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Technological Institute of the PhilippinesCollege of Engineering & Architecture

Industrial Engineering Department

Laboratory Manual

Advanced perations !esearch"IEP#$$L%

Ma.Teodora E. Gutierrez

Copyright 2014

ADVANCED OPERATON RE!EARC" #A$ORATOR% E&ERC!E!



PRE'ACE

The course focused on the advanced tools in operations research' The course is about

optimi(ation of the resources in the systems such as man) materials) methods) e*uipment)

energy) etc' It involves mathematical modeling of the systems being investigated and

identified the appropriate advanced tools in the field of operations research' It covers the

integer programming pure and binary) scheduling problem) *ueueing theories +ith different

model situations'

The laboratory e,ercises uses LI-D .oft+are and .imulation .oft+are to betterunderstand the systems being investigated' /sing the LI-D and Promodel .imulation

.oft+are) a +hat if scenario could be easily applied in the system'

The ob0ective of this manual is for the students to optimi(e the resources in the organi(ation

given the desired level of ob0ectives of the company' Moreover) it is the aim of the e,ercises

to ac*uire s1ills in modeling the system through LI-D and .imulation soft+are and be

able to interpret the results'

T2E A/T2!



CONTENTS

Preface PA3E -'

Laboratory E,ercise -o' % $a(i) Mode**i+g !tate,e+t4444444''' #

Laboratory e,ercise no' $ Mode**i+g !tate,e+t i+ +teger Progra,,i+g Pro-*e,( 44' 10

Laboratory exercise no. 3 Integer Programming Problems (Product Mix )

…………. 15

Laboratory exercise no. 4 . Pure $i+ary +teger Progra,,i+g Mode*( /+e(t,e+t

!e*e)tio+($5

Laboratory e,ercise no' 6 Pure +teger Progra,,i+g Mode*( Per(o++e* !)hedu*+g''''''$6

Laboratory exercise no. 6 $+ary +teger Progra,,i+g Mode* ''''''''''''''''

7%

Laboratory e,ercise no' 8 3ueue Theory 44444 78

Laboratory e,ercise no' 9 3ueue Theory 44444 ##

Laboratory e,ercise no' : 3ueue Theory 44444 #:

Laboratory e,ercise no' %5 3ueue Theory 44444 6#

Laboratory e,ercise no' %% 'i+a* Proe)t 5 Ca(e !tudy Co,pa+y 44444 ;5



#a-oratory E6er)i(e 1$A!C MODE##NG !TATEMENT

1. O-e)tie(7The activity aims to introduce the basic modeling statement in advanced operations research'

2. +te+ded #ear+i+g Out)o,e( #O(7The students shall be able to<

$'% Enter data into LI-D soft+are

$'$ Display optimum solutions

$'7 /pdate results graph by changing data in the model

8. Di()u((io+7LINDO (Linear, INteractive, and Discrete Optimizer) is a convenient, bt po!er"#

too# "or so#vin$#inear, inte$er, and %adratic pro$rammin$ prob#ems. &'ese prob#ems occr in

areas o" bsiness,

indstr, researc' and $overnment. peci*c app#ication areas !'ere LINDO 'as

proven to be o" $reatse !o#d inc#de prodct distribtion, in$redient b#endin$, prodction and

personne# sc'ed#in$,inventor mana$ement ,etc.I" o do not speci" ot'er!ise, LINDO assmes t'at variab#es are continos,

bonded be#o! b zero and nbonded "rom above. &'at is, variab#es can be an

non+ne$ative "ractiona# nmber increasin$ inde*nite#. In man app#ications, t'is

assmption ma not be rea#istic. ppose or "aci#ities #imit t'e %antit prodced

o" an item. In t'is case, t'e variab#e t'at represents t'e %antit prodced is

bonded "rom above. &'e "o##o!in$ sma## examp#es i##strate t'e se o" mode#in$ statements "or indo!s

versions and command+#ine versions o" LINDO'

4. Re(our)e(7Lindo /ser=s Manual



9. Pro)edure7Pro-*e, 1 7

The follo+ing small e,amples illustrate the use of the >!EE statement for the ?indo+s version and

command@line versions of LI-D<

< MI- 6 B .T B6 @8 E-D< >!EE Pro-*e,27

The follo+ing small e,amples illustrate the use of the 3I- statement for ?indo+s versions and


