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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