RISET OPERASIONAL teknik sipil

BAHAN KULIAH RISET OPERASIONAL

Disusun oleh:

BAMBANG YUWONO, ST, MT PUTRI NUR ISTIANI (!"#"#"$

%URUSAN TEKNIK IN&ORMATIKA IN&ORMATIKA &AKULT AKULTAS TEKNOLOGI INDUSTRI UPN 'ETERAN) YOGYAKARTA !##*

BAHAN KULIAH RISET OPERASIONAL TEK INFORMATIKA UPN YOGYAKARTA BAMBANG YUWONO

SILABUS M++-uli+h :Rise O.e/+sion+l (O.e/+ion Rese+/0h$ M+e/i : 

PENDAHULUAN Pe/-e12+n3+n Rise O.e/+si A/i Rise O.e/+si

!

PROGRAM LINEAR : Meo4e GRA&IK

"

PROGRAM LINEAR : Meo4e Si1.le-

5

DUALITAS DAN ANALISA SENSITIIT SENSITI ITAS AS PERSOALAN PENUGASAN (ASSIGNMENT$

6

PERSOALAN TRANSPORTASI TRANSPORTASI

*

ANALISA NETWORK

7

TEORI ANTRIAN

8

De1o P/o3/+1 1en33un+-+n POM 9 LINDO 9 M

Bu-u : ;D+s+/ Rise O.e/+si , BP&E, 87", Yo3?+-+/+ ";P/insi. Rise O.e/+si , E/l+n33+, !##< PENILAIAN :  ! " 5

UTS UAS KUIS TUGAS


@ ONTOH PERMASALAHAN PERMASALAHAN RISET OPERASI SOAL 1 (MAKSIMASI) BAYU FURNITURE memproduksi 2 jeis produk !"i#u mej" d" kursi !"$ %"rus diproses me&"&ui per"ki#" d" 'iis%i$ roses per"ki#" memi&iki *+ j"m kerj" sed"$ proses 'iis%i$ memi&iki ,- j"m kerj" U#uk me$%"si&k" s"#u mej" di.u#u%k" , j"m per"ki#" d" 2 j"m 'iis%i$/ sed"$k" s"#u kursi mem.u#u%k" 2 j"m per"ki#" d" , j"m 'iis%i$ L"." 0u#uk 0#i"p 0mej" - 0d" 0#i"p 0kursi * 0Sek"r"$ 0ki#" 0%"rus mee#uk" kom.i"si #er."ik d"ri jum&"% mej" d" kursi !"$ %"rus diproduksi/ "$"r me$%"si&k" &"." m"ksim"&

SOAL 2 (MAKSIMASI) erus"%"" #"s ANIF 3 mem.u"# 2 m"4"m #"s !"i#u #"s merk

5ORA

d" 0merk SON6EBOB  0U#uk 0mem.u"# 0#"s 0#erse.u# 0perus"%"" memi&iki 7 mesi Mesi 1 k%usus u#uk mem.eri &o$o 5ORA/ mesi 2 k%usus u#uk mem.eri &o$o SON6EBOB d" mesi 7 u#uk mej"%i# #"s d" mem.u"# ri#s&e#i$ Se#i"p &usi #"s merk 5ORA mu&"8mu&" dikerj"k" di mesi 1 se&"m" 2 j"m/ kemudi" #"p" me&"&ui mesi 2 #erus dikerj"k" di mesi 7 se&"m" * j"m Sed"$ u#uk #"s merk SON6EBOB #id"k diproses di mesi 1/ #e#"pi per#"m" k"&i dikerj"k" di mesi 2 se&"m" 7 j"m kemudi" kemudi" di mesi mesi 7 se&"m" se&"m" 9 j"m :"m kerj" m"ksimum m"ksimum se#i"p se#i"p %"ri %"ri u#uk mesi 1;- j"m/ mesi 2;19 j"m/ d" mesi 7;7+ j"m Sum."$" #er%"d"p &"." u#uk se#i"p &usi #"s merk 5ORA 07/ sed"$ merk SON6EBOB

9 0M"s"&"%!" 0"d"&"% 0mee#uk" 0.er"p" 0&usi

se."ik!" #"s merk 5ORA d" merk SON6EBOB !"$ di.u"# "$"r .is" mem"ksimumk" &"."

!


