Linear Programming Simplex Methode

7/23/2019 Linear Programming Simplex Methode

http://slidepdf.com/reader/full/linear-programming-simplex-methode 1/78

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Jan 4, 2016 LINEAR PROGRAMMING 1

LINEAR PROGRAMMINGLINEAR PROGRAMMINGSIMPLEX METHODSIMPLEX METHOD

Simplex Method- An algebraic, iterative method to solve linear programming problems.

The simplex method requires that the problem is expressed as a standard LP problem. This implies that all the constraints are

expressed as equations by adding slac variables. (a!ia"#$ %angm$&a'i#i ing'a )$nganggu!an aau 'a)asias %ang m$!u)a'an"aasan* +1 , +2, +,--+m

The method uses !aussian elimination "s#eep out method$ to solve the linear simultaneous equations generated in the

process.



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+#ac' a!ia"#$s+#ac' a!ia"#$s

In$/ua#i% cns!ains can "$ cn$!$d $/ua#ii$s

"% in!ducing s#ac' a!ia"#$s3

Misa#

512525454 ≤ 27, "isa diu#is s$"agai

512525454+ 8 27d$ngan s ≥ 0 (s#ac' a!ia"#$*

Aau

512525454 ≥ 27, "isa diu#is s$"agai 512525454 9 + 8 27

d$ngan s ≥ 0 (s#ac' a!ia"#$*



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Jan 4, 2016 LINEAR PROGRAMMING

Objective function: 8 ;1 51 ;2 52 ; 5 ;4 54 ............. ;n 5n

Constraints 1*. a11 51 a12 52 a1 5 ......... a1n 5n < "1

2*. a21 51 a22 52 a2 5 ......... a1n 5n < "2

*. a1 51 a2 52 a 5 ......... an 5n < "

4*. a41 51 a42 52 a4 5 ......... a4n 5n < "4

. . . . . . . m*. am1 51 am2 52 am 5 ......... amn 5n < "m

5i = 0 , i 8 1,2,-..n " > ≥0 , >81,2,-,m

5 ≡ a!ia"#$s? c ≡ ">$ci$ )a!am$$!s?

a ≡ cns!ains )a!am$$!s?" ≡ !igh hand sid$ a#u$ cns!ains

Lan!a" #en$e%esaian &'(Lan!a" #en$e%esaian &'(

') %epresent the LP problem in standard &orm



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Lan!a" #en$e%esaian &*(Lan!a" #en$e%esaian &*(

;0=−∑ j

j jc z χ

dengan z adalah nilai dari fungsi tujuan

'. (yataan persamaan &ungsi tu)uan dalam bentu

*. %ubah semua pertidasamaan batasan "all inequality constrains$ e dalam bentu persamaan batasan "equality constrains$ dengan memasuan variable Slac

+. Susun persamaan &ungsi tu)uan dan persamaan batasan e dalam tabel simplex.



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Lanah .

MaimiB$ :8451752 (0*

Caasan

51 252 ≤ 40 (1*

451 52 ≤ 120 (2*

;ONDO Perusahaan barang tembikar

Langah ' dan *

Fungsi u>uan : 9 451 9 752 8 0 (0*

51 252 +1 8 40 (1*

451 52 +2 8 120 (2*

Lan!a" #en$e%esaian &+(Lan!a" #en$e%esaian &+(

Caasan



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Lan!a" #en$e%esaian &,(Lan!a" #en$e%esaian &,(

Langah + Menyusun table Simplex a#al

-asic.aria

b%e

E/t)

;$ici$n R+(s#*: 51 52 +1 +2

0 "$ -+ -/

S' "$ ' +

S*"'$ + * '

0asic 1ariable "1ariable dasar$ ada#ah a!ia"#$ %ang ni#ain%asama d$ngan sisi 'anan da!i )$!samaan.

Pada persamaan (1) dan (2) bila belum ada kegiatan maka X 1 = 0

dan X 2 = 0, sehingga nilai S1 = 40 dan nilai S2 = 120 Pada tabel

a!al diatas nilai "asi# $ariable (S1 dan S2 ) pada fungsi tujuan harus

0 (n%l)



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Lan!a" #en$e%esaian &1(Lan!a" #en$e%esaian &1(

Langah / Setelah data tersusun dalam tabel simplex a#al,lauan iterasi sehingga dihasilan titi optimal

7.1. M$n$nu'an entering variable Hica!i a!ia"$# %ang )a#ing s$nsii $!hada) ungsi u>uan (ma :*.

Ha!i a"$# ("a!is :* $!#iha "ah&a ni#ai a"s#u '$isi$n 52 $!"$sa!

%aiu #7#, >adi di)i#ih 52 s$"agai $n$!ing a!ia"#$. #m 52 dis$"u

)i c#umn (P;*. Ci#a )ada a"$# sudah ida' m$m)un%ai #agi!oeffisien $an berni%ai neatif )ada "a!is ungsi u>uan, ma'aa"$# ini ida' "isa #agi di )ima#'an (sudah )ima#*.

+$#an>un%a hiung ni%ai ratio, Ni#ai '#m R+ di"agi d$ngan ni#ai)ada )i c#umn %ang "$!'$s$suaian.

7.2. M$n$nu'an Leaving variable di$nu'an "$!dasa!'an ni#ai !ai minimum, di)i#ih +1

s$"agai #$aing a!ia"#$. Ca!is +1dis$"u Pi Ra& (PR*



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Jan 4, 2016 LINEAR PROGRAMMING K

Iteration

-asic.aria

b%e

E/t)

;$ici$n R+(s#* Rai: 51 52 +1 +2

0

0 "$ -+ -/

S' "$ ' + '

S*"'$ + * ' +

Lan!a" #en$e%esaianLan!a" #en$e%esaian

Pic#umn

Ratio 2

,34* 5 *36ini6u6

Pi$#$m$n

Pi Ra&



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1)+) Me6buat tab%e si6#%e7 !e8ua

Cagi s$mua $#$m$n )ada Pi Ra& (PR* d$ngan Pi E#$m$n.Hihasi#'an $#$m$n )i !a& "a!u.

