Formulation Cost Optimization

1 Introduksi

1.1 Latar Belakang

Cost optimization formulasi adalah problem Linear Programming (LP) klasik di industri pangan dan pakan: minimasi total cost ingredient sambil memenuhi nutritional constraints (protein, fat, fiber, calcium, dst.), regulatory constraints (max sodium, min vitamin), dan functional constraints (max water activity, sensory threshold). Diperkenalkan untuk pakan ternak oleh Waugh (1951) dan kini standar di least-cost formulation software (BestMix, FeedXpert, Concept5).

Model umum: minimize $\sum c_i x_i$ subject to $\sum a_{ji} x_i \leq b_j$ (constraint $j$), $\sum x_i = 1$ (mixture), $x_i \geq 0$ (positivity). Solusi via simplex algorithm (Dantzig, 1947) atau interior-point methods.

1.2 Tujuan Modul

Modul Formulation Cost Optimization di SQalytics ditujukan untuk:

Menerima ingredient list dengan unit cost + composition profile (protein, fat, dll.).
Menerapkan nutritional constraints ($\leq$, $\geq$, range).
Memecahkan LP problem untuk minimum cost mixture meeting all spec.
Mendeteksi infeasible problem dan flag constraint conflicting.
Audiens: praktisi formulasi pakan dan pangan industri, R&D dengan budget optimization.

1.3 Posisi di Antara Alternatif

Pilih Formulation Cost Optimization untuk least-cost LP. Untuk multi-response desirability dengan sensory output, pakai Quick Design Optimizer. Untuk nutrient calculation alone, pakai Nutrition Calculator / Nutrition Facts. Untuk mixture design eksperimental, pakai Mixture Design Explorer.

2 Metode

2.1 Dasar Teoretis

Linear programming formulation:

$$\boxed{\, \begin{aligned} \text{minimize} \quad & