Operations Research an Introduction 10th Edition Taha Solutions Manual

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2-6

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2-8



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LINEAR PROGRAMMIN G

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2-9



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( 4)

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3



Chapter 2

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2-11



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r+ ·'l r .r = <:'t .r <:'t r I r+

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+ · r .! .! � ( M/t M.J

continued ...

2-12



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r

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0,DOOO

0,0000

0.0000

0,0000 ::::: O,OIIOCI

0.0000

0.0000

0.0000 1.0150

I 0000 40°.D41

O.DOOO o.oaoo 0.0000 1,0790

0.0000

••

•

• •

••

•

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o.oooa

0.0000

O ,DOOO

o

.::: 4,2111 0.0000 Z.304 7

o.oooa 1.11850

Sl1ck ( l/ Sur plutl J •

o.oooa

Z.DOOO Z.5000 5000 Z.5000 l.DODD l.5000 J.5000 4.0DDD 4.DOOO

O,INIOO 0.0000 0.00 0000 00 O 0000 0:0000

o

5.0000

0:0000

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n

•

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•

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Optimum solution: Invest S 100,00 000 0 in A in yr I and SI 70,000 in B in yr 2. Alternative o p pttimum: Invest $100,000 in Bin yr I and $300,000 in A in yr 3. X ( ( ..

&UreJj ,I ,I � � t 14,. , 14.S .:. �Z ,� ¥

:

}/ =??? �� X 3+ 1!. Xy -3>f ,+4 X , -7

ma:x ,mi2c

>f 1 -3 X z.z. + '/ X 3+ 4 X v 5 >f v

'Z' : .........,,,

{

s x, -

X 1

x , , X 3 ,

,

Xy

Y y Y ._ -1-t o x3-f 'i X ._

� o

cam m ,fe�.L .Lr r ��ca .. ��
2-13



Chapter 2

'·

-3

x; -t 4 )( ,_,_ -7 X 3

S- X'
!f•

S X 3· i + a � +- 4 X � � y.. I - X I

3 x x , - "i x2 + 10 X-.3-.f' xv -::, d• s,,o o x; -t X i-i- -t X 1 + Xv � s,, .

3 X ,X Xu x x x x 'f �" 'f ,_�

?! �

SOLlfTI lfTION ON SUMMARY - OPTIMUM SOL Titl itle e: Flnal iter ation No: 5

Ob j jective ective value (max) = 1175.0000

Variable x1

x2 xJ x4 x5y

value 0.0000 0.0000 287.5000 212.5000 1175.0000

Constraint

1 (>) 2 (>) 3 (>) 4 (<)

RHS 0.0000 0.0000 0.0000 500.0000 ·

.

·

Obj Coeff Obj Val Contrib 0.0000 0.0000 0.0000 0.0000 1.0000

:?,oo, z = - / / :?, 'F ,1, = /; / I .56 I .56 2 '1 7 = &3-1 / 7 tJO-l ��= 0 tJO /36 ·l

0.000 0 0.0000 0.0000 0.0000 1175.000 0

c, ,r r r : Do/!- , Do/!-

YSurp rplus( +) Slacl<(-YSu 0.0000 + 2262.5000+ 0.0000+ 0.0000-

t:

,t; x: ,

X?,t

I

0

0

2

0

,.q s :ir

X31:;

0

zoo

0

313. 53

0

37 .t/ t/ 3 5 37

0 0

3

1

5

5' - CJ O $ / 17 5'

J l/ l/ . 37 !J J

.30 z '3 7 .3 Y 7 3'1.6 Y

7

0 0

8

0

'l-7 Cf .

q

0

8 9�.I C, C,

0

0

b

l / ocafi.., f' .J. .J. g7 . S-0 --u cJ.q;<-<-- 3 /J l (1,-s,,.(/ $ .2. / :2. s O � ck,z :2 ,c. f . /Z ., w,i,,v = c.e J e J

0

10

sr .:»

0 0

C, 0

0

/2

2- 1 4

0

·

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently

exist. No portion of this mat erial may be reproduced, in any form or by any means, without permission in writing from the pub lisher.

