erseB the total will in$rease ) U3/ to U&&1.



rea7even&.ost-?o%u!e 0na%5sis

'ostvolume analsis is used in several different areas of PO# and 5#Bese$iall $aa$it lanning and lo$ation analsis. 'ostvolume analsis is used to

find the oint of indifferen$e )etween two otions )ased on fi=ed and varia)le$osts. 6 )rea:even oint is $omuted in terms of units or dollars. 9rea:even issiml a se$ial $ase of $ostvolume analsis where there is one fi=ed $ostB onevaria)le $ostB and revenue erunit.

.ost-?o%u!e 0na%5sis

In $ostvolume analsis we $omare two or more otions to determine what otionis least $ostl at an volume. The $osts $onsist of two tes fi=ed $osts andvaria)le $ostsB )ut there ma )e several individual $osts that $omrise the fi=ed

$osts or the varia)le $osts. In the e=amle that followsB we are indi$ating that thereare five different individual $osts and two otions.

Data

ost t&!e. +a$h te of $ost must )e identified as either a fi=ed $ost or a varia)le$ost. The default is that the first $ost in the list is fi=ed and that all other $osts arevaria)le. These values $an )e $hanged ) using the drodown )o= in that $ell.

osts. The se$ifi$ $ost for ea$h otion gets listed in the two $olumns in the ta)le.

3ol'me. If a volume analsis is desiredB enter the volume at whi$h this analsis

should )e erformed. The volume analsis will $omute the total $ost 7revenue8 atthe $hosen volume. If the volume is /B no volume analsis will )e erformed otherthan for the )rea:even oint. 0e have as:ed for a volume analsis at &!/ units.



So%ution

The solution s$reen is ver straightforward. In the re$eding s$reen there are five$osts with some fi=ed and some varia)le. The rogram dislas the followingresultsC

Total fi4ed costs. ,or ea$h of the two otionsB the rogram ta:es the fi=ed $ostsBsums themB and lists them )elow the ta)le. In this e=amleB the total fi=ed $osts forOtion 1 are U13// 7//V!//8B while the total fi=ed $osts for Otion & are U-//7%//V&//8.

Total ariable costs. The rogram identifies the varia)le $ostsB sums them uB andlists them. In this e=amleB the total varia)le $osts for Otion 1 are U1/ er unitBwhile for Otion & the are U1& er unit.

1reaeen !oint in 'nits. The )rea:even oint is the differen$e )etween the fi=ed$osts divided ) the differen$e )etween the varia)le $ostsB and this is dislaed inunits. In the e=amleB it is &// units.

1reaeen !oint in dollars. The )rea:even oint $an also )e e=ressed in dollars.

6 volume analsis has )een erformed for a volume of &!/ units. The total fi=ed$osts and total varia)le $osts have )een $omuted for ea$h otion and these have )een summed to ield the total $ost for ea$h otion.

6 grah is availa)leB as follows.



49a!p%e *: rea7even ana%5sis

One standard te of )rea:even analsis has revenue versus $osts.

*ata entr for this otion is slightl different in that the rogram $reates a $olumnfor $osts and a $olumn for revenues. The fi=ed and varia)le $osts get entered in the$ost $olumn and the revenueerunit is la$ed in the revenue $olumn.

This model requires e=a$tl 3 inuts. The first is for the fi=ed $ost of U1/B///B these$ond for the varia)le $ost of U&/ er unitB and the third for the 7varia)le8 revenueof U&! er unit. The rogram will $omute a )rea:even volume of &/// units orU!/B/// 7not shown8.

This e=amle $ould also have )een solved ) using the $ostvolume su)model.Sele$t two otions and let one )e the $osts and one )e the revenues. Pla$e thefi=ed $osts and varia)le $osts in their o)vious $ellsB use no fi=ed $ost for therevenue and use the revenue er unit as a varia)le $ost. 0e disla this )elow.

49a!p%e (: rea7even point with !ore than two options



The )rea:even module $an erform a )rea:even analsis for u to five otions. Inthe following we demonstrate the outut for a threeotion )rea:even. Thesolution s$reen )elow is indi$ating three )rea:even oints as it ma:es $omarisonsfor 'omuter 1 vs. 'omuter &B 'omuter 1 vs. 'omuter 3B and 'omuter & vs.'omuter 3. Of $ourseB while there are three )rea:even ointsB onl two of them

are relevant.

This is seen a little more easil ) loo:ing at the )rea:even grah given )elow.The )rea:even oint at 4/B/// units does not matter sin$e at 4/B/// units the two$omuters that )rea:even have higher $osts than the 'omuter & otion.



.apita% Invest!ent

This module $an )e used for finding the net resent value of a $ash flow or forfinding the internal rate of return of a $ash flow. The data for this e=amle $onsist

of a stream of inflows and a stream of outflows. In additionB for finding the net resent value an interest rate must )e given.

Net Present ?a%ue

'onsider the following e=amle. 6 $oman is going to ur$hase new equimentthat $osts U1//B///. 9e$ause of the use of the new equiment the $oman wille=erien$e savings over the ne=t " ears of U&&B///F U&!B///F U&&B///F U&1B///FU1-B///F and U1B///. 6t the end of si= ears the $oman anti$iates )eing a)leto salvage the ma$hine for U&!B///. The $oman would li:e to :now the net

resent value using an interest rate of 1/ er$ent. The data s$reen aears )elowC

The s$reen has two $olumns for data. One $olumn is la)eled inflow and the other$olumn is la)eled outflow. 0e had indi$ated at the time of ro)lem $reation thatthis was a si=eriod ro)lem and the data ta)le in$ludes the si= eriods lus the$urrent eriod 7/8. The ur$hase $ost of U1//B/// is an outflow that o$$urs at the )eginning of the ro)lemB so this is la$ed in the outflow for eriod /. The si=savings in the list a)ove are inflowsB and the are la$ed in the inflow $olumn forPeriods 1 through ". The salvage value $ould )e handled two wasB and we have

$hosen the wa that we thin: gives a )etter disla. 0e $ould have added thesalvage value of U&!B/// to the inflow in Period ". InsteadB we $hose to reresentit as a negative outflow. This :ees the meaning of the num)ers $learer. The lastitem to )e entered is the interest rate in the te=t )o= a)ove the data. The resultsaear )elowC



6 $olumn has )een $reated that gives the resent value fa$tors for single aments. To the right of thisB the inflows and outflows are multilied ) these resent value fa$torsB and the far right $olumn $ontains the resent values for thenet inflow 7inflowoutflow8 on a eriod)eriod )asis. The )ottom row gives thetotals for ea$h $olumn and the solution to our ro)lem is a net resent value of

U%"/3.&!.

Interna% Rate of Return

The $omutation of the internal rate of return is ver simle. The data is set u thesame wa )ut the method )o= is $hanged from net resent value to internal rate ofreturn. The results aear )elow where ou $an see that the internal rate of returnfor the same data is 1&.3 er$ent andB of $ourseB the net resent value 7)ottomright8 when using this rate is U/.



49a!p%e /: 0 Decision a%e

The following e=amle resents three de$ision otions whi$h are to su)$ontra$tB touse overtimeB or to use arttime hel. The ossi)le s$enarios 7states of nature8 arethat demand will )e lowB normalB or highF or that there will )e a stri:e or a wor:

slowdown. The ta)le $ontains rofits as indi$ated. The first row in the ta)lereresents the ro)a)ilit that ea$h of these states will o$$ur. The remaining threerows reresent the rofit that we a$$rue if we ma:e that de$ision and the state ofnature o$$urs. ,or e=amleB if we sele$t to use overtime and there is high demandBour rofit will )e 1/.

So%ution

The results s$reen that follows $ontains )oth the data and the results for thise=amle.

04!ected al'es. The e=e$ted values for the otions have )een $omuted andaear in a $olumn la)eled M+#L 7e=e$ted monetar value8B whi$h has )een

aended to the righthand side of the data ta)le.

ow minim'm. ,or ea$h rowB the minimum element has )een found and listed.This element is used to find the ma=imin or minimin.

ow ma4im'm. ,or ea$h rowB the ma=imum element in the row has )een foundand listed. This num)er is used for determining the ma=ima= or minima=.



$'rwic5. These reresent 4/ er$ent multilied ) the )est out$ome lus "/ er$ent multilied ) the worst out$ome for ea$h row. ,or e=amleB forsu)$ontra$ting the Hurwi$? isC

.4 Q 14/ V ." Q 1// W 11".

Ma4im'm e4!ected al'e. 9e$ause this is a rofit ro)lem we are interested infinding the ma=imum values. The ma=imum e=e$ted value is the largest num)erin the e=e$ted value $olumnB whi$h in this e=amle is 1&4.!.

Ma4imin. The ma=imin is the largest 7#6Imum8 num)er in the #I(imum$olumn. In this e=amleB the ma=imin is 1//.

Ma4ima4. The ma=ima= is the largest value in the ta)le or the largest value in thema=imum $olumn. In this e=amleB it is 1-/.

Perfect Infor!ation

6 se$ond s$reen of results resents the $omutations for the e=e$ted value of erfe$t information as shown )elow.

Perfect information. 6n e=tra row la)eled MPerfe$t InformationL has )een added )elow the original data. In this rowB we have listed the )est out$ome for ea$h$olumn. ,or e=amleB for the low demand s$enario the )est out$ome is the 1&/given ) using overtime.

Perfect6!robabilit& (04!ected al'e 'nder certaint&). The e=e$ted value under

$ertaint is $omuted as the sum of the rodu$ts of the ro)a)ilities multilied )the )est out$omes. In the e=amleB this isC

+7'ertaint8W .&Q1&/ V .3Q1!/ V .&!Q1-/ V .1!Q1&/ V .1Q13/ W 14%.!/The row dislas the individual multili$ations in the equation a)ove 7&4B 4!B4%.!B 1B and 138 and the sum 714%.!8 dislaed on the right hand side of )oth theequation and the row.



04!ected al'e of !erfect information. The e=e$ted value of erfe$t information7+PI8 is the differen$e )etween the )est e=e$ted value 71&4.!8 and the e=e$tedvalue under $ertaint 714%.!8B whi$h in this e=amle is &3.

Reret&Opportunit5 Aoss

6 third availa)le outut disla is that of regret or oortunit loss as dislaed )elow.

Table al'es. The values in the ta)le are for ea$h $olumn $omuted as the $ellvalue su)tra$ted from the )est value in the $olumn in the data. ,or e=amleB underlow demand the )est out$ome is 1&/. If we su)$ontra$t and get 1// then our regretis &/ while if we use art time hel our regret is 1&/1/! W1!. The two $olumnson the right ield two sets of results. In the $olumn la)eled ma=imum regretB wedetermine the worst 7highest8 regret for ea$h de$ision and then find the minima=regret 7!/8 ) loo:ing at the )est 7lowest8 of these regrets. In the $olumn la)eled

+=e$ted RegretB we siml multil the regrets in ea$h row ) the ro)a)ilities.

There also is a window 7not dislaed in this manual8 that ields Hurwi$? valuesfor alha ranging from / to 1 ) ./1 for ea$h de$ision otion.

Decision rees

*e$ision trees are used when sequen$es of de$isions are to )e made. The trees$onsist of )ran$hes that $onne$t either de$ision ointsB oints reresenting $han$eBor final out$omes. The ro)a)ilities and rofits or $osts are entered and thede$isions that should )e made and the values of ea$h node are $omuted. 6llde$ision ta)les $an )e ut in the form of a de$ision tree. The $onverse is not true.

Note: ersion 3 of the software in$ludes two different inut stles for de$isiontrees$ The first model has ta)ular data entr while the se$ond model is easier touse )e$ause it has grahi$al data entr. The first model has )een maintained in thesoftware for $onsisten$ with revious versions.

49a!p%e * - 0 Decision ree - non raphica%



The general framewor: for de$ision trees is given ) the num)er of )ran$hes orthe num)er of nodes in the tree. The num)er of )ran$hes is alwas one less thanthe num)er of nodes. +a$h node alwas has e=a$tl one )ran$h going into it. Thenum)er of )ran$hes going out of an node $an )e /B 1 or &F the nodes are of threetes. There are de$ision nodesB $han$e nodesB and final nodes. Ti$allB the

de$ision nodes are reresented ) re$tanglesB and the $han$e nodes are reresented ) $ir$les. Our e=amle is given ) a ti$al de$ision tree diagram. The figure has1& )ran$hes. Profits are to the right of the terminal nodes. (oti$e that there is aU1// $ost in the middle for sele$ting a $ertain 7mar:et resear$h8 )ran$h.

In order to use the de$ision tree moduleB two things must o$$ur. ,irstB nodes must )e added to the right of the ending )ran$hes. 7Te$hni$allB it is illegal to draw atree that ends with )ran$hes rather than nodes.8 Se$ondB the nodes must )enum)ered. The figure that follows shows the added nodes and the fa$t that allnodes have )een given num)ers. The most $onvenient wa to num)er the nodes isfrom left to right and to to )ottom.

The initial data s$reen is generated ) answering that there are 1& )ran$hes and

those we wish to ma=imi?e rofits. The following s$reen $ontains )oth the dataand the solution.



Start and end node. 9ran$hes are $hara$teri?ed ) their start and end nodes. 6nadded )ran$h named MStartL aears in order to reresent the final out$ome. Thenode values are shown in the far right $olumn. In this e=amleB the value of thede$ision tree is U4"!.

1ranching !robabilities. These o$$ur in $olumn 4 and are the ro)a)ilit of goingfrom the start node on the )ran$h to the end node. The ro)a)ilities out of anindividual $han$e )ran$h should sum to 1.

Profits or costs. The rofit 7$ost8 for ea$h ending node that is terminal is to )eentered. In additionB it is ossi)le to enter a rofit or $ost for an )ran$h. ,ore=amleB noti$e that in )ran$h 1/ 7node " to 118 we have entered a $ost of U1// ) la$ing 1// in that $ell.

he Decision ree So%ution

The solution data areC

1ranch 'se. ,or those )ran$hes that are de$ision )ran$hes and should alwas )e$hosenB an M6lwasL is dislaed. In our e=amleB we should $hoose 7138 ratherthan 71&8. ,or those )ran$hes that we should $hoose if we get thereB we dislaMPossi)l.L ,or e=amleB if we get to node " we should sele$t 7"-8 rather than 7"8. HoweverB there is no guarantee that we will get to node " due to the ro)a)ilisti$ nature of the de$ision tree. The last te of )ran$h is one that weshould sele$t if we get thereB )ut we should not get there. These are mar:ed asM9a$:wards.L 2oo: at )ran$h % 7node 4 to node 8. If we get to node B we should

use this )ran$h. HoweverB sin$e we will sele$t 1 to 3 at the )eginningB we shouldnot end u at node 4.



0nding node. The ending node is reeated to ma:e the outut easier to read.

0nding node t&!e. ,or ea$h ending nodeB the rogram identifies it as either a finalnodeB a de$ision nodeB or a $han$e node.

04!ected al'e. The e=e$ted value for ea$h node is listed. ,or final nodesB thee=e$ted value is identi$al to the inut. ,or $han$e nodesB the e=e$ted value isthe weighted $om)ination of the values of the nodes that follow. ,or de$isionnodesB the e=e$ted value is the )est value availa)le from that )ran$h. 9oth $han$enodes and de$ision nodes will have an $osts su)tra$ted from the node values. ,ore=amleB the value of node 11 is U!!/. HoweverB the value of node " is U4!/ dueto the U1// $ost of going from node " to node -.

6 grah of the tree stru$ture $an )e dislaed ) the rogram.

49a!p%e ( - 0 Decision ree - 2raphica% 6ser Interface

One of the models allows for de$ision trees to )e entered grahi$all rather than inthe ta)le as given a)ove. 0e will use this model to e=amine the same e=amle @ust$omleted.

6fter sele$ting the modelB the interfa$e will aear as dislaed )elow. This is theonl model in the software that has an inut interfa$e that is not the usual datata)le interfa$e.



The grah is dislaed in the large area on the left and $reated using the tools onthe right. In the )eginning there is onl one node. The ne=t ste is to add two eventnodes to node 1. The tool on the right is set to node 1. The default for node 1 isthat it is a de$ision node as we need in this $ase. 6 )utton is availa)le to $hangethe node if this )e$omes ne$essar. Sin$e the default num)er of )ran$hes to add is&B the first ste is to $li$: on the M6dd n )ran$hesL )utton. Our new tree aears asfollows.

(oti$e that two )ran$hes have )een added. The $urrent node is node & whi$h isindi$ated ) )oth the fa$t that the node num)er in the uer right is node & )utalso ) the fa$t that the )ran$h to node & is highlighted in a different $olor. (oti$ethat the )ran$hes have )een given default names of M*e$ision 1 and M*e$ision &.These $an )e $hanged ) using the )ran$h information area at the )ottom of theinut tool area.



6t this oint we need to add two )ran$hes to node &. The default is to add de$ision )ran$hes to events and vi$eversa. The te of node $an alwas )e $hanged later.'li$: on M6dd n )ran$hesLB then enter the ro)a)ilities for )oth new nodes andenter the rofit of U4// for the se$ond node. Then add two )ran$hes to node 3 andfill in the ro)a)ilities and the U!// rofit. This ields the following.

NO4: (odes ma )e sele$ted ) either $li$:ing on them or using thes$roll)ar;te=t)o= $om)ination at the to of the tools se$tion on the right.

'omlete the data inut ) adding de$ision )ran$hes and data at nodes 4 and "and an event at node 11. 6lsoB in$lude the U1// $ost at de$ision " 7nodes "118.6fter all data has )een enteredB $li$: on the Solve )utton on the tool)ar. The data

is in )la$: and the solution is in )lue as usual. (oti$e that )ran$hes that should )eused are indi$ated in )lue.

49a!p%e 1 - Sin%e period inventor5



This model $an )e used to $reate a de$ision ta)le for single eriod inventor7sul;demand8 situations. 'onsider the following e=amle. In the astB an airlinehas o)served a demand for meals that are sold on the lane as given in the ta)le )elow. +a$h meal $osts the airline U4 and sells for U1/. If the airline is short amealB the give the assengers a vou$her worth U! for food at the airort of arrival.

How man meals should the airlane sto$: er flightX

#eals Pro)a)ilit

1/1!&/&!3/

.1

.&

.!

.1!

./!

9eginB ) $reating a ta)le with ! demands.

The rogram is requesting three rofits as well as the o)vious demands and ro)a)ilities.

Profit !er 'nit . This is the normal rofit for units )ought and sold. In this $ase the rofit isB U1/U4WU".

Profit !er 'nit e4cess is the rofit for units that are overordered. In some $asesBwhere there is a salvage value that e=$eeds the $ost of the unit this will )e a rofitwhile in other $ases this will )e a loss. In this $ase there is a loss whi$h is equal to

the $ost of an unsold meal or U4.

