18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare...

12
= 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ ëÉã= ^ÄëÅÜåáíí= ÉáåáÖÉ= ïÉëÉåíäáÅÜÉ= ^åïÉåÇìåÖëÑ®ääÉ= ÇÉê= äáåÉ~êÉå= ^äÖÉÄê~= ÑΩê= ÇáÉ= _ÉíêáÉÄëJ ïáêíëÅÜ~ÑíëäÉÜêÉ=Ç~êÖÉëíÉääí=ïÉêÇÉåK= 18.1 Rohstoffe und Endprodukte ^ìÑÖ~ÄÉW= wìê= mêçÇìâíáçå= îçå= ÇêÉá= båÇéêçÇìâíÉå= b N I=b O =ìåÇ=b P = ïÉêÇÉå= ÑΩåÑ= oçÜëíçÑÑÉ= o N I= o O I=o P I=o Q =ìåÇ=o R =ÄÉå∏íáÖíK=aáÉ=sÉêÄê~ìÅÜëåçêãÉå=ëáåÇ=áå=ÑçäÖÉåÇÉê=§ÄÉêëáÅÜí=~åÖÉÖÉÄÉåW== sçã=oçÜëíçÑÑ=o N =ïÉêÇÉå=ÄÉå∏íáÖíW== cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b N W= U=jb= cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb= îçã=båÇéêçÇìâí=b O W= T=jb= cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b P W= NM=jb= sçã=oçÜëíçÑÑ=o O =ïÉêÇÉå=ÄÉå∏íáÖíW== cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b N W= NO=jb= cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb= îçã=båÇéêçÇìâí=b O W= S=jb= cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b P W= Q=jb= sçã=oçÜëíçÑÑ=o P =ïÉêÇÉå=ÄÉå∏íáÖíW== cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b N W= V=jb= cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb= îçã=båÇéêçÇìâí=b O W= V=jb= cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b P W= NQ=jb= sçã=oçÜëíçÑÑ=o Q =ïÉêÇÉå=ÄÉå∏íáÖíW== cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b N W= R=jb= cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb= îçã=båÇéêçÇìâí=b O W= NM=jb= cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b P W= OM=jb= sçã=oçÜëíçÑÑ=o R =ïÉêÇÉå=ÄÉå∏íáÖíW== cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b N W= OT=jb= cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb= îçã=båÇéêçÇìâí=b O W= NU=jb= cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê= jb=îçã=båÇéêçÇìâí=b P W= OM=jb= ~F=tÉäÅÜÉ=oçÜëíçÑÑãÉåÖÉå=ïÉêÇÉå=ÄÉå∏íáÖíI=ïÉåå=îçå=ÇÉã=båÇéêçÇìâí=b N =ÖÉå~ì=OMM=jbI= îçå=b O =ÖÉå~ì=QMM=jb=ìåÇ=îçå=b P =ÖÉå~ì=PMM=jb=ÜÉêÖÉëíÉääí=ïÉêÇÉå=ëçääÉå\= ÄF=aáÉ=òìê=sÉêÑΩÖìåÖ=ëíÉÜÉåÇÉå=oçÜëíçÑÑãÉåÖÉå=ëáåÇ=ÇìêÅÜ=ÇÉå=sÉâíçê=Ä=ÖÉÖÉÄÉå=ãáí= ENUKMNF= = táÉ= îáÉäÉ= báåÜÉáíÉå= ÇÉê= bêòÉìÖåáëëÉ= b N I= b O = ìåÇ= b P = â∏ååÉå= éêçÇìòáÉêí= ïÉêÇÉåI= ïÉåå= ÇáÉ= îçêÜ~åÇÉåÉå=oçÜëíçÑÑãÉåÖÉå=~ìëÖÉëÅÜ∏éÑí=ïÉêÇÉå=ëçääÉå\== ( ) 4520 2600 2300 1480 1780 = b

Transcript of 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare...

Page 1: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

=

18 Lineare Algebra und Betriebswirtschaftslehre

få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

ëÉã=^ÄëÅÜåáíí=ÉáåáÖÉ=ïÉëÉåíäáÅÜÉ=^åïÉåÇìåÖëÑ®ääÉ=ÇÉê= äáåÉ~êÉå=^äÖÉÄê~= ÑΩê=ÇáÉ=_ÉíêáÉÄëJ

ïáêíëÅÜ~ÑíëäÉÜêÉ=Ç~êÖÉëíÉääí=ïÉêÇÉåK=

18.1 Rohstoffe und Endprodukte

^ìÑÖ~ÄÉW=wìê=mêçÇìâíáçå=îçå=ÇêÉá=båÇéêçÇìâíÉå=bN

I=bO

=ìåÇ=bP

=ïÉêÇÉå=ÑΩåÑ=oçÜëíçÑÑÉ=oN

I=

oO

I=oP

I=oQ

=ìåÇ=oR

=ÄÉå∏íáÖíK=aáÉ=sÉêÄê~ìÅÜëåçêãÉå=ëáåÇ=áå=ÑçäÖÉåÇÉê=§ÄÉêëáÅÜí=~åÖÉÖÉÄÉåW==

sçã=oçÜëíçÑÑ=oN

=ïÉêÇÉå=ÄÉå∏íáÖíW==

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bN

W=

U=jb=

cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb=

îçã=båÇéêçÇìâí=bO

W=

T=jb=

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bP

W=

NM=jb=

sçã=oçÜëíçÑÑ=oO

=ïÉêÇÉå=ÄÉå∏íáÖíW==

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bN

W=

NO=jb=

cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb=

îçã=båÇéêçÇìâí=bO

W=

S=jb=

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bP

W=

Q=jb=

sçã=oçÜëíçÑÑ=oP

=ïÉêÇÉå=ÄÉå∏íáÖíW==

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bN

W=

V=jb=

cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb=

îçã=båÇéêçÇìâí=bO

W=

V=jb=

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bP

W=

NQ=jb=

sçã=oçÜëíçÑÑ=oQ

=ïÉêÇÉå=ÄÉå∏íáÖíW==

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bN

W=

R=jb=

cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb=

îçã=båÇéêçÇìâí=bO

W=

NM=jb=

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bP

W=

OM=jb=

sçã=oçÜëíçÑÑ=oR

=ïÉêÇÉå=ÄÉå∏íáÖíW==

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bN

W=

OT=jb=

cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb=

îçã=båÇéêçÇìâí=bO

W=

NU=jb=

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bP

W=

OM=jb=

~F=tÉäÅÜÉ=oçÜëíçÑÑãÉåÖÉå=ïÉêÇÉå=ÄÉå∏íáÖíI=ïÉåå=îçå=ÇÉã=båÇéêçÇìâí=bN

=ÖÉå~ì=OMM=jbI=

îçå=bO

=ÖÉå~ì=QMM=jb=ìåÇ=îçå=bP

=ÖÉå~ì=PMM=jb=ÜÉêÖÉëíÉääí=ïÉêÇÉå=ëçääÉå\=

ÄF=aáÉ=òìê=sÉêÑΩÖìåÖ=ëíÉÜÉåÇÉå=oçÜëíçÑÑãÉåÖÉå=ëáåÇ=ÇìêÅÜ=ÇÉå=sÉâíçê=Ä=ÖÉÖÉÄÉå=ãáí=