SOAL 7 (MINIMASI) Se.u"% 0#oko TO MIN6 SE 3 me!edi"k" du" 0merk 0pupuk/ 0!"i#u S#"d"rd d" Super Se#i"p jeis me$"du$ 4"mpur" ."%" i#ro$e d" 'os'"# d"&"m jum&"% #er#e#u :eis S#"d"rd Super

K"du$" B"%" Kimi" Ni#ro$e (k$
Seor"$ pe#"i mem.u#u%k" p"&i$ sediki# 1* k$ i#ro$e d" 2, k$ 'os'"# u#uk &"%" per#"i"!" "r$" pupuk S#"d"r d" Super m"si$8 m"si$ 07 d" 0* e#"i #erse.u# i$i me$e#"%ui .er"p" s"k m"si$8 m"si$ 0jeis 0pupuk 0%"rus 0di.e&i 0"$"r 0#o#"& 0%"r$" 0pupuk 0me4"p"i miimum d" ke.u#u%" pupuk u#uk &"%"!" #erpeu%i

SOAL * (MAKSIMASI) M: Tekik Tekik I'orm"#ik" UN "k" memproduksi du" jeis j"ke#/ !"i#u j"ke# S#"d"rd S#"d"rd d" d" j"ke# super super se#i"p jeis jeis j"ke# me$$u"k me$$u"k" " sum.er sum.er d"!" se."$"i .eriku# = sum.er d"!" B"%" ."ku jum&"% j"m

jeis j"ke# j"ke# S#"d"rd Super , * , 2

K"p"si#"s 12++ -++

5iperkir"k" permi#"" roduk s#"d"rd m"ksimum 29+ ui# per .u&"/ sed"$ produk super 7++ ui# per .u&" Sum."$" keu#u$" u#uk produk s#"d"rd se.es"r Rp ,++ per ui# sed"$k" produk Super Rp 7++ per ui# Ber"p" k"p"si#"s produksi op#imum u#uk kedu" jeis produk #erse.u# sup"!" dipero&e% keu#u$" m"ksimum >

"


BAB I; PENDAHULUAN PENDAHULUAN

; Pen3e/i+n Rise O.e/+si Rise h+/i seh+/i>h+/i 2+i- 1en3en+i 1en3en+i 2isnis, 2isnis, e-ono1i, sosi+l sosi+l 1+u.un 2i4+n3 2i4+n3 l+inn?+ -e 4+l+1 .e1o4el+n 1+e1+is unu- 1en4+.+-+n solusi ?+n3 o.i1+l;

!; Pe1o4el+n M+e1+is B+3i+n <e/.enin3 <4+/i h+/i h+/i <-e <4+l+1 <4+l+1 <1o4el <1o4el <1+e1+is; <1+e1+is; <&+-o/>+-o/ <&+-o/>+-o/
5


BAB II; PROGRAM LINEAR P/o3/+1 line+/ +4+l+h s+l+h s+u 1o4el 1+e1+i-+ ?+n3 4i3un+-+n unu1en?eles+i-+n 1+s+l+h o.i1is+si, ?+iu 1e1+-si1u1-+n ++u 1e1ini1u1-+n  un3si un3si uu+n ?+n3 2e/3+nun3 .+4+ seu1l+h C+/i+2el in.u; H+l e/.enin3 ?+n3 .e/lu -i+ l+-u-+n +4+l+h 1en0+/i +hu uu+n .en?eles+i+n 1+s+l+h 4+n +.+ .en?e2+2 1+s+l+h e/se2u; Du+ 1+0+1 un3si P/o3/+1 Line+/: ♦ &un3si

uu+n : 1en3+/+h-+n +n+lis+ unu- 1en4ee-si uu+n .e/u1us+n

1+s+l+h ♦ &un3si

-en4+l+ : unu- 1en3e+hui su12e/ 4+?+ ?+n3 e/se4i+ 4+n .e/1in++n

++s su12e/ 4+?+ e/se2u; ; M+s+l+h M+-si1is+si M+-si1is+si 4+.+ 2e/u.+ 1e1+-si1+l-+n -eunun3+n ++u h+sil; @onoh: PT LAUNATEKSTIL 1e1ili-i se2u+h .+2/i- ?+n3 +-+n 1e1./o4u-si ! enis ./o4u-, ?+iu ?+iu -+in sue/+ 4+n 4+n -+in =ol; =ol; Unu- 1e1./o4u-si 1e1./o4u-si -e4u+ ./o4u- 4i.e/lu-+n 4i.e/lu-+n 2+h+n 2+-u 2+-u 2en+n3 sue/+, sue/+, 2+h+n 2+-u 2en+n3 2en+n3 =ol 4+n 4+n en+3+ -e/+; M+-si1u1 .en?e4i++n 2en+n3 sue/+ +4+l+h 6# -3 .e/ h+/i, 2en+n3 =ol "# -3 .e/ h+/i 4+n 4+n en+3+ -e/+ 5# +1 .e/ h+/i; Ke2uuh+n Ke2uuh+n sei+. sei+. uni ./o4u- +-+n 2+h+n 2+-u 4+n +1 en+3+ -e/+ 4+.+ 4ilih+ 4+l+1 +2el 2e/i-u: %enis 2+h+n 2+-u Ten+3+ -e/+ !

