METODE RESPONSE SURFACE (RSM)
RESPONSE SURFACE METHODOLOGY (RSM) • Merupakan suatu metode gabungan antara teknik matematika dan teknik statistik yang digunakan untuk membuat model dan menganalisa suatu respon y yang dipengaruhi oleh beberapa variabel x yang tujuannya untuk mengoptimalkan respon tersebut.
Hubungan antara respon Y dan variabel bebas X adalah: Y = f(X1, X2,...., Xk) + ε dimana: Y = variabel respon Xi = variabel bebas/faktor (i = 1, 2,.., k ) ε = error Jika ekspektasi response dinotasikan dengan E(y) = f(X1, X2, .. , Xk) = ŋ, maka surface dinyatakan dengan: ŋ = f(X1, X2, …, Xk)
MODEL ORDE-PERTAMA Langkah pertama dari RSM adalah menemukan fungsi pendekatan yang tepat untuk melihat hubungan antara respon y dan faktor x melalui persamaan polinomial orde pertama (firstorder model): y= β0 + β1x1 + β2x2 + … + βkxk +ε
MODEL ORDE-KEDUA Jika hubungan tidak linier, maka fungsi polinomial dengan orde yang lebih tinggi digunakan seperti fungsi polinomial orde kedua (second-order model): k k y= β0 + ∑ βixi + ∑ βiixi2 + … +∑ ∑ βijxi xj+ε i=1
i=1
i<j
Contoh : y = β 0 + β 1 x1 + β 2 x2 + β 3 x1 2 + β 4 x2 2 + β 5 x1 x2
KURVA RESPONSE SURFACE
Sumber : Montgomery (2007)
• RSM adalah prosedur yang bertahap/berurutan • Pada titik diluar daerah optimum, bentuk surface tidak terlalu curve, sehingga yang digunakan polinomial orde-1 • Pada daerah optimum, polinomial orde-2 yg digunakan. • Analisis “climbing the hill”
Models Far away from optimum: first order model
Models
Near optimum: second order model
Metode Steepest Ascent • Steepest Ascent adalah metode bergerak secara bertahap melalui suatu jalur yang menaik, dimana nilai response meningkat untuk mencapai maksimum. • Kebalikannya : Steepest Descent minimum. • Besar tahapan (step) proporsional terhadap nilai koefisien regresi (βi). • Ukuran step dihitung oleh pembuat eksperimen berdasarkan pengetahuan proses atau pertimbangan praktis.
Metode Steepest Ascent perpendicular to contour line
direction of steepest ascent
contour lines of first-order model
region where 1eorder-model has been determined
Contoh • Seorang ahli teknik kimia ingin mengetahui kondisi proses yang dapat memaksimumkan hasil proses kimia. Variabel yang mempengaruhi proses : waktu reaksi dan temperatur. Si ahli pada saat ini mengoperasikan proses pada lama waktu 35 menit dan temperatur 155oF, yang hasilnya adalah lebih kurang 40 %. Karena sepertinya kondisi ini belum yang optimum, ia ingin mengetahui kondisi optimum dengan mengaplikasikan model orde pertama dan metode steepest ascent. Si ahli membuat range percobaan (30,40) menit untuk waktu reaksi, dan (150, 160)oF untuk suhu reaksi.
Untuk penyederhanaan, variabel waktu reaksi dan suhu dikodekan dengan interval (-1,1). Jadi, jika ξ1 adalah variabel waktu reaksi yang aktual dan ξ2 adalah variabel temperatur yang aktual, maka variabel yang dikodekan : x1 = _ξ1- 35 dan x2 = ξ2 – 155 5 5 Desain eksperimen dilakukan dengan desain 22 faktorial ditambah dengan 5 kali percobaan pada nilai tengah. Pengulangan pada nilai tengah digunakan untuk mengestimasi error dan mengecek kecukupan model orde-1. Hasilnya dapat dilihat pada tabel dibawah ini.