Profit !er 'nit short. This is the rofit for units when we donEt order enough. Itwill )e a rofit if we $an ur$hase units to sell after the fa$t at a $ost less than theselling ri$e. Otherwise it will )e a / or ossi)l a loss. In this $aseB )e$ause wegive a vou$her we have a loss equal to the $ost of the vou$herB U!. If we did notgive the vou$her there would )e no rofit or loss for units for whi$h we $ould notsatisf the demand.



e=onential smoothing with trendB there is a slightl different outut disla. ,orthe regression modelsB there is another set of outut. The first availa)le method isthe (aive method whi$h siml uses the data for the most re$ent eriod as thefore$ast for the ne=t eriod. This is a se$ial $ase of a moving average with nW1 ore=onential smoothing with W 1 so we do not disla the naive method here.

0e )egin with the moving averages.

49a!p%e /: Movin averaes

0e are using a twowee: 7n W &8 moving average. The main outut is a summarta)le of results.

The $omutations for all of these results $an )e seen on the details windowdislaed )elow.

7orecasts. The first $olumn of outut data is the set of fore$asts that would )emade when using the te$hnique. (oti$e that sin$e this is a twowee: movingaverageB the first fore$ast $annot )e made until the third wee:. This value is the11/B whi$h aears as the first entr in the M,ore$astL $olumn. The 11/ is$omuted as 71//V1&/8;&. The following three num)ers 11!B 1/%.!B and 1/%.!



Standard error. One more error measure is imortant. This is the standard error.*ifferent )oo:s have different formulas for the standard error. That isB some use n Y1 in the denominatorB and some use n Y&. This rogram uses n Y&. Thedenominator is dislaed in the summar outut as shown a)ove. In this e=amleBthe standard error is 11.4!"4.

NO4: The (ormal distri)ution $al$ulator $an )e used to find $onfiden$eintervals and address other ro)a)ilisti$ questions related to fore$asting.

One more s$reen is availa)le for all of these methods. It is a s$reen that gives thefore$ast $ontrol 7tra$:ing signals8 results.

,or moving averages there is a summar s$reen of error measures versus the JnE inthe moving average.

1One of the outut dislas 7not shown in this manual8 resents error measures asa fun$tion of n. 6lsoB the moving average grah has a s$roll )ar whi$h ena)les ou

to easil see how the fore$asts $hange as ν varies.



49a!p%e *: Weihted !ovin averaes

If the weighted moving average method is $hosenB two new $olumns will aearon the data ta)le as shown in the following s$reen. The far right $olumn is wherethe weights are to )e la$ed. The weights ma )e fra$tions that sum to one as in

this e=amle 7." and .48B )ut the do not have to sum to 1. If the do notB the will )e res$aled. ,or e=amleB weights of & and 1 will )e $onverted to &;3 and 1;3. Inthis e=amleB weights of ." and .4 have )een used to erform the fore$asting. ,ore=amleB the fore$ast for wee: % is ."Q1&/ V .4Q11/ W 11".

6 7se$ondar8 solution s$reen aears )elow. 6s )eforeB the errors and the errormeasures are $omuted.

49a!p%e (: 49ponentia% s!oothin

/l!ha for e4!onential smoothing . In order to use e=onential smoothingB a valuefor the smoothing $onstantB alhaB must )e entered. This num)er is )etween / and1. 6t the to of the s$reen a s$roll)ar;te=t )o= $om)ination will aearB ena)lingou to enter the value for the smoothing $onstantB alha as shown in the followings$reen. The smoothing $onstant alha is .! in this e=amle.



NO4: If ou sele$t alha W /B the software will find the )est 7lowest #6*8value for alha

/ starting forecast for e4!onential smoothing . In order to erform e=onentialsmoothingB a starting fore$ast is ne$essar. 0hen e=onential smoothing issele$tedB the $olumn la)el Mfore$astL will aear on the s$reen. Anderneath will )e

a )lan: $olumn. If ou wantB ou ma enter one num)er in this $olumn as thefore$ast. If ou enter no num)erB the starting fore$ast is ta:en as the startingdemand.

The results s$reen has the same $olumns and aearan$e as the revious twomethodsB as shown ne=t.

One of the outut dislas 7not shown in this manual8 resents error measures as afun$tion of alha. 6lsoB the grah for e=onential has a s$roll )ar whi$h ena)les

ou to easil see how the fore$asts $hange as varies.

49a!p%e 1: 49ponentia% s!oothin with trend



+=onential smoothing with trend requires two smoothing $onstants. 6 smoothing$onstantB )etaB for the trend is added to the model.

1eta, for e4!onential smoothing . In order to erform e=onential smoothing withtrendB a smoothing $onstant must )e given 7in addition to alha8. If )eta is /B single

e=onential smoothing is erformed. If )eta is ositiveB e=onential smoothingwith trend is erformed as shown.

*nitial trend . In this modelB the trend will )e set to / unless it is initiali?ed. Itshould )e set for the same time eriod as the initial fore$ast.

The solution s$reen for this te$hnique is different from the s$reens for the reviousl des$ri)ed te$hniques. The fore$ast $omutations aear in the $olumnla)eled Munad@usted fore$ast.L These num)ers are the same as in the reviouse=amle 7)e$ause we used the same value for alha8. The trend fore$asts aear in

the $olumn la)eled Mtrend.L The trend is the differen$e )etween the dou)lsmoothed fore$asts from eriod to eriod 7weighted ) )eta8. The fore$asts aearin the $olumn mar:ed Mad@usted fore$ast.L

Note: AnfortunatelB there are several different e=onential smoothing with trendmethods. 0hile the are all similarB the results will var. ThereforeB it is ossi)lethat the results given ) POM-QM for Windows will not mat$h the results of ourte=t. This is ver unfortunate )ut unavoida)le. If ou are using a Prenti$e Hallte=tB )e $ertain that the software is registered 7"e%p, 6ser Infor!ation8 for thatte=t in order to get the mat$hing results.



49a!p%e +: rend ana%5sis

6s mentioned reviouslB the solution s$reen for regression differs from thesolution s$reens for the other fore$asting te$hniques. 6 samle summar oututusing regression for the same data aears )elow.

3al'es for inde!endent (4) ariable. ,or timeseries regressionB the default valuesare set to 1 through n and $annot )e $hanged. ,or aired regressionB the a$tualvalues of the deendent varia)le need to )e entered 7see +=amle "8.

The s$reen is set u in order that the $omutations made for finding the sloe andthe inter$et will )e aarent. In order to find these values it is ne$essar to$omute the sum of the 4& and the sum of the 4&. These two $olumns are resented.*eending on the )oo:B either the sum of these $olumns or the average of these$olumnsB as well as the first two $olumnsB will )e used to generate the regression

line. The line is given ) the sloe and the inter$etB whi$h are listed at the )ottomleft of the s$reen. In this e=amleB the line that fits the data )est is given )C

: W 1/4.33 V 1.!%Q ;

whi$h is read as MSales has a )ase of 1/4 with an in$rease of 1.!% er wee:.L

If the data is sequentialB the ne=t eriod fore$ast is dislaed. This is given )inserting one more than the num)er of eriods into the regression line. In thee=amleB we would insert % into the re$eding equationB ielding 11%.33B as shown

on the s$reen at the )ottom of the fore$ast $olumn.

The standard error is $omuted and shown as with all other methods. In thise=amleB it is ./"-"B whi$h is )etter than an other method seen et. 6lso noti$ethat the mean squared error is dislaed 743.41 in this e=amle8. The )ias isB of$ourseB /B as linear regression is un)iased. 0e disla the summar s$reen asfollows.



(oti$e that the $orrelation $oeffi$ient and rsquared 7r[&8 $oeffi$ient are dislaedas outut. In the summar are the fore$asts for the ne=t several eriodsB sin$e thiswas a trend analsis 7time series regression8.

The trend analsis grah has s$roll)ars whi$h ma:e it ver eas to modif thesloe and inter$et of the line.



49a!p%e 3: Non ti!e series reression

Regression $an )e used on data that is $ausal. In the ne=t s$reenB we resent thesales of um)rellas as a fun$tion of the num)er of in$hes of rain in the last fourquarters of the ear. The interretation of the solution s$reen is that the line that

)est fits this data is given ) um)rella sales W 4-.-3 V &%.43 Q num)er of in$hes ofrain.

6)ove the data is a te=t)o= that ena)les us to la$e in a value for = to enter intothe regression equation. The solution aears in the summar ta)le 7notdislaed8. In our e=amleB if =W1/B then the summar ta)le indi$ates that W3&4.

49a!p%e G: Deco!position and Deseasona%i8ation

The following s$reen dislas an e=amle with seasonal data. 6s $an )e seen inthe s$reenB there are 1& data oints.

ou must enter the num)er of seasons su$h as 4 quarters or 1& months or ! or %das. In additionB ou must enter the )asis for smoothing. ou ma use either the$entered moving average 7whi$h is $ommon8 or the average of all of the data. InadditionB ou $an have the software s$ale the seasonal fa$tors if ou li:e.The solution s$reen $ontains several $olumns.



entered moing aerage. The data is smoothed using a moving average that is as

long as the time eriod 4 seasons. 9e$ause the num)er of seasons is evenB theweighted moving average $onsists of onehalf of the end eriods and all of thethree middle eriods. ,or e=amleB for summer 1--"B the weighted average isC

\.!7-"8 V " V -! V -4 V .!7-38];4 W %.%!

This average $annot )e ta:en for the first n;& eriods and )egins in eriod 3.

Demand to moing aerage ratio. ,or all of the data oints that have movingaverages $omutedB the ratio of the a$tual data to the moving average is $omuted.,or e=amleB for summer &//&B the ratio is -!;%.%! W 1./11.

Seasonal factors. The seasonal fa$tors are $omuted as the average of all of theratios. ,or e=amleB the summer seasonal fa$tor is the average of 1./11 7summer&//&8 and .--%1"% 7summer &//38B whi$h ields 1./3-1B as shown for summer&//&B summer &//3B and summer &//4.

Seasonal 7actor Scaling. The four seasonal fa$tors are 1./""%B .13&B 1./3-1 and1.1/4" whi$h sum to 4./&3" 7rather than 48. If we sele$t the otion in the areaa)ove the data to s$ale the fa$tors then our seasonal fa$tors will )e res$aled7multilied ) 4;4./&3"8 and )e$ome 1./"/4B ./4B 1./33 and 1./-&Brese$tivel.

Smoothed data. The original data is divided ) its seasonal fa$tor in order to ta:eout the seasonal effe$ts and $omute the smoothed data.+nad2'sted forecast . 6fter smoothing the data the software finds the trend line forthe smoothed data. This $olumn reresents the Jfore$astsE using this trend line.The trend line itself $an )e found on the summar results s$reen.



/d2'sted forecast . The final $olumn 7)efore the error analsis8 ta:es the fore$astsfrom the trend line and then multilies them ) the aroriate seasonal fa$tors.The errors are )ased on these ad@usted fore$asts versus the original data.

The summar ta)le $ontains the fore$asts for the $oming eriods.

0dditive Deco!position

0e do not disla the outut here. The additive model uses differen$es rather thanratios to determine the seasonal fa$tors that are additive rather than multili$ative.

10dditive Deco!position

The last method availa)le is userdefined. This allows ou to enter the fore$astsand let the software erform the error analsis. The same module is availa)le asthe fourth su)model when New is sele$ted.



Mu%tip%e Reression

6s noted earlierB the fore$asting module $an erform multile regression. Thereare two inuts to the data. The num)er of eriods of data must )e givenF inadditionB the num)er of indeendent varia)les must )e given. In this first e=amleB

we will e=tend the regression ro)lem in +=amle ". (ote that for simleregression 7one indeendent varia)le8 there are two su)models that $an )e used tosolve the ro)lem. +ither timeseries analsis using the regression method or theregression su)model.

49a!p%e J: Mu%tip%e Reression

In this e=amleB we have used two indeendent varia)les and therefore multileregression must )e used. 0e have entered 4 for the num)er of eriods and & forthe num)er of indeendent varia)les.

0e have filled in the dataB and the solution s$reen aears ne=t. The inut has four$olumnsC one for the name of the time eriodF one for the deendent varia)leBum)rellasF one for the indeendent varia)leB rainF and one for the indeendentvaria)leB time 71 through 48. The outut disla is somewhat different from )efore.The $omutations are not shown. The regression equation is not shown e=li$itlon this s$reen )ut $an )e found ) loo:ing at the )eta $oeffi$ients )elow the ta)le.That isB the equation is Am)rella sales W -.&31 V &".!&3 Q Rain 11.-31Qtime.This is shown e=li$itl on the summar s$reen that we do not disla.

ProKectin

The third model in fore$asting allows us to ta:e a regression equation and ro@e$tit. 'onsider the e=amle )elow.



0hen we set u the ro)lemB we indi$ated that there were five indeendentvaria)les and that we wanted to $reate three fore$asts. The regression line is given ) the first $olumn 7W / V 3=1 V %=& V&1=3 "=4 V &=!8. The three $olumns$ontain the data for =1 through =! for ea$h of the three fore$asts to )e made. Row 1$ontains a 1 sin$e this is for the inter$et. ,inallB the )ottom row $ontains thefore$asts whi$h are -4&B 1/1 and 1/! for the three s$enarios.

14rror ana%5sis

1The last model $an )e used to enter )oth fore$asts and data in order to erform a$omlete error analsis. The error analsis is identi$al to the ones dislaed )efore. The differen$e is that the software allows the user to enter the fore$ast$olumn rather than reling on one of the availa)le methods.



PO#5# for 0indows

strategies 1B 3B or 4. If the follow these mi=esB the 7e=e$ted8 value of the game isthat $olumn will a row &/.&!44. That isB if the laed this game a large num)erof times following their otimal mi=esB the aoffs would )e 1&B "%B !%B and 33Band would average &/.&!44.

Ma9i!in and Mini!a9

0hen e=amining gamesB we usuall )egin ) finding the ma=imin and minima=.To find the ma=imin for the row laerB e=amine ea$h row and find the worst7minimum8 out$ome. These aear in the $olumn la)eled Mrow minimumL as 1&and 33 in the following ta)le. Then find the )est of theseB 1&B whi$h is thema=imum of the minima or the ma=imin.

To find the minima= for the $olumn laer e=amine ea$h $olumn and find theworst 7ma=imum sin$e $olumn is aing8 aoff. These aear in the row namedM'olumn #a=imumL and are 3B !%B -"B %!B and "%. The minima= is the )est

7lowest8 of theseB or 3. The value of the game is )etween the ma=imin andminima= as aears in this gameB with a value of &/.&!44B whi$h is )etween 1&and 3.

49pected ?a%ues for Row

The ta)le )elow dislas the $omutations 7multili$ations8 e=e$ted value forea$h of rowEs strategies. Sin$e row should use )oth strategiesB the e=e$ted valuesare the same and mat$h the e=e$ted value of the game.

1//



'hater "C #odules

49pected ?a%ues for .o%u!n

SimilarlB if $olumn las $olumn & or !B $olumn will a$hieve the value of thegame. HoweverB if $olumn sele$ts $olumn 1B 3B or 4B then he or she will a morethan the value of the game as shown ) the e=e$ted values in the ta)le )elow.

Grahs are availa)le if either or )oth laers have at most two strategies.

1/1



2oa% Prora!!in

Goal rogramming models are ver similar to linear rogramming modelsB )utwhereas linear rograms have one o)@e$tiveB goal rograms $an have several

o)@e$tives. 'onsider the following e=amle.

Suose that a $oman manufa$tures two rodu$ts 7 41 and 4&8. The resour$erequirements and rofit are rovided in the ta)le )elow

Produ$t 1 7 418 Produ$t & 7 4&8 6vaila)le

Profit er unit2a)or hours er unit#aterial er unit

1"3&

1&"1

%&3/

In additionB the $oman has the following goalsC

1. The total rofit should )e at least &!/.&. It ta:es time to set u rodu$tion for Produ$t &B so we li:e to rodu$e in )at$hesof at least !.3. The $urrent demand for Produ$t 1 is 14. ThereforeB we would li:e to rodu$ee=a$tl 14.

This ro)lem aears similar to a linear rogramB )ut now we have three goalsrather than one o)@e$tive.

Data

6n goal rogram is defined ) the num)er of varia)les and the num)er of$onstraints or goals. *o not $ount the nonnegativit restri$tions as $onstraints.ThusB in this e=amleB we have two varia)les and five $onstraints;goals 7two$onstraints and three goals8. The information is entered as shown in the followings$reenC

Some of the information is identi$al to linear rogrammingB )ut there are somedifferen$es. ,irst note that there is no o)@e$tive fun$tion. Se$ondB noti$e that thereare four e=tra $olumns at the )eginning 7left8 of the ta)le )efore the de$ision



varia)les. These e=tra $olumns are used for the goals and not for the $onstraints7where ou $an see that the are ?ero8.

%oals<onstraints. In ea$h line of the ta)leB we enter either a $onstraint or a goal.The first two lines reresent $onstraints. Sin$e these are $onstraintsB the first four

$olumns are not used 7/s are entered8. The $onstraints are entered in the usualfashion.

The ne=t three lines reresent goalsB and there are two ase$ts to these goals. Sin$ethese are goalsB noti$e that the sign in the row is MWL. The values under 41 and 4&serve to $reate the goal in $on@un$tion with the varia)les d V and d B indi$ating )how mu$h we overa$hieve or undera$hieve the goal. ,or e=amleB line 3 in theta)le stands forC

41 7d 1V8 V 7d 18 W 14.

If 41 is )elow 14B d 1 reresents the amount )elowB )ut if 41 is a)ove 14B d 1Vreresents the amount ) whi$h we go over.

SimilarlB the ne=t line 7goal 48 reresentsC

1" 41 V 1& 4& 7d &V8 V 7d &8 W &%/

ThusB d &V and d & reresent the amount of rofit )eond &%/ and )elow &%/Brese$tivel.

#athemati$allB the goals are d 1VB d 1B d &VB d &B d 3VB d 3. The question isC Howdo we want to order or weight these goalsX That isB how do we $ontrast theimortan$e of ea$h of these si= goalsX

0e do this using the riorities and weights on the line.

Priorities and weightsC ,irstB there are si= goals 7 d 1VB d 1B d &VB d &B d 3VB d 38 inthis e=amleB )ut we do not $are if we overa$hieve our rofit goal of &%/B nor dowe $are if we rodu$e more than five units of Produ$t &. ThereforeB )oth theweights and riorities of these two goals have )een set to /. The riorities for theother four goals range from 1 to 3. The meaning of different riorities is the order

in whi$h the goals are satisfied. In other wordsB goals with riorit 1 must )esatisfied )efore goals with riorit &B whi$h must )e satisfied )efore goals with riorit 3B and so on. In this e=amleB we first want to ma:e e=a$tl 14 units ofProdu$t 1B then we want to guarantee our minimum rofit level of &%/B then wewant to tr to guarantee our minimum )at$h level of five for Produ$t &.

