Penyelesaian Program Bilangan Bulat Campuran Dua Kriteria dengan Menggunakan Metode Branch and Cut

(1)

DAFTAR LAMPIRAN

Lampiran 1 Pembahasan Masalah Program Linier Dua Kriteria Menggunakan Parametric Simplex Algorithm

Tabel Simpleks Program Linier Dua Kriteria Iterasi 1

s = t, uuuu

cE

-5

2

0

cE

'

1

-4

0

Variabel Basis

_'

,

_'

,

_J

,

_K

Solusi

,

-1

2

1

0 ₀

₀

3

,

_'

3

1

₀

1

₀

8

,

J

5

0

1

0

6

,

K

0

3

₀

0

1

4

Keterangan:

g = &

,1

,

,2

,

,3

,

,4

(

` = & ,

_'

(

[ = &0, 0, 3, 8, , 6, 4(

V = j &0,0000, 0,1667, 0,6667( = 0,6667

, = j &0,0000, 0,1667, 0,6667( = 0,6667 =

'

(2)

s = u, wwwx

cE

-5

0

-0,6667

-2,6667

cE

'

1

0

1,3334

5,3334

Variabel Basis

_'

,

_'

,

_J

,

_K

Solusi

,

-1

₀

1

0 ₀

-0,6667

-0,3334

,

'

3

₀

1

₀

0,3334

6,6667

,

J

5

0

1

0,0000

6,0000

'

0

1

0

0,3334

1,3334

Keterangan:

g = &

,1

,

,2

,

,3, 2

(

` = & (

[

'

= &0, 1,3334, −0,3334, 6,6667, 6, 0(

V

'

= j &0,1667( = 0,16667

, = j &0,1667( = 0,16667 =

(3)

s = u, twwx

cE

0

1

-0,6667

3,3334

cE

'

0

-0,2000

1,3334

4,1334

Variabel Basis

_'

,

_'

,

_J

,

_K

Solusi

,

0

1

0

0,2000

-0,6667

1,5334

,

'

0

1

-0,6000

0,3334

3,0667

1

₀

0,2000

0

1,2000

'

0

1

0

0,3334

1,3334

Keterangan:

g = &

,1

,

,2

,

1, 2

(

` = &∅(

[

J

= &1,2000, 1,3334, 1,5334, 3,0667, 0, 0(

karena

` = &∅(

maka permasalahan telah optimum, diperoleh

= 1,2000

,

_'

= 1,3334

,

, =

1,5334

,

,'=

3,0667

,

,J=

0

,

, =

0

dengan

$ =

3,3334

dan

$' =

4,1334

, hasil optimum diatas harus berupa

bilangan bulat. Untuk memperoleh hasil bilangan bulat maka digunakan metode branch and cut, dengan ini terlebih dahulu kita terapkan percabangan (branch) yaitu pada bagian A dan bagian B, terlebih dahulu

(4)

Lampiran 2 Pembahasan Masalah Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)

Tabel Simpleks Program Linier Dua Kriteria pada Bagian A Iterasi 1

s = t, uuuu

cE

-5

2

0

cE

'

1

-4

0

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

Solusi

,

-1

2

1

0 ₀

₀

3

,

'

3

1

₀

1

₀

8

,

J

5

0

1

0

6

,

K

0

3

0

1

0

4

,

_L

0

1

0

1

Keterangan:

g = &

,1

,

,2

,

,3

,

,4, ,5

(

` = & ,

'

(

[ = &0, 0, 3, 8, 6, 4, 1(

V = j &0,1667, 0,6667( = 0,6667

, = j &0,1667, 0,6667( = 0,6667 =

'

(5)

s = u, wwwx

cE

-5

0

-2

cE

'

1

0

4

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

Solusi

,

-1

₀

1

0 ₀

₀

-2

1

,

'

3

₀

1

₀

-1

7

,

J

5

0

1

0

6

,

K

0

1

-3

1

'

0

1

0

1

Keterangan:

g = &

,1

,

,2

,

,3

,

,4, '

(

` = & (

[

'

= &0, 1, 1, 7, 6, 1,0(

V

'

= j &0,1667( = 0,1667

, = j &0,1667( = 0,1667 =

(6)

s = u, twwx

cE

0

1

-2

4

cE

'

0

-0,8000

4

0,8000

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

Solusi

,

0

1

0

0,2000

-2

2,2000

,

'

0

1

0

-0,6000

-1

3,4000

1

₀

1

0,2000

0

1,2000

,

K

0

-3

1

'

0

1

0

1

Keterangan:

g = &

,1

,

,2

,

1

,

,4, '

(

` = &∅(

[

J

= &1,2000, 1, 2,2000, 3,4000, 0, 1, 0(

karena

` = &∅(

= 1,2000

,

_'

= 1

,

, =

2,2000

,

,'=

3,4000

,

,J=

0

,

,K=

1

,

,L=

0

dengan

(7)

Lampiran 3 Pembahasan Masalah Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)

Tabel Simpleks Program Linier Dua Kriteria pada Bagian B Iterasi 1

s = t, uuuu

cE

-5

2

0

cE

'

1

-4

0

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_M

Solusi

,

-1

2

1

0 ₀

₀

3

,

'

3

1

₀

1

₀

8

,

J

5

0

1

0

6

,

K

0

3

0

1

0

4

,

_L

0

1

0

-1

1

2

Keterangan:

g = &

,1

,

,2

,

,3

,

,4, ,5

(

` = & ,

'

(

[ = &0, 0, 3, 8, 6, 4, 2(

V = j &0,1667, 0,6667( = 0,6667

, = j &0,1667, 0,6667( = 0,6667 =

'

(8)

Tabel Simpleks Program Linier Dua Kriteria pada Bagian B Iterasi 2

s = u, wwwx

cE

-5

0

-0,6667

0

-2,6667

cE

'