%+1 en+3+ -e/+ K+in =ol " ! 

M+-si1u1 .en?e4i++n 6# -3 "# -3 5# +1

Ke4u+ enis ./o4u- 1e12e/i-+n -eunun3+n se2es+/ R. 5# u+ unu- -+in sue/+ 4+n R. "# u+ unu- -+in =ol; M+s+l+hn?+ +4+l+h 2+3+i1+n+ 1enenu-+n u1l+h uni sei+. enis ./o4u- ?+n3 +-+n 4i./o4u-si sei+. h+/i +3+/ -eunun3+n ?+n3 4i.e/oleh 2is+ 1+-si1+l;

<


L+n3-+h>l+n3-+h: $
!F ! ≤ "# (2en+n3 =ol$

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F! 5#

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!# < E < A #

D

!

@ B !#

F

"#

4+e/+h .en?eles+i+n .en?eles+i+n

6


@+/+ 1en4+.+-+n solusi o.i1+l: ; Den3+n 1en0+/i nil+i  sei+. ii- e-s/i1;B Tii- A F#, F#  ! 1+su--+n nil+i F4+n F !-e    5# ; # J "# ; #  #< Tii- B F!#, F#  ! 1+su--+n nil+i F4+n F !-e    5# ; !# J "# ; #  7## Tii- @ Men0+/i ii- .oon3 (<$4+n ( "$ !FJ "F  ! 6# !FJ F !  5#

!F!# !

F# !

M+su--+n F!-e -en4+l+ ($ !FJ "F  ! 6# !FJ " ; #  6# !FJ "#  6# !F "#

F <

1+su--+n nil+i F4+n F !-e  5#FJ "#F  ! 5# ; < J "# ; #  6## J "##  8## (o.i1+l$ Tii- D !F! "# F! < 1+su--+n F!-e -en4+l+ ($ !FJ " ; <  6# !FJ 5<  6# !F <

F *,<

1+su--+n nil+i F4+n F !-e 

*


  5# ; *,< J "# ; <  "## J 5<# *<# *<# Tii- E F! < F # 1+su--+n nil+i F4+n F !-e    5# ; # J "# ;<  5<# Kesi1.ul+n : unu- 1e1.e/oleh -eunun3+n o.i1+l, 1+-+ F  <  4+n F  # 4en3+n ! -eunun3+n se2es+/ R. 8## u+; !; Den3+n 0+/+ 1en33ese/ 3+/is un3si uu+n;B Solusi o.i1+l +-+n e/0+.+i .+4+ s++ 3+/is un3si uu+n 1en?in33un3 4+e/+h  e+si2le e+si2le (4+e/+h ?+n3 4ili.ui oleh se1u+ -en4+l+$ ?+n3 e/+uh 4+/i ii- o/i3in; P+4+ 3+12+/, solusi o.i1+l e/0+.+i .+4+ ii- <@?+iu .e/sil+n3+n 3+/is -en4+l+ ($ 4+n ("$; Tii- @ Men0+/i ii- .oon3 (<$4+n ( "$ !FJ "F  ! 6# !FJ F !  5#

!F!# !

F# !

M+su--+n F!-e -en4+l+ ($ !FJ "F  ! 6# !FJ " ; #  6# !FJ "#  6# !F "#

F <

1+su--+n nil+i F4+n F !-e  5#FJ "#F  ! 5# ; < J "# ; #  6## J "##  8##

7


! ; M+s+l+h Mini1is+si Mini1is+si 4+.+ 2e/u.+ 1e1ini1u1-+n 2i+?+ ./o4u-si; Solusi o.i1+l e/0+.+i .+4+ s++ 3+/is un3si un3si uu+n 1en?in33un3 1en?in33un3 4+e/+h 4+e/+h +si2le ?+n3 ?+n3 e/4e-+ 4en3+n ii- o/i3in; @onoh : Pe/us+h++n 1+-+n+n ROYAL ROYAL 1e/en0+n+-+n 1e/en0+n+-+ n unu- 1e12u+ 4u+ enis 1+-+n+n ?+iu Ro?+l Bee 4+n Ro?+l %ell?; Ke4u+ enis 1+-+n+n e/se2u 1en3+n4un3 Ci+1in 4+n ./oein; Ro?+l Bee .+lin3 se4i-i 4i./o4u-si ! uni 4+n Ro?+l %ell? .+lin3 se4i-i se4i-i 4i./o4u-si  uni; T+2el T+2el 2e/i-u 1enunu--+ 1enunu--+n n u1l+h Ci+1in Ci+1in 4+n ./oein 4+l+1 sei+. sei+. enis 1+-+n+n: 1+-+n+n: %enis 1+-+n+n

i+1in (uni$

P/oein (uni$

Bi+?+ .e/ uni (/i2u /u.i+h$

Ro?+l Bee

!