0ithin ea$h rioritB it is ossi)le to assign different weights to the goals. This isshown in the ne=t e=amle.



The constraint sign. This is a drodown )o= that $an )e used to $hange the$onstraint te from Mless than or equal toL to Mequal toL to Mgreater than or equalto.L 6s stated a)oveB goals must have the sign MWL.

ight-hand side coefficients. The values on the right hand side of the $onstraintsare entered here. ,or $onstraintsB these are the usual $oeffi$ientsB while for goalsBthese are the goals that are set.

he So%ution

The following s$reen dislas the summar solution 7the simle= goal ta)leau isalso availa)le for dislaB as is a grah for twodimensional ro)lems8.

The otimal solution is to rodu$e 14 units of Produ$t 1 and & units of Produ$t &.Priorit 1 will )e a$hieved 7the nona$hievement is /8B while we have failed toa$hieve riorities & and 3. Remem)erB d V is the amount ) whi$h we havee=$eeded the goalB and d is the amount ) whi$h we have $ome u short. The$onstraint analsis indi$ates that we used 1 fewer hours than we had availa)leBe=a$tl the amount of material that we hadB rea$hed goal 3 e=a$tlB undera$hieved rofit 7goal;$onstraint 48 ) && and )at$h si?e 3 7goal;$onstraint !8 ) 3. 6 grah7not shown8 is also availa)le if the num)er of varia)les is two.



49a!p%e *: 6sin weihts

0e have revised our goal riorities as shown in the following e=amle. This timewe ma:e meeting the rofit level our highest goal and everthing else isse$ondar. HoweverB we have given twi$e the weight to undera$hieving our fiveunits of Produ$t &B $omared with missing our goal of 14 for Produ$t 1.

The results follow. 0e should rodu$e 1&.! units of Produ$t 1 and ! units ofProdu$t &. 0e will a$hieve our first riorit )ut miss our se$ond riorit. 7The 1.!reresents 14 1&.!8. (oti$e that we had onl two riorities.

The goal;$onstraint analsis shows us that we have 4.! la)or hours left overB usede=a$tl the 3/ ounds of material that we hadB undera$hieved our Produ$t 1demand ) 1.!B overa$hieved our rofit goal ) 1/B and met the goal of ma:ing atleast ! units of Produ$t & e=a$tl.



Inteer and Mi9ed Inteer Prora!!in

6n integer or mi=ed integer linear rogram is defined ) the num)er of varia)lesand the num)er of $onstraints. 6s with linear rogrammingB do not $ount the non

negativit restri$tions as $onstraints. #ost student linear rogramming a$:ages7e=$et +=$elEs Solver8 assume that the varia)les must )e nonnegative.

'onsider the following integer rogramming e=amleC

ma=imi?esu)@e$t to

3!/ 41 41 41

41B 4& 41B 4&

V !// 4&V 1.! 4&

V 4 4&^W /

integer

_W^W

1!/

The $omonents and the data entr are nearl the same as for linear rogramming.The differen$e is that the inut s$reen has one e=tra row for identifing the te ofvaria)le as either realB integer or /;1.

Ob2ectie f'nction. The $hoi$e of minimi?ation or ma=imi?ation is made in theusual wa at the time of ro)lem $reationB )ut it $an )e $hanged on the data s$reenusing the o)@e$tive otions a)ove the data.

Ob2ectie f'nction coefficients. The $oeffi$ients 7ti$all referred to as c @8 areentered as numeri$al values.

onstraint coefficients. The main )od of information $ontains the $onstraint$oeffi$ientsB whi$h ti$all are $alled the ai@ s. These $oeffi$ients ma )e ositive

or negative.The constraint sign. This $an )e entered in one of two was. It is ermissi)le to ress the N_ :eB the N^ :eB or the NW :e. 0hen ou go to a $ell with the$onstraint signB a drodown arrow aears in the $ell and $an )e used.

ight-hand side coefficients. The values on the righthand side of the $onstraintsare entered here. These are also termed the bi s. The must )e nonnegative.



The ariable t&!e. This is a drodown )o= that will $hange the varia)le te fromMintegerL to MrealL to M/;1. ou $an $hange all varia)les at on$e ) $li$:ing onthe leftmost $olumn. This is ver useful for $aital )udgeting ro)lems.

Ma4im'm n'mber of iterations and Ma4im'm leel (de!th)C If ou re$eive a

message regarding the num)er of iterations or deth rather than a solution ou main$rease these num)ers

he So%ution

The solution is given ) a simle s$reen with the varia)les and their values.

Iterations

The iterations $an )e found in another s$reen. The 2P solution to the original ro)lem 7see iteration 1B level /8 had 41 and 4& )oth as nonintegers. 0e then )ran$hed on 41 ) adding the $onstraint 41_W1/. The 2P solution to this ro)lemhad 41 as an integer )ut 4& was nonintegerB so we )ran$hed on 4& ) adding the$onstraint 7 4&_W38. This ields an integer solution 7iteration 38. 0e $reate theother )ran$h ) adding 7 4&^W48 and this ields an even )etter integer solution. 0ego )a$: to the original node and $reate the )ran$h 41^11B whi$h ields an

infeasi)le solution. ThereforeB we are done.

6 grah 7not shown8 is availa)le for this module.

49a!p%e of a !i9ed inteer prora!



'onsider the following e=amleC

ma=imi?e 3/ 4 V 33 & V !/ 5 su)@e$t to &3 4V 43 & V 1" 5 _W 1/// 3& 4 V 33 & V &! 5 _W &!// 43 4 V !3 & V &" 5 _W 1!// 4, &, 5 ^W /

4 integerB 5 /;1

6gainB the $omonents are identi$al to linear and integer rogramming e=$et thatthere is one e=tra row of information that needs to )e given indi$ating the te ofea$h varia)le 7realB integerB or /;18.

he So%ution

The varia)le tes and their values are dislaed.

NO4C 2inear rogram $an )e entered and solved as mi=ed integer rograms )utthe ranging ta)le and linear rogramming iterations will not )e availa)le.



Inventor5

These models use different variations of the e$onomi$ order quantit 7+O58model in order to determine roer order or rodu$tion quantities. 9esides the

standard +O5 modelB we in$lude the e$onomi$ rodu$tion quantit 7+P58 model.,or )oth the +O5 and +P5 modelB we allow shortages to )e in$luded. ,inallB weallow quantit dis$ounts for the +O5 model.

6 se$ond te of model is 69' analsis.

The last two models are used for $omuting reorder oints for (ormaldistri)utions and dis$rete distri)utions.

4OQ-5pe Mode%s

The following s$reen $ontains an e=amle that in$ludes )oth the data and thesolution.

he Data

Demand rate. The rate of demand or usage is to )e entered here. Ti$allB thisdemand rate is an annual rate )ut it does not need to )e. The time units for thisdemand rate must mat$h the time units for the holding $ost.

Set'! cost . This is the fi=ed $ost of la$ing ea$h order or ma:ing ea$h rodu$tionrun.



$olding cost rate. This is the $ost of holding or $arring one unit of inventor forone time eriod. This $ost is either given as a arti$ular dollar amount or given asa er$entage of the ri$e of the item.

NO4: If ou want the holding $ost to )e a er$entage of the unit $ostB enter a

er$ent signB MZLB after the num)er. ,or e=amleB M&/ means &/ dollarsB )utM&/ZL means &/ er$ent of the unit $ost. If the holding $ost is a er$entage of theunit $ostB ou must enter the unit $ost.

+nit cost. This is sometimes ne$essarB )ut man timesB it is not )e$ause the +O5is indeendent of the unit $ost.

eorder !oint . The otion )o= a)ove the data ena)les us to $al$ulate the reorder oint. Three lines of inut are added in these $ases. 0e must either enter a daildemand rate or enter the num)er of das in the ear so that the dail demand rate$an )e $omuted from the annual demand rate. In additionB we must enter the

num)er of das for the lead time.

Order 9'antit&. 6)ove the data is a te=t)o=;s$roll )ar $om)ination that allows outo enter a value for the order quantit. If ou enter a num)er other than /B then twosets of results will )e dislaed. One $olumn will )e for the +O5B while the other$olumn will )e for the se$ified order quantit.

he So%ution

The outut s$reen aears in the re$eding s$reen. In +=amle 1B we have solved astandard +O5 model andB in additionB found results when using an order quantit

of &/ units. The model results are as followsC

O!timal order 9'antit&. This is the most e$onomi$al order quantit. If there is noquantit dis$ountB this is the +O5. HoweverB when a quantit dis$ount is availa)le7as in +=amle 38B this is either the +O5 or a dis$ount oint a)ove the +O5. Inthis e=amleB the otimal order quantit is 1".33 units er order.

Ma4im'm inentor& leel . It is useful to :now the largest amount that will )e ininventor. In the standard +O5 modelB this is siml the amount that is orderedF ina rodu$tion or shortage modelB this is less. In this e=amleB the inventor will

never e=$eed 1".33 units when using the +O5 or &/ units if &/ is the orderquantit.

/erage inentor& leel . If there are no )a$:ordersB the average inventor is half ofthe ma=imum inventor. 6nnual holding $osts are )ased on the average inventor.Orders !er &ear . The assumed time eriod is one earB and the num)er of orders isdislaed. In this e=amleB it is 1&.&! for the +O5 and 1/ for an order quantit of&/ units.



The solution aears ne=t. The dail demand rate has )een found to )e 4/. Theremaining results are the same as in the first e=amle. (oti$e that the holding $osthas )een $omuted as U1 )ased on &/Z of the U-/ unit $ost.

49a!p%e (: Quantit5 discounts

6 s$reen for quantit dis$ounts aears in the following illustration. The usualinformation is la$ed at the to. In additionB the num)er of ri$e ranges must )egiven at the time of ro)lem $reation..

6 detailed analsis of the order quantities and $osts at ea$h ri$e range isavaila)leB as shown )elowC



ac7order Mode%s

The software also has the $aa)ilit to $omute the +O5 or the rodu$tion modelwith )a$:orders. These models do not aear in all te=t)oo:sB so we do notdisla them in this manual. If ou have the software set for one of the Renderte=tsB then these models will not show u in the model su)menu.

0. 0na%5sis

The goal of 69' analsis is to identif the most imortant items that are :et ininventor. Imortan$e is measured ) dollar volume. 6n e=amle aears )elowfor a ro)lem with si= items.

,or ea$h itemB the information to )e entered isC

*tem name. 6s usualB a name $an )e entered on ea$h line.

Demand . The demand rate for ea$h item is to )e given.

*tem !rice. The $ost or ri$e of ea$h item is to )e given

Percentage of / and 1 items. In the e=amleB we want &/ er$ent of the items to )e 6 items and 3/ er$ent to )e 9 items. 6fter the rogram sorts the items )dollar volumeB the first &/ er$ent of " items 7." items rounded to 18 will )e$lassified as an 6 item and then 3/ er$ent of " 71. items rounded to &8 will )e$lassified as 9 items.



(oti$e that the items are sorted a$$ording to their dollarvolume er$entages. ThatisB the do not aear in the same order as on the original s$reen of inut. The

outut $omuted for ea$h item isC

Dollar-ol'me. This is the demand multilied ) the ri$e for ea$h item

Dollar-ol'me !ercentage. This is the item dollarvolume divided ) the;totaldollar volume.

'm'latie dollar-ol'me !ercentage. This is a running total of dollar volumewhen the items are sorted from highest dollar volume to lowest dollar volume.

ategor&. This is the $lassifi$ation as e=lained a)ove.

Reorder Points for the Nor!a% distriution

9elow is the solution s$reen for $al$ulating the safet sto$: and reorder oint forthe $ase where demand during lead time is given ) a (ormal distri)ution. Thesolution s$reen in$ludes the inut on the left.

Dail& demand . This is the dail demand rate during the lead time.



Demand standard deiation. This is the standard deviation for the dail demandrate. If the dail demand rate is fi=ed enter a standard deviation of /.

Serice =eel . This is the er$entage of demands whi$h should )e met.

=ead time in da&s. This is the lead time in das.

=ead time standard deiation. This is the standard deviation of the lead time. If thelead time is fi=ed enter / for the standard deviation.

NO4: The disla for Hei?er;Render is )ased on a given lead time demand andstandard deviation and has onl three inuts. In generalB to use the disla a)oveBif the ro)lem has a given lead time demand and standard deviation then set thelead time das to 1 and lead time standard deviation to /.

Reorder Points for a Discrete distriution

eorder !oint w<o safet& stoc . This is the reorder oint rior to $onsideration ofsafet sto$:. In this e=amleB our initial lan is to reorder when the inventor fallsto %/. ThusB if the demand during the lead time does not e=$eed %/ we will nothave an sto$:outs. Sin$e the ma=imum demand is -/ and our reorder oint is %/our ma=imum safet sto$: will )e &/ units.

arr&ing cost !er &ear . This is the usual $ost of $arring inventor. In this

e=amle it is U% er unit er ear.

Stoco't cost. This is the $ost er unit of not )eing a)le to meet the demand. Inthis $ase it is U1 ea$h time that we are short a unit.

Orders !er &ear . This is the num)er of times er ear we erform the ordering ro$ess. In this e=amle we la$e orders twi$e a ear.



Probabilit& distrib'tion. This is the $olumn of lead time demands and theirasso$iated ro)a)ilities.

The results are )elowB indi$ating that the minimal $ost safet sto$: is 1/ and thusthe revised reorder oint is the original %/ lus the 1/ for a total of / units.

Sin$e our original reorder oint is !/ and our lead time demands var from%/ to11/ we need a ma=imum of 7-/%/8 &/ units as safet sto$: so the results aregiven for values )etween / and &/ units. In ea$h of these 3 $asesB the $arring$ostsB sto$:out $osts and total $osts are $omuted and dislaed. 9elow the$omutations are the summar results.



#o Shop Schedu%in BSeuencinC

The @o) sho s$heduling models are used to solve oneand twoma$hine @o) sho ro)lems. ,or the onema$hine ro)lem the availa)le methods are shortest

ro$essing timeB first $ome first serveB due date s$hedulingB #ooreEs methodB sla$:timeB sla$: er oerationB longest ro$essing timeB and $riti$al ratio. ,or twoma$hine s$hedulingB >ohnsonEs method is used to minimi?e the ma:esan.

One-Machine Schedu%in

'onsider the following onema$hine s$heduling ro)lem. ,ive emloees are to )etrained to oerate different ma$hines ) a single trainer who $an train onl one erson at a time. The time to train ea$h erson varies and is given in thea$$omaning ta)le along with due dates and the num)er of oerations involved.

>o) Time *ue *ate (um)er of Oerations

>anet9arr6le=isSamm2isa+rnie

3 das!4%-&

4%131/1!!

&41&13

9oth the data and a solution aear in the ne=t s$reen.

Methods (!riorit& r'les). The rules availa)le for s$heduling in$ludeC

1. Shortest ro$essing time 7SPT8&. ,irst $ome first serve 7,',S8



3. +arliest due date 7*ue *ate84. Sla$: time 7Sla$:8!. Sla$: er oeration 7Sla$:;o8". #ooreEs method 7#oore8%. 2ongest ro$essing time 72PT8

. 'riti$al ratio 7'rit rat8The data to )e entered areC

Starting da& n'mber. 6n otional starting da num)er ma )e given. +=amle &will disla the use of this otion.

Date receied. It is ossi)le to list the date ea$h @o) is re$eived. This informationwill )e used in s$heduling when ,irst 'ome ,irst Served is used. If re$eit datesare given then the will )e used in the $omutation of the flow times 7see e=amle&8. The re$eit das must )e less than or equal to the starting da. That isB all @o)smust )e re$eived )efore the starting date.

>ob names. (ames $an )e entered for ea$h @o).

Machine name. The word Mma$hine 1L at the to of the $olumn $an )e $hanged togive the name of the te of ma$hine. In this e=amleB the ro$ess has )eenrenamed MTrainingL.

Processing time. The amount of time that ea$h @o) will ta:e on ea$h ma$hine isentered in the $olumn la)eled with the ma$hine name.

D'e date. In some instan$esB due dates are used. These are entered here.

8'mber of o!erations. In order to use the sla$: er oeration ruleB it is ne$essarto give a ositive num)er of oerations. ,or an other method this $olumn $an )eignored.

49a!p%e /: Shortest processin ti!e

The results deend on the rule that is $hosen. In our first e=amleB we have $hosenshortest ro$essing timeB )ut in going through the e=amles all of the informationthat will )e dislaed is e=lained. The outut for our first e=amle is shown in

the re$eding s$reen.



>ob order . 6 $olumn is dislaed that shows when ea$h erson 7@o)8 will )etrained 7ro$essed8. In +=amle 1B the $olumn shows that >anet will )e se$ondB9arr fourthB 6le=is thirdB Samm fifthB 2isa si=thB and +rnie will )e first.

Se9'ence. The same sequen$e is dislaed )ut in a different manner at the )ottom

of the s$reen. In this e=amleB the sequen$e is +rnie followed ) >anetB 6le=isB9arr SammB and 2isa.

7low time. The time at whi$h ea$h @o) ends is given in a $olumn of flow times. Inthe e=amleB +rnie is the first one trained and ends after the ro$essing time of &das. >anet is the se$ond trained and ends after 3 more das at time !. The last @o) erformedB 2isaB ends after 3/ das.

om!letion time. If the starting da is not /B a $olumn of $omletion times is giventhat in$ludes the starting da 7see +=amle 8.

Tardiness or lateness. If due dates are givenB the differen$e )etween the flow timeand the due date is dislaed. 7On the s$reen the disla is in red.8. This differen$ewill never )e )elow /. There is generall no su$h thing as earl in s$heduling.

Totals. ,or )oth the flow time and the latenessB the totals are $omuted anddislaed.

/erages. #ore relevant than the totals are the averages. The average flow timereresents the seed with whi$h @o)s leave the sstem after the have entered. Theaverage lateness 7tardiness8 reresents how )adl the s$hedule is erforming withrese$t to our romised due dates.

NO4: The average lateness is $omuted )ased on all @o)sB not @ust the @o)s thatare late. In the e=amleB it is 34;"B even though +rnie was trained on time.

/erage n'mber of 2obs in the s&stem. This is $omuted as the total flow time 7onor after the starting da8 divided ) the ma=imum flow time.

2antt .hart

6 Gantt $hart illustrating the s$heduling on the ma$hine is availa)leB as seen in the

following s$reenC



Su!!ar5

One of the outut windows for one ma$hine s$heduling is a summar of results forall methods as dislaed )elow.

49a!p%e *: First co!e first served

In this e=amleB we have $hanged our rule to ,irst 'ome ,irst Served and in

addition we have la$ed re$eit das for the @o)s and a starting da of 1//.

The results are )elow.



Sin$e the first $ome first served 7,',S8 otion is sele$tedB the rogram wills$hedule the @o)s a$$ording to the re$eit da.