1

0

1,3334

0

5,3334

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_M

Solusi

,

-1

₀

1

0 ₀

-0,6667

₀

0,3334

,

'

3

₀

1

₀

-0,3334

₀

3,3334

,

J

5

0

1

0

6

'

0

1

0

0,3334

0

1,3334

,

L

0

-0,3334

-1

1

0,6667

Keterangan:

g = &

,1

,

,2

,

,3

,

', ,5

(

` = & (

[

'

= &0, 1,3334, 0,3334, 3,3334, 6, 0, , 0,6667(

V

'

= j &0,1667( = 0,1667

, = j &0,1667( = 0,1667 =

(9)

Tabel Simpleks Program Linier Dua Kriteria pada Bagian B Iterasi 3

s = u, twwx

cE

0

1

-0,6667

0

3,3334

cE

'

0

-0,2000

1,3334

0

0,1334

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_M

Solusi

,

0

1

0

0,2000

-0,6667

0

1,5334

,

'

0

1

-0,6000

-0,3334

0

3,0667

1

0 ₀

₀

0,2000

0 ₀

₀

1,2000

'

0

1

0

0,3334

0

1,3334

,

L

0

-0,3334

-1

1

0,6667

Keterangan:

g = &

,1

,

,2

,

1

,

',

,

L

(

` = &∅(

[

J

= &1,2000, 1,3334, 1,5334, 3,0667,0, 0, 0,6667(

karena

` = &∅(

= 1,2000

,

_'

= 1,3334

,

, =

1,5334

,

,'=

3,0667

,

,J=

0

,

,K=

0

,

(10)

Lampiran 4 Pembahasan Masalah Program Linier Dua Kriteria dengan Menerapkan Pemotongan (Cut)

Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Itersi 1

s = u, twwx

cE

0

1

-2

0

4

cE

'

0

-0,8000

4

0

0,8000

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_G

Solusi

,

0

1

0

0,2000

-2

0

2,2000

,

'

0

1

0

-0,6000

-1

0

3,4000

1

₀

1

0,2000

0 ₀

1,2000

,

K

0

-3

0

1

'

0

1

0

1

0

1

,

G

0

-0,2000

0

1

-0,2000

Keterangan:

g = y

,1, ,2,

,

,4, _'

, ,

G

z

` = &,

J

, ,

L

(

[ = &1,2000, 1, 2,2000,3,4000, 0, 1, 0, −0,2000(

V = j &0, 0,6667( = 0,6667

(11)

e = j)f &−1,1000, −3,4000, ∞, −0,3334, 1, ∞( = 1 =

'

s = u, twwx

cE

0

2

0

1

2

0

6

cE

'

0

-4

0

-0,8000

0

-3,2000

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_G

Solusi

,

0

2

1

0

0,2000

0

4,2000

,

'

0

1

0

1

0

-0,6000

0

4,4000

1

0 ₀

₀

1

0,2000

₀

1,2000

,

K

0

3

0

4

,

L

0

1

0

1

0

1

,

G

0

-0,2000

0

1

-1,2000

Keterangan:

g = y

,1, ,2,,

,

,4,

,

L

, ,

G

z

` = &,

J

(

[

'

= &1,2000, 0, 4,2000, 4,4000, 0, 4, 1, −1,2000(

V

'

= j &0( = 0

(12)

e = j)f &∞, ∞, 1,2000, ∞, ∞, ∞ ( = 1,2000 =

s = u

cE

0

2

0

2

0

6

cE

'

0

-4

0

-0,8000

0

3,2000

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_G

Solusi

,

0

2

1

0

0,2000

0

4,2000

,

'

0

1

0

1

0

-0,6000

0

4,4000

,

J

1

0

1

0,2000

0

1,2000

,

K

0

3

0

4

,

L

0

1

0

1

0

1

,

G

0

-0,2000

0

1

-0,2000

Keterangan:

g = &

,1

,

,2

,

1

,

',

,

L

(

` = &∅(

(13)

karena

` = &∅(

maka permasalahan telah optimum. diperoleh

= 0

,

_'

= 0

,

, =

4,2000

,

,'=

4,4000

,

,J= 1,200

0

,

,K=

4,

,

,L=

1

,

G

= −0,2000

dengan

$ =

6

dan

$'=

3,2000

, dikarenakan hasil optimum pada bagian A setelah penambahan kendala gomory belum

diperoleh berupa bilangan bulat

pada bagian A, maka kita lanjut menerapkan pemotongan (cut) pada bagian B.

Lampiran 5 Pembahasan Masalah Program Linier Dua Kriteria dengan Menerapkan Pemotongan (Cut)

Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian B Itersi 1

s = u, twwx

cE

0

1

-0,6667

0

3,3334

cE

'

0

-0,2000

1,3334

0

0,1334

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_M

,

_G

Solusi

,

0

1

0

0,2000

-0,6667

0

1,5334

,

'

0

1

-0,6000

-0,3334

0

3,0667

1

0 ₀

₀

0,2000

0 ₀

₀

1,2000

'

0

11

0

0,3334

0

1,3334

,

L

0

-0,3334

-1

1

0

0,6667

,

G

0

-0,3334

0

1

-0,3334

Keterangan:

g = y

,1, ,2,, , '

,

,5,

,

G

z

(14)

[ = &1,2000, 1,3334, 1,5334, 3,0667, 0, 0, 0,6667, −0,3334(

V = j &0,1667, 0,6667( = 0,6667

, = j &0,1667, 0,6667( = 0,6667 = ,

K

e = j)f &−2,9999, −9,1983, ∞, 3,9994, 1,9997, 1( = 1 = ,

G

Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian B Itersi 2

s = u, twwx

cE

0

1

0

-2

4

cE

'