Ohapterz'

[4 7

X 1,1, ,::, 4'1 1' cu , / u-lt! U4v,j/ 4'1 NU 1' J J , , ,v-v---h' 111111< 1,::, # i = #�/ X w w i ..._.:..... ""'- � � ""' / � wlc.""tj � wlc. Xwz = if Xe ,=# c� / wl. � �u/4 � X c c i i =.� c � w,f. '""""J � w,f. / 1
+ .1-2. xc .. + �-z X c c 3

f wto s" � f

1o12 e.,1 :S S" SO , X 1o1 X e.,

�

2

x , , , + X c c a.a.

� 2

+

c3 J( c

·i:,

P ft l1J :F": � ' f'r,.wr,f I=

t-1�')1./mize. z

'lo

s.1;.

w 3 X w

� / S oo S

K r r 3

� 12.oo

QPR�

= ru.

·

+ 3 X �

tit Xl(

ll- X .3 .3 + 2Xy T ll-

SDo

:!!;

o x 4 '!,,! '!,,!f 3x1 + 2x2 + X 3 +z. x 7 >< ><, + s x z, + .l.X.3 + X'4 S4SO

w 3- )( c c ,- X r x ..,..,, +:V <1,1 z +� X w r ,_,_ - Xe 1 .:i

-1-

/ 9 X ,:t 2 X 3 J. )( « / ,t

.:1 )( 1

"'1..

T

7 S- :J, X l lo -a x S-7,:'f

r l
W l l + X w w 3 + )( W

x.1.J ,+ Xi.)L Xe, +- x,2

�

=

o Z t

.X c , ,� , z o , X.,2 s

X l>/ l>/ 1

"' .1 .1

i

j= =�� i X j p � .w.c,J- j' J -:.-:. , , , -z, 3, 'f

So

X,

,l(u

XJ • X.t, ;ao

To!eP, $al,,1,jp,.. :

><,: z

.,ao

o, >
StJ

;( q=

so ;.:1. 7

s,

FJ : /-- T OR FJ
X e,

Z (b)

=

(I O (I l )(c3 = Z l

S 1< 1
:fl. !4, 918

J,,,.�

�

M 111 ,c 1,cirn , , z ' l!. l!.

� x, +ax. :r s x.3

© I?? ��

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:S. I .S .S 1111 0

•

. x.1-3 x ,.,.

� M.,l. �

=o

-;s:,ot. s x ,. -2.)(j =

x, � :loo, X z z;;: ; ;: i , o, -

"n'l �o�� , P d d �

eu/ . Aa.'o/ eu/

:i:. 4110

�7)(3 !.6 0 00

1.1. di. :r - X Ti x .... - x.,

-UIUd � -UIUd ��M

��

S O K .3.3

x, + · � X z + L x 3 .J x

�� ./2�- �·�

Zi ....-........ �1ar.Jf AU!v..' c/J '1>1. ..,. c/J '1>1.

z = � x , + J O.Y z r :::,0

CPTUIM SOL IIT IClt SIJtlMT

� �1.,0 -

z ... •12 f ltle1 Pr o ob bl mn IDI 4 f l l lt11• tf mn CbJ-=:tl n Ylh c-..J a410l1.DIZO

•

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v,t.. .Jl44] l24.Jl 216.ZT6Z S40 .S405

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•

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continued ... © 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as t hey currently


�traln t

2-15 I �:) Z

J

<> •

l•J

4 C J •

s ,.,

....... , U•az

....x3

IJIS

tlK k k C•> ISUrphat: ;

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4&.4165 ·

0.0000 ZDD..1111111 ZDD ZIIO.DDOD 151.0000

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0.0000

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o.oooa



Chapter 2 / i,

152

,1 t .e , j = �i ,1

J = £ ..J J I ..J � '7- Jj j i fT"arlLc.t fT"ar lLc.t i "'" ,_.B.

t

i 4ao

100

so j u

i

z

4a <1

411 ()

0 200

0

;l.00

....