ENUKMNF= =

táÉ= îáÉäÉ= báåÜÉáíÉå=ÇÉê= bêòÉìÖåáëëÉ= bN

I= bO

= ìåÇ=bP

= â∏ååÉå=éêçÇìòáÉêí=ïÉêÇÉåI=ïÉåå=ÇáÉ=

îçêÜ~åÇÉåÉå=oçÜëíçÑÑãÉåÖÉå=~ìëÖÉëÅÜ∏éÑí=ïÉêÇÉå=ëçääÉå\==

( )45202600230014801780=

Tb

Page 2: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NU==iáåÉ~êÉ=^äÖÉÄê~=ìåÇ=_ÉíêáÉÄëïáêíëÅÜ~ÑíëäÉÜêÉ=OSQ=

i∏ëìåÖW==

wì=~F=jáí=eáäÑÉ=ÉáåÉê=ëç=ÖÉå~ååíÉå=sÉêÄê~ìÅÜëã~íêáñ=^=ä®ëëí=ëáÅÜ=ÇáÉ=çÄáÖÉI=ëÉÜê=ìãÑ~åÖJ

êÉáÅÜÉ=ìåÇ=îÉêïáêêÉåÇÉ=sÉêÄ~äÄÉëÅÜêÉáÄìåÖ=ÇÉê=sÉêÄê~ìÅÜëåçêãÉå=ïÉëÉåíäáÅÜ=âΩêòÉê=ìåÇ=

~ìëë~ÖÉâê®ÑíáÖÉê=ÑçêãìäáÉêÉåW==

ENUKMOF= =

bÄÉåëç= â~åå= ÇáÉ= wìë~ããÉåëíÉääìåÖ= ÇÉê= ÖÉéä~åíÉå= mêçÇìâíáçåëãÉåÖÉå= ÇÉê= ÇêÉá= bêòÉìÖJ

åáëëÉ=áå=ÉáåÉê=j~íêáñ=_=ãáí=ÉáåÉê=pé~äíÉ=ìåÇ=ÇêÉá=wÉáäÉåI=~äëç=áå=ÇÉã=pé~äíÉåîÉâíçê==

ENUKMPF= =

ÉêÑçäÖÉåK=aìêÅÜ=ÇáÉ=j~íêáòÉå=^=ìåÇ=_=áëí=ÇáÉ=^ìÑÖ~ÄÉ=âä~ê=ÄÉëÅÜêáÉÄÉåK==

aáÉ=åìå=~åëíÉÜÉåÇÉ=i∏ëìåÖ=ÇÉë=mêçÄäÉãëI=ÇK=ÜK=ÇáÉ=bêãáííäìåÖ=ÇÉê=ÉêÑçêÇÉêäáÅÜÉå=jÉåÖÉå=

ÇÉê=oçÜëíçÑÑÉ=ÑΩê=ÇáÉ=ÖÉéä~åíÉå=mêçÇìâíáçåëãÉåÖÉåI=ÑΩÜêí=ìåë=~ìÑ=ÇáÉ=^ìÑÖ~ÄÉI=ÇáÉ=j~íêáJ

òÉå=^=ìåÇ=_=Eáå=ÇáÉëÉê=oÉáÜÉåÑçäÖÉF=å~ÅÜ=ÇÉå=oÉÖÉäå=ÇÉê=j~íêáòÉåãìäíáéäáâ~íáçå=ãáíÉáå~åJ

ÇÉê=òì=ãìäíáéäáòáÉêÉåK==

wìê= mêΩÑìåÖI= çÄ= ÇáÉëÉ= jìäíáéäáâ~íáçå= ΩÄÉêÜ~ìéí= ã∏ÖäáÅÜ= áëíI= ëçïáÉ= òìê= å~ÅÜÑçäÖÉåÇÉå=

aìêÅÜÑΩÜêìåÖ= ÇÉê=jìäíáéäáâ~íáçå= îÉêïÉåÇÉå=ïáê= ~ã= ÄÉëíÉå= Ç~ë= pÅÜÉã~= îçå= c~äâ= EëáÉÜÉ=

pÉáíÉ=OOV=áã=^ÄëÅÜåáíí=NRKRKNFK=a~ë=bêÖÉÄåáë=áëí=áå=_áäÇ=NUKN=~ÄòìäÉëÉåK=

=

_áäÇ=NUKNW=j~íêáòÉåãìäíáéäáâ~íáçå=áã=pÅÜÉã~=îçå=c~äâ=

wìÉêëíW=aáÉ=jìäíáéäáâ~íáçå=îçå=^=ãáí=_= áëí=ã∏ÖäáÅÜK=^äë=mêçÇìâí=^⋅_=ÉåíëíÉÜí=ÉáåÉ=j~íêáñ=

ãáí=ÉáåÉê=pé~äíÉ=ìåÇ=ÑΩåÑ=wÉáäÉåI=~äëç=Éáå=pé~äíÉåîÉâíçêK==

=

201827

20105

1499

4612

1078

A

=

300

400

200

B

Page 3: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NUKN==oçÜëíçÑÑÉ=ìåÇ=båÇéêçÇìâíÉ= OSR=

aÉå=êÉëìäíáÉêÉåÇÉå=pé~äíÉåîÉâíçê=â~åå=ã~å= áå=_áäÇ=NUKN=êÉÅÜíë=ìåíÉå=~ÄäÉëÉåK=bê= áëí=Ççêí=

ÇìåâÉä=ÜÉêîçêÖÉÜçÄÉåK==

=

a~ãáí=ÉêÜ~äíÉå=ïáê=ÇáÉ=i∏ëìåÖ=ÇÉê=^ìÑÖ~ÄÉ=~FW==

cΩê=ÇáÉ=ÖÉïΩåëÅÜíÉ=mêçÇìâíáçåëãÉåÖÉå=ÇÉê=bêòÉìÖåáëëÉ=bN

I=bO

=ìåÇ=bP

=ïÉêÇÉå=áåëÖÉë~ãí=

TQMM=jb=ÇÉë=oçÜëíçÑÑë=oN

I=SMMM=jb=ÇÉë=oçÜëíçÑÑë=oO

I=VSMM=jb=ÇÉë=oçÜëíçÑÑë=oP

I=NNMMM=

jb=ÇÉë=oçÜëíçÑÑë=oQ

=ìåÇ=NUSMM=jb=ÇÉë=oçÜëíçÑÑë=oR

=ÄÉå∏íáÖíK===

wì=ÄFW=aáÉëÉ=^ìÑÖ~ÄÉåëíÉääìåÖ=ÑΩÜêí=ìåë=~ìÑ=Éáå=äáåÉ~êÉë=däÉáÅÜìåÖëëóëíÉãI=ÇÉåå=ïáê=ãΩëJ

ëÉå=ÇêÉá=ìåÄÉâ~ååíÉ=mêçÇìâíáçåëãÉåÖÉå=ÉêãáííÉäåW==

kÉååÉå=ïáê=ëáÉ==

ñN

=Ó=ÑΩê=ÇáÉ=ìåÄÉâ~ååíÉ=^åò~Üä=ÇÉê=jÉåÖÉåÉáåÜÉáíÉå=ÇÉë=bêòÉìÖåáëëÉë=bN

I===

ñO

=Ó=ÑΩê=ÇáÉ=ìåÄÉâ~ååíÉ=^åò~Üä=ÇÉê=jÉåÖÉåÉáåÜÉáíÉå=ÇÉë=bêòÉìÖåáëëÉë=bO

I===

ñP

=Ó=ÑΩê=ÇáÉ=ìåÄÉâ~ååíÉ=^åò~Üä=ÇÉê=jÉåÖÉåÉáåÜÉáíÉå=ÇÉë=bêòÉìÖåáëëÉë=bP

=K===

^ìë=ÇÉê=_áä~åòI=Ç~ëë=ÇáÉ=mêçÇìâíÉ=~ìë=ÇÉå=råÄÉâ~ååíÉå=ñN

I=ñO

=ÄòïK=ñP

=ìåÇ=ÇÉã= àÉïÉáäë=

éêç=jÉåÖÉåÉáåÜÉáí=EjbF=ÉáåòìëÉíòÉåÇÉå=oçÜëíçÑÑ=ÇáÉ=oÉëëçìêÅÉå=~ìëëÅÜ∏éÑÉå=ëçääÉåI=ÑçäÖí=

