lpp formulation & graphical new (2).ppt

7/23/2019 LPP formulation & graphical new (2).ppt

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LP Model FormulationLP Model Formulation

Decision variables mathematical symbols representing levels of activity of an

operation

Objective function

a linear relationship reflecting the objective of an operation most frequent objective of business firms is to maximize profit

most frequent objective of individual operational units such as aproduction or pac!aging department" is to minimize cost

#onstraint a linear relationship representing a restriction on decision

ma!ing


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Steps of Mathematical Formulation of Linear Programming

Problem:

1. Detect the decision variables (which are to be determined in theproblem and assign s!mbol to them ( "1# "$# "%&.

$. 'dentif! all the constraints in the problem and e"press them in

the form of linear euations or in eualities# as the case ma! be.

%. )"press the non negativit! condition of decision variables such

as "1*# "$ * etc.

+. 'dentif! the ob,ective function of the problem and e"press it as

linear function of decision variable -b,ective function isgenerall! denoted b! . /lso note whether the ob,ective

function is to be ma"imi0ed or to be minimi0ed.


3/21

LP Model FormatLP Model Format

Ma$%min & ' c($() c*$*) +++ ) cn$n

subject to,

a(($() a(*$*) +++ ) a(n$n-. '. /" b(a*($() a**$*) +++ ) a*n$n-. '. /" b*

, am($( ) am*$*) +++ ) amn$n-. '. /" bm

$j' decision variables

bi' constraint levels

cj ' objective function coefficientsaij' constraint coefficients

0nd1he decision variables / 2


4/21

LP Model, 3$ampleLP Model, 3$ample

Labor Clay Revenue

P4OD5#1 (hr/unit) (lb/unit) (Rs./unit)

6o7l ( 8 82

Mug * 9 :2

1here are 82 hours of labor and (*2 pounds of clay

available each day

Decision variables

x(' number of bo7ls to producex*' number of mugs to produce

43;O54#3 431;43;O54#3 431;


5/21

LP Formulation, 3$ampleLP Formulation, 3$ample

Ma$imi&e Z' 82x() :2x*

;ubject tox( ) *x* 82 hr labor constraint"

8x( ) 9x* (*2 lb clay constraint"

x( .x* 2


6/21

?raphical ;olution Method?raphical ;olution Method

(+ Plot model constraint on a set of coordinates

in a plane

*+ =dentify the feasible solution space on the

graph 7here all constraints are satisfied

simultaneously

9+ Find the values corresponding to it and

hence find the solution by using the objectivefunction+


7/21

?raphical ;olution, 3$ample?raphical ;olution, 3$ample

88xx(() 9) 9xx** (*2 lb(*2 lb

xx(() *) *xx** 82 hr82 hr

0rea common to0rea common toboth constraintsboth constraints

:2:2

8282

9292

*2*2

(2(2

22

@

(2(2

@

A2A2

@

:2:2

@

*2*2

@

9292

@

8282 xx11

xx22


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#omputing Optimal Balues#omputing Optimal Balues

xx(( )) **xx** '' 8282

88xx(( )) 99xx** '' (*2(*2

88xx(( )) CCxx** '' (A2(A2

88xx(( 99xx** '' (*2(*2

::xx** '' 8282

xx** '' CC

xx(( )) *C"*C" '' 8282

xx(( '' *8*8

88xx(() 9) 9xx** (*2 lb(*2 lb

xx(() *) *xx** 82 hr82 hr

8282

9292

*2*2

(2(2

22

@

(2(2

@

*2*2

@

9292

@

8282

xx11

xx22

ZZ' 4s+:2*8" ) 4s+:2C" ' 4s+(.9A2' 4s+:2*8" ) 4s+:2C" ' 4s+(.9A2

*8

C


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3$treme #orner Points3$treme #orner Points

x1= 24 bowls

x2 =

8 mugs

Z= Rs.1,360 x1= 30 bowls

x2 =0 mugs

Z= 1,200

x1= 0 bowls

x2 =20 mugs

Z= Rs.1,000

AA

BB

CC|

2020

|

3030

|

4040

|

1010 xx11

xx22

4040

3030

2020

1010

00


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;olution isx(' *8 bo7ls x* ' C mugs

