Quick Design Optimizer

1 Introduksi

1.1 Latar Belakang

Setelah eksperimen DOE selesai dengan beberapa response variables, peneliti perlu menemukan kombinasi faktor optimal yang menyeimbangkan semua respons sekaligus — yield maksimum + cost minimum + kualitas sensori ≥ threshold, misalnya. Solusi klasik: Derringer-Suich desirability function (Derringer & Suich, 1980; Costa, Lourenço, & Pereira, 2011).

Desirability $d_i$ adalah skala 0–1 yang men-translate setiap response ke "desirability score": $d_i = 0$ unacceptable, $d_i = 1$ ideal. Overall desirability $D$ adalah geometric mean semua $d_i$ — implicit weighting bahwa semua response harus acceptable (geometric mean sensitive ke nilai rendah).

1.2 Tujuan Modul

Modul Quick Design Optimizer di SQalytics ditujukan untuk:

Membaca fitted response surface (dari RSM Studio) atau model regresi (dari Regression Studio).
Menerapkan Derringer-Suich desirability untuk multi-response optimization.
Mendukung 3 target types: maximize, minimize, target (nominal).
Memberikan optimal factor settings dan predicted response values.
Menghasilkan desirability contour plot untuk visualisasi trade-off.
Audiens: praktisi R&D yang menyeimbangkan multi-attribute optimization.

1.3 Posisi di Antara Alternatif

Pilih Quick Design Optimizer untuk multi-response optimization setelah RSM atau full factorial. Untuk single response optimization, langsung pakai RSM Studio dengan canonical analysis. Untuk overlay plot manual, pakai Response Overlay / Sweet Spot Finder. Untuk mixture optimization, gunakan Mixture Design Explorer.

2 Metode

2.1 Dasar Teoretis

Individual desirability $d_i$ untuk tiap response $y_i$:

Maximize (yield, strength, etc.):

$$ d_i = \begin{cases} 0 & y_i < L \\ \left(\dfrac{y_i - L}{T - L}\right)^r & L \leq y_i \leq T \\ 1 & y_i > T \end{cases} $$

dengan $L$ = lower limit (unacceptable), $T$ = target (ideal), $r$ = weight exponent (default 1).

Minimize (impurity, cost, defect):

$$ d_i = \begin{cases} 1 & y_i < T \\ \left(\dfrac{U - y_i}{U - T}\right)^r & T \leq y_i \leq U \\ 0 & y_i > U \end{cases} $$

dengan $U$ = upper limit unacceptable.

Target (nominal-the-best) dengan target $T$ dan limit $L$, $U$:

$$ d_i = \begin{cases} \left(\dfrac{y_i - L}{T - L}\right)^{r_L} & L \leq y_i \leq T \\ \left(\dfrac{U - y_i}{U - T}\right)^{r_U} & T \leq y_i \leq U \\ 0 & \text{otherwise} \end{cases} $$

Overall desirability (Derringer & Suich, 1980):

$$ \boxed{\, D = \left(\prod_{i=1}^{m} d_i^{w_i}\right)^{1/\sum w_i} \,} $$

dengan $w_i$ weight masing-masing response (default $w_i = 1$ → geometric mean unweighted).

Property kunci: jika salah satu $d_i = 0$, maka $D = 0$ (semua response harus acceptable). Inilah yang membuat desirability lebih konservatif daripada weighted average.

Optimization algorithm: maximize $D$ atas factor levels via Nelder-Mead simplex atau gradient descent. Modul SQalytics pakai global optimizer untuk avoid local maxima.

Confidence interval optimal via bootstrap (1000 iterations) atau delta method dari covariance matrix coefficient model.

2.2 Persamaan Inti

$d_i$ maximize: $((y_i - L)/(T - L))^r$ untuk $L \leq y_i \leq T$

$d_i$ minimize: $((U - y_i)/(U - T))^r$ untuk $T \leq y_i \leq U$

Overall $D$: $(\prod d_i^{w_i})^{1/\sum w_i}$

Optimization: $\arg\max_x D(x)$ atas factor space

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
Response model fits well ($R^2 > 0.9$)	Optimum bias	Cek $R^2$ + diagnostics pada modul RSM
Limits $L$, $T$, $U$ realistic	Optimum tidak feasible	Domain expert review
Weights $w_i$ sesuai prioritas	Wrong trade-off	Sensitivity analysis dengan multiple weight
Factor space dalam range design	Extrapolasi bias	Modul flag bila optimum di edge
Independent responses (atau correlation captured by model)	Optimal mismatch real test	Confirmatory run wajib

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka Quick Design Optimizer dari domain Desain Eksperimen.
Pilih source: Use model dari RSM Studio, Upload coefficients, atau Manual entry.
Daftarkan responses: Nama (mis. Yield, Cost, Color_dE); Type: maximize, minimize, target; Limits: $L$, $T$, $U$; Weight $w_i$; Exponent $r$ (default 1).
Pilih optimizer: Nelder-Mead, Gradient descent, Genetic algorithm.
Klik Run Optimization.
Tinjau hasil: Tab Optimal Settings — factor levels + predicted responses; Tab Desirability Surface — 2D/3D contour plot $D$ atas faktor; Tab Sensitivity — robustness optimum ke perturbasi; Tab Confirmatory Plan — design replikasi optimal.

