Operations Research Lab Report

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LAB REPORT # 1

OBJECTIVE: Introduction to Operation Research

LITERATURE REVIEW:

THE ORIGINS OF OPERATIONS RESEARCH

Since the advent of the industrial revolution, the world has seen a remarkablegrowth in the size and complexity of organizations. The artisans small shops of anearlier era have evolved into the billion-dollar corporations of today. n integral partof this revolutionary change has been a tremendous increase in the division of laborand segmentation of management responsibilities in these organizations. The results

have been spectacular. !owever, along with its blessings, this increasing specializationhas created new problems, problems that are still occurring in many organizations."ne problem is a tendency for the many components of an organization to grow intorelatively autonomous empires with their own goals and value systems, thereby losingsight of how their activities and ob#ectives mesh with those of the overallorganization. $hat is best for one component fre%uently is detrimental to another, sothe components may end up working at cross purposes. related problem is that asthe complexity and specialization in an organization increase, it becomes more andmore difficult to allocate the available resources to the various activities in a waythat is most effective for the organization as a whole. These kinds of problems andthe need to find a better way to solve them provided the environment for the

emergence of operations research &commonly referred to as OR'.

The roots of "( can be traced back many decades, when early attempts weremade to use a scientific approach in the management of organizations. !owever, thebeginning of the activity called operations research has generally been attributed tothe military services early in $orld $ar )). *ecause of the war effort, there was anurgent need to allocate scarce resources to the various military operations and to theactivities within each operation in an effective manner. Therefore, the *ritish andthen the +.S. military management called upon a large number of scientists to apply ascientific approach to dealing with this and other strategic and tactical problems. )neffect, they were asked to do research on &military' operations. These teams of

scientists were the first "( teams. *y developing effective methods of using the newtool of radar, these teams were instrumental in winning the ir *attle of *ritain.Through their research on how to better manage convoy and antisubmarineoperations, they also played a ma#or role in winning the *attle of the orth tlantic.Similar efforts assisted the )sland ampaign in the acific.


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$hen the war ended, the success of "( in the war effort spurred interest inapplying "( outside the military as well. s the industrial boom following the war wasrunning its course, the problems caused by the increasing complexity andspecialization in organizations were again coming to the forefront. )t was becomingapparent to a growing number of people, including business consultants who hadserved on or with the "( teams during the war, that these were basically the sameproblems that had been faced by the military but in a different context. *y the early

/012s, these individuals had introduced the use of "( to a variety of organizations inbusiness, industry, and government.The rapid spread of "( soon followed. t least two other factors that played a keyrole in the rapid growth of "( during this period can be identified.

/- "ne was the substantial progress that was made early in improving thetechni%ues of "(

3- second factor that gave great impetus to the growth of the field was theonslaught of the computer revolution.

INTRODUCTION TO OPERATIONS RESEARCH

Operations research &"('4 is an interdisciplinary branch of applied mathematics thatuses methods such as mathematical modeling, statistics, and algorithms to arrive atoptimal or near optimal solutions to complex problems. )t is typically concerned withdetermining the maxima &of profit, assembly line performance, crop yield,bandwidth, etc' or minima &of loss, risk, etc.' of some ob#ective function. "perationsresearch helps management achieve its goals using scientific methods.

The term "perations (esearch &"(' describes the discipline that is focused on theapplication of information technology for informed decision-making. )n other words,"( represents the study of optimal resource allocation. The goal of "( is to providerational bases for decision making by seeking to understand and structure complexsituations, and to utilize this understanding to predict system behavior and improve


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system performance. 5uch of the actual work is conducted by using analytical andnumerical techni%ues to develop and manipulate mathematical models oforganizational systems that are composed of people, machines, and procedures. Thisarticle introduces some of the methods and application that are affiliated with "(,and elaborates on some of the benefits that may be gained by incorporating "( intothe actual business framework.

INTRODUCTION TO LINEAR PROGRAMMING

The development of linear programming has been ranked among the most importantscientific advances of the mid-32th century, and we must agree with this assessment.)ts impact since #ust /012 has been extraordinary. Today it is a standard tool that hassaved many thousands or millions of dollars for most companies or businesses of evenmoderate size in the various industrialized countries of the world6 and its use in othersectors of society has been spreading rapidly. The most common type of applicationinvolves the general problem of allocating limited resources among competingactivities in a best possible &i.e., optimal' way. 5ore precisely, this problem involvesselecting the level of certain activities that compete for scarce resources that arenecessary to perform those activities. The choice of activity levels then dictates howmuch of each resource will be consumed by each activity. The variety of situations towhich this description applies is diverse, indeed, ranging from the allocation ofproduction facilities to products to the allocation of national resources to domesticneeds, from portfolio selection to the selection of shipping patterns, from agriculturalplanning to the design of radiation therapy, and so on. !owever, the one commoningredient in each of these situations is the necessity for allocating resources toactivities by choosing the levels of those activities.

7inear programming uses a mathematical model to describe the problem of concern.The ad#ective linear means that all the mathematical functions in this model arere%uired to be linear functions. The word programming does not refer here tocomputer programming6 rather, it is essentially a synonym for planning. Thus, linearprogramming involves the planning of activities to obtain an optimal result, i.e., aresult that reaches the specified goal best &according to the mathematical model'among all feasible alternatives.

BASIC CONCEPT OF LINEAR PROGRAMMING


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"b#ective 8unction4 The "b#ective 8unction is a linear function of variables whichis to be optimized i.e., maximized or minimized. e.g., profit function, cost functionetc. The ob#ective function may be expressed as a linear expression.

onstraints4 linear e%uation represents a straight line. 7imited time, labor etc.may be expressed as linear in e%uations or e%uations and are called constraints.

"ptimization4 decision which is considered the best one, taking into

consideration all the circumstances is called an optimal decision. The process ofgetting the best possible outcome is called optimization.

Solution of a 74 set of values of the variables x/, x3,9.xnwhich satisfy all theconstraints is called the solution of the 7..

8easible Solution4 set of values of the variables x/, x3, x:,9.,xnwhich satisfy allthe constraints and also the non-negativity conditions is called the feasible solution ofthe 7.

"ptimal Solution4 The feasible solution, which optimizes &i.e., maximizes orminimizes as the case may be' the ob#ective function is called the optimal solution.)mportant terms onvex (egion and on-convex Sets.

MATHEMATICAL FORMULATION OF LINEAR PROGRAMMING PROBLEMS

There are mainly four steps in the mathematical formulation of linear programmingproblem as a mathematical model. $e will discuss formulation of those problems

which involve only two variables.

/- )dentify the decision variables and assign symbols x and y to them. Thesedecision variables are those %uantities whose values we wish to determine.

3- )dentify the set of constraints and express them as linear e%uations;ine%uations in terms of the decision variables. These constraints are the givenconditions.

:- )dentify the ob#ective function and express it as a linear function of decisionvariables. )t might take the form of maximizing profit or production orminimizing cost.


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LINEAR PROGRAMMING PROBLEMS (LLP)

EXAMPLE 1:

company produces two types of hats. !at = !at *, !at re%uires twice the labor

as !at *. )f the company produces only !at *, then it can produce a total of 122 hats

a day. The market limit for the daily sales of hat and !at * is /12 = 312 respectively.rofit on !at > ? rupees = !at * > 1 rupees.

SOLUTION:

1- ssume that @a> !at

@b > !at *

3- "b#ective 8unction4

5aximize profit4 A> ? @a B 1@b

:- onstraints4

@a C /12@b C 312 which implies that @a B @b C :22

E- 5aximum rofit4

A > ?&/22' B 1&:22'A > 3:22 rupees

EXAMPLE 2:


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7eather limited manufacturers produces two types of belts, *elt = *elt *, Fach

type re%uires / s%uare yard of leather. *elt * re%uires / hour of labor and *elt

re%uires 3 !our of labor. Fach week : rupees and profit on > < rupees

SOLUTION:

/- ssume that @a> *elt = @b > *elt *


5aximize rofit4 A> < @a B :@b

:- onstraints4

@a B @b C 32

E- 5aximum rofit4

Substituting values into A to get

A> 140

EXAMPLE 3:


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company owns a small paint factory that produces both interior and exterior paints

for whole sale distribution6 *asic raw material = * are used. The daily re%uirements

are shown in the table.

5arket survey shows that the daily demand for ). cannot exceed the F. by more

than / ton. The survey also revealed that the maximum demand for ). is limited to 3

ton only. The whole sale price per Ton is :222 for exterior paint and 3222 for interior

paint.

Fxterior aint

&F.'

)nterior aint

&).'

5aximum daily

availability &Tons'

(aw 5aterial, 5/ / 3 E

(aw 5aterial, 53 3 / ?

rofit er ton :222 3222

So!tion:

/- ssume that @e> Fxterior paint = @i > )nterior paint


5aximize rofit4 A> :222 @e B 3222 @i

:- onstraints4

@e B 3@i C E --- /

3@e B @i C ? --- 3

@i C @e B / --- :

@i C 3 --- /2;:, @i > /;:

E- 5aximum rofit4

Substituting values into A to get

A> :222&/;:' B 3222&


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LAB REPORT # 2

OBJECTIVE: Introduction to Linear Programming Problems(LPP

)

LITRATURE REVIEW:

INTRODUCTION TO LINEAR PROGRAMMING PROBLEMS

Linear pro%ra&&in%uses a mathematical model to describe the problem of concern.

The ad#ective linear means that all the mathematical functions in this model are

re%uired to be linear functions. The word programming does not refer here tocomputer programming6 rather, it is essentially a synonym for planning. Thus, linear

programming involves the planning of activities to obtain an optimal result, i.e., a

result that reaches the specified goal best &according to the mathematical model'

among all feasible alternatives.

Intro'!ction

The mathematical model which tells to optimize &minimize or maximize' the

ob#ective function A sub#ect to certain condition on the variables is called a 7inearprogramming problem &7'.

Linear Pro%ra&&in% Pro(e&s )LPP*

The standard form of the linear programming problem is used to develop theprocedure for solving a general programming problem.

general 7 is of the form

5ax &or min' A > c/x/B c3x3B 9 Bcnxnx/, x3, ....xnare called decision variable.

MATHEMATICAL FORMULATION OF LINEAR PROGRAMMING PROBLEMS
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There are mainly four steps in the mathematical formulation of linearprogramming problem as a mathematical model. $e will discuss formulation of thoseproblems which involve only two variables.