Variabel aktual
Variabel dikodekan
Response
ξ1
ξ2
x1
x2
y
30
150
-1
-1
39.3
30
160
-1
1
40.0
40
150
1
-1
40.9
40
160
1
1
41.5
35
155
0
0
40.3
35
155
0
0
40.5
35
155
0
0
40.7
35
155
0
0
40.2
35
155
0
0
40.6
Model orde-1 berdasarkan hasil regresi least square : y = 40.44 + 0.775 x1 + 0.325 x2
Sebelum melakukan metode steepest ascent: 1. Estimasi nilai error : σ2 = (40.32 + 40.52 + 40.72 + 40.22+40.62) – (202.32/5) 5-1 = 0.043 2. Periksa interaksi dalam model : β12 = ¼[(1x39.3)+(1x41.5)+(-1x40.0)+(-1x40.9)] = ¼ (-0.1) = - 0.025. Sum square interaksi : SS interaksi = (-0.1)2/4 = 0.0025 Hitung nilai F statistik : F = SS interaksi/ σ2 = 0.0025/0.043 = 0.058. Kesimpulan F statistik kecil, sehingga interaksi diabaikan.
3. Periksa efek kuadratik : β11 + β22 = ȳF – ȳC = 40.425 – 40.46 = -0.035 Sum square kuadratik murni : SS kuadratik = (nFnC(ȳF – ȳC )2 )/nF +nC) = 4(5)(-0.035)2 = 0.0027 9 Hitung nilai F statistik : F = SS kuadratik/ σ2 = 0.0027/0.043 = 0.063. Kesimpulan F statistik kecil, sehingga interaksi diabaikan.
Metode steepest ascent • Berdasarkan model orde-1 untuk bergerak dari nilai tengah (x1=0, x2=0) diperlukan step 0.775 unit x1 untuk setiap 0.325 unit x2. • Kemiringan jalur = 0.325/0.775. • Besarnya step Δx1 = 1 ; Δx2 = 0.325/0.775 = 0.42 • Lakukan prosedur step ascent dengan menambahkan nilai tengah dengan besarnya step sampai didapatkan response yang menurun. •
Step
Variabel dikodekan
Variabel aktual
Response
x1
x2
ξ1
ξ2
y
Origin
0
0
-1
-1
39.3
Δ
1.0
0.42
-1
1
40.0
Origin+Δ
1.0
0.42
1
-1
40.9
Origin+2Δ
2.0
160
1
1
41.5
Origin+3Δ
3.0
155
0
0
40.3
Origin+4Δ
4.0
155
0
0
40.5
Origin+5Δ
5.0
155
0
0
40.7
Origin+6Δ
6.0
155
0
0
40.2
Origin+7Δ
35
155
0
0
40.6
Origin+8Δ Origin+9Δ Origin+10Δ Origin+11Δ Origin+12Δ
FACTORS TO CONSIDER • CRITICAL FACTORS ARE KNOWN • REGION OF INTEREST , WHERE FACTOR LEVELS INFLUENCING PRODUCT IS KNOWN • FACTORS VARY CONTINUOUSLY THROUGHOUT THE EXPERIMENTAL RANGE TESTED • A MATHEMATICAL FUNCTION RELATES THE FACTORS TO THE MEASURED RESPONSE • THE RESPONSE DEFINED BY THE FUNCTION IS A SMOOTH CURVE
LIMITATIONS TO RSM • LARGE VARIATIONS IN THE FACTORS CAN BE MISLEADING (ERROR, BIAS, NO REPLICATION) • CRITICAL FACTORS MAY NOT BE CORRECTLY DEFINED OR SPECIFIED • RANGE OF LEVELS OF FACTORS TO NARROW OR TO WIDE --OPTIMUM CAN NOT BE DEFINED • LACK OF USE OF GOOD STATISTICAL PRINCIPLES • OVER-RELIANCE ON COMPUTER -- MAKE SURE THE RESULTS MAKE GOOD SENSE
