Contoh Soal: 1. Peta Kendali X-bar dan Range (R) PT XYZ adalah suatu perusahaan pembuatan suatu produk industri. Ditetapkan spesifikasi adalah : 2,40 ± 0,05 mm. Untuk mengetahui kemampuan proses dan mengendalikan proses itu bagian pengendalian pen gendalian PT XYZ telah melakukan pengukuran p engukuran terhadap 20 sampel. Masing-masing berukuran 5 unit (n = 5 ) Hasil Pengukuran
Sampel X1
X2
X3
X4
X5
1
2.38
2.45
2.40
2.35
2.42
2
2.39
2.40
2.43
2.34
2.40
3
2.40
2.37
2.36
2.36
2.35
4
2.39
2.35
2.37
2.39
2.38
5
2.38
2.42
2.39
2.35
2.41
6
2.41
2.38
2.37
2.42
2.42
7
2.36
2.38
2.35
2.38
2.37
8
2.39
2.39
2.36
2.41
2.36
9
2.35
2.38
2.37
2.37
2.39
10
2.43
2.39
2.36
2.42
2.37
11
2.39
2.36
2.42
2.39
2.36
12
2.38
2.35
2.35
2.35
2.39
13
2.42
2.37
2.40
2.43
2.41
14
2.36
2.38
2.38
2.36
2.36
15
2.45
2.43
2.41
2.45
2.45
16
2.36
2.42
2.42
2.43
2.37
17
2.38
2.43
2.37
2.39
2.38
18
2.40
2.35
2.39
2.35
2.35
19
2.39
2.45
2.44
2.38
2.37
20
2.35
2.41
2.45
2.47
2.35
Perhitungan :
Sampel
Perhitungan Rata-rata
Range
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2.40 2.39 2.37 2.38 2.39 2.40 2.37 2.38 2.37 2.39 2.38 2.36 2.41 2.37 2.44 2.40 2.39 2.37 2.41 2.41
0.10 0.09 0.05 0.04 0.07 0.05 0.03 0.05 0.04 0.07 0.06 0.04 0.06 0.02 0.04 0.07 0.06 0.05 0.08 0.12
Jumlah
47.78
1.19
Rata-rata
2.39
0.06
Untuk n =5: A2 = 0,577 D4 = 2,114 D3 = 0
X=
( X) 47,78 = = 2,39 k 20
R=
( R) 1,19 = = 0, 06 k 20
Peta kendali X :
CL = X = 2,39 UCL = X + (A2 x R) = 2,39 + (0,577 x 0,06) = 2,42 LCL = X – (A2 x R) = 2,39 – (0,577 x 0,06) = 2,36 Peta kendali R :
CL = R = 0,06 UCL = D4 x R = 2,114 x 0,06 = 0,12 LCL = D3 x R = 0 x 0,06 = 0
RATA-RATA 2.46 2.44 2.42
Rata-rata
2.4
UCL
2.38
CL
2.36
LCL
2.34 2.32 2.3 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
RANGE 0.14 0.12 0.1
Range UCL
0.08
CL 0.06
LCL
0.04 0.02 0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Dari hasil Plotting diketahui bahwa ada data yang out of control yaitu data sampel 15, maka data pada sampel tersebut dihilangkan dan dilakukan revisi dengan menghitung ulang CL, UCL dan LCL tanpa menggunakan data-data yang out of control.
Berikut perhitungan X bar dan Range setelah data dengan nilai X bar yang out of control dihilangkan:
Sampel
Perhitungan Rata-rata
Range
1
2.40
0.10
2
2.39
0.09
3
2.37
0.05
4
2.38
0.04
5
2.39
0.07
6
2.40
0.05
7
2.37
0.03
8
2.38
0.05
9
2.37
0.04
10
2.39
0.07
11
2.38
0.06
12
2.36
0.04
13
2.41
0.06
14
2.37
0.02
16
2.40
0.07
17
2.39
0.06
18
2.37
0.05
19
2.41
0.08
20
2.41
0.12
Jumlah
45.34
1.15
Rata-rata
2.386
0.0605
X=
( X) 45,34 = = 2,386 k 19
R=
( R) 1,15 = = 0, 0605 k 19
Peta kendali X :
CL = X = 2,386 UCL = X + (A2 x R) = 2,386 + (0,577 x 0,0605) = 2,4209 LCL = X - (A2 x R) = 2,386 - (0,577 x 0,0605) = 2,3511 Peta kendali R :
CL = R = 0,0605 UCL = D4 x R = 2,114 x 0,0605 = 0,1280 LCL = D3 x R = 0 x 0,06 = 0
RATA-RATA 2.44 2.42 2.4 Rata-rata
2.38
CL 2.36
UCL LCL
2.34 2.32 2.3 1
2
3
4
5
6
7
8
9
10 11 12 13 14 16 17 18 19 20
RANGE 0.14 0.12 0.1 Range 0.08
CL UCL
0.06 LCL 0.04 0.02 0 1
2
3
4
5
6
7
8
9
10 11 12 13 14 16 17 18 19 20
2. Peta Kendali X-bar dan Standar Deviasi (S) Misalnya, Perusahaan ASA melakukan monitoring terhadap produk yang dihasilkan oleh bagian produksi selama 25 hari. Setiap hari diambil 5 (harusnya n > 10, namun untuk kepentingan kemudahan pembahasan dimisalkan n = 5) produk untuk diukur panjangnya. Selengkapnya tersaji dalam tabel berikut:
Jumlah Observasi
Hasil Pengukuran
x
S
1
20, 22, 21, 23, 22
21,60
1,14
2
19, 18, 22, 20, 20
19,80
1,48
3
25, 18, 20, 17, 22
20,40
3,21
4
20, 21, 22, 21, 21
21,00
0,71
5
19, 24, 23, 22, 20
21,00
2,07
6
22, 20, 18, 18, 19
19,40
1,67
7
18, 20, 19, 18, 20
19,00
1,00
8
20, 18, 23, 20, 21
20,40
1,82
9
21, 20, 24, 23, 22
22,00
1,58
10
21, 19, 20, 20, 20
20,00
0,71
11
20, 20, 23, 22, 20
21,00
1,41
12
22, 21, 20, 22, 23
21,60
1,14
13
19, 22, 19, 18, 19
19,40
1,52
14
20, 21, 22, 21, 22
21,20
0,84
15
20, 24, 24, 21, 23
22,80
1,64
16
21, 20, 24, 20, 21
21,20
1,64
17
20, 18, 18, 20, 20
19,20
1,10
18
20, 24, 23, 23, 23
22,40
1,52
19
20, 19, 23, 20, 19
20,20
1,64
20
22, 21, 21, 24, 22
22,00
1,22
21
23, 22, 22, 20, 22
21,80
1,10
22
21, 18, 18, 17, 19
18,60
1.52
23
21, 24, 24, 23, 23
23,00
1,22
24
20, 22, 21, 21, 20
20,80
0,84
25
19, 20, 21, 21, 22
20,60
1,14
Untuk n = 5: A3 = 1,427 B4 = 2,089 B3 = 0 Peta Kendali X bar
X bar = ∑2 =
=
2,4 2
= 20,186
CL = X bar = 20,186 UCL X bar = X bar + (A3. S bar) = 20,186 + (1,427) (1,395) = 22,807 LCL X bar = X bar - (A3. S bar) = 20,186 + (1,427) (1,395) = 18,825
Jumlah Observasi 1
x
S
S bar
X bar
21.6
1.14
1.395
20.816
UCL X bar 22.807
2
19.8
1.48
1.395
20.816
22.807
18.825
3
20.4
3.21
1.395
20.816
22.807
18.825
4
21
0.71
1.395
20.816
22.807
18.825
5
21
2.07
1.395
20.816
22.807
18.825
LCL X bar
18.825
6
19.4
1.67
1.395
20.816
22.807
18.825
7
19
1
1.395
20.816
22.807
18.825
8
20.4
1.82
1.395
20.816
22.807
18.825
9
22
1.58
1.395
20.816
22.807
18.825
10
20
0.71
1.395
20.816
22.807
18.825
11
21
1.41
1.395
20.816
22.807
18.825
12
21.6
1.14
1.395