3.2 Template Tabel Input + Contoh Data Sintetis

Setup multi-response optimization (ekstraksi flavonoid):

Response	Type	$L$	$T$	$U$	$w$
Yield_mg_per_g	maximize	10	25	—	2
Time_min	minimize	—	20	60	1
Ethanol_pct	minimize	—	50	90	1

3.3 Contoh Luaran

Optimal settings:

Factor	Optimal Value	Code Level
Temperature (°C)	73	+0.53
Time (min)	28	−0.13
Solvent ratio	15	+0.33
Ethanol (%)	62	−0.40

Predicted responses:

Response	Predicted	$d_i$	$w_i$
Yield	23.5 mg/g	0.90	2
Time	28 min	0.80	1
Ethanol	62%	0.70	1

Overall desirability:

$$ D = (0.90^2 \cdot 0.80 \cdot 0.70)^{1/4} = (0.4536)^{0.25} = 0.821 $$

Kesimpulan ringkas: "Quick optimizer menemukan optimal combination: T = 73 °C, t = 28 min, ratio = 15 mL/g, ethanol = 62%. Overall desirability $D$ = 0.821 (excellent; 0.8+ = highly desirable per Derringer-Suich 1980). Trade-off realistis: yield 23.5 mg/g (90% from max ideal), waktu 28 min (acceptable), ethanol relatif rendah 62% (cost-saving). Sensitivity analysis menunjukkan optimum robust ke perturbasi ±5% faktor (D tetap > 0.7). Confirmatory plan: jalankan 3 replikasi pada kombinasi optimal di laboratorium; bila yield observasi within 95% PI dari model, optimum divalidasi."

Derringer-Suich desirability function untuk multi-response optimization: kurva individual d_i tiga responses dan kontur overall D dengan star marker optimal size 180 dibedakan dari marker biasa size 8 — Gambar 1. Analisis desirability Derringer-Suich untuk optimasi multi-response ekstraksi flavonoid (3 responses, data sintetis). **(a)** Kurva individual desirability $d_i$ untuk Yield (maximize, $L=10$, $T=25$, $w=2$ — biru), Time (minimize, $T=20$, $U=60$, $w=1$ — oranye), Ethanol (minimize, $T=50$, $U=90$, $w=1$ — hijau); sumbu-x = nilai response aktual, sumbu-y = $d_i$ [0–1]; fungsi piecewise linear menunjukkan transisi dari unacceptable ke ideal. *Legend* faktor di lower-right. **(b)** Contour plot overall desirability $D$ atas factor space (T vs Time, dengan Ratio = 15 dan Ethanol = 62% held constant): iso-$D$ contours dari 0.4 hingga 0.9; *marker* star "star" (size 180) menandai titik optimal $D = 0.821$ di (73 °C, 28 min) — dibedakan jelas dari *marker* observasi biasa (size 8); *label* "$D = 0.821$" dan "(rank #1)" ditempatkan di **kanan bar** dengan clear margin; **optimal combo callout** "T = 73 °C, t = 28 min, D = 0.821" di **bottom dalam navy bbox**; *legend* faktor di lower-right. Referensi: Derringer & Suich (1980); Costa et al. (2011).

4 Kesimpulan

4.1 Relevansi Real-World

Multi-attribute formulasi pangan — yield + cost + sensory + nutrition balance.
Process scale-up — productivity + energy cost + waste minimization.
Pharmaceutical excipient selection — disolusi + stability + manufacturability.
Six Sigma DMAIC Improve phase — quality + cost.

4.2 Where to Go from Here

RSM Studio — fit quadratic model untuk sumber desirability.
Response Overlay / Sweet Spot Finder — alternative visualisasi.
Compare Many Groups — uji statistik confirmatory di optimum.

⚙ Troubleshooting Cepat

$D = 0$. Salah satu $d_i = 0$ (response di luar limit). Cek limits realistis.

Optimal di edge factor space. Extrapolasi — extend design atau confirm dengan domain knowledge.

Multiple local maxima. Pakai global optimizer (genetic algorithm).

Weight $w_i$ tidak yakin. Sensitivity analysis dengan multiple weight scenarios.

i Riwayat Revisi

Tanggal	Revisi	Penulis
2026-05-12	Migrasi MD v2 → HTML final dengan figure publikasi + caption Elsevier-style (W2 batch 12 Mei)	Claude
2026-05-12	Draft v2 publikasi (KaTeX Derringer-Suich + overall $D$ + APA Derringer-Suich/Costa/Myers)	Claude

4 Referensi

Derringer, G., & Suich, R. (1980). Simultaneous optimization of several response variables. Journal of Quality Technology, 12(4), 214–219. https://doi.org/10.1080/00224065.1980.11980968
Costa, N. R., Lourenço, J., & Pereira, Z. L. (2011). Desirability function approach: A review and performance evaluation in adverse conditions. Chemometrics and Intelligent Laboratory Systems, 107(2), 234–244. https://doi.org/10.1016/j.chemolab.2011.04.004
Myers, R. H., Montgomery, D. C., & Anderson-Cook, C. M. (2016). Response surface methodology: Process and product optimization using designed experiments (4th ed.). John Wiley & Sons.
Harrington, E. C. (1965). The desirability function. Industrial Quality Control, 21(10), 494–498.
Box, G. E. P., & Draper, N. R. (2007). Response surfaces, mixtures, and ridge analyses (2nd ed.). John Wiley & Sons. https://doi.org/10.1002/9780470181812