�

0 l, Z.00

7 , 4 �o · trkl C!JR.f = :f 16 7 7 trkl

�. " Pr,� · annvu,,vr ..,.; ru.rk,, "· I SJ ..,.; r ru.rk,, ,= , =Gd ��· I . ·

i

S"too

loo

i

7Sl

� O

0

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p so e .,*: *: ,,

( 4500 .Cl

Sto

J

.3

/ OQ OQO

i

jl i. oo ':, j

I Z.. Z..oo

� .

I Z o-

6 X ,( -1- ?.'f �+ Z 6 ff\1 ,;.,;,,,, �c Z= �O X 1+ z. z. xl -1·

I, +- Z .._ + 3) Z .._ 3.!:( + I )

s. s. s

= Sao + I 1 ,-,. x'l,-= 4- fl fl o-to-t-[ i. I

0

X 3 = 4so-+ .L3 z r Z z 1j -t X y y = ,,,t �o

X· 6 400 , i.=S';,4 L • f: / / Oo , t -��3 .I;_

I o= z., = o

.......

ci , ,1r """m """m � :

d, ....4

l

•

300

t d /d t d CAAi" .,_

t � l

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S" O 3 �.Z S" t;3 t;

2-16 © 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as t hey currently


2 Chapter 2 X;

·

= Uni,(;

IJ �c,1- i .

ISS-44

J ·: ,, ...

, }·.,I 2

�:-- =�./,;w...'r macl.-i / = � -/,w.,.1-n-, "Jd...:. i, ·

1

l

/

;c;..,,'ze Z=- II O X 1 + I .M1111;c;.. I oo ( ;j.;j._� l K X 1-t oo + ) , S

f .

,

+

)( I

S

I

x , + x' ,.,. 5

;J.,- - J: -t

X'z. +

$

'7[

.::,

8'

I

+- "' - <7 1.1.

a ,

.,-1- = 8.

o;.

i + � 4 , cfz+ =
i-, · , .;;! 1'

> �- >

�-, y

RCArl,,,,,NJ... ><, = z.

=

;J...-#' � ()

:/, .,.

5"6 , ;J_

+

t; 2 s z

= 4 fu

I of

4.,

,.,- :. :x- , ,

,. =

o

C1

2-17 © 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as t hey currently


Chapter 2 2

155

� L.

..

�""' ' ' :Z

, "

?.,, JS:,?.

J<;::- 4,

=-

4L¥c,1.4,.

.,...,,a"...i .i � --..,,

32 -,.

l( 'I 'I :

·

' , �z If t;, JC,;_ -=.z, x ,. 7 q , X io" ' ' '

0

¥ "' � "'k'_ /,/,;. $cao•·d. � AAJI, ¥ Mf ".". Mf ( < •I : 111"' � . < � 7: sv.-. "', ) M ,;,; ,,, ,'z, C: Z = � + x.,. + � + Yy r X .r .r +l' , 'f

x , , ",

s.f . X ._

�

X3

Y7

Xy

Xs-

I

I

I

I

I

I

I

I

I

:!: IO

I

S �I S

1

I I

I

I

IS&

I

I

� 2o

� Jl / .ic;

= 4. x.. = �

.>

.x.,:: � J<.r """""'J � = "' .. 0

;/.��= 0 � "- (., � • ll .=- ;,,, : ';}..,. • ;/.

�-z.

8:0 1

I

'f:01

I

I

101 1

I

l

I

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I

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•

ll• l •

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I

?:4

I

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�3

I \ \ \

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4:o\

salMJ.tm: z = 'I X,:l,

2-18

e.rll; ,.d e.rll; ,.d sl

�= i,X.,== 3, lr 7=?

