Penyelesaian Program Bilangan Bulat Campuran Dua Kriteria dengan Menggunakan Metode Branch and Cut
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
cE
cE
'
-5
2
0
0
0
0
0
1
-4
0
0
0
0
0
,
,'
,J
,K
Solusi
'
Variabel Basis
,
,'
,J
,K
-1
2
1
0
0
0
3
3
1
0
1
0
0
8
5
0
0
0
1
0
6
0
3
0
0
0
1
4
Keterangan:
g = &,1 , ,2 , ,3 , ,4 (
` = & ,
'(
V = j
&0,0000, 0,1667, 0,6667( = 0,6667
[ = &0, 0, 3, 8, , 6, 4(
, = j
&0,0000, 0,1667, 0,6667( = 0,6667 =
e = j)f &1,5000, 8, ∞, 1,3334( = 1,3334 = ,K
'
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria Iterasi 2 s = u, wwwx
cE
cE
cE '
-5
0
0
0
0
-0,6667
-2,6667
1
0
0
0
0
1,3334
5,3334
,
,'
,J
,K
Solusi
'
Variabel Basis
,
-1
0
1
0
0
-0,6667
-0,3334
3
0
0
1
0
0,3334
6,6667
,J
5
0
0
0
1
0,0000
6,0000
'
0
1
0
0
0
0,3334
1,3334
,'
Keterangan:
g = &,1 , ,2 , ,3 ,
` = & (
2(
[ ' = &0, 1,3334, −0,3334, 6,6667, 6, 0(
V' = j
, = j
&0,1667( = 0,16667
&0,1667( = 0,16667 =
e = j)f &−0,3334, 2,2223,1,2000, ∞( = ,J
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria Iterasi 3 s = u, twwx
cE
cE
cE '
0
0
0
0
1
-0,6667
3,3334
0
0
0
0
-0,2000
1,3334
4,1334
,
,'
,J
,K
Solusi
'
Variabel Basis
,
,'
'
0
0
1
0
0,2000
-0,6667
1,5334
0
0
0
1
-0,6000
0,3334
3,0667
1
0
0
0
0,2000
0
1,2000
0
1
0
0
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
mengerjakan pada bagian A.
Universitas Sumatera Utara
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
cE
cE '
-5
2
0
0
0
0
0
0
1
-4
0
0
0
0
0
0
,
,'
,J
,K
,L
Solusi
'
Variabel Basis
,
,'
,J
,K
,L
-1
2
1
0
0
0
0
3
3
1
0
1
0
0
0
8
5
0
0
0
1
0
0
6
0
3
0
0
0
1
0
4
0
1
0
0
0
0
1
1
Keterangan:
g = &,1 , ,2 , ,3 , ,4 , ,5 (
` = & ,
'(
V = j
&0,1667, 0,6667( = 0,6667
[ = &0, 0, 3, 8, 6, 4, 1(
, = j
&0,1667, 0,6667( = 0,6667 =
'
e = j)f &1,5000, 8,0000, ∞, 1,3334, 1,0000( = 1,0000 = ,5
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian A Iterasi 2 s = u, wwwx
cE
cE
cE '
-5
0
0
0
0
0
-2
-2
1
0
0
0
0
0
4
4
,
,'
,J
,K
,L
Solusi
'
Variabel Basis
,
-1
0
1
0
0
0
-2
1
3
0
0
1
0
0
-1
7
5
0
0
0
1
0
0
6
,K
0
0
0
0
0
1
-3
1
'
0
1
0
0
0
0
1
1
,'
,J
Keterangan:
g = &,1 , ,2 , ,3 , ,4 ,
` = & (
'(
[ ' = &0, 1, 1, 7, 6, 1,0(
V' = j
, = j
&0,1667( = 0,1667
&0,1667( = 0,1667 =
e = j)f &−1, 2,3334, 1,2000, 6,6667, ∞, ∞( = 1,2000 = ,3
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian A Iterasi 3 s = u, twwx
cE
cE
cE '
0
0
0
0
0
1
-2
4
0
0
0
0
0
-0,8000
4
0,8000
,
,'
,J
,K
,L
Solusi
'
Variabel Basis
,
0
0
1
0
0
0,2000
-2
2,2000
0
0
0
1
0
-0,6000
-1
3,4000
1
0
0
0
1
0,2000
0
1,2000
,K
0
0
0
0
0
0
-3
1
'
0
1
0
0
0
0
1
1
,'
Keterangan:
g = &,1 , ,2 ,
` = &∅(
1 , ,4 ,
'(
[ J = &1,2000, 1, 2,2000, 3,4000, 0, 1, 0(
karena ` = &∅( maka permasalahan telah optimum, diperoleh
= 1,2000,
'
= 1, , = 2,2000, ,' = 3,4000, ,J = 0, ,K = 1, ,L = 0 dengan
$ = 4 dan $' = 0,8000, dikarenakan hasil optimum pada bagian A belum diproleh bilangan bulat, maka kita lanjut mengerjakan pada bagian
B.
Universitas Sumatera Utara
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
cE
cE '
-5
2
0
0
0
0
0
0
0
1
-4
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
Solusi
'
Variabel Basis
,
,'
,J
,K
,L
-1
2
1
0
0
0
0
0
3
3
1
0
1
0
0
0
0
8
5
0
0
0
1
0
0
0
6
0
3
0
0
0
1
0
0
4
0
1
0
0
0
0
-1
1
2
Keterangan:
g = &,1 , ,2 , ,3 , ,4 , ,5 (
` = & ,
'(
V = j
&0,1667, 0,6667( = 0,6667
[ = &0, 0, 3, 8, 6, 4, 2(
, = j
&0,1667, 0,6667( = 0,6667 =
'
e = j)f &1,5000, 8, ∞, 1,3334, 2( = 1,3334 = ,4
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian B Iterasi 2 s = u, wwwx
cE
cE
cE '
-5
0
0
0
0
-0,6667
0
0
-2,6667
1
0
0
0
0
1,3334
0
0
5,3334
,
,'
,J
,K
,L
,M
Solusi
'
Variabel Basis
,
-1
0
1
0
0
-0,6667
0
0
0,3334
3
0
0
1
0
-0,3334
0
0
3,3334
,J
5
0
0
0
1
0
0
0
6
'
0
1
0
0
0
0,3334
0
0
1,3334
0
0
0
0
0
-0,3334
-1
1
0,6667
,'
,L
Keterangan:
g = &,1 , ,2 , ,3 ,
` = & (
' , ,5 (
[ ' = &0, 1,3334, 0,3334, 3,3334, 6, 0, , 0,6667(
V' = j
, = j
&0,1667( = 0,1667
&0,1667( = 0,1667 =
e = j)f &−0,3334, 2,2223,1,2000, ∞, ∞( = 1,2000 = ,3
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian B Iterasi 3 s = u, twwx
cE
cE
cE '
0
0
0
0
1
-0,6667
0
0
3,3334
0
0
0
0
-0,2000
1,3334
0
0
0,1334
,
,'
,J
,K
,L
,M
Solusi
'
Variabel Basis
,
,'
'
,L
0
0
1
0
0,2000
-0,6667
0
0
1,5334
0
0
0
1
-0,6000
-0,3334
0
0
3,0667
1
0
0
0
0,2000
0
0
0
1,2000
0
1
0
0
0
0,3334
0
0
1,3334
0
0
0
0
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 ` = &∅( maka permasalahan telah optimum, diperoleh
= 1,2000,
'
= 1,3334, , = 1,5334, ,' = 3,0667, ,J = 0, ,K = 0,
,L = 0,6667 dengan $ = 3,3334 dan $' = 0,1334, dikarenakan hasil optimum pada bagian A dan bagian B belum diproleh bilangan bulat,
maka kita lanjut menerapkan pemotongan (cut) pada bagian A.