Gani#ah "asic a!ia"#$ )ada "a!is iu d$ngan a!ia"#$ %ang$!da)a diaas )i c#umn

Iteration

-asic.ariab%e

E/t)

;$ici$n R+

(s#*: 51 52 +1 +2

1

0 "$

X* "$ ,/ ,/ '

S* "'$



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iung ni#ai $#$m$n )ada "a!is %ang #ain (ida' $!masu' $#$m$nPi Ra&* d$ngan ca!a

0 -+ -/

X* ,/ ,/ '

0 -,/ ',/

E#$m$n "a!is "a!u 8 (E#$m$n "a!is #ama* ('$isi$n )ada

)i c#umn* (ni#ai $#$m$n "a!u )i !a&*

&enghitung nilai baru element baris dari '

E#$m$n "a!is #ama

$#$m$n "a!u )i !a&

E#$m$n "a!is "a!u

9oeffisien#a8a PC



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S* + * '

X* ,/ ,/ '

S* ',/ -,/ 2

&enghitung nilai baru element baris dari S2

E#$m$n "a!is #ama

$#$m$n "a!u )i !a&

E#$m$n "a!is "a!u

9oeffisien#a8a PC


Hihasi#'an a"$# sim)#$ 2



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Iterat

ion

-asic

.ariab%e

E/t)

;$ici$n R+

(s#*: 51 52 +1 +2

1

0 "$ -,/ ',/

X* "$ ,/ ,/ '

S*"'$ ',/ -,/ 2


Da"$# +im)#$ 2

Pada "a!is ungsi u>uan masih ada '$isi$n "$!ha!ga n$gai, %aiu

'$isi$n a!ia"#$ 51.

D$!us'an )!s$s i$!asi, dimu#ai da!i #ang'ah '$ 7



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Iteration

-asic.aria

b%e

E/t)

;$ici$n R+(s#* Rai

: 51 52

+1 +2

1

0 "$ -,/ ',/

X* "$ ,/ ,/ ' +

S*"'$ ',/ -,/ 2 '+


Pic#umn

Ratio 2:34*;1 5 *,

6ini6u6

Pi$#$m$n

Pi Ra&

Lang'ah 7.1 dan 7.2



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Lan!a" 1)+) Me6buat tab%e si6#%e7 !etia

Cagi s$mua $#$m$n )ada Pi Ra& (PR* d$ngan Pi E#$m$n.Hihasi#'an $#$m$n )i !a& "a!u.

Gani#ah "asic a!ia"#$ )ada "a!is iu d$ngan a!ia"#$ %ang$!da)a diaas )i c#umn

Iteration

-asic.ariab

%eE/t)

;$ici$n R+

(s#*: 51 52 +1 +2

1

0 "$

X* "$

X' "'$ -,2 ,+ '+



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iung ni#ai $#$m$n )ada "a!is %ang #ain (ida' $!masu' $#$m$nPi Ra&* d$ngan ca!a

0 -,/ ',/

X' -,2 ,+ '+

0 ,2 ,2 *2

E#$m$n "a!is "a!u 8 (E#$m$n "a!is #ama* ('$isi$n )ada

)i c#umn* (ni#ai $#$m$n "a!u )i !a&*

&enghitung nilai baru element baris dari '

E#$m$n "a!is #ama

$#$m$n "a!u )i !a&

E#$m$n "a!is "a!u

9oeffisien

#a8a PC



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X* ,/ ,/ '

X'

-,2 ,+ '+

X* ,3 -' 3

&enghitung nilai baru element baris dari X 2

E#$m$n "a!is #ama

$#$m$n "a!u )i !a&

E#$m$n "a!is "a!u

9oeffisien#a8a PC

Hihasi#'an a"$# sim)#$



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Iterat

ion

-asic

.ariab%e

E/t)

;$ici$n R+

(s#*: 51 52 +1 +2

2

0 "$ ,2 ,2 *2

X* "$ ,3 -' 3

X'"'$ -,2 ,+ '+


Da"$# +im)#$

Pada "a!is ungsi u>uan ida' ada #agi '$isi$n "$!ha!ga n$gai,

Hihasi#'an ii' )ima# )ada 51 8 24 dan 52 8 K d$ngan ni#ai'$unungan : 8 16.



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Jan 4, 2016 LINEAR PROGRAMMING 1K

Iteration

-asic.aria

b%eE/t.

;$ici$n R+(+#*: 51 52 +1 +2

0

0 "$ -+ -/

S'"$ ' +

S*"'$ + * '

1

0 "$ -,/ ',/

X*"$ ,/ ,/ '

S*"'$ ',/ -,/ 2

2

0 "$ ,2 ,2 *2

X*"$ ,3 -' 3

X'"'$ -,2 ,+ '+

Lan!a" #en$e%esaianLan!a" #en$e%esaian SIMPLEX METHODSIMPLEX METHOD



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SIMPLEX METHODSIMPLEX METHOD

') -i%a a8a 8ua atau %ebi" variabe% non basis 6e6#un$ai !oefisienneatif terbesar $an sa6a; 6a!a #e6i%i"an enterin variab%e8a#at 8ija%an!an secara bebas)

Mana $an %ebi" ce#at 6enca#ai o#ti6a% ti8a! 8a#at 8i#re8i!si)

4ontoh 2 Fungsi u>uan )ada cnh )a"!i' $m"i'a! di!u"ah m$n>adi

MaimiB$ :8451452 (0*

Caasan

51 252 ≤ 40 (1*

451 52 ≤ 120 (2*

+$#an>un%a disusun a"$# sim)#$ a&a#



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-asic.ariab%

e

E/t)

;$ici$n R+

(s#*: 51 52 +1 +2

0 "$ -+ -+

S'"$ ' +

S*"'$ + * '


Da"$# sim)#$ a&a#

Ha!i a"$# sim)#$ a&a#, am)a' a!ia"#$ nn "asis 51 dan 52

m$m)un%ai ni#ai '$isi$n n$gai %ang sama %aiu 94. O#$h 'a!$na iu)ada )$n%$#$saian a&a# $n$!ing a!ia"#$ da)a di)i#ih 51 aau 52 .