!

##

Ro?+l %ell?



"

7#

1ini1u1 -e2uuh+n

7

!

B+3+i1+n+ 1enenu-+n -o12in+si -e4u+ enis 1+-+n+n +3+/ 1e1ini1u1-+n 2i+?+ ./o4u-si; ./o4u-si; L+n3-+h  l+n3-+h: ;
(Ci+1in$

!$
(./oein$

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8


F <, # F ! < 7 F< <, F< # 5< !  !$
("$ 7

(!$

4+e/+h .en?eles+i+n .en?eles+i+n @

5

B

 !

(5$

A 5

F

6

Solusi !F! >5

F! !

1+su--+n F!-e -en4+l+ ($ !FJ F  ! 7 !FJ !  7 ! F 6

F "

1+su--+n nil+i F4+n F !-e   1in  ##F J 7#F< ! ## ; " J 7# ; !  "## J 6#  56# Kesi1.ul+n :

#


Unu- 1e1ini1u1-+n 2i+?+ ./o4u-si, 1+-+ F  " 4+n F ! ! 4en3+n 2i+?+ ./o4u-si 56# /i2u /i2u /u.i+h;

SOAL LATIHAN ;
!

Ken4+l+ : $ !F < 7  ≤ !$ "F!<≤< "$ 6FJ
!

Ken4+l+: $ 6F J 6  F 1+sin3 R. <#, R. 5## 4+n R. 6##; Sei+. 1in33u 4ie+.-+n 1ini1u1 ./o4u-si /oi .i+ !< uni, 2olu-is1is 2olu-is1is "# uni 4+n 0o-l+-eu

uni; Kei3+ enis enis /oi 1e1e/lu-+n 1e1e/lu-+n

.e1/oses+n i3+ i3+ -+li ?+iu .en?i+.+n .en?i+.+n 2+h+n, 2+h+n, .e/+0i-+n 4+n .en3oCen+n .en3oCen+n se.e/i e/lih+ .+4+ +2el 2e/i-u: Pe1/oses+n

%enis /oi .i+

Pen?e4i++n Pen?e4i++n 1+

2olu-is1is 2olu-is1is <0o-l+-eu <0o-l+-eu

(+1$

.en?i+.+n 2+h+n 2+h+n

5

!

6

"#

.e/+0i-+n

"

5

8

*#

.en3oCen+n



!

5

!

B+3+i1+n+ o/1ul+si ./o3/+1 line+/ 1+s+l+h PT B+-e/? e/se2u 4+n hiun3 solusi o.i1+ln?+ o.i1+ln?+


BAB III; METODE SIMPLEF Meo4e 3/+i- i4+- 4+.+ 1en?eles+i-+n 1en?eles+i-+n .e/so+l+n line+/ ./o3/+1 ?+n3 1e1il-i C+/i+2el <-e.uus+n n?+ RHS>n?+ ne3+i, 4i-+li-+n l+3i 4en3+n  4+n 4i+12+h +:P-1--+M2n+:i+2el (M$; ;<&un3si -en4+l+ 4en3+n +n4+ ') h+/us 4i+12+h +:P-1--+M2n+:-+3RM (M$; Pe12u++n T+2el Si1.le @onoh so+l:   "FJ
"F!

"$ 6F
≤

7

≤

<

≤

"#

L+n3-+h>l+n3-+h: ;ol-C2( +#s5(C2us0(6(+Cs3+Cs5(C2usi-C3+"+s(l +#s5(C2us0(6(+Cs3+Cs5(C2usi-C3+"+s(lih+ ih+ 2e2e/+.+ -eenu+n ?+n3 h+/us 4i.e/h+i-+n 4i ++s$ &un3si uu+n   "FJ
  > "F>
!


&un3si -en4+l+ $ !F !$

"F!

"$ 6F
≤

7

≤

< <

≤

"#

 !F

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BAB III; DUALITAS D+l+1 se2u+h .e1o4el+n Pe1/o3/+1+n Line+/, e/4+.+ 4u+ -onse. ?+n3 s+lin3 2e/l+=+n+n; Konse. Konse. ?+n3 .e/+1+ .e/+1+ -i+ se2u P/i1+l 4+n ?+n3 ?+n3 -e4u+ Du+l;BenuDu+l;BenuDu+l +4+l+h -e2+li-+n 4+/i 2enu- P/i1+l; Hu2un3+n P/i1+l 4+n Du+l se2+3+i 2e/i-u: M+s+l+h P/i1+l (++u Du+l$