< MA %% B %5 .T $ B F %$ @ 7 % E-D< 3I- < 3I- < LG ALLProblem 7<7' The follo+ing small e,amples illustrate the use of the I-T statement for ?indo+s versions and


< MA @%55 B $5A B %$H .T A @ %5 F 5 A B H F %% H F 8 E-D< I-T Ma1e 5J%< LG ALL

:. Data a+d Re(u*t(7;ith 'ree %MI- 6 B .ub0ect ToBK6@K8



E-Dfree y

;ithout 'ree %MI- 6 B .ub0ect ToBK6

@K8E-D'or Pro-*e, 17

%' ?hat is the optimal solutionVARA$#E VA#<E

& :.000000% 51.000000

$' ?hat is the value of ob0ective function

If you remove the synta, statement >ree y and Loo1 all

O$=ECTVE '<NCTON VA#<E1 89.00000

7' ?hat is the optimal solution

VARA$#E VA#<E

& >.000000

% 0.000000#' ?hat is the value of ob0ective function

O$=ECTVE '<NCTON VA#<E

1 2?.00000

6' ?hat is your analysis on this

The >ree statement allo+s you to remove all bounds on a variable so that it may ta1e on any real

value ) positive or negative '

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Data 'or Pro-*e, 2

;ith GN !tate,e+tMA %% B %5



.T$ B F %$ @ 7 %E-D3I- 3I- ;ithout GN !tate,e+t

MA& 11& @ 10%!T2& @ % 12& 5 8% B 1END

'or Pro-*e, 2 7

%' ?hat is the optimal solution

VARA$#E VA#<E

& :.000000

% 0.000000


If you remove the synta, statement 3I- and 3I- and LG ALL


1 >2.429>


VARA$#E VA#<E

& 9.29>14

% 1.429>1

#' ?hat is the value of ob0ective function


1 ::.00000




According to LI-D) Hy default LI-D assumes that all variables are continuous) fractional

values may be of little use' 3I- statement restricts the value of the variable to the nonnegative

integers "5)%)$)4

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

Data 'or Pro-*e, 8

;ith NT !tate,e+t

MA @%55 B $5A B %$H.TA @ %5 F 5A B H F %%H F 8E-D

I-T Ma1e 5J%

;ithout NT !tate,e+t

MA @%55 B $5A B %$H.TA @ %5 F 5A B H F %%H F 8

E-D

'or Pro-*e, 8 7

%' ?hat is the optimal solution

VARA$#E VA#<E

& 1.000000

A 10.000000$ 1.000000


If you remove the synta, statement I-T and MAGE 5J%




1 124.0000


VARA$#E VA#<E

& 0.400000

A 4.000000

$ >.000000

#' ?hat is the value of ob0ective function


1 112.0000


I-T .tatements restricts a variable to being either 5 or %' These variables are often referred to

as binary variables'



>. Data A+a*y(i( a+d Co+)*u(io+7



Re*e)tio+ o+ the Attai+,e+t o +te+ded #ear+i+g Out)o,e( #O(7



8. ssessment (!ubric "or #aborator$ Per"ormance)%



#a-oratory E6er)i(e 2MODE##NG !TATEMENT N NTEGER PROGRAMMNG PRO$#EM!

1. O-e)tie(7The activity aims to construct integer programming problems of a sample problem"s and evaluate its optimal

solutions'


$'# Construct mathematical model of an integer programming problem"s

$'6 .et up and solve Integer Programming models using LI-D soft+are

$'; Display optimum solutions

$'8 /nderstand use integer variables in formulating problems'

$'9 Evaluate results of the model'

8. Di()u((io+7Inte$er pro$rammin$ is a so#tion met'od "or man discrete optimization prob#ems.

ome bsiness prob#ems can be so#ved on# i" variab#es 'ave inte$er va#es. -ir#ine

decides on t'e nmber o" i$'ts to operate in a $iven sector mst be an inte$er or

!'o#e nmber amont.Ot'er examp#es/&'e nmber o" aircra"t prc'ased t'is

ear0&'e nmber o" mac'ines needed "or prodction

&'e nmber o" trips made b a sa#es person0&'e nmber o" po#ice ocers assi$ned

to t'e ni$'t s'i"t0

P!O&#EM '%

&'e 2a#i"ornia an"actrin$ 2ompan is considerin$ expansion b bi#din$ a ne!

"actor in eit'er Los -n$e#es or an rancisco, or per'aps even bot' cities.It a#so

considerin$ bi#din$ at most one ne! !are'ose+(constraints no 1.) bt t'e c'oice

o" #ocation is restricted to a cit !'ere a ne! "actor is bein$ bi#t.(constraints no )



&'e net present va#e (tota# pro*tabi#it considerin$ time va#e o" mone) o" eac'

a#ternatives is s'o!n in t'e "ort' co#mn o" &ab#e 1. &'e ri$'tmost co#mn $ives

t'e capita# re%ired (a#read inc#ded in t'e net present va#e) "or t'e respective

investments, !'ere t'e tota# capita# avai#ab#e is 17 mi##ion. &'e ob8ective is to *nd

t'e "easib#e combination o" a#ternatives t'at maximizes t'e tota# net present va#e.

Table % Data of the e,ample problem

Decision

Nmber

9es or No

:estion

Decision

;ariab#e

Net <resent

;a#e

2apita#

=e%ired

1 >i#d actor

in Los

-n$e#es?

@1 A 6

>i#d actorin an

rancisco?

@ 5 3

3 >i#d

!are'ose in

Los -n$e#es

@3 6 5

4 >i#d

!are'ose in

san rancisco

@4 4

4. Re(our)e(7Introdction to Operation =esearc' b/ Bi##er and Lieberman

9. Pro)edure7%' Construct mathematical model of an integer programming problem"s

$' .et up and solve Integer Programming models using LI-D soft+are



;' Data and !esults<

"Copy and paste in this section the graphs and diagrams of the above problem"s +ith corresponding labels'

%' ?hat is the optimal solution$' ?hat is the value of ob0ective function


8' Data Analysis and Conclusion<







#aborator$ Exercise



INTEE! P!O!MMIN P!O&#EMS (P!O*+CT MI, )

'. Ob-ectie(s)%

T/e actiit$ aims to construct integer 0rogramming 0roblems o" a

sam0le 0roblem(s) and ealuate its o0timal solutions.

1. Intended #earning Outcomes (I#Os)%

T/e students s/all be able to%

2.? Construct mat/ematical model o" an integer 0rogramming

0roblem(s)

2.10 Set u0 and sole Integer Programming models using

#IN*O so"t2are

2.11 *is0la$ o0timum solutions

2.12 +se integer ariables in "ormulating 0roblems.

2.18 Ealuate results o" t/e model.. *iscussion%

Integer 0rogramming is a solution met/od "or man$ discrete

o0timi3ation 0roblems. Some business 0roblems can be soled onl$

i" ariables /ae integer alues. irline decides on t/e number o"

4ig/ts to o0erate in a gien sector must be an integer or 2/ole

number amount.Ot/er exam0les%T/e number o" aircra"t 0urc/ased

t/is $ear5T/e number o" mac/ines needed "or 0roduction

T/e number o" tri0s made b$ a sales 0erson5T/e number o" 0olice



o6cers assigned to t/e nig/t s/i"t5

P!O&#EM%

7$ndor lass Com0an$ 2ants to deelo0 t/ree 0ossible ne2

0roducts.

Product 1 Product 2 Product 3 Available

Hours per

week

Plant 1 3 4 37

Plant 2 4 6 47

Unit Proft 5 C 3

Sales

Potential

C 5 A

igure '9 *ata "rom 7$ndor lass Com0an$

ind t/e o0timum combination o" 0roducts to maximi3e t/eir 0ro:t

;. !esources%

Introduction to O0eration !esearc/ b$% <iller and #ieberman

=. Procedure%

'. Construct mat/ematical model o" an integer 0rogramming

0roblem(s)

1. Set u0 and sole Integer Programming models using #IN*O

so"t2are



>. *ata and !esults%

Max =x?@$?3

Sub-ect to% x?;$?1ABD

;x?>$?13B;D

xD

$D

#IN*O SOT7!E

ax 5xC3z

b8ect to

3x4EFG374x6zFG47

xHG7

HG7

zHG7

ND

!ES+#TS

O>J2&I; KN2&ION ;-LK

1) 57.77777

;-=I->L ;-LK =DK2D 2O& @ 7.777777 7.777777 9 5.777777 7.777777 E 5.777777 7.777777(Co0$ and 0aste in t/is section t/e gra0/s and diagrams o" t/e

aboe 0roblem(s) 2it/ corres0onding labels)

'. 7/at is t/e o0timal solutionF;-=I->L ;-LK

@ 7.777777 9 5.777777E 5.777777



1. 7/at is t/e alue o" ob-ectie "unctionF

HNECTIOE >/-CTI- OAL/E

% 65'55555

8. ;hat i( your a+a*y(i( o+ thi(

LI-D is a po+erful soft+are) has the ability to compute variables even the optimal solution is integer)

through commanding ,)y)( K5'

@. *ata nal$sis and Conclusion%









#a-oratory E6er)i(e 4P<RE $/NAR% NTEGER PROGRAMM/NG MODE#! /NVE!TMENT !E#ECT/ON!


solutions'


$'%# Construct mathematical model of an integer programming problem"s

$'%6 .et up and solve Integer Programming models using LI-D soft+are

$'%; Display optimum solutions

$'%8 /nderstand use integer variables in formulating problems'

$'%9 Evaluate results of the model'

8. Di()u((io+7Inte$er variab#es ma be re%ired !'en t'e mode# represents a onetime decision

(not an on$oin$ operation).

P!O&#EM '%

>ive pro0ects are being evaluated over a three year planning hori(on' The table gives the e,pected returns

for each pro0ect and associated yearly e,penditures'




9. Pro)edure77' Construct mathematical model of an integer programming problem"s

#' .et up and solve Integer Programming models using LI-D soft+are

Table %@ Data of >ive Pro0ects



:. Data a+d Re(u*t(7"Copy and paste in this section the graphs and diagrams of the above problem"s +ith corresponding labels'

;' ?hich pro0ect should be selected over the 7@year hori(on

8' ?hat is the optimal solution9' ?hat is the value of ob0ective function:' ?hat is your analysis on this










#a-oratory E6er)i(e 9P<RE NTEGER PROGRAMM/NG MODE#! PER!ONNE# !C"ED<#NG


solutions'


$'%: Construct mathematical model of an integer programming problem"s

$'$5 .et up and solve Integer Programming models using LI-D soft+are

$'$% Display optimum solutions

$'$$ /nderstand use integer variables in formulating problems'

$'$7 Evaluate results of the model'

8. Di()u((io+7Inte$er variab#es ma be re%ired !'en t'e mode# represents personne# sc'ed#in$.



P!O&#EM




9. Pro)edure76' !ead and understand the problem' .olve it in LI-D and ma1e a +rite up based on the result'

;' Construct mathematical model of an integer programming problem"s

8' .et up and solve Integer Programming models using LI-D soft+are













#a-oratory E6er)i(e :$NAR% NTEGER PROGRAMM/NG MODE#


solutions'


$'$# Construct mathematical model of an integer programming problem"s

$'$6 .et up and solve Integer Programming models using LI-D soft+are

$'$; Display optimum solutions

$'$8 /nderstand use integer variables in formulating problems'

$'$9 Evaluate results of the model'

8. Di()u((io+7Binary (or Zero - ne! inte"er pro"ra##in" proble#s concerns t'at a##

decision variab#es mst 'ave so#tion va#es o" eit'er 7 or 1 sc' as personne#

assi$nment.

P!O&#EM %






9. Pro)edure79' !ead and understand the problem' .olve it in LI-D and ma1e a +rite up based on the result'

:' Construct mathematical model of an integer programming problem"s

%5' .et up and solve Integer Programming models using LI-D soft+are













#a-oratory E6er)i(e >G+E+E T<EO!H

1. O-e)tie(7 &'e activit aims to i##strate t'e app#ications o" %ee t'eor in service sstem


.1 Desi$n sim#ation mode# o" t'e samp#e prob#em

. va#ate t'e res#ts o" t'e sim#ated mode# vs. t'e mana#

comptation

8. Di()u((io+7Gueuing T/eor$

O0erations !esearc/ (O!) tec/niues can also be used to a00roximate

0er"ormance metrics. I" t/e

sstem is simp#e eno$', a sim#ation ma not be necessar to even mae. Ksin$

tec'ni%es sc' as

%ein$ t'eor ma be an option a"ter $at'erin$ data "rom t'e sstem. - %ein$sstem is described

as a sstem !'ere a cstomer enters t'e sstem and 8oins a %ee. -t certain

times, a member o" t'e

%ee is se#ected to receive a service b some r#e, no!n as a %ee discip#ine.

