0% found this document useful (0 votes)
18 views2 pages

3

Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
0% found this document useful (0 votes)
18 views2 pages

3

Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

 /LQHDU,QGHSHQGHQFH 

 )LQGWKHJHQHUDOÁRZSDWWHUQRIWKHQHWZRUNVKRZQLQWKH VKRZQLQWKHÀJXUH )ORZUDWHVDUHLQFDUVPLQXWH


ÀJXUH $VVXPLQJWKDWWKHÁRZVDUHDOOQRQQHJDWLYH ZKDWLV E 'HVFULEHWKHJHQHUDOWUDIÀFSDWWHUQZKHQWKHURDGZKRVH
WKHVPDOOHVWSRVVLEOHYDOXHIRU x4 " ÁRZLV x5 LVFORVHG
B F :KHQ x5 D 0 ZKDWLVWKHPLQLPXPYDOXHRI x4 "
x2 100
A A x1 B
x1 x3
80 100
x4 80 x5 x4 x2
C

 D )LQGWKHJHQHUDOÁRZSDWWHUQRIWKHQHWZRUNVKRZQLQWKH 90 90
D x3 C
ÀJXUH
E $VVXPLQJWKDWWKHÁRZPXVWEHLQWKHGLUHFWLRQVLQGL  ,QWHUVHFWLRQVLQ(QJODQGDUHRIWHQFRQVWUXFWHGDVRQHZD\
FDWHG ÀQGWKHPLQLPXPÁRZVLQWKHEUDQFKHVGHQRWHG ´URXQGDERXWVµ VXFKDVWKHRQHVKRZQLQWKHÀJXUH $VVXPH
E\ x2  x3  x4  DQG x5  WKDWWUDIÀFPXVWWUDYHOLQWKHGLUHFWLRQVVKRZQ )LQGWKH
JHQHUDO  VROXWLRQ  RI  WKH  QHWZRUN  ÁRZ )LQG  WKH  VPDOOHVW
30 40
SRVVLEOHYDOXHIRU x6 

A C x1
80 x2 x5 100
B A B
x1 x6 60 70
x6 x2
60 x3 x4 90 80 100
E D
F C
x5 x4 x3
E D
20 40

 D )LQGWKHJHQHUDOWUDIÀFSDWWHUQRIWKHIUHHZD\QHWZRUN 80 90

40-65*0/4 50 13"$5*$& 130#-&.4

 :ULWHWKHSHUFHQWDJHVDVGHFLPDOV 6LQFHDOORXWSXWPXVWEHWDNHQLQWRDFFRXQW HDFK


FROXPQPXVWVXPWR 7KLVIDFWKHOSVWRÀOOLQDQ\PLVVLQJHQWULHV

'LVWULEXWLRQRI2XWSXWIURP
$JULFXOWXUH 0LQLQJ 0DQXIDFWXULQJ 3XUFKDVHGE\
   $JULFXOWXUH
   0LQLQJ
   0DQXIDFWXULQJ

 6LQFH x5 ' 500 WKH  HTXDWLRQV  ' DQG  $ IRU x1 DQG x2 LPSO\  WKDW x1 & 100
DQG x2 ' 700 7KHIDFWWKDW x5 & 0 LPSOLHVWKDW x1 ' 600 DQG x2 & 200 6R
100 ' x1 ' 600 DQG 200 ' x2 ' 700

 -*/&"3 */%&1&/%&/$&


7KHKRPRJHQHRXVHTXDWLRQVLQ6HFWLRQ FDQEHVWXGLHGIURPDGLIIHUHQWSHUVSHFWLYH
E\ZULWLQJWKHPDVYHFWRUHTXDWLRQV ,QWKLVZD\ WKHIRFXVVKLIWVIURPWKHXQNQRZQ
VROXWLRQVRI A[ D  WRWKHYHFWRUVWKDWDSSHDULQWKHYHFWRUHTXDWLRQV
 $)"15&3  /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD

)RULQVWDQFH FRQVLGHUWKHHTXDWLRQ
2 3 2 3 2 3 2 3
1 4 2 0
x1 4 2 5 C x2 4 5 5 C x3 4 1 5 D 4 0 5 
3 6 0 0
7KLV  HTXDWLRQ  KDV  D  WULYLDO  VROXWLRQ RI  FRXUVH ZKHUH x1 D x2 D x3 D 0 $V  LQ
6HFWLRQ  WKHPDLQLVVXHLVZKHWKHUWKHWULYLDOVROXWLRQLVWKH RQO\RQH