(oti$e that there is an e=tra $olumn of results named M'omletion timeL. This isthe flow time lus the starting da minus 1. ,or e=amleB 2isaEs @o) )egan at the )eginning of da 1//B was wor:ed on for - das and therefore was finished at theend of da 1/. 'omletion times are at the end of that da.

49a!p%e (: Schedu%e accordin to s%ac7

Sla$: is defined as the due date minus the time required to ro$ess a @o). In orderto use sla$:B the due date must )e given.

The sla$: $olumn did not aear )efore )ut does now. It is the differen$e )etweenthe due $olumn and the MtrainingL $olumn. ,or e=amleB >anet must )e trained )da 4 )ut it ta:es 3 das to trainB so there is one da of sla$:. The @o)s are



s$heduled a$$ording to in$reasing order of sla$:. >anet has the least and iss$heduled firstB while 6le=is has the most 7-8 and is s$heduled last. The solutionaears )elow.

49a!p%e 1: S%ac7 ti!e per operation

0e have used the data in the num)er of oerations $olumn. The outut 7notshown8 $ontains a new $olumn titled sla$:;oB whi$h is generated ) dividing thesla$: ) the num)er of oerations. ,or e=amleB the 1 da of sla$: for @o) 1 isdivided ) the & oerationsB ielding a sla$: er oeration value of .!. >o)s ares$heduled a$$ording to in$reasing order of sla$: er oeration. ThereforeB >anet isfirst 7.!8 and 6le=is is last 7-8. 7Ties are )ro:en ar)itraril.8

49a!p%e +: Due date schedu%in

>anet is the first one due and is s$heduled firstB while 2isa is the last one due and is

s$heduled last.

49a!p%e 3: Moores !ethod

#ooreEs method minimi?es the num)er of late @o)s. In the e=amle shown ne=tB#ooreEs method leads to the sequen$e >anetB +rnieB 2isaB SammB 6le=isB and9arrB whi$h has three @o)s late. (o s$hedule will have fewer than three @o)s lateas $an )e seen in the summar ta)le dislaed at the end of e=amle 1.



49a!p%e G: Aonest processin ti!e

The 2PT method s$hedules @o)s from longest to shortest. This is ti$all theworst wa to erform s$heduling. In the e=amle 7not shown8B 2PT ields thesequen$e 2isaB SammB 9arrB 6le=isB >anetB +rnieB whi$h is e=a$tl oosite theSPT s$hedule of $ourse. This s$hedule has an average flow time of &1.!. (os$hedule will have a larger average flow time. This s$hedule has 4.3 @o)s in thesstem on average. (o s$hedule will have a larger average num)er of @o)s in thesstem.

49a!p%e J: .ritica% ratio

The $riti$al ratio is defined as 7due date toda8;ro$essing time. This is the firste=amle in whi$h we have used the starting da num)er a)ove the data. >o)s ares$heduled in as$ending order of the $riti$al ratio. In this e=amleB the s$hedule is>anetB 9arrB SammB +rnieB 2isaB 6le=is. (oti$e in the s$reen that there is an e=tra

$olumn of oututB $omletion time. 9e$ause @o)s do not start at time /B the flowtime and the $omletion time are different. ,or e=amleB >anet is the first @o) doneand )egins toda on da 3. Sin$e it ta:es 3 dasB we wor: on >anet on das 3B 4Band !B whi$h )e$omes the $omletion time. The @o) is one da late.



wo-Machine Schedu%in

'onsider the following ro)lem. 6 ting $enter needs to te and rint @o)s forseven $ustomers. The length of time that ea$h @o) requires for final $orre$tions7ting8 and for rinting is dislaed in the following ta)le. +a$h @o) must first

have the ting finished )efore it $an )e duli$ated.

'ustomer Ting *uli$ating

Harr*e)2eah*ara6rtSharonRiv:a

&/ minutes433%"&/1&&!

1- minutes&%311!&3"41

In the ne=t s$reenB we demonstrate the twoma$hine ro)lem. >ohnsonEs method isusedB and the order and sequen$e are listed as follows.

In additionB the time at whi$h ea$h @o) ends on ea$h ma$hine is dislaed. Thelargest of all of these times is the ma:esanB or time at whi$h all wor: is$omletedF it is dislaed at the )ottom. In this e=amleB it will ta:e &-/ minutesto finish the wor:.

#ohnsons Method Steps

6 se$ondar outut for this su)model is the disla of the order in whi$h @o)swere $hosen a$$ording to >ohnsonEs method. This is dislaed in the followings$reen.



The smallest time among the 14 times was 11 minutes for *ara on *uli$ating.ThusB *ara is s$heduled last sin$e duli$ating is the se$ond ma$hine. The ne=tsmallest time among all $ustomers e=$et *ara is 1& for Sharon on Ting. Sin$eTing is the first ro$essB Sharon is s$heduled first. The method $ontinues to findthe smallest time of all uns$heduled $ustomers and s$hedules the $ustomer as soon

as ossi)le if the time is on Ting and as late as ossi)le if the time is on*uli$ating.

6 twoma$hine Gantt $hart $an )e dislaed also as shown )elow. If the name ofthe @o) is too long for the )ar in the $hartB then it will )e trun$ated. ,or e=amleBsee Sharon on ma$hine 1.



Aa5out

The fa$ilit laout model is used to la$e deartments in rooms in order tominimi?e the total distan$e traveled as a fun$tion of the distan$es )etween the

rooms and the flow )etween deartments. In some $asesB it is ne$essar to fi=$ertain deartments to )e lo$ated in se$ifi$ rooms. *istan$es )etween rooms maor ma not )e smmetri$. 7Asuall the areB )ut this is not required.8

Data

The framewor: for laout is given ) the num)er of deartments or the num)er ofrooms that we assume to )e the sameB sin$e ea$h deartment must )e assigned toone and onl one room.

The data that follows essentiall $onsists of two ta)les of num)ersB one for theflows and one for the distan$es.

Method. There are two methods availa)le. The default method is e=li$itenumeration. This is guaranteed to find the otimal solution. AnfortunatelB if the ro)lem si?e is too large this method will ta:e too mu$h time. 6 se$ond methodB airwise $omarison is availa)le. AnfortunatelB this method is not guaranteed to

alwas find the )est laout.

*nterde!artmental flows. The num)er of tris from one deartment to another isindi$ated in a ta)le termed the flow matri=.

Distance matri4. The distan$e )etween rooms is entered in this ta)le. Ti$all thedistan$e matri= will )e smmetri$. The $hoi$e is made at the )eginning of themodule. That isB the distan$e from room i to room 2 is the same distan$e as forroom 2 to room i.



6 samle set of data aears in the re$eding s$reen. (oti$e in this e=amle thatthe distan$es are smmetri$.

So%ution

The solution siml is to assign the deartments to the aroriate rooms. Thetotal movement is also noted.

oom assignments. On the right of ea$h deartment row will aear the room inwhi$h the deartment should )e la$ed. In our e=amle #aterials should )e la$edin room &B 0elding in room 1B et$. Total moement. The sum of the rodu$ts of the num)er of tris multilied ) the

distan$e is listed at the to. This is what we are tring to minimi?e. (oti$e that forour e=amle the minimum total movement is 13B///.

It is ossi)le to disla the individual multili$ations of roomtoroom distan$es ) ro$esstoro$ess flows. This is shown 7artiall8 in the s$reen that follows.



In order to demonstrate airwise $omarison we have ta:en the same e=amle )ut$hanged the method and solved it. The results are dislaed )elow. (oti$ethatB indeedB airwise $omarison did not find the otimal solution sin$e themovement under airwise $omarisonB 1!&//B is larger than the movementunder e=li$it enumeration.



Fi9in Depart!ents in Specific Roo!s

If the room name aears in the $olumn la)eled M,i=ed RoomBL that deartmentwill )e fi=ed in that room. Suose that in our revious e=amle we are requiredto have #aterials la$ed in room 1. ThenB to a$$omlish thisB we la$e Mroom 1in the row for #aterials in the M,i=ed RoomL $olumn using the drodown )o=.

The solution follows. The room assignments areB of $ourseB differentB and the totalmovement isB of $ourseB greater than the otimal solution for the unrestri$ted ro)lem.



Aearnin B49perienceC .urves

Two models are availa)le for learning $urves. In the first modelB it is assumed thatthe learning $oeffi$ient is :nownF in the se$ond modelB it is assumed that the

rodu$tion time for two units is :nownB and that the learning $urve $oeffi$ient is$omuted )ased on these. In either $aseB we $an find the rodu$tion times for unitsfrom 1 to a se$ified num)er and the $umulative rodu$tion time for these units.In additionB for either model the learning $urve $an )e grahed.

he Data

'onsider a situation in whi$h unit 1 too: 1/ hoursB the learning $urve $oeffi$ient is-/ er$ent and we are interested in the first &/ units. ,ollowing is a s$reen that$ontains )oth the data and the one line of rimar oututC

+nit n'mber of base 'nit . This is usuall 1 as in our e=amleB )ut it $an )e set toan num)er.

Time for base 'nit. This is the length of time that it ta:es to manufa$ture the unitnum)er as se$ified a)ove. In our e=amleB it is 1/ hours.

8'mber of the last 'nit. This is the item num)er for the last unit whi$h will )edislaed and;or used for $omutations. In the e=amleB we are interested in unit&/ or the first &/ units.

=earning c're coefficient. This is a num)er )etween / and 1. It is the er$entage

of the first unitEs time that it ta:es to ma:e the se$ond unit and also the er$entageof the se$ond unitEs time that it ta:es to ma:e the fo'rth unit. The learning $urve$oeffi$ient is onl entered for the first model. The se$ond model will determinethe learning $urve $oeffi$ient )ased on the ne=t data inut item.Time to mae last 'nit. (ot shown on this s$reenB )ut shown in the ne=t e=amleBthe last ie$e of information for the se$ond model is the time it ta:es to rodu$ethe last unit rather than the learning $urve $oeffi$ient. 9ased on this ie$e ofinformationB the learning $urve $oeffi$ient will )e determined 7see +=amle &8.



49a!p%e /: .o!putin ti!es and cu!u%ative ti!es

The samle ro)lem aears in the re$eding s$reen. The four lines of inut dataindi$ate that the first unit 7unit num)er 18 ta:es 1/ minutes to manufa$tureB the lastunit is unit num)er &/B and the learning $oeffi$ient is -/ er$ent. That isB thede$rease in time is su$h thatB with the dou)ling of the unit num)erB the time is -/ er$ent of the revious time.

The solution in the re$eding s$reen is that the last unit 7num)er &/8 ta:es ".34minutes.

6n additional ta)le of times and $umulative times $an )e dislaed. The outut$onsists of three $olumns.

+nit n'mber . This runs from 1 to the last unitB whi$h in our e=amle is &/. Thetwo additional $olumns are as followsC

Time to !rod'ce a single 'nit (!rod'ction time). This $olumn $ontains the time to rodu$e a unit. ,or e=amleB it ta:es 1/ minutes to rodu$e unit 1 7as se$ified )the inut8B - minutes to rodu$e unit & 7as $omuted using the learning$oeffi$ient8B .1 minutes to rodu$e unit 4 7-/ er$ent of - minutes8B %.&- minutesto rodu$e unit B and so on. The interesting num)ers are the ones that are not owers of &. ,or e=amleB it ta:es ".34 units to rodu$e unit &/.

'm'latie time. The last $olumn $ontains the amount of time to rodu$e all ofthe units u to and in$luding that unit num)er. O)viouslB it ta:es 1/ minutes to rodu$e unit 1. It ta:es 1- minutes to rodu$e units 1 and &F 3!.&/ minutes to



rodu$e the first four unitsF and 14"./ minutes to rodu$e the first &/ units. 6grah of the unit rodu$tion times $an )e dislaed. The s$roll)ar $an )e used tosee the effe$ts of the learning $oeffi$ient on the grah.

49a!p%e *: Findin the %earnin curve coefficient

,ollowing is the solution s$reen for an e=amle of the se$ond model. In this $aseBwe :now that unit 4 too: %3 minutes and unit 3% too: " minutes. The rogramhas $omuted thatB )ased on these two timesB the learning $urve $oeffi$ient is .-%1.



Ainear Prora!!in

6n linear rogram is defined ) the num)er of varia)les and the num)er of$onstraints. *o not $ount the nonnegativit restri$tions as $onstraints. #ost linear

rogramming a$:ages 7)ut not +=$elEs Solver8 assume that unless told otherwisethe varia)les must )e nonnegative.

'onsider the following e=amle with two $onstraints and two varia)lesC

ma=imi?e 3 4 V 3 &su)@e$t to 3 4 V 4 & _W 14 7la)or hours8 " 4 V 4 & _W 1! 7l)s material8 4B & ^W /

The data s$reen for this aears ne=t. 0e show the entire s$reen so that we $an oint out that a ST+P tool now aears on the tool)ar )efore the SO2+ tool.6lsoB Step is ena)led in the Fi%e menu.

Ob2ectie f'nction. The $hoi$e of minimi?ation or ma=imi?ation is made in theusual wa at the time of ro)lem $reationB )ut it $an )e $hanged on the data s$reenusing the o)@e$tive otions a)ove the data.

Ob2ectie f'nction coefficients. The $ost;rofit $oeffi$ients 7ti$all referred to asc @8 are entered as numeri$al values. These $oeffi$ients ma )e ositive or negative.

onstraint coefficients. The main )od of information $ontains the $onstraint$oeffi$ientsB whi$h ti$all are $alled the ai@ s. These ma )e ositive or negative.

ight-hand side ($S) coefficients. The values on the righthand side of the$onstraints are entered here. These are also termed the bi s. These must )e nonnegative.



The constraint sign. This $an )e entered in one of two was. It is ermissi)le to ress the N_ :eB the N^ :eB or the NW :e. 6lternativelB when ou go to a $ellwith the $onstraint signB a drodown arrow aears in the $ellB as shown in thefollowing s$reen in $onstraint & in the $olumn with the $onstraint signs.

ou $an $li$: on the arrow )ringing in a drodown )o= as shown ne=tC

09'ation form. The $olumn on the far right dislas the equation form of the$onstraint and $an not )e dire$tl edited )ut $hanges as the $oeffi$ientsB $olumnnameB sign or right hand side $hange.

he So%ution

,ollowing is the solution to our e=amle. Please note that the disla variessomewhat a$$ording to the te=t)oo: otion sele$ted in "e%p, 6ser Infor!ation.

O!timal al'es for the ariables. Anderneath ea$h $olumnB the otimal values forthe varia)les are given. In this e=amleB 4 should )e .33 and & should )e 3.&!.

O!timal cost<!rofit. In the lower right hand $orner of the ta)leB the ma=imum rofit or the minimum $ost is given. In this e=amleB the ma=imum rofit isU1/.%!.



Shadow !rices. The shadow 7or dual8 ri$es aear on the right of ea$h $onstraint.In this e=amleB we would a .! more for one more unit of resour$e 1 and .&!more for one more unit of resour$e &.

he raph

One of the other outut dislas is a grah as shown in the following s$reen. Thefeasi)le region is shaded. On the right is a ta)le of all of the feasi)le $orner ointsand the value of the o)@e$tive fun$tion 7 ? 8 at those oints. In additionB the$onstraints and o)@e$tive fun$tion $an )e highlighted in red ) $li$:ing on theotion )uttons on the right under M'onstraint *isla.L



a%e of Ranes

In addition to listing the valuesB we have rovided additional information a)out thevaria)les. The interretation of the additional information is left for our te=t)oo:.In the e=amleB ou $an see the redu$ed $ostB original o)@e$tive value $oeffi$ientBand the lower and uer limit 7the range8 over whi$h the solution will )e the same.That isB the varia)les will ta:e on the same values of .333 and 3.&!F onl theo)@e$tive fun$tion value 7rofit or $ost8 will $hange.NO4: Some te=ts and other rograms give the allowa)le de$rease and in$rease7from the original value8 rather than the uer and lower limits on the ranges.

Iterations

The iterations $an also )e dislaed. The ta)leau stle varies a$$ording to thete=t)oo: sele$ted.



So%ution Aist

It is also ossi)le to disla the solution in a listB as shown ne=tC

he dua% pro%e!

6nother window of results dislas the dual ro)lem.

Steppin

If ou loo: at the first s$reen at the to of this se$tionB ou will noti$e that to theleft of the SO2+ tool a ST+P tool aears.

0hile the iterations are availa)le in the iteration outut s$reenB it also is ossi)leto ste through and see the iterations one at a time. The ma@or advantage ofsteing is that &o' $an sele$t the entering varia)le. 0e have ressed ST+P andthe s$reen aears as follows. 7The disla varies a$$ording to the te=t set in "e%p,6ser Infor!ation.8



The software has $reated a simle= ta)leau adding two sla$: varia)les. The first$olumn is highlighted sin$e it has the highest rofit $ontri)ution. If ou want this$olumnB ress the ST+P tool. If ou want to $hange the ivot $olumnB siml $li$:on a different $olumn and then ress ST+P. 16fter one iteration the s$reen aearsas )elow.

0hen the otimal solution is foundB a message to that effe$t will aear in theinstru$tion )ar as shown )elow. Sin$e the software allows ou to iterate een after

finding the o!timal sol'tionB when ou are done ou must ress the ,I(ISH tool.

Aocation



There are four fa$ilit lo$ation models. The first module is the standardqualitative;su)@e$tive weighting sstem. Several fa$tors are identified that are$onsidered to )e imortant for the lo$ation de$ision. 0eights are assigned to thesefa$torsB and s$ores for these fa$tors are determined for the various ossi)le sites.The rogram $omutes the weighted sum of the s$ores 7and identifies the site withthe highest s$ore8.

The se$ond and third methods are quantitative methods for lo$ation on a line 7onedimensional8 or a lane 7two dimensional8. In the onedimensional $aseB the$oordinate or street num)er must )e givenF in the twodimensional $aseB )oth ahori?ontal $oordinate and a verti$al $oordinate must )e given. In either $aseB the rogram will have a default weight of 1 tri er lo$ationB )ut this ma )e $hangedto refle$t different num)ers of tris or different weights of materials. The rogramwill find the median lo$ation and the mean lo$ation and total weighted andunweighted distan$es from ea$h lo$ation.

NO4C Some of the lo$ation models $alled one and two dimensional lo$ationare also :nown as M$enter of gravitL models in some te=t)oo:s

The last model is siml )rea:even analsis alied to lo$ation ro)lems.

he Qua%itative BWeihtinC Mode%

If the qualitative model is $hosenB the general framewor: is given ) the num)erof fa$tors and the num)er of otential sites. In the s$reen followingB we show an

e=amle with seven fa$tors and three otential sites.

7actor weights. 0eights should )e given for ea$h fa$tor. The weights $an )e givenas whole num)ers or fra$tions. GenerallB weights sum to one or one hundredB )utthis is not a requirement.

Scores. The s$ore of ea$h $it on ea$h fa$tor should )e given.



