Mixture Design Explorer

1 Introduksi

1.1 Latar Belakang

Mixture experiments adalah keluarga DOE khusus di mana proportions of components sum to 1 (atau 100%) — semua komponen adalah fraksi dari total massa. Contoh: formulasi cookie (tepung + gula + lemak + telur = 100%), blending wine (4 varieties = 100%), formulasi cat (5 pigmen = 100%). Karena constraint $\sum x_i = 1$, faktor mixture tidak independent — ANOVA/RSM klasik tidak applicable; perlu simplex design (Scheffé, 1958; Cornell, 2002).

Tiga design utama: simplex-lattice ($\{q, m\}$ design untuk $q$ komponen + $m$ derajat polinomial), simplex-centroid (semua kombinasi pure + pair + triplet centroid + overall centroid), dan augmented simplex (tambah interior points untuk model orde lebih tinggi). Modul ini memandu konstruksi design + fit Scheffé canonical polynomial.

1.2 Tujuan Modul

Modul Mixture Design Explorer di SQalytics ditujukan untuk:

Membangun simplex-lattice $\{q, m\}$ ($q$ komponen, $m$ derajat).
Membangun simplex-centroid design dengan optional axial + center points.
Mendukung constrained mixture ($L_i \leq x_i \leq U_i$) — pseudo-component coordinate.
Memfit Scheffé canonical polynomial (linear, quadratic, special cubic).
Visualisasi ternary plot contour ($q = 3$).
Audiens: praktisi R&D formulasi pangan/coating/farmasi.

1.3 Posisi di Antara Alternatif

Pilih Mixture Design Explorer ketika components sum to constant. Untuk independent process variables, pakai Factorial Design Builder atau RSM Studio. Untuk mixture + process variables combined, pakai augmented design (advanced, akan datang). Untuk single-factor multi-level, pakai Taguchi Quick Design.

2 Metode

2.1 Dasar Teoretis

Constraint mixture untuk $q$ komponen:

$$ \sum_{i=1}^{q} x_i = 1, \quad 0 \leq x_i \leq 1 $$

Geometric: titik dalam $(q-1)$-simplex (segitiga untuk $q=3$, tetrahedron untuk $q=4$, dst).

Simplex-lattice $\{q, m\}$ design — semua titik di mana setiap $x_i \in \{0, 1/m, 2/m, ..., 1\}$. Jumlah titik:

$$ N = \binom{q + m - 1}{m} $$

Contoh $\{3, 2\}$: 6 titik = 3 pure ($x_i = 1$) + 3 binary midpoint ($x_i = x_j = 0.5$).

Simplex-centroid design ($q$ komponen):

$$ N = 2^q - 1 $$

Termasuk: $q$ pure + $\binom{q}{2}$ binary midpoint + $\binom{q}{3}$ ternary centroid + ... + 1 overall centroid (semua $x_i = 1/q$).

Scheffé canonical polynomial (Scheffé, 1958) — model tanpa intercept karena $\sum x_i = 1$:

Linear (only main effects):

$$ \eta = \sum_{i=1}^{q} \beta_i x_i $$

Quadratic (Scheffé):

$$ \boxed{\, \eta = \sum_{i=1}^{q} \beta_i x_i + \sum_{i < j} \beta_{ij} x_i x_j \,} $$

Koefisien $\beta_i$ = predicted response pada pure component $i$. Koefisien $\beta_{ij}$ menangkap synergistic ($\beta_{ij} > 0$) atau antagonistic ($\beta_{ij} < 0$) interaction.

Special cubic (3-way interaction):

$$ \eta = \sum_i \beta_i x_i + \sum_{iPseudo-component untuk constrained mixture ($L_i \leq x_i \leq U_i$): transformasi ke pseudo-component $x'_i = (x_i - L_i)/(1 - \sum L)$ dengan $\sum x'_i = 1$. Design dibangun di pseudo-space, lalu di-decode.

2.2 Persamaan Inti

Mixture constraint: $\sum x_i = 1$

Simplex-lattice $\{q, m\}$ size: $N = \binom{q+m-1}{m}$

Simplex-centroid size: $N = 2^q - 1$

Scheffé quadratic: $\eta = \sum \beta_i x_i + \sum_{i<j} \beta_{ij} x_i x_j$

Pseudo-component: $x'_i = (x_i - L_i) / (1 - \sum L)$

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
$\sum x_i = 1$ exactly	Constraint violated	Modul auto-normalize
$q \leq 6$ untuk visualisasi praktis	Ternary plot impossible	Pakai slice/projection
Replikasi center / pure untuk $\sigma^2$	Error tidak teridentifikasi	$\geq 2$ replicate centroid
Constraint $L_i$, $U_i$ tidak terlalu ketat	Pseudo-space tiny	Cek $\sum L < 1$

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka Mixture Design Explorer.
Daftarkan komponen ($q$): nama, opsional $L_i$ dan $U_i$.
Pilih design type: Simplex-lattice (set $m$); Simplex-centroid (rekomendasi $q = 3-4$); Augmented simplex (tambah axial + check points).
Atur replicates dan center points.
Klik Generate Mixture Design.
Setelah eksekusi, input response value per run.
Pilih model order: linear / quadratic / special cubic.
Klik Fit Scheffé Model.
Tinjau: Tab Design Matrix + ternary plot; Tab Model Coefficients ($\beta_i$, $\beta_{ij}$); Tab Contour Plot ternary dengan iso-response; Tab Diagnostics (residual, $R^2$).