0

-0,2000

0

4

2,8000

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_M

,

_G

Solusi

,

0

1

0

0,2000

0

-2,0000

2,2000

,

'

0

1

-0,6000

0

-1,0000

3,4000

1

0

0,2000

0

1,2000

'

0

1

0

1

,

L

0

-1

1

-1

1

,

K

0

1

0

-3

1

Keterangan:

g = Q, , ,

',

, ,

'

, ,

K

, ,

L

S

(15)

[

'

= &1, 1,2000, 2,2000, 3,4000, 0, 1, 1, 0(

V

'

= j &0,1667( = 0,1667

, = j &0,1667( = 0,1667 = ,

J

e = j)f &11, −5,6667, 6, ∞, ∞, ∞( = 6 =

Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian B Itersi 3

s = u, twwx

cE

-5

0

-2

2

cE

'

₀

0

4

-4

Variabel Basis

_'

,

_'

,

_J

,

_K

,

_L

,

_M

,

_G

Solusi

,

-1

₀

1

0 ₀

₀

-2

1

,

'

4

₀

1

₀

7

,

_J

5

0 ₀

₀

1

₀

6

'

0

1

0

1

,

L

0 ₀

₀

-1

1

-1

1

,

K

0

1

0

-3

1

Keterangan:

g = Q, , ,

',

, ,

J,

,

'

, ,

K

, ,

L

S

(16)

[

J

= &0, 1, 1, 7, 6, 0,1,1(

karena

` = &∅(

= 0

,

_'

= 1

,

, = 1

,

_'

= 7

,

_J

= 6

,

_K

= 1

,

_L

= 1

,

_G

= 0,

dengan

$ = 2

dan

$

_'

= −4

, dikarenakan hasil optimum pada bagian B setelah penambahan kendala gomory sudah diperoleh bilangan bulat, maka

sudah diperoleh solusi optimumnya pada bagian B setelah penambahan kendala gomory.

Lampiran 6 Pembahasan Contoh Kasus Program Linier Dua Kriteria Menggunakan Parametric Simplex Algorithm

Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 1

s = t, uuuu

cE cE -2287,2380 -1789,1180 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

cE' _7712,7620 _8210,2920 _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀

Variabel

Basis ' J , ,' ,J ,K ,L ,M ,N ,O ,P , l , , ' , J , K , L , M Solusi

, 3 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2500

,' 2 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,K 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 800

,L 0 0,0600 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 10

,M 0,2000 0,2000 0,2000 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18750

,N 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6000

,O 5 5 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 22500

(17)

Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 1

s = t, uuuu

cE cE -2287,2380 -1789,1180 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

cE' _7712,7620 _8210,2920 _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀

Variabel

, l 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 259200

, 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2250

, ' 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 3000

, J 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 24000

, K 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 12600

, L 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 10800

, M 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1800

Keterangan:

g = &, , ,

_'

, ,

_J

, ,

_K

, ,

_L

, ,

_M

, ,

_N

, ,

_O

, ,

_P

, ,

_l

, , , ,

_'

, ,

_J

, ,

_K

, ,

_L

, ,

_M

(

` = & ,

'

,

J

(

[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750,259200, 2250, 3000, 24000, 12600, 10800, 1800(

V = j &0,7713, 0,8211, 0,6848( = 0,8211

(18)

e = j)f &833,3333, 500, ∞, 800, 166,6667, 93750, 6000, 4500, 93750, 259200, 4500, 3000, 48000, 25200, 21600, 3600(

= 166,6667 = ,

_L

s = u, {|tt

cE cE -2287,2380 0 -4097,2380 0 0 0 0 29818,6333 0 0 0 0 0 0 0 0 0 0 0 298186,3333

cE' _7712,7620 ₀ _8902,7620 ₀ ₀ ₀ ₀ _-136838,2000 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _1368382

Variabel

, 3 0 3 1 0 0 0 -50 0 0 0 0 0 0 0 0 0 0 0 2000

,' 2 0 0 0 1 0 0 -33,3333 0 0 0 0 0 0 0 0 0 0 0 666,6667

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,K 1 0 1 0 0 0 1 -16,6667 0 0 0 0 0 0 0 0 0 0 0 633,3333

' 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 166,6667

,M 0,2000 0 0,2000 0 0 0 0 -3,3333 1 0 0 0 0 0 0 0 0 0 0 18716,6667

,N 1 0 1 0 0 0 0 -16,6667 0 1 0 0 0 0 0 0 0 0 0 5833,3333

,O 5,0000 0 5 0 0 0 0 -83,3333 0 0 1 0 0 0 0 0 0 0 0 21666,6667

,P 0,2000 0 0,2000 0 0 0 0 -3,3333 0 0 0 1 0 0 0 0 0 0 0 18716,6667

, l 3 0 3 1 0 0 0 -50 0 0 0 0 0 0 0 0 0 0 0 2000

(19)

s = u, {|tt

cE cE -2287,2380 0 -4097,2380 0 0 0 0 29818,6333 0 0 0 0 0 0 0 0 0 0 0 298186,3333 cE' _7712,7620 ₀ _8902,7620 ₀ ₀ ₀ ₀ _-136838,2000 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _1368382 Variabel

, ' ₁ ₀ ₁ ₀ ₀ ₀ ₀ _-16,6667 ₀ ₀ ₀ ₀ ₀ ₀ ₁ ₀ ₀ ₀ ₀ _2833,3333

, J _0,5000 ₀ _0,5000 ₀ ₀ ₀ ₀ _-8,3333 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁ ₀ ₀ ₀ _23916,6667

, K _0,5000 ₀ _0,5000 ₀ ₀ ₀ ₀ _-8,3333 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁ ₀ ₀ _12516,6667

, L _0,5000 ₀ _0,5000 ₀ ₀ ₀ ₀ _-8,3333 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁ ₀ _10716,6667