49a!p%e /: Weihted %ocation Bua%itativeC ana%5sis

In the following s$reenB we disla a filledin samleB along with the solution. (oti$e that the $ities and the fa$tors have )een named.

The outut is ver straightforward and $onsists of the followingC

Total weighted score. ,or ea$h $itB the weights are multilied ) the s$ores forea$h fa$tor and summed. The total is rinted at the )ottom of ea$h $olumn. ,ore=amleB the s$ore for Philadelhia has )een $omuted asC

1/Q-/ V 3/Q/ V !Q"/ V 1!Q-/ V &/Q!/ V 1/Q4/ V !Q3/ W "!//

whi$h is listed at the )ottom of the Philadelhia $olumn.

The weighted average 7total s$ore;total weight8 is also dislaed for ea$h lo$ation.

One-Di!ensiona% Sitin

If onedimensional siting is $hosenB the general framewor: $onsists of a $olumn ofweights or tris and a single $oordinate or address $olumn. The requiredinformation in order to get started is the num)er of sites to )e in$luded in theanalsis.

The solution s$reenB whi$h in$ludes the data for a foursite analsisB is given in thefollowing s$reen.

The information to )e filled in isC

Weight<tri!s. The weight or num)er of tris to and;or from ea$h site. The defaultvalue is 1 for ea$h lo$ation. This is what should )e used when all



$ustomers;lo$ations are $onsidered to )e equal. If more tris are made to one$ustomer than anotherB this $an )e in$luded in the weight;tri $olumn. If thenum)er of tris is the same )ut the weight of the materials differsB this should )ein$luded.

4 coordinate. The $oordinate of the lo$ations must )e given. This $an )e e=ressedin several different was. These ma )e street addresses 7on the same street sin$ethis is one dimensional8B the ma )e floors in a )uilding 7it is ossi)le for thedimension to go u instead of a$ross8B or the ma )e eastwest or northsouth$oordinates where a negative num)er means west and a ositive num)er meanseast or a negative num)er means south and a ositive num)er means north.

Our samle ro)lem with a solution aears a)ove. The outut again is verstraightforward.

Total weight or n'mber of tri!s. In order to find the mean or median lo$ationB it isne$essar to determine the total num)er of tris or total weight. In the e=amleBthere are 13 total tris so the middle tri is the seventh.

The mean location. This is the lo$ation that minimi?es the sum of the squares ofthe distan$es of the tris.

The median tri!. The median tri is identified as tri num)er % and o$$urs fromthe lo$ation at &//.

In generalB an interesting question is whether a manager should minimi?e totaldistan$e or total distan$e squared. (oti$e in this e=amle that one ields an answerat )lo$: &// and the other ields an answer at )lo$: 31//3&//.

wo-Di!ensiona% Sitin

The information for twodimensional siting is analogous to the informationrequired for onedimensional siting. 6gainB the onl setu information is thenum)er of lo$ations. The following s$reen $ontains the data and solution for a



twodimensional siting ro)lem. The onl differen$e )etween the data for oneand twodimensional siting is the e=tra $olumn that now aears for the se$ond$oordinate data. *ata is to )e entered in the same wa as for one dimensionalsiting.

49a!p%e (: wo-di!ensiona% %ocation

The solution s$reen for twodimensional siting has a large amount of informationas e=hi)ited in the re$eding s$reen. Some of the information is the same as that ofone dimensional siting and some is e=tra.

Weighted 4-coordinate. This is siml the $oordinate multilied ) the num)er oftris. In the e=amleB the num)er of tris is ositive for ea$h of the first fivelo$ations )ut / for the last two. The multili$ations ) the weights $an )e seen.

Weighted &-oordinate. This is identi$al to the revious $olumn e=$et that it isthe & $oordinate that is multilied. These weighted $olumns demonstrate the$omutations that lead to the answers )elow the data. The averages of these$olumns are the answers. (oti$e in this e=amle that dividing ) % ields the first7unweighted8 average and dividing ) the sum of the weightsB whi$h is 3&/B ieldsthe se$ond 7weighted8 average.

Median. The median tri is 1"/ 7there is no 1"/.!8 and the median 4 $oordinate is13& while the median & $oordinate is %!.

/erages. The unweighted and weighted averages of the $oordinates are dislaed.

a%e of Distances

Ta)les of distan$es from oint to oint $an )e dislaed.



Total Distance. The row named MTotalL $ontains the total distan$e from ever siteto this site. The means for $omuting the distan$e deends on the ta)le 7airdistan$e vs. $it )lo$: distan$e8. The num)er --%.""! for raw material 1 meansthat the site of raw material 1 at $oordinates 13&B 1&3 has a total distan$e of

--%."% from ea$h of the other si= sites. That isB if ou made one tri from ea$h ofthe si= sites to the site of raw material 1B it would $over a distan$e of --%."%.6nother wa to view this $olumn is to sa that otential site 1 is more $entral than otential site & )e$ause it has a distan$e of -&1.%1B whi$h is smaller than the1/1.-- of otential site &.

Weighted total . The num)ers in the distan$e row do not ta:e into a$$ount thatdifferent num)ers of tris are made )etween oints or that different amounts ofmaterial are moved )etween oints. This $olumn multilies the distan$e times thenum)er of tris or amount of materials moved. 6gainB thoughB otential site 1seems to have the advantage over otential site &.

NO4: It is ossi)leB and ma)e even usefulB to solve the onedimensional ro)lem ) using the twodimensional model with one $oordinate equal to / for allsites.



rea7-even 0na%5sis

One more model is availa)le. This is siml a )rea:even analsis model sin$e )rea:even is alied to lo$ation ro)lem. 6n e=amle aears )elow. The$rossover oints will )e found and a grah is availa)le.



Aot Si8in

This model will erform lot si?ing for determining total holdingB setuB andsto$:out $osts when demands are not equal in ea$h eriod. Standard methods

in$lude the e$onomi$ order quantit 7+O58B eriod order quantit 7PO58B lotforlotB arteriod )alan$ing methodB and 0agner0hitin whi$h finds the otimals$hedule. 2ot si?ing is almost invaria)l dis$ussed in asso$iation with #RPsstems.

he data


0ee: *emand

>ul 11>ul 1>ul &!6ugust 16ugust 6ugust 1!

!&4-3

Holding $osts U& er unit er wee: and the $ost to set u a rodu$tion run is U&1.There is no initial inventorB nor is there a lead time.

6 data s$reen for our ro)lem aears ne=t. The data to )e given in$ludesdemands on the left and $osts and other information on the right of the ta)le.

Si= methods are availa)le in the method )o= a)ove the data.

1. 0agner0hitin finds the rodu$tion s$hedule whi$h minimi?es the total$osts 7holding V setu8.&. 2otforlot is the traditional #RP wa of ordering e=a$tl what is neededin ever eriod. 7This is otimal if setu $osts are /.8



3. The +O5 method $omutes the +O5 )ased on the average demand overthe eriod and orders in lots of this si?e. +nough lots are ordered to $overthe demand.

4. The eriod order quantit 7PO58 translates the +O5 into time units

7num)er of eriods8 rather than an order quantit. The PO5 is the length oftime an +O5 order will $overB rounded off to an integer. ,or e=amleB ifthe demand rate averages 1// units er eriod and the +O5 is &/ units erorderB the PO5 is 1//;&/ W ! eriods.

!. Parteriod )alan$ing. This is a well:nownB widelused heuristi$ that is$overed in man )oo:s.

". Aser defined. The user ma define the rodu$tion quantities.

Demands. The demands in ea$h eriod are to )e given. The demands are integers.

Prod'ce. This $olumn is used onl for the userdefined otion. +nter the num)erof units to )e rodu$ed. If an otion other than userdefined is $hosenB the rogramwill revise this $olumn and disla it as outut.

The information on the right in$ludesC

$olding cost. The $ost of holding one unit for one eriod is to )e entered here.The holding $ost is $harged against the inventor at the end of the eriod.

Shortage cost. The $ost of )eing short one unit for one eriod is to )e entered here.The shortage $ost is $harged against the inventor at the end of the eriod if theinventor is negative. *ue to lead time or under the userdefined otion it is ossi)le for the inventor to )e negative. 7,or e=amleB the user $ould define rodu$tion to )e / in ever eriod8.

NO4C 0e generall assume that the holding and shortage $osts are $hargedagainst the inventor that is on hand at the end of the eriod.

Set'! cost. This is the $ost of ea$h rodu$tion run. It is $harged onl in the eriodsthat have ositive rodu$tion.

*nitial inentor&. It is ossi)le to allow for a situation where there is )eginninginventor.



=ead time. This will offset the requirements and rodu$e n eriods earlier. 7See+=amle 38

49a!p%e /: 0 si9-period %ot si8in pro%e!

The solution for our e=amle is dislaed in the re$eding s$reen. The orderre$eit $olumn has )een derived ) the rogram. The e=tra $olumns that arederived $ontain the following informationC

*nentor&. This is the amount of inventor on hand at the end of the eriod. In thee=amleB there are si= units left after eriod 1B four units left after eriod &B andthree units on hand after eriod !. The holding $ost is $harged against these

amounts.

$olding cost . This is the $ost of holding inventor at the end of this eriod. It issiml the num)er of units on hand multilied ) the holding $ost er unitB whi$hin this e=amle is U&.

Set'! cost. This is U/ if no rodu$tion o$$urs or the setu $ost if rodu$tion o$$ursduring this eriod. In the e=amleB setus o$$ur in eriods 1B 4B and !B so the setu$ost of U&1 is listed in these three eriods )ut not in the other three eriods.Totals. The total inventorB holding $ostsB and setu $osts are listed at the )ottomof ea$h $olumn. Thirteen units were held for one month at a $ost of U&".//. Threesetus o$$urred at a total $ost of U"3.Total cost. The sum of the setu and holding $osts are dislaed in the )ottom lefthand $orner. The total $ost in this e=amle is U-. Sin$e we used 0agner0hitinBthis solution is otimal.

49a!p%e *: 6sin the 4OQ



One of the otions for la$ing orders is to use the e$onomi$ order quantit. The+O5 is $omuted )ased on the average demand over the eriods. In the e=amleBthe +O5 is )ased on the demand rate of 31 units er " eriods 731;" W !.1"%8.Asing the holding $ost and setu $ost with this demand generates an +O5 of 1/7after rounding8B as shown near the )ottom of the s$reen. The rogram will la$e

an order for 1/ units ever time that the inventor is insuffi$ient to $over thedemand. ,or e=amleB the first order for 1/ units is la$ed in eriod 1. This $oversthe demand in eriod 1 and the demand in eriod &. In eriod 3B we need anotherorder of 1/ units. Asing this method in the e=amle generates four orders 7whi$htotal 4/ unitsB not 31 units8 and a total $ost of U14&.

(ote that the +O5 method will li:el order more units than needed and thereforehave higher holding $osts than ne$essar.

49a!p%e (: 6sin the POQ

0e have modified our revious two e=amles ) adding an initial inventor of "units and a lead time of 1 wee:. 0e also have $hanged the method to the PO5.

One of the otions for la$ing orders is to use the eriod order quantit. The PO5is the +O5 )ut e=ressed in time rather than units. In our e=amleB the PO5 is the1/ units divided ) the average demand rate and rounded offB whi$h is two eriodsB as seen in the following s$reen. The rogram will la$e an order to $overever two eriods.



9e$ause there is a lead timeB the results s$reen in$ludes an e=tra $olumn for theorder release. ,or e=amleB the order due on >ul 1 must )e released on >ul 11due to this lead time. The order quantities are the same as without the lead timeB )ut the orders are released earlier due to the lead time. (oti$e that if we had used aone wee: lead time )ut not added the initial inventor to $over the first eriodBthen there would have )een an unavoida)le shortage in eriod 1.

49a!p%e 1: Aot-for-%ot orderin

2otforlot ordering 7not shown8 is ver straightforward and a $ommon wa for#RP sstems to oerate. The e=a$t amount demanded is alwas ordered. This isotimal if there is no setu $ost.



Mar7ov 0na%5sis

6 #ar:ov 'hain is des$ri)ed ) a transition matri= that gives the ro)a)ilit ofgoing from state to state. ,or e=amleB $onsider the followingC

,rom;To State 1 State & State 3

State 1State &State 3

.%

./!

./!

.1

.!

./!

.&

.1

.-

If we are in state 1B there is a %/ er$ent $han$e that we will )e in state 1 at thene=t stageB a 1/ er$ent $han$e that we move to state &B and a &/ er$ent $han$ethat we move to state 3. There are essentiall two tes of questions that need to )e answered for #ar:ov 'hains. One isC 0here will we )e after a small num)er of

stesX The other isC 0here will we )e after a large num)er of stesX Often timesthis deends on the state in whi$h we start.

The data s$reen for this e=amle is shown ne=t. The first $olumn 7MinitialL8 isindi$ating that we have an equal $han$e of starting in an of the three states. This$olumn does not have to $ontain ro)a)ilities as we show in +=amle &. The e=tradata a)ove the data ta)le 7num)er of transitions8 indi$ates that we want to loo: atthe results after 3 transitions.

Resu%ts

The results s$reen $ontains three different tes of answers. The to 3)3 ta)le$ontains the threeste transition matri= 7whi$h is indeendent of the starting

state8. The ne=t row gives the ro)a)ilit that we end in state 1 or & or 3B whi$h isa fun$tion of the initial state ro)a)ilities. The last row gives the longrun ro)a)ilit 7stead state ro)a)ilit8 or the er$entage of time we send in ea$hstate.



The following s$reen dislas the multili$ations through three transitions 7asrequested in the e=tra data )o= a)ove the data8.

49a!p%e *: 0 co!p%ete ana%5sis

'onsider the #ar:ov 'hain that is dislaed ne=t. The $hain $onsists of threedifferent tes of states. State 1 is a)sor)ingB states 3 and 4 together form a $losedBre$urrent $lassB while state & is transient. ,urthermoreB we are indi$ating that at the )eginning of this ro)lem there are "/B /B1//B and "/ 7total W 3//8 items in states1B &B 3B and 4 rese$tivel. 6s reviousl statedB the initial $olumn does not haveto $ontain ro)a)ilities.



The first outut ta)le isB as )eforeB des$ri)ing longrun )ehavior. The to of theta)le $ontains the longrun ro)a)ilities. The ending num)er row indi$ates thee=e$ted num)er 7in the statisti$al sense8 of how man of the original 3// itemswill end u in ea$h state. In this e=amleB we :now that the "/ that started in state1 will end in state 1 and the 1"/ that started in states 3 and 4 will end in thosestates 7divided evenl8. Of the / that started in state &B &.!% er$ent 7&&.!%8will end u in state 1B while the others will )e slit evenl over states 3 and 4.

The )ottom row of stead state ro)a)ilities all need to )e interreted as$onditional on the $losed re$urrent $lass that the states are in. ,or e=amleB firstnote that the do not sum to 1. These $lasses are identified in a se$ond oututs$reenB as shown )elowC

,inallB there is one more outut s$reen. This s$reen $ontains the usual #ar:ovmatri$es that are generated when erforming a #ar:ov 'hain analsis. The tomatri= is a sorted version of the original #ar:ov 'hain. It is sorted so that allstates in the same re$urrent $lass are ad@a$ent 7see states 3 and 48 and so that thetransient states are last 7state &8.



The 9 matri= is the su)set of the original matri= $onsisting of onl the transientstates.

The , matri= is given ) the equationC

,W 7I981

where I is the identit matri=.

,inallB the ,6 matri= is the rodu$t of the , matri= and the matri= formed )$ells that reresent going from a transient state to an nontransient state.



Materia% Reuire!ents P%annin

The material requirements lanning 7#RP8 model is used to determine rodu$tionrequirements for items whi$h are deendent.

he Data


The num)ers on the left of ea$h item indi$ate the num)er of su)$omonents thatmust )e used in the arent $omonent. 2ead times are 1 wee:B e=$et for item bBwhi$h has a &wee: lead time.

The framewor: for #RP is given ) the num)er of lines in the indented )ill ofmaterials and the num)er of time eriods. In our s$reenB whi$h followsBreresenting the e=amle a)oveB we show a ro)lem with % 9O# lines and

eriodsB as seen at the to of the s$reen. 17This is a good module to use the *o not*isl ero otion from the tool)ar or For!at menu8.

*tem names. The item names are entered in this $olumn. The same name willaear in more than one row if the item is used ) two arent itemsB su$h as iteme. (ote that as a ruleB names are unimortantB )ut in #RP names are e=tremelimortant. 'ase 7uer;lower8 does not matterB )ut sa$es matter ver mu$h.

*tem leel . The level in the indented 9O# must )e given here. The item $an not )e la$ed at a level more than one )elow the item immediatel a)ove. *o not use low



level $odes. 6lsoB lease note that it is ermissi)le to have more than one item atlevel / 7more than one end item8 as shown in +=amle &.

=ead time. The lead time in order to get the item is entered here. The default is 1.

8'mber 7`8 er arent. The num)er of units of this su)assem)l needed for its arent is entered here. The default is 1.

On-hand . The $urrent inventor onhand at the )eginning of the ro)lem is listedhere. If a su)assem)l is listed twi$eB it ma:es sense for the $urrent inventor toaear onl one time. HoweverB if it aears twi$eB the starting inventor will )ethe sum of the listed amounts 7see +=amle &8.

=ot si5e. The lot si?e $an )e se$ified here. 6 / or a 1 will erform lotforlotordering. If another num)er is la$ed hereB all orders for that item will )e in lotsthat are integer multiles of that num)er 7see +=amle &8.

Minim'm Q'antit&. It is ossi)le to se$if minimum order si?es 7see +=amle &8.

Demands 7entered under eriod 1 through eriod 8. The demands are entered foran& level / itemB in the wee: in whi$h the items are demanded.

Sched'led recei!ts. If units are s$heduled to )e delivered in the futureB the should )e listed in the aroriate time eriod 7$olumn8 and item 7row8 7see +=amle &8.

49a!p%e /: 0 si!p%e MRP e9a!p%e

6 samle data s$reen that e=resses the ro)lem aears in the re$edingillustration. The levels indi$ate that we have an item termed a, whi$h has two7level 18 su)$omonents named b and c. Su)$omonent b has two 7level &8su)$omonents named e and f. Su)$omonent c has two su)$omonents named dand e. (oti$e that e is a su)$omonent of )oth b and c.

The demand for the end itemB aB is 1&/ units in wee: % and 14/ units in wee: .The num)er of su)$omonents used is given in the num)ererarent $olumn. ,ore=amleB end item a $onsists of two su)$omonents b, whi$h in turn $onsists of1 e and & f s. 6t the )eginning of the ro)lemB there are no units of an :ind of

inventor onhand.MRP Product ree ?iewer



#RP has a rodu$t tree viewerB as shown )elow. 6n indented )ill of materials isdislaedC

Resu%ts

6 ortion of the results is dislaed in the following s$reen.

Total re9'ired . The total num)er of units required in ea$h wee: is listed in the firstrow. ,or the end itemB this is the demand s$hedule that was inut on the datas$reen. ,or other itemsB this is $omuted.