1. Identif t!e deci"ion $ri$%&e" $nd $""i'n "(%o&" ) $nd to t!e(.T!e"e deci"ion $ri$%&e" $re t!o"e *u$ntitie" +!o"e $&ue" +e +i"! todeter(ine.

2. Identif t!e "et of con"tr$int" $nd e),re"" t!e( $" &ine$re*u$tion"ine*u$tion" in ter(" of t!e deci"ion $ri$%&e". T!e"e con"tr$int"$re t!e 'ien condition".

. Identif t!e o%/ectie function $nd e),re"" it $" $ &ine$r function ofdeci"ion $ri$%&e". It (i'!t t$0e t!e for( of ($)i(iin' ,rot or ,roductionor (ini(iin' co"t.

3. Add t!e non-ne'$tiit re"triction" on t!e deci"ion $ri$%&e"4 $" in t!e,!"ic$& ,ro%&e("4 ne'$tie $&ue" of deci"ion $ri$%&e" !$e no $&idinter,ret$tion.

GRAPHICAL METHOD SOLUTION OF LINEAR PROGRAMMING PROBLEMS

The graphical method is applicable to solve the 7 involving two decisionvariables x/, and x3, we usually take these decision variables as x, y instead of x/, x3.To solve an 7, the graphical method includes two ma#or steps.

$5 T!e deter(in$tion of t!e "o&ution ",$ce t!$t dene" t!e fe$"i%&e

"o&ution 67ote t!$t t!e "et of $&ue" of t!e $ri$%&e )14 )24 )4....)n +!ic!"$ti"f $&& t!e con"tr$int" $nd $&"o t!e non-ne'$tie condition" i" c$&&ed t!efe$"i%&e "o&ution of t!e LPP5.

%5 T!e deter(in$tion of t!e o,ti($& "o&ution fro( t!e fe$"i%&e re'ion.

T!ere $re t+o tec!ni*ue" to nd t!e o,ti($& "o&ution of $n LPP. CornerPoint 8et!od $nd I9O- PROIT 6OR I9O-CO9T5.

Some Fxceptional ases

We ($ co(e $cro"" LPP +!ic! ($ !$e no fe$"i%&e 6infe$"i%&e5"o&ution or ($ !$e un%ounded "o&ution. If t!e inter"ection of t!econ"tr$int" i" e(,t $nd t!e ,ro%&e( !$" no fe$"i%&e "o&ution. T!erefore t!e'ien L.P.P !$" no "o&ution.
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LINEAR PROGRAMMING PROBLEMS SOLVED USING GRAPHICAL

METHODS


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CO88E7T94 OB9ERVATIO79 ; RE8ARr$,!ic$& "o&ution ,roed to %e c!$&&en'in' $t r"t %ut t!en $fter"o&in' ,ro%&e(" it %ec$(e er c&e$r t!$t >r$,!ic$& "o&ution of $Line$r Pro'r$((in' Pro%&e( i" $n e=cient +$ to nd t!e o,ti(u(

"o&ution to $ ,ro%&e(.

CONCLUSIONS:

T!e 'r$,!ic$& (et!od of "o&in' $n LPP i" ,o""i%&e on& if t!ere $re t+odeci"ion $ri$%&e" 6"$ ) $nd 5. T!i" (et!od i" not "uit$%&e if t!ere $ret!ree or (ore deci"ion $ri$%&e". In t!i" c$"e4 t!ere i" $ ,o+erfu&(et!od c$&&ed ?"i(,&e) (et!od?. T!e +ide u"$'e of &iner ,ro'r$((in'!e&," in %u"ine"" $nd econo(ic"4 to u"e t!e re"ource" $$i&$%&e in $

,&$nned $nd econo(ic$& +$. We !$e /u"t &e$rnt t!e %$"ic" of LPP@t!ere i" in f$ct $ &ot to &e$rn. A &ot of re"e$rc! +or0 i" c$rried $&& oert!e +or&d +!ic! i" %$"ed on LPP.

T!ere $re $ fe+ &i(it$tion" of &ine$r ,ro'r$((in'

1- Line$r ,ro'r$((in' i" $,,&ic$%&e on& to ,ro%&e(" +!ere t!econ"tr$int" $nd o%/ectie function $re &ine$r i.e.4 +!ere t!e c$n%e e),re""ed $" e*u$tion" +!ic! re,re"ent "tr$i'!t &ine". In re$&&ife "itu$tion"4 +!en con"tr$int" or o%/ectie function" $re not

&ine$r4 t!i" tec!ni*ue c$nnot %e u"ed.2- $ctor" "uc! $" uncert$int4 +e$t!er condition" etc. $re not t$0eninto con"ider$tion.


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LAB REPORT #

OBJECTIVE: Introduction to Simplex Method

LITRATURE REVIEW:

THE ESSENCE OF THE SIMPLEX METHOD

We c$n "o&e t+o $ri$%&e" LP (ode&" e$"i& u"in' t!e 'r$,!ic$&

(et!od out&ined in t!e ,reiou" "ection %ut +!$t "!ou&d +e do in c$"e of

t!ree $ri$%&e ,ro%&e("4 i.e. +!en our co(,$n ($0e" t!ree ,roduct" +e

!$e to ($0e deci"ion" $%out. W!$t $%out four or e $ri$%&e ,ro%&e("

T!i" i" +!ere t!e "i(,&e) (et!od co(e" in. It i" $n iter$tie (et!od +!ic!

% re,e$ted u"e 'ie" u" t!e "o&ution to $n n $ri$%&e LP (ode&.

T!e simplex method i" $ (et!od for "o&in' ,ro%&e(" in &ine$r

,ro'r$((in'. it !$" ,roed to %e $ re($r0$%& e=cient (et!od t!$t i" u"edroutine& to "o&e !u'e ,ro%&e(" on tod$" co(,uter". E)ce,t for it" u"e ontin ,ro%&e("4 t!i" (et!od i" $&+$" e)ecuted on $ co(,uter4 $nd"o,!i"tic$ted "oft+$re ,$c0$'e" $re +ide& $$i&$%&e. E)ten"ion" $nd$ri$tion" of t!e "i(,&e) (et!od $&"o $re u"ed to ,erfor( post optimalityanalysis 6inc&udin' "en"itiit $n$&"i"5 on t!e (ode&.

T!i" (et!od4 inented % >eor'e $nti' in 13D4 te"t" $d/$cent

ertice" of t!e fe$"i%&e "et in "e*uence "o t!$t $t e$c! ne+ erte) t!e

o%/ectie function i(,roe" or i" unc!$n'ed. T!e "i(,&e) (et!od i" er

e=cient in ,r$ctice4 'ener$&& t$0in' 2mto3miter$tion" $t (o"t 6+!ere m i"t!e nu(%er of e*u$&it con"tr$int"54 $nd coner'in' in e),ected ,o&no(i$&

ti(e for cert$in di"tri%ution" of r$ndo( in,ut". o+eer4 it" +or"t-c$"e

co(,&e)it i" e),onenti$&4 $" c$n %e de(on"tr$ted +it! c$refu&& con"tructed

e)$(,&e".

The advantages of the simplex method are


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Con"tr$int" .2PH.1P 1.2

.PH.P 2.3

P1 ; P2

Solution ()!

P-P1-P2G

P1HP2-PG fro( ro+ 65

.2P1H.P2H1PG 1.K fro( ro+ 615

.2P1H.1P2H1P3G 1.2 fro( ro+ 625

.P1H.P2H1PG2.3 fro( ro+ 65

Put non %$"ic $ri$%&e" P1;P2Go in $&& e*u$tion"4 to 'et

PG

P1G1.K4 P2G1.24 PG2.3

T!e ,rot i" ero4 t!e "o&ution i" not o,ti($&.

Solution ()!

ST"P (i)

P-P1-P2G Ro+

1P1H1P2H1PG Ro+ 1

1P1H11P2H1P3G Ro+ 2

1P1H1P2H1PG1 Ro+

ST"P (ii) ro( Ro+65 "e&ect t!$t non-%$"ic $ri$%&e +!ic! !$" ($)i(u(

ie $&ue i.e MP2N$nd t!en ,ut

P1GPGP3GPG in $&& ro+" e)ce,t ro+ 6

P2G P2G QQQ.fro( ro+ 615

11P2G P2G12 QQQfro( ro+ 625

1P2G12 P2GK QQQfro( ro+ 65

ST"P (iii) C!oo"e ro+ t!$t 'ie" (ini(u( $&ue of MPN i.e Ro+ 615 $nd

t!en ($0e coe=cient of MPN unit in t!$t ro+


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6115P1H1P2 H 615 PG 615 QQQ. 1 Ro+

615

625P1H1P2H 615 PG QQQ. Ro+

615

Re+rite $&& e*u$tion" re,&$cin' ro+ 615 % t!i" ne+ ro+

P-P1-P2G QQQQQ Ro+ 65

625P1HP2H 615 PG QQQQ. Ro+ 615

615PH 615 PHPG QQQQ Ro+ 625

615PH 65 1PHPG12 QQQ. Ro+ 65

ST"P (#) E&i(in$te c!oo"e $ri$%&e i.e MPN fro( $&& Ro+" e)ce,t Ro+615

P-P1-P2G QQ.. Ro+ 615

6125P1HP2H2PG QQ. Ro+ 615

P-P1 H2PG

P-P1H2PG QQQRo+ 65S

And 2P1 H P2 H 1P3G12 QQ.. Ro+ 6251

625 P1 P2 615 PG QQQ. Ro+ 615

635P1 - 615 P H 1P3G QQQQQQ Ro+ 625S

And P1HP2H1PG 23 QQQ.Ro+ 651

2P1P21PG1K

1P1-1PH1PG QQQ.Ro+ 65S

7o+ P-P1H2PG Q...Ro+ 65S

625P1H 615 PHP2G QQ.Ro+ 615S

635P1-615PH1P3G QQRo+ 625S

1P1-1PH1PG Q..Ro+ 65S

7o+ ,ut

P1GPG@ in $&& t!e"e Ro+" to 'et

PG@ P2G@ P3G@ PG


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T!e "o&ution i" not o,ti($& $" t!e o%/ectie function cont$in" ie $ri$%&e i.e MP1N

Solution ()

ST"P () P-P1H2PG QQ.QRo+ 65S

625P1H 615 PHP2G QQ.Ro+ 615S

635P1-615PH1P3G QQRo+ 625S

1P1-1PH1PG Q..Ro+ 65S

ST"P () "e&ect MPN $" it !$" ($) ie $ri$%&e in o%/ectie function $nd

t!en ,ut

P2GPGP3GPG in $&& ro+" e)ce,t Ro+ 65S

625P1G P1 G fro( ro+ 615S

635P1G P1 G3. fro( Ro+ 625S

P1G P1 G fro( Ro+ 65S

ST"P () C!oo"e Ro+ t!$t 'ie" (ini(u( $&ue of MPN i.e Ro+ 625S

$nd t!en ($0e coe=cient of MPN unit in t!$t Ro+ i.e Ro+ 625S

i.e

6335P1-6315 PH 6315 P 3G3 Q.... 635Ro+ 625S

P1-625 PH 6125 P3G 2 Q..Q..Ro+ 625S

Re+rite $&& t!e e*u$tion re,&$cin' Ro+ 625S % t!i" ne+ Ro+.

P-P1H2PG QQQRo+ 65S

P1-625 PH 6125 P3G 2 QQQ.Ro+ 625S@

625P1HP2H 615 PG QQQRo+ 615S

P1-1PH1PG ....QQRo+ 65S

ST"P (#) E&i(in$te c!o"en $ri$%&e i.e MP1N fro( $& ro+" e)ce,t Ro+ 625S

P-P1H2PG Q.ro+ 65S

P1-625 PH 6125 P3G2 ....Ro+ 625S

PH 625 PH 6125 P3GK12 Q..Ro+ 65N


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6225P1H 625 P2H 6215 PG 2 QQQ.Ro+ 615 )

2

P1-H 625 P 6125 P3G2 QQQQRo+ 62S5

625PH 6125 P-6125 PG2 QQ..Ro+ 65N

P1-1PH1PG Q..Ro+ 65S

P1-625 PH 6125 P3 G2 Q..Ro+ 625S

-6125 P-6125 P3H1 PG2 QQ..Ro+ 65Nno+

PH 625 PH 6125 P3GK12 QQRo+ 65N

625P2H 6125 P-6125 P3G2 QQRo+ 615N

-6125P-6125P3H615PG2 QQQRo+ 65N

P1-625PH6125P3G2 QQ..Ro+ 625N

Put PGP3G in $&& e*u$tion" +e 'et

PG3.@ PG@ PG3.@ PG.1

9ince o%/ectie function cont$in" no ie coe=cient "o it i" t!e o,ti($& "o&ution

T!e co(,$n +i&& 'et ($)i(u( ,rot of 3. if it ,urc!$"e" 3.3. ton fro( "ource

6I5 ; ton fro( "ource 6II5

7OTE:- 7ote t!$t "o&ution" continue in t!e "$(e ($nner unti& t!e o%/ectie function

cont$in" no ie ter(

"$AMPL" %O (&)!'

9u,,o"e t!$t $ co(,$n !$" t!e o,tion of c!oo"in' one (ore dierent t,e" of

,roduction ,roce""e". T!e 1"t; 2nd,roce"" ie&d" ite(" ,roduct MAN. rd;3t!ie&d

ite(" of ,roduct MBN. t!e in,ut of e$c! ,roce"" i" (e$"ured in ,ound of ($teri$& MN

$nd $rie" in it" in,ut re*uire(ent"4 t!e ,rot" ,roducin' t!e "$(e ite(. T!e

($nuf$cture decidin' on $ +ee0S" ,roduction "c!edu&e i" &i(ited in t!e r$n'e of

,o""i%i&itie" % t!e $$i&$%&e $(ount" of ($n,o+er $nd of %ot! t,e" of r$( +($teri$&. T!e fu&& tec!no&o' $nd in,ut de"cri,tion $re 'ien in t!e t$%&e %e&o+@

Ite( One ite( of ,roduct MAN One ite( of ,roduct

MBN

P1 P2 P P3

Tot$&

$$i&$%&e

8$n +ee0" 1 1 1 1 1


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&% of MN D 2 12

1 Bo) $" of

MXN

1 1 1

Unit ,rot 3 11

9u,,o"ed,roduction

&ee&

F1 F2 F F3

9o&ution:-

o%/ectie function ($)i(ie 3)1H)2H)H11)3

)1-)2H)H)3 1

D)1H)2H)H2)3 12 con"tr$int

)1H)2H1)H1)31

)14 )2 4) 4)3 Y

SOLUTION (I)

3)1H)2H)H11)3G) QQ Ro+ 65

)1H)2H)H)3H)G1 QQ.Ro+ 615

D)1H)2H)H2)3H)G 12 Q.Q Ro+ 625

)1H)2H1)H1)3H)DG 1 Q Ro+ 65

Put )1G)2G)G)3G in $&& e*u$tion" to 'et

)G4 )G14 )G124 )DG1

7o ,rot4 "o t!e "o&ution i" not o,ti($&

SOLTO% ()

Step (i) )-3)1-)2-)-11)3G ...Ro+ 65

F1H)2H)H)3H)G1 ...Ro+ 615

D)1H)2H)H2)3H)G12 ...Ro+ 625

)3H)2H1)H1)3H)DG1 QRo+ 65


20/73

Step () ro( Ro+ 65 "e&ect t!e non-%$"ic $ri$%&e +!ic! !$" ($)i(u(

ie $&ue i.e M)3N $nd t!en

Put )1G)2G)G)3G)G)G)DG in $&& ro+" e)ce,t ro+ 65

)3G1 fro( Ro+ 615

2)3G12 )3G fro( ro+ 625

1)3G1 )3G2 fro( ro+ 65

Step () C!oo"e t!e Ro+ t!$t 'ie" (ini(u( $&ue of M)3N i.e Ro+ 65 $nd t!en

($0e coe=cient of M)3N unit in t!$t ro+ i.e

615 )1H 615)2H6115)H)3H6115)DG11 6115 )

Ro+ 65

615)1H615)2H625)H)3H6115)DG2

Q..Ro+ 65

Re+rite $&& e*u$tion re,&$cin' Ro+ 65 % t!i" ne+ Ro+

)-3)1-)2-)-11)3G QQ...Ro+ 65

)1H)2H)H)3H)G1 QQQRo+ 615

D)1H)2H)H2)3H)G12 .QQ. Ro+ 625

615)1H615)2H625)H)3H6115)DG2 QQQ..Ro+ 65

ST"P (#) e&i(in$te c!o"en $ri$%&e i.e M)N fro( $&& Ro+" e)ce,t Ro+ 65

)-3)1-)2-)-11)3G QQQ.

ro+ 65

6115)1H6115)2H6225)H6115)3H61115)DG22 QQQ. 6115

) ro+ 65

)-65F1H635F2-65FH61115FDG 22 QQQ Ro+

65S

)1H ) 2H )H )3H )G 1 QQQ.Ro+ 615

615)1H615)2H625)H)3H615)DG 2 Q..QQ

Ro+ 65

Z 635)1H625)2H615)H )-6115)DG 2 QQ....Ro+ 615S


21/73

D)1H)2H)H2)3H)G 12 QQQQRo+

625

625)1H625)2H635)H2)3H625)DG3 QQQQ.2 )

Ro+65

65)1H615)2H65)H )-6215)DG 2

QQQ.Ro+62S5

635)1H625)2H615)H )-6115)DG 2 QQ....Ro+ 615S

7o+ )-65F1H635F2-65FH61115FDG 22 QQQ Ro+

65S

65)1H615)2H65)H )-6215)DG 2

QQQ.Ro+62S5

615)1H615)2H625)H)3H6115)DG2 QQQ..Ro+ 65

Put )1G)2G)G)DG in $&& Ro+" to 'et )G22 4 )G2 4

)G2 4 )3G2

T!e "o&ution i" not o,ti($& %ec$u"e o%/ectie function cont$in" ne'$tie ter("

Solution III

STEP I

F-65)1-635)2-65)H61115)DG 22 QQQ.Ro+6S5

635)1H625)2H615)-615)DH )G 2 QQQQ.Ro+61S5

65)1H615)2H65)-6215)DH )G 2 QQQQ.Ro+62S5

615)1H615)2H625)H6115)DH )3G 2 QQQQ.Ro+6S5

STEP II

ROE8 Ro+6S5 "e&ect t!e non-%$"ic $ri$%&e !$in' ($)i(u( ne'$tie $&ue i.e )

$nd t!en ,ut )2G)G)3G)G)G)DG

in $&& ro+" e)ce,t ro+ 6S5

635)1G2 )1G1212 fro( Ro+61S5

65)1G2 )1G1 fro( Ro+62S5

615)1G2 )1G1 fro( Ro+6S5

STEP III C!oo"e t!e Ro+ t!$t 'ie" (ini(u( $&ue of M)1N i.e Ro+61S5 $nd t!e

($0e coe=cient of M)1N unit in t!$t Ro+ i.e


22/73

635 [ 635)1H625)2H615)-615)DH615)G635 625

QQQ 635 ) Ro+61S5

)1H65)2H6125)H635)-61125)DG 1212

QQQ..Ro+61S5

Re+rite $&& e*u$tion" re,&$cin' Ro+61S5 % t!i" ne+ Ro+61S5

)-65)1-635)2-65)H61115)DG 22 QQQ.Ro+6S5

)1H65)2H6125)H635)-61125)DG 1212

QQQ..Ro+61S5

65)1H615)2H65)-6215)DH )G 2 QQQQ.Ro+62S5

615)1H615)2H625)H6115)DH )3G 2 QQQQ.Ro+6S5

STEP IV E&i(in$te c!o"en $ri$%&e i.e M)1N fro( $&& Ro+" e)ce,t Ro+ 61S5

)-65)1-635)2-65)H61115)DG 22 QQQ.Ro+6S5

65)1H625)2H635)H635)-625)DGD3

QQ........65 ) Ro+61S5

FH615)2-611125)H635)H6D125)DG1112 QQQ

Ro+6N5

65)1H 615) 2H 65) H )- 6215) DG2 QQ..

Ro+62S5

65)1H61125)2H61135)H635)-H61125)DG2D3 QQ.

65 ) Ro+61S5

-6D5)2-61125)-635)H )H6125)DG312

QQ..Ro+62N5

615)1H615)2H625)H )3H6115)DG2 QQQ

Ro+6S5

615)1H615)2H61125)H6135)-H615)DG212 QQQ615 )

Ro+61S5

615)2H6D125)H )3-6135)H61125)DG12 QQQ

Ro+6N5

7o+ FH615)2-611125)H635)H6D125)DG1112 QQQ

Ro+6N5

)1H65)2H6125)H635)-61125)DG 1212 QQ..Ro+61S5


23/73

-6D5)2-61125)-635)H )H6125)DG312

QQ..Ro+62N5

615)2H6D125)H )3-6135)H61125)DG12 QQQRo+6N5

Put )2G)G)G)DG in $&& Ro+" to 'et

)G 1112 4 )1G 1212 4 )G 312 4 )3G 12

9ince t!e o%/ectie function "ti&& cont$in" ne'$tie coe=cient "o t!e "o&ution i" not

o,ti($&.

SOLUTI0N IV

Re,e$t t!e "$(e ,roce""

A79WER )GD4 )1GD 4 )GD 4 )G2D

t!i" i" t!e o,ti($& "o&ution.

EXAMPLE NO 3:-

MINIMIZE G )1H3)2

"u%/ect to 3)1H)2 @ D)1H2)2 23

WERE )1; )2Y

Solution I; T!e MLPN (ode& of ,% i" t!e o%/ectie function

)1-3)2G QQQ..Ro+65

3)1H)2H)G QQ..Ro+ 615

D)1H2)2H)3G 23 QQQ..Ro+625

Put )1G)2G in $&& e*u$tion"

G

)G ; )3G 23

T!e "o&ution i" not o,ti($&4 $" no ,rot

Solution ii;

-)1-3)2G QQQ..Ro+65

3)1H)2H)G QQQ.Ro+615

D)1H2)2H)3G 23 QQQ.. Ro+ 625


24/73

STEP( I) ro( Ro+ 65 "e&ect t!e 7on-%$"ic $ri$%&e !$in' 8$)i(u( ne'$tie

V$&ue I.e M)2N

And t!en ,ut

)1G)G)3G in $&& Ro+" e)ce,t Ro+65

)2G fro( Ro+615

)2G12 fro( Ro+625

STEP (II); C!oo"e t!e Ro+ t!$t 'ie" (ini(u( $&ue of M)2N i.e Ro+ 615

7o+ ($0e t!e coe=cient of M)2N unit in Ro+ 615 i.e

635)1H )2H615)G

7o+ re+rite $&& t!e e*u$tion" 4 re,&$cin' Ro+615 % t!i" ne+ Ro+

-)1-3)2G QQQ.Ro+65

635)1H )2H615)G QQQQ..Ro+615

D)1H2)2H)3G23 QQQRo+625

STEP (III); "liminate t!e c!o"en $ri$%&e fro( $&& ro+" e)ce,t Ro+615

-)1-3)2G

615)1H3)2H635)G23 QQQ3 ) Ro+615

XH615)1H635)G 23 QQQRo+6S5

D)1H 2)2H ) 3 G23 QQQRo+625

65)1H2)2H625) G 12 QQQ2 ) Ro+615

62D5 )1- 625)H )3G 12 QQQQ. Ro+ 62S5

ence H615)1H 635)G 23 QQQ.Ro+6S5

635)1H )2H615 ) G QQQ..Ro+61S5

62D5)1-625)H ) 3G12 QQQQRo+62S5

STEP IV; ,ut )1G)G in $&& Ro+" to 'et

G234 )2G4 )3G12

T!e "o&ution i" o,ti($&4 "ince o%/ectie function cont$in" no ne'$tie $ri$%&e

EXAMPLE NO (4)

8I7I8IXE )1H2)2H2)


25/73

9u%/ect to

)1H)2H) 2

3)1H2)2H) 13

W!en )1; )2; )Y

SOLUTION (I);

)1H2)2H2)G) QQQQRo+65

)1H)2H)H)3G2 QQQQ..Ro+615

3)1H2)2H)H)G13 QQQQRo+625

Put 7on-%$"ic $ri$%&e )1G)2G)G in $&& Ro+" to 'et

)G4 )3G24 )G13

T!e "o&ution i" not o,ti($&4 $" no ,rot

Solution ()@

F-)1-2)2-2)G QQQQ..Ro+65

)1H)2H)H)3G2 QQQQQRo+615

3)1H2)2H)H)G13 QQQQ..Ro+625

STEP (I);ro( Ro+ 65 "e&ect $ 7on-%$"ic $ri$%&e !$in' ($)i(u( ne'$tie $&ue

i.e M)1N $nd t!en

)2G)G)3G)G in $&& e*u$tion e)ce,t Ro+ 65 to 'et

)1G2GK. fro( Ro+ 615

)1G133G. fro( Ro +625

STEP(II)@ C!oo"e t!e Ro+ t!$t 'ie" (ini(u( $&ue of M)1N i.e Ro+ 625

7o+ ($0e t!e coe=cient of M)1N unit in Ro+ 625 i.e

)1H6125)2H635)H6135)G133

Re+rite $&& t!e e*u$tion" re,&$cin' Ro+ 625 % t!i" ne+ Ro+

)-)1-2)2-2)G QQQ.Ro+65

)1H)2H)H)3G2 QQQ.Ro+615

F1H6125)H635)H6135)G133 QQQ..Ro+625

STEP (III); E&i(in$te t!e c!o"en $ri$%&e i.e M)1N fro( $&& ro+" e)ce,t Ro+625


26/73

F-)1-2)2-2)G QQQQ.Ro+65

)1H1)2H6D25)H6125)G1 QQQQ. ) Ro+625

F- K)2H61D25)H6125)G1 QQQQ.Ro+6S5

)1H)2H)H)3G2 QQQRo+615

)1H)2H6125)H625)G21 QQQ. ) Ro+625

2)2-625)H ) 3625)G1 QQQ. Ro+61S5

ence F- K)2H61D25)H6125)G1 QQQQ.Ro+6S5

2)2-625)H ) 3625)G1 QQQ. Ro+61S5

F1H6125)H635)H6135)G133 QQQ..Ro+625

,ut )2G)G)G in $&& Ro+" to 'et

)G14 )3G14 )1G133

t!e "o&ution i" not o,ti($& $" t!e o%/ectie function cont$in" ne'$tie

coe=cient

SOLUTION (3);

STEP (I); 9e&ect t!e $ri$%&e !$in' ne'$tie coe=cient fro( o%/ectie function

i.e M)2N

Put )1G)G)3G)G in $&& Ro+" e)ce,t Ro+6S5 to 'et

F2G12G1. fro( Ro+61S5

F2G61335 2 fro( Ro+62S5

STEP (II)@ C!oo"e t!e Ro+ t!$t ie&d" (ini(u( $&ue of M)2N i.e Ro+ 62S5

7o+ ($0e t!e coe=cient of M)2N unit in t!e c!o"en Ro+ 62S5 i.e

2)1H ) 2H 625)H6125)GD

7o+ re+rite $&& e*u$tion" re,&$cin' Ro+ 62S5 % t!i" ne+ Ro+

F- K)2H 61D25)H6125)G1 QQ..Ro+6S5

2)2-625 ) H ) 3-625)G1 QQQRo+61S5

2)1H ) 2H 625)H6125)GD QQQ.Ro+62S5

STEP (III); E&i(in$te t!e c!o"en $ri$%&e fro( $&& ro+" E)ce,t Ro+ 62S5

F- K)2H61D25)H6125)G1 QQQQ...Ro+6S5


27/73

1)1HK)2H2)H3)G QQQQQ. K ) Ro+ 62S5

FH1)1H 6D25)H6225)G11 QQQQ Ro+ 6N5

2)2-625) H )3-625)G1 QQQQQ. Ro+ 61S5

F1H2)2H)H)G13 QQQQQ. 2 ) Ro+ 62S5

3)1-6125)H ) 3-625)G1D QQQQQQ Ro+61N5

ence FH1)1H 6D25)H6225)G11 QQQQ Ro+ 6N5

3)1-6125)H ) 3-625)G1D QQQQRo+61N5

2)1H ) 2H 625)H6125)GD QQQ..Ro+62S5

STEP IV; Put )1G)G)G@ in $&& Ro+" to 'et

FG114 )3G1D4 )2GD

T!e "o&ution i" o,ti($&4 $" t!e o%/ectie function cont$in" no ne'$tie coe=cient.

LAB REPORT # 3


28/73

OBJECTIVE: Introduction to t!eig M Method

LITRATURE REVIEW:

THE BIG M METHOD TO SOLVE LINEAR PROGRAMMING PROBLEMS

In Preiou" ,ro%&e(" +e !$e "een t!$t $&& t!e con"tr$int" $re 65 +it!non-ne'itie ri'!t !$nd "ide"4 $nd oer $ conenient $&&-"&$c0 "t$rtin' %$"icfe$"i%&e "o&ution. 8ode&" cont$inin' 6G5 $ndor 65 con"tr$int" do not.

T!e ,rocedure for "t$rtin' "uc! 0ind of LP" i" to u"e arti*cial variables t!$t,&$ t!e ro&e of "&$c0 $t t!e r"t iter$tion4 $nd t!en +e di",o"e t!e( of $t $&$ter iter$tion.

T!e Bi' 8 8et!od

If $n LP !$" $n 65 or 6G5 con"tr$int"4 $ "t$rtin' %$"ic fe$"i%&e "o&ution ($not %e re$di& $,,$rent.T!e Bi' 8 (et!od i" $ er"ion of t!e 9i(,&e) A&'orit!( t!$t r"t nd" $%$"ic fe$"i%&e "o&ution % $ddin' \$rtici$&\ $ri$%&e" to t!e ,ro%&e(. T!eo%/ectie function of t!e ori'in$& LP (u"t4 of cour"e4 %e (odied to en"uret!$t t!e $rtici$& $ri$%&e" $re $&& e*u$& to $t t!e conc&u"ion of t!e "i(,&e)$&'orit!(.

9te,"

1- 8odif t!e con"tr$int" "o t!$t t!e R9 of e$c! con"tr$int i"nonne'$tie 6T!i" re*uire" t!$t e$c! con"tr$int +it! $ ne'$tie R9%e (u&ti,&ied % -1. Re(e(%er t!$t if ou (u&ti,& $n ine*u$&it %$n ne'$tie nu(%er4 t!e direction of t!e ine*u$&it i" reer"ed5.After (odic$tion4 identif e$c! con"tr$int $" $ 4 4 or G con"tr$int.

2- Conert e$c! ine*u$&it con"tr$int to "t$nd$rd for( 6If con"tr$int i i" $con"tr$int4 +e $dd $ "&$c0 $ri$%&e "4 $nd if con"tr$int i i" $ con"tr$int4 +e "u%tr$ct t!e "&$c0 $ri$%&e.

- Add $n $rtici$& $ri$%&e R1 to t!e con"tr$int" identied $" or Gcon"tr$int" $t t!e end of 9te, 1. A&"o $dd t!e "i'n re"triction R1 .

3- If t!e LP i" $ ($)i(i$tion ,ro%&e(4 $dd 6for e$c! $rtici$& $ri$%&e5

8R1 to t!e o%/ectie function@ +!ere 8 denote $ er &$r'e ,o"itienu(%er.- If t!e LP i" $ (ini(i$tion ,ro%&e(4 $dd 6for e$c! $rtici$& $ri$%&e5

8R1 to t!e o%/ectie function.- 9o&e t!e tr$n"for(ed ,ro%&e( % t!e "i(,&e) (et!od.

If $&& $rtici$& $ri$%&e" $re e*u$& to ero in t!e o,ti($& "o&ution4 +e !$efound t!e o,ti($& "o&ution to t!e ori'in$& ,ro%&e(.


29/73


30/73

7e)t "$(e $" 9I8PLEF 8ETO .

+st solution!

F G-24 1G 3 2G

T!e "o&ution i" not o,ti($& "ince o%/ectie function "ti&& cont$in" e co-e=cient.

&nd solution!

9te, I:

ro( R65SS "e&ect t!e non %$"ic $ri$%&e !$in' ($)i(u( e co-e=cient i.e. ]S)SS

; t!en ,ut )1 G)2G 1 G 2in $&& ro+" e)ce,t R65SS to 'et

F G3 fro( R615

F G 13 fro( R625

9te, II:

9e&ect t!e ro+ t!$t 'ie" (ini(u( $&ue" of ]S) ]Si.e. R625

7o+ ($0e t!e co-e=cient of ]S )1 ]S $" unit in t!i" ro+ i.e.

13 )1 H 13 )2 H ) H13 2 G ^

Re+rite $&& e*u$tion" re,&$cin' ro+625 % t!i" ro+

F -232)1 H K )2 -1 ) G-2 ------------R65SS

)1 - )2 H1 ) H 1 G3 ------------R615

13)1 H 13 )2 H ) H 132 G13 ------------R625

Step !

E&i(in$te t!e c!o"en $ri$%&e fro( $&& e)ce,t R625.

F -232)1 H K )2 -1 ) G-2 ------------

R65SS

113)1 H 13 )2 H1 ) H13 1 G13

------------R615_1

-D3)1 H 13 )2 H13 1 G-2 ------------R65SSS

-)1 -)2 H1 ) H1 G3 ------------R615

)1 H13)2 H1 ) H132 G13 ------------1_R625

)1 -13)2 -31 G13 ------------R615S


31/73

-131 -1)2 H1 -132 G13 ------------R615S

F -D3)1 H 13 )2 H 32 G-2 ------------R65SS

131 -1)2 -31 G13 ------------R615S

13)1 H13)2 H ) H132 G13 ------------R625S

Step #!

,ut )1 G)2G 2 G in $&& ro+" 'et

) G-24 )G 13.4 1 G-

t!e "o&ution i" not o,ti($& "ince t!e o%/ectie function "ti&& cont$in" e ter(.

,rd Solution!

9te, I:

ro( R65SS "e&ect t!e non %$"ic $ri$%&e !$in' ($)i(u( e co-e=cient i.e. ]S)SS

; t!en ,ut )2 G 1G) G in $&& ro+" e)ce,t R65SS to 'et

F1 G11 fro( R615S

F1 G 33 G1 fro( R625S

9te, II:

9e&ect t!e ro+ t!$t 'ie" (ini(u( $&ue" of ]S)1 ]SSi.e. R615S

7o+ ($0e t!e co-e=cient of ]S )1 ]S $" unit in t!i" R615S i.e.

)1 - 11 )2 1 1 G 11

Re+rite $&& e*u$tion" re,&$cin' ro+615S % t!i" ne+ ro+

F -D3)1 H 13 )2 -13 2 G-2 ------------R65SS

)1 -1 1 )2 -1 1 G11 ------------R615S

13)1 H 13 )2 H ) H 132 G13 ------------R625S

Step !

E&i(in$te t!e c!o"en $ri$%&eN)1N fro( $&& e)ce,t R615S.

F -D3)1 H 13 )2 H13 2 G-2 ------------R65SS

D3)1 -2D )2 - 2DD 1 G1222D1D ------------

R615S_D3


32/73

F - 11D )2 -2DD 1 H13 2 GKK1D ------------R65SSS

13)1 H13)2 H ) H131 G13 ------------R625S

13)1 -1D)2 -D1 G13D ------------

R615S_13

3)2 H) HD1HH132 G1D ------------R625SS

ence

F - 11D )2 -2DD 1 H13 2 GKK1D ------------R65SSS

6135)1 -1)2 -31 G13 ------------R615S

6135 )1 H13)2 H ) H132 G13 ------------R625S

635 )2 H) H6D51HH61352 G1D

------------R625SS

Step #!

,ut )2 G1G 2 G in $&& ro+" to 'et

) G-KK1D4 )1G 11.4 ) G1D

T!e "o&ution i" not o,ti($& "ince t!e o%/ectie function "ti&& cont$in" e coe=cient".

SOLUTION (4)

o our "e&f


33/73

EXAMPLE NO ()@

8$)i(ie -)1-2)2

9u%/ect to )1H )2G1

)1Y 3 +!ere )1; )2Y

SOLUTION; T!e "t$nd$rd MLPN fro( i"

-)1-2)2G)

)H)1H2)2G

)1H)2G 1

)1)G3

7o+ % 8-(et!od

)H)1H2)2H81H82G

)1H)2H1G1

)1-)H2G3

7o+ "ee +!ic! coe=cient or con"t$nt i" ($)i(u( in R615 i.e 1

9o 8G1

)H)1H2)2H11H2G QQQ.R65

7o+ e&i(in$te 1; 2fro( R65

)H)1H2)2H11H2G QQQ.R65

-1)1H- 1)2H-11G-1 QQ..1R615

F-D)1-K)2H12G-1 QQQR65S

F-D)1-K)2H12G-1 QQQR65S

1)1-1)H12G3 QQQ1R625

F-1D)1H1)-K)2G-13 QQQR65N

ence )-1D)1-K)2H1)G-13 QQQR65N

F1H)2H1G1 QQ.R615

F1-)H2G3 QQ.R625

1ST SOLUTION @ Put )1G)2G)G@ in $&& ro+" to 'et


34/73

FG-13@ 1G1 ; 2G3

9ince t!e ,rot i" ie@ 9o&ution i" not o,ti($& 6-ie ,rot "!o+" &o""5

&%- SOLTO%.

9te,615 ro( R65N "e&ect t!e 7on %$"ic $ri$%&e !$in' ($)4 -ie coe=cient i.e

M)1N $nd t!en ,ut

)2G)G1G2G@ in $&& ro+" e)ce,t R65N to 'et

)1G1 QQ. ro( R615

)1G3 QQ. ro( R625

ST"P(&)! C!oo"e t!e Ro+ t!$t 'ie" (ini(u( $&ue i.e R625

7o+ ($0e coe=cient of M)1Nunit in t!i" ro+4 +!ic! $&re$d e)i"t"

F1-)H2G3

ST"P(,): E&i(in$te t!e c!o"en $ri$%&e M)1N fro( $&& ro+" e)ce,t R625

F-1D)1-K)2H1)G-13 QQR65N

1D)1-1D)2H1DGK QQ1DR625

F-K)2-D)H1D2G-D2 QQ..R65N

F1H)2H1G1 QQ.R615

F1-)H2G3 QQ.R625

F2H)H1-2G QQR615S

ence )-K)2-D)H1D2G-D2 QQ.R65N

F2H)H1-2G QQR615S

F1-)H2G3 QQ.R625

ST"P(/)! Put )2G)G2G@ in $&& ro+" to 'et

FG-D24 1G4 )1G3

t!e "o&ution i" not o,ti($&4 "ince t!e o%/ectie function "ti&& cont$in" ie ter(

SOLTO%(,)!

Step(+) ro( R65N "e&ect t!e non %$"ic $ri$%&e !$in' ($)i(u( ie

coeecient i.e )2$nd t!en ,ut

F1G)G1G2G@ in $&& ro+" e)ce,t R65N to 'et


35/73

F2G Q..fro( R615S

F2G doe" not e)i"t in R625

ST"P(&) T!e coeecient of M)2N in R615S i" $&re$d unit

F2H)H1-2G

ST"P(,) E&i(in$te t!e c!o"en $ri$%&e M)2N fro( $&& ro+" e)ce,t R615 i.e

F-K)2-D)H1D2G-D2 QQ.R65N

K)2HK)HK1-K2G3K QQQKR615S

FH)HK1H2G-23 QQ...R65N

ence FH)HK1H2G-23 QQ...R65N

F2H)H1-2G QQR615S

F1-)H2G3 QQ.R625S

ST"P(/) Put )G1G2G@ in $&& ro+" t, 'et

FG-234 )2G4 )1G3

T!e "o&ution i" o,ti($&4 $" t!e o%/ectie function cont$in" no ie ter(

#eri*cation Put deter(ined $&ue" in e*u$tion R65

FH)1H2)2H11H12 G

-23H635H265H165H165 G

-23H23 G

G

ence eried


36/73


37/73

THE TRANSPORTATION ALGORITHM MODEL

T!e (ode& for $ tr$n",ort$tion ,ro%&e( ($0e" t!e fo&&o+in' $""u(,tion$%out t!e"e "u,,&ie" $nd de($nd".