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*) -i%a a8a 8ua atau %ebi" variabe% basis 6e6#un$ai ni%ai RATIO6ini6u6 $an sa6a; 6a!a #e6i%i"an %eavin variab%e 8a#at8ija%an!an secara bebas)

Mana $an %ebi" ce#at 6enca#ai o#ti6a% ti8a! 8a#at 8i#re8i!si)

4ontoh 2 "aasan (1* )ada cnh )a"!i' $m"i'a! di!u"ah m$n>adi

MaimiB$ :8451752 (0*

Caasan

51 252 ≤ 40 (1*

4X 1 4X 2 ≤ 0 (2*

+$#an>un%a disusun a"$# sim)#$ a&a#



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Iteration

-asic.ariab

%eE/t)

;$ici$n R+

(s#* Rai: 51 52 +1 +2

0

0 "$ -+ -/

S' "$ ' + *3

S*"'$ + + 3 '

Da"$# sim)#$ a&a#

Pic#umn

Ratio6ini6u6

Ha!i a"$# sim)#$ a&a#, am)a' a!ia"#$ "asis+1 dan +2 m$m)un%ai ni#ai !ai minimum

%ang sama %aiu 20. O#$h 'a!$na iu )ada

)$n%$#$saian a&a# #$aing a!ia"#$ da)adi)i#ih +1 aau +2 .



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P$n%$#$saian sim)#$ m$hd "agi 'asus %ang m$n%im)ang da!i"$nu' sanda!d

His$#$sai'an d$nganm$ngin!du'si s#ac'a!ia"#$ s$"agai a!ia"#$

"asis %ang ha!gan%a samad$ngan !uas 'anan()sii*

∑ χ= j

j jczMaimiB$

+u">$c ai> 5 > "i ("i 0 *

5 > 0

Ci#a ada )$n%im)angan da!i "$nu' sanda!d di#a'u'an

)$n%$suaian9)$n%$suaian di #ang'ah a&a#. +$$#ah iu m$d$sim)#$ dis$#$sai'an s$)$!i s$"$#umn%a.



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P$nd$'aan sanda!d D$'ni' m$ngguna'an a!ia"#$ "uaan(a!iicia# a!ia"#$*

M$masu''an dumm* $ariable (dis$"u a!iicia# a!ia"#$* '$da#am s$ia) "aasan (cns!ains* %ang m$m$!#u'an.

a!ia"#$ %ang "a!u a'an m$n>adi a!ia"#$ "asis )ada )ada)$n%$#$saian a&a# "agi "aasan (cnsains* %ang "$!sang'uan

I$!asi m$d$ sim)#$ a'an m$m"ua a!iicia# a!ia"#$ m$n>adin#, s$hingga a'hi!n%a sau )$!sau hi#ang.



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') Persa6aan batasan &constraints( jenis 5 &sa6a 8enan(

;nh asus a&a#

MaimiB$ . : 8 51 7 52 (0*

;ns!ains 51 4 (1*

2 52 12 (2*

51 2 52 1K (*

Misa#'an )ada "aasan (* di!u"ah m$n>adi >$nis 8 (samad$ngan*

51 2 52 8 1K (*



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Pada "aasan P$!samaan (* ida' $!da)a a!ia"#$ "asis. Hiam"ah'an artificia% variab%e A & 3(; hasi# !$isi

dihasi#'an

Fungsi u>uan : 9 51 9 7 52 8 0 (0*

;ns!ains 51 +1 8 4 (1*

2 52 +2 8 12 (2*

51 2 52 8 1K (*

: 9 51 9 7 52 8 0(0*

51 +1 8 4 (1*

2 52 +2 8 12 (2*

51 2 52 A 8 1K (*



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Lang'ah s$#an>un%a m$ma'sa ni#ai a!iicia# a!ia"#$ A m$n>adi n#.

Ha)a di#a'u'an d$ngan m$d$ Te!ni! M m$d$ )$na#%.Pada )$nd$'aan ini ungsi u>uan di!u"ah du#u m$n>adi

Pada hasi# !$isi ada )$!samaan d$ngan 7 a!ia"#$.

Ada dua a!ia"#$ nn "asis 51 dan 52 %ang )ada )$n%$#$saian #a%a'

a&a# ha!gan%a 8 0.

Ha!i )$!samaan (1*, (2* dan (* dida)a'an ni#ai a!ia"#$ "asis

S' 5 , ; S* 5 '* 8an A 5 '<

: 8 51 7 52 MA

H$ngan M ada#ah "i#angan )sii %ang sanga "$sa! "$!hingga.

P$!samaan (0* da!i ungsi u>uan a'an m$n>adi

: 9 51 9 7 52 MA 8 0



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Pada )$!samaan (0* %ang di!$isi $!da)a a!ia"#$ "asis d$ngan

'$isi$n M. a!ia"#$ "asis ha!us dihi#ang'an da!i )$!samaan (0*. Ca!is : %ang dihasi#'an (!$isi* di'u!angi d$ngan M 'a#i s$ia) "a!is"aasan %ang s$suai.

9 97 0 0 M 0Ca!is : )$!s (0* !$isi

2 0 0 1 1K= M

Ca!is : )$!s (0* "a!u 9M9 92M97 0 0 0 9 1KM

Hisusun a"$# sim)#$ a&a#.

i$!asi unu' m$nda)a'an ni#ai)ima#. Hihasi#'an 51 8 2, 52 8 6 dan : 8 6



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I$!ain

Casic

a!ia

"#$

E/.

;$ici$n R+

: 51 52 +1 +2 A

(+#*

0

: (0* 1 (9M9* (92M97* 0 0 0 91KM

+1 (1* 0 1 0 1 0 0 4

+2 (2* 0 0 2 0 1 0 12

A (* 0 2 0 0 1 1K


Da"$# sim)#$ a&a#

Pic#umn

Pi !a&51 $n$!ing a!ia"#$

+1 #$aing a!ia"#$



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Pe6i%i"an enterin variab%e $isi$n a!ia"#$ nn "asis )ada

)$!samaan u>uan m$m)un%ai "$nu' ungsi #in$a! (aM "*a a'! )$ngganda

" a'! )$nam"ah

a!$na M sanga "$sa!, ma'a " s$#a#u '$ci# di"anding'an $!hada) aM.