M+s+l+h Du+l (++u P/i1+l$

Koeisien un3si uu+n 
@onoh : P/i1+l Mini1u1-+n  

&un3si 2++s+n:

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SOAL LATIHAN ;

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BAB I; MASALAH PENUGASAN (ASSIGNMENT PROBLEM$ S+l+h s+u 1eo4e ?+n3 4i3un+-+n unu- Penu3+s+n +4+l+h Meo4e Hun3+/i+n; P+4+ Meo4e Hun3+/i+n, u1l+h su12e/>su12e/ ?+n3 4iu3+s-+n h+/us s+1+ .e/sis <4en3+n <u1l+h <u1l+h <u3+s <u3+s 2+/isn?+ 1enunu--+n 1enunu--+n su12e/>su12e/ 4+n -olo1> -olo1n?+ 1enunu--+n u3+s>u3+s; ; M+s+l+h Mini1is+si @onoh: Se2u+h .e/us+h++n -e0il 1e1.un?+i 5 .e-e/++n ?+n3 2e/2e4+ unu- 4iseles+i-+n oleh 5 -+/?+=+n; Bi+?+ .enu3+s+n seo/+n3 -+/?+=+n unu- .e-e/++n ?+n3 2e/2e4+ +4+l+h 2e/2e4+ -+/en+ si+ si+ .e-e/++n 2e/2e4+>2e4+; 2e/2e4+>2e4+; Sei+. Sei+. -+/?+=+n 1e1.un?+i in3-+ -e/+1.il+n, .en3+l+1+n -e/+ 4+n l++/ 2el+-+n3 .en4i4i-+n se/+ l+ih+n ?+n3 2e/2e4+ .ul+; Sehin33+ 2i+?+ .en?eles+i+n .e-e/++n ?+n3 s+1+ oleh .+/+ -+/?+=+n ?+n3 2e/l+in+n u3+ 2e/2e4+; T+2el 2i+?+ se2+3+i 2e/i-u: .e-e/++n -+/?+=+n R+ih+n H+14+n H+s+n D+-=+n

I

II

III

I

R. <# R. 5# R. !<# R. *#

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M+s+l+hn?+ <+4+l+h <2+3+i1+n+ <1enu3+s-+n <-ee1.+ <-+/?+=+n l+n3-+h: ;
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2; -u/+n3-+n se1u+ se1u+ 2i+?+ 4en3+n 4en3+n 2i+?+ e/-e0il e/-e0il sei+. 2+/is Sehin33+ 1en3h+sil-+n :RGiRG2< P26+P:-*2/1+/i- 2i+?+ ?+n3 el+h 4i-u/+n3i; .e-e/++n -+/?+=+n R+ih+n H+14+n H+s+n D+-=+n

I (<#><#$ # (5#>5#$ # (!<#>!##$  <# (*#>6#$  #

II (!##><#$ <# (6#>5#$  !# (!##>!##$ # (7#>6#$  !#

III (7#><#$  "# (!#>5#$ *# (!"#>!##$  "# (7#>6#$  !#

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";-olo1 e/se2u; P+4+ 0onoh 4i ++s h+n?+ 4il+-u-+n .+4+ -olo1 III -+/en+ se1u+ -olo1 l+inn?+ el+h 1e1.un?+i ele1en ?+n3 2e/nil+i nol (#$; %i-+ l+n3-+h -e4u+ el+h 1en3h+sil-+n .+lin3 se4i-i s+u nil+i nol .+4+ sei+. -olo1, 1+-+ l+n3-+h -ei3+ 4+.+ 4ihil+n3-+n; Be/i-u 1+/i o+l o..o/uni? 0os, 4i1+n+ sei+. 2+/is 4+n -olo1 e/4+.+ .+lin3 se4i-i s+u nil+i nol; T+2el o+l o..o/uni? 0os 1+/i .e-e/++n -+/?+=+n R+ih+n H+14+n H+s+n D+-=+n

I

II

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I

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5; Me12enu- .enu3+s+n o.i1u1 P/ose4u/ ./+-is unu- 1el+-u-+n es o.i1+lis+si +4+l+h 4en3+n 1en+/iseu1l+h 1ini1u1 3+/is ho/ison+l 4+n9 ++u Ce/i-+l unu- 1eli.ui selu/uh ele1en 2e/nil+i nol 4+l+1 o+l o..o/uni? 0os 1+/i; %i-+ u1l+h 3+/is s+1+ 4en3+n u1l+h 2+/is9 -olo1 1+-+ .enu3+s+n el+h o.i1+l; %i-+ i4+- 1+-+ h+/us 4i/eCisi;

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I

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Be/i-u +2el .enu3+s+nn?+ Penu3+s+n R+ih+n H+14+n H+s+n D+-=+n

> > > >

III I II I

Bi+?+ R. 7# R. 5# R. !## R. 6# R. 67#

!; %u1l+h Pe-e/++n Ti4+- S+1+ Den3+n %u1l+h K+/?+=+n Bil+ u1l+h .e-e/++n le2ih 2es+/ 4+/i u1l+h -+/?+=+n, 1+-+ h+/us 4i+12+h-+n 4i+12+h-+n -+/?+=+n se1u (4u11? =o/-e/$; Bi+?+ se1u s+1+ 4en3+n nol -+/en+ i4+- +-+n e/+4i 2i+?+ 2il+ su+u .e-e/++n 4iu3+s-+n -e -+/?+=+n se1u; Bil+ u1l+h -+/?+=+n le2ih 2+n?+- 4+/i.+4+ .e-e/++n, 1+-+ 4i+12+h-+n .e-e/++n se1u

!<


(4u11? o2$; Se2+3+i Se2+3+i 0onoh, 2il+ u1l+h .e-e/++n le2ih 2es+/ 4+/i u1l+h -+/?+=+n 4+.+ 4ilih+ .+4+ +2el 2e/i-u: .e-e/++n -+/?+=+n R+ih+n H+14+n H+s+n D+-=+n Du11? F

I R. <# R. 5# R. !<# R. *# R. #

II

III

R. !## R. 6# R. !## R. 7# R. #

I

R. 7# R. !# R. !"# R. 7# R. #

R. !!# R. *# R. !## R. 6# R. #

P/ose4u/ .en?eles+i+n s+1+ 4en3+n l+n3-+h>l+n3-+h se2elu1n?+; "; M+s+l+h M+-si1is+si D+l+1 <1+s+l+h <1+-si1is+si, ele1en <1+/i-s <1enunu--+n <in3-+ -eunun3+n; -eunun3+n; Ee-iCi+s E e-iCi+s .el+-s+n++n u3+s oleh -+/?+=+n 4iu-u/ 4en3+n u1l+h -on/i2usi -eunun3+n; @onoh: T+2el -eunun3+n Pe-e/++n -+/?+=+n Ai B+4? D+-? &+//+s Gh+?

I

II

III

I



R. ### R. 5## R. 8## R. "## R. ###

R. !## R. ### R. 7## R. <## R. "##

R. ### R. 8## R. *## R. 7## R. 5##

R. 7## R. <## R. 7## R. 6## R. ##

R. <## R. "## R. !## R. ## R. *##

L+n3-+h>l+n3-+h: +;
I

II

III

I



## ## "## "## *##

"## ## 5## ## 5##

## 6## ## 7## "##

*## # 5## # 6##

# !## # ## #

!6


2;loss o..o/uni?>loss 4en3+n 4en3+n 0+/+ 1en3u/+n3i 1en3u/+n3i selu/uh ele1en ele1en 4+l+1 sei+. -olo1 (?+n3 2elu1 +4+ nol>n?+$ 4en3+n ele1en e/-e0il 4+/i -olo1 e/se2u; M+/i-s o+l o..o/uni? loss Pe-e/++n -+/?+=+n Ai B+4? D+-? &+//+s Gh+?

I

II

III

I



5## # !## !## 6##

!## 5## "## # "##

!## "## !## ## #

*## # 5## # 6##

# !## # ## #

D+/i 1+/i-s 4i ++s 4+.+ 4ilih+ 2+h=+ selu/uh ele1en ?+n3 2e/nil+i nol 2+/u 4+.+ 4ili.u oleh 5 3+/is; %+4i 1+/i-s h+/us 4i/eCisi; 0; Me/eCisi 1+/i-s Pe-e/++n -+/?+=+n Ai B+4? D+-? &+//+s Gh+?

I

II

III

I



!## # # !## 6##

# 5## ## # "##

# "## # ## #

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# 5## # *## !##

S0he4ul <.enu3+s+n
-eunun3+n

Penu3+s+n +le/n+i !

-eunun3+n

Ai B+4? D+-? &+//+s Gh+?

R. !## R. 5## R. !## R. 6## R. 5## R. 67##

Ai B+4? D+-? &+//+s Gh+?

R. <## R. <## R. 8## R. <## R. I(++B R. 67##

> II >I > > I > III

!*

> > I >I > II > III


SOAL LATIHAN ;
M !# 5# <# !##

Mesin < M! <# 6# *< <

M" 7# !## !!# 6#

M5 "# 8# !## 8#

M+s+l+hn?+ <+4+l+h <2+3+i1+n+ <1enu3+s-+n <-ee1.+ <1esin 1+sin3 -+/?+=+n i+. 2ul+nn?+ 4+.+ 4ilih+ .+4+ +2el 2e/i-u: K+/?+=+i < Ul+h S+l1+ R+n+ N+2il+

@el+n+ .+n+n3 6 ! 7 *

P/o4u- < Ro* 7 8 

He1 # * !