3.2 Template Tabel Input + Contoh Data Sintetis

Setup formulasi cookie (3 komponen, simplex-centroid):

Component	$L$	$U$	Default fraction
Flour	0.40	0.70	0.55
Sugar	0.10	0.30	0.20
Butter	0.20	0.40	0.25

(Constrained mixture; $\sum L = 0.70$, $\sum U = 1.40$.)

3.3 Contoh Luaran

Simplex-centroid design ($2^3 - 1 = 7$ pseudo-component points + 3 replicates centroid = 10 runs):

Run	Flour	Sugar	Butter	Type
1	0.70	0.10	0.20	pure Flour vertex
2	0.40	0.30	0.30	pure Sugar vertex
3	0.40	0.10	0.40	pure Butter vertex
4	0.55	0.20	0.25	overall centroid
...	(sisa 3 binary midpoints + 2 replicate centroid)

Fit Scheffé quadratic for Texture_score:

Term	Coefficient	SE	$p$-value
Flour ($\beta_1$)	7.5	0.3	< 0.001
Sugar ($\beta_2$)	6.2	0.3	< 0.001
Butter ($\beta_3$)	8.1	0.3	< 0.001
Flour × Sugar ($\beta_{12}$)	+3.5	0.8	0.005
Flour × Butter ($\beta_{13}$)	−1.2	0.8	0.18
Sugar × Butter ($\beta_{23}$)	+2.8	0.8	0.012
$R^2$	0.96	—	—

Kesimpulan ringkas: "Simplex-centroid design 3-komponen menemukan Butter pure ($\beta_3 = 8.1$) memberikan texture tertinggi di antara pure components, tetapi Flour × Sugar synergistic ($\beta_{12} = +3.5$) — kombinasi binary memberi additional texture boost. Optimal formulasi prediksi: Flour 55%, Sugar 20%, Butter 25% (centroid) untuk balance. Trade-off: pure Butter terlalu lemak, pure Sugar terlalu manis — sweet spot di interior simplex. Lanjut ke Quick Design Optimizer untuk multi-response (Texture + Sweetness + Cost) optimization."

Simplex-centroid design 3-komponen pada ternary plot dan contour Scheffé quadratic model untuk texture score cookie formulasi — Gambar 1. Mixture design simplex-centroid 3-komponen (Flour, Sugar, Butter) untuk optimasi Texture_score cookie, data sintetis. **(a)** Ternary plot design points: 7 titik simplex-centroid (3 pure vertex, 3 binary edge midpoint, 1 overall centroid) + 3 replicate centroid (total 10 runs); setiap titik diberi *label* proporsi Flour/Sugar/Butter; gradasi warna biru→merah pada titik merepresentasikan urutan run (run-number kecil italic abu-abu di pojok kiri-atas tiap titik tidak mengganggu *label* proportion utama); sisi segitiga berlabel nama komponen dengan satuan fraksi. ***Legend*** design point type di **bottom-center**. **(b)** Contour plot Scheffé quadratic $\hat{\eta}$ (Texture_score) di atas ternary space: iso-contour lines dari 6.0 hingga 8.5 dengan interval 0.5; **Butter vertex** menunjukkan puncak ($\beta_3 = 8.1$, warm *color*); region Flour–Sugar binary edge menunjukkan synergistic interaction ($\beta_{12} = +3.5$, lighter warm). Centroid optimal (55/20/25) ditandai *marker* "+" dengan *label* "Texture = 7.7". Reproduced via `design_mixture_simplex`, Scheffé quadratic fit ($R^2 = 0.96$; Scheffé, 1958; Cornell, 2002).

4 Kesimpulan

4.1 Relevansi Real-World

Formulasi cookie/cake/bread — tepung + sugar + fat + air.
Wine/spirit blending — 3–5 varieties blend.
Cat dan coating — pigment ratio.
Pharmaceutical excipient — binder + filler + disintegrant.
Sausage formulation — meat + fat + binder + spice.
Concrete mix — cement + sand + aggregate + admixture.

4.2 Where to Go from Here

Quick Design Optimizer — multi-response desirability.
Response Overlay / Sweet Spot Finder — ternary contour visualization.
Cornell (2002) untuk advanced mixture topics.

⚙ Troubleshooting Cepat

Sum ≠ 1. Modul auto-normalize; cek input.

Pseudo-component pakai $L$ tinggi. Bila $\sum L > 0.9$, factor space sangat kecil — re-set $L$.

Model $R^2$ rendah dengan linear. Coba quadratic (Scheffé) untuk capture binary interaction.

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 Scheffé canonical + simplex-lattice/centroid + APA Scheffé/Cornell/Smith)	Claude

4 Referensi

Scheffé, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society: Series B, 20(2), 344–360. https://doi.org/10.1111/j.2517-6161.1958.tb00299.x
Cornell, J. A. (2002). Experiments with mixtures: Designs, models, and the analysis of mixture data (3rd ed.). John Wiley & Sons. https://doi.org/10.1002/9781118204221
Smith, W. F. (2005). Experimental design for formulation. SIAM-ASA. https://doi.org/10.1137/1.9780898718539
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.