Universitas Sumatera Utara
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
cE
cE '
0
0
0
0
0
1
-2
0
4
0
0
0
0
0
-0,8000
4
0
0,8000
,
,'
,J
,K
,L
,G
Solusi
'
Variabel Basis
,
0
0
1
0
0
0,2000
-2
0
2,2000
0
0
0
1
0
-0,6000
-1
0
3,4000
1
0
0
0
1
0,2000
0
0
1,2000
,K
0
0
0
0
0
0
-3
0
1
'
0
1
0
0
0
0
1
0
1
0
0
0
0
0
-0,2000
0
1
-0,2000
,'
,G
Keterangan:
g = y,1 , ,2 ,
, ,4 ,
' , ,G
z
` = &,J , ,L (
[ = &1,2000, 1, 2,2000,3,4000, 0, 1, 0, −0,2000(
V = j
, = j
&0, 0,6667( = 0,6667
&0, 0,6667( = 0,6667 = ,L
Universitas Sumatera Utara
e = j)f &−1,1000, −3,4000, ∞, −0,3334, 1, ∞( = 1 =
'
Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Itersi 2 s = u, twwx
cE
cE
cE '
0
2
0
0
1
2
0
0
6
0
-4
0
0
0
-0,8000
0
0
-3,2000
,
,'
,J
,K
,L
,G
Solusi
'
Variabel Basis
,
,'
,K
,L
,G
0
2
1
0
0
0,2000
0
0
4,2000
0
1
0
1
0
-0,6000
0
0
4,4000
1
0
0
0
1
0,2000
0
0
1,2000
0
3
0
0
0
0
0
0
4
0
1
0
0
0
0
1
0
1
0
0
0
0
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
, = j
&0( = 0
&0( = 0 = ,J
Universitas Sumatera Utara
e = j)f &∞, ∞, 1,2000, ∞, ∞, ∞ ( = 1,2000 =
Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Itersi 3 s = u
cE
cE
cE '
0
2
0
0
0
2
0
0
6
0
-4
0
0
0
-0,8000
0
0
3,2000
,
,'
,J
,K
,L
,G
Solusi
'
Variabel Basis
,
,'
,J
,K
,L
,G
0
2
1
0
0
0,2000
0
0
4,2000
0
1
0
1
0
-0,6000
0
0
4,4000
1
0
0
0
1
0,2000
0
0
1,2000
0
3
0
0
0
0
0
0
4
0
1
0
0
0
0
1
0
1
0
0
0
0
0
-0,2000
0
1
-0,2000
Keterangan:
g = &,1 , ,2 ,
` = &∅(
1,
' , ,L (
[ J = &0, 0, 4,2000, 4,4000, 1,2000, 4, 1, −4,2000(
Universitas Sumatera Utara
karena ` = &∅( maka permasalahan telah optimum. diperoleh
= 0,
'
= 0, , = 4,2000, ,' = 4,4000, ,J = 1,2000, ,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
cE
cE '
0
0
0
0
1
-0,6667
0
0
0
3,3334
0
0
0
0
-0,2000
1,3334
0
0
0
0,1334
,
,'
,J
,K
,L
,M
,G
Solusi
'
Variabel Basis
,
,'
'
,L
,G
0
0
1
0
0,2000
-0,6667
0
0
0
1,5334
0
0
0
1
-0,6000
-0,3334
0
0
0
3,0667
1
0
0
0
0,2000
0
0
0
0
1,2000
0
11
0
0
0
0,3334
0
0
0
1,3334
0
0
0
0
0
-0,3334
-1
1
0
0,6667
0
0
0
0
0
-0,3334
0
0
1
-0,3334
Keterangan:
g = y,1 , ,2, ,
,
' , ,5 , ,G
z
` = &,J , ,K (
Universitas Sumatera Utara
[ = &1,2000, 1,3334, 1,5334, 3,0667, 0, 0, 0,6667, −0,3334(
V = j
, = j
&0,1667, 0,6667( = 0,6667
&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
cE
cE '
0
0
0
0
1
0
0
0
-2
4
0
0
0
0
-0,2000
0
0
0
4
2,8000
,
,'
,J
,K
,L
,M
,G
Solusi
'
Variabel Basis
,
,'
'
,L
,K
0
0
1
0
0,2000
0
0
0
-2,0000
2,2000
0
0
0
1
-0,6000
0
0
0
-1,0000
3,4000
1
0
0
0
0,2000
0
0
0
0
1,2000
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
-1
1
-1
1
0
0
0
0
0
1
0
0
-3
1
Keterangan:
g = Q, , ,', ,
` = &,J (
,
' , ,K , ,L S
Universitas Sumatera Utara
[ ' = &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
cE
cE '
-5
0
0
0
0
0
0
0
-2
2
0
0
0
0
0
0
0
0
4
-4
,
,'
,J
,K
,L
,M
,G
Solusi
'
Variabel Basis
,
-1
0
1
0
0
0
0
0
-2
1
4
0
0
1
0
0
0
0
0
7
,J
5
0
0
0
1
0
0
0
0
6
'
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
-1
1
-1
1
0
0
0
0
0
1
0
0
-3
1
,'
,L
,K
Keterangan:
g = Q, , ,', , ,J, ,
` = &∅(
' , ,K , ,L S
Universitas Sumatera Utara
[ J = &0, 1, 1, 7, 6, 0,1,1(
karena ` = &∅( maka permasalahan telah optimum, diperoleh
= 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
Ec '
-2287,2380
-1789,1180
-4097,2380
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
,
,'
,J
,K
,L
,M
,N
,O
,P
l
'
,
J
,
K
,
L
,
'
J
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
0
0
3
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1000
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
800
0
0,0600
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
10
0,2000
0,2000
0,2000
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
18750
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
6000
5
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
22500
0,2000
0,2000
0,2000
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
18750
Basis
M
Solusi
Universitas Sumatera Utara
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
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
,
,
,
,
,
,
,
'
J
,
K
,
L
,
J
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
l
l
,
'
Basis
M
Solusi
Keterangan:
g = &, , ,' , ,J , ,K , ,L , ,M , ,N , ,O , ,P , , l , , , , ' , , J , , K , , L , , M (
` = & ,
', J(
V = j
&0,7713, 0,8211, 0,6848( = 0,8211
[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750, 259200, 2250, 3000, 24000, 12600, 10800, 1800(
, = j
&0,7713, 0,8211, 0,6848( = 0,8211 =
'
Universitas Sumatera Utara
e = j)f &833,3333, 500, ∞, 800, 166,6667, 93750, 6000, 4500, 93750, 259200, 4500, 3000, 48000, 25200, 21600, 3600(
= 166,6667 = ,L
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 s = u, {|tt
cE
cE
Ec '
-2287,2380
0
-4097,2380
0
0
0
0
29818,6333
0
0
0
0
0
0
0
0
0
0
0
298186,3333
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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
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
0,2000
0
0,2000
0
0
0
0
-3,3333
1
0
0
0
0
0
0
0
0
0
0
18716,6667
1
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
5833,3333
5,0000
0
5
0
0
0
0
-83,3333
0
0
1
0
0
0
0
0
0
0
0
21666,6667
0,2000
0
0,2000
0
0
0
0
-3,3333
0
0
0
1
0
0
0
0
0
0
0
18716,6667
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
,M
,N
,O
,P
,
,
l
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 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
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
'
Basis
,
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
'
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(
V' = j
&0,7713, 0,6848( = 0,7713
, , ', , J, , K, , L, , M(
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
, = j
&0,7713, 0,6848( = 0,7713 =
Universitas Sumatera Utara
666,6667, 333,3333, ∞, 633,3333, ∞, 93583,3333, 5833,3333, 4333,3333, 93583,3333,
~
e = j)f }
259033,3333, 4333,3333, 2833,3333,4783,3333,25033,3333, 21433,3333, 3433,3333
= 333,3333 = ,'
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
0
0
-4097,2380
0
1143,6190
0
0
-8302
0
0
0
0
0
0
0
0
0
0
0
1060599
0
0
8902,7620
0
-3856,3810
0
0
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
3939302,6667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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
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
0
0
0,2000
0
-0,1000
0
0
0
1
0
0
0
0
0
0
0
0
0
0
18650
0
0
1
0
-0,5000
0
0
0
0
1
0
0
0
0
0
0
0
0
0
5500
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
l
0
0
1
0
-0,5000
0
0
0
0
0
0
0
1
0
0
0
0
0
0
258700
,J
,M
,N
,O
,
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
0
0
-4097,2380
0
1143,6190
0
0
-8302
0
0
0
0
0
0
0
0
0
0
0
1060599
0
0
8902,7620
0
-3856,3810
0
0
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
3939302,6667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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 (
, ,J , ,K ,
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M(