The re1uirements assumption! E$c! "ource !$" $ )ed "u,,& ofunit"4 +!ere t!i" entire "u,,& (u"t %e di"tri%uted to t!e de"tin$tion".6We &et "idenote t!e nu(%er of unit" %ein' "u,,&ied % "ource i4 for i G14 2 ... (.5 9i(i&$r&4 e$c! de"tin$tion !$" $ )ed de($nd for unit"4

+!ere t!i" entire de($nd (u"t %e receied fro( t!e "ource". 6We &et d/denote t!e nu(%er of unit" %ein' receied % de"tin$tion /4 for /G 142 ... n.5T!i" $""u(,tion t!$t t!ere i" no &ee+$ in t!e $(ount" to %e "ent orreceied (e$n" t!$t t!ere need" to %e $ %$&$nce %et+een t!e tot$&"u,,& fro( $&& "ource" $nd t!e tot$& de($nd $t $&& de"tin$tion".The feasible solutions propert2! A tr$n",ort$tion ,ro%&e( +i&& !$efe$"i%&e "o&ution" if $nd on& if t!e tot$& de($nd e*u$&" t!e tot$&"u,,&. If t!e (ode& i" un%$&$nced4 +e c$n $&+$" $dd $ du(( "ourceor $ du(( de"tin$tion to re"tore %$&$nce.The cost assumption! T!e co"t of di"tri%utin' unit" fro( $n

,$rticu&$r "ource to $n ,$rticu&$r de"tin$tion i" direct& ,ro,ortion$& tot!e nu(%er of unit" di"tri%uted. T!erefore4 t!i" co"t i" /u"t t!e unit co"tof di"tri%ution ti(e" t!e nu(%er of unit" di"tri%uted. 6We &et ci/denotet!i" unit co"t for "ource i $nd de"tin$tion /.5T!e on& d$t$ needed for $tr$n",ort$tion ,ro%&e( (ode& $re t!e "u,,&ie"4 de($nd"4 $nd unitco"t". T!e"e $re t!e ,$r$(eter" of t!e (ode&. A&& t!e"e ,$r$(eter"c$n %e "u(($ried conenient& in $ "in'&e ,$r$(eter t$%&e $" "!o+n%e&o+The model!An ,ro%&e( 6+!et!er ino&in' tr$n",ort$tion or not5 t"t!e (ode& for $ tr$n",ort$tion ,ro%&e( if it c$n %e de"cri%edco(,&ete& in ter(" of $ ,$r$(eter t$%&e "!o+n %e&o+ $nd it "$ti"e"

%ot! t!e re*uire(ent" $""u(,tion $nd t!e co"t $""u(,tion. T!eo%/ectie i" to (ini(ie t!e tot$& co"t of di"tri%utin' t!e unit". A&& t!e,$r$(eter" of t!e (ode& $re inc&uded in t!i" ,$r$(eter t$%&e.


38/73

SUMMARY OF THE TRANSPORTATION MODEL

T!e "te," of t!e tr$n",ort$tion $&'orit!( $re e)$ct ,$r$&&e&" of t!e "i(,&e)$&'orit!(.

9te, 1. eter(ine $ starting %$"ic fe$"i%&e "o&ution4 $nd 'o to "te, 2.9te, 2. U"e t!e o,ti($&it condition of t!e "i(,&e) (et!od todeter(ine t!e entering variable fro( $(on' $&& t!e non %$"ic$ri$%&e". If t!e o,ti($&it condition i" "$ti"ed4 "to,. Ot!er+i"e4 'o to"te, .9te, . U"e t!e fe$"i%i&it condition of t!e "i(,&e) (et!od todeter(ine t!e leaving variable fro( $(on' $&& t!e current %$"ic$ri$%&e"4 $nd nd t!e ne+ %$"ic "o&ution.Return to "te, 2.

DETERMINATION OF A BASIC FEASIBLE SOLUTION

T!e ",eci$& "tructure of t!e tr$n",ort$tion ,ro%&e( $&&o+" "ecurin' $ non$rtici$& "t$rtin' %$"ic "o&ution u"in' one of t!ree (et!od":1. 7ort!+e"t-corner (et!od2. Le$"t-co"t (et!od. Vo'e&S" $,,ro)i($tion (et!od

T!e t!ree (et!od" dier in t!e \*u$&it\ of t!e "t$rtin' %$"ic "o&utiont!e ,roduce4 in t!e "en"e t!$t $ %etter "t$rtin' "o&ution ie&d" $ "($&&ero%/ectie $&ue. In 'ener$&4 t!ou'! not $&+$"4 t!e Vo'e& (et!od ie&d" t!e%e"t "t$rtin' %$"ic "o&ution4 $nd t!e nort!+e"t-corner (et!od ie&d" t!e+or"t. T!e tr$deo i" t!$t t!e nort!+e"t-corner (et!od ino&e" t!e &e$"t$(ount of co(,ut$tion".

+' %orth3est'4orner MethodT!e (et!od "t$rt" $t t!e nort!+e"t-corner ce&& 6route5 of t!e t$%&e$u6$ri$%&e )115.9te, 1: A&&oc$te $" (uc! $" ,o""i%&e to t!e "e&ected ce&&4 $nd $d/u"t t!e$""oci$ted $(ount" of "u,,& $nd de($nd % "u%tr$ctin' t!e $&&oc$ted$(ount.9te, 2: Cro"" out t!e ro+ or co&u(n +it! ero "u,,& or de($nd to indic$tet!$t no furt!er $""i'n(ent" c$n %e ($de in t!$t ro+ or co&u(n. If %ot! $ ro+$nd $ co&u(n net to ero "i(u&t$neou"&4 cross out one only, $nd &e$e $ero "u,,& 6de($nd5 in t!e uncro""ed-out TOW 6co&u(n5.

9te, : If exactly one ro+ or co&u(n i" &eft uncro""ed out4 "to,. Ot!er+i"e4(oe to t!e ce&& to t!e ri'!t if $ co&u(n !$" /u"t %een cro""ed out or %e&o+ if$ ro+ !$" %een cro""ed out. >o to "te, 1.

&' Least 4ost MethodT!e &e$"t-co"t (et!od nd" $ %etter "t$rtin' "o&ution % concentr$tin' ont!e c!e$,e"t route". T!e (et!od $""i'n" $" (uc! $" ,o""i%&e to t!e ce&& +it!t!e "($&&e"t unit co"t 6tie" $re %ro0en $r%itr$ri&5. 7e)t4 t!e "$ti"ed ro+ or


39/73

co&u(n i" cro""ed out $nd t!e $(ount" of "u,,& $nd de($nd $re $d/u"ted$ccordin'&.If %ot! $ ro+ $nd $ co&u(n $re "$ti"ed "i(u&t$neou"&4 only one is crossedout, t!e "$(e $" in t!e nort!+e"t-corner (et!od. 7e)t4 &oo0 for t!euncro""ed-out ce&& +it! t!e "($&&e"t unit co"t $nd re,e$t t!e ,roce"" unti&e)$ct& one ro+ or co&u(n i" &eft uncro""ed out.

,' #ogel5s Approximation Method (#AM)VA8 i" $n i(,roed er"ion of t!e &e$"t-co"t (et!od t!$t 'ener$&&4 %ut not$&+$"4 ,roduce" %etter "t$rtin' "o&ution".9te, 1: or e$c! ro+ 6co&u(n54 deter(ine $ ,en$&t (e$"ure % "u%tr$ctin't!e smallest unit co"t e&e(ent in t!e ro+ 6co&u(n5 fro( t!e next smallestunit co"t e&e(ent in t!e "$(e ro+ 6co&u(n5.9te, 2: Identif t!e ro+ or co&u(n +it! t!e &$r'e"t ,en$&t. Bre$0 tie"$r%itr$ri&. A&&oc$te $" (uc! $" ,o""i%&e to t!e $ri$%&e +it! t!e &e$"t unitco"t in t!e "e&ected ro+ or co&u(n. Ad/u"t t!e "u,,& $nd de($nd4 $nd cro""out t!e "$ti"ed ro+ or co&u(n. If $ ro+ $nd $ co&u(n $re "$ti"ed"i(u&t$neou"&4 on& one of t!e t+o i" cro""ed out4 $nd t!e re($inin' ro+6co&u(n5 i" $""i'ned ero "u,,& 6de($nd5.9te, :6$5 If e)$ct& one ro+ or co&u(n +it! ero "u,,& or de($nd re($in"uncro""ed out4 "to,.6%5 If one ro+ 6co&u(n5 +it! positive "u,,& 6de($nd5 re($in" uncro""edout4 deter(ine t!e %$"ic $ri$%&e" in t!e ro+ 6co&u(n5 % t!e &e$"t-co"t(et!od. 9to,.6c5 If $&& t!e uncro""ed out ro+" $nd co&u(n" !$e 6re($inin'5 ero "u,,&$nd de($nd4 deter(ine t!e zero %$"ic $ri$%&e" % t!e &e$"t-co"t (et!od.9to,.6d5 Ot!er+i"e4 'o to "te, 1.

"xample!

e"tin$tion"

9ource" 1 2 3 "u,,&

1 1 2 2 11 1

2 12 D 2 2

3 13 1 1K 1

e($nd" 1 1 1e($nd"G"u,,&

T!i" E)$(,&e i" "o&ed u"in' $&& t!ree (et!od" (entioned $%oe


40/73

+' %orth3est 4orner Method

CO9T PER U7IT I9TRIBUTE

E9TI7ATIO7

9OURCE 1 2 3 9UPPLa

1

1 2 2 11 1

212 D 2 2

3 13 1 1K 1

E8A7 1 1 1 678&9

Le$"t Co"t G 1H12HDH1H2H11KG8&9

&' Least 4ost Method


E9TI7ATIO7

9OURCE 1 2 3 9UPPLa

1

1 start 2 2 11 1

212 D "nd 2 2

1

1

1

1

1


41/73

3 13 1 1K 1

E8A7 1 1 1 67/:8

Le$"t Co"tG 12H3H1H11H11KH2 G /:8

,' #AM (#ogel5s Approximation Method)


E9TI7ATIO7

9OURCE 1 2 3 9UPPLa ,en$&tie"

1

1

(b) 2 2

111

1-2GK11-2G

2

12

D (c) (d) 2 2 -DG2-DG2

(a) 3 1

3

1

(e) 1K1

13-3G1

1-13G2

E8A7 1 1 1 67/:8,en$&tie" 1-3G

-D-2GD-2G

1-GD1-GD

1K-11GD2-1KG2

Le$"t Co"t G 3H12H1H12H1K G /:8

CO88E7T9:

T!e &ine$r ,ro'r$((in' (ode& enco(,$""e" $ +ide $riet of ",ecict,e" of ,ro%&e(". T!e 'ener$& "i(,&e) (et!od i" $ ,o+erfu& $&'orit!( t!$tc$n "o&e "ur,ri"in'& &$r'e er"ion" of $n of t!e"e ,ro%&e(". o+eer4"o(e of t!e"e ,ro%&e( t,e" !$e "uc! "i(,&e for(u&$tion" t!$t t!e c$n %e"o&ed (uc! (ore e=cient& % streamlined $&'orit!(" t!$t e),&oit t!eirspecial structure. T!e"e "tre$(&ined $&'orit!(" c$n cut do+n tre(endou"&

1

1

1

1


42/73

on t!e co(,uter ti(e re*uired for &$r'e ,ro%&e("4 $nd t!e "o(eti(e" ($0eit co(,ut$tion$&& fe$"i%&e to "o&e !u'e ,ro%&e(". T!e ",eci$&-,ur,o"e$&'orit!(" $re inc&uded in "o(e &ine$r ,ro'r$((in' "oft+$re ,$c0$'e" &i0eTORA.