:

� .s I �3


exist. No portion of t his material may be reproduced, in any form or by any means, without permission in writing from the publisher.

.

•

Chapter 2

60 Let x , = Nbr . starting on day i and lasting f or 7 7 days st art art in g their 2 days off on day j yj, # and st da j Y u= Nbr . starting shift on day i and Thus, of the x1 workers who start on Monday, Y io will take T and W off , y13 will take W and Th off , and so on, on, as the f ollowing table shows.

x ,

X

:

I

X 3

x,

Y1:+Yu

. y13+ y y. ,

X 5 5

. y14+ y ,

2

y2-l+y25

3

y3l+y37

4

y-ll+y-17

y4l+y42

5

y5l+y57

y5l+y52

y52+y53

y53

6

y6l+y67

y6l+y62

y62+y63

y63+y64

y 64

7

y71

y7l+y72

y72+y73

y73+y74

y74+y75

Minimize Min imize

x.

x ,

s+ + y y , s y , ,.

Y • •

y2 5+ y26

y26+y2 7

y35+y36

y36 y3 6+y37

45+ + y4 6 y45

y46+y47

y5 6

y56+ y5 y57 7

�� y67 .,,--�. ,;(;,.�.,, y75

�'· f..' .Su'!�. ,"f ', '"rtr ·

•

z = x 1+x �+ x 3+ X 4+x s+x i;+x i;+x i i

Each Eac h employee has 2 days off : x , = sum{j in I.. 7 J# l lY u Y

y27 7 +y31+ y37 + y-ll+y-17+ y51+ y57 +y6l+y67 + y71)>= 12 aint: s-(y2 Mon Mo n (1) constr ai aint: s -(yl2 + y3 l + y-1 1+ y-12 + y5l+y52 + y 6l+ y6 2 + y7HJ72>= 1 8 Tuee (2) constr ai Tu str aint: s -(yl2+ yl3 + y23 + y-12 + y52 r ai Wed (3) const y52+y53 +y53 + y62+y63 + y72ry ·73>= 20

aint: s -(y13ry·l4 + y23 +y24+ y2-I + y53 Th (4) constr ai y53 + y63ry ·64 + y73 +y7-1>= 28

-l+ Fri (5) constraint: s -(yl-l +yl5 + y24 ry ·25+ y34 ry·JS + y-15+ y64 + y7-l+y75>= 32 Sat(6) const r ai y36 + y45+y46+ y56+y75>= 40 aint: s -(y 15ry·J6+ y25 +y26+ y35 + y36

aint: s -( yl6+ y26 +y27+ y36 + y37 ry 46+y47+ y56ry ·57+ y67)>= 40 Sun(?) constr ai ·

continued

2-19



Chapter 2 Solution: 42 employees Starting

On

Nbr Nb r off

Nbr

M

M

16

Tu

Wed

i _6

16

-

l;

'.'8

� �..),,,,�

Tu

8

Wed

8

Th

0

Fri

6

.8,..

.

'"'; "';"-]

Sun

2

.

Sun

-

C

' )

8

,._................

n

6

2

-

Sat

"'

� Sat

Fr i

Th

..:.

..e,

'

r:7 ,2

··

2

2

N br off

10

24

22

14

10

2

2

Nbr at work

32

18

20

28

32

40

40

.