Éáå=äáåÉ~êÉë=däÉáÅÜìåÖëëóëíÉã=ãáí=ÑΩåÑ=däÉáÅÜìåÖÉå=ÑΩê=ÇáÉ=ÇêÉá=råÄÉâ~ååíÉ=ñN

I=ñO

=ìåÇ=ñP

W=

===

_áäÇ=NUKOW=§ÄÉêÄÉëíáããíÉë=äáåÉ~êÉë=däÉáÅÜìåÖëëóëíÉã=

aáÉëÉë=däÉáÅÜìåÖëëóëíÉãI= ÇáÉ=_áäÇìåíÉêëÅÜêáÑí= îçå=_áäÇ= NUKO= ë~Öí= Éë= ëÅÜçå= ~ìëI= áëí= äáåÉ~ê=

ìåÇ=ΩÄÉêÄÉëíáããíW=eáÉê=ãΩëëÉå=ïáê=ìåëI= ®ÜåäáÅÜ=ïáÉ=ÄÉá=ìåíÉêÄÉëíáããíÉå= äáåÉ~êÉå=däÉáJ

ÅÜìåÖëëóëíÉãÉå=EëáÉÜÉ=^ÄëÅÜåáíí=NTKQKN=~ìÑ=pÉáíÉ=ORUF=Ç~ê~ìÑ=ÉáåëíÉääÉåI=Ç~ëë=Éë=âÉáåÉI=ÉáåÉ=

çÇÉê=ìåÉåÇäáÅÜ=îáÉäÉ=i∏ëìåÖÉå=ÖÉÄÉå=â~ååK==

i~ëëÉå=ïáê=ìåë=ïáÉÇÉê=ãáí=eáäÑÉ=ÇÉë=d~ì≈ëÅÜÉå=^äÖçêáíÜãìë=ÇìêÅÜ=oÉÅÜåìåÖ=òì=ÉáåÉê=bêJ

âÉååíåáë=ÑΩÜêÉåK==

_Éîçê=ïáê=ÄÉÖáååÉåI=ëçääíÉå=ïáê=ÑΩê=ÇáÉ=_ÉèìÉãäáÅÜâÉáí=ÇÉê=oÉÅÜåìåÖ=ãáåÇÉëíÉåë=ÉáåÉ=báåë=

áêÖÉåÇïç=ÄÉëÅÜ~ÑÑÉåW==

a~ë=â∏ååÉå=ïáê=ÉêêÉáÅÜÉåI=áåÇÉã=ïáê=ÇáÉ=îçêäÉíòíÉ=wÉáäÉ=ÇìêÅÜ=ÑΩåÑ=ÇáîáÇáÉêÉåK=

a~åå=ëíÉÜí=Ççêí= áå=ÇÉê=ÉêëíÉå=pé~äíÉ=ÉáåÉ=báåëI=ìåÇ=Ç~ãáí=Ü~ÄÉå=ïáê=Éáå= áÇÉ~äÉë=máîçíÉäÉJ

ãÉåí=ÑΩê=ÇÉå=ÉêëíÉå=d~ì≈JpÅÜêáíí=ÖÉÑìåÇÉå=EëáÉÜÉ=^ÄëÅÜåáíí=NTKPKO=~ìÑ=pÉáíÉ=ORPFK==

aáÉ=îáÉêíÉ=wÉáäÉ=ãáí=ÇáÉëÉê=báåë=ïáêÇ=~äë=máîçíòÉáäÉ=ã~êâáÉêíI=ìåÇ=Ç~åå=ä®ëëí=ëáÅÜ=ÇìêÅÜ=ÖÉJ

òáÉäíÉ=hçãÄáå~íáçå=ÇÉê=máîçíòÉáäÉ=ãáí= àÉÇÉê= ~åÇÉêÉå=wÉáäÉ= ÉêêÉáÅÜÉåI=Ç~ëë= îáÉê=åÉìÉ=däÉáJ

ÅÜìåÖÉå=ÉåíëíÉÜÉåI=ÇáÉ=~ääÉ=ìåíÉê=ÇÉê=~äíÉå=máîçíëé~äíÉ=åìê=kìääÉå=ÄÉëáíòÉå=E_áäÇ=NUKPFK==

Page 4: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NU==iáåÉ~êÉ=^äÖÉÄê~=ìåÇ=_ÉíêáÉÄëïáêíëÅÜ~ÑíëäÉÜêÉ=OSS=

=

=

_áäÇ=NUKPW=bêëíÉê=bäáãáå~íáçåëëÅÜêáíí==

få=ÇÉã=Éåíëí~åÇÉåÉå= póëíÉã=îçå= îáÉê=däÉáÅÜìåÖÉå=ÖáÄí= Éë= äÉáÇÉê= âÉáåÉ=ïÉáíÉêÉ=báåëI= Ç~=

ãΩëëÉå=ïáê=ìåë=Éáå= ÑΩê=ÇáÉ=oÉÅÜåìåÖ=ÄÉèìÉãÉë=bäÉãÉåí=ï®ÜäÉåK=kÉÜãÉå=ïáê=~äëç=òìã=

_ÉáëéáÉä=ÇáÉ=jáåìë=åÉìå=áå=ÇÉê=ÉêëíÉå=wÉáäÉ=ÇÉê=òïÉáíÉå=pé~äíÉ=~äë=máîçíÉäÉãÉåíK=

a~åå= ã~êâáÉêÉå= ïáê= ÇáÉ= åÉìÉ= máîçíòÉáäÉI= ëìÅÜÉå= ÇáÉ= ÖÉÉáÖåÉíÉå= bäáãáå~íáçåëâçÉÑÑáòáÉåíÉå=

ìåÇ=âçãÄáåáÉêÉå=å~ÅÜÉáå~åÇÉê=ÇáÉ=åÉìÉ=máîçíòÉáäÉ=ãáí=ÇÉå=~åÇÉêÉå=ÇêÉá=wÉáäÉåK==

pç=ÉåíëíÉÜÉå= ëÅÜäáÉ≈äáÅÜ=ÇêÉá=åÉìÉ=wÉáäÉåI=ïáÉ=ÄÉ~ÄëáÅÜíáÖí=ãáí=kìääÉå=ìåíÉê=ÇÉê=~äíÉå=ìåÇ=

åìå=~ìÅÜ=ìåíÉê=ÇÉê=åÉìÉå=máîçíëé~äíÉ=E_áäÇ=NUKQFK===

=

_áäÇ=NUKQW=aÉê=^äÖçêáíÜãìë=ÄêáÅÜí=~Ä=Ó=Éë=ä®ëëí=ëáÅÜ=âÉáå=åÉìÉë=máîçíÉäÉãÉåí=ÑáåÇÉå=

açÅÜ=ï~ë=ÉêÖáÄí=ëáÅÜ=åÉÄÉåÄÉá=åçÅÜ=Ó=ÇáÉ=ÇêÉá=åÉìÉå=wÉáäÉå=ÉåíÜ~äíÉå=ΩÄÉêÜ~ìéí=åìê=kìäJ