B+u s++/i 8 7 ! "

Bu+ .enu3+s+n +3+/ u1l+h ./o4u- ?+n3 4ih+sil-+n 2is+ 1+-si1u1

!7


BAB ; METODE TRANSPORTASI

Meo4e T/+ns.o/+si 1e/u.+-+n su+u 1eo4e ?+n3 4i3un+-+n unu- 1en3+u/ 4is/i2usi 4+/i su12e/>su12e/ ?+n3 1en?e4i+-+n ./o4u- ?+n3 s+1+ -e e1.+> e1.+ ?+n3 1e12uuh-+n se0+/+ o.i1+l 4en3+n 2i+?+ ?+n3 e/1u/+h ; Alo-+si ./o4u- ini h+/us h+/us 4i+u/ se4e1i-i+n se4e1i-i+n /u.+ -+/en+ -+/en+ e/4+.+ .e/2e4++n .e/2e4++n 2i+?+>2i+?+ 2i+?+>2i+?+ +lo-+si 4+/i s+u su12e/ ++u 2e2e/+.+ su12e/ -e e1.+ uu+n ?+n3 2e/2e4+; T+2el +=+l 4+.+ 4i2u+ 4en3+n 4u+ 1eo4e, ?+iu: ; Meo4e No/h Wes @o/ne/ (NW@$

 4+/i .oo- -i/i ++s -e .oo-

-+n+n 2+=+h Kele1+h+n : i4+- 1e1.e/hiun3-+n 2es+/n?+ 2i+?+ sehin33+ -u/+n3 eisien; !; Meo4e 2i+?+ e/-e0il

 1en0+/i 4+n 1e1enuhi ?+n3 2i+?+n?+ e/-e0il

4ulu; Le2ih eisien 4i2+n4in3 1eo4e NW@;

Seel+h +2el +=+l 4i2u+, +2el 4+.+ 4io.i1+l-+n l+3i 4en3+n 1eo4e: ; Se..in3 Sone (2+u lon0++n$ !;Mo4iie4 Dis/i2uion Meho4 (MODI$

Sel+in 1eo4e>1eo4e 4i ++s 1+sih +4+ s+u 1eo4e ?+n3 le2ih se4e/h+n+ .en33un++nn?+ .en33un++nn?+ ?+iu 1eo4e o3els A../oi1+ion Meho4 (AM (AM

$;

@onoh 1+s+l+h /+ns.o/+si:

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Pen?eles+i+n: ; Meo4e NW@

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Men3o.i1+l-+n +2el: ; Meo4e Se..in3 Sone , 1is+l +2el +=+l 1en33un+-+n ?+n3 NW@ -e

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!; Meo4e MODI L+n3-+h>l+n3-+h: +; Mis+l +2el +=+l ?+n3 4i3un+-+n +4+l+h +2el NW@ 2; Bu+ C+/i+2el Ri 4+n K unuunu- 1+sin3>1+sin3 1+sin3>1+sin3 2+/is 4+n -olo1; -olo1; 0; Hiun3 sel ?+n3 2e/isi (nil+i i+. -olo1 4+n i+. 2+/is$ 4en3+n /u1us: Ri 2+/is

J

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@i > Ri > K ;A  <  <  !#  > !# !;
A  !<  <  !# < #<  > 5 "; W>@ < 7  #  5 5;@  #  <  5  > 8 (o.i1+l i-+ .+4+ sel ?+n3 -oson3, in4e- .e/2+i-+nn?+ ≥< #, i-+ 2elu1 1+-+ .ilih ?+n3 ne3+in?+ ne3+in?+ 2es+/$ e; Me1ilih ii- ol+- .e/u2+h+n Pilih nil+i ?+n3 ne3+in?+ 2es+/ ?+iu H>A  ; Bu+ +lu/ e/uu. Be/il+h +n4+ .osii .+4+ H>A; Pilih  sel e/4e-+ ?+n3 isi 4+n se2+/is (H>B$,  sel ?+n3 isi e/4e-+ 4+n se-olo1 (W>A$, 2e/il+h +n4+ ne3+i .+4+ 4u+ sel e/e2u; Ke1u4i+n .ilih s+u sel ?+n3 se2+/is ++u se-olo1 4en3+n 4u+ sel 2e/+n4+ ne3+i ne3+i +4i (W>B$ (W>B$ 4+n 2e/i +n4+ +n4+ .osii; Sel+nun?+ Sel+nun?+ .in4+h-+n .in4+h-+n isi 4+/i sel 2e/+n4+ ne3+i -e ?+n3 2e/+n4+ .osii se2+n?+- isi e/-e0il 4+/i sel ?+n3 2e/+n4+ .osii (<#$; %+4i, H>A -e1u4i+n 2e/isi <#, H>B 2e/isi 6#><##, W>B 2e/isi 5#J<#8# 4+n W>A i4+- 2e/isi; -e 4+/i