LAB REPORT #

OBJECTIVE: Introduction to t!e 4PM (4ritical Path Method)and Transshipment Model

LITRATURE REVIEW:

NETWORK MODELS

T!e net+or0 (ode&" inc&ude t!e tr$dition$& $,,&ic$tion" of ndin' t!e (o"te=cient +$ to &in0 $ nu(%er of &oc$tion" direct& or indirect&4 ndin' t!e"!orte"t route %et+een t+o citie"4 deter(inin' t!e ($)i(u( `o+ in $,i,e&ine net+or04 deter(inin' t!e (ini(u(-co"t `o+ in $ net+or0 t!$t"$ti"e" "u,,& $nd de($nd re*uire(ent" $t dierent &oc$tion"4 $nd"c!edu&in' t!e $ctiitie" of $ ,ro/ect.7et+or0" $ri"e in nu(erou" "ettin'" $nd in $ $riet of 'ui"e".Tr$n",ort$tion4 e&ectric$&4 $nd co((unic$tion net+or0" ,er$de our d$i&&ie". 7et+or0 re,re"ent$tion" $&"o $re +ide& u"ed for ,ro%&e(" in "uc!dier"e $re$" $" ,roduction4 di"tri%ution4 ,ro/ect ,&$nnin'4 f$ci&itie" &oc$tion4re"ource ($n$'e(ent4 $nd n$nci$& ,&$nnin'bto n$(e /u"t $ fe+ e)$(,&e".In f$ct4 $ net+or0 re,re"ent$tion ,roide" "uc! $ ,o+erfu& i"u$& $ndconce,tu$& $id for ,ortr$in' t!e re&$tion"!i," %et+een t!e co(,onent" of""te(" t!$t it i" u"ed in irtu$&& eer e&d of "cientic4 "oci$&4 $ndecono(ic ende$or.One of t!e (o"t e)citin' dee&o,(ent" in o,er$tion" re"e$rc! 6OR5 in recente$r" !$" %een t!e unu"u$&& r$,id $d$nce in %ot! t!e (et!odo&o' $nd$,,&ic$tion of net+or0 o,ti(i$tion (ode&". A nu(%er of $&'orit!(ic%re$0t!rou'!" !$e !$d $ ($/or i(,$ct4 $" !$e ide$" fro( co(,uter


43/73

"cience concernin' d$t$ "tructure" $nd e=cient d$t$ ($ni,u&$tion.Con"e*uent&4 $&'orit!(" $nd "oft+$re no+ $re $$i&$%&e and are being usedto "o&e !u'e ,ro%&e(" on $ routine %$"i" t!$t +ou&d !$e %een co(,&ete&intr$ct$%&e t+o or t!ree dec$de" $'o.8$n net+or0 o,ti(i$tion (ode&" $ctu$&& $re ",eci$& t,e" of linearprogramming ,ro%&e(". or e)$(,&e4 %ot! t!e tr$n",ort$tion ,ro%&e( $ndt!e $""i'n(ent ,ro%&e( di"cu""edin t!e ,recedin' L$% re,ort" f$&& into t!i"c$te'or %ec$u"e of t!eir net+or0 re,re"ent$tion"

THE TERMINOLOGY OF NETWORKS

A net+or0 con"i"t" of $ "et of points $nd $ "et of lines connectin' cert$in,$ir" of t!e ,oint". T!e ,oint" $re c$&&ed nodes 6or ertice"5. T!e &ine" $rec$&&ed arcs 6or &in0" or ed'e" or %r$nc!e"5@ Arc" $re &$%e&ed % n$(in' t!enode" $t eit!er end If ò+ t!rou'! $n $rc i" $&&o+ed in on& one direction6e.'.4 $ one +$ "treet54 t!e $rc i" "$id to %e $ directed arc;T!e direction i" indic$ted % $ddin' $n $rro+!e$d $t t!e end of t!e &inere,re"entin' t!e $rc. W!en $ directed $rc i" &$%e&ed % &i"tin' t+o node" it

connect"4 t!e from node $&+$" i" 'ien %efore t!e to node@ e.'.4 $n $rc t!$ti" directed from node A to node B (u"t %e &$%e&ed $" AB r$t!er t!$n BA.A&tern$tie&4 t!i" $rc ($ %e &$%e&ed $"A B. If ò+ t!rou'! $n $rc i"$&&o+ed in eit!er direction 6e.'.4 $ ,i,e&ine t!$t c$n %e u"ed to ,u(, ùid ineit!er direction54 t!e $rc i" "$id to %e $n undirected arc; To !e&, oudi"tin'ui"! %et+een t!e t+o 0ind" of $rc"4 +e "!$&& fre*uent& refer toundirected $rc" % t!e "u''e"tie n$(e of lin


44/73

THE TRANSSHIPMENT MODEL

T!e tr$n""!i,(ent (ode& reco'nie" t!$t it ($ %e c!e$,er to "!i, t!rou'!inter(edi$te or transient node" %efore re$c!in' t!e n$& de"tin$tion. T!i"conce,t i" (ore 'ener$& t!$n t!$t of t!e re'u&$r tr$n",ort$tion (ode&4 +!eredirect "!i,(ent" on& $re $&&o+ed %et+een $ "ource $nd $ de"tin$tion.

CRITICAL PATH METHOD

CP8 6Critic$& P$t! 8et!od5 $nd PERT 6Pro'r$( E$&u$tion $nd Reie+Tec!ni*ue5 $re net+or0-%$"ed (et!od" de"i'ned to $""i"t in t!e ,&$nnin'4"c!edu&in'4 $nd contro& of ,ro/ect". A ,ro/ect i" dened $" $ co&&ection ofinterre&$ted $ctiitie" +it! e$c! $ctiit con"u(in' ti(e $nd re"ource". T!eo%/ectie of CP8 $nd PERT i" to ,roide $n$&tic (e$n" for "c!edu&in' t!e$ctiitie". i'ure "!o+n %e&o+ "u(($rie" t!e "te," of t!e tec!ni*ue".

ir"t4 +e dene t!e $ctiitie" of t!e ,ro/ect4 t!eir ,recedencere&$tion"!i,"4 $nd t!eir ti(e re*uire(ent". 7e)t4 t!e ,recedence

re&$tion"!i," $(on' t!e $ctiitie" $re re,re"ented % $ net+or0. T!e t!ird"te, ino&e" ",ecic co(,ut$tion" to dee&o, t!e ti(e "c!edu&e for t!e,ro/ect. urin' t!e $ctu$& e)ecution of t!e ,ro/ect t!in'" ($ not ,roceed $",&$nned4 $" "o(e of t!e $ctiitie" ($ %e e),edited or de&$ed. W!en t!i"!$,,en"4 t!e "c!edu&e (u"t %e rei"ed to re`ect t!e re$&itie" on t!e 'round.T!i" i" t!e re$"on for inc&udin' $ feed%$c0 &oo, %et+een t!e ti(e "c!edu&e,!$"e $nd t!e net+or0 ,!$"e4 $" "!o+n in i'ure %e&o+

T!e t+o tec!ni*ue"4 CP8 $nd PERT4 +!ic! +ere dee&o,ed inde,endent&4dier in t!$t CP8 $""u(e" deter(ini"tic $ctiit dur$tion" $nd PERT


45/73

$""u(e" ,ro%$%i&i"tic dur$tion". T!i" ,re"ent$tion +i&& "t$rt +it! CP8 $ndt!en ,roceed +it! t!e det$i&" of PERT.

%et3or< Representation in 4PM

E$c! $ctiit of t!e ,ro/ect i" re,re"ented % $n $rc ,ointin' in t!e directionof ,ro're"" in t!e ,ro/ect. T!e node" of t!e net+or0 e"t$%&i"! t!e ,recedencere&$tion"!i," $(on' t!e dierent $ctiitie". T!ree ru&e" $re $$i&$%&e forcon"tructin' t!e net+or0.

Rule +! "ac activity is represented by one, and only one, arc.Rule &! "ac activity must be identi#ed by two distinct end nodes.Rule ,!$o maintain te correct precedence relationsips, te followingquestions must be answered as eac activity is added to te networ%&

6$5 'at activities must immediately precede te current activity(6%5 'at activities must follow te current activity(6c5 'at activities must occur concurrently wit te current activity(

4PM 4omputations

T!e end re"u&t in CP8 i" t!e con"truction of t!e ti(e "c!edu&e for t!e ,ro/ect6"ee i'ure $%oe5. To $c!iee t!i" o%/ectie conenient&4 +e c$rr out",eci$& co(,ut$tion" t!$t ,roduce t!e fo&&o+in' infor($tion:+;Tot$& dur$tion needed to co(,&ete t!e ,ro/ect.&. C&$""ic$tion of t!e $ctiitie" of t!e ,ro/ect $" critical $nd noncritical.

An $ctiit i" "$id to %e critic$& if t!ere i" no \&ee+$\ in deter(inin' it""t$rt $nd ni"! ti(e". A noncritic$& $ctiit $&&o+" "o(e "c!edu&in' "&$c04 "ot!$t t!e "t$rt ti(e of t!e $ctiit c$n %e $d$nced or de&$ed +it!in &i(it"+it!out $ectin' t!e co(,&etion d$te of t!e entire ,ro/ect.