333,3333, 166,6667, 0, 1000, 0, 1000, 300, 0, 166,6667, 18650, 5500,
~
[J = }
20000, 18650, 258700, 1000, 333,3333, 1000, 300, 166,6667,18650
VJ = j
, = j
&0,6848( = 0,6848
&0,6848( = 0,6848 =
J
e = j)f &333,3333, ∞, 333,3333, 300, ∞, 93250, 5500, 4000, 93250, 258700, 4000, 2500, 47500, 24700, 21100, 3100( = 300 = ,K
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 4 s = u, w{€{
cE
cE
Ec '
0
0
0
0
-905
0
4097,2380
-8302,0000
0
0
0
0
0
0
0
0
0
0
0
2289770,4000
0
0
0
0
595
0
-8902,7620
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
6610131,2667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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
0
0
0
0
0
0
-0,2000
0
1
0
0
0
0
0
0
0
0
0
0
18590
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
5200
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
J
0
0
0
0
0
0
-0,5000
0
0
0
0
0
0
0
0
1
0
0
0
23600
,M
,N
,O
,
,
,
,
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 4 s = u, w{€{
cE
cE
Ec '
0
0
0
0
-905
0
4097,2380
-8302,0000
0
0
0
0
0
0
0
0
0
0
0
2289770,4000
0
0
0
0
595
0
-8902,7620
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
6610131,2667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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 ` = &∅( maka permasalahan telah optimum, diperoleh
= 333,3333,
,K = 0, ,L = 0, ,M = 18950, ,N = 5200, ,O = 18500, ,P = 18950, ,
12200, ,
L
= 10400, ,
M
l
'
= 166,6667,
= 258400, ,
J
= 300, , = 100, ,' = 0, ,J = 100,
= 1850, ,
'
= 2200, ,
J
= 23600, ,
K
=
= 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
(branch) yaitu pada bagian A dan bagian B, terlebih dahulu mengerjakan pada bagian A.
Universitas Sumatera Utara
Lampiran 7 Pembahasan Contoh Kasus Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 1 s = t, uuuu
cE
cE
Ec '
-2287,2380
-1789,1180
-4097,2380
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
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
0
0
3
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1000
1
1
1,0000
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
800
0
0,0600
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
10
0,2000
0,2000
0,2000
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
18750
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
6000
5
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
22500
0,2000
0,2000
0,2000
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
18750
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
J
0,5000
0,5000
0,5000
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
24000
,L
,M
,N
,O
,P
,
,
,
,
l
L
M
,
3
,K
K
,
3
,J
J
,
3
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
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
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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(
V = j
&0,7713, 0,8211, 0,6848( = 0,8211
[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750, 259200, 2250, 3000, 24000, 12600, 10800, 1800, 333(
, = 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(
= 166,6667 = ,L
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 s = u, {|tt
cE
cE
Ec '
-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
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M ,N
,O
,P
,
,
,
Variabel
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
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
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
1
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
0
5833,3333
5
0
5
0
0
0
0
-83,3333
0
0
1
0
0
0
0
0
0
0
0
0
21666,6667
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
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
J
0,5000
0
0,5000
0
0
0
0
-8,3333
0
0
0
0
0
0
0
1
0
0
0
0
23916,6667
,N
,O
,P
,
,
,
,
l
L
M
,
3
,M
K
,
0
,J
J
,
3,0000
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 s = u, {|tt
cE
cE
Ec '
-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
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M ,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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(
V' = j
&0,7713, 0,6848( = 0,7713
, , ', , J, , K, , L, , M, , N(
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
, = j
&0,7713, 0,6848( = 0,7713 =
666,6667, 333,3333, ∞, 633,3333, ∞, 93583,3333, 5833,3333, 4333,3333,93583,3333,
~
e = j)f }
259033,3333, 4333,3333, 2833,3333,47833,3333, 25033,333321433,3333,3433,3333, 333
= 333 = ,
N
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
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
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
0
0
0,2000
0
0
0
0
-3,3333
1
0
0
0
0
0
0
0
0
0
0
0
18650,0667
0
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
-1
5500,3333
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
J
0
0
0,5000
0
0
0
0
-8,3333
0
0
0
0
0
0
0
1
0
0
0
-1
23750,1667
,N
,O
,
,
,
,
L
M
,
3
,M
K
,
0
,J
J
,
0
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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 ,
` = & J (
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M,
(
333, 166,6667, 0, 1001, 0,6667, 1000, 300,3333, 0, 18650,0667, 5500,3333, 20001,6667, 18650,0667,
~
[J = }
18650,0667, 258700,3333,2000,16667, 2500,3333, 23750,1667, 12350,1667, 10550,1667,1550,1667, 0
VJ = j
, = j
&0,6848( = 0,6848
&0,6848( = 0,6848 =
J
333,6667, ∞, 333,3333, 300,3333, ∞, 93250,3333, 5500,3333, 4000,3333, 93250,3333,
~
e = j)f }
258700,3333, 4000,3333,2500,3333, 47500,3333, 24700,3333, 21100,3333, 3100,3333, ∞
= 300,3333 = ,K
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 4 s = u, w{€{
cE
cE
Ec '
0
0
0
0
0
0
4097,2380
-38468,6667
0
0
0
0
0
0
0
0
0
0
0
-1810
2290373,7333
0
0
0
0
0
0
-8902,7620
11541,1667
0
0
0
0
0
0
0
0
0
0
0
1190
6610527,9333
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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
0
0
0
0
0
0
-0,2000
0
1
0
0
0
0
0
0
0
0
0
0
0
18590
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
0
5200
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
J
0
0
0
0
0
0
-0,5000
0
0
0
0
0
0
0
0
1
0
0
0
0
23600
,'
,M
,N
,O
,
,
,
,
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
0
0
0
0
0
0
4097,2380
-38468,6667
0
0
0
0
0
0
0
0
0
0
0
-1810
2290373,7333
0
0
0
0
0
0
-8902,7620
11541,1667
0
0
0
0
0
0
0
0
0
0
0
1190
6610527,9333
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
,
M
N
Solusi
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 ` = &∅( maka permasalahan telah optimum, diperoleh
= 333,
99, ,K = 0, ,L = 0, ,M = 18590, ,N = 5200, ,O = 18500, ,P = 18950, ,
12200, ,
L
= 10400, ,
M
= 1400, ,
N
'
l
= 166,6667,
= 258400, ,
J
= 300,333, , = 100, ,' = 0,6667, ,J =
= 1850, ,
'
= 2200, ,
J
= 23600, ,
K
=
= 0 dengan $ = 2290373,7333dan $' = 6610527,9333, dikarenakan hasil optimum pada
bagian A belum diproleh bilangan bulat, maka kita lanjut mengerjakan pada bagian B.