&'e service is

per"ormed b a server or service mec'anism, a"ter !'ic', t'e cstomer #eaves t'e

sstem. :ein$

t'eor ses simi#ar data to sim#ation #ie interarriva# time data (#amda or M) andservice time data (m

or ) to determine sstem metrics sc' as t'e "o##o!in$/

LG xpected nmber o" entities in t'e sstem

L%Gxpected nmber o" entities in t'e %ee



G xpected tota# time in t'e sstem

%Gxpected !ait time in t'e %ee

<G<robabi#it t'at a speci*ed nmber, n, entities are in t'e sstem

-not'er metric t'at can be ca#c#ated sin$ M , is t'e tra""ic intensit "actor (r'o

or P).

&'ere are diQerent !as o" c#assi"in$ %ein$ t'eor mode#s. <er'aps t'e most

common mode# is t'e

1 mode#. &'is notation stands "or interarriva# distribtion tpe service

distribtion tpe nmber

o" servers. &'e RS notation stands "or a arovian or exponentia# distribtion.

Ot'er notations are RTS

"or $enera# distribtions or RDS "or deterministic distribtions. Ksin$ an 1 mode#

re%ires data

!it' an exponentia# distribtion and on# one server in t'e sstem. I" t'is is t'e

case, as !e## as a IO

%ein$ discip#ine, t'ese e%ations can be sed to estimate e metrics/

Exam0le Problem%

ppose parts arrive at a mi##in$ mac'ine ever 1 mintes, exponentia##distribted. &'e parts are

processed at a mean time o" 17 mintes, exponentia## distribted. 'at is t'e

expected nmber o"

cstomers t'e sstem and in t'e %ee? 'at is t'e expected !aitin$ time "or

cstomers, bot' tota#

time and in t'e %ee? 'at is t'e probabi#it t'at t'ere !i## be exact# 4

cstomers in t'e sstem?



4. Re(our)e(71.<romode# im#ation o"t!are0 . >oo / Introdction to Operations =esearc' b

Bi##ier and Lieberman

9. Pro)edure71. om t'e $iven prob#em, sim#ate t'e scenario sin$ <romode# im#ation.

. nter t'e data $iven "rom t'e prob#em into sim#ation sntax

3. 2ompare t'e compted res#ts vs. t'e sim#ation res#ts













#a-oratory E6er)i(e G+E+E T<EO!H




.4 va#ate t'e res#ts o" t'e sim#ated mode#.

8. Di()u((io+7 &'e "o##o!in$ are t'e sstems process in <o# rnitre actor

• ooden #o$s are received at t'e =eceivin$ Doc at t'e rate o" one #o$ ever

17mintes.

• Lo$s $o to t'e sp#itter !'ere "or pieces are made "rom eac' #o$. &'e

sp#ittin$ time is norma## distribted !it' a mean o" 4 mintes and a standard

deviation o"1 minte.

• &'e individa# pieces comin$ oQ o" t'e sp#itter are t'en sent to t'e Lat'e

!'eret'e are trned "or a time. &'e #at'e time is trian$#ar# distribted

!it' aminimm va#e o" 3 mintes, most #ie# va#e o" 6 mintes, and

maximm va#eo" A mintes. -"ter trnin$, t'e pieces are ca##ed =onds.

• =onds t'en $o to t'e <aintin$ >oot'. Kp to ten ronds can be p#aced in

t'epaintin$ boot'. ac' one is operated pon independent# and

• <ainted #o$s t'en $o on t'e tora$e area and t'e paintin$ time "or eac' one

isexponentia## distribted !it' a mean o" 5 mintes.

Transportation time from one area to another is considered to be 1 minute.






5. nter t'e data $iven "rom t'e prob#em into sim#ation sntax




1. 'at is t'e impact on <o# rnitre actorUs prodction rate and !or in

process b addin$ t'e drin$ Oven? Is t'is !'at o expected? ' or !' not?

ae sre o exp#ain t'e impact b identi"in$ t'e nmerica# c'an$es in

t'eres#ts.