Pada umumn%a )$mi#ihan $n$!ing a!ia"#$ didasa!'an )ada ni#ai

fa!tor #enan8a a)

;nh Pada a"$# a&a# '$isi$n 51 ada#ah (9M9*, unu' 52 ada#ah (9

2M97*. Fa'! )$ngganda + > *, s$hingga di)i#ih X' s$"agai $n$!ing

a!ia"#$.

Ci#a )ada '$isi$n $!s$"u ni#ai a'! )$ngganda sama, ma'a

)$mi#ihan $n$!ing a!ia"#$ didasa!'an )ada fa!tor #ena6ba" b



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I$!ai

n

Casica!ia

"#$

E/.

;$ici$n R+

(+#*

: 51 52 +1 +2 A

1

: (0* 1 0 (92M97* (M* 0 0 96M12

51 (1* 0 1 0 1 0 0 4

+2 (2* 0 0 2 0 1 0 12

A (* 0 0 2 9 0 1 6


2

: (0* 1 0 0 92 0 (M72* 2

51 (1* 0 1 0 1 0 0 4

+2 (2* 0 0 0 1 91 6

52 (* 0 0 1 92 0 12

: (0* 1 0 0 0 2 (M1* 6

51 (1* 0 1 0 0 91 1 2

+1 (2* 0 0 0 1 1 91 252 (* 0 0 1 0 12 0 6



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*) Perti8a!sa6aan jenis

Hi!u"ah m$n>adi ≤ d$ngan ca!a m$nga#i'an '$dua !uas

)$!ida'samaan d$ngan (91*.;nh 0,6 51 0,4 52 ≥ 6 m$n>adi

90,6 51 9 0,4 52 ≤ 96

Ruas 'i!i diam"ah s#ac' a!ia"#$

90,6 51 9 0,4 52 + 8 96

Ni#ai s#ac' a!ia"#$ + 8 96 n$gai, ida' m$m$nuhi s%a!a.a!us di'a#i'an (91*, dihasi#'an

0,6 51 0,4 52 9 + 8 6

Ruas 'anan )$!samaan $!a'hi! sudah )sii, namun '$isi$ns#ac' a!ia"#$ $a) n$gai s$#$sai'an s$)$!i 'asus i)$ 8(sama d$ngan*. Hiam"ah a!iicia# a!ia"#$ A.




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0,6 51 0,4 52 + A 8 6

A!iicia# a!ia"#$ A di)a'ai s$"agai a!ia"#$ "asis a&a# (A86*.

H$ngan d$mi'ian + m$mu#ai s$"agai a!ia"#$ nn "asis.

H$ngan m$ngin!du'si A!iicia# a!ia"#$ A ; berarti m$d$te!ni! M >uga di)$!#a'u'an disini.

+) Me6ini6u6!an 8iruba" 6enja8i 6e6a!si6u6!an $an e/uiva%ent

∑=

=n

j j j X C Z

1

MinimiB$

∑=

−=− n

j j j X C Z

1

)()(

E/uia#$n d$ngan

MaimiB$

M$n%$#$sai'an)ima# %ang

sama




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MinimiB$ : 8 0,4 51 0,7 52 (0*

+u">$c 0, 51 0,1 52 ≤ 2, (1*

0,7 51 0,7 52 8 6 (2*

0,6 51 0,4 52 ≥ 6 (*

51 ≥ 0 , 52 ≥ 0

Conto" 2

MinimiB$ : 8 0,4 51 0,7 5

MaimiB$ (9:* 8 90,4 51 9 0,7 52 Pen$e%esaian 2

Masu''an a!iicia# a!ia"#$ A1 dan A2 )ada )$!s (2* dan (*, dan

$!a)'an m$d$ D$'ni' M, ma'a

MinimiB$ : 8 0,4 51 0,7 52 MA1 MA2

MaimiB$ (9:* 8 90,4 51 9 0,7 52 9 MA1 9 MA2




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9: 0,4 51 0,7 52 MA1 MA2 8 0 (0*

0, 51 0,1 52 +1 8 2, (1*

0,7 51 0,7 52 A1 8 6(2*

0,6 51 0,4 52 9 +2 A2 8 6 (*

+1, A1 dan A2 ada#ah a!ia"#$ "asis unu' )$n%$#$saian dasa!a&a#.

Siste6 #ersa6aan Ma7i6i?e &=0(

0,4 0,7 0 M 0 M 0Ca!is :, )$!s (0* !$isi

0,7 0,7 0 1 0 0 6

Ca!is :, )$!s (0* "a!u 91,1M0,4 90,M0,7 0 0 M 0 912M

9M

0,6 0,4 0 0 91 1 69M




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I$!ain Casica!ia"#$

E/uain ;$ici$n R+(+#*: 51 52 +1 A1 +2 A2

0

: (0* 91 91,1M0,4 90,M0,7 0 0 M 0 912M

+1 (1* 0 0, 0,1 1 0 0 0 2,

A1 (2* 0 0,7 0,7 0 1 0 0 6

A2 (* 0 0,6 0,4 0 0 91 1 6

Tabe% si6#%e7 a@a%

Pic#umn

PiRa& En$!ing a!ia"#$ 51

L$aing a!ia"#$ +1




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I$

!ain

Casic

a!ia"#$

E/ua

in

;$ici$n R+

(+#*: 51 52 +1 A1 +2 A2

1

: (0* 91 0 9160M110 11M4 0 M 0 92,1M9,6

51 (1* 0 1 1 1 0 0 0

A1 (2* 0 0 1 0 1 0 0 1,7

A2 (* 0 0 0,2 0 0 91 1 0,6


T i D %it 8 %iTeo i D a%itas 8an ana%isa



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Jan 4, 2016 LINEAR PROGRAMMING K

Teori Dua%itas 8an ana%isaTeori Dua%itas 8an ana%isasensitivitassensitivitas

Conto" 2 Masa%a" Diet)

Da"$# "$!i'u m$m"$!i'an gam"a!an >um#ah min$!a# dan iamin %ang ha!usdi'nsumsi #$h )as$n. Min$!a# dan iamin "$!asa# da!i dua >$nis ma'anandaging dan sa%u!an.

andungan Ma'anan $"uuhan minimum)$! ha!iHaging sa%u!an

Min$!a# 2 4 40

iamin 2 70

a!gauni 2,7

P$!sa#an M$n$nu'an >um#ah )$m"$#ian daging dan sa%u!an

s$d$mi'ian s$hingga '$"uuhan min a'an min$!a# dan iamin$!)$nuhi.