Surplus above minimum

20

0

0

0

0

0

0

2- 20



- �"'

Xe..,. tJ br: 6b 'M .-a »

lw!; lw!;

b.-. 1 ,d. ,d... .. r xd = IJ b.-. xd lxu lx 7' - H � x ,.= N hr:1_ 7' x� = lld,;,)� ..,.. ft �

Chapter 2 161

·

tl ::: ( o x ,Nl � x x� z ::: l oo\ o xG+7 so x ,.,-12ou{ t e-r . X e !:S-a11 , K , J � soo,>
. , X 3�· S , )l ,,><�,,.. 1, X Y s } Y �I oX .3.3.,. 7 1111 Xy Xy ?:- .,_ ea 8U x , + ' O X z + S" X

l n,n, "111: Z -,. O p f f J lf1 ,"1 / -,. 3({. J / i1111 $ 1( = X .=I , X 3= X v v =·S" ' 1 J .f.J. .f.J.r)- � ,.1. continued .. .

2-21



Chapter 2 . x � 1·3 )( .,. , z

ss»; v

n,

,(

O,el-i' m ,-.:

Z =

K + I +·S U

v« T r r

•

2.
X l Z 7 l ,.,-=-Z Z 7 .o'f s- , x ,., =l oo,- X ,,,:.12. s-

" r r::: : J' oo Xe::

I

>
'I , �::: :i.s 3:/..S 'I

o

J,/e.,i j ja,,w} IAIU �.' � .1. .X, +3 )( ._._+ � x3 + X'f 1::. 8 5 5 ( �011+ I Ouj ·

i?>..... ..... ..rdwlr m : e.,1 O' p�i?> t/ e.,1 1 z •.J 3 p s;-,4t{ / . ss: , J s;.>(, ==

br-. if -4.: ><1=;.t br-. -4.: f, f,-{ �� l(i."

� 1 ol tnJ. tnJ.1 H ,.-

N ,,.

�

3,. N N �•-1 ¥ -( � � X . ,. 'iq

1,u� �

�"' I,I,,.

/,j�,m, ze z: t o,o, t oo�+ 12.ooo x,. + t .3 s- 011 0 K .3 ·

J./..

S x ' t t oo :t >< ><1+-i1:><.+4 X 3 +Xy,;;, 8 S' ·

x ,

x ,+ x ,-· X z --r 11.3 11.3?:o ._._+ x 3 �. s err ·S>t •

x p p-.,' "t "t -'2.\-3' x 1.:o x4 � X ,+:vr 3-+�
. « � t s s r t

)(._ =- x 3 = a x ,.,. ..,. / . 91 �

t1z =

;

�1�,'1 2 f '16 6 / .3S- -3,3"1�S 2 f, 2 o

-,

3

- 1>423, 7 lf-lf-0 .3� so , oao» ,,JI,.� s A z < � esn..f 'l

1 (! 1

a or e» &r
� "b d,.

'>'INI a �

.-A

�� -

!LO' O

d o }( 'f d qo }(3 + f 'f . � ( 0 '!, OtT O l a o '(, -,1 z.00 X 1 _+t ',t / l ',t qo

400 X, + 6 a" K i' s: Zoo, oa � f 1 X ' i' o Xa +4' S S 1' K l.l.+ B'

X, , , X',. , )(7 x., �o

,

X, = 339. JS" how,e s

. x ,.

0

X3

:. o

x'f

= / . ( 'l 'l

s: ..

t. 3,

a..u•

3 '7 �s 2.1 . zo



2-22



Chapter 2

1, 1

x s ...

�� ,..� l brf � � ,..

x b

t b � .t&; � 'f"l4.�

.

2

Xn =- lb

, ,, ,,

f . � � �clz-�

, ,. .. / /, "'9- � -p-v. r-�

M ,� z o

&

s· ,r.6 + 7 o x : + .!:!! x. I X 5 +l w n J<"q 30

/,

·

wY =- � + X .1, + X n -t X w-

, , J /. /.,

Xs

;i,

,/

Y

s Y , X 1:,1:, x , � . � s x , O

)( n �

Xw �

'f �

tUt

l Y , , ·I t'

so

!::

x .."",,lXi. � Y. n o

I

"VM,'.''� jllM. � "VM,'.