äÉåK= pçïçÜä= äáåâë=ìåíÉê=ÇÉå=råÄÉâ~ååíÉå= ~äë= ~ìÅÜ= êÉÅÜíë=ìåíÉê=ÇÉã=däÉáÅÜÜÉáíëòÉáÅÜÉåK=

táê=â∏ååÉå=åáÅÜí=ïÉáíÉê=êÉÅÜåÉåI=ÇÉê=d~ì≈ëÅÜÉ=^äÖçêáíÜãìë=ÄêáÅÜí=~ÄK==

^ÄÉê=Éê=ÉåÇÉí=çÜåÉ=táÇÉêëéêìÅÜK==

Page 5: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NUKN==oçÜëíçÑÑÉ=ìåÇ=båÇéêçÇìâíÉ= OST=

cçäÖäáÅÜ=ÖáÄí=Éë=ìåÉåÇäáÅÜ=îáÉäÉ=i∏ëìåÖÉåK=rã=ëáÉ=òì=ÑáåÇÉåI=ëíÉääÉå=ïáê=ÇáÉ=ÉáåòáÖÉå=ÄÉáJ

ÇÉå=máîçíòÉáäÉåI=ÇáÉ=ïáê=~ìëï®ÜäÉå=âçååíÉåI=àÉíòí=òìë~ããÉåW==

=

_áäÇ=NUKRW=_ÉáÇÉ=máîçíòÉáäÉå=ÄáäÇÉå=Ç~ë=aêÉáÉÅâëëÅÜÉã~=

lÑÑÉåÄ~ê=â∏ååÉå=ïáê=åìå=ÉáåÉ=ÇÉê=ÇêÉá=råÄÉâ~ååíÉå=ÑêÉá=ï®ÜäÉåI=Éë=ÄáÉíÉí=ëáÅÜ=ÜáÉê=ñP

=~åK=

aáÉ=äÉíòíÉ=wÉáäÉ=ïáêÇ=ÇÉëÜ~äÄ=ëç=ìãÖÉÑçêãíI=Ç~ëë=ñO

=~ääÉáå=~ìÑ=ÇÉê=äáåâÉå=pÉáíÉ=ëíÉÜíW==

ENUKMQF= =

pÉíòÉå=ïáê=ÇÉå=ÉêÜ~äíÉåÉå=tÉêí=ÑΩê=ñO

=áå=ÇáÉ=ÉêëíÉ=wÉáäÉ=ÉáåI=ëç=â∏ååÉå=ïáê=ïÉáíÉê=ñN

=ÇìêÅÜ=

ñP

=~ìëÇêΩÅâÉåW=

ENUKMRF= =

qÜÉçêÉíáëÅÜ=Ó=ïÉåå=ïáê=åìê=ÄÉá=ÇÉê=êÉáå=ã~íÜÉã~íáëÅÜÉå=rãëÉíòìåÖ=ÇÉë=bêÖÉÄåáëëÉë=ÄäÉáJ

ÄÉå=Ó=â∏ååíÉå=ïáê=åìå=ÑΩê=ñP

=~ìÑ=ÇÉå=êÉÅÜíÉå=pÉáíÉå=îçå=ENUKMQF=ìåÇ=ENUKMRF=àÉÇÉå=ÄÉäáÉÄáJ

ÖÉå= w~ÜäÉåïÉêí= ÉáåëÉíòÉåK= fããÉê=ïΩêÇÉå=ïáê= Ç~åå= Ç~òì= é~ëëÉåÇÉ=tÉêíÉ= ñN

=ìåÇ=ñO

=ÄÉJ

âçããÉåI=ëç=Ç~ëë=ÇáÉ=ÉêÜ~äíÉåÉ=ñN

JñO

JñP

Jwìë~ããÉåëíÉääìåÖ=Ç~ë=däÉáÅÜìåÖëëóëíÉã=ÉêÑΩääíK==

açÅÜ=ëÉíòÉå=ïáê=åìê=ñP=Z=M=ìåÇ=áåíÉêéêÉíáÉêÉå=ïáê=Ç~ë=bêÖÉÄåáë=ÑΩê=ìåëÉêÉ=^ìÑÖ~ÄÉåëíÉääìåÖK=

cΩê=ÇáÉëÉë=ñP=

Z=

M=ÉêÖ®ÄÉå=ëáÅÜ=å®ãäáÅÜ=ëçÑçêí=ñO=

Z=

OPUMLV=

Z=

OSQIQQ=ìåÇ=ñN=

Z=

ÓUMLV=

Z=ÓUIUVK=

táê=ãΩëëíÉå=åÉÖ~íáîÉ=jÉåÖÉå=~å=bN

=éêçÇìòáÉêÉåI=ìã=ÇáÉ=oÉëëçìêÅÉå=~ìëòìëÅÜ∏éÑÉåK==

a~ë=áëí=å~íΩêäáÅÜ=∏âçåçãáëÅÜÉê=råëáååK==

cçäÖäáÅÜ=ãΩëëÉå=ïáê=ÇÉå=_ÉêÉáÅÜ=ÑΩê=ñP

=åçÅÜ=ëç=ÉáåëÅÜê®åâÉåI=Ç~ëë=ëçïçÜä=ÇáÉ=mêçÇìâíáçåëJ

ãÉåÖÉ=ñN

=~äë=~ìÅÜ=ñO

=åáÅÜí=åÉÖ~íáî=ïÉêÇÉå=â~ååI=ÇK=ÜK=ïáê=ãΩëëÉå=ìåÄÉÇáåÖí=Ç~ÑΩê=ëçêÖÉåI=

Ç~ëë=ñP

=åìê=ëç=ÖÉï®Üäí=ïÉêÇÉå=Ç~êÑI=Ç~ëë=ÇáÉ=ÄÉáÇÉå=råÖäÉáÅÜìåÖÉå=

ENUKMSF= =

ÖäÉáÅÜòÉáíáÖ=ÉêÑΩääí=ëáåÇK=a~òì=ãìäíáéäáòáÉêÉå=ïáê=òìÉêëí=ÄÉáÇÉ=råÖäÉáÅÜìåÖÉå=àÉïÉáäë=ãáí=ÇÉê=

éçëáíáîÉå=w~Üä=VI=Ç~ÄÉá=ÄäÉáÄí=Ç~ë=oÉä~íáçåëòÉáÅÜÉå=ÉêÜ~äíÉå=EëáÉÜÉ=^ÄëÅÜåáíí=PKOKQ=~ìÑ=pÉáíÉ=

PVFK==

aìêÅÜ=Éêä~ìÄíÉ=ïÉáíÉêÉ=rãÑçêãìåÖÉå=ÉêÜ~äíÉå=ïáê=Ç~åå=

ENUKMTF= =

)222380(

9

1

32xx −=

)808(

9

1

)]222380(

9

1

[24520

31

331

−=

−−−=

xx

xxx

0)222380(

9

1

0)808(

9

1

3

3

≥−

≥−

x

x

18,108

22

2380

2380220222380

108080808

3333

333

≤⇔≤⇔≤⇔≥−

≥⇔≥⇔≥−

xxxx

xxx

Page 6: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NU==iáåÉ~êÉ=^äÖÉÄê~=ìåÇ=_ÉíêáÉÄëïáêíëÅÜ~ÑíëäÉÜêÉ=OSU=