, M _0,5000 ₀ _0,5000 ₀ ₀ ₀ ₀ _-8,3333 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁ _1716,6667

Keterangan:

g = &, , ,

'

, ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

(

` = & ,

J

(

[

'

= } 0, 166,6667, 0, 2000, 666,6667, 1000, 633,3333, 0, 18716,6667, 5833,3333, 21666,6667,

18716, 6667, 2000, 666,6667, 2833,3333, 23916,6667, 12516,6667, 10716,6667, 1716,6667 ~

V

'

= j &0,7713, 0,6848( = 0,7713

(20)

e = j)f }666,6667, 333,3333, ∞, 633,3333, ∞, 93583,3333, 5833,3333, 4333,3333, 93583,3333,

259033,3333, 4333,3333, 2833,3333,4783,3333,25033,3333, 21433,3333, 3433,3333~

= 333,3333 = ,

_'

s = u, xxt•

cE cE 0 0 -4097,2380 0 1143,6190 0 0 -8302 0 0 0 0 0 0 0 0 0 0 0 1060599

cE' ₀ ₀ _8902,7620 ₀ _-3856,3810 ₀ ₀ _-8292,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _3939302,6667

Variabel

, 0 0 3 1 -1,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

1 0 0 0 0,5000 0 0 -16,6667 0 0 0 0 0 0 0 0 0 0 0 333,3333

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,K 0 0 1 0 -0,5000 0 1 0 0 0 0 0 0 0 0 0 0 0 0 300

' 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 166,6667

,M 0 0 0,2000 0 -0,1000 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18650

,N 0 0 1 0 -0,5000 0 0 0 0 1 0 0 0 0 0 0 0 0 0 5500

,O 0 0 5 0 -2,5000 0 0 0 0 0 1 0 0 0 0 0 0 0 0 20000

,P 0 0 0,2000 0 -0,1000 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18650

(21)

s = u, xxt•

cE cE 0 0 -4097,2380 0 1143,6190 0 0 -8302 0 0 0 0 0 0 0 0 0 0 0 1060599

cE' ₀ ₀ _8902,7620 ₀ _-3856,3810 ₀ ₀ _-8292,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _3939302,6667

Variabel

, 0 0 3 1 -1,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

, ' 1 0 0 0 0,5000 0 0 -16,6667 0 0 0 0 0 0 0 0 0 0 0 333,3333

, J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

, K 0 0 1 0 -0,5000 0 1 0 0 0 0 0 0 0 0 0 0 0 0 300

, L 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 166,6667

, M 0 0 0,2000 0 -0,1000 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18650

Keterangan:

g = &, , , ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

(

` = &

_J

(

[

J

= }333,3333, 166,6667, 0, 1000, 0, 1000, 300, 0, 166,6667, 18650, 5500,

20000, 18650, 258700, 1000, 333,3333, 1000, 300, 166,6667,18650 ~

V

J

= j &0,6848( = 0,6848

, = j &0,6848( = 0,6848 =

J

(22)

s = u, w{€{

cE cE 0 0 0 0 -905 0 4097,2380 -8302,0000 0 0 0 0 0 0 0 0 0 0 0 2289770,4000

cE' ₀ ₀ ₀ ₀ ₅₉₅ ₀ _-8902,7620 _-8292,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _6610131,2667

Variabel

, 0 0 0 1 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 100

1 0 0 0 0,5000 0 0 -16,6667 0 0 0 0 0 0 0 0 0 0 0 333,3333

,J 0 0 0 0 1,5000 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 100

J 0 0 1 0 -0,5000 0 1 0 0 0 0 0 0 0 0 0 0 0 0 300

' 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 166,6667

,M 0 0 0 0 0 0 -0,2000 0 1 0 0 0 0 0 0 0 0 0 0 18590

,N 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 5200

,O 0 0 0 0 0 0 -5,0000 0 0 0 1 0 0 0 0 0 0 0 0 18500

,P 0 0 0 0 0 0 -0,2000 0 0 0 0 1 0 0 0 0 0 0 0 18590

, l 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 258400

, 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 1 0 0 0 0 0 1850

, ' 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 2200

(23)

Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 4

s = u, w{€{

cE cE 0 0 0 0 -905 0 4097,2380 -8302,0000 0 0 0 0 0 0 0 0 0 0 0 2289770,4000

cE' ₀ ₀ ₀ ₀ ₅₉₅ ₀ _-8902,7620 _-8292,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _6610131,2667

Variabel

, K 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 1 0 0 12200

, L 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 0 1 0 10400

, M 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 0 0 1 1400

Keterangan:

g = &, , , ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

(

` = &∅(

[

K

= &333,3333, 166,6667, 300, 100, 0, 100,0, 0, 18590, 5200, 18500, 18950, 258400,1850, 2200,23600,12200,10400,1400(

karena

` = &∅(

= 333,3333

,

_'

= 166,6667

,

_J

= 300

,

, = 100

,

_'

= 0

,

_J

= 100

,

K

= 0

,

L

= 0

,

M

= 18950

,

N

= 5200

,

O

= 18500

,

P

= 18950

,

l

= 258400

,

, = 1850

,

'

= 2200

,

J

= 23600

,

K

=

12200

,

_L

= 10400

,

_M

= 1400

dengan

$ = 2289770,4000

dan

$

_'

= 6610131,2667

, hasil optimum diatas harus bilangan bulat.

Untuk mendapatkan hasil bilangan bulat maka digunakan metode branch and cut, dengan ini terlebih dahulu menerapkan percabangan

(24)

Lampiran 7 Pembahasan Contoh Kasus Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)

s = t, uuuu

cE cE -2287,2380 -1789,1180 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

cE' _7712,7620 _8210,2920 _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀

Variabel

Basis ' J , ,' ,J ,K ,L ,M ,N ,O ,P , l , , ' , J , K , L , M , N Solusi

, 3 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2500

,' 2 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,K 1 1 1,0000 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 800

,L 0 0,0600 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 10

,M 0,2000 0,2000 0,2000 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 18750

,N 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6000

,O 5 5 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 22500

,P 0,2000 0,2000 0,2000 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18750

, l 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 259200

, 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2250

, ' 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3000

(25)

s = t, uuuu

cE cE -2287,2380 -1789,1180 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

cE' _7712,7620 _8210,2920 _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀

Variabel

, K 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 12600

, L 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 10800

, M 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1800

, N 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 333

Keterangan:

g = &, , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, ,

N

(

` = & ,

'

,

J

(

[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750,259200, 2250, 3000, 24000, 12600, 10800, 1800, 333(

V = j &0,7713, 0,8211, 0,6848( = 0,8211

, = j &0,7713, 0,8211, 0,6848( = 0,8211 =

'

e = j)f &833,3333, 500, ∞, 800, 166,6667, 93750, 6000, 4500, 93750, 259200, 4500, 3000, 48000, 25200, 21600, 3600(

(26)

s = u, {|tt

cE cE -2287,2380 0 -4097,2380 0 0 0 0 29818,6333 0 0 0 0 0 0 0 0 0 0 0 0 298186,3333

cE' _7712,7620 ₀ _8902,7620 ₀ ₀ ₀ ₀ _{-136838,2000 0 0} ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _1368382

Variabel

, 3,0000 0 3 1 0 0 0 -50 0 0 0 0 0 0 0 0 0 0 0 0 2000

,' 2 0 0 0 1 0 0 -33,3333 0 0 0 0 0 0 0 0 0 0 0 0 666,6667

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,K 1 0 1 0 0 0 1 -16,6667 0 0 0 0 0 0 0 0 0 0 0 0 633,3333

' 0 1 0,0000 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 0 166,6667

,M 0,2000 0 0,2000 0 0 0 0 -3,3333 1 0 0 0 0 0 0 0 0 0 0 0 18716,6667

,N 1 0 1 0 0 0 0 -16,6667 0 1 0 0 0 0 0 0 0 0 0 0 5833,3333

,O 5 0 5 0 0 0 0 -83,3333 0 0 1 0 0 0 0 0 0 0 0 0 21666,6667

,P 0,2000 0 0,2000 0 0 0 0 -3,3333 0 0 0 1 0 0 0 0 0 0 0 0 18716,6667

, l 1 0 1 0 0 0 0 -16,6667 0 0 0 0 1 0 0 0 0 0 0 0 259033,3333

, 0,5000 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 1 0 0 0 0 0 0 2166,6667

, ' 1 0 1 0 0 0 0 -16,6667 0 0 0 0 0 0 1 0 0 0 0 0 2833,3333

(27)

s = u, {|tt

cE cE -2287,2380 0 -4097,2380 0 0 0 0 29818,6333 0 0 0 0 0 0 0 0 0 0 0 0 298186,3333

cE' _7712,7620 ₀ _8902,7620 ₀ ₀ ₀ ₀ _{-136838,2000 0 0} ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _1368382

Variabel

, K 0,5000 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 0 0 0 1 0 0 0 12516,6667

, L 0,5000 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 0 0 0 0 1 0 0 10716,6667

, M 0,5000 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 0 0 0 0 0 1 0 1716,6667

, N 1,0000 0 0,0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 333

Keterangan:

g = &, , ,

'

, ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, ,

N

(

` = & ,

J

(

[

'

= }0, 166,6667, 0, 2000, 6666,6667, 1000, 633,3333, 0, 18716,6667, 5833,3333, 21666,6667, 18716,6667,

259033,3333, 2166,6667,2833,3333, 23916,6667, 12516,6667, 10716,6667,1716,6667, 333

~

V

'

= j &0,7713, 0,6848( = 0,7713

, = j &0,7713, 0,6848( = 0,7713 =

e = j)f } 666,6667, 333,3333, ∞, 633,3333, ∞, 93583,3333, 5833,3333, 4333,3333,93583,3333,

259033,3333, 4333,3333, 2833,3333,47833,3333, 25033,333321433,3333,3433,3333, 333 ~

(28)

Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3

s = u, xxt•

cE cE 0 0 -4097,2380 0 0 0 0 29818,6333 0 0 0 0 0 0 0 0 0 0 0 2287,2380 1059836,5873

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ _-136838,2000 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 _3936731,7460

Variabel

, 0 0 3 1 0 0 0 -50 0 0 0 0 0 0 0 0 0 0 0 -3 ₁₀₀₁

,' 0 0 0 0 1 0 0 -33,3333 0 0 0 0 0 0 0 0 0 0 0 -2 _0,6667

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ₁₀₀₀

,K 0 0 1 0 0 0 1 -16,6667 0 0 0 0 0 0 0 0 0 0 0 -1 _300,3333

' 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 0 166,6667

,M 0 0 0,2000 0 0 0 0 -3,3333 1 0 0 0 0 0 0 0 0 0 0 0 _18650,0667

,N 0 0 1 0 0 0 0 -16,6667 0 1 0 0 0 0 0 0 0 0 0 -1 _5500,3333

,O 0 0 5 0 0 0 0 -83,3333 0 0 1 0 0 0 0 0 0 0 0 -5 _20001,6667

,P 0 0 0 0 0 0 0 -3,3333 0 0 0 1 0 0 0 0 0 0 0 0 18650,0667

, l 0 0 1 0 0 0 0 -16,6667 0 0 0 0 1 0 0 0 0 0 0 -1 _258700,3333

, 0 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 1 0 0 0 0 0 -1 _2000,16667