Universitas Sumatera Utara
Lampiran 8 Pembahasan Contoh Kasus Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 1 s = t, uuuu
cE
cE
Ec '
-2287,2380
-1789,1180
-4097,2380
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
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
0
0
3
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1000
1
1
1,0000
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
800
0
0,0600
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
10
0,2000
0,2000
0,2000
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
18750
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
6000
5
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
22500
0,2000
0,2000
0,2000
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
18750
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
J
0,5000
0,5000
0,5000
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
24000
,L
,M
,N
,O
,P
,
,
,
,
l
L
M
,
3
,K
K
,
3
,J
J
,
3
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
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
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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(
V = j
&0,7713, 0,8211, 0,6848( = 0,8211
[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750, 259200, 2250, 3000, 24000, 12600, 10800, 1800, 333(
, = 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(
= 166,6667 = ,L ,
Universitas Sumatera Utara
= 333,3333,
setelah diselasaikan pada bagian B diperoleh penyelesaian yang sama dengan hasil pada solusi optimum awal yaitu
'
= 166,6667,
258400,
,
J
= 300, , = 100, ,' = 0, ,J = 100, ,K = 0, ,L = 0, ,M = 18950, ,N = 5200, ,O = 18500, ,P = 18950, ,
= 1850,
,
'
= 2200,
,
J
= 23600,
,
K
= 12200,
,
L
= 10400,
,
M
= 1400,
,
N
= −0,6667
l
=
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
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
0
0
0
0
0
0
-0,2000
0
1
0
0
0
0
0
0
0
0
0
0
0
0
18590
,'
,M
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 1 s = u, w{€{
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,N
J
l
'
,
J
,
K
,
L
,
M
,
N
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
5200
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
0
0
0
0
0
0
0
-0,6667
0
0
0
0
0
0
0
0
0
0
0
0
1
-0,3333
,O
,
,
,
,
,
,
,
,G
Universitas Sumatera Utara
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
, = j
&0,6648, 0,2308, 0,3967( = 0,6848
&0,6648, 0,2308, 0,3967( = 0,6848 = ,K
−33,3333, ∞, −33, 300,3333, ∞, −92950, −5200, −3700, −92950,
e = j)f }
~ = 300,3333 =
−258400, −3700, −2200, −47200, −24400, −20800, −2800, ∞, ∞
J
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Iterasi 2
s = u, w{€{
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
N
,G
Solusi
Variabel
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
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
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
0
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
-1
0
5500,3333
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
J
0
0
0,5000
0
0
0
0
-8,3334
0
0
0
0
0
0
0
1
0
0
0
-0,5000
0
23750,1667
,N
,O
,
,
,
,
L
M
,
3
,M
K
,
0
,J
J
,
0
,'
'
,
J
,
l
,
'
Basis
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 2 s = u, w{€{
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
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
,G
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
, = j
&0,8211, 0,7713( = 0,8211
&0,8211, 0,7713( = 0,8211 = ,L
Universitas Sumatera Utara
−20,0200, −0,0200, −9999, −18,02000, 10, −5595, 0088, −330, 0193, −240,0195, −5569, 0088,
~ = 0,4999 = ,G
e = j)f }
−15521,9890, 240,0195, −150, 0197, −2850, 0143, −1482, 0170, −1266, 0175, −186,0196, ∞, 0,4999
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Iterasi 3
s = u, {|tt
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
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
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
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
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
-1
-24,9988
5508,6654
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
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
-1
-24,9988
2508,6654
,N
,O
,
,
,
'
L
M
,
3
,M
K
,
0
,J
J
,
0
,'
'
,
J
,
l
,
'
Basis
N
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 3 s = u, {|tt
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
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
,L
Keterangan:
g = &, , ,' , ,J , ,K ,
` = &, N (
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M,
, ,L (
333, 158, 0, 1025,9962, 17,3309,999,9999,308,6654, 18651,7331, 5508,6654, 20043,3270, 18651,7331,
~
[J = }
258708,6654, 2004,3327, 2508,6654, 23754,3327, 12354,3327, 10554,3327, 1554,3327, 0, 0
VJ = j
, = j
& 0,7713( = 0,7713
& 0,7713( = 0,7713 = ,
N
Universitas Sumatera Utara
−341,9987, 8,6654, ∞, −308,6654, ∞, −93258,6653, −5508,6654, −4008,6654, −93258,6653, −258708,6654
~
e = j)f }
, −4008,6655, −2508,6654, −47508,6655, −24708,6655, −21108,6655, −3108,6655, 333, ∞
= 8,6654 = ,'
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Iterasi 4
s = u, xxt•
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
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
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
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
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
-1
-24,9988
5508,6654
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
,M
,N
,O
,
L
M
,
3
,J
K
,
0
N
J
,
0
,
'
,
J
,
l
,
'
Basis
N
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 4 s = u, {|tt
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
,
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
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
,L
Universitas Sumatera Utara
Keterangan:
g = &, , ,' , ,J , ,K ,
` = &, N (
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M,
, ,L (
333, 158, 0, 1025,9962, 0,999,9999,308,6654, 18651,7331, 5508,6654, 20043,3270, 18651,7331,
~
[J = }
258708,6654, 2004,3327, 2508,6654, 23754,3327, 12354,3327, 10554,3327, 1554,3327, 17,3309, 0
karena ` = &∅( maka permasalahan telah optimum, diperoleh
= 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, ,
2004,3327, ,
'
= 2508,6654, ,
J
= 23754,3327, ,
K
= 12354,3327, ,
L
= 10554,3327, ,
M
l
= 258708,6654, ,
= 1554,3327, ,
M
= 17,3309, ,
=
N
=
1554,3327, ,G = 0 dengan $ = 1044929 dan $' = 3868323 terlihat hasil optimum pada bagian A setelah penambahan kendala
gomory sudah diperoleh bilangan bulat, maka permasalahannya diberhentikan.