_ÉáÇÉ=_ÉÇáåÖìåÖÉå=ä~ëëÉå=ëáÅÜ=ëÅÜäáÉ≈äáÅÜ=áå=ÉáåÉê=råÖäÉáÅÜìåÖ=òìë~ããÉåÑ~ëëÉåW=

ENUKMUF= =

a~ãáí=áëí=ÇáÉ=^ìÑÖ~ÄÉ=ÄF=ÖÉä∏ëíK==

c~ëëÉå=ïáê=Ç~ë=bêÖÉÄåáë=åçÅÜ=áå=ÉáåÉã=~ÄëÅÜäáÉ≈ÉåÇÉå=^åíïçêíë~íò=òìë~ããÉåW=

tÉåå=ã~å=ëáÅÜ=ÉáåÉ=mêçÇìâíáçåëãÉåÖÉ=ÇÉë=bêòÉìÖåáëëÉë=bP

=òïáëÅÜÉå=NM=ìåÇ=NMU=báåJ

ÜÉáíÉå=ÄÉäáÉÄáÖ=îçêÖáÄí=ìåÇ=~åëÅÜäáÉ≈ÉåÇ=ÇáÉ=Ç~òì=ÖÉÜ∏êáÖÉå=mêçÇìâíáçåëãÉåÖÉå=ÑΩê=ÇáÉ=

bêòÉìÖåáëëÉ=bN

=ìåÇ=bO

=~ìë=ÇÉå=cçêãÉäå=ENUKMQF=ìåÇ=ENUKMRF=ÄÉêÉÅÜåÉíI=Ç~åå=ïÉêÇÉå=ÑΩê=

àÉÇÉ=ÇÉê~êíáÖÉ=sçêÖ~ÄÉ=ÇáÉ=îçêÜ~åÇÉåÉå=oÉëëçìêÅÉå=îçääëí®åÇáÖ=~ìëÖÉëÅÜ∏éÑíK==

18.2 Mehrstufige Produktion

^ìÑÖ~ÄÉW=báå=_ÉíêáÉÄ=ëíÉääí=~ìë=ÇÉå=oçÜëíçÑÑÉå=oN

=ìåÇ=oO

=áå=ÇÉê=ÉêëíÉå=mêçÇìâíáçåëëíìÑÉ=ÇáÉ=

wïáëÅÜÉåéêçÇìâíÉ=wN

I=wO

=ìåÇ=wP

=ÜÉêX= áå=ÇÉê=òïÉáíÉå=mêçÇìâíáçåëëíìÑÉ=ïÉêÇÉå=~ìë=ÇáÉëÉå=

wïáëÅÜÉåéêçÇìâíÉå=ÇáÉ=båÇéêçÇìâíÉ=bN

=ìåÇ=bO

=ÖÉÑÉêíáÖíK==

aáÉ=^ìÑï~åÇëâçÉÑÑáòáÉåíÉå=ÇÉê=oçÜëíçÑÑÉ= àÉ=báåÜÉáí=wïáëÅÜÉåéêçÇìâí=ëáåÇ=áå=ÇÉê=j~íêáñ=^I=

ÇáÉ=ÑΩê=ÇáÉ=wïáëÅÜÉåéêçÇìâíÉ=àÉ=báåÜÉáí=båÇéêçÇìâí=áå=ÇÉê=j~íêáñ=_=ÉåíÜ~äíÉåW==

ENUKMVF= =

~F=dÉëìÅÜí=áëí=ÇáÉ=j~íêáñI=ÇáÉ=ÇÉå=oçÜëíçÑÑîÉêÄê~ìÅÜ=àÉ=báåÜÉáí=båÇéêçÇìâí=ÉåíÜ®äíK==

ÄF= få=ÉáåÉã=ÄÉëíáããíÉå=wÉáíê~ìã=ëáåÇ=NM=báåÜÉáíÉå=îçå=bN

=ìåÇ=ORM=báåÜÉáíÉå=îçå=bO

=òì=

ÑÉêíáÖÉåK=^ì≈ÉêÇÉã=ëáåÇ=~äë=bêë~íòíÉáäÉ=PM=báåÜÉáíÉå=îçå=wN

=ìåÇ=àÉïÉáäë=OM=báåÜÉáíÉå=îçå=wO

=

ìåÇ=wP

=ÄÉêÉáíòìëíÉääÉåK=tÉäÅÜÉ=oçÜëíçÑÑãÉåÖÉå=ëáåÇ=Ç~ÑΩê=ÉêÑçêÇÉêäáÅÜ\=

ÅF=tÉäÅÜÉ=oçÜëíçÑÑâçëíÉå=ÉåíëíÉÜÉåI=ïÉåå=ÇáÉ=oçÜëíçÑÑéêÉáëÉ=NOIRM=db=ÑΩê=ÉáåÉ=báåÜÉáí=oN

=

ìåÇ=NTIOM=db=ÑΩê=ÉáåÉ=báåÜÉáí=oO

=ÄÉíê~ÖÉå\=

i∏ëìåÖW=aáÉ=ÖÉëìÅÜíÉ=j~íêáñI=ÇáÉ=òìë~ããÉåÑ~ëëÉåÇ=ÄÉëÅÜêÉáÄíI=ïÉäÅÜÉê=oçÜëíçÑÑîÉêÄê~ìÅÜ=

ëáÅÜ=ÑΩê=àÉÇÉ=báåÜÉáí=àÉÇÉë=båÇéêçÇìâíë=ÉêÖáÄíI=ÉêÜ®äí=ã~å=ÇìêÅÜ=jìäíáéäáâ~íáçå=ÇÉê=ÄÉáÇÉå=

ÖÉÖÉÄÉåÉå=j~íêáòÉå=^=ìåÇ=_K=aáÉ=jìäíáéäáâ~íáçå=ÑáåÇÉí=òïÉÅâã®≈áÖ=ïáÉÇÉê=áã=pÅÜÉã~=îçå=

c~äâ=ëí~ííW==

=

_áäÇ=NUKSW=oçÜëíçÑÑîÉêÄê~ìÅÜ=éêç=båÇéêçÇìâí===

18,108103≤≤ x

=⎟⎟

⎜⎜

=

14

23

21

,

132

541

BA

Page 7: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NUKO==jÉÜêëíìÑáÖÉ=mêçÇìâíáçå= OSV=

oÉÅÜíë=ìåíÉå=áã=ÜÉêîçêÖÉÜçÄÉåÉå=_ÉêÉáÅÜ=ÑáåÇÉí=ëáÅÜ=Ç~åå=ÇáÉ=bêÖÉÄåáëã~íêáñ=

ENUKNMF= =

ÇáÉ=òìë~ããÉåÖÉÑ~ëëí=ÇáÉ=sÉêÄê~ìÅÜëåçêãÉå=áå=ÑçäÖÉåÇÉê=tÉáëÉ=ÄÉëÅÜêÉáÄíW==

sçã=oçÜëíçÑÑ=oN

=ïÉêÇÉå=ÄÉå∏íáÖíW==

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bN

W=

PP=jb=

cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb=

îçã=båÇéêçÇìâí=bO

W=

NR=jb=

sçã=oçÜëíçÑÑ=oO

=ïÉêÇÉå=ÄÉå∏íáÖíW==

cΩê= ÇáÉ= mêçÇìâíáçå= ÉáåÉê=

jb=îçã=båÇéêçÇìâí=bN

W=

NR=jb=

cΩê=ÇáÉ=mêçÇìâíáçå=ÉáåÉê=jb=

îçã=båÇéêçÇìâí=bO

W=

NN=jb=

wì=ÄFW=píÉääí=ã~å=ÇáÉ=ÖÉÑçêÇÉêíÉå=mêçÇìâíáçåëãÉåÖÉå=îçå=bN

=ìåÇ=bO

=áå=ÉáåÉã=pé~äíÉåîÉâíçê==

é=ìåÇ=ÇáÉ=ÄÉêÉáíòìëíÉääÉåÇÉå=bêë~íòíÉáäãÉåÖÉå=ÇÉê=wïáëÅÜÉåéêçÇìâíÉ=wN

I=wO

=ìåÇ=wP

= áå=ÉáJ

åÉã=ïÉáíÉêÉå=pé~äíÉåîÉâíçê=w=òìë~ããÉå=

ENUKNNF= =

Ç~åå=ÉêÖáÄí=ëáÅÜ=~ìë=ÇÉå=ÄÉëÅÜêáÉÄÉåÉå=cçêÇÉêìåÖÉå=ÇáÉ=^ìÑÖ~ÄÉI=òïÉá=j~íêáñéêçÇìâíÉ=òì=