, ' 0 0 1 0 0 0 0 -16,6667 0 0 0 0 0 0 1 0 0 0 0 -1 2500,3333

(29)

s = u, xxt•

cE cE 0 0 -4097,2380 0 0 0 0 29818,6333 0 0 0 0 0 0 0 0 0 0 0 2287,2380 1059836,5873

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ _-136838,2000 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 _3936731,7460

Variabel

, K 0 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 0 0 0 1 0 0 -0,5000 12350,1667

, L 0 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 0 0 0 0 1 0 -0,5000 10550,1667

, M 0 0 0,5000 0 0 0 0 -8,3333 0 0 0 0 0 0 0 0 0 0 1 -0,5000 1550,1667

1 0 0,0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 333

Keterangan:

g = &, , ,

'

, ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, (

` = &

J

(

[

J

= } 333, 166,6667, 0, 1001, 0,6667, 1000, 300,3333, 0, 18650,0667, 5500,3333, 20001,6667, 18650,0667,

18650,0667, 258700,3333,2000,16667, 2500,3333, 23750,1667, 12350,1667, 10550,1667,1550,1667, 0~

V

J

= j &0,6848( = 0,6848

, = j &0,6848( = 0,6848 =

J

e = j)f } 333,6667, ∞, 333,3333, 300,3333, ∞, 93250,3333, 5500,3333, 4000,3333, 93250,3333,

258700,3333, 4000,3333,2500,3333, 47500,3333, 24700,3333, 21100,3333, 3100,3333, ∞~

(30)

s = u, w{€{

cE cE 0 0 0 0 0 0 4097,2380 -38468,6667 0 0 0 0 0 0 0 0 0 0 0 -1810 2290373,7333

cE' ₀ ₀ ₀ ₀ ₀ ₀ _-8902,7620 _11541,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁₁₉₀ _6610527,9333

Variabel

, 0 0 0 1 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 100

,' 0 0 0 0 1 0 0 -33,3333 0 0 0 0 0 0 0 0 0 0 0 -2 0,6667

,J 0 0 0 0 0 1 -3 50 0 0 0 0 0 0 0 0 0 0 0 3 99

J 0 0 1 0 0 0 1 -16,6667 0 0 0 0 0 0 0 0 0 0 0 -1 300,3333

' 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 0 166,6667

,M 0 0 0 0 0 0 -0,2000 0 1 0 0 0 0 0 0 0 0 0 0 0 18590

,N 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 5200

,O 0 0 0 0 0 0 -5 0 0 0 1 0 0 0 0 0 0 0 0 0 18500

,P 0 0 0 0 0 0 -0,2000 0 0 0 0 1 0 0 0 0 0 0 0 0 18590

, l 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 258400

, 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 1 0 0 0 0 0 0 1850

, ' 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 2200

(31)

Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3

s = u, xxt•

cE cE 0 0 0 0 0 0 4097,2380 -38468,6667 0 0 0 0 0 0 0 0 0 0 0 -1810 2290373,7333

cE' ₀ ₀ ₀ ₀ ₀ ₀ _-8902,7620 _11541,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁₁₉₀ _6610527,9333

Variabel

, K 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 1 0 0 0 12200

, L 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 0 1 0 0 10400

, M 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 0 0 1 0 1400

1 0 0 0 0 0 0,0000 0 0 0 0 0 0 0 0 0 0 0 0 1 333

Keterangan:

g = &, , ,

'

, ,

J

,

J

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, (

` = &∅(

[

K

= &333, 166,6667, 300,3333, 100,0,6667,99, 0, 0,18590, 5200, 18500,18590, 258400,1850, 2200,23600 , 12200, 10400, 1400, 0 (

karena

` = &∅(

= 333

,

_'

= 166,6667

,

_J

= 300,333

,

, = 100

,

_'

= 0,6667

,

_J

=

99

,

_K

= 0

,

_L

= 0

,

_M

= 18590

,

_N

= 5200

,

_O

= 18500

,

_P

= 18950

,

_l

= 258400

,

, = 1850

,

_'

= 2200

,

_J

= 23600

,

_K

=

(32)

Lampiran 8 Pembahasan Contoh Kasus Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)

s = t, uuuu

cE cE -2287,2380 -1789,1180 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

cE' _7712,7620 _8210,2920 _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀

Variabel

, 3 3 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2500

,' 2 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

,K 1 1 1,0000 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 800

,L 0 0,0600 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 10

,M 0,2000 0,2000 0,2000 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 18750

,N 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6000

,O 5 5 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 22500

,P 0,2000 0,2000 0,2000 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18750

, l 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 259200

, 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2250

, ' 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3000

(33)

s = t, uuuu

cE cE -2287,2380 -1789,1180 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

cE' _7712,7620 _8210,2920 _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀

Variabel

, K 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 12600

, L 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 10800

, M 0,5000 0,5000 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1800

, N -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -334

Keterangan:

g = &, , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, ,

N

(

` = & ,

'

,

J

(

[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750,259200, 2250, 3000, 24000, 12600, 10800, 1800, 333(

V = j &0,7713, 0,8211, 0,6848( = 0,8211

, = j &0,7713, 0,8211, 0,6848( = 0,8211 =

'

e = j)f &833,3333, 500, ∞, 800, 166,6667, 93750, 6000, 4500, 93750, 259200, 4500, 3000, 48000, 25200, 21600, 3600(

(34)

setelah diselasaikan pada bagian B diperoleh penyelesaian yang sama dengan hasil pada solusi optimum awal yaitu

= 333,3333

,

'

= 166,6667

,

J

= 300

,

, = 100

,

'

= 0

,

J

= 100

,

K

= 0

,

L

= 0

,

M

= 18950

,

N

= 5200

,

O

= 18500

,

P

= 18950

,

l

=

258400

,

, = 1850

,

_'

= 2200

,

_J

= 23600

,

_K

= 12200

,

_L

= 10400

,

_M

= 1400

,

_N

= −0,6667

dengan

$ = 2289770,4000

dan

$

_'

= 6610131,2667

, dikarenakan pada bagian B mengulang penyelesaian yang sama pada solusi optimum

awal, maka tidak diperoleh solusi yang layak yang bernilai bilangan bulat pada branch bagian B, oleh karena itu lanjut menerapkan

pemotongan (cutting) pada bagian A.