Universitas Sumatera Utara
Lampiran 1 Pembahasan Masalah Program Linier Dua Kriteria Menggunakan Parametric Simplex Algorithm
Tabel Simpleks Program Linier Dua Kriteria Iterasi 1 s = t, uuuu
cE
cE
cE
'
-5
2
0
0
0
0
0
1
-4
0
0
0
0
0
,
,'
,J
,K
Solusi
'
Variabel Basis
,
,'
,J
,K
-1
2
1
0
0
0
3
3
1
0
1
0
0
8
5
0
0
0
1
0
6
0
3
0
0
0
1
4
Keterangan:
g = &,1 , ,2 , ,3 , ,4 (
` = & ,
'(
V = j
&0,0000, 0,1667, 0,6667( = 0,6667
[ = &0, 0, 3, 8, , 6, 4(
, = j
&0,0000, 0,1667, 0,6667( = 0,6667 =
e = j)f &1,5000, 8, ∞, 1,3334( = 1,3334 = ,K
'
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria Iterasi 2 s = u, wwwx
cE
cE
cE '
-5
0
0
0
0
-0,6667
-2,6667
1
0
0
0
0
1,3334
5,3334
,
,'
,J
,K
Solusi
'
Variabel Basis
,
-1
0
1
0
0
-0,6667
-0,3334
3
0
0
1
0
0,3334
6,6667
,J
5
0
0
0
1
0,0000
6,0000
'
0
1
0
0
0
0,3334
1,3334
,'
Keterangan:
g = &,1 , ,2 , ,3 ,
` = & (
2(
[ ' = &0, 1,3334, −0,3334, 6,6667, 6, 0(
V' = j
, = j
&0,1667( = 0,16667
&0,1667( = 0,16667 =
e = j)f &−0,3334, 2,2223,1,2000, ∞( = ,J
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria Iterasi 3 s = u, twwx
cE
cE
cE '
0
0
0
0
1
-0,6667
3,3334
0
0
0
0
-0,2000
1,3334
4,1334
,
,'
,J
,K
Solusi
'
Variabel Basis
,
,'
'
0
0
1
0
0,2000
-0,6667
1,5334
0
0
0
1
-0,6000
0,3334
3,0667
1
0
0
0
0,2000
0
1,2000
0
1
0
0
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
mengerjakan pada bagian A.
Universitas Sumatera Utara
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
cE
cE '
-5
2
0
0
0
0
0
0
1
-4
0
0
0
0
0
0
,
,'
,J
,K
,L
Solusi
'
Variabel Basis
,
,'
,J
,K
,L
-1
2
1
0
0
0
0
3
3
1
0
1
0
0
0
8
5
0
0
0
1
0
0
6
0
3
0
0
0
1
0
4
0
1
0
0
0
0
1
1
Keterangan:
g = &,1 , ,2 , ,3 , ,4 , ,5 (
` = & ,
'(
V = j
&0,1667, 0,6667( = 0,6667
[ = &0, 0, 3, 8, 6, 4, 1(
, = j
&0,1667, 0,6667( = 0,6667 =
'
e = j)f &1,5000, 8,0000, ∞, 1,3334, 1,0000( = 1,0000 = ,5
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian A Iterasi 2 s = u, wwwx
cE
cE
cE '
-5
0
0
0
0
0
-2
-2
1
0
0
0
0
0
4
4
,
,'
,J
,K
,L
Solusi
'
Variabel Basis
,
-1
0
1
0
0
0
-2
1
3
0
0
1
0
0
-1
7
5
0
0
0
1
0
0
6
,K
0
0
0
0
0
1
-3
1
'
0
1
0
0
0
0
1
1
,'
,J
Keterangan:
g = &,1 , ,2 , ,3 , ,4 ,
` = & (
'(
[ ' = &0, 1, 1, 7, 6, 1,0(
V' = j
, = j
&0,1667( = 0,1667
&0,1667( = 0,1667 =
e = j)f &−1, 2,3334, 1,2000, 6,6667, ∞, ∞( = 1,2000 = ,3
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian A Iterasi 3 s = u, twwx
cE
cE
cE '
0
0
0
0
0
1
-2
4
0
0
0
0
0
-0,8000
4
0,8000
,
,'
,J
,K
,L
Solusi
'
Variabel Basis
,
0
0
1
0
0
0,2000
-2
2,2000
0
0
0
1
0
-0,6000
-1
3,4000
1
0
0
0
1
0,2000
0
1,2000
,K
0
0
0
0
0
0
-3
1
'
0
1
0
0
0
0
1
1
,'
Keterangan:
g = &,1 , ,2 ,
` = &∅(
1 , ,4 ,
'(
[ J = &1,2000, 1, 2,2000, 3,4000, 0, 1, 0(
karena ` = &∅( maka permasalahan telah optimum, diperoleh
= 1,2000,
'
= 1, , = 2,2000, ,' = 3,4000, ,J = 0, ,K = 1, ,L = 0 dengan
$ = 4 dan $' = 0,8000, dikarenakan hasil optimum pada bagian A belum diproleh bilangan bulat, maka kita lanjut mengerjakan pada bagian
B.
Universitas Sumatera Utara
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
cE
cE '
-5
2
0
0
0
0
0
0
0
1
-4
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
Solusi
'
Variabel Basis
,
,'
,J
,K
,L
-1
2
1
0
0
0
0
0
3
3
1
0
1
0
0
0
0
8
5
0
0
0
1
0
0
0
6
0
3
0
0
0
1
0
0
4
0
1
0
0
0
0
-1
1
2
Keterangan:
g = &,1 , ,2 , ,3 , ,4 , ,5 (
` = & ,
'(
V = j
&0,1667, 0,6667( = 0,6667
[ = &0, 0, 3, 8, 6, 4, 2(
, = j
&0,1667, 0,6667( = 0,6667 =
'
e = j)f &1,5000, 8, ∞, 1,3334, 2( = 1,3334 = ,4
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian B Iterasi 2 s = u, wwwx
cE
cE
cE '
-5
0
0
0
0
-0,6667
0
0
-2,6667
1
0
0
0
0
1,3334
0
0
5,3334
,
,'
,J
,K
,L
,M
Solusi
'
Variabel Basis
,
-1
0
1
0
0
-0,6667
0
0
0,3334
3
0
0
1
0
-0,3334
0
0
3,3334
,J
5
0
0
0
1
0
0
0
6
'
0
1
0
0
0
0,3334
0
0
1,3334
0
0
0
0
0
-0,3334
-1
1
0,6667
,'
,L
Keterangan:
g = &,1 , ,2 , ,3 ,
` = & (
' , ,5 (
[ ' = &0, 1,3334, 0,3334, 3,3334, 6, 0, , 0,6667(
V' = j
, = j
&0,1667( = 0,1667
&0,1667( = 0,1667 =
e = j)f &−0,3334, 2,2223,1,2000, ∞, ∞( = 1,2000 = ,3
Universitas Sumatera Utara
Tabel Simpleks Program Linier Dua Kriteria pada Bagian B Iterasi 3 s = u, twwx
cE
cE
cE '
0
0
0
0
1
-0,6667
0
0
3,3334
0
0
0
0
-0,2000
1,3334
0
0
0,1334
,
,'
,J
,K
,L
,M
Solusi
'
Variabel Basis
,
,'
'
,L
0
0
1
0
0,2000
-0,6667
0
0
1,5334
0
0
0
1
-0,6000
-0,3334
0
0
3,0667
1
0
0
0
0,2000
0
0
0
1,2000
0
1
0
0
0
0,3334
0
0
1,3334
0
0
0
0
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 ` = &∅( maka permasalahan telah optimum, diperoleh
= 1,2000,
'
= 1,3334, , = 1,5334, ,' = 3,0667, ,J = 0, ,K = 0,
,L = 0,6667 dengan $ = 3,3334 dan $' = 0,1334, dikarenakan hasil optimum pada bagian A dan bagian B belum diproleh bilangan bulat,
maka kita lanjut menerapkan pemotongan (cut) pada bagian A.