(. If (.

o pt ;,, ;,,"""" �' =-1, Y =Ci,q,1

r )(. =·I "' .,,--' x = s , , X,-=- ·Z r J A A W W =- �/. /2 X. 5

X0, ( A,.S , C = )

/ I, sr o-zvl .r .r 4'i, � It B c ll

' '

,a

,

-�

A, t:) Xr, ( A,

)(

c ,( 8 ,c

= //, /) � .,,;. w.u.t,, 11- , , C ) =- lb 9-ccrcon...io .....; Cui.Id,, 13. C

xa,CA,�c) "- / I, '1 o/ mrrv(,.,;, mrrv(,.,;, aMJ.

'

A, � c

� " {,t -t �B + "8c

x ,. C

� ... XrA +

X i.i. : j l.

-

-

,'1

.

c

B

4S' DOO

7 .s;,11<1 .s;

3-

4-s; 011 f1

7r, Of1CI

It

0 f tJ,O d0 tJ,O

'

1s + ic

Ya

=

x;; R X a B B + \c Rt X

"?!

•

� �-+

1-A + .l\;11

� � 1e:+{s-t1s

J

5

J

Ye.

/ S-0 1 Otl S-0 d d 3' t1 t111

00

o

� - x; c -+ \c + � c + � c

0

0 0

O

S' d S' d f )(' s = 80 fr ns, )(, = klo r ,11J , -Ka =I ms

/ d � Z = / 4 3� 0011 /

2-23 © 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all copyright laws as they currently


Chapter 2 2

continued ...

2 -2 4



Chapter 2 l d d 't/ Z N' R = 1, /, /, / 't/ 'I na,,l,m �..; "#JA, 1- Z

!,I p,, 1:, 1,1 )1 )1,y f r.y A!. � 4.i � ,'d 1716f " M.J if= 1,1, ,; , "1A1 A ..uetfl M.J if-

ur

Lfl = ;,1.1I d 1 d

fJO X z.z. + z / m , z z H ox9 + s s-1s, SO x, +t/ fJO n'J4')( ·z e z = / S H x s t. .

"J 1Ji!. ..u,wf"'" "fj..u-. .

�""4Uf' ..- F""",,.,,_ f � M.C M.C..Ul ..Ul � J 'e.1'e.1t(s""/ ;de"� _. / � ); / °"' ?,-. ); - /{ p f' 1) +111( ! _ 1) ! -i) )r 7o ( 14� ,;, , , ze z « 5 of? of?S- P -+ c:o:r - ) S J * ) l o f?.--t-l l l + f t )-{ z z t "3 ,. , ,'I -( l

.,,_

� ""'

� + -:iF ):!S' . !1 x qooo /x x , + ( - . I' -J .1 , Xi

.,.._

S , f

- ..,

R-,. le -R ;:.

s- 11 110 0

F+ f -- p .,..,,

110 7 11

oo

XI :. :/ S

/;n.s � w ,,t,k

X3

,,

.

= J ( ('J .'J.S" ian-o,,..,.. w�ck..

r r -

t>J .. •

ft..

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�

R +L'R R =N R 'P +L P P= Jv 'P

Ya i "

.,., N J" + L J' :[ .,.,

,,"""4:7�

.

= :/', 7 / / 7 3. 3. 17

7 o R ,q:

= :2 s: /ins '!"" we.• le..

. ...,. x. :: 400 -f d d....,. .r. �� , .s ·o ,1 , 11 s z

::: 11/ R R-1 -1 Af P -t - N t J · 3
Z

2.r

.r .64irn .64irn·cfo'm

x.

-=- "t oo --11" -=:r + J --

p/, ._., c:!r4t,:.,.,,: tf

"?:

/ , / z.

$:

O�

·

at.{ � a.M.