T i D %it 8T i D %it 8



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F!mu#asi md$# LP Misa# 51 >um#ah daging dan 52

>um#ah sa%u!an

Fungsi u>uan MinimiB$ :8512,752 (0* Caasan

251 452 ≥ 40 (1*

51 252 ≥ 70 (2*

51 ≥ 0 , 52 ≥ 0

+$'a!ang )i'i!'an masa#ah %ang "$!"$da %ang masih "$!hu"ungand$ngan masa#ah %ang as#i (dis$"u primal *.

+$"uah H$a#$! m$n>ua# Min$!a# dan iamin

R$s!an s$$m)a m$m"$#i min$!a# dan iamin da!i d$a#$! dan

m$m"ua daging dan sa%u!an i!uan %ang m$ngandung min$!a# daniamin s$)$!i %ang $!u#is )ada a"$#

Teori Dua%itas 8an )Teori Dua%itas 8an )

T i D %it 8T i D %it 8



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Persoa%an bai 8ea%er M$n$a)'an ha!ga >ua# min$!a# dan

iamin )$! unin%a %ang maimum s$d$mi'ian s$hinggam$nghasi#'an ha!ga daging dan sa%u!an i!uan ida' m$#$"ihi ha!ga)asa! %ang ada.

H$a#$! m$muus'an ha!ga )$! uni Min$!a# 1 dan iamin 2

$"uuhan min$!a# 40 ha!ga a# 40 1

$"uuhan iamin 70 ha!ga a# 70 2

a!ga )$! uni daging ( m$ngandung 2 min$!a# dan iamin*

ada#ah 2 1 2 ≤

a!ga )$! uni sa%u!an ( m$ngandung 4 min$!a# dan 2 iamin*

ada#ah 4 1 2 2 ≤ 2,7


T i D %it 8Teori Dua%itas 8an



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Fungsi u>uan

MaimiB$ 840 1 70 2 (0*

Caasan

2 1 2 ≤ (1*

4 1 2 2 ≤ 2,7 (2*

1 ≥ 0 , 2 ≥ 0

P$!umusan masa#ah da#am "$nu' md$# LP

His$"u "$nu' Hua#.

1 dan 2 dinama'an a!ia"#$ dua#


Teori Dua%itas ))Teori Dua%itas ))



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Dua%itas B

Ha#am '$n%aaan $!n%aa dis$ia) "$nu' LP $!da)a 2 "$nu'

1. C$nu' I aau "$nu' as#i dan dinama'an PRIMAL

2. C$nu' II %ang "$!hu"ungan dan dinam'an HQAL d$mi'ians$hingga suau s#usi $!hada) LP %ang as#i >uga m$m"$!i'ans#usi )ada "$nu' dua# n%a.

Asumsi da#am $!i dua#ias ada#ah "ah&a masa#ah )!ima# da#am"$nu' sanda!d.

∑=

=n

j j j X C Z

1

MaimiB$

;ns!ains mib X a i

n

j jij .....,3,2,1

1

=≤∑=

0≥ j X




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Fungsi u>uan

MaimiB$ 840 1 70 2 (0*

Caasan

21 2 ≤ (1*

4 1 2 2 ≤ 2,7 (2*1 ≥ 0 , 2 ≥ 0

P$!"andingan masa#ah P!ima# dan Hua#

Fungsi u>uan

MinimiB$ :8512,752 (0*

Caasan

251 452 ≥ 40 (1*

51 252 ≥ 70 (2*

51 ≥ 0 , 52 ≥ 0

P R

I M

A L

D A L

Teori Dua%itas 8anTeori Dua%itas 8an



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1. $isi$n ungsi u>uan masa#ah )!ima# m$n>adi 'nsana sisi'anan masa#ah Hua#

2. nsana sisi 'anan )!ima# m$n>adi '$isi$n ungsi u>uanmasa#ah Hua#

. Danda )$!ida'samaan di"a#i'

4. Du>uan diu"ah da!i minimB$ (maimiB$* da#am )!ima#m$n>adi maimiB$ (minimB$* da#am dua#

7. +$ia) '#m )ada )!ima# "$!hu"ungan d$ngan suau "a!is('$nda#a* da#am dua#. "an%a'n%a '$nda#a da#am dua#samad$ngan "an%a'n%a a!ia"#$ )!ima#

6. +$ia) "a!is '$nda#a )ada )!ima# "$!hu"ungan d$ngan suau'#m da#am dua#. ada sau a!ia"#$ dua# unu' s$ia)

'$nda#a )!ima#





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Da"$# P!ima# dua# unu' LP

Ma7i6i?e : 8 ;1 51 ;2 52 ; 5 ..... ;n 5n

Constraints 1*. a

11

51

a12

52

a1

5

......... a1n

5n

< "1

2*. a21 51 a22 52 a2 5 ......... a1n 5n < "2

. . . . ......... ... . m*. am1 51 am2 52 am 5 ......... amn 5n < "m

5 > = 0 , > 8 1,2,-..n

Mini6i?e 8 "1 1 "2 2 " ..... "m m

Constraints 1*. a11 1 a21 2 a1 ........ am1 m ≥ ;1

2*. a12 1 a22 2 a2 ........ am2 m ≥ ;2

. . . . ......... .... n*. a1n 1 a2n 2 an ........ amn m ≥ ;n

> = 0 , i 8 1,2,-..m

P r i 6

a %

n . a r ; 6

C o n

s t r

D A L

6

v a r ; n c o n s t r

h d




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PRIMAL DAL

∑=

=n

j

j jC Z

1

χ Ma.

