unit i linear models

8/17/2019 Unit I Linear Models

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Introduction to OR

Pre world war II

During world war II

Post world war II


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Pre world war II

• Fredric W Taylor 1885 – Application ofscientic analysis to !et"ods ofproduction

• #enry $ %antt – &o' sc"eduling( )apping eac" *o' fro! !ac"ine to!ac"ine to !ini!i+e delay,

• F W #aris 1-15 – In.entory control

• A / 0rlang 1-1 – 2ueue


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During world war II

First OR Team ( Blackette’s circus)

• P ) 3 4lacette )anc"ester uni.ersity –)ilitary !anage!ent of 0ngland approac"ed

• 6 P"ysiologists7 !at"e!atical p"ysicists7 1astrop"ysicist7 1 ar!y o9cer7 1 sur.eyor7 1general p"ysicist and !at"e!aticians

Objective : !ost e;ecti.e allocation of

li!ited resources to t"e .arious !ilitaryoperations and acti.ities

T"e na!e OR coined in 1-


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Post world war II

• Industries application

• 1-


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3cope of OR

• Agriculture

• Finance

•

)areting• Personnel )anage!ent

• Production )anage!ent

•

Pu'lic utilities – #ospitals7 transport7$I? and

depart!ental

store etc


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Agriculture

• Opti!u! allocation of land to .ariouscrops in accordance wit" t"e cli!aticconditions

• Opti!u! distri'ution of water fro!.arious resources lie canal forirrigation


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Finance

• )a@i!i+e t"e per capita inco!e wit"!ini!u! resources a.aila'ility

• Find t"e prot plan of t"e co!pany

• Deter!ine t"e 'est replace!entpolicies


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Industry

• Opti!u! allocation of resources

)en7 !ac"ines7 !aterials7 !oneyand ti!e

• )a@i!i+e t"e prot

• )ini!i+e t"e production cost


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)areting

• Opti!u! distri'ution of products forsale

• 3i+e of stoc to !eet t"e futurede!and

• 3election of 'est ad.ertising !ediawit" respect to ti!e and cost

• #ow7 w"en and w"at to purc"ase att"e !ini!u! possi'le cost


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Personnel !anage!ent

• Appoint suita'le candidates on!ini!u! salary

• Deter!ine t"e 'est age of retire!entfor t"e e!ployees

• Find t"e nu!'er of persons to 'eappointed on full ti!e 'asis w"en t"eworload is seasonal (notcontinuous,


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Production !anage!ent

• Find t"e nu!'er si+e of ite!s to 'eproduced

• 3c"eduling and seBuencing t"eproduction run 'y proper allocationof !ac"ines

• ?alculating t"e opti!u! product !i@

• 3elect 7 locate and design t"e sitesfor t"e production plants>


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Pu'lic utilities

• To reduce waiting ti!e of out door patients

• To regulate t"e train arri.als and t"eirrunning ti!es

• W"at s"ould 'e t"e pre!iu! rates for.arious !odes of policies

• #ow 'est t"e prots could 'e distri'uted int"e cases of wit" prot policies

• 0!ploying additional sales persons7 use oftransport .ec"icle etc


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P"ases of OR

• For!ulating t"e pro'le!

• ?onstructing a !at"e!atical !odel

•

Deri.ing t"e solutions• Testing t"e !odel and its solutions

• ?ontrolling t"e solution

•

I!ple!enting t"e solution


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For!ulate t"e Pro'le!

• Dene t"epro'le!

• Deli!it t"esyste!