POLYNOMIAL MODELS • SECOND DEGREE - ONE INDEPENDENT VARIABLE Y = bo +b1x1 + b11x12 constant term,+ linear term + quadratic term
• FOR p FACTORS, THERE WILL BE ONE CONSTANT TERM, p LINEAR TERMS p QUADRATIC TERMS AND p(P-1) CROSS PRODUCT TERMS
USES OF RSM • TO DETERMINE THE FACTOR LEVELS THAT WILL SIMULTANEOUSLY SATISFY A SET OF DESIRED SPECIFICATIONS • TO DETERMINE THE OPTIMUM COMBINATION OF FACTORS THAT YIELD A DESIRED RESPONSE AND DESCRIBES THE RESPONSE NEAR THE OPTIMUM • TO DETERMINE HOW A SPECIFIC RESPONSE IS AFFECTED BY CHANGES IN THE LEVEL OF THE FACTORS OVER THE SPECIFIED LEVELS OF INTEREST
USES -CONTINUED • TO ACHIEVE A QUANTITATIVE UNDERSTANDING OF THE SYSTEM BEHAVIOR OVER THE REGION TESTED • TO PRODUCT PRODUCT PROPERTIES THROUGHOUT THE REGION - EVEN AT FACTOR COMBINATIONS NOT ACTUALLY RUN • TO FIND CONDITIONS FOR PROCESS STABILITY = INSENSITIVE SPOT
PROCESS MODELS • Ym = fm(x1, x2, ….,xp) • Polynomials with a small number of terms are most desirable • Most process outputs are some sort of smooth function of the inputs • Second-degree polynomials are generally adequate
POLYNOMIAL MODELS • POLYNOMIAL MODEL DOES A POOR JOB OF PREDICTING RESPONSE OUTSIDE THE REGION OF EXPERIMENTATION
SIZE OF RESPONSE SURFACE MODELS NUMBER OF
RUNS IN 3
COEF. IN
TRIALS IN
TRIALS IN
FACTORS
LEVEL FACT.
FULL QUAND
FACE-
BOX-
CENTERED
BEHNKEN
2
9
6
11
11
3
27
10
17
15
4
81
16
27
27
5
243
21
45
46
6
729
28
81
54
7
2187
36
62
DESIGNS • PREDICTIONS ALWAYS HAVE SOME DEGREE OF UNCERTAINTY • SHOULD HAVE REASONABLE PREDICTION THROUGHOUT THE EXPERIMENTAL RANGE • UNIFORM PREDICTIONS ERROR IS OBTAINED BY USING A DESIGN THE FILLS OUT THE REGION OF INTEREST • THE CHOICE OF EXPERIMENTAL DESIGN IS AFFECTED BY THE SHAPE OF THE EXPERIMENTAL REGION
DESIGNS - CONTINUED • IN MOST CASES, THE REGION IS DETERMINED BY THE RANGES OF THE INDEPENDENT VARIABLE. IN THIS CASE THE REGION IS CUBICAL (IN CODED VALUES OF X) AND THE BEST DESIGN IN FACE CENTERED • IF “STANDING THE THE CENTER” AND ONE IT IS DESIRED THAT THE PRECISION OF PREDICATIONS BE INDEPENDENT OF DIRECTION FROM CENTER THEN THE REGION IS SPHERICAL AND DESIGN OF CHOICE IS BOX-BEHNKEN