On-hand . The num)er onhand is listed here. This starts as given on the datas$reen and is redu$ed a$$ording to needs. 6 later e=amle will demonstrate onhand inventor.

Sched'led recei!t . This is the amount that was s$heduled in the original datas$reen 7see +=amle &8.

8et re9'ired . The net amount required is the amount needed after the onhandinventor is used. 6gainB $omonent c illustrates the su)tra$tion 7see +=amle &8.

Planned recei!t. This is the amount that will )e re$eived. It will )e the same as the

net required man timesB )ut it also ma )e larger due to minimum order si?e andlot si?e requirements 7see +=amle &8.

Order release. This is the net required )ut offset ) the lead time.



Printin

0e show a ortion of the rintout for the ro)lem for one main reason. The rintout of the inut is in slightl different form than the s$reen disla. (oti$ethat the software rints an indented )ill of materials.

+r!est Stu"e!t Stat 0%/MS&M 0$ Hoar" eiss

M,2+xamles2examle1.basi.mat 1%-15-%004 13,3%,%0

Mo"ule/submo"el, Material Reuireme!ts 'la!!i!

'roblem title, +xamle 1

)!"e!te" 6&M a!" Results ----------

)!"e!te" 6ill o7 Materials

)tem 8umber er &! ha!" 9ot Size

Mi!imum

)D 9ea"time are!t )!:e!tory (i7 ; 1#

ua!tity

a 1 1

b % %

e 1 1

7 1 % 1 3

" 1 5

e 1 4

Dema!"s 7or le:el 0 items

)tem )" = a

'erio" Dema!"

1 0

% 0

3 0

4 0

5 0



'roblem title, +xamle %

)!"e!te" 6&M a!" Results ----------

)!"e!te" 6ill o7 Materials

)tem 8umber er &! ha!"

9ot Size Mi!imum

)D 9ea" time are!t )!:e!tory

(i7 !ot lot ua!tity 7or lot#

a 1 1

b % %

300

e 1 1 10

7 1 %

144

1 3

" 1 5

e 1 4 %0

Dema!"s 7or le:el 0 items

)tem )" = a

'erio" Dema!"

1%0

140

)tem )" = "

'erio" Dema!"

$5

She"ule" reeits 7or all items hih are !ot e!" (le:el 0# items (i7

a!y#

)tem )" = 'erio" Reeit

% 00

a(lo le:el = 0#

<= " 0 "1 "% "3 "4 "5 "$ "

"

-------------------------------------------------------------------------

---------

&.R+. 1%0

140

&8 H?8D

Sh"R+>.

8+ R+ 1%0

140'la!R+>. 1%0

140

&RD R+9. 1%0 140

"(lo le:el = 0#

<= " 0 "1 "% "3 "4 "5 "$ "

"

-------------------------------------------------------------------------

---------



&.R+.

$5

&8 H?8D

Sh"R+>.

8+ R+

$5

'la!R+>.

$5&RD R+9. $5

b(lo le:el = 1#

<= " 0 "1 "% "3 "4 "5 "$ "

"

-------------------------------------------------------------------------

---------

&.R+. %40 %0

&8 H?8D $0

0

Sh"R+>.

8+ R+ %40 %%0

'la!R+>. 300 300

&RD R+9. 300 300

(lo le:el = 1#

<= " 0 "1 "% "3 "4 "5 "$ "

"

-------------------------------------------------------------------------------

---

&.R+. 3$0 4%0

&8 H?8D 00 00 00 00 440 %0

Sh"R+>. 00

8+ R+

'la!R+>.

&RD R+9.

e(lo le:el = %# <= " 0 "1 "% "3 "4 "5 "$ "

"

-------------------------------------------------------------------------------

---

&.R+. 300 300 %$0

&8 H?8D 30 30 30 30 30

Sh"R+>.

8+ R+ %0 300 %$0

'la!R+>. %0 300 %$0

&RD R+9. %0 300 %$0

7(lo le:el = %#

<= " 0 "1 "% "3 "4 "5 "$ "

"

----------------------------------------------------------------------------------

&.R+. $00 $00

&8 H?8D 1%0 @$ @$ @$

Sh"R+>.

8+ R+ $00 40

'la!R+>. %0 5$

&RD R+9. %0 5$



(oti$e the onhand inventor for item e. It )egins at 3/ and remains until it isneeded. (oti$e that for item c, a s$heduled deliver arrives in eriod & and thengoes into inventor until it is needed. (oti$e that for item f in eriod 4B "// unitsare neededB )ut the order is la$ed for %&/ units )e$ause it must )e a multile of144. (oti$e that for bin eriod "B the amount required is &4/B )ut 3// units are

ordered sin$e this is the minimum order si?e.



PO#5# for 0indows

Networ7s

Three models are availa)le in this moduleC minimum sanning treeB shortest athB

and ma=imal flow. 0e will use the following diagram for ea$h of our threee=amles. In order to start an of the three su)modelsB it is ne$essar to indi$atethe num)er of )ran$hes. In our e=amleB there are 14 )ran$hes. In order to enterea$h )ran$hB its starting node and its ending node must )e given.

Mini!u! Spannin ree

In the minimum sanning treeB we tr to $onne$t n nodes to ea$h other using Y1 ofthe availa)le ar$s. 6r$s have $ostsB and the goal is to minimi?e the total $ost. Thedata and solution to our e=amle aear in the following s$reenC

The data is the standard data of the ar$ or )ran$hB e=ressed as from and to nodenum)ers and the $ost of using the ar$. 6)ove the data is a )o= that ena)les the userto se$if the starting node num)er. If ou leave it as /B the lowest node num)er

1"&



PO#5# for 0indows

6)ove the dataB the networ: $an )e set to )e dire$ted or undire$ted. If it isundire$tedB the distan$e from node 2 to node i is set equal to the distan$e fromnode i to node 2. ,or e=amleB the distan$e from node & to node 1 is set to &/.

The solution is as followsC

,our )ran$hes should )e in$luded in the shortest athB $reating the ath 13"-Bwith a total distan$e of 113. In additionB the rogram $omutes the minimum totaldistan$e from ever node to ever other node as follows. To see the athB set thevalues for the start and end a)ove the data.

Ma9i!a% F%ow

In this situationB we want to ma=imi?e the flow from the )eginning 7sour$e8 to theend 7sin:8. The num)er along ea$h ar$ reresents its $aa$ities and the se$ondnum)er reresents its reverse $aa$it 7$aa$it in the oosite dire$tion8. 6t the

toB the sour$e and sin: $an )e set. If the are left at /B the sour$e is the node withthe lowest num)erB and the sin: is the node with the highest num)er. 9efore resenting our solution we remind ou that oftentimes more than one solutione=ists. 6lsoB there ma )e more than one wa to derive the solution. The ma=imalflow is "1B and the flows along the )ran$hes $an )e seen in the figure.

1"4



'hater "C #odules

The iterations are given in the following s$reen. Please note that there are generallseveral different iteration stes that $ould )e ta:en to arrive at the same ma=imalflow.

1"!



Productivit5

Produ$tivit is defined as the ratio of outut to inut. This software ena)les ou to$omute the rodu$tivit for an num)er of inuts and for an num)er of time

eriods. The software will $omute )oth the rodu$tivit measures for ea$h inutand also the $hanges in rodu$tivit from eriod to eriod for ea$h inut. InadditionB the software allows for $reating a $ommon denominator so that ou willend u with one rodu$tivit measure )ased on multile inuts. 6 samle s$reenthat in$ludes )oth the data and solution aears ne=t. The initiali?ation for this ro)lem requested three inuts 7la)or hoursB materials and inse$tion hours8 andtwo time eriods.

Data

@<+nit . This $olumn is used to aggregate all of the inuts into one meaningfulmeasure. ThusB we will $onvert our )asi$ inuts using rates of U;hour for la)orBU&;l) for materialB and U1&;hour for inse$tion $osts. ,or e=amleB in eriod 1 the

aggregate measure is Q4B///V&Q!B///V1&Q1B/// W !4B///.

Period A, B. The outut is entered in the first rowB and the inuts are entered in theremaining rows for ea$h eriod.

So%ution

Prod'ctiit&. ,or ea$h inutB the ratio of outut to inut is dislaed as the rodu$tivit for ea$h eriod. In additionB an aggregate measure is $reated using the$onversion fa$tors in $olumn 1. ThusB as mentioned a)oveB the denominator for eriod 1 isC

Q4/// V &Q!/// V 1&Q1/// W !4B/// whi$h ields a rodu$tivit of 1/B///;!4B/// or .1!&.

hange. 6 $olumn has )een added that relates the $hange from eriod to eriod forea$h of the rodu$tivit measures. ,or e=amleB the aggregate rodu$tivit hasde$reased ) %.1- er$ent from eriod 1 to eriod &.



ProKect Schedu%in

The ro@e$t s$heduling models are used to find the 7e=e$ted8 ro@e$t $omletiontime for either a P+RT 7a$tivit on ar$B 6O68 networ: or a 'P# 7a$tivit on nodeB

6O(8 networ:. ,or )oth networ:sB either one or threetime estimate ro)lems $an )e reresented or ro)lems with means and standard deviations for ea$h a$tivitma )e entered.

There are five models that are $ommon. These models $an )e $hanged withoutstarting anew ) using the method )o=.

Sin%e i!e 4sti!ate and rip%e i!e 4sti!ate P4R

'onsider a small ro@e$t given ) the following re$eden$e diagram and the ta)le

of times that follows the grah.

Tas: Start (ode +nd (ode Otimisti$ #ost 2i:el Pessimisti$

69'*+

1&334

&34""

&41/34

1&!&3!%

&!"&%-

Data

The following s$reen $ontains a triletime estimate P+RT data s$reen for oure=amle. In P+RT reresentationsB the networ: is defined ) giving the startingnode and ending node for ea$h tas:. The networ: te is given ) the )o= on the

left a)ove the data. In this first e=amleB we are using the start node;end nodereresentation for a$tivities.



Time. If the three time estimate version is usedB a single time estimate is $omutedand rinted for ea$h a$tivit. The formula used is the traditional formula of

t W 7a V 4b V c8;"B

where a is the otimisti$ timeB b is the most li:el timeB and c is the essimisti$time. ,or e=amleB in the s$reen the time used for a$tivit 1& is

7& V 4Q1& V &!8;" W %!;" W 1&.!.

0arl& start (0S). ,or ea$h a$tivitB its earl start is $omuted. ,or e=amleB theearl start for a$tivit 3" is 1%.!. The $olumn named MtimeL is used for this$omutation.

0arl& finish (07). ,or ea$h a$tivitB its earl finish is $omuted. In the e=amleB theearl finish for a$tivit 3" is &&.!. The earl finish isB of $ourseB the earl start lus

the a$tivit time. ,or e=amleB the earl finish of 3" is its earl start of 1%.! lusthe ! from the time $olumn.

=ate start (=S). ,or ea$h a$tivitB its late start is $omuted. In the e=amleB the latestart for a$tivit 3" is 3".

=ate finish (=7). ,or ea$h a$tivitB its late finish is $omuted.

Slac. ,or ea$h a$tivitB its sla$: 7late start earl start or late finish earl finish8is $omuted. In the e=amleB the sla$: for a$tivit 3" isC

41&&.! W 1.! or 3"1%.! W 1.!.

Standard deiation. ,or the threetime estimate model the standard deviation ofea$h a$tivit is listed. The standard deviation is given ) essimisti$otimisti$divided ) ". In the e=amleB the standard deviation of 3" is 7%38;"W."%.

Pro2ect com!letion time. The 7e=e$ted8 time at whi$h the ro@e$t should )e$omleted is given. In the e=amleB this time is 41.



Pro2ect standard deiation. If the threetime estimate module is $hosenB the ro@e$tstandard deviation is rinted. It is $omuted as the square root of the ro@e$tvarian$eB whi$h is $omuted as the sum of the varian$es of all $riti$al a$tivities.

(ote that in general there are ro)lems in defining the varian$e of ro@e$t

$omletion time. In additionB some )oo:s var in the manner of $omutation. This rogram will overestimate the standard deviation if there is more than one $riti$al ath. our te=t li:el does not e=lain what to do when more than one $riti$al athe=ists.

There is availa)le a ta)le that dislas the $omutations of the tas:sE timesBstandard deviationsB and varian$esB as illustrated in the following s$reenC

It is ossi)le to disla Gantt 'harts for the ro@e$tB as shown ne=tC

.PM BPrecedence %istC

The $riti$al ath module has the data inut in a fashion nearl identi$al to theassem)l line )alan$ing module. 'onsider the e=amle given in the following ta)leC



Tas: Time Pre$eden$es

design rogramdo$umenttestadvertise

&!3/&&1/3/

designdesign rogramdesign

The initial data s$reen aears as given )elowC

Tas names. Tas:s $an )e given names. The usual naming $onventions are true.That isB uer and lower$ase do not matter )ut sa$es within a name do.

Tas times. The tas: times are entered here.

Predecessors. The rede$essors are listed here. +nter one rede$essor ersreadsheet $ell with u to seven rede$essors er a$tivit. It is suffi$ient to enteronl the immediate rede$essors.

Rather than dislaing the solution to this ro)lemB we show the re$eden$e grahthat the software $an disla. 1On the s$reenB the $riti$al ath is dislaed in red.

.rashin



,ollowing is an e=amle of ro@e$t management with $rashing. The four $olumnsof data are the standard $olumns for this te of ro)lem the normal time andnormal $ost for ea$h a$tivit as well as the $rash time and $rash $ost for ea$ha$tivit. The $rash time must )e less than or equal to the normal timeB and the $rash$ost must )e greater than or equal to the $rash $ost.

The results are as follows. The software finds the normal time of 1" das and theminimum time of 1& das. ,or ea$h a$tivit the $omuter finds the $ost of $rashing er eriod 7$rash $ostnormal $ost8;7normal time$rash time8B whi$h a$tivitiesshould )e $rashed and ) how mu$hB and the rorated $ost of $rashing.

6 da)da $rash s$hedule is availa)le.



udetin

The software has a model for determining the amount of mone that will )e sentover a ro@e$tEs lifetime. The data is the a$tivit $ost as shown ne=t. 6n earlstart )udget and a latestart )udget $an )e $omuted.



The regular solution s$reen is givenB )ut two others are also availa)le. In the ne=ts$reenB we show a art of the earlstart )udget.

In additionB the grah $ontains the earlstart and latestart )udget for the entire ro@e$t.

Nor!a% Distriution

Pro@e$t management is an area where the normal distri)ution $al$ulator is useful asshown in the following s$reen. The mean and standard deviation from the ro@e$t7+=amle 18 are automati$all filled in. 0e have several otions in terms of whatwe ma want to $omute. ,or e=amleB we $an $omute the ro)a)ilit of finishingwithin !/ dasB or a -!Z $onfiden$e interval for finishing the ro@e$t.6lternativelB we $an $omute how man das to allow to )e -/ er$ent sure offinishing within that time. 0e have $hosen a -!Z $onfiden$e interval.



6fter ressing the ;.o!pute< )uttonB the solution aears as shown )elowC

0e are -! er$ent $onfident that the ro@e$t will )e $omleted in 31 to !1 das.



Qua%it5 .ontro%

This module $an )e used for the three ma@or areas of statisti$al qualit $ontrol a$$etan$e samlingB $ontrol $hartsB and ro$ess $aa)ilit. ,or a$$etan$e

samlingB )oth attri)utes and varia)les lans $an )e develoed. 6ttri)utes lans areused when the measurement is a defe$tive;nondefe$tive te of measurementB whilevaria)les lans are used for ta:ing a numeri$al result rather than siml a es;no. InadditionB the model $an )e used to $omute the rodu$erEs and $onsumerEs ris:under a given samling lan and;or to ma:e a $rude lot of the oerating$hara$teristi$ 7O'8 $urve. ,or $ontrol $hartsB it is ossi)le to develo !$harts forthe er$entage defe$tiveB =)ar $harts for the meanB or c$harts for the num)er ofdefe$ts. The s$reens for the first three otions are similarB and the s$reens for the$ontrol $harts are also similar.

0cceptance Sa!p%inThe e=a$t elements in the data s$reen deend on whether an attri)utes samling lan or a varia)les samling lan is sele$ted. In either $aseB the tes of datas$reens are ver similar. 0e )egin with the des$rition of the attri)utes data s$reenand resent the varia)les data s$reen later.

0ttriutes Sa!p%in

6 samle s$reen that in$ludes )oth the data and solution aears ne=tC

Data

/Q=. ,or a$$etan$e samlingB the 6$$eta)le 5ualit 2evel must )e given. The

652 must )e 7stri$tl8 greater than / and must )e less than 1. The interretationof ./1 is an 652 of 1 er$ent defe$tive.

=TPD. The 2ot Toleran$e Per$ent *efe$tive must )e entered. This has$hara$teristi$s similar to the 652. It must )e )etween / and 1.



/=P$/-The !rod'cerCs ris. The ro)a)ilit of a te 1 error $an )e set using thedrodown )o= to )e either 1 er$ent or ! er$ent in attri)utes samling. ,orvaria)les samlingB this entr is numeri$alB with a ma=imum allowa)le value of .--.

10T/ -The cons'merCs ris. The ro)a)ilit of a te & error $an )e set to )e 1

er$entB ! er$entB or 1/ er$ent for attri)utes samling. This entr is numeri$alBwith a ma=imum value of .1 for varia)les samling.

So%ution

6 samle ro)lem and solution s$reen aears in the re$eding s$reen. In thise=amleB we are tring to determine the aroriate samling lan when the 652is se$ified at 1 er$entB the 2TP* is se$ified at ! er$entB alha is ! er$entB and )eta is 1/ er$ent.

The sam!le si5e. The minimum samle si?e that meets the requirements a)ove is

determined and dislaed. In this e=amleB the aroriate si?e is 13%.

The critical al'e. The ma=imum num)er of defe$tive units 7attri)utes samling8 orthe ma=imum varia)le average 7varia)les samling8 is dislaed. In this e=amleBthe ma=imum allowa)le num)er of defe$ts in the & units is 3.

Two additional outut values aear on the rightB indi$ating that the a$tual ris:sdiffer from the se$ified ris:s. The rogram is designed to find the minimumsamle si?e that meets the requirements. The requirements $an )e more than metdue to the integer nature of the samle si?e and $riti$al value.

NO4: The $omutation of the a$tual ris:s is )ased on the )inomial distri)ution.

/ct'al !rod'cerCs ris. The rodu$erEs ris: in the inut is the uer level for theallowa)le rodu$erEs ris:. The a$tual rodu$erEs ris: $an )e less and is dislaed.In this e=amleB it haens to )e ./4-! whi$h is nearl the same as the ./! that wasset as inut.