• 3elect !easures

Formulate the

Problem

Problem

Statement

Data

Situation

• Deter!ine.aria'les

• Identifyconstraints


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?onstruct a

)odel

•

)at"> Progra!!ing )odel – 3toc"astic )odel

– Deter!inistic )odel

– 3i!ulation )odel

•

O'*ecti.e function• ?onstraints (or, restrictions

• Decision .aria'les andpara!eters

Construct

a Model

Model

Formulate the

Problem

Problem

Statement

Data

Situation


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Find a

3olution

• $inear

Progra!!ing• Conlinear

Progra!!ing

• Regression• Direct 3earc"

• 3toc"astic

Opti!i+ation

Construct

a Model

Model

Formulate the

Problem

Problem

Statement

Data

Situation

Solution

Find

a Solution

Tools


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0sta'lis" a

Procedure• Production

software

•

0asy to use• 0asy to

!aintain

• Accepta'leto t"e user

Solution

Establish

a Procedure

Solution

Finda Solution

Tools

Construct

a Model

Model

Formulate the

Problem

Problem

Statement

Data

Situation


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?ontrolling t"e solution

• It !ust 'e esta'lis"ed to indicatet"e li!its wit"in w"ic" t"e !odeland its solution can 'e considers as

relia'le>


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I!ple!ent

t"e 3olution• ?"ange for t"e

organi+ation

•

?"ange is di9cult• 0sta'lis" controls

to recogni+ec"ange in t"e

situation

Solution

Establisha Procedure

Solution

Finda Solution

Tools

Construct

a Model

Model

Formulate the

Problem

Problem

Statement

Data

Situation

Implementthe Solution


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Deterministic Models Stochastic Models

• Linear Programming• Discrete-Time Markov Chains

• Network Optimization• Continuous-Time Markov Chains• Integer Programming• Queuing

• Nonlinear Programming• Decision Analysis

Operations Research Models

D i i i 3 " i


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Deterministic models – 60% of course

Stochastic (or probabilistic) models – 40% of course

Deterministic models assume all data are known withcertainty

Stochastic models

explicitly represent uncertain data viarandom variables or stochastic processes

Deterministic models involveoptimization

Stochastic modelscharacterize/estimate

system performance.

Deter!inistic .s> 3toc"astic)odels

0 l f OR


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0@a!ples of ORApplications

•Resc"eduling aircraft in response togroundings and delays

• Planning production for printed circuit

'oard asse!'ly• 3c"eduling eBuip!ent operators in

!ail processing distri'ution centers

•

De.eloping routes for propanedeli.ery

• Ad*usting nurse sc"edules in lig"t ofdaily Euctuations in de!and


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Data

T"e Process: Recogni+e t"e

Pro'le!• )anufacturing

– Planning

–

Design – 3c"eduling

– Dealing wit"Defects

–

Dealing wit"aria'ility

– Dealing wit"In.entory

–

G

Situation


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Ot"er applications

• 3er.ice Industries

• $ogistics

•

Transportation• 0n.iron!ent

• #ealt" ?are

•3ituations wit"co!ple@ity

• 3ituations wit"

uncertainty

Data

Situation


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25

–

In any organi+ation7 li!ited resources – 'estallocation of resources – !a@i!i+e prot7 !ini!i+e

loss and !a@i!i+e t"e utili+ation of production

capacity>

–

A $inear Progra!!ing !odel sees to !a@i!i+e or!ini!i+e a linear function7 su'*ect to a set of linear

constraints>

– De.eloped and applied $P 'y Prof A / Dant+ig

– T"e linear !odel consists of t"e followingco!ponents

A set of decision .aria'les>An o'*ecti.e function>

A set of constraints>

Introduction to $inear Progra!!ing


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$P contd>7• $inear – Relations"ip a!ong two or !ore

.aria'les w"ic" are directly proportioned

i.e. Increase the production capacity increasesthe proft

• Progra!!ing – Planning of acti.ities in a !anner

t"at ac"ie.es opti!u! results

Denition

A !et"od of opti!i+ing(!a@ (or, !in, a linearfunction wit" a nu!'er of constraints(li!itations,

e@pressed in t"e for! of ineBualities>


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$PP for!ulation

A furniture !anufacturing co!pany plansto !ae c"airs and ta'les fro! itsa.aila'le resources w"ic" consist of


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Pro'le!

A !anufacturer produces two types of!odels H > 0ac" H !odel reBuires < "rs ofgrinding and "rs of polis"ing w"ereas eac"

!odel reBuires "rs of grinding and 5 "rsof polis"ing> T"e !anufacturer "as grinders and 6 polis"ers> 0ac" grinder worsfor Prot of an H !odel is Rs>6 and !odel is Rs> For!ulate a $P!odel to !a@i!i+e a prot in a wee>


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Pro'le!