Universitas Sumatera Utara
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
cE
cE '
0
0
0
0
0
1
-2
0
4
0
0
0
0
0
-0,8000
4
0
0,8000
,
,'
,J
,K
,L
,G
Solusi
'
Variabel Basis
,
0
0
1
0
0
0,2000
-2
0
2,2000
0
0
0
1
0
-0,6000
-1
0
3,4000
1
0
0
0
1
0,2000
0
0
1,2000
,K
0
0
0
0
0
0
-3
0
1
'
0
1
0
0
0
0
1
0
1
0
0
0
0
0
-0,2000
0
1
-0,2000
,'
,G
Keterangan:
g = y,1 , ,2 ,
, ,4 ,
' , ,G
z
` = &,J , ,L (
[ = &1,2000, 1, 2,2000,3,4000, 0, 1, 0, −0,2000(
V = j
, = j
&0, 0,6667( = 0,6667
&0, 0,6667( = 0,6667 = ,L
Universitas Sumatera Utara
e = j)f &−1,1000, −3,4000, ∞, −0,3334, 1, ∞( = 1 =
'
Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Itersi 2 s = u, twwx
cE
cE
cE '
0
2
0
0
1
2
0
0
6
0
-4
0
0
0
-0,8000
0
0
-3,2000
,
,'
,J
,K
,L
,G
Solusi
'
Variabel Basis
,
,'
,K
,L
,G
0
2
1
0
0
0,2000
0
0
4,2000
0
1
0
1
0
-0,6000
0
0
4,4000
1
0
0
0
1
0,2000
0
0
1,2000
0
3
0
0
0
0
0
0
4
0
1
0
0
0
0
1
0
1
0
0
0
0
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
, = j
&0( = 0
&0( = 0 = ,J
Universitas Sumatera Utara
e = j)f &∞, ∞, 1,2000, ∞, ∞, ∞ ( = 1,2000 =
Tabel Simpleks Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Itersi 3 s = u
cE
cE
cE '
0
2
0
0
0
2
0
0
6
0
-4
0
0
0
-0,8000
0
0
3,2000
,
,'
,J
,K
,L
,G
Solusi
'
Variabel Basis
,
,'
,J
,K
,L
,G
0
2
1
0
0
0,2000
0
0
4,2000
0
1
0
1
0
-0,6000
0
0
4,4000
1
0
0
0
1
0,2000
0
0
1,2000
0
3
0
0
0
0
0
0
4
0
1
0
0
0
0
1
0
1
0
0
0
0
0
-0,2000
0
1
-0,2000
Keterangan:
g = &,1 , ,2 ,
` = &∅(
1,
' , ,L (
[ J = &0, 0, 4,2000, 4,4000, 1,2000, 4, 1, −4,2000(
Universitas Sumatera Utara
karena ` = &∅( maka permasalahan telah optimum. diperoleh
= 0,
'
= 0, , = 4,2000, ,' = 4,4000, ,J = 1,2000, ,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
cE
cE '
0
0
0
0
1
-0,6667
0
0
0
3,3334
0
0
0
0
-0,2000
1,3334
0
0
0
0,1334
,
,'
,J
,K
,L
,M
,G
Solusi
'
Variabel Basis
,
,'
'
,L
,G
0
0
1
0
0,2000
-0,6667
0
0
0
1,5334
0
0
0
1
-0,6000
-0,3334
0
0
0
3,0667
1
0
0
0
0,2000
0
0
0
0
1,2000
0
11
0
0
0
0,3334
0
0
0
1,3334
0
0
0
0
0
-0,3334
-1
1
0
0,6667
0
0
0
0
0
-0,3334
0
0
1
-0,3334
Keterangan:
g = y,1 , ,2, ,
,
' , ,5 , ,G
z
` = &,J , ,K (
Universitas Sumatera Utara
[ = &1,2000, 1,3334, 1,5334, 3,0667, 0, 0, 0,6667, −0,3334(
V = j
, = j
&0,1667, 0,6667( = 0,6667
&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
cE
cE '
0
0
0
0
1
0
0
0
-2
4
0
0
0
0
-0,2000
0
0
0
4
2,8000
,
,'
,J
,K
,L
,M
,G
Solusi
'
Variabel Basis
,
,'
'
,L
,K
0
0
1
0
0,2000
0
0
0
-2,0000
2,2000
0
0
0
1
-0,6000
0
0
0
-1,0000
3,4000
1
0
0
0
0,2000
0
0
0
0
1,2000
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
-1
1
-1
1
0
0
0
0
0
1
0
0
-3
1
Keterangan:
g = Q, , ,', ,
` = &,J (
,
' , ,K , ,L S
Universitas Sumatera Utara
[ ' = &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
cE
cE '
-5
0
0
0
0
0
0
0
-2
2
0
0
0
0
0
0
0
0
4
-4
,
,'
,J
,K
,L
,M
,G
Solusi
'
Variabel Basis
,
-1
0
1
0
0
0
0
0
-2
1
4
0
0
1
0
0
0
0
0
7
,J
5
0
0
0
1
0
0
0
0
6
'
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
-1
1
-1
1
0
0
0
0
0
1
0
0
-3
1
,'
,L
,K
Keterangan:
g = Q, , ,', , ,J, ,
` = &∅(
' , ,K , ,L S
Universitas Sumatera Utara
[ J = &0, 1, 1, 7, 6, 0,1,1(
karena ` = &∅( maka permasalahan telah optimum, diperoleh
= 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
Ec '
-2287,2380
-1789,1180
-4097,2380
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
,
,'
,J
,K
,L
,M
,N
,O
,P
l
'
,
J
,
K
,
L
,
'
J
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
0
0
3
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1000
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
800
0
0,0600
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
10
0,2000
0,2000
0,2000
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
18750
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
6000
5
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
22500
0,2000
0,2000
0,2000
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
18750
Basis
M
Solusi
Universitas Sumatera Utara
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
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
,
,
,
,
,
,
,
'
J
,
K
,
L
,
J
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
l
l
,
'
Basis
M
Solusi
Keterangan:
g = &, , ,' , ,J , ,K , ,L , ,M , ,N , ,O , ,P , , l , , , , ' , , J , , K , , L , , M (
` = & ,
', J(
V = j
&0,7713, 0,8211, 0,6848( = 0,8211
[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750, 259200, 2250, 3000, 24000, 12600, 10800, 1800(
, = j
&0,7713, 0,8211, 0,6848( = 0,8211 =
'
Universitas Sumatera Utara
e = j)f &833,3333, 500, ∞, 800, 166,6667, 93750, 6000, 4500, 93750, 259200, 4500, 3000, 48000, 25200, 21600, 3600(
= 166,6667 = ,L
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 s = u, {|tt
cE
cE
Ec '
-2287,2380
0
-4097,2380
0
0
0
0
29818,6333
0
0
0
0
0
0
0
0
0
0
0
298186,3333
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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
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
0,2000
0
0,2000
0
0
0
0
-3,3333
1
0
0
0
0
0
0
0
0
0
0
18716,6667
1
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
5833,3333
5,0000
0
5
0
0
0
0
-83,3333
0
0
1
0
0
0
0
0
0
0
0
21666,6667
0,2000
0
0,2000
0
0
0
0
-3,3333
0
0
0
1
0
0
0
0
0
0
0
18716,6667
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
,M
,N
,O
,P
,
,
l
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 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
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
'
Basis
,
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
'
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(
V' = j
&0,7713, 0,6848( = 0,7713
, , ', , J, , K, , L, , M(
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
, = j
&0,7713, 0,6848( = 0,7713 =
Universitas Sumatera Utara
666,6667, 333,3333, ∞, 633,3333, ∞, 93583,3333, 5833,3333, 4333,3333, 93583,3333,
~
e = j)f }
259033,3333, 4333,3333, 2833,3333,4783,3333,25033,3333, 21433,3333, 3433,3333
= 333,3333 = ,'
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
0
0
-4097,2380
0
1143,6190
0
0
-8302
0
0
0
0
0
0
0
0
0
0
0
1060599
0
0
8902,7620
0
-3856,3810
0
0
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
3939302,6667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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
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
0
0
0,2000
0
-0,1000
0
0
0
1
0
0
0
0
0
0
0
0
0
0
18650
0
0
1
0
-0,5000
0
0
0
0
1
0
0
0
0
0
0
0
0
0
5500
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
l
0
0
1
0
-0,5000
0
0
0
0
0
0
0
1
0
0
0
0
0
0
258700
,J
,M
,N
,O
,
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
0
0
-4097,2380
0
1143,6190
0
0
-8302
0
0
0
0
0
0
0
0
0
0
0
1060599
0
0
8902,7620
0
-3856,3810
0
0
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
3939302,6667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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 (
, ,J , ,K ,
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M(