To c$rr out t!e nece""$r co(,ut$tion"4 +e dene $n eent $" $ ,oint in ti(e$t +!ic! $ctiitie" $re ter(in$ted $nd ot!er" $re "t$rted. In ter(" of t!enet+or04 $n eent corre",ond" to $ node. ene

/G E$r&ie"t occurrence ti(e of eent //G L$te"t occurrence ti(e of eent /

i/G ur$tion of $ctiit 6i4 /5

T!e denition" of t!e earliest $nd latest occurrence" of eent / $re ",eciedre&$tie to t!e "t$rt $nd co(,&etion d$te" of t!e entire ,ro/ect. T!e critic$&,$t! c$&cu&$tion" ino&e t+o ,$""e": T!e for+$rd ,$"" deter(ine" t!eearliest occurrence ti(e" of t!e eent"4 $nd t!e %$c0+$rd ,$"" c$&cu&$te"t!eir latestoccurrence ti(e".


46/73

"xample +!

ACTIVITa URATIO7 EPE7E7CaA D -B -C A BE 4 2 B> C 2 E4

1

2 3

D

K


47/73

EXAMPLE NO. 2

Activity Predecessor ES LS EF LF Slack

time

Critical

path

A 0 0 9 9 0 *

B 0 5 15 20 5 -

C A 9 9 14 14 0 *

D C 14 14 20 20 0 *

E D 20 20 24 24 0 *F B & E 24 24 27 27 0 *

G E & F 27 27 45 45 0 *

PER !

> 4 1K

4

E 4 3

23 23 2D 2

D2D 2D 3

3

2

2 23 23

2

3

D

2 1


48/73

B 41

4

A 4 C 4

Critical path A-C-D-E-F-G 9 ! 5 ! " ! 4 ! # ! 1$

45 %a'

CPM!

9TART

Critical path A-C-D-E-F-G 9!5!"!4!#!1$ 45 %a'

EXAMPLE NO. "!

1

3

1

3

2

2

1

3

1

3

1

2

3

2

3

2

3

2

D

2

D

2

D

2

D

3

3

1

3

1

3

2

2

2

2

2

3

2

3

1

3

1

3

A4

4 B4

1

E4 3

>4

1K

C4

4


49/73


50/73

Critical path A-E-F-G-- +r 1 2 # 4 5 " 7 $ 9 10 11 12

9 ! 2$ ! 0 ! 15 ! 10 ! 20 ! 0 ! 27

109 /+th'

4PM 9t$rt

1 2

2

2

2

1

3

D

D

D

2

2

2

D

D

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Critic$& ,$t! G A-E-->-J-< or or 1 2 D K 11 12

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52/73

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54/73

LAB REPORT # D

OBJECTIVE: Introduction to Simulation Modeling

Sim#latio$=i/,lati+ i' a %'cripti( tchi?, i 6hich a /+%l +< a pr+c'' i' %(l+p% a% th

@pri/t' ar c+%,ct% + th /+%l t+ (al,at it' ;ha(i+r ,%r (ari+,' c+%iti+'li: /a +< th +thr /+%l' %'cri;% i th t@t 'i/,lati+ i' +t a +pti/ii tchi?,

3t %+' +t pr+%,c a '+l,ti+ pr ' 3'ta% 'i/,lati+ a;l' %ci'i+ /a:r' t+ t't their

'+l,ti+' + a /+%l that ra'+a;l %,plicat' a ral pr+c'' 'i/,lati+ /+%l' a;l %ci'i+/a:r' t+ @pri/t 6ith %ci'i+ altrati(' ,'i a what if appr+ach

8h ,' +< 'i/,lati+ a' a %ci'i+-/a:i t++l i'


55/73

Ithr ra'+'


56/73

c+/pari th r',lt' +< 'i/,lati+ r,' 6ith :+6 pr


57/73

2 +r: +,t a a''i/t '+ that itr(al' +< ra%+/ ,/;r' 6ill c+rr'p+% t+ th

pr+;a;ilit %i'tri;,ti+# I;tai th ra%+/ ,/;r' %%


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8h pr+c'' +< 'i/,lati+ 6ill ;c+/ clarr a' 6 6+r: thr+,h '+/ 'i/pl pr+;l/'

E)ample

8h /aar +< a /achi 'h+p i' c+cr% a;+,t /achi ;ra:%+6' ha' /a% a

%ci'i+ t+ 'i/,lat ;ra:%+6'


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A''i ra%+/-,/;r itr(al' t+ c+rr'p+% 6ith th c,/,lati(


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0(.0) H 16.5 H 26.25 H 6.25 H 36.15 H 6.5 G 2. ,er d$

=(ral p+it' ar 6+rth +tiJ

1 8hi' 'i/pl @a/pl i' it%% t+ ill,'trat th ;a'ic c+cpt +< +t Carl+ 'i/,lati+

3< +,r +l +al 6r t+ 'ti/at th a(ra ,/;r +< ;ra:%+6' 6 6+,l% +t ha(t+ 'i/,lat 6 c+,l% ;a' th 'ti/at + th hi't+rical %ata al+

2 8h 'i/,lati+ 'h+,l% ; (i6% a' a 'a/pl it i' ?,it li:l that a%%iti+al r,' +< 10,/;r' 6+,l% pr+%,c %i


61/73

8h 'i/,lati+ r',lt' ar 'h+6 i th


62/73

b. ' t6+-%iit ra%+/ ,/;r'


63/73

.2 I;tai ra%+/ ,/;r' c+(rt t+ %/a% ,p%at i(t+r acc+r%il a% r+r%r 6h

c''arJ

Simulating Theoretical Distributions

3 /ai'tac' a 'i/,lati+ 6ill i(+l( th ,' +< th+rtical %i'tri;,ti+' A/+ th /+'t


64/73

8hat i'J

=i/,lat% (al, a ! a%+/ ,/;r P =ta%ar% %(iati+

3


65/73

K@t +;tai thr-%iit ,/;r'


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=igure &! A negative exponential distribution

3t i' i/p+rtat t+ rc+i that E@a/pl 4 i(+l(' a c+ti,+,' (aria;l 6hra' th pr(i+,'@a/pl' ha( i(+l(% %i'crt (aria;l' .//;r that %i'crt (aria;l' tpicall ta: ++l itr (al,' 6hra' c+ti,+,' (aria;l' ca ta: + itr a% + itr (al,'

h(r p+''i;l a /+%l +< a c+ti,+,' (aria;l 'h+,l% ; a;l t+ 'i/,lat + itr

(al,' a' 6ll a' itr (al,' A+thr c+ti,+,' tp +< %i'tri;,ti+ 6 ca c+'i%r i' th,i


67/73

ith a ati( @p+tial %i'tri;,ti+ th pr+;a;ilit i'


68/73

8h /a,

i' 5 h+,r' 8h ra%+/ ,/;r' ar $4 a% 05 'i F+r/,la .2 th 'i/,lat%

ti/' arJ

or K34 t G -[&n6.K35Z G -[-.1D33Z G .KD2!our".

or 4 t G -[&n6.5Z G -[-2.DZ G 13.D!our".

7ote t!$t t!e "($&&er t!e $&ue of t!e r$ndo( nu(%er4 t!e &$r'er t!e "i(u&$ted

$&ue of t.

COMPUTER SIMULATIONAlth+,h th /pha'i' i thi' ',ppl/t ha' ; + /a,al 'i/,lati+ i +r%r t+ c+( th

/ai c+cpt' /+'t ral-li


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Fi#re "! Choosi$ a sol#tio$ tech$i/#e

Crtai li/itati+' ar al'+ a''+ciat% 6ith 'i/,lati+ Chi< a/+ th' arJ

1 =i/,lati+ %+' +t pr+%,c a +pti/,/ '+l,ti+ it /rl i%icat' a approximate;ha(i+r


70/73

b. =i/,lati+' ar ;a'% + /+%l' a% /+%l' ar +l appr+@i/ati+' of ralit

2 F+r lar-'cal 'i/,lati+ it ca r?,ir c+'i%ra;l


71/73

LAB REPORT # K

OBJECTIVE: Introduction to TORA

LITRATURE REVIEW:

INTRODUCTION

TORA i" $ ,ro(inent "oft+$re ,$c0$'e for &ine$r ,ro'r$((in' $nd it"

e)ten"ion". It i" "i(,&e $nd !$" $n e$" to u"e interf$ce for r"t ti(e u"er".

A&(o"t $&& t!e +or0 t!$t !$" %een done ($nu$&& in t!e ,reiou" &$% re,ort"

c$n %e done on TORA.

TORA i" $ ,o+erfu& "oft+$re dee&o,ed % $($d A. T$!$4 t!e $ut!or of t!e

O,er$tion" Re"e$rc! %oo0. 7eed&e"" to "$ t!$t t!ere $re ,&ent of ot!er

"oft+$re in t!e ($r0et $" +e&&. 9o(e of t!e ($/or $re CPLEF4 LI7O $nd t!e

(o"t ,o+erfu& of $&& E)ce&. 9,re$d"!eet %$"ed "o&er" $re %eco(in'incre$"in'& ,o,u&$r for &ine$r ,ro'r$((in' ,ro%&e(" $nd it" e)ten"ion".

Unfortun$te& due to &$c0 of ti(e4 t!i" &$% re,ort concentr$te" on& on TORA.

SOLVING LLP!S USING TORA

T!i" i" t!e ($in TORA "creen4 +!ere +e c!oo"e t!e t,e of ,ro%&e( to %e "o&ed. C&ic0Line$r Pro'r$((in'


72/73

7o+ C&ic0 on R>O TO I7PUT 9CREE7to "t$rt $ ,ro%&e( fro( "cr$tc!4 C&ic0 R9e&ect

E)i"tin' i&eif ou !$e $&re$d "$ed t!e d$t$ $nd +$nt to reo,en it.

7o+ enter t!e d$t$ $" "een in t!e i($'e $%oe ; t!en c&ic0 on R9OLVE (enu4 $

"creen ,o," u,4 "$in' Ro ou +i"! to "$e d$t$4 C!oo"e t!e de"ired o,tion


73/73

Documents

Operations Research Lab Report