A r! produces t"ree products> T"ese products are processed

on t"ree di;erent !ac"ines> T"e ti!e reBuired to !anufactureone unit of eac" of t"e t"ree products and daily capacity of t"et"ree !ac"ines are gi.en in t"e ta'le 'elow

It is reBuired to deter!ine t"e daily nu!'er of units to 'e

!anufactured for eac" product> T"e prot per unit for product17 and 6 is Rs>6 and Rs> J respecti.ely> It is assu!ed t"atall t"e a!ounts produced are consu!ed in t"e !aret>For!ulate t"e !at"e!atical ($P, !odel t"at will !a@i!i+e t"edaily prot

Machi

ne

Time per unit (min)Machine

capacity

(min/day)

Produ

ct 1

Produ

ct 2

Produ

ct

)1 6


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Pro'le!

• An ani!al feed co!pany !ustproduce == g of !i@ture consistingof ingredients H17 H daily> H1 costs

Rs 6Lg and H Rs 8Lg> Co !oret"an 8= g of H1 can 'e used and atleast J= g of H !ust 'e used>

For!ulate a $P !odel to !ini!i+et"e cost


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Pro'le!

A co!pany "as t"ree operational depart!ents na!elywea.ing7 processing and pacing> T"e capacity toproduce t"ree di;erent types of clot"es na!elysuiting7 s"irting and woolens yielding a prot of Rs> 7<and 6 per !eter respecti.ely> One !eter of suiting

reBuires 6 !in in wea.ing7 !in of processing and 1!in of pacing> 3i!ilarly 1 !eter of s"irting reBuires <!in in wea.ing7 1 !in of processing and 6 !in ofpacing w"ereas 1 ! of woolen reBuires 6 !inutes ineac" depart!ent> In a wee7 t"e total runti!e of eac"

depart!ent is J=7


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Answer

)a@ M N H1 < H8 6H6

3T

6H1


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3ol.ing of $PP

• If t"e no> of .aria'les (H≤ ,

%rap"ical !et"od

• If t"e no> of .aria'les H N 1776 G> n

3i!ple@ !et"od

3i!ilarity N no> of .aria'les

Di;erence N ineBuality constraints

33


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Pro'le!

)in M N


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Pro'le!

)a@ M N =H1 1=H

3T

1=H1 5H ≤ 5=

JH1 1=H ≤ J=


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Pro'le!•

An auto!o'ile !anufacture !aes auto!o'iles andtrucs in a factory t"at is di.ided into two s"ops A 4> 3"op A7 w"ic" perfor!s t"e 'asic asse!'lyoperation !ust wor 5 !anKdays on eac" truc 'utonly !anKdays on eac" auto!o'ile> 3"op 47 w"ic"

perfor!s t"e nis"ing operation !ust wor 6 !anKdays on eac" truc(or,auto!o'ile t"at it produces>4ecause of !en and !ac"ine li!itations s"op A "as18= !anKdaysLwee a.aila'le w"ile s"op 4 "as 165

!anKdaysLwee> If t"e !anufacturer !aes a protof Rs>6== on eac" truc and Rs> == on eac"auto!o'ile7 "ow !any of eac" s"ould "e produce to!a@i!i+e "is prot


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?ontd>

)a@i!i+e M N 6== @1 == @

3u'*ect to t"e constraints

5@1@

Q 18= 6@

1 6@

Q 165

@1 7@ > =

Ans: II iteration:)a@ M N 1===

H1 N 6=7 H N 15


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Pro'le!

)a@i!i+e M N 5 @1 J @ @6


-@16@ K @6 Q 5


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Pro'le!