4evenue ' 4s+(.9A2

;ince the objective function 7as to

ma$imi&e+


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;pecial ;ituation in LP;pecial ;ituation in LP

Infeasibility

Ehen no feasible solution e$ists thereis no feasible region"


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(*(*

1

No oint, simultaneously,

lies bot! abo"e line an#

below lines an#

+

1

2 32

3

=nfeasible Model=nfeasible Model


13/21

$nboun#e# %olutions 7hen nothing

prevents the solution from becominginfinitely large

Ma$ *( ) **;ubject to,

*( ) 9* G A

(. * G 2

0 1 2 3 &1

&2

2

1

0

D

irection

ofsolutio

n


14/21

(8(8

5nbounded solution5nbounded solution

'!ef

easible(egion

)a*imi+e

t!e,b-e.ti"e/un.tion

Ma$ *( ) **

;ubject to,

*( ) 9* G A

1. # G 2


15/21

;pecial ;ituation in LP;pecial ;ituation in LP

lte(nate timal %olutions 7hen there ismore than one optimal solution

Ma$ *( ) **;ubject to,

( ) * H (2

( H :

*H A (. * G 2

0 10 &1

&2

10

6

0

2

&1 2&2=20

0ll points on

4ed segment

are optimal


16/21

)": / firm manufactures three products /##2 the profits are 3s

%# 3s $# 4 3s + respectivel!. 5he firm has two machines M14 M$

and below given is the reuired processing time in minutes for

each machine on each product.

Machine Product

0 6 #

M( 8 9 :

M* * * 8

Machines M1 4 M$ have $*** 4 $6** machine minutes

respectivel!. 5he firm must manufacture 1** /7s# $** 7s 4 6*

27s# but not more than 16* /7s. Set up an LPP to ma"imi0e

profits.


17/21

Ex :5he ob,ective of a diet problem is to ascertain that uantities

of certain food should be eaten to meet certain nutritional

reuirements at a minimum cost. 5he consideration is limited to

mil8# beef 4 eggs and to vitamins / 4 2. the number ofmiligrams of each vitamins contain with a unit of each food is

given below.

Bitamin Mil! 6eef 3ggs Minimumdaily

requirementsmg"

0

6

#

(

(22

(2

(

(2

(22

(2

(2

(2

(

:2

(2

cost 4s ( 4s(+(2 4s +:2

Formulate this LPP.


18/21

)" : Solve the LPP b! graphical method:

Minimi0e 9 %1 ; $$

Sub,ect to 61 ; $ 1*

1 ; $ < =

1 ; +$ 1$

1# $ *


19/21

0 toy company manufactures t7o types of

dolls+ 6asic Bersion 0 and Delu$e version 6+

3ach doll of 1ype 6 ta!es t7ice as long as to

produce as of type 0 and the company have

time to ma!e a ma$imum of *222 per day if it

produces only the basic version+ 1he supplyof plastic is sufficient to produce (:22 dolls

per day both 0 I 6" 1he delu$e version

requires a fancy dress of 7hich A22 per day is

available+ 1he profit is 4s+9 0nd 4s+ :respectively for doll 0 and 6+ Jo7 many dolls

must be manufactured to ma$imi&e the profit+


20/21

)" : Find the ma"imum value of 9 61 ;> $

Sub,ect to 1 ; $ +

%1 ; ?$ $+

1*1 ; >$ %6

1# $ @*

)" : Ma" 9 ?1 ; 6$

Sub,ect to $1 ; $ 6**

1 16*

$ $6*

1# $ *


21/21

)" : Solve the following graphicall!.

Minimi0e 9 $*1 ; 1* $Sub,ect to# 1 ; $ $ +*

% 1 ; $ %*

+ 1 ; %$ =*

1# $ *.

lpp formulation & graphical new (2).ppt

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