·

.x 3 � 2.)- , Yi, ?l O

l s )) -t: 9r l ll

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L + .X,3 + 'I .S.S- X

x , � « s , x. s

·

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- ( 3o A

� 4fl0 <7 <7 ,c • I

Xq

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bl. / ./,,.. I ./,,..

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c!o.ttl,.i. t

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s,s.

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;;( = 1G /l �.21

·

, B=- o ,

� =- se», P = 7 ot7 , J"

<7 o <7 'l continued ...



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'&,}

Z : :f,

/ O ,

I/

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/ "Z ( "Z ( 1 . cw � 2+Y e ) �/

sr s:

�c1 � oe1r ,,. � ,

.4,,,

� AM-

J.. / J 13.13. 1,. , / e;e; B = 7 o» , ..,, "' �t ,, Ga s� I p-,- ,..t,,..,+,+,..,,

' G'� !2 �c./, i,,,-=. '17!2,--t )Bi -'if 1� W ' ' ' c./,i,,, 1nr =3ta" ,8'1+'1 t 6· U - � 6

L/S°I

J

2-25



2 Ghap Gh apter 2

iJ

x ,"J · = �

.,ii... .,ii...

aJ/.qai,fJ/ 1o J!.:y A

i.

176

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s.u,b jeeci: t-o j

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+

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·

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A + 1<.36 � a o , ,o x .,., A -

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?/ A f a.ft ?/ A

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2 -2 6



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-

Chapter 2 X L· : N b ....

1y- o.tl. ,tfa- � (.: i = ��'f 78

/11 ,n ,' m ,it:. z. s» ,

= s-+ s-+ S'.-+ s •L

I

f{
J

-

+ s; -

•

T

+ S' 1 :·SIX 910,tJ • II

t'o,•••1" 3' 3' �000 -1 1,, ( t'o, )( � + !{ - s..-t = ·S" / 1<1<4 �0 011 , ••J 4o, 040+ 1ci.ooa ) K + ) s.-- s+ • s-J x <1011,od o ( 4o, •

3 • ;, � + ( f o , ,ooo -:t s ,o ,,o ) xy r s;- Sy,,. .,., x \l o� .uo 1 soo( x , + X i + x s + xy J :s;, 1001 tJ t1 t10 •

x , . iCJ , Xv

X 11

.;;. o

�·;

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- '

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19 «r

a.'1( 1°ml M a. z e Z : z

s s - s .

?f

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s> J h+ rJ � X,a + X z z3 XL1

80

2-27



Chapter 2

.fl r X .fl � 2q + � ' r

0 10

.r-,0 , xu , .s,. 1.r-, 0,0 x , 4 .,. x

r l r x,r -1 X l -1 r� 3S:to -t- X �r -t..,.'Lv r ..,.

)<,, -t X z. , ,-t � ,+:ry , , >- 3.S-20 .s .41,,,;,.

-,, , ,..,: z.c z-=- d,+<1,+· ·-tq10 M , ,- ,, ·

s ) . .

d ;

<1,_.-11i-b

!S

i._ .-

?:- d ,:,:

at = ]»

Q ,b,

�cU

d , >-.o .J;,�:

0 Al - A Z : � s = 17 6 60 ( 1,,, ..-M : X n;= o AJ ..-M A3..;. A A:i.: X z z s=- o •u ,-= 1 7 60 6 113 � Al( : x

.;.,.. .;.,

c... Y d d )

p1 ...... A 0 Az , 'Ss =- 17 { { 0 P t .... t� A'I : X 3 , '" 0 f'2 .... A z. : x"s = a f' A'f J � A 'f : X46 = ,1 ,0 ,10 , , 0 u1- .. , 0 3.2. , , ao»

/ Z "' :t '1S1 / I./ Z 'i! I./ i. + b .f 'i! o7

;ffe W L �W ;ffe � �!Zwl W 3

Y a'a' 6 0 c� - /J Y I = :i r /J I r, uo.,oa /f sa := 17 6 7 6o ., 4 = 3S :l 0 6o A .20 , 113-= t 7 /l;. � .:I S" :l 0 S" .2 , ,ee ,.kr i-o ce.Ji,.. ) ).,,;. 1) i sh,,.,c.u ( sh,,., .,,;. m ;,&,,, , Al A
8

P.1t 7

�

7 3

'

continued ...

continued ...