/ct'al cons'merCs ris. The $onsumerEs ris: in the inut is the uer level for theallowa)le ris:. The a$tual $onsumerEs ris: $an )e less and is dislaed. In thise=amleB it haens to )e ./44B whi$h is less than the setting of .1 that was entered

as inut.



6n Oerating 'hara$teristi$ 7O'8 'urve $an )e dislaed as illustrated ne=tC

6n 6verage Outgoing 5ualit 76O58 'urve is also availa)le.

49a!p%e *: ?aria%es sa!p%in

(e=t we resent the data and outut for a varia)les samling lan. 0e want toa$$et the lot if the mean is &// ounds )ut re@e$t the lot if the mean is 1/ ounds.The standard deviation of the items rodu$ed is 1/ ounds. 0e are using ! er$entand 1/ er$ent for alha and )etaB rese$tivel. ,or varia)les samlingB alha and )eta are numeri$al rather than reset ) the drodown )o=. The ne=t e=amle willillustrate this more $learl.



The outut is ver similar to the outut for attri)utes samling. In this e=amleB weshould samle 3 items and weigh them. If the average weight is less than 1-.-"4 oundsB we should re@e$t the lot.

.ontro% .harts

The fourthB fifthB and si=th otions from the su)menu are used to develo $ontrol

$harts. Otion 4 is used when the er$entage of defe$ts is of interestB otion ! whenthere is a varia)le measurement and an 4)ar or range 7 9ar8 $hart is required. Thelast otion is for the num)er of defe$ts 7distri)uted as a Poisson random varia)le8.In an of these $asesB it is ne$essar to indi$ate how man samles there are.

49a!p%e (: 0 p-ar chart

The module )egins ) as:ing for the num)er of samles. In +=amle 3B whi$h isshown in the following s$reenB we filled in a $olumn of data indi$ating the num)erof defe$ts in ea$h of 1/ samles. 6lsoB we have as:ed for a 3sigma $ontrol $hart atthe to. The to indi$ates that the samle si?e for ea$h of these samles was 1!/.,inallB ou $an sele$t the $enter line ) using the s$roll)ar;te=t)o= $om)ination orleave it at / in whi$h $ase the software will use the mean.

The rogram has $omuted the average er$entage of defe$tsB whi$h is dislaed as3. er$ent. The standard deviation of !)ar is shown at the uer right as ./1!".



6 $ontrol $hart $an also )e dislaed and is e=hi)ited )elowC

49a!p%e 1 - an x -ar and rane chart

6fter the num)er of samles is enteredB there are two otions. +ither the raw data$an )e entered or the mean and range for ea$h samle $an )e entered. 0e willdisla )oth. In the e=amleB we disla the data and outut for the mean 7 4)ar8and range 7 )ar8 $harts. Si= samles of five items have )een ta:en and theirweights have )een re$orded. The first samle had an average weight of !"1.- ounds and a range of &!.3 ounds. The $ontrol $harts are set u )ased on therange. 7Some authors set u $ontrol $harts )ased on standard deviations rather thanranges.8 The mean and range $hart on the right are )ased on three standarddeviations. (oti$e that rather than setting the $enter line of the mean $harta$$ording to the overall mean we have set to a se$ifi$ation of !!! using thes$roll)ar;te=t)o= a)ove the data.



49a!p%e +: 6sin raw data in x ar and rane charts

The following s$reen resents the data for an e=amle with raw dataC

The software $omutes the mean and range for ea$h samle and then $omutes the$ontrol $harts.

49a!p%e 3: c-charts

The following s$reen $ontains a samle c$hart. The num)er of samles is enteredfollowed ) the num)er of defe$ts in ea$h samle. The rogram $omutes anddislas the defe$t rate 74.48B its standard deviation 7&./-%"8B and the $ontrol limits.



Process .apai%it5

,or ro$ess $aa)ilitB the uer and lower toleran$es for a ro$ess must )e set.OtionallB the mean ma )e set. If the mean is not set then the $enter oint

)etween the uer and lower toleran$es will )e used. ,inallB a standard deviationmust )e given. 9oth an uer and lower inde= are $omutedB and ro$ess $aa)ilitis the minimum of these two indi$es.



Re%iai%it5

The relia)ilit module will $omute the relia)ilit of simle sstems. If it is usedreeatedlB $omle= sstems $an )e develoed. This module $an easil )e used to

determine the aroriate num)er of )a$:u 7stand)8 ie$es of $omonents. Themodule has five su)models. The general framewor: for the first three is identi$alBand the framewor: for the last two is identi$al.

Data

The general framewor: for relia)ilit is given ) the num)er of simle sstemsthat are in series and the ma=imum num)er of $omonents in an simle sstem.9 a simle sstemB we mean that it is a set of arallel $omonents witho't anseries. In the following e=amleB we are setting u for a sstem with four simle

sstems in series. The largest num)er of $omonents in an of these four simlesstems is si=. There is onl one te data that needs to )e entered.

om!onent reliabilit&. The required information is the relia)ilit of ea$h$omonent. It is used for $omuting the relia)ilit of the simle arallel seriesreresented ) the $olumn.

NO4: This is a module where using the otion to not disla ?eros will ma:ethe data disla more reada)le.

So%ution

6 samle solution s$reen that also $ontains the data is given ne=t. (oti$e that therow in whi$h the ro)a)ilit is entered does not matterB as e=emlified ) the fa$tthat sstems 1 and & ea$h have the same relia)ilit of -- er$ent.

Sim!le s&stem reliabilit&. 9elow ea$h arallel sstem 7$olumn8B its relia)ilit is resented. The relia)ilitB r B of n $omonents in arallel is given )C



r W 1 71 r 1871 r &8...71 r n8

where r @ is the relia)ilit of the 2th individual $omonent. In the e=amleB the first arallel set has a relia)ilit of its one $omonentB whi$h is .--F the same is true forthe se$ond setF the third has an overall relia)ilit of .---!"B as listed at the

)ottom of the third $olumnF and the fourth has a relia)ilit of .---%&.

S&stem reliabilit&. The overall sstem relia)ilit is given at the )ottom. The overallrelia)ilit is the rodu$t of the individual arallel series relia)ilities. The e=amlehas a sstem relia)ilit of .-%-34.

49a!p%e *: Deter!inin the nu!er of ac7ups

It is ossi)le to $omute the num)er of )a$:us required in order to ensurese$ified sstem relia)ilit for a arallel sstem. ,or e=amleB suose that therelia)ilit of an individual $omonent is !/ er$ent and the desired sstem

relia)ilit is -- er$ent. ThenB ) $reating a ta)le with no )a$:usB 1 )a$:uB & )a$:usB et$.B the aroriate num)er of )a$:us $an )e found. 7This is anenumeration method8. ,or the relia)ilities se$ified as !/ er$ent and -- er$entwe see from )elow that the aroriate num)er of $omonents is seven 7si= )a$:us8.



Identica% Para%%e% or Identica% Seria% .o!ponents

6lternativelB a different su)model $ould )e used. The fourth and fifth su)modelsin the relia)ilit module ma )e used for $omuting the relia)ilit of arallelsstems with identi$al $omonents or sstems in series with identi$al $omonents.9elowB we show a s$reen for identi$al arallel $omonents. There are onl twoitems to )e entered.

8'mber in !arallel . In the e=tra data anel a)ove the data ta)leB there is as$roll)ar;te=t)o= $om)ination into whi$h we la$e the num)er of $omonents. Inthis e=amleB we are indi$ating that there are 1/ identi$al $omonents 7oneoriginal and nine )a$:us8.

om!onent reliabilit&. The data ta)le requires one ie$e of information. This isthe relia)ilit of the $omonents. In the e=amleB we have la$ed a .! indi$atingthat ea$h of the 1/ arallel $omonents has a relia)ilit of !/ er$ent.

So%ution

The solution indi$ates that the overall relia)ilit is .---/&3 whi$h agrees with themore detailed disla in the revious s$reen in $olumn 1/.



Si!u%ation

The simulation model is used to generate values from dis$rete ro)a)ilitdistri)utions or frequen$ ta)les. A to ten $ategories $an )e simulatedB and u to

1/B/// num)ers $an )e generated in ea$h e=eriment. The num)er and er$entageof o$$urren$es of ea$h $ategor are dislaedB and the generation of the num)ers$an )e viewed on a ste)ste )asis.

In order to generate a simulation ro)lemB it is ne$essar to rovide the num)er of$ategories for the data. In the following s$reenB we show a s$reen with )oth thedata and solution for a simulation of 1/ $ategories.

The elements of data areC

The n'mber of trials. This is the num)er of random num)ers to )e generated. Ato 1/B/// trials $an )e generated.

Seed. 0hen using simulationB a seed for the random num)er generator must )egiven. The default seed for the $omuter is /. If ou use the same seed or row or$olumn two timesB the same set of random num)ers will )e generated. In otherwordsB to run different e=eriments ou must reset the random num)er generation

ro$ess ) $hanging the seed.

NO4: ,or Hei?er;RenderB Talor or Render;Stair;Hanna users the followingrandom num)er generation method is availa)leC

andom n'mber generation method. There are two )asi$ was that the randomnum)ers $an )e generated. It is ossi)le to have the software generate random



num)ers and then $onvert them to the desired frequen$iesB or it is ossi)le to usethe random num)ers from a ta)le in a )oo:.

3al'e. The values for the varia)les are given here. In the e=amleB the are 1through 1/B )ut the $an )e an set of values. The are used for the $omutation

of the e=e$ted value.

ategor& fre9'encies. The frequen$ies for ea$h $ategor are entered here. Thesemust )e nonnegative )ut do not need to )e integerB nor do the need to sum toanthing se$ial 7su$h as 1 or 1//8B sin$e the rogram will total this $olumn ands$ale the results.

49a!p%e: Si!u%atin a freuenc5 ta%e

In the re$eding s$reenB we disla a 1/$ategor ro)lem and its solution. 6t thetoB it $an )e seen that we as:ed for !/ trialsB with the $omuter generating

random num)ers and the seed )eing 3. The solution in$ludes the followingC

Total. This is the total of the frequen$ $olumnB and as mentioned )eforeB is usedfor s$aling. In this $aseB we will divide the frequen$ies ) 3- in order to determinethe relative frequen$ies or ro)a)ilities.

Probabilit&. This $olumn reresents the s$aled frequen$ for ea$h $ategor given ) the frequen$ divided ) the total frequen$. ,or e=amleB $ategor & has arelative frequen$ of divided ) 3-B or &/.!1 er$ent.

'm'latie. The $umulative ro)a)ilit is needed to $onvert the uniform randomnum)er from the $omuter or )oo: to the aroriate relative frequen$. The$umulative ro)a)ilit is siml the running sum of ro)a)ilities. ,or e=amleB the$umulative ro)a)ilit for $ategor 3 is ./&!" V .&/!1 V ./%"- W .3/%%.

3al'e6fre9'enc&. This $olumn is used to $omute the weighted average ore=e$ted value of the given frequen$ distri)ution. In this e=amleB the $olumntotal is !.&/!1B whi$h is the weighted average of the two $olumns or the e=e$tedvalue of the distri)ution.

Occ'rrences. This is the $ount of the num)er of times this $ategor was generated.

The individual o$$urren$es $an )e seen ) dislaing the histor. In thise=erimentB $ategor 4 was generated three times. Percentages This is the o$$urren$es divided ) the total num)er of trials. ,ore=amleB the three o$$urren$es of $ategor 4 reresent " er$ent of the total of !/trials.



Occ'rrences6al'e. This $olumn is used to $omute the weighted average of thesimulated frequen$ distri)ution. In this e=amleB the $olumn total is &1B whi$hdivided ) the !/ runs ields a weighted average of !."&.

6 list of the !/ individual num)ers $an )e dislaed.

The first uniform num)er generated was .3% and this falls )etween .3/%% and .41/3B the $umulative for $ategor 4B so $ategor 4 is $hosen. The se$ond randomnum)er generated was .!33% and this falls )etween .4"1! and .!-%B so $ategor "is designated.



Statistics

The statisti$s module is used to $omute the mean or e=e$ted value or weightedaverage and standard deviation of a samle or oulation or a frequen$ ta)le or a

ro)a)ilit distri)ution. In additionB (ormal distri)ution $al$ulators ma )e doneusing this module. The (ormal distri)ution $al$ulator toolB dislaed in the Pro@e$t#anagement se$tion erforms the same $al$ulations as this model. It has a slightl )etter inut design 7)e$ause it is not restri$ted to )eing a ta)le8 )ut it $an not savefiles as this module $an.

0hen $reating a data set ou will )e as:ed in the lower left $orner of the $reations$reen a)out the te of data set that ou want. 0e will show e=amles for ea$hof the tes.

49a!p%e /: .o!putin statistics on raw data

6 samle s$reen that in$ludes a list of 1/ items and the solution aears )elow.3al'e. The first $olumn $ontains the numeri$al values 7 4i8

The ne=t two $olumns disla the $omutations that are used for the varian$e andstandard deviation. The last $olumn dislas the data in sorted order.

The results in$lude the meanB medianB modeB oulation 7divide ) n8 and samle7divide ) Y18 varian$es and standard deviationsB the minimumB ma=imum and

range. There also is a grah availa)le and a histogram whi$h $an )e $omuted forthe data.



49a!p%e *: Freuencies

In the e=amle )elowB we have entered groued data.

Mid!oint or al'e. This is the value to )e used for $omutations.



7re9'enc& or !robabilit&. The frequen$ies for ea$h $ategor are entered here.These must )e nonnegative )ut do not need to )e integerB nor do the need to sumto anthing se$ial 7su$h as 1 or 1//8B sin$e the rogram will total this $olumn ands$ale the results.

Total. This is the total of the frequen$ $olumn andB as mentioned )eforeB is usedfor s$aling.

Percent . This $olumn reresents the s$aled frequen$ for ea$h $ategorB given )the frequen$ divided ) the total frequen$.

3al'e6fre9'enc&. This $olumn is used to $omute the weighted average ore=e$ted value of the given frequen$ distri)ution. In this e=amleB the $olumntotal is 1&&B&1" whi$h we divide ) the num)er of o)servations 71B!/8 to derivethe mean of 1./4!1.

4i-4bar. In order to $omute the standard deviation we need to $omute the values 4i minus 4 )ar.

4i-4bar [&. The revious value is squared.

(4-4barB)6fi. The squared values are weighted ) the ro)a)ilities and summed.

.o!putin statistics for a proai%it5 distriution

9oth the data and the results are on the results s$reen )elow.

3al'e. This is the value to )e used for $omutations.



Probabilit&. The ro)a)ilities for ea$h $ategor are entered here. These must )enonnegative and must sum to 1.

Total. This is the total of the ro)a)ilit $olumn and )etter )e 1.

'm'latie. The $umulative ro)a)ilit is resented. The $umulative ro)a)ilit issiml the running sum of ro)a)ilities. ,or e=amleB the $umulative ro)a)ilitfor value 3 is .4 V.&! V.1! W ..

3al'e6!robabilit&. This $olumn is used to $omute the weighted average ore=e$ted value of the given frequen$ distri)ution. In this e=amleB the $olumntotal is &.3B whi$h is the weighted average of the two $olumns or the e=e$tedvalue or mean of the distri)ution.

-m'. In order to $omute the standard deviation we need to $omute the values 4i

minus 4 )ar.

-m'[&. The revious value is squared.

(-m'B)6!(4). The squared values are weighted ) the ro)a)ilities and summed.The varian$e for this data is &.11

he Nor!a% Distriution

ou need to indi$ate on the $reation s$reen whether ou want to $omute $utoffsgiven a ro)a)ilit or $omute ro)a)ilities given a $utoff or $utoffs

In our ro@e$t management e=amle we $omuted $utoffs given a -!Z ro)a)ilitso in this e=amle we will $omute the ro)a)ilit given $utoffs. The data and theresults aear in the results s$reen.



On$e againB a grah is availa)le.



he ransportation Mode%

This module is used to solve transortation ro)lems. Of $ourseB this module $an )e used to solve other ro)lems su$h as assignment ro)lems and rodu$tion

lanning ro)lems.

NO4: The 6ggregate lanning module $ontains a transortation model otion.

Data

The transortation ro)lem is stru$tured a$$ording to the num)er of origins in the ro)lem and the num)er of destinations.

Ob2ectie f'nction. 0hile minimi?ation is the usual o)@e$tive in transortationB

either minimi?e or ma=imi?e $an )e $hosen at the time that the data set is $reatedorB as usual at the edit s$reenB through the o)@e$tive )o= a)ove the data set.

'onsider the following e=amle. Our initial data s$reen for this !)" samle ro)lem followsC

(oti$e that a ST+P )utton aears on the tool)ar.

Ob2ectie. The o)@e$tive fun$tion $an )e $hanged in the usual otion method.

Shi!!ing costs. The main )od of information is the shiing $ost from ea$h originto ea$h destination.

(OT+C If ou enter an J=E for a $ell then a large shiing $ost 7U----;unit8 will )e la$ed in the $ell whi$h will effe$tivel eliminate the $ell from $onsideration inthe solution.



S'!!lies. The $olumn on the far right $ontains the sul at ea$h origin.

Demands. The demand row $ontains the demand at ea$h destination.

Starting method. ,our otions are availa)le in the method drodown )o=. TheareC

1. 6n method 7the software a$tuall uses ogelEs aro=imation method8&. (orthwest $orner method3. ogelEs aro=imation method4. #inimum $ost method. 76lso :nown as the intuitive method8

The otimal $ost isB of $ourseB indeendent of the initial methodB as is the otimalshiing s$hedule when there are no alternative solutions.

So%ution

6 solution to the samle ro)lem follows. The main solution s$reen shows theshiments that are to )e made and $ontains the total $ost in the uerleft $orner. Ifa dumm row or $olumn needs to )e addedB it will aear in this ta)le.

Total cost or !rofit. The total $ost or rofit aears in the uerleft $orner.

Marina% .osts

6 ta)le of marginal $osts is availa)le.



Steppin

0e $onsider the same ro)lem )ut $hange the solution method to (orthwest$orner. The first ta)le aears as )elowC

(oti$e that some of the num)ers are en$losed in arentheses while others are not.7On the s$reen the num)ers also disla in two different $olors.8 The num)erswithout a sign reresent the shimentsB while the num)ers with a sign reresent

the marginal $osts. The largest 7a)solute value8 marginal $ost is - in the $ell>en:intown to *umm 7whi$h is the $urrent $ell sele$ted ) the software8. 6lsonoti$e that the total $ostB whi$h is U31B/%B is dislaed at the to of the ta)le. oudo not have to use the entering $ell suggested ) the software. ou $an use thedire$tion :es to $hange the entering $ell.

Reeating this ro$ess five more times )rings us to the s$reen dislaed ne=t. Inthis s$reenB there is a message after the $ost indi$ating that the solution is otimaland we need to ress ,I(ISH. +verthing after this is as )efore if we ress a :eone more time. That isB we $an disla the shimentsB the marginal $ostsB or )othin one ta)le.

6nother otimal solution is given ) steing again.



