1.operations research.ppt

8/15/2019 1.Operations Research.ppt

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OperationsResearch


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Operations Research may be

described as a scientifc approach to

decision making that involves the

operations o organizational systems.


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Linear ProgrammingProblems (LPP)


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LPP in general are concerned with the

allocation o resources such as labour,materials, machinery, capital etc in the

best possible manner, so that costs are

minimized or profts are maimized.

LP is a mathematical techni!ue or

fnding the optimal use or an

organization"s scarce resources.


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#he LP model includes $ basic elements%Decision variables that we seek to

determine.

Objective &goal' that we seek to

optimize.

Constraints that we need to satisy.


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( home decorator manuactures two types

o lamps, ( and ;. ;oth lamps go through

two technicians, frst a cutter and then a

fnisher. ( lamp o type ( re!uires hours

o the cutter"s time and 5 hour o the

fnisher"s time. ( lamp o type ; re!uires 5

hour o the cutter"s time and hours o

the fnisher"s time.


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#he cutter has 56/ hours and the fnisher

has => hours o available time each month.

#he proft on type ( lamp is Rs >6?+ and on

type ; is Rs 556?+. @ow many lamps o

type ( and ; should be manuactured so as

to maimize the proftA *ormulate the

problem. (ssume that the manuacturer

can sell all the lamps he manuactures.


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#ype owork

#imere!d. ortype (

#imere!d. ortype ;

(vailabletime

8utting 5 56/

*inishing 5 =>

Proft?unit Rs>6?+ Rs556?+


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)aimize B + >6- 5560

Cub1ect to%

- 0 2 56/

- 0 2 =>

-,0 7 6.

-

Do. o lamps o type (.

0 Do. o lamps o type ;.


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( company sells two diEerent products, (

and ;. #he company makes a proft o

Rs/6?+ and Rs$6?+ per unit on product (

and ; respectively. #he products are

produced in a common production process

and sold in two diEerent markets. #he

production has a capacity o $6,666 man

hours per month.


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Ft takes $ hours to produce one unit o (

and 5 hour to produce one unit o ;. (

market survey revealed that only a

maimum o G,666 units o ( and 5,666

units o ; can be sold per month. @ow

many units o product ( and ; should be

manuactured so as to maimize the

proftA *ormulate the problem.


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)aimize B + /6- $60

Cub1ect to%

$- 0 2 $6,666

- 2 G,666

0 2 5,666

-,0 7 6.

- Do. o units o product (.

0 Do. o units o product ;.


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( person re!uires a minimum o 56, 5

and 5 units o chemicals (, ; and 8

respectively or his garden. ( li!uid

product contains 3, and 5 units o (,;

and 8 respectively in a 1ar. ( dry

product contains 5, and / units o (, ;

and 8 respectively per carton.


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F the li!uid product sells at Rs $66?+

per 1ar and the dry product sells at Rs

66?+ per carton. @ow much o each

should be purchased to minimize the

cost and to meet the re!uirementsA

*ormulate the problem.


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#ype ochemical

8hemicalsre!d?unito li!uid

prod.

8hemicalsre!d?unit

o dryprod.

)inimumre!uireme

nt ochemicals

( 3 5 56

; 5

8 5 / 5

8ost Rs $66?+ Rs 66?+


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)inimize B + $66- 660

Cub1ect to%

3- 0 7 56

- 0 7 5

- /0 7 5

-,0 7 6.

- Do. o 1ars o li!uid product.

0 Do. o cartons o dry product.


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Graphical Method


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Hraphical )ethod can be used to

solve a Linear Programming

Problem with two decision

variables.


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)aimize B + $- 30

Cub1ect to%

- 2 /

0 2 5

$- 0 2 5G

-,0 7 6.

Eample !"

- + /

0 + >

$- 0 + 5G


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0+>

-+/$ - . , 0 +

5 G &,>'


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I;right PaintsJ produces interior and

eterior paints rom two raw materials

)5 and ). the ollowing table provides

the basic data o the problem.


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#ons o raw materialsper ton.

)a dailyavailabilit

y

&tons' / /

) 5 >Proft perton &Rs5666'

3 /


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( market survey restricts the maimum

daily demand o interior paint to tons.(lso the daily demand or interior paint

cannot eceed that o eterior paint by

more than 5 ton. #he company wants to

determine the optimum product mi o

eterior and interior paints that

maimizes the total daily proft.


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- tons produced o eterior paints

0 tons produced o interior paints

)aimize B + 3- /0

Cub1ect to%

>- /0 2 /

- 0 2 >

4- 0 2 5

0 2 -,0 7 6.


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0+

4- 0+5

$- 0+5

- 0+>

&$,5.3'


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IDew lieJ arm daily uses at least

G66kg o special eed. #he special eed

is a miture o corn and soybean with

the ollowing !uantities.


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Kg per kg o eed 8ost

&Rs?kg'Protein *iber

8orn 6.6 6.6 $6

Coybean 6.> 6.6> 6

#he dietary re!uirements o the eed mi

stipulates at least $6M protein and at most

3M fber. *ind the optimum solution.


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)in B + $6- 60

Cub1ect to%

- 0 7 G66

6.6- 6.>0 7 &- 0'6.$

6.6- 6.6>0 2 &- 0'6.63-,0 7 6

- Do. o kgs o 8orn

0 Do. o kgs. o Coybean


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)in B + $6- 60

Cub1ect to% - 0 7 G66

6.5- N 6.$0 2 6

6.6$- N 6.650 7

6

-,0 7 6

)in B + $6- 60

Cub1ect to% - 0 7 G66

=- N 560 2 6

$- N 0 7 6

-,0 7 6


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-0+G66

=-4560+6

$-40+6


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@ome ork

9uestion 5%

)aimize B + 666- $660

Cub1ect to%

/66- >660 2 >666

/66- 660 2 /666

-,0 7 6.


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9uestion %

)aimize B + -

0

Cub1ect to%

- 0 2 3

- 4 0 2

-,0 7 6.

9uestion $%

)inimize B + 4/- >0

Cub1ect to%

4- >0 7 /

- 4 0 2 =

- G0 2 G6

-,0 7 6.


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Fn some LP models, the values o the

variables maybe increased indefnitely

without violating any o the constraints. Fe.

the solution space is unbounded in at least

one direction. (s a result the ob1ective

unction value may increase &in a

maimization problem' or decrease &in a

minimization problem' indefnitely.

nbounded Colution


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nboundedColution

1.operations research.ppt

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