;ns!ains

i

n

j jij b X a ≤∑

=1

miuntuk

o X j

....3,2,1=

≥

∑=

=m

i

iiY bW

1

Min.

j

m

iiij C Y a ≥∑

=1

;ns!ains

nuntuk

oY i

....3,2,1 j =

≥

Min 8 b

;ns!ain

a C 3

Ma : 8 C X

;ns!ain

a X b

X 3

h @ id Teori Dua%itas ))Teori Dua%itas ))



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conto"

Ma. [ ]

=

21 53 X

X Z

;ns!ains

≤

18

12

4

23

20

01

2

1

X

X

≥

0

0

2

1

X

X

Min. [ ]

=18

12

4

321 Y Y Y W

;ns!ains

[ ] [ ]53

23

20

01

321 ≥

Y Y Y

[ ] [ ]000321 ≥Y Y Y

PRIMAL

HQAL

suhanan@ugm ac idTeori Dua%itas ))Teori Dua%itas ))



[email protected]



∑=

=n

j j jC Z

1

χ Ma.

;ns!ains

i

n

j jij b X a ≤∑

=1

miuntuk

o X j

....3,2,1=

≥

∑=

=m

iiiY bW

1

Min.

j

m

iiij C Y a ≥∑

=1

;ns!ains

nuntuk

oY i

....3,2,1 j=

≥

Masa%a" Pri6a% 8ua% si6etris +$mua a!ia"#$ di"aasi nnn$gai dan s$mua "aasan "$!u)a )$!ida'samaan

suhanan@ugm ac id Teori Dua%itas ))Teori Dua%itas ))



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conto"

Ma. [ ]

=

21 53 X

X Z

;ns!ains

≤

18

12

4

23

20

01

2

1

X

X

≥

0

0

2

1

X

X

Min. [ ]

=18

12

4

321 Y Y Y W

;ns!ains

[ ] [ ]53

23

20

01

321 ≥

Y Y Y

[ ] [ ]000321 ≥Y Y Y

PRIMAL

HQAL

[email protected] Dua%itas ))Teori Dua%itas ))



[email protected]


Teori Dua%itas ))eo ua tas

MA+ALA PRIMAL

RQA+ANAN

$isi$n da!i

51 52 . . 5n

1 a11 a12 a1 . a1n "1

2 a21 a22 a2 . a2n "2

a1 a2 a . an "

. . . . . . .

. . . . . . .

m am1 am2 am . amn "m

.

;1 ;2 ; . ;n

$isi$n ungsi u>uan (ma.*

DACEL PRIMAL HQAL

J 0 $ @ @ i s i $ n

@ u n g s i

u > u a n (

M i n . *

M A + A L

A E

H Q A L

J 0 $ @ i s i $ n

d a ! i

R Q A +

J

A N A N

[email protected] Teori Dua%itas ))Teori Dua%itas ))



@ g


eo ua tas

51 52

1 2 4 40

2 2 70

2,7

;nh DACEL PRIMAL HQAL

Fungsi u>uan MinimiB$ :8512,752 (0*Caasan

251 452 ≥ 40 (1*

51 252 ≥ 70 (2*

51 ≥ 0 , 52 ≥ 0





Soa% $!>a'an dan 'um)u#'an

+$"uah )$!usahaan m$m)!du'si >a'$ dan as 'u#i. +$"uah >a'$ m$m$!#u'anK m$$! )$!s$gi 'u#i, s$m$na!a s$"uah as han%a m$ngguna'an m$$!)$!s$gi. Qnu' m$n%$#$sai'an s$"uah >a'$ dan as di)$!#u'an &a'u masing9masing 12 dan 4 >am. a!ga )$m"$#ian 'u#i ada#ah K )$! m$$! )$!s$gi dan"ia%a $naga '$!>a di)$!'i!a'an 17 )$! >am. P$!s$diaan 'u#i dan >am $naga'$!>a mingguan di"aasi 1200 m$$! )$!s$gi dan 1K00 >am. P$!usahaan m$n>ua#

>a'$ dan as masing9masing d$ngan ha!ga 70 dan 120. Du>uan )$!usahaan

m$n$nu'an )!du'si mingguan >a'$ dan as unu' m$ma'simum'an)$nda)aan "$!sih.

1. Cua !mu#asi md$# da!i 'asus diaas

2. C$!a)a )$nda)aan "$!sih mingguan

. Cua !mu#asi md$# da!i "$nu' dua# 'asus $!s$"u

4. dan "ua#ah a"$# masa#ah P!ima#9Hua# n%a.





u"ungan P!ima# Hua# unu' s$mua masa#ah LP,"$nu' P!ima# masa#ah MaimiB$

PRIMAL HQAL

Ma7i6i?e Mini6i?e

ih cns!ain %)$ Hua# a!. i 0


ih cns!ain 8 %)$ i un!$s!ic$d

5 > 0 >h cns!ain %)$

5 > 0 >h cns!ain %)$

5 > un!$s!ic$d >h cns!ain 8 %)$





u"ungan P!ima# Hua# unu' s$mua masa#ah LP,"$nu' P!ima# masa#ah MinimiB$

PRIMAL HQAL

Mini6i?e Ma7i6i?e



ih cns!ain 8 %)$ i un!$s!ic$d

5 > 0 >h cns!ain %)$

5 > 0 >h cns!ain %)$

5 > un!$s!ic$d >h cns!ain 8 %)$





Pri6a% Prob%e62

&a+ B 8 + 1 10 + 2 9 +

su">$c

7 + 1 4 + 2 24

2 + 1 7 + 2 + 8 1

+ 1 9 2 + 2 + 7

+ 2 2 + 10

+ 1 0, + 2 0 , + un!$s!ic$d

Dua% Prob%e6 2

Min 8 24 1 1 2 7 - 10 4

7 1 2 2 -

4 1 7 2 2 - 4 10

su">$c

2 - 2 4 8 9 1

1 0, 2 un!$s!ic$d, 0 ,4

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P i P i # S H # P "#




P!)$!i$s P!ima# S Hua# P!"#$ms

1. Hua# da!i dua# ada#ah )!ima#

2. Da"$# sim)#$ )ima# %ang "$!'aian d$ngan sau masa#ah()!ima# aau dua#* s$ca!a #angsung m$m"$!i'an in!masi#$ng'a) $nang )$m$cahan )ima# unu' masa#ah #ainn%a.