A ?o!pany !aes inds of leat"er 'elts> 4elt A is"ig" Buality and 4 is low Buality> T"e respecti.eprots are Rs>< and Rs>6 per 'elt> 0ac" type of AreBuires twice as !uc" as a 'elt type of 4 and if all

'elts were of type 47 t"e co!pany could !ae 1===per day> T"e supply of leat"er is su9cient for only8== 'elts per day ('ot" A and 4 co!'ined,> 4elt AreBuires a fancy 'ucle and only T"ere are only == 'uclesa.aila'le per day for 'elt 4> W"at would 'e t"e dailyproduction of eac" type of 'elt For!ulate t"e $Ppro'le! and sol.e using t"e si!ple@ !et"od>


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)a@i!i+e M N < @1 6 @


@1@ Q 1=== @1 @ Q 8==

@1 Q =

Ans: III iteration:)a@ M N J==

H1 N ==7 H N J==


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Degeneracy in $PP

Tie 'etween two (or, !ore 'asic .aria'lesfor lea.ing t"e 'asis

Pro'le! 1

)a@i!i+e M N @1 @ @6


@1@K @6 Q K@1@K 5@6 KJ

=


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)a@i!i+e M N @1 @ @6 =31 =3 =36


@1@K @6 31 N @1K@

5@6 3 N J =

Ans: II iteration:)a@ M N 1=

H N


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3uggested conditions for 0ntering

.aria'le

• Tie 'etween two (or, !ore decision.aria'les7 c"oose ar'itrarily

• Tie 'etween decision .aria'les

slacLsurplus .aria'le7 c"oose onlydecision .aria'le

• Tie 'etween two (or, !ore

slacLsurplus .aria'les7 c"oosear'itrarily


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$ea.ing .aria'le

• Pertur'ation !et"od

?alculation


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Pro'le!

)a@i!i+e M N @1 @



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For!ati.e Assess!ent

Operations Researc" pro.ides

a, 0arliest solution', Feasi'le solutionsc, 3cientic approac" to solutionsd, 3tatistical approac" to solution

$inear progra!!ing deals wit" t"e opti!i+ation of a function of .aria'les nown as

a, ?onstraints

', O'*ecti.e function

c, 0;ecti.eness

d, ProtLloss

Operations Researc" approac"es pro'le! sol.ing and decision !aing fro!

a, Indi.idualSs .iew

', Depart!ental .iew

c, Tec"nical .iew

d, T"e total syste!Ss .iew


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In t"e grap"ical !et"od of linear progra!!ing pro'le!7 t"e opti!u! solution would lie in t"efeasi'le polygon at

a> its one corner

'> its center

c> !iddle of any side

d> 'otto!

Opti!al solution always occurs

a> Wit"in t"e feasi'le region

'> On t"e 'oundaries of feasi'le region

c> At corner points of feasi'le region

d> Anyw"ere

Product A taes 5 )Lc "ours and Product 4 taes J la'our "ours> T"e total ti!e a.aila'le for )Lc"ours is 6J> T"e constraint eBuation for t"is is

a> 5H J N 6J

'> 5H JUN6J

c> 5H JVN6J

d> Data inco!plete


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A 'asic feasi'le solution in si!ple@ !et"od is one7 w"en

a> all t"e decision .aria'les are in t"e 'ase

'> all t"e decision .aria'les and slacLsurplus .aria'les are assigned +ero .alues

c> all t"e 'ase .aria'les are non – negati.e

d> all t"e 'ase .aria'les satisfy t"e constraint eBuations

$inear progra!!ing can 'e applied to

a> steel industry

'> oil industry

c> c"e!ical industry

d> all of t"e a'o.e

In t"e si!ple@ !et"od7 t"e .aria'les w"ic" "a.e not 'een assigned t"e .alue +ero7 during aniteration7 are called as

a> 'asic .aria'le

'> articial .aria'le

c> actual .aria'le

d> slacLsurplus .aria'le


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3i!ple@ !et"od is not applied w"en nu!'er of decision .aria'les is eBual to

a, <

',

c, 1

d, J

Opti!ality is reac"ed w"en all t"e inde@ .alues are

a> +ero

'> negati.e

c> positi.e

d> none of t"e a'o.e

T"e si!ple@ !et"od is t"e 'asic !et"od for

a>.alue analysis

'>operation researc"

c> linear progra!!ing

d> !odel analysis


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Articial .aria'les

• Purpose is to *ust o'tain an initial'asic feasi'le solution

aria'les once dri.en out in an

iteration7 !ay reKenter t"esu'seBuent iteration> 4ut an A oncedri.en can ne.er reKenter

4ig ) !et"od (A ?"arnes, Two p"ase !et"od (Dant+ig7Orden and Wolfe,


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Pro'le!