Lampiran 9 Pembahasan Contoh Kasus Program Linier Dua Kriteria dengan Menerapkan Pemotongan (Cut)

Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Iterasi 1

s = u, w{€{

cE cE 0 0 0 0 0 0 4097,2380 -38468,6667 0 0 0 0 0 0 0 0 0 0 0 -1810 0 2290373,7333

cE' ₀ ₀ ₀ ₀ ₀ ₀ _-8902,7620 _11541,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁₁₉₀ ₀ _6610527,9333

Variabel

Basis ' J , ,' ,J ,K ,L ,M ,N ,O ,P , l , , ' , J , K , L , M , N

,

G _Solusi

, 0 0 0 1 0 0 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100

,' 0 0 0 0 1 0 0 -33,3333 0 0 0 0 0 0 0 0 0 0 0 -2 0 0,6667

,J 0 0 0 0 0 1 -3 50 0 0 0 0 0 0 0 0 0 0 0 3 0 99

J 0 0 1 0 0 0 1 -16,6667 0 0 0 0 0 0 0 0 0 0 0 -1 0 300,3333

' 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 0 0 166,6667

(35)

Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A

Iterasi 1

s = u, w{€{

cE cE 0 0 0 0 0 0 4097,2380 -38468,6667 0 0 0 0 0 0 0 0 0 0 0 -1810 0 2290373,7333

cE' ₀ ₀ ₀ ₀ ₀ ₀ _-8902,7620 _11541,1667 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₁₁₉₀ ₀ _6610527,9333

Variabel

Basis ' J , ,' ,J ,K ,L ,M ,N ,O ,P , l , , ' , J , K , L , M , N ,G Solusi

,N 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 5200

,O 0 0 0 0 0 0 -5 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18500

,P 0 0 0 0 0 0 -0,2000 0 0 0 0 1 0 0 0 0 0 0 0 0 0 18590

, l 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 258400

, 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1850

, ' 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2200

, J 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 1 0 0 0 0 0 23600

, K 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 1 0 0 0 0 12200

, L 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 0 1 0 0 0 10400

, M 0 0 0 0 0 0 -0,5000 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1400

1 0 0 0 0 0 0,0000 0 0 0 0 0 0 0 0 0 0 0 0 1 0 333

(36)

Keterangan:

g = y, , ,

_'

, ,

_J

,

_J

,

_'

, ,

_M

, ,

_N

, ,

_O

, ,

_P

, ,

_l

, , , ,

_'

, ,

_J

, ,

_K

, ,

_L

, ,

_M

, , ,

_G

z

` = &,

K

, ,

L

, ,

N

(

[ = } 333, 166,6667, 300,3333, 100, 0,6667, 99, 0,0,18590, 5200, 18500,

18590, 258400, 1850, 2200, 23600, 12200, 10400,1400, 0, −0,3333 ~

V = j &0,6648, 0,2308, 0,3967( = 0,6848

, = j &0,6648, 0,2308, 0,3967( = 0,6848 = ,

K

(37)

s = u, w{€{

cE cE 0 0 -4097,2380 0 0 0 0 29818,7699 0 0 0 0 0 0 0 0 0 0 0 2287,2380 0 1059836,7239

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ _-136838,4967 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 ₀ _3936732,0427

Variabel

, 0 0 3 1 0 0 0 -50,0001 0 0 0 0 0 0 0 0 0 0 0 -3 0 1000,9999

,' 0 0 0 0 1 0 0 -33,3333 0 0 0 0 0 0 0 0 0 0 0 -2 0 0,6667

,J 0 0 3 0 0 1 0 -0,0001 0 0 0 0 0 0 0 0 0 0 0 0 0 999,9999

,K 0 0 1 0 0 0 1 -16,6667 0 0 0 0 0 0 0 0 0 0 0 -1 0 300,3333

' 0 1 0 0 0 0 0 16,6667 0 0 0 0 0 0 0 0 0 0 0 0,0000 0 166,6667

,M 0 0 0,2000 0 0 0 0 -3,3333 1 0 0 0 0 0 0 0 0 0 0 -0,2000 0 18650,0667

,N 0 0 1 0 0 0 0 -16,6667 0 1 0 0 0 0 0 0 0 0 0 -1 0 5500,3333

,O 0 0 5 0 0 0 0 -83,3335 0 0 1 0 0 0 0 0 0 0 0 -5 0 20001,6665

,P 0 0 0,2000 0 0 0 0 -3,3333 0 0 0 1 0 0 0 0 0 0 0 -0,2000 0 18650,0667

, l 0 0 1 0 0 0 0 -16,6667 0 0 0 0 1 0 0 0 0 0 0 -1 0 258700,3333

, 0 0 0,5000 0 0 0 0 -8,3334 0 0 0 0 0 1 0 0 0 0 0 -0,5000 0 2000,1667

, ' 0 0 1 0 0 0 0 -16,6667 0 0 0 0 0 0 1 0 0 0 0 -1 0 2500,3333

(38)

Iterasi 2

s = u, w{€{

cE cE 0 0 -4097,2380 0 0 0 0 29818,7699 0 0 0 0 0 0 0 0 0 0 0 2287,2380 0 1059836,7239

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ _-136838,4967 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 ₀ _3936732,0427