Waitin Aines

There are man different waiting line situations that are des$ri)ed in PO# and5# te=t)oo:s. 0e $onsider standard models that isB singlehase queueing

models that do not allow feed)a$:B )at$h arrivalsB )at$h servi$eB )al:ing orreneging. #odels of this te are des$ri)ed ) a standard notation termedendallEs notationB although man te=t)oo:s avoid this ver $ommon notation.

Some queueing models allow for the determination of the average $ost ofoerating a queueing sstem where the $ost is the sum of the la)or $osts and thewaiting $osts as $harged against either the sstem time 7num)er in sstem8 or thewaiting time 7num)er waiting8.

Data

The framewor: for waiting lines deends on the se$ifi$ model to )e used. 0e$onsider nine modelsB and ea$h of these models $an )e used with or without $osts.In generalB the e=a$t data required will var as the model $hanges. The models are$hosen at the )eginning.

There are nine models availa)le. Some models are se$ial $ases of other models.In arti$ularB all of the singleserver models are se$ial $ases of the $orresondingmultileserver models. The models are listed )elow with their aliases. 6ll modelsassume a Poisson arrival ro$ess.

M<M<A e=onential servi$e timesB 1 server 7a.:.a. the single server model8

M<D<A $onstant servi$e timesB 1 server 7a.:.a. the $onstant servi$e model8 M<%<A general servi$e timesB 1 server M<0 <A +rlang: servi$e timesB 1 server M<M<s e=onential servi$e timesB 1 or more servers 7a.:.a. the multileserver model8

M<M<A with a finite queue 7or finite sstem8 si?e M<M<s with a finite queue 7or finite sstem8 si?e M<M<A with a finite oulation. M<M<s with a finite oulation

The first arameter in the notation refers to the arrival ro$ess. The M stands for#emorlessnessB whi$h means a Poisson arrival ro$ess. The se$ond arameterrefers to the servi$e ro$ess. The M again stands for memorless whi$h means thatthe servi$e times follow an e=onential distri)ution. The D stands fordeterministi$B whi$h is used when servi$e times are $onstant 7alwas the same8.The % stands for generalB and the 0 stands for +rlang: distri)ution. The third arameter is the num)er of servers 7also $alled $hannels8. (ote that s $an )e set toone in the M<M<s models to solve the M<M<A model.



Data

6 samle data s$reen aears ne=t.

/rrial rate (lambda8. +ver queueing sstem must have a $ustomer arrival rate.This num)er is a rate, whi$h means that a time unit 7hourB daB et$.8 is asso$iatedwith the arrival rate. This is $riti$al )e$ause the time unit must mat$h the time unit

of the ne=t arameter.

Serice rate (m'8. The num)er to )e entered is the rate at whi$h individual servers ro$ess $ustomers. (ote that this is a rate. That isB it is $ommon to :now theservi$e time. 9ut this time must )e $onverted to a rate and the time 'nit of this ratem'st match the time 'nit of the arrial rate.

8'mber of serers. The minimum and default value for the num)er of servers is 1.There are other inut arameters for the other models whi$h will )e e=lained inthe e=amles.

Time 'nit . There is a drodown )o= for the time unit. This serves two uroses.One is to remind ou that the arrival rate and servi$e rate must )oth )e )ased onthe same time unit. Se$ondB if ou sele$t hoursB then the outut disla will showMminutesL and Mse$ondsL. If notB the outut disla will show M"/Qtimes answer.L

49a!p%e /: he M/M/1 !ode%

'ustomers arrive at a rate of &" er hour a$$ording to a Poisson arrival ro$ess.There is one server who serves $ustomers in an average time of & minutesa$$ording to an e=onential distri)ution.The outut s$reen for this sstem follows. (oti$e that the arrival rate is entered as&"B as given in the ro)lem statement. The servi$e time of & minutes must )e$onverted to a rate of 3/ er hour.



/erage serer 'tili5ation. This is the er$entage of time that ea$h server is )uson average. In the e=amleB the one server is )us % er$ent of the time.

/erage n'mber in the 9'e'e (line). This is the average num)er of $ustomers whoare in the sstem waiting for servi$e. That isB the have not et )egun their servi$e.In the e=amleB there are !."3 $ustomers waitingB on average.

/erage n'mber of c'stomers in the s&stem. This is the average num)er of$ustomers who are either in line or )eing served. In the e=amleB there are ".!$ustomers in the sstemB on average.

/erage time in the 9'e'e (line). This is the average time that a $ustomer sendswaiting )efore servi$e )egins. The time unit is the same as that of the arrival andservi$e rates. In the e=amleB it is .&1"% ho'rs.

/erage time in the s&stem. This is the average time that a $ustomer sendswaiting and )eing served. In the e=amle it is .&! ho'rs

#an times we want to $onvert the average waiting and sstem times from hoursto minutes or from minutes to se$onds. The average times will )e multilied ) "/or 3"//B and the answers will show )eside the original averages. The num)ersthere e=ress the same time )ut with a unit of min'tes, sin$e the original timeswere in hours.

0e $an list the ro)a)ilities 7er$entage of time8 of e=a$tl $ustomers in thesstemB the $umulative ro)a)ilities of or fewer $ustomers )eing in the sstemand the de$umulative ro)a)ilit of stri$tl more than $ustomers in the sstem.

The s$reen will aear as follows. ,or e=amleB the ro)a)ilit that e=a$tl 3$ustomers are in the sstem is ./"B while the ro)a)ilit that 3 or fewer$ustomers are in the sstem is .43!. The ro)a)ilit that 7stri$tl8 more than 3$ustomers are in the sstem is .!"4&. (ote that these ro)a)ilities are availa)le forall models that have e=onential 7memorless8 servi$e times.



49a!p%e *: he M/D/1 !ode%$

0e have left the data the same as the revious e=amle )ut have $hanged themodel and solved the ro)lem. The solution s$reen followsC

The outut format is the same. 9e$ause the model has $hangedB some of the resultshave $hanged. In arti$ularB the num)er of $ustomers in line is &.1"% rather thanthe !."3 from the M<M<A sstem. 7The num)er in line and the time in line in an

M<D<A sstem are alwas half of those in an M<M<A sstem.8 Pro)a)ilities are notavaila)le sin$e the servi$e times are not e=onential.

49a!p%e (: he M/G/1 !ode%

In this modelB servi$e times ma have an distri)ution. The inut to the routine isnot onl the mean servi$e rate )ut also the standard deviation of the servi$e time.The following s$reen $ontains all of the information for this e=amle. (oti$e that



there is one e=tra row for the inut. The outut is the same. In the e=amleB themean rate is still 3/ $ustomers er hourB as )eforeB )ut the servi$e time standarddeviation is ./! ho'rs or 3 minutes.

6ll servi$e distri)utions are a se$ial $ase of the general distri)ution. 6 standarddeviation of 1;rate ields the e=onential distri)ution. ,or e=amleB sin$e theservi$e rate is 3/B if the servi$e time standard deviation is 1;3/ W ./3333B the

model has an e=onential servi$e time distri)ution.

This is shown in the re$eding s$reen. (oti$e that the answers are identi$al tothose in +=amle 1 e=$et for roundoff 7sin$e we used ./333 rather than 1;3/the=a$tl8.

6 standard deviation of / will ield the $onstant servi$e time 7 M<D<A8 model. Thisis dislaed ne=t. 'omare the results with +=amle &B whi$h dislaed the

M<D<A model.

49a!p%e 1: he M/E k/ 1 !ode%$



6nother availa)le servi$e time distri)ution is the +rlang distri)ution. The s$reen )elow e=hi)its the M<0 <A model and solution. The onl differen$e in the inut isthat the value for must )e given 7rather than no value or a standard deviation8.

0hen is oneB as in the following s$reenB then we have an e=onentialdistri)ution. 'omare the results with our first e=amle.

49a!p%e +: he M/M/s Queue

The most )asi$ question in queueing is what will haen if the num)er of serversis in$reased. In the s$reen )elow we show the outut for the original situatione=$et with two servers. 0aiting time is now .//%% hours rather than the .&1%hours in the original des$rition. To dou)le $he$: the original e=amleB ou $anuse the #;#;s model and enter 1 server.

49a!p%e 3: he M&M&/ s5ste! with a finite ueue



In this sstemB the num)er of waiting sa$es is finite. The ti$al e=amle is atelehone sstem. The e=tra line of inut to this model is the ma=imum allowa)lesstem si?e. (oti$e that we said s&stem not waiting . In the following e=amle wehave indi$ated that at most two $ustomers $an )e in the sstem. This means that nomore than one $an )e waiting with the se$ond )eing served. 7This is a singleline

hone with $all waiting.8 If there were two serversB it would mean that no one $an )e waiting. 9e $areful when $onsidering the sstem si?e versus the waiting areasi?e.

NO4C The model is termed a finite queue )ut it is the sstem si?eB not the queuesi?eB whi$h is entered into the rogram.

9e$ause the sstem si?e is limitedB it is ossi)le that $ustomers will arrive at thesstem )ut )e )lo$:ed from entering. ThereforeB we define the effe$tive arrivalrate as the a$tual num)er of $ustomers who enter the store rather than arrive at thestore. ,urthermoreB the outut dislas the er$entage of time 7ro)a)ilit8 that the

sstem is full.

In the e=amleB onl %1 er$ent of the $ustomers who show u enter the sstemB$ustomers show u at a rate of &" er hour )ut the effe$tive arrival rate is 1.!3- er hour. 0hen we disla the ro)a)ilities as followsB we see that &."- er$entof the time the sstem is full 7 W&8. That isB &."- er$ent of the time when a hone $all is made it re$eives a )us signal.

49a!p%e G: he M/M/1 s5ste! with a finite popu%ation



Ti$all we assume that the oulation is infinite. In the following s$reenB we aree=hi)iting a oulation of 13 otential $ustomersB each arriving at a rate of & times er hour 7for a net otential arrival rate of &"B as in the revious e=amles8. This isthe arrival rate when the are not in the sstem. HoweverB from the outut it $an )e seen that ea$h $ustomer averages ./ hours ea$h time he or she arrives. The

effe$tive arrival rate is the 1 arrival er hour times the average num)er of the 13who are not in the sstem. In this e=amleB the effe$tive arrival rate is onl &&.1$ustomers er hour 7rather than the otential of &" arrivals er hour8. If servi$ewere )etterB then these $ustomers $ould arrive more frequentl. The s$reenin$ludes the ro)a)ilit that a $ustomer waits. 7This is not the ro)a)ilit that allservers are )usB sin$e arrival rates var deending on the num)er in the sstem.8

NO4: In this modelB the arrival rate to )e entered into the rogram is the arrivalrate ,OR 6( I(*II*A62 'ASTO#+R. In man te=t)oo:sB the time )etweenarrivals is given. This time must )e $onverted to an arrival rate. ,or e=amleB itmight )e that ea$h of ! $ustomers shows u on average ever 3/ minutes. This

must )e $onverted to a rate of "/;3/ W & er hour 7er $ustomer8. The rogram willautomati$all ad@ust for the num)er of $ustomers. (oti$e that we enter the num)er& as the arrival rate. It is temting to enter &Q! W1/B )ut this is in$orre$t

49a!p%e J: he M /M/s ueue with costs

The ne=t s$reen $ontains an e=amle with $osts. 'ustomer $osts $an )e $hargedagainst either the time a $ustomer sends in the sstem or $harged against onl thetime waiting. 0e $harge U& for ea$h hour the $ustomer waits. This ields a total$ost of U&Q".! $ustomers in the sstem lus the U4; hour la)or $ost for a totalsstem $ost of U1% er hour 7)ottom line in ta)le8. 6lternativelB we might $hargeagainst the time a $ustomer waits. In this $aseB we have !."33 $ustomers waitingon average multilied ) U& for a su)total of U11.&" to whi$h we add the U4 server$harge ielding U1!.&%B as dislaed in the se$ond line from the )ottom.



Wor7 Measure!ent

This module $an )e used for the three ma@or areas of wor: measurementC time

studB $omutation of samle si?e for time studB and wor: samling

i!e Stud5

6 samle s$reen that in$ludes the data aears ne=t. Our ro$ess $onsists of threeelements and we have ta:en ! o)servations of ea$h.

Data

Performance rating . ,or ea$h elementB its erforman$e rating must )e given. Thenormal time will )e $omuted as the average time multilied ) the erforman$erating.

Obserations. The time o)served for ea$h element must )e entered. In some $asesBo)servations will )e )ad 7outliers8. In order to e=$lude them from $omutationsBenter a /B as in the $ase of o)servation & for element &.

/llowance factor. The overall allowan$e fa$tor is given. This allowan$e fa$torad@usts the final time for the sum of all three normal times.

So%ution

The solution s$reen for our e=amle aears )elow.

/erage. The average for ea$h element is $omuted. (oti$e that the average for

elements 1 and 3 are ta:en over ! valuesB )ut that the average for element & ista:en over 4 values sin$e o)servation & was given as / and this is not in$luded inthe averaging ro$ess.

Standard deiation. The standard deviation for ea$h element is $omutedBalthough it is not used for an further $omutations in this su)model.



8ormal time. The normal time is $omuted ) multiling the average of theo)servations ) the erforman$e rating for that element.

8ormal !rocessing time. The normal ro$essing time is the sum of the normaltimes.

Standard time. The standard time is $omuted in one of two was deending on

the te=t)oo:. Some authors useC

Standard time W normal ro$essing time Q 71 V allowan$e fa$tor8

while others useC

standard time W normal ro$essing time ;71 allowan$e fa$tor8

If ou are using a Prenti$e Hall te=t)oo: then the aroriate formula should )e inuse. If notB lease $he$: "e%p, 6ser Infor!ation to )e $ertain that the software is

listed as using the $orre$t te=t)oo:.

49a!p%e *: .o!putin the sa!p%e si8e

0e resent the data for our se$ond e=amle )elowC

The inut is similar to the time stud a)oveB )ut the goal is different. 0e want tofind the minimum samle si?e to )e --.4! er$ent $onfident of our results. Theinut for this su)model is as followsC



/cc'rac& leel . 0ithin what er$entage do we want our results to holdX ,ore=amleB for element & we want to )e -!.4! er$ent $onfident that our results arewithin 1 er$ent of the true time.

Obserations. This is the same as a)ove. The time o)served for ea$h element must

)e entered. In some $asesB o)servations will )e )ad 7outliers8. In order to e=$ludethem from $omutationsB enter a /B as in the $ase of o)servation & for element &.

onfidence. Si= otions a)out the $onfiden$e level are resented in the drodown )o= a)ove the data.

The outut aears )elow. The samle si?es for the three elements are &/B &-%Band 3 rese$tivel. GenerallB this means that we use the largest and have &-%o)servations of ea$h element.

49a!p%e (: Wor7 Sa!p%in

6n e=amle of )oth the inut and outut for wor: samling aears )elowC

Pro!ortion. This is the estimated roortion of time )eing sent in the tas:.

/cc'rac& leel . This is similar to the a$$ura$ level a)ove. 0ithin what er$entage do we want our results to holdX ,or e=amleB we want to )e --.%3 er$ent $onfident that our results are within ! er$ent of the true roortion.

onfidence. Si= otions a)out the $onfiden$e level are resented in the drodown )o= a)ove the data

The result is siml the samle si?e. In this $aseB we must samle %!" time units.



6endi$es0ppendi9 0$ .usto!i8ation due to te9too7

#odule 'ustomi?ation

*e$ision 6nalsis Row namesHurwi$? not in$luded in all te=t)oo:s

,ore$asting In$lusion of eriod 1 in error $omutation for

e=onential smoothing methods'omutations for e=onential smoothing withtrend model varies ) te=t)oo:

Inventor +O5 with shortages model not in$luded in allte=t)oo:sSafet sto$: model for (ormal distri)ution varies ) te=t)oo:

>o) sho s$heduling #ethod names (um)er of oerations $olumn not in$luded in all

te=t)oo:s2inear rogramming Simle= ta)leau disla

2o$ation #odel names

#aterials RequirementsPlanning

Ta)le disla 7order of rows8

5ualit $ontrol 6$$etan$e samling model not in$luded in allte=t)oo:s

Simulation Random num)er ta)le

0aiting 2ine #odel availa)ilit and names (otation

0or: measurement 'omutation of standard time varies ) te=t)oo:

0ppendi9 $ 6sefu% hints for !odu%es

Modu%e "e%pfu% "ints



6ggregate Planning This is a model where it ma )e useful to use themonths as the row name otion on the $reations$reenB rather than the default names of eriod 1Beriod &B et$. 6lsoB the 'o down )utton is usefulfor entering $onstant $aa$ities. In order to use the

transortation su)model for aggregate lanning the$osts $an not var from eriod to eriod. If the $ostsdo varB ou must set u the model ourself using thetransortation model.

6ssem)l 2ine9alan$ing

9e sure the time unit for the tas:s is set roerl.

6ssignment +nter = to have a large $ost 7----8 la$ed in the $ellto re$lude the assignment from )eing made.

9rea:even ,or ro)lems with & otions 6(* revenue simltreat revenue as a third otion.

*e$ision Ta)les Ase W in the ro)a)ilit row to set all ro)a)ilitiesto )e equal for the de$ision tree. (ote that in the 1 eriod inventor modelB the rofits on e=$ess units orshorted units $an )e negativeB that isB a loss.

,ore$asting The standard error is $omuted using n& in thedenominator.

Integer < mi=ed integer rogramming

It is not ne$essar to in$lude the nonnegativitrestri$tions 7e.g.B =^W /8 when $ounting the num)erof $onstraints. The )ottom left $ell has a dro down )o= whi$h $an )e used to set all varia)les to the samete.

Inventor The holding $ost in the +O5 )ased models $an )eentered as a num)er or as a er$ent. The entr .3/means 3/ $ents while the entr 3/Z means 3/Z ofthe unit $ost.

2inear rogramming It is not ne$essar to in$lude the nonnegativitrestri$tions 7e.g.B =^W /8 when $ounting the num)erof $onstraints. This is one of two modules where it is ossi)le to ste through the solution ro$edure.

#RP *ou)le$li$:ing on the data ta)le will disla the#RP Produ$t Tree. 6lsoB the do not disla /s

)utton $an )e ver useful in this module5ualit ,or $ontrol $hartsB ou ma use the average as the

$enter line or set the $enter line ourself.

Statisti$s This module in$ludes a (ormal *istri)ution$al$ulation su)model. The same $al$ulations $an )e erformed with the (ormal distri)ution tool 7tool)aror main menu TOO2S8 whi$h has a different 7)etter8user interfa$e.



Transortation +nter = to have a large $ost 7----8 la$ed in the $ellto re$lude units )eing shied from the original tothe destination. This is one of two modules where itis ossi)le to ste through the solution ro$edure.

0aiting 9oth arrivals and servi$e are given ) R6T+s rather

than TI#+s. 9e sure that the time unit for the arrivalrate and servi$e rate are the same.

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