. In t'e -rriva#s !indo! t'ere are t!o items, occrrences and "re%enc. 'at is

t'e diQerence and t'e re#ations'ip bet!een t'e t!o? 'at is t'e si$ni*cance o" t'e

R:t eac'S co#mn?

3. 'at are t'e varios time nits t'at o can se !'i#e deve#opin$ a mode#?

'ere do o provide t'is in"ormation?

4. &'e >i#dV<rocessin$ 'as t!o tab#es or !indo!s t'at need to be edited. 'at are

t'eir names and t'eir "nctions?

5. 'en an entit 'as comp#eted a## t'e processin$ in t'e sstem, !'at s'o#d !e

do !it' it? 'ere s'o#d it be roted?

6. Bo! are entities and #ocations diQerent?










#a-oratory E6er)i(e ?G+E+E T<EO!H




.6 va#ate t'e res#ts o" t'e sim#ated mode#.

8. Di()u((io+7 &'is is t'e "o##o!in$ contination o" t'e sstems process in <o# rnitre actor

!it' t'e "o##o!in$ restrictions/

1. Instead o" sendin$ re8ects bac "or re!or (!'ic' does not acta## mae

sense),at'orize more prodction b creatin$ additiona# Waxi#iarW !or

orders. Bo! do o mode# t'is? (<rovide t'e command(s) and an exp#anation

"or 'o! to mode# t'is).

. &r to better ba#ance t'e sstem b ad8stin$ capacities, arriva# patterns,

operator !or#oadtass, etc. 'o! or experiments and res#ts in tab#es. (xp#ain

reasons "or t'e steps o too in ba#ancin$ t'e sstem).

3. &reat t'e <o# rnitre actor sstem as a non+terminatin$ sstem and per"orm

experiments to/ determine t'e !arm+p period, ca#c#ate t'e avera$e I< and t'e

con*dence interva# at A5X con*dence #eve# using bot/ multi0le re0lications

and batc/ run. (ae sre o test t'e ato+corre#ation "or samp#es "rom batc'

rns.)

4. Determine t'e nmber o" rep#ications t'e mode# needs to be rn in order to



s'rin t'e con*dence interva# b 57 percent. (ae sre to s'o! t'e ca#c#ations

and veri" t'e res#ts b rernnin$ t'e mode#. Describe t'e steps o too i" t'e

process invo#ved iterative ca#c#ations and rns)




C. nter t'e data $iven "rom t'e prob#em into sim#ation sntax













#a-oratory E6er)i(e 10G+E+E T<EO!H



.C Desi$n sim#ation mode# o" t'e samp#e prob#em

.Y va#ate t'e res#ts o" t'e sim#ated mode#.

8. Di()u((io+7







9. Pro)edure7Y. om t'e $iven prob#em, sim#ate t'e scenario sin$ <romode# im#ation.

A. nter t'e data $iven "rom t'e prob#em into sim#ation sntax













#a-oratory E6er)i(e 11'i+a* Proe)t 5 Ca(e !tudy Co,pa+y

Cour(e7 E6peri,e+t No.7Group No.7 !e)tio+7Group Me,-er(7 Date Peror,ed7

Date !u-,itted7+(tru)tor7

1. O-e)tie(7The activity aims to apply several techni*ues of Advanced peration !esearch in an actual operations

of an e,isting company


$'% Analy(e the e,isting operations of the selected company

$'$ Apply Advanced peration !esearch techni*ues to the identified problem"s or goal"s'

$'7 Apply design of e,periments methodologies in the identified problem"s or goal"s$'# Communicate problems and solutions through +ritten and oral presentation

'

8. Di()u((io+%' .tudents +ill choose a practical problem for their final pro0ect' They +ill be e,pected in each

case to define a problem) collect data and present recommendations' >inal pro0ects +ill be

presented in +ritten and oral presentations'

$' Prepare a report paper in %6 $5 pages double space

4. Re(our)e(7

9. Pro)edure

7' Choose a company and select an area of study

#' Identified problemsJgoals

6' Collect pertinent data

;' /se appropriate modern tools in Advanced perations !esearch to solve the identified

problemsJgoals

8' Discuss in +ritten and oral presentations the detailed measurement of the identified system

:. Data a+d Re(u*t(7












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