333,3333, 166,6667, 0, 1000, 0, 1000, 300, 0, 166,6667, 18650, 5500,
~
[J = }
20000, 18650, 258700, 1000, 333,3333, 1000, 300, 166,6667,18650
VJ = j
, = j
&0,6848( = 0,6848
&0,6848( = 0,6848 =
J
e = j)f &333,3333, ∞, 333,3333, 300, ∞, 93250, 5500, 4000, 93250, 258700, 4000, 2500, 47500, 24700, 21100, 3100( = 300 = ,K
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 4 s = u, w{€{
cE
cE
Ec '
0
0
0
0
-905
0
4097,2380
-8302,0000
0
0
0
0
0
0
0
0
0
0
0
2289770,4000
0
0
0
0
595
0
-8902,7620
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
6610131,2667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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
0
0
0
0
0
0
-0,2000
0
1
0
0
0
0
0
0
0
0
0
0
18590
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
5200
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
J
0
0
0
0
0
0
-0,5000
0
0
0
0
0
0
0
0
1
0
0
0
23600
,M
,N
,O
,
,
,
,
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 4 s = u, w{€{
cE
cE
Ec '
0
0
0
0
-905
0
4097,2380
-8302,0000
0
0
0
0
0
0
0
0
0
0
0
2289770,4000
0
0
0
0
595
0
-8902,7620
-8292,1667
0
0
0
0
0
0
0
0
0
0
0
6610131,2667
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
M
Solusi
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 ` = &∅( maka permasalahan telah optimum, diperoleh
= 333,3333,
,K = 0, ,L = 0, ,M = 18950, ,N = 5200, ,O = 18500, ,P = 18950, ,
12200, ,
L
= 10400, ,
M
l
'
= 166,6667,
= 258400, ,
J
= 300, , = 100, ,' = 0, ,J = 100,
= 1850, ,
'
= 2200, ,
J
= 23600, ,
K
=
= 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
(branch) yaitu pada bagian A dan bagian B, terlebih dahulu mengerjakan pada bagian A.
Universitas Sumatera Utara
Lampiran 7 Pembahasan Contoh Kasus Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 1 s = t, uuuu
cE
cE
Ec '
-2287,2380
-1789,1180
-4097,2380
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
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
0
0
3
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1000
1
1
1,0000
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
800
0
0,0600
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
10
0,2000
0,2000
0,2000
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
18750
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
6000
5
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
22500
0,2000
0,2000
0,2000
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
18750
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
J
0,5000
0,5000
0,5000
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
24000
,L
,M
,N
,O
,P
,
,
,
,
l
L
M
,
3
,K
K
,
3
,J
J
,
3
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
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
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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(
V = j
&0,7713, 0,8211, 0,6848( = 0,8211
[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750, 259200, 2250, 3000, 24000, 12600, 10800, 1800, 333(
, = 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(
= 166,6667 = ,L
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 s = u, {|tt
cE
cE
Ec '
-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
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M ,N
,O
,P
,
,
,
Variabel
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
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
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
1
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
0
5833,3333
5
0
5
0
0
0
0
-83,3333
0
0
1
0
0
0
0
0
0
0
0
0
21666,6667
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
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
J
0,5000
0
0,5000
0
0
0
0
-8,3333
0
0
0
0
0
0
0
1
0
0
0
0
23916,6667
,N
,O
,P
,
,
,
,
l
L
M
,
3
,M
K
,
0
,J
J
,
3,0000
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 2 s = u, {|tt
cE
cE
Ec '
-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
7712,7620
0
8902,7620
0
0
0
0
-136838,2000
0
0
0
0
0
0
0
0
0
0
0
0
1368382
,
,'
,J
,K
,L
,M ,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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(
V' = j
&0,7713, 0,6848( = 0,7713
, , ', , J, , K, , L, , M, , N(
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
, = j
&0,7713, 0,6848( = 0,7713 =
666,6667, 333,3333, ∞, 633,3333, ∞, 93583,3333, 5833,3333, 4333,3333,93583,3333,
~
e = j)f }
259033,3333, 4333,3333, 2833,3333,47833,3333, 25033,333321433,3333,3433,3333, 333
= 333 = ,
N
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
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
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
0
0
0,2000
0
0
0
0
-3,3333
1
0
0
0
0
0
0
0
0
0
0
0
18650,0667
0
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
-1
5500,3333
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
J
0
0
0,5000
0
0
0
0
-8,3333
0
0
0
0
0
0
0
1
0
0
0
-1
23750,1667
,N
,O
,
,
,
,
L
M
,
3
,M
K
,
0
,J
J
,
0
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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 ,
` = & J (
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M,
(
333, 166,6667, 0, 1001, 0,6667, 1000, 300,3333, 0, 18650,0667, 5500,3333, 20001,6667, 18650,0667,
~
[J = }
18650,0667, 258700,3333,2000,16667, 2500,3333, 23750,1667, 12350,1667, 10550,1667,1550,1667, 0
VJ = j
, = j
&0,6848( = 0,6848
&0,6848( = 0,6848 =
J
333,6667, ∞, 333,3333, 300,3333, ∞, 93250,3333, 5500,3333, 4000,3333, 93250,3333,
~
e = j)f }
258700,3333, 4000,3333,2500,3333, 47500,3333, 24700,3333, 21100,3333, 3100,3333, ∞
= 300,3333 = ,K
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 4 s = u, w{€{
cE
cE
Ec '
0
0
0
0
0
0
4097,2380
-38468,6667
0
0
0
0
0
0
0
0
0
0
0
-1810
2290373,7333
0
0
0
0
0
0
-8902,7620
11541,1667
0
0
0
0
0
0
0
0
0
0
0
1190
6610527,9333
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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
0
0
0
0
0
0
-0,2000
0
1
0
0
0
0
0
0
0
0
0
0
0
18590
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
0
5200
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
J
0
0
0
0
0
0
-0,5000
0
0
0
0
0
0
0
0
1
0
0
0
0
23600
,'
,M
,N
,O
,
,
,
,
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 3 s = u, xxt•
cE
cE
Ec '
0
0
0
0
0
0
4097,2380
-38468,6667
0
0
0
0
0
0
0
0
0
0
0
-1810
2290373,7333
0
0
0
0
0
0
-8902,7620
11541,1667
0
0
0
0
0
0
0
0
0
0
0
1190
6610527,9333
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
,
M
N
Solusi
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 ` = &∅( maka permasalahan telah optimum, diperoleh
= 333,
99, ,K = 0, ,L = 0, ,M = 18590, ,N = 5200, ,O = 18500, ,P = 18950, ,
12200, ,
L
= 10400, ,
M
= 1400, ,
N
'
l
= 166,6667,
= 258400, ,
J
= 300,333, , = 100, ,' = 0,6667, ,J =
= 1850, ,
'
= 2200, ,
J
= 23600, ,
K
=
= 0 dengan $ = 2290373,7333dan $' = 6610527,9333, dikarenakan hasil optimum pada
bagian A belum diproleh bilangan bulat, maka kita lanjut mengerjakan pada bagian B.