ÄÉêÉÅÜåÉå=ìåÇ=ÇáÉ=bêÖÉÄåáëã~íêáòÉå=òì=~ÇÇáÉêÉåW==

ENUKNOF= =

_áäÇ=NUKT=òÉáÖí=ÇÉå=oÉÅÜÉåïÉÖ=ÇÉê=ÄÉáÇÉå=jìäíáéäáâ~íáçåÉå=áã=pÅÜÉã~=îçå=c~äâK==

=

_áäÇ=NUKTW=pÅÜÉã~=îçå=c~äâ==

aáÉ=ÄÉáÇÉå=ÜÉêîçêÖÉÜçÄÉåÉå=bêÖÉÄåáëã~íêáòÉå=Ü~ÄÉå=~äë=pé~äíÉåîÉâíçêÉå=ãáí=àÉ=òïÉá=bäÉJ

ãÉåíÉå=Ç~ëëÉäÄÉ=cçêã~í=ìåÇ=ÇΩêÑÉå=ÇÉëÜ~äÄ=~ÇÇáÉêí=ïÉêÇÉåW=

=

1115

1533

C

⎟⎟

⎜⎜

=⎟

=

20

20

30

,

250

120

Zp

ZApCr ⋅+⋅=

Page 8: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NU==iáåÉ~êÉ=^äÖÉÄê~=ìåÇ=_ÉíêáÉÄëïáêíëÅÜ~ÑíëäÉÜêÉ=OTM=

=

ENUKNPF= =

wìê=i∏ëìåÖ=ÇÉê=^ìÑÖ~ÄÉ=ÄF=â∏ååÉå=ïáê=Ç~ãáí=ÇÉå=ÑçäÖÉåÇÉå=^åíïçêíë~íò=ÑçêãìäáÉêÉåW=

wìê=cÉêíáÖìåÖ=ÇÉê=ÖÉÑçêÇÉêíÉå=báåÜÉáíÉå=ÇÉê=ÄÉáÇÉå=båÇéêçÇìâíÉ= ëçïáÉ=ÇÉê=bêë~íòíÉáäÉ=

ïÉêÇÉå=TVOM=báåÜÉáíÉå=ÇÉë=oçÜëíçÑÑë=oN

=ìåÇ=QSVM=báåÜÉáíÉå=ÇÉë=oçÜëíçÑÑë=oO

=ÄÉå∏íáÖíK==

wì= ÅF=aáÉ= ÉåíëíÉÜÉåÇÉå= oçÜëíçÑÑâçëíÉå= â∏ååÉå= ÉäÉãÉåí~ê= ÄÉêÉÅÜåÉí= ïÉêÇÉåI= áåÇÉã= ÇáÉ=

jÉåÖÉå=~ìë=ÇÉê=j~íêáñ=o=~ÄÖÉäÉëÉå=ïÉêÇÉå=ìåÇ=ÉáåÑ~ÅÜ=ãáí=ÇÉå=ÉåíëéêÉÅÜÉåÇÉå=mêÉáëÉå=

ãìäíáéäáòáÉêí=ïÉêÇÉåW==

ENUKNQF= =

táê=â∏ååÉå=~ÄÉê=~ìÅÜ=ÇáÉ=ÄÉáÇÉå=ÖÉÖÉÄÉåÉå=mêÉáëÉ=áå=ÉáåÉã=mêÉáëîÉâíçê=Ö=òìë~ããÉåÑ~ëJ

ëÉå=ìåÇ=Ç~åå=ÇáÉ=hçëíÉå=~ìë=ÇÉã=j~íêáñJmêçÇìâí=ÇÉë=íê~åëéçåáÉêíÉå=oçÜëíçÑÑãÉåÖÉåîÉâJ

íçêë=êq

=ãáí=ÇÉã=mêÉáëîÉâíçê=Ö=ÉêãáííÉäåW==

ENUKNRF= =

få=ÄÉáÇÉå=c®ääÉå=ÉêÖáÄí=ëáÅÜW=aáÉ=dÉë~ãíâçëíÉå=ÄÉíê~ÖÉå=NTVSSU=dbK==

18.3 Maschinenzeitfonds

^ìÑÖ~ÄÉW=báå=_ÉíêáÉÄ= ÑÉêíáÖí=îáÉê=îÉêëÅÜáÉÇÉåÉ=bêòÉìÖåáëëÉ=~ìÑ=îáÉê=j~ëÅÜáåÉåÖêìééÉåK= få=

ÇÉê=q~ÄÉääÉ=îçå=_áäÇ=NUKU=áëí=ÇÉê=^êÄÉáíëòÉáí~ìÑï~åÇ=áå=jáåìíÉå=éêç=píΩÅâ=ÇÉë=bêòÉìÖåáëëÉë=

ìåÇ=ÇÉê=îÉêÑΩÖÄ~êÉ=j~ëÅÜáåÉåòÉáíÑçåÇë=ÉåíÜ~äíÉåK=

=

_áäÇ=NUKUW=^êÄÉáíëòÉáí~ìÑï~åÇ=ìåÇ=j~ëÅÜáåÉåòÉáíÑçåÇë=

cΩê=ïÉäÅÜÉ=píΩÅâò~ÜäÉå=ïáêÇ=ÇÉê=j~ëÅÜáåÉåòÉáíÑçåÇë=îçää=~ìëÖÉä~ëíÉí\=

i∏ëìåÖW=tÉåå=ïáê=ãáí=ÇÉå=råÄÉâ~ååíÉå=ñN

=Äáë=ñQ

=ÇáÉ=^åò~Üä=ÇÉê= àÉïÉáäë=ÜÉêòìëíÉääÉåÇÉå=

píΩÅâÉ=ÇÉê=îáÉê=bêòÉìÖåáëëÉ=ÄÉòÉáÅÜåÉå=ìåÇ=òìë®íòäáÅÜ=ÇáÉ=oÉëëçìêÅÉå=ÄÉíê~ÅÜíÉåI=ÇáÉ=~ìëJ

ÖÉëÅÜ∏éÑí=ïÉêÇÉå=ëçääÉåI=ÉêÜ~äíÉå=ïáê=ëçÑçêí=Éáå=äáåÉ~êÉë=däÉáÅÜìåÖëëóëíÉãK==

86697102,71096405,210297 =⋅+⋅=K

=⎟

+⎟

=

4690

7920

140

210

4550

7710

r

( ) 179668

20,17

50,12

46907920

20,17

50,12

=⎟

⋅=⋅=⇒⎟

= grKg

T

Page 9: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NUKP==j~ëÅÜáåÉåòÉáíÑçåÇë= OTN=

