Metode Response Surface (RSM) adalah metode yang menggabungkan teknik matematika dan statistik untuk membuat model dan menganalisis respon (Y) yang dipengaruhi oleh beberapa variabel (X) dengan tujuan mengoptimalkan respon tersebut. RSM menggunakan model polinomial orde pertama dan kedua untuk memodelkan hubungan antara respon dan variabel. Desain eksperimen seperti factorial dan Box-Behnken digunakan untuk mengestimasi koefisien model.

1. METODE RESPONSE
SURFACE (RSM)

2. 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.

3.  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)

4. 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 (first-
order model):
y= β0 + β1x1 + β2x2 + … + βkxk +ε

5. 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 + ∑ βiixi
2 + … +∑ ∑ βijxi xj+ε
i=1 i=1 i<j
Contoh :
y = β0 + β1 x1 + β2 x2 + β3 x1
2 + β4 x2
2 + β5 x1 x2

6. KURVA RESPONSE SURFACE
Sumber : Montgomery (2007)

7. • 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”

8. Models
Far away from
optimum:
first order model

9. Models
Near optimum:
second order
model

10. 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.

11. Metode Steepest Ascent
direction of
steepest ascent
contour lines of
first-order model
perpendicular
to contour line
region where
1eorder-model
has been determined

12. 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.

13. 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.

14. 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

15. 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.

16. 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.

17. 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.
•

18. 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

19. FACTORS TO CONSIDER
• CRITICAL FACTORS ARE KNOWN
• REGION OF INTEREST , WHERE FACTOR
LEVELS INFLUENCING PRODUCT IS KNOWN
• FACTORS VARY CONTINUOUSLY THROUGH-
OUT THE EXPERIMENTAL RANGE TESTED
• A MATHEMATICAL FUNCTION RELATES THE
FACTORS TO THE MEASURED RESPONSE
• THE RESPONSE DEFINED BY THE FUNCTION
IS A SMOOTH CURVE

20. 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

21. 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

22. 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

23. 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

24. 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

25. POLYNOMIAL MODELS
• POLYNOMIAL MODEL DOES A POOR
JOB OF PREDICTING RESPONSE
OUTSIDE THE REGION OF
EXPERIMENTATION

26. SIZE OF RESPONSE SURFACE MODELS
NUMBER OF
FACTORS
RUNS IN 3
LEVEL FACT.
COEF. IN
FULL QUAND
TRIALS IN
FACE-
CENTERED
TRIALS IN
BOX-
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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. THREE FACTOR FACE CENTEREDCUBE
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 *

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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”

42. 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??????