DESIGNS - CONTINUED • BOX-BEHNKEN DESIGNS EXCLUDE THE CORNERS, WHERE ALL VARIABLE ARE SIMULTANEOUSLY AT THE MAXIMUM LEVELS THEREFORE BOX-BEHNKEN DESIGN PERMITS A WIDER RANGE OF INDIVIDUAL RANGES. • IF THE SHAPE OF THE EXPERIMENT IS NEITHER SPHERICAL OR CUBICAL AND HAS STRONG CONSTRAINTS - THEN THE REGION MAY BE AN IRREGULAR TETRAHEDRON AND WILL REQUIRE A SPECIAL DESIGN
FACE CENTERED CUBE FOR 3 FACTORS • TWO-LEVEL FACTORIAL • TWO FACE CENTERED POINTS FOR EACH FACTOR • THREE OR MORE CENTER POINTS • WHEN RUN IN BLOCKS, CENTER POINTS ARE RUN WITH EACH BLOCK • FACE POINTS ARE RUNS FOR WHICH ALL FACTORS EXCEPT ONE ARE AT THE MIDDLE SETTING - AND PROVIDE THE INFORMATION NEEDED TO DETERMINE CURVATURE
BLOCKING • IN LARGE SIZES, BOTH FACE-CENTERED CUBE AND BOX-BEHNKEN PERMIT BLOCKING. • DIFFERENCE (OR BIASES) IN THE LEVEL OF THE RESPONSES BETWEEN BLOCKS WITH NOT AFFECT ESTIMATES OF COEFFICIENTS NOR ESTIMATES OF THE FACTOR AND INTERACTION EFFECTS
FACE CENTERED CUBE • THE MAIN PART OF THE FACE-CENTERED CUBEDESIGN IS A TWO-LEVEL FACTORIAL, WHICH FILLS OUT A CUBIC REGION • THE FACE POINTS CONSTITUTE A SEPARATE BLOCK - SO THAT THE FIRST TWO BLOCKS, WHICH COMPRISE A TWO LEVEL FACTORIAL, CAN BE RUN FIRST. • THE FACE POINTS ARE ADDED IF SERIOUS CURVITURE IS FOUND • “PIGGY BACK” APPROACH GIVES FLEXIBILITY
FACE CENTERED CUBE • CENTER POINTS ARE NEED TO PROVIDE GOOD PREDICTORS OF CENTER OF REGION • FOR 3 OR MORE FACTORS, IT IS BEST TO USE BLOCKS -FIRST HALF-FRACTION -SECOND HALF-FRACTION -FACE POINTS
THREE FACTOR FACE CENTERED CUBE TRIAL
X1
X2
X3
BLOCK
1
-
-
-
1
2
+
-
-
2
3
-
+
-
2
4
+
+
-
1
5
-
-
+
2
6
+
-
+
1
7
-
+
+
1
8
+
+
+
2
9
-
0
0
3
10
+
0
0
3
11
0
-
0
3
12
0
+
0
3
13
0
0
-
3
14
0
0
+
3
15
0
0
0
*
16
0
0
0
*
17
0
0
0
*
BOX-BEHNKEN DESIGN • THE BOX-BEHNKEN DESIGN FILLS OUT A POLYHEDRON, APPROXIMATING A SPHERE • FOR 3 FACTORS (15 RUNS) THE DESIGN CONSIST OF THREE FOUR-RUN, TWO-LEVEL FACTORIALS IN TWO FACTORS, WITH THE THIRD FACTOR AT ITS MID-LEVEL AND THREE CENTER POINT - RUN IN THREE BLOCKS OF 10 RUNS
BOX-BEHNKEN DESIGN • FOR A 3 FACTOR EXPERIMENT, THE 15 RUNS CONSIST OF THREE FOUR-RUN, TWO-LEVEL FACTORIALS IN TWO FACTORS - WITH THE THIRD FACTOR AT ITS MID-LEVEL, AND THREE CENTER POINTS. • BOX-BEHNKEN AND FACE-CENTERED CUBIC DESIGNS ARE SUBSETS OF THE FULL THREE LEVEL FACTORIAL DESIGNS. EXCEPT FOR CENTER POINTS, THEY ARE COMPLETMENTARY FRACTIONS IN THAT NO POINT IN ONE DESIGN IS IN THE OTHER DESIGN