=

_áäÇ=NUKVW=iáåÉ~êÉë=däÉáÅÜìåÖëëóëíÉã=ÇÉê=^ìÑÖ~ÄÉ=

aáÉëÉë=póëíÉã=áëí=èì~Çê~íáëÅÜI=ïáê=ãΩëëÉå=~äëç=ÖÉã®≈=^ÄëÅÜåáíí=NTKO=~ìÑ=pÉáíÉ=OQU=Ç~ãáí=

êÉÅÜåÉåI=Ç~ëë=Éë=âÉáåÉ=çÇÉê=ÖÉå~ì=ÉáåÉ=çÇÉê=ìåÉåÇäáÅÜ=îáÉäÉ=i∏ëìåÖÉå=ÖÉÄÉå=â~ååK==

pí~êíÉå=ïáê=ÇÉëÜ~äÄ=ÇÉå=d~ì≈ëÅÜÉå=^äÖçêáíÜãìëI=ìã=ìåë=ï®ÜêÉåÇ=ÇÉê=oÉÅÜåìåÖ=ë~ÖÉå=òì=

ä~ëëÉåI= ïÉäÅÜÉ= i∏ëìåÖëëáíì~íáçå= ÜáÉê= îçêäáÉÖí= ìåÇ= ïÉäÅÜÉ= pÅÜäìëëÑçäÖÉêìåÖÉå= ïáê= òáÉÜÉå=

ãΩëëÉåK==

iáåâë=çÄÉå=áã=däÉáÅÜìåÖëëóëíÉã=ëíÉÜí=ÖäÉáÅÜ=ÉáåÉ=báåë=Ó=Éë=ëéêáÅÜí=åáÅÜíë=Ç~ÖÉÖÉåI=ëáÉ=~äë=

ÉêëíÉë=máîçíÉäÉãÉåí=~ìëòìï®ÜäÉåK==

aáÉ=ÉêëíÉ=wÉáäÉ=ïáêÇ=~äëç=~äë=máîçíòÉáäÉ=ã~êâáÉêíI=ìåÇ=Éë=ïÉêÇÉå=êÉÅÜíë=ÇáÉ=bäáãáå~íáçåëâçJ

ÉÑÑáòáÉåíÉå=ÑÉëíÖÉäÉÖíI=ãáí=ÇÉêÉå=eáäÑÉ=ÇáÉ=máîçíòÉáäÉ=ãáí=~ääÉå=~åÇÉêÉå=wÉáäÉå=ëç=âçãÄáåáÉêí=

ïáêÇI=Ç~ëë=ìåíÉê=ÇÉê=máîçíëé~äíÉ=ä~ìíÉê=kìääÉå=ÉåíëíÉÜÉåK=

_áäÇ=NUKNM=òÉáÖí=ÇÉå=ÉêëíÉå=bäáãáå~íáçåëëÅÜêáíí=ìåÇ=ëÉáå=bêÖÉÄåáëW=

=

_áäÇ=NUKNMW=bêëíÉê=bäáãáå~íáçåëëÅÜêáíí=

bë=ÉåíëíÉÜÉåI=ïáÉ=ÄÉ~ÄëáÅÜíáÖíI=ÇêÉá=åÉìÉ=wÉáäÉå=ÑΩê=ÇáÉ=îáÉê=råÄÉâ~ååíÉåI=~ì≈Éê=áå=ÇÉê=ÉêëJ

íÉå= EkìääÉåJF= pé~äíÉ= â~åå= áã= ìåíÉêÉå= pÅÜÉã~= åìå= Éáå= åÉìÉë= máîçíÉäÉãÉåí= ÄÉäáÉÄáÖ= ~ìëJ

ÖÉï®Üäí=ïÉêÇÉåK==

_áäÇ=NUKNN=òÉáÖí=ÉëW=wìê=_ÉèìÉãäáÅÜâÉáí=ïáêÇ=ïáÉÇÉê=ÉáåÉ=báåë=~ìëÖÉï®ÜäíI=Ç~ãáí=ÉêÖáÄí=ëáÅÜ=

ÇáÉ= ãáííäÉêÉ= ÇÉê= ÇêÉá= wÉáäÉå= ~äë= åÉìÉ= máîçíòÉáäÉK= a~ë= bêÖÉÄåáë= å~ÅÜ= ÇÉã= òïÉáíÉå= d~ì≈J

pÅÜêáííW=bë=ÉåíëíÉÜÉå=åìê=åçÅÜ=kìääÉåK==

bë=â~åå=~äëç= áå=ÇÉå=ÉåíëíÉÜÉåÇÉå=êÉëíäáÅÜÉå=òïÉá=wÉáäÉå=âÉáå=åÉìÉë=máîçíÉäÉãÉåí=~ìëÖÉJ

ï®Üäí=ïÉêÇÉåI=ÇÉê=d~ì≈ëÅÜÉ=^äÖçêáíÜãìë=ÄêáÅÜí=~ÄK==

^ÄÉê=Éê=ÄêáÅÜí=åáÅÜí=ãáí=ÉáåÉã=táÇÉêëéêìÅÜ=~ÄI=Ç~=~ìÅÜ=ìåíÉê=ÇÉã=däÉáÅÜÜÉáíëòÉáÅÜÉå=kìäJ

äÉå=ÉêëÅÜÉáåÉåK=táê=ÉêÜ~äíÉå=òïÉá=âçãéäÉííÉ=kìääòÉáäÉåK===

Page 10: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NU==iáåÉ~êÉ=^äÖÉÄê~=ìåÇ=_ÉíêáÉÄëïáêíëÅÜ~ÑíëäÉÜêÉ=OTO=

=

_áäÇ=NUKNNW=bêëíÉê=ìåÇ=òïÉáíÉê=bäáãáå~íáçåëëÅÜêáíí===

cçäÖäáÅÜ=Ü~í=ìåëÉê=äáåÉ~êÉë=èì~Çê~íáëÅÜÉë=däÉáÅÜìåÖëëóëíÉã=ìåÉåÇäáÅÜ=îáÉäÉ=i∏ëìåÖÉåK==

aáÉëÉ=ãΩëëÉå=ïáê=àÉíòí=ÄÉëÅÜêÉáÄÉåK=a~òì=ëíÉääÉå=ïáê=òìÉêëí=ÇáÉ=ÄÉáÇÉå=máîçíòÉáäÉå=òìë~ãJ

ãÉå=E_áäÇ=NUKNOFK=a~ãáí=ÇáÉ=aêÉáÉÅâëÑçêã=EëáÉÜÉ=^ÄëÅÜåáíí=NTKQKO=~ìÑ=pÉáíÉ=ORUF=ÉêâÉååÄ~ê=

ïáêÇI=ïáêÇ=~åëÅÜäáÉ≈ÉåÇ=ÇáÉ=ÇêáííÉ=ãáí=ÇÉê=òïÉáíÉå=pé~äíÉ=îÉêí~ìëÅÜí=E_áäÇ=NUKNPFK===

======= =

_áäÇ=NUKNOW=máîçíòÉáäÉå=

========= =

_áäÇ=NUKNPW=aêÉáÉÅâëÑçêã==

rã=ëÅÜäáÉ≈äáÅÜ=òìê= ëçÑçêí=~ÄäÉëÄ~êÉå=i∏ëìåÖ=òì=âçããÉåI= ëíêÉÄÉå=ïáê=~ÄëÅÜäáÉ≈ÉåÇ=ëçÖ~ê=