2-28



Chapter 2 .

I

I

I

II

'

.

III

gullar 200 r egu 200 r eserves

P1«� / rn rn, "%.�

s.e.

z .,.. T

T

x i.-= ef f, c t f u.� t u.

t , , ,, + t ,-...,..,. t ,3

"!,

·

c"

=

l e t;l l

S + K , G

·

�. +

°'3

�

t

+3• •• ( 1-. ) (1 )+ ,'l 'f , 91/z. ( S

:

,o

f . r l O .s /

•

+ >J >J33

ZI )(Z )(

3o

IM3

,,

.J.u

6

•

20 .

-..

O

= 8 7./ �ti �

Zo

-

•

:s><, �. 99

s x3

'l 't't

!: . �

•

'/ 'I'I

""= l
P !t>J I !t>J I

'If w,..':f. " ,

a sr r,ar,a-1,,.,,,.., :

QJJ,W.. �-..,

'f . (!{" Z 'f

·

11 :$ J<',

=- / .s; +9 ,l X z z33 + Y.1 •

z.( z. ( 300 ) { ,....,., )t ,'7 V l " ) , +·'l 'l 'l 'f'f ( .S' .S' oo )( l V( f ( ooo )C I -'!;, ) t ->f 4J .s ,ooo z xi!. I I O, O, • + t ooo( t 0

40

1"" _

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•

3a

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1<,)

x200 , ,, �.004 z o,,

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·

n.,. t. 3 + 3 / t ,... x .L-+ Y ,... . x t l l 3

�"' 2zo, ood

·•••

·

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t.7 t. / 7

U �aoo s-oa( t - ,<, ) ,;,, ,a ao{f !; !; X U

( t !:' o a ) )( !:' • '14( t ", )

.!!.l.!... .3o

S +10 z 13 " ·S S .z X',3

s:i . s:

11 11 2-0

7 S S +7, 'l x ,.+.Y,•

·

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+ I · S / oo o ) )( y S ( 6 6 ••• )1C.3 + ,18(

+ x ,..+{ X z , +- �2.-t- X i3 s; z oo 3 -r X

a1., + �3 +

65 165

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, ,, ( soo ) x;+-2s · ; ,.;u. z" 2 ( M

T � °t: i, + t:.... + -t.._ 3 X 11

'6 t' .,,.;t .,,.;t

99

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l�S"-1



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eff ww- � =

·

91.2 1.29 9

$1: K'l con co ntinued ...

2-29



2 Chapter 2 W ;. =- C ..,-. ..,-.�

if tJok. � { /
!Z t t = R� ...; � <' 1.1. = R� <' !

...,. �

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-

It-

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)+ 1 100 z :: / 110( .J x , , )

( .Z z ) . J( Z , z

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+s,

=

1000

) z +3 • ( 3'1' u ) + Z •C X .11 A X, z .11 ) ) + .s1 ,. Sb( 3 X u Aooo So ( �,t, ,J +J o( )+2 ° .1 ) 3) 4 Jt .1 ° f ll( l 3 l ( r v ) v + +S 'l oo so ( • + 3o( r 't o( a x x :,., ) +sv =.12 00 :,., l i l ,,,,,.,,/ .sj· � o Al ,t ,,,,,.,,/ •

=

"O

.

•



z

4

3

I

K ).5

3

.2.

7 ·S'

� du,,d � ....... 4,1-� �• '/Pt? C«zl = /; 121 , , '/Pt? continued ...

.

2-30

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Operations Research an Introduction 10th Edition Taha Solutions Manual

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