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Bi+?+ /+ns.o/+si : (8# ; <$ J (<# ; <$ J (# ; #$ J (!# ; #$ J ("# ; 8$  !#*# Hiun3 sel ?+n3 ?+n3 2e/isi: W>B < R J K!  < P>B  R" J K!  # P>@  R" J K"  8 H>@  R! J K"  # H>A  R! J K  <

 #< J K!  < ,  R" J <  # ,  < J K"  8 ,  R! J 5  # ,  > 5 J K  < ,

Pe/2+i-+n in4e-s (sel -oson3$ : W>A < !#  #  #  !# W>@ < W>@ < 7  #  5  > 6 H>B < !#  <  <  # P>A < !<  <  #  !#

"<

K!  < R"  < K"  5 R!  > 5 K  8


-e

Gu4+n3 A

Gu4+n3 B

Gu4+n3 @

K+.+si+

K  8

K!  <

K"  5

s .+2/i-

4+/i P+2/i-W

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J

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#

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Bi+?+ /+ns.o/+si : (7# ; <$ J (# ; 7$ J (<# ; <$ J (# ; #$ J ("# ;#$ J (!# ; 8$ !## Sel 2e/isi: W>B < R J K!  < W>@ < R J K"  7 H>@  R! J K"  # H>A  R! J K  < P>B  R" J K!  #

 #< J K!  < ,  # J K"  7 ,  R! J 7  # ,  ! J K  < ,  R" J <  # ,

K!  < K"  7 R!  ! K  " R"  <

In4e-s .e/2+i-+n: W>A < !#  #  8   H>B  !#  (>5$  <  8 P>A  !<  <  8   In4e-s .e/2+i-+n su4+h .osii se1u+, 2e/+/isu4+h 2e/+/isu4+h o.i1+l ; -e 4+/i

Gu4+n3 A

Gu4+n3 B

Gu4+n3 @

K+.+si+

K  "

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K"  7

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"; Meo4e AM AM Meo4e AM 1e/u.+-+n 1eo4e ?+n3 le2ih 1u4+h 4+n le2ih 0e.+ unu1en3+u/ +lo-+si 4+/i 2e2e/+.+ su12e/ -e 4+e/+h uu+n; L+n3-+h 1eo4e 1eo 4e AM: AM: ;<@+/i .e/2e4++n 4u+ 2i+?+ e/-e0il, ?+iu e/-e0il .e/+1+ 4+n -e4u+ (-olo1 4+n 2+/is$ !;<
B

@

K+.+si+s

Pe/2e4++n 2+/is

7

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F< PB <# Hil+n3-+n 2+/is P

Pe/2e4++n 2+/is

 6# Pe/2e4++n

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Bi+?+ /+ns.o/+si : (# ; <#$ J (< ; 6#$ J (7 ; "#$ J (< ; <#$ J (# ; #$  78# (o.i1+l$

"7


SOAL LATIHAN ; -e

Gu4+n3 A

Gu4+n3 B

Gu4+n3 @

R. "!## R. "6## R. "5## !!

R. ""## R. 5!## R. "*## 
R. "5## R. "7## R. 5### 8

4+/i P+2/i-  P+2/i- ! P+2/i- " Ke2uuh+n 3u4+n3

K+.+si+s .+2/i#6 "! !* "6<

Seles+i-+n 4en3+n 1eo4e: +;
K+.+si+s ./o4u-si i+. 2ul+n <# on 5# on 7# on !*# on

Gu4+n3 .+2/i- e/se2u 1e1.un?+i -+.+si+s se2+3+i 2e/i-u: Gu4+n3 H I % u1l+h

Ke2uuh+n ./o4u-si i+. 2ul+n # on *# on 8# on !*# on

Bi+?+ unu- 1en4is/i2usi-+n 2+/+n3 4+/i .+2/i- -e 3u4+n3 : D+/i P+2/i- A P+2/i- B P+2/i- @

Bi+?+ i+. on Ke Gu4+n3 H !*### #### "####

(R.$ Ke Gu4+n3 I !"### 5<### 5###

Ke Gu4+n3 % "### 5#### "<###

+;
"8


DA&TAR PUSTAKA ;Hill, Sin3+.o/e; ";D+s+/ O.e/+ion Rese+/0h, Yo3?+-+/+ : PT; BP&E>Yo3?+-+/+, !###; 5;P/insi. P/insi.>P/insi. Rise O.e/+si , E/l+n33+, !##< ;
5#

RISET OPERASIONAL teknik sipil

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