Universitas Sumatera Utara
Lampiran 8 Pembahasan Contoh Kasus Program Linier Dua Kriteria dengan Menerapkan Percabangan (Branch)
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria Iterasi 1 s = t, uuuu
cE
cE
Ec '
-2287,2380
-1789,1180
-4097,2380
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Variabel
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
0
0
3
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1000
1
1
1,0000
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
800
0
0,0600
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
10
0,2000
0,2000
0,2000
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
18750
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
6000
5
5
5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
22500
0,2000
0,2000
0,2000
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
18750
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
J
0,5000
0,5000
0,5000
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
24000
,L
,M
,N
,O
,P
,
,
,
,
l
L
M
,
3
,K
K
,
3
,J
J
,
3
,'
'
,
J
,
l
,
'
Basis
N
Solusi
Universitas Sumatera Utara
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
0
7712,7620
8210,2920
8902,7620
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
Ec '
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
Solusi
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(
V = j
&0,7713, 0,8211, 0,6848( = 0,8211
[ = &0, 0, 0, 2500, 1000, 1000, 800, 10, 18750, 6000, 22500, 18750, 259200, 2250, 3000, 24000, 12600, 10800, 1800, 333(
, = 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(
= 166,6667 = ,L ,
Universitas Sumatera Utara
= 333,3333,
setelah diselasaikan pada bagian B diperoleh penyelesaian yang sama dengan hasil pada solusi optimum awal yaitu
'
= 166,6667,
258400,
,
J
= 300, , = 100, ,' = 0, ,J = 100, ,K = 0, ,L = 0, ,M = 18950, ,N = 5200, ,O = 18500, ,P = 18950, ,
= 1850,
,
'
= 2200,
,
J
= 23600,
,
K
= 12200,
,
L
= 10400,
,
M
= 1400,
,
N
= −0,6667
l
=
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
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
0
0
0
0
0
0
-0,2000
0
1
0
0
0
0
0
0
0
0
0
0
0
0
18590
,'
,M
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 1 s = u, w{€{
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,N
J
l
'
,
J
,
K
,
L
,
M
,
N
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
5200
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
0
0
0
0
0
0
0
-0,6667
0
0
0
0
0
0
0
0
0
0
0
0
1
-0,3333
,O
,
,
,
,
,
,
,
,G
Universitas Sumatera Utara
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
, = j
&0,6648, 0,2308, 0,3967( = 0,6848
&0,6648, 0,2308, 0,3967( = 0,6848 = ,K
−33,3333, ∞, −33, 300,3333, ∞, −92950, −5200, −3700, −92950,
e = j)f }
~ = 300,3333 =
−258400, −3700, −2200, −47200, −24400, −20800, −2800, ∞, ∞
J
Universitas Sumatera Utara
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Iterasi 2
s = u, w{€{
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
N
,G
Solusi
Variabel
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
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
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
0
0
1
0
0
0
0
-16,6667
0
1
0
0
0
0
0
0
0
0
0
-1
0
5500,3333
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
J
0
0
0,5000
0
0
0
0
-8,3334
0
0
0
0
0
0
0
1
0
0
0
-0,5000
0
23750,1667
,N
,O
,
,
,
,
L
M
,
3
,M
K
,
0
,J
J
,
0
,'
'
,
J
,
l
,
'
Basis
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 2 s = u, w{€{
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
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
,G
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
, = j
&0,8211, 0,7713( = 0,8211
&0,8211, 0,7713( = 0,8211 = ,L
Universitas Sumatera Utara
−20,0200, −0,0200, −9999, −18,02000, 10, −5595, 0088, −330, 0193, −240,0195, −5569, 0088,
~ = 0,4999 = ,G
e = j)f }
−15521,9890, 240,0195, −150, 0197, −2850, 0143, −1482, 0170, −1266, 0175, −186,0196, ∞, 0,4999
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Iterasi 3
s = u, {|tt
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
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
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
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
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
-1
-24,9988
5508,6654
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
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
-1
-24,9988
2508,6654
,N
,O
,
,
,
'
L
M
,
3
,M
K
,
0
,J
J
,
0
,'
'
,
J
,
l
,
'
Basis
N
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 3 s = u, {|tt
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
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
,L
Keterangan:
g = &, , ,' , ,J , ,K ,
` = &, N (
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M,
, ,L (
333, 158, 0, 1025,9962, 17,3309,999,9999,308,6654, 18651,7331, 5508,6654, 20043,3270, 18651,7331,
~
[J = }
258708,6654, 2004,3327, 2508,6654, 23754,3327, 12354,3327, 10554,3327, 1554,3327, 0, 0
VJ = j
, = j
& 0,7713( = 0,7713
& 0,7713( = 0,7713 = ,
N
Universitas Sumatera Utara
−341,9987, 8,6654, ∞, −308,6654, ∞, −93258,6653, −5508,6654, −4008,6654, −93258,6653, −258708,6654
~
e = j)f }
, −4008,6655, −2508,6654, −47508,6655, −24708,6655, −21108,6655, −3108,6655, 333, ∞
= 8,6654 = ,'
Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A Iterasi 4
s = u, xxt•
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
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
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
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
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
-1
-24,9988
5508,6654
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
,M
,N
,O
,
L
M
,
3
,J
K
,
0
N
J
,
0
,
'
,
J
,
l
,
'
Basis
N
Universitas Sumatera Utara
Lanjutan Tabel Simpleks Contoh Kasus Program Linier Dua Kriteria dengan Penambahan Kendala Gomory 1 pada Bagian A
Iterasi 4 s = u, {|tt
cE
cE
Ec '
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
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
,
,'
,J
,K
,L
,M
,N
,O
,P
,
,
,
,G
Solusi
Variabel
'
Basis
,
,
,
,
,
,
J
l
'
,
J
,
K
,
L
,
M
,
N
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
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
,L
Universitas Sumatera Utara
Keterangan:
g = &, , ,' , ,J , ,K ,
` = &, N (
' , ,M , ,N , ,O , ,P , , l , ,
, , ', , J, , K, , L, , M,
, ,L (
333, 158, 0, 1025,9962, 0,999,9999,308,6654, 18651,7331, 5508,6654, 20043,3270, 18651,7331,
~
[J = }
258708,6654, 2004,3327, 2508,6654, 23754,3327, 12354,3327, 10554,3327, 1554,3327, 17,3309, 0
karena ` = &∅( maka permasalahan telah optimum, diperoleh
= 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, ,
2004,3327, ,
'
= 2508,6654, ,
J
= 23754,3327, ,
K
= 12354,3327, ,
L
= 10554,3327, ,
M
l
= 258708,6654, ,
= 1554,3327, ,
M
= 17,3309, ,
=
N
=
1554,3327, ,G = 0 dengan $ = 1044929 dan $' = 3868323 terlihat hasil optimum pada bagian A setelah penambahan kendala
gomory sudah diperoleh bilangan bulat, maka permasalahannya diberhentikan.
Universitas Sumatera Utara