. +$ia) )asangan )$m$cahan )!ima# dan dua# %ang #a%a'

Ni#ai u>uan da#am

masa#ah ma, T

Ni#ai u>uan da#am

masa#ah min,

4. Ha#am )$m$cahan )ima# unu' '$dua masa#ah

Ni#ai u>uan da#am

masa#ah ma, :

8Ni#ai u>uan da#am

masa#ah min,

[email protected]!)$!i$s P!ima# S Hua# --..




;nh

Fungsi u>uan MinimiB$ :8 7512 52 (0*

Caasan

51 9 52 ≥ (1*

251 52 ≥ 7 (2*

51 ≥ 0 , 52 ≥ 0PRIMAL

P$m$cahan PRIMAL #a%a' 51 8 dan 52 8 0

Ni#ai u>uan PRIMAL MinimiB$ : 87 2 0 8 17





Fungsi u>uan MaimiB$ 8172 (0*Caasan

1 22 7 (1*

91 2 2 (2*

1 ≥ 0 , 2 ≥ 0

HQAL

P$m$cahan HQAL #a%a' 1 8 dan 2 8 1

Ni#ai u>uan HQAL maimiB$ 8 7 1 8 14

14 (NILAI MA5* 17 (NILAI MIN*





P$m$cahan Hua# )ima#

Ha!i a"$# sim)#$ a'hi! #ri6a% o#ti6a% da)a dihasi#'ans#usi 8ua% o#ti6a%.

Ni%ai !oefisien 8ari s%ac! atau artificia% variab%e )ada "a!isungsi u>uan da!i a"$# sim)#$ a'hi! )!ima# )ima#6eru#a!an ni%ai 8ua% o#ti6a% %ang "$!'$s$suaian.

;nh masa#ah P!ima#MaimiB$ : 8 25152 (0*

Caasan 51752 10 (1*

5152 6 (2*

251252 K (*

51 ≥ 0 , 52 ≥ 0





Masa#ah Hua#

MinimiB$ 8 10 1 6 2 K (0*

Caasan 1 2 2 2 (1*

71 2 2 1 (2*

1 ≥ 0 , 2 ≥ 0 , ≥ 0

Iteration

-asic.ariab

%e

E/t)

;$ici$n R+

(s#*

: 51 52 +1 +2 +

0

0 "$ -' -

S' "$ /

S* "'$ * 2

S+ "*$ ' ' 3

Da"$# sim)#$ a&a#)!ima#

[email protected]

Da"$# +im)#$ a'hi! Masa#ah P!ima# )ima#




Iterat

ion

-asic.aria

b%e

E/t)

;$ici$n R+

(s#*

: 51 52 +1 +2 +

-

: "$ 3

+ "$ 4 -5' 2

+2 "'$ ' -5' '

51 "*$ 5' +

Da"$# +im)#$ a'hi! Masa#ah P!ima# )ima#

Ni#ai O)ima# 51 8 4 dan 52 8 0.d$ngan ungsi u>uan Ma. : 8 K

Ni#ai )ima# ada#ah '$isi$na!. S

1 8 2 8 0, 8 1 d$ngan 8

K





In$!)!$asi $'nmi da#am masa#ah Hua#

ARGA HQAL (HQAL PRI;E*

∑=

=n

j j jC Z

1

χ Ma.

;ns!ains

i

n

1 j jij bXa =∑

=

m321i untuk

oX j

....,,=

≥

∑=

=m

iiiY bW

1

Min.

j

m

iiij C Y a ≥∑

=1

;ns!ains

n321 juntuk

edunrestrict Yi

....,,=

PRIMAL HQAL





. j m$&a'i#i #a"a ma!gina# da!i '$giaan > %ang ing'a

'$giaann%a > uni.

∑=

n

1 j j jX

∑=

m

1i jijXa

m$&a'i#i #a"a da!i s$mua '$giaan

m$&a'i#i )$nggunaan sum"$! da%a

Qnu' )$m$cahan O)ima# : 8

∑∑==

=m

1iii

n

1 j j j Y bX

Me@a!i%i ni%ai uan#ene6ba%ian

Fu6%a" &unit( su6ber i

Ni%ai uan #er unit su6ber

i





i)sumberunit!i)("sumberunit(ian)(#en$emba%

m

1i∑=

=$

a!ia"#$ dua# i m$&a'i#i ni#ai )$! uni sum"$! i.

His$"u ha!ga dua# aau ha!ga "a%angan (shad%! pri#e*

[email protected]+igniicanc$ Hua# P!"#$m




1. Mah$maica##% $!% im)!an

2. ;m)uaina##% On$ md$# (&ih $&$! cns!ains* is $asi$! s#$

. Ecnmic in$!)!$ain h$ dua# a!ia"#$ (shad& )!ic$*

+had& )!ic$ cns!ain i gi$s h$ !a$ chang$ in h$">$ci$ uncin )$! uni chang$ in h$ R+ a#u$ h$cns!ain.

Sensitivity Analysis




Why sensitivity analysis is important?

• Many coefficients are estimated

• Want to know the sensitivity of the optimal solution with respect tothese coefficients.

Sensitivity AnalysisThere are three kinds of such analysis:

1. Obective function ran!in! "coefficient ran!in!#

$. %&' value ran!in!(. )onstraint coefficient ran!in!

Sensitivity analysis is to answer the following question:

How does the optimal solution change as a coefficient

is varied from its given value?