DESIGN CHOICE • FACE CENTERED CUBE AND BOX-BEHNKEN TAKE ABOUT THE SAME NUMBER OF EXPERIMENTS • IF TIME OR MONEY DICTATES FEWER THAT THE REQUIRED NUMBER OF INDEPENDENT VARIABLES, THEN CONSIDER -REDUCE NUMBER OF FACTORS -TRY A SIMPLEX DESIGN -CONSIDER RUNNING A TWO-LEVEL FACTORIAL DESIGN THAT IS THE FIRST TWO BLOCKS OF THE FACE-CENTERED CUBE AND COMPLETE THE LAST BLOCKS WHEN ADDITIONAL EXPERIMENTATION IS POSSIBLE
DESIGN CHOICES • UNREPLICATED RESPONSE SURFACE DESIGNS CAN DETECT EFFECTS ABOUT 1-2 TIMES EXPERIMENTAL ERROR. • A FEW RUNS MAY BE INCLUDED IN THE PROGRAM TO TEST HUNCHES, SPECIAL CASES, “POLITICAL PREFERENCES” OR STANDARD OR REFERENCE RUNS. UP TO 20% OF THE NUMBER OF RUNS AVAILABLE MAY BE USED FOR THIS PURPOSE - IF A GOOD STATISTICAL DESIGN IS AT THE HEART OF THE PROGRAM
OPERABILITY REVIEW • RUNS SHOULD BE REVIEWED FOR OPERABILITY. • RUNS THAT SET ALL THE “DRIVING FORCE” VARIABLES AT MINIMUM OR MAXIMUM VALUES MAY NOT WORK • RANDOMIZATION CAN BE ALTERED TO SCHEDULE THESE RUNS EARLY TO ALLOW FOR LATTER ADJUSTMENTS • EXPLORATORY TESTING OF POTENTIAL TROUBLESOME RUNS BEFORE EXPERIMENTATION SHOULD BE CONSIDERED
OPERABILITY REVIEW • YOU MAY FIND, PART-WAY THROUGH THE EXPERIMENT THAT SOME DESIGN POINTS WILL NOT RUN. THIS IS TRUE IS A BOUNDARY CURVE PASSES THROUGH THE EXPERIMENTAL REGION. • IF ONLY ONE OR A FEW POINTS ARE INVOLVED, THEY MAY BE MOVED TOWARDS THE CENTER, JUST ENOUGH TO BECOME OPERABLE • ALL STANDARD RESPONSE SURFACE DESIGNS ARE ROBUST AGAINST MODEST DISPLACEMENT OR A FEW DATA POINTS
AVOIDING BLUNDERS • EXECUTE EXPERIMENT WITH CARE. SMALL STATISTICAL DESIGNS ARE SUSCEPTIBLE TO ERRORS BECAUSE EVERY RUN ESTIMATES MORE THAN ONE EFFECT • RECORD RESULTS FOR ALL RUNS • PLAN FOR ANALYSIS FROM THE BEGINNING • A COMPUTER IS GENERALLY REQUIRED FOR ANALYSIS - AND REGRESSION ANALYSIS IS THE BASIS FOR MOST ANALYTICAL PROCEDURES • MAKE SURE THE RESULTS “MAKE SENSE”
TAKE-AWAYS • SURFACE RESPONSE SURFACE ANALYSIS PROVIDES A MEANS FOR OPTIMIZATION OF FORMULATION AND PROCESS • SELECTION OF VARIABLES AND VARIABLE LEVELS ARE CRITICAL • EACH DIFFERENT APPROACH HAS DIFFERENT ADVANTAGES AND DISADVANTAGES • MOST LARGE COMPANIES INSIST ON YOU USING THEIR TRAINED STATISTICIANS • BOTTOM LINE - DOES IN MAKE SENSE??????