ÇáÉ=â~åçåáëÅÜÉ=cçêã=~åW==

táê= ÉêòÉìÖÉå= ìåíÉê= ÇÉå= ÄÉáÇÉå= _~ëáëî~êá~ÄäÉå= ñN

= ìåÇ= ñP

= ÇáÉ= báåÜÉáíëã~íêáñ= EëáÉÜÉ= ^ÄJ

ëÅÜåáíí=NTKQKO=~ìÑ=pÉáíÉ=OSMFK==

a~ë=ÉêêÉáÅÜÉå=ïáêI= áåÇÉã=ïáê=Ç~ë=açééÉäíÉ=ÇÉê=òïÉáíÉå=wÉáäÉ=îçå=ÇÉê=ÉêëíÉå=wÉáäÉ=~ÄòáÉJ

ÜÉå=ìåÇ=ÇáÉ=òïÉáíÉ=wÉáäÉ=~ÄëÅÜêÉáÄÉå=E_áäÇ=NUKNQFK==

gÉíòí=ëÉÜÉå=ïáê=Éë=ëçÑçêíW=cΩê=ÇáÉ=råÄÉâ~ååíÉ=ñN

=ÉêÖáÄí=ëáÅÜ=ÇÉê=w~ÜäÉåïÉêí=ñN

=Z=NMMK=aÉê=

tÉêí=ÇÉê= òïÉáíÉå=_~ëáëî~êá~ÄäÉå=ñP

= Ç~ÖÉÖÉå=ÉêÖáÄí= ëáÅÜI=ïÉåå=ÇáÉ=káÅÜíÄ~ëáëî~êá~ÄäÉå=ñO

=

ìåÇ=ñQ

=áå=áêÖÉåÇÉáåÉê=tÉáëÉ=ÄÉäáÉÄáÖ=îçêÖÉÖÉÄÉå=ëáåÇK=wìã=_ÉáëéáÉä=ñO

Z=ñQ

=Z=M=→=ñP

Z QRMK=

a~ë=ï®êÉ=Ç~åå=ëçÖ~ê=ÉáåÉ=ÇÉê=_~ëáëä∏ëìåÖÉåK==

Page 11: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NUKP==j~ëÅÜáåÉåòÉáíÑçåÇë= OTP=

=

_áäÇ=NUKNQ=h~åçåáëÅÜÉ=cçêã=

fëí=Éë=~ÄÉê=ïáêâäáÅÜ=ëçI=Ç~ëë=ïáê=ÑΩê=ÇáÉ=káÅÜíÄ~ëáëî~êá~ÄäÉå=ñO

=ìåÇ=ñQ

=ÄÉäáÉÄáÖÉ=w~ÜäÉåïÉêíÉ=

îçêÖÉÄÉå=ÇΩêÑíÉå\==

k~íΩêäáÅÜ=åáÅÜí=Ó=ëÅÜçå=åÉÖ~íáîÉ=sçêÖ~ÄÉå=ÑΩê=ñO

=ìåÇ=ñQ

=ëáåÇ=îÉêÄçíÉåI=ÇÉåå=Ç~åå=ïΩêÇÉå=

ïáê=åÉÖ~íáîÉ=píΩÅâò~ÜäÉåI=~äëç=∏âçåçãáëÅÜÉå=råëáååI=îçêÖÉÄÉåK===

a~êΩÄÉê=Üáå~ìë=ëáåÇ=~ÄÉê=~ìÅÜ=ïÉáíÉêÉ=sçêÖ~ÄÉå=îÉêÄçíÉåI=òìã=_ÉáëéáÉä=ñO=

Z=

OMMI=

ñQ=

Z=

OMM.

aÉåå=Ç~åå=ÉêÖ®ÄÉ=ëáÅÜ=ÑΩê=ÇáÉ=~ÄÜ®åÖáÖÉ=_~ëáëî~êá~ÄäÉ=ñP=

Éáå=åÉÖ~íáîÉê=tÉêíW==

= = ñP=

Z=

QRM=

Ó=

PñO=

Ó=

ñQ=

Z=

QRM=

Ó=

SMM=

Ó=

OMM=

Z=

Ó=

PRMK==

^ìÅÜ=Ç~ë=ï®êÉ=ìåëáååáÖK==

=

_áäÇ=NUKNRW=_ÉêÉáÅÜ=ÇÉê=òìä®ëëáÖÉå=sçêÖ~ÄÉå=ÑΩê=ñO

=ìåÇ=ñQ

==

wìë~ããÉåÖÉÑ~ëëí=ãΩëëÉå=ïáê=~äëç=ÑΩê=ÇáÉ=i∏ëìåÖ=ìåëÉêÉê=^ìÑÖ~ÄÉ=ÑçêãìäáÉêÉåW=

aáÉ=îçêÖÉÖÉÄÉåÉå=oÉëëçìêÅÉå=ïÉêÇÉå=~ìëÖÉëÅÜ∏éÑíI=ïÉåå=îçã=bêòÉìÖåáë=bN

=ÖÉå~ì=NMM=

báåÜÉáíÉå=ÜÉêÖÉëíÉääí=ïÉêÇÉå=ìåÇ=ÇáÉ=jÉåÖÉå=ñO

=ìåÇ=ñQ

=ëç=îçêÖÉÖÉÄÉå=ïÉêÇÉåI=Ç~ëë=ÇáÉ=

ÇêÉá=_ÉÇáåÖìåÖÉå=ENUKNSF=ÉêÑΩääí=ëáåÇK==

Page 12: 18 Lineare Algebra und Betriebswirtschaftslehrerenesse/LA09/... · 2011-10-19 · 18 Lineare Algebra und Betriebswirtschaftslehre få=cçêã=îçå=ÇêÉá=íóéáëÅÜÉå=^ìÑÖ~ÄÉåëíÉääìåÖÉå=Eãáí=~ìëÑΩÜêäáÅÜÉå=i∏ëìåÖÉåF=ëçääÉå=áå=ÇáÉJ

NU==iáåÉ~êÉ=^äÖÉÄê~=ìåÇ=_ÉíêáÉÄëïáêíëÅÜ~ÑíëäÉÜêÉ=OTQ=

=

ENUKNSF= ==========================K=

=

a~åå=ïáêÇ=ÇáÉ=jÉåÖÉ=ñP

=ÇÉê=ÜÉêòìëíÉääÉåÇÉå=báåÜÉáíÉå=îçå=bP

=~ìë=ÇÉê=cçêãÉä==

ENUKNTF= =

ÄÉêÉÅÜåÉíK==

aáÉ=jÉåÖÉ=~ääÉê=ñO=

J=

ñQ=

J=

m~~êÉI=ÇáÉ=Ç~ë=råÖäÉáÅÜìåÖëëóëíÉã=ENUKNSF=ÉêÑΩääÉåI=ä®ëëí=ëáÅÜ=áå=ÉáJ

åÉã=ñO=

J=

ñQ=

J=

hççêÇáå~íÉåëóëíÉã=Öê~ÑáëÅÜ=Ç~êëíÉääÉåK=få=_áäÇ=NUKNR=ÄÉëÅÜêÉáÄí=ÇÉê=ëÅÜê~ÑÑáÉêíÉ=

_ÉêÉáÅÜ=ÖÉå~ì=ÇáÉ=jÉåÖÉ=ÇÉê=Éêä~ìÄíÉå=m~~êÉK==

táÉ=ã~å= ëáÅÜ= ëçäÅÜÉ= òìä®ëëáÖÉå= _ÉêÉáÅÜÉ= ÄÉëÅÜ~ÑÑíI= Ç~ë=ïáêÇ= áã= ÑçäÖÉåÇÉå=^ÄëÅÜåáíí= NV=

~ìëÑΩÜêäáÅÜ=ÄÉëÅÜêáÉÄÉåK==

=

=

4233450 xxx −−=

03450

0

0

42

4

2

≥−−

xx

x

x