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Range Obective !unction "oefficients




"ase #: $on%basic variable

What happens if the coefficient of a non*basic obectivefunction+ c j + is chan!ed by an amount of ,-

What ran!e of values for , is the current solution remains

optimal-

)onsider the followin! / problem:

max 0 $2 x 13 12 x $

subect to

4 x 13 5 x $ 6 $5

$ x 13 4 x $ 6 1(

x 1 + x $ 7 2

[email protected]$ O">$ci$ FuncinRang$ O">$ci$ Funcin;$ici$ns(;ninu$d*;$ici$ns(;ninu$d*




;$ici$ns(;ninu$d*;$ici$ns(;ninu$d*

Iniia# +im)#$ a"#$au

-asic.ariab%

e

E/t)

;$ici$n R+

(s#*

Rai

: 51 52 +1 +2

0 "$ -' -

S'"$ / + '+ 247

S*"'$ ' / * 12

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Rang$ O">$ci$ FuncinRang$ O">$ci$ Funcin;$ici$ns(;ninu$d*;$ici$ns(;ninu$d*




Fina# +im)#$ a"#$au

-asic.ariab%

e

E/t)

;$ici$n R+

(s#*

: 51 52 +1 +2

0 "$ 2 + 62

X'"$ +5/ 5/ '+5/

S*"'$ 75/ -'5/ 75/

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Rang$ O">$ci$ FuncinRang$ O">$ci$ Funcin;$ici$ns (;ninu$d*;$ici$ns (;ninu$d*




I # 2= 10 U, &ha &u#d ha))$n h$ c$ici$ns

in h$ ">$ci$ uncin !&V

R&(0* a!$ u)da$d !m h$ iniia# a"#$au

-asic.ariab%

e

E/t)

;$ici$n R+

(s#*

Rai

: 51 52 +1 +2

0 "$ -' --8

S'"$ / + '+ 247

S*"'$ ' / * 12

[email protected]

Rang$ O">$ci$ FuncinRang$ O">$ci$ Funcin;$ici$ns (;ninu$d*;$ici$ns (;ninu$d*




I ##&s ha h$ ina# +im)#$ a"#$au is gi$n "%

-asic

.ariab%e

E/t)

;$ici$n R+(s#*

: 51 52 +1 +2

0 "$ 2-8 + 62

X' "$ +5/ 5/ '+5/

S*"'$ 75/ -'5/ 75/

$nc$, h$ $ising "asic &i## !$main )ima# as #ng as6 9 U =0 U<6.In h$! &!ds, h$ s#uin &i## !$main )ima# as #ng as # 2 < 16.

I # 2 16, h$n R&(0* c$ici$n ! + 2 "$cm$s n$gai$ and + 2 &u#d $n$! h$ s#uin and s2 &u#d "$cm$ nn"asic.

[email protected]$ O">$ci$ FuncinRang$ O">$ci$ Funcin;$ici$ns (;ninu$d*;$ici$ns (;ninu$d*




( *( *

;as$ 2 Casic a!ia"#$ha ha))$ns i h$ c$ici$n a "asic ">$ci$ uncin,#j , is chang$d "% an amun UV

I # 1 8 20U, &ha &u#d ha))$n h$ c$ici$ns in h$

">$ci$ uncin !&V

-asic.ariab%

eE/t)

;$ici$n R+

(s#*

Rai

: 51 52 +1 +2

0 "$ -'- 8 -

S'"$ / + '+ 247

S*"'$ ' / * 12






;$ici$ns (;ninu$d*( *Dh$ s$cnd +im)#$ a"#$au n& "$cm$s

-asic.ariab%

eE/t)

;$ici$n R+

(s#*

: 51 52 +1 +2

0 "$ - 8 2 + 62

X'"$ +5/ 5/ '+5/

S*"'$ 75/ -'5/ 75/

N$ ha h$ c$ici$n in c#umn + 1and R&(0* shu#d "$ $#imina$d.

Dhis !$duc$s h$ ##&ing +im)#$ a"#$au





( *Dh$ ina# +im)#$ a"#$au n& "$cm$s

-asic.ariab%

e

E/t)

;$ici$n R+

(s#*

: 51 52 +1 +2

0 "$ 29"+5/$ 8 +9 85/ 629"'+5/$ 8

X'"$ +5/ 5/ '+5/

S*"'$ 75/ -'5/ 75/

$nc$, in !d$! ! h$ cu!!$n s#uin !$main )ima#, &$ n$$d

6(47*U = 0

4 U7 = 0

I ##&s haU = 9.7U = 920

Dhis gi$s U = 9.7 ! # 1 = 12.7.

[email protected]$ R+ a#u$Rang$ R+ a#u$




ha ha))$ns i bi , is chang$d "% an amun UV

I b1 8 24U,


-asic

.ariab%e E/t)

;$ici$n R+

(s#*

Rai

: 51 52 +1 +2

0 "$ -' -

S'"$ / + '+ 98 (24 8 *7

S* "'$ ' / * 12


h$n h$ ina# +im)#$ a"#$au n& "$cm$s




-asic

.ariab%e

E/t)

;$ici$n R+

(s#*: 51 52 +1 +2

0 "$ 2 + 629+8

X'"$ +5/ 5/ '+5/ 9 85/

S* "'$ 75/ -'5/ 75/ :"'5/$ 8 Dha is, h$ R+ c#umn is !$)#ac$d "% h$ sum h$ R+ c#umn andU im$s h$ s1 c#umn.

N&, in !d$! ! h$ cu!!$n s#uin !$mains )ima#, i mus "$$asi"#$. Dha is

Dhis gi$s 924 < U < K.7 ! 0 < b1 < 2.7.247 U7 = 017 (27*U = 0


+imi#a!#% i " 8 1 8 h$n h$ ina# +im)#$ a"#$au n&




-asic.ariab%

e

E/t)

;$ici$n R+

(s#*: 51 52 +1 +2

0 "$ 2 + 62

X'"$ +5/ 5/ '+5/

S*"'$ 75/ -'5/ 75/ 9 8

+imi#a!#%, i "2 8 1 8, h$n h$ ina# +im)#$ a"#$au n&

"$cm$s

I ##&s ha

='41 or b* J):

[email protected]@or! #rob%e6sHo6e@or! #rob%e6s



&ontana 'ood roducts manuacturers t*o+i$ -ua%it #roducts, tab%es and cairs. /ts #roit is "15 #er cair and "21 #er tab%e.

'eek% #roduction is constrained b aai%ab%e %abor and *ood. ac

cair re-uires 4 %abor ours and 8 board eet o *ood *i%e eac tab%e

re-uires 3 %abor ours and 12 board eet o *ood. ai%ab%e *ood is

2400 board eet and aai%ab%e %abor is 20 ours. &ana$ement a%so

re-uires at %east 40 tab%es and at %east 4 cairs be #roduced or eer

tab%e #roduced. o maimize #roits, o* man cairs and tab%es

sou%d be #roduced6

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Linear Programming Simplex Methode

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