20.816
22.807
18.825
13
19.4
1.52
1.395
20.816
22.807
18.825
14
21.2
0.84
1.395
20.816
22.807
18.825
15
22.8
1.64
1.395
20.816
22.807
18.825
16
21.2
1.64
1.395
20.816
22.807
18.825
17
19.2
1.1
1.395
20.816
22.807
18.825
18
22.4
1.52
1.395
20.816
22.807
18.825
19
20.2
1.64
1.395
20.816
22.807
18.825
20
22
1.22
1.395
20.816
22.807
18.825
21
21.8
1.1
1.395
20.816
22.807
18.825
22
18.6
1.52
1.395
20.816
22.807
18.825
23
23
1.22
1.395
20.816
22.807
18.825
24
20.8
0.84
1.395
20.816
22.807
18.825
25
20.6
1.14
1.395
20.816
22.807
18.825
Terdapat data yang out of control, yaitu data ke-22 dan data ke-23 Peta Kendali X bar 25
20
x
15
X bar UCL X bar 10
LCL X bar
5
0 1 2
3 4
5
6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Peta Kendali S bar
4,88
2
S bar = ∑2 = =
= 1,395
CL S bar = S bar = 1,395 UCL S bar = B4 . S bar = 2,089(1,395) = 2,915 LCL = B3 . S bar = 0 (1,395) = 0 Jumlah Observasi
S
S bar
UCL S bar
LCL S bar
1
1.14
1.395
2.915
0
2
1.48
1.395
2.915
0
3
3.21
1.395
2.915
0
4
0.71
1.395
2.915
0
5
2.07
1.395
2.915
0
6
1.67
1.395
2.915
0
7
1
1.395
2.915
0
8
1.82
1.395
2.915
0
9
1.58
1.395
2.915
0
10
0.71
1.395
2.915
0
11
1.41
1.395
2.915
0
12
1.14
1.395
2.915
0
13
1.52
1.395
2.915
0
14
0.84
1.395
2.915
0
15
1.64
1.395
2.915
0
16
1.64
1.395
2.915
0
17
1.1
1.395
2.915
0
18
1.52
1.395
2.915
0
19
1.64
1.395
2.915
0
20
1.22
1.395
2.915
0
21
1.1
1.395
2.915
0
22
1.52
1.395
2.915
0
23
1.22
1.395
2.915
0
24
0.84
1.395
2.915
0
25
1.14
1.395
2.915
0
Terdapat data yang out of control, yaitu data ke-3
Peta Kendali S bar 3.5 3 2.5
S
2
S bar UCL S bar
1.5
LCL S bar 1 0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Revisi X bar dan S bar Jumlah Observasi 1
x
S
S bar
X bar
20.836
UCL X bar 22.713
LCL X bar 18.960
UCL S bar 2.747
LCL S bar 0
21.6
1.14
1.315
2
19.8
1.48
1.315
20.836
22.713
18.960
2.747
0
4
21
0.71
1.315
20.836
22.713
18.960
2.747
0
5
21
2.07
1.315
20.836
22.713
18.960
2.747
0
6
19.4
1.67
1.315
20.836
22.713
18.960
2.747
0
7
19
1
1.315
20.836
22.713
18.960
2.747
0
8
20.4
1.82
1.315
20.836
22.713
18.960
2.747
0
9
22
1.58
1.315
20.836
22.713
18.960
2.747
0
10
20
0.71
1.315
20.836
22.713
18.960
2.747
0
11
21
1.41
1.315
20.836
22.713
18.960
2.747
0
12
21.6
1.14
1.315
20.836
22.713
18.960
2.747
0
13
19.4
1.52
1.315
20.836
22.713
18.960
2.747
0
14
21.2
0.84
1.315
20.836
22.713
18.960
2.747
0
15
22.8
1.64
1.315
20.836
22.713
18.960
2.747
0
16
21.2
1.64
1.315
20.836
22.713
18.960
2.747
0
17
19.2
1.1
1.315
20.836
22.713
18.960
2.747
0
18
22.4
1.52
1.315
20.836
22.713
18.960
2.747
0
19
20.2
1.64
1.315
20.836
22.713
18.960
2.747
0
20
22
1.22
1.315
20.836
22.713
18.960
2.747
0
21
21.8
1.1
1.315
20.836
22.713
18.960
2.747
0
24
20.8
0.84
1.315
20.836
22.713
18.960
2.747
0
25
20.6
1.14
1.315
20.836
22.713
18.960
2.747
0
Peta Kendali X bar