%&'*/*5*0/ $QLQGH[HGVHWRIYHFWRUV fY1 ; : : : ; Yp g LQ Rn LVVDLGWREH OLQHDUO\LQGHSHQGHQW


LIWKHYHFWRUHTXDWLRQ
x1 Y1 C x2 Y2 C ! ! ! C xp Yp D 
KDVRQO\WKHWULYLDOVROXWLRQ 7KHVHW fY1 ; : : : ; Yp g LVVDLGWREH OLQHDUO\GHSHQGHQW
LIWKHUHH[LVWZHLJKWV c1 ; : : : ; cp  QRWDOO]HUR VXFKWKDW
c1 Y1 C c2 Y2 C ! ! ! C cp Yp D  

(TXDWLRQ  LVFDOOHGD OLQHDUGHSHQGHQFHUHODWLRQ DPRQJ Y1 ; : : : ; Yp ZKHQWKH


ZHLJKWVDUHQRWDOO]HUR $QLQGH[HGVHWLVOLQHDUO\GHSHQGHQWLIDQGRQO\LILWLVQRW
OLQHDUO\LQGHSHQGHQW )RUEUHYLW\ ZHPD\VD\WKDW Y1 ; : : : ; Yp DUHOLQHDUO\GHSHQGHQW
ZKHQ  ZH  PHDQ  WKDW fY1 ; : : : ; Yp g LV  D  OLQHDUO\  GHSHQGHQW  VHW :H  XVH  DQDORJRXV
WHUPLQRORJ\IRUOLQHDUO\LQGHSHQGHQWVHWV
2 3 2 3 2 3
1 4 2
&9".1-&  /HW Y1 D 4 2 5 Y2 D 4 5 5 DQG Y3 D 4 1 5
3 6 0
D 'HWHUPLQHLIWKHVHW fY1 ; Y2 ; Y3 g LVOLQHDUO\LQGHSHQGHQW
E ,ISRVVLEOH ÀQGDOLQHDUGHSHQGHQFHUHODWLRQDPRQJ Y1  Y2  DQG Y3 
40-65*0/
D :HPXVWGHWHUPLQHLIWKHUHLVDQRQWULYLDOVROXWLRQRIHTXDWLRQ  DERYH 5RZRSHU
DWLRQVRQWKHDVVRFLDWHGDXJPHQWHGPDWUL[VKRZWKDW
2 3 2 3
1 4 2 0 1 4 2 0
42 5 1 0 5 % 4 0 "3 "3 0 5
3 6 0 0 0 0 0 0
&OHDUO\ x1 DQG x2 DUHEDVLFYDULDEOHV DQG x3 LVIUHH (DFKQRQ]HURYDOXHRI x3
GHWHUPLQHVDQRQWULYLDOVROXWLRQRI   +HQFH Y1 ; Y2 ; Y3 DUHOLQHDUO\GHSHQGHQW DQG
QRWOLQHDUO\LQGHSHQGHQW 
E 7RÀQGDOLQHDUGHSHQGHQFHUHODWLRQDPRQJ Y1  Y2  DQG Y3  FRPSOHWHO\URZUHGXFH
WKHDXJPHQWHGPDWUL[DQGZULWHWKHQHZV\VWHP
2 3
1 0 "2 0 x1 " 2x3 D 0
40 1 1 05 x2 C x3 D 0
0 0 0 0 0D0
7KXV x1 D 2x3  x2 D "x3  DQG x3 LVIUHH &KRRVHDQ\QRQ]HURYDOXHIRU x3 ³VD\
x3 D 5 7KHQ x1 D 10 DQG x2 D "5 6XEVWLWXWHWKHVHYDOXHVLQWRHTXDWLRQ  DQG
REWDLQ
10Y1 " 5Y2 C 5Y3 D 
7KLVLVRQH RXWRILQÀQLWHO\PDQ\ SRVVLEOHOLQHDUGHSHQGHQFHUHODWLRQVDPRQJ Y1 
Y2  DQG Y3 

You might also like