Variabel

, K 0 0 0,5000 0 0 0 0 -8,3334 0 0 0 0 0 0 0 0 1 0 0 -0,5000 0 12350,1667

, L 0 0 0,5000 0 0 0 0 -8,3334 0 0 0 0 0 0 0 0 0 1 0 -0,5000 0 10550,1667

, M 0 0 0,5000 0 0 0 0 -8,3334 0 0 0 0 0 0 0 0 0 0 1 -0,5000 0 1550,1667

1 0 0,0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 333

,G 0 0 0,0000 0 0 0 0 -0,6667 0 0 0 0 0 0 0 0 0 0 0 0 1 -0,3333

Keterangan:

g = y, , ,

'

, ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, , ,

G

z

` = &,

L

, ,

N

(

[

'

= }333, 166,6667, 0, 1000,9999, 0,6667, 999,9999, 300,3333, 0, 18650,0667, 5500,3333, 20001,6665, 18650,066720001,6665,

18650,0667, 258700,3333, 2000,1667, 2500,3333, 23750,1667, 12350,166710550,1667, 1550,1667, 333, −0,3333

~

V

'

= j &0,8211, 0,7713( = 0,8211

(39)

e = j)f } −20,0200, −0,0200, −9999, −18,02000, 10, −5595, 0088, −330, 0193, −240,0195, −5569, 0088,

−15521,9890, 240,0195, −150, 0197, −2850, 0143, −1482, 0170, −1266, 0175, −186,0196, ∞, 0,4999~ = 0,4999 = ,

G

s = u, {|tt

cE cE 0 0 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2287,2380 44725,9186 1044929

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 _-205247,4827 _3868323

Variabel

, 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3 -74,9964 1025,9962

,' 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -49,9975 17,3309

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,0001 999,9999

,K 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -24,9988 308,6654

' 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24,9988 158

,M 0 0 0,2000 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -0,2000 -4,9997 18651,7331

,N 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -24,9988 5508,6654

,O 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -5 -124,9940 20043,3270

,P 0 0 0,2000 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -0,2000 -4,9997 18651,7331

, l 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -24,9988 258708,6654

, 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -0,5000 -12,4995 2004,3327

(40)

Iterasi 3

s = u, {|tt

cE cE 0 0 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2287,2380 44725,9186 1044929

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 _-205247,4827 _3868323

Variabel

, J 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -0,5000 -12,4995 23754,3327

, K 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -0,5000 -12,4995 12354,3327

, L 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -0,5000 -12,4995 10554,3327

, M 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -0,5000 -12,4995 1554,3327

1 0 0,0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 333

,L 0 0 0,0000 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1,4999 0,4999

Keterangan:

g = &, , ,

'

, ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, , ,

L

(

` = &,

_N

(

[

J

= }333, 158, 0, 1025,9962, 17,3309,999,9999,308,6654, 18651,7331, 5508,6654, 20043,3270, 18651,7331,

258708,6654, 2004,3327, 2508,6654, 23754,3327, 12354,3327, 10554,3327, 1554,3327, 0, 0

~

V

J

= j & 0,7713( = 0,7713

(41)

e = j)f }−341,9987, 8,6654, ∞, −308,6654, ∞, −93258,6653, −5508,6654, −4008,6654, −93258,6653, −258708,6654

, −4008,6655, −2508,6654, −47508,6655, −24708,6655, −21108,6655, −3108,6655, 333, ∞

~

= 8,6654 = ,

_'

s = u, xxt•

cE cE 0 0 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2287,2380 44725,9186 1044929

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 _-205247,4827 _3868323

Variabel

, 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3 -74,9964 1025,9962

, N 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -49,9975 17,3309

,J 0 0 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0,0001 999,9999

,K 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -24,9988 308,6654

' 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24,9988 158

,M 0 0 0,2000 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -0,2000 -4,9997 18651,7331

,N 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -24,9988 5508,6654

,O 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -5 -124,9940 20043,3270

,P 0 0 0,2000 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -0,2000 -4,9997 18651,7331

(42)

Iterasi 4

s = u, {|tt

cE cE 0 0 -4097,2380 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2287,2380 44725,9186 1044929

cE' ₀ ₀ _8902,7620 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ _-7712,7620 _-205247,4827 _3868323

Variabel

, 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -0,5000 -12,4995 2004,3327

, ' 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -24,9988 2508,6654

, J 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -0,5000 -12,4995 23754,3327

, K 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -0,5000 -12,4995 12354,3327

, L 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -0,5000 -12,4995 10554,3327

, M 0 0 0,5000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -0,5000 -12,4995 1554,3327

1 0 0,0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 333

(43)

Keterangan:

g = &, , ,

'

, ,

J

, ,

K

,

'

, ,

M

, ,

N

, ,

O

, ,

P

, ,

l

, , , ,

'

, ,

J

, ,

K

, ,

L

, ,

M

, , ,

L

(

` = &,

N

(

[

J

= } 333, 158, 0, 1025,9962, 0,999,9999,308,6654, 18651,7331, 5508,6654, 20043,3270, 18651,7331,

258708,6654, 2004,3327, 2508,6654, 23754,3327, 12354,3327, 10554,3327, 1554,3327, 17,3309, 0 ~

karena

` = &∅(

= 333

,

_'

= 158

,

_J

= 0

,

, = 1025,9962

,

_'

= 0

,

_J

= 999,9999

,

K

= 308,6654

,

_L

= 0

,

_M

= 18651,7331

,

_N

= 5508,6654

,

_O

= 20043,3270

,

_P

= 18651,7331

,

_l

= 258708,6654

,

, =

2004,3327

,

_'

= 2508,6654

,

_J

= 23754,3327

,

_K

= 12354,3327

,

_L

= 10554,3327

,

_M

= 1554,3327

,

_M

= 17,3309

,

_N

=

1554,3327

,

_G

= 0

dengan

$ = 1044929

dan

$

_'

= 3868323

terlihat hasil optimum pada bagian A setelah penambahan kendala