4ig ) )et"od (or, ?"arnes )tec"niBue (or, )et"od of penalties

)a@i!i+e M N @1 6 @ < @6


6@1@< @6 Q J==

@1< @ @6


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)a@i!i+e M N @1 6 @ < @6 =31 =3K !A1 – !A


6@1@< @6 31 N J==

@1< @ @6 – 3 A1 N


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Ans : II iteration• 31 N J=

• HNJ=

• H6 N 1=

•)a@ M N JJ=


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Pro'le!

Food H contains J units of .ita!inA per gra! and units of .ita!in 4per gra! and costs 1 paise per

gra!> Food contains 8 units of.ita!in A per gra! and 1 units of.ita!in 4 per gra! and costs =

paise per gra!> T"e !ini!u! dailyreBuire!ents of .ita!in A 4 is 1== 1= units respecti.ely> Find t"e!ini!u! cost of product !i@ 'y t"e


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Pro'le!

)in M N 1H1 =H

3T

JH1 8H ≥ 1==

H1 1H ≥ 1= H1 7 H > =

Ans : II iteration

H1N157 H N 1>5 )in M N

=5


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Pro'le!

)a@ M N 6H1 H

3T

H1 H Q

6H1 =

Ans : Pseudo opti!al solution (In feasi'le

solution,I iteration – 3olution wit" presence of articial

.aria'le


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Pro'le!

During festi.al season7 a co!pany co!'inestwo ite!s A and 4 to fro! gift pacs> 0ac"pac !ust weig" 5 g and s"ould contain atleast g of A and not !ore t"an < g of 4>

T"e net contri'ution to t"e co!pany is Rs>1=per g of A and Rs>1 per g of 4> T"eco!pany wants to deter!ine t"e opti!u!!i@>

For!ulate t"e a'o.e as a $PP to !a@i!i+e netcontri'ution per pac and sol.e t"e sa!e 'yusing t"e si!ple@ !et"od>


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Answer

)a@ M N 1=H1 1H

3T

H1 H N 5

H1 ≥

H ≤ <

H1 7 H

>

=

Ans : II iteration

H1N7 H N 6 )a@ M N 5J


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Pro'le!

)in M N H1 5H

3T

H1 Q =

Ans : III iteration H1N


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Two P"ase !et"od

)ini!i+e M N 5 @1 K J@ K @6


@15@ K 6 @6 155@1 K J@1= @6 ≤ =

@1 @@6 N 5

@1 7@7 @6 > =

Phase I


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Phase I

I step : con.ert ineBuality into eBuality

II step : For!ation of re.ised o'*ecti.e function 'y assigning K1 to articial

.aria'les and = to ot"er .aria'les (decision LslacLsurplus,

III step : sol.e it 'y si!ple@ !et"od until t"e following conditions are

satised

• )a@ M


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3olution

P"ase I – II iteration ( = K.e,

P"ase II – H8 N 6>5

38 N 6=

H6 N 1>5

)a@ M N 61>5

)in M N K61>5


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Pro'le!

)a@i!i+e M N 5 @1 6 @


@1@ Q 1@1< @ J

@1 7@ > =


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3olution

• All .alues of ? *KM * is = negati.e> 3o

opti!al solution is reac"ed

• 4ut )a@ M U = i>e>7 K also onearticial .aria'le (A1 N , appear int"e 'asis at t"e .e le.el

3o $PP doesnSt "a.e feasi'le

solution


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Pro'le!

)ini!i+e M N @1 @


@1 @ <@1 @

@1 7@ >=


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3olution

P"ase I – II iteration ( = K.e,

P"ase II – H1 N 1L16

H N 1=L16 )a@ M N 61L16

)in M N K61L16

unit i linear models

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