CL = X bar = 20,836 UCL X bar = X bar + (A3. S bar) = 20,836 + (1,427) (1,315) = 22,713 LCL X bar = X bar - (A3. S bar) = 20,836 - (1,427) (1,315) = 18,960 Terdapat data yang out of control, yaitu data ke-15
Peta Kendali X bar 25 20
x X bar
15
UCL X bar 10
LCL X bar
5 0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22
Peta Kendali S bar
CL = S bar = 1,315 UCL S bar = B4 . S bar = 2,089(1,315) = 2,747 LCL = B3 . S bar = 0 (1,315) = 0 Tidak ada data yang out of control
3
Peta Kendali S bar
2.5 S 2
S bar
1.5
UCL S bar
1
LCL S bar
0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Revisi X bar dan S bar Jumlah Observasi 1
x
S
S bar
X bar
20.743
UCL X bar 22.597
LCL X bar 18.888
UCL S bar 2.715
21.6
1.14
1.300
2
19.8
1.48
1.300
20.743
22.597
18.888
2.715
0
4
21
0.71
1.300
20.743
22.597
18.888
2.715
0
5
21
2.07
1.300
20.743
22.597
18.888
2.715
0
6
19.4
1.67
1.300
20.743
22.597
18.888
2.715
0
7
19
1
1.300
20.743
22.597
18.888
2.715
0
8
20.4
1.82
1.300
20.743
22.597
18.888
2.715
0
9
22
1.58
1.300
20.743
22.597
18.888
2.715
0
10
20
0.71
1.300
20.743
22.597
18.888
2.715
0
11
21
1.41
1.300
20.743
22.597
18.888
2.715
0
12
21.6
1.14
1.300
20.743
22.597
18.888
2.715
0
13
19.4
1.52
1.300
20.743
22.597
18.888
2.715
0
14
21.2
0.84
1.300
20.743
22.597
18.888
2.715
0
16
21.2
1.64
1.300
20.743
22.597
18.888
2.715
0
17
19.2
1.1
1.300
20.743
22.597
18.888
2.715
0
18
22.4
1.52
1.300
20.743
22.597
18.888
2.715
0
19
20.2
1.64
1.300
20.743
22.597
18.888
2.715
0
20
22
1.22
1.300
20.743
22.597
18.888
2.715
0
21
21.8
1.1
1.300
20.743
22.597
18.888
2.715
0
24
20.8
0.84
1.300
20.743
22.597
18.888
2.715
0
25
20.6
1.14
1.300
20.743
22.597
18.888
2.715
0
LCL S bar
0
Peta Kendali X bar
CL = X bar = 20,743 UCL X bar = X bar + (A3. S bar) = 20,743 + (1,427) (1,300) = 22,597 LCL X bar = X bar - (A3. S bar) = 20,743 - (1,427) (1,300) = 18,888 Tidak terdapat data yang out of control Peta Kendali X bar 23 22
x
21
X bar
20
UCL X bar
19
LCL X bar
18 17 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21
Peta Kendali S bar
CL = S bar = 1,300 UCL S bar = B4 . S bar = 2,089(1,300) = 2,715 LCL = B3 . S bar = 0 (1,300) = 0 Tidak ada data yang out of control Peta Kendali S bar 3 2.5 S
2
S bar 1.5
UCL S bar LCL S bar
1 0.5 0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21
3. Peta Kendali Individual Moving Range Pengendalian viskositas cat, dengan data sampling sebagai berikut Batch 1
Viskositas 33.75
Moving Range
2 3 4 5 6 7 8 9 10 11 12 13
33.05 34 33.81 33.46 34.02 33.68 33.27 33.49 33.2 33.62 33 33.54
0.7 0.95 0.19 0.35 0.56 0.34 0.41 0.22 0.29 0.42 0.62 0.54
14 15 Total Rata-rata
33.12 33.84 502.85 33.52
0.42 0.72 6.73 0.48
Penyelesaian: Untuk n = 2 : D3 = 0 D4 = 3.267 d2 = 1.128 2.8 ̅ = ∑ = 33.52 = =
6.7 ̅ = ∑4 = = 0.48 = −
4
Peta Kendali CL = ̅ = 33.52 ̅
UCL = ̅ + 3 = 33.52 + 3 d2
.48
̅
.48
d2
.28
LCL = ̅ - 3 = 33.52 - 3
= 34.80
.28
= 32.24
Peta Kendali ̅
̅ = 0.48 CL = ̅ = (3.267) (0.48) = 1.52 UCL = D4 ̅ = (0) (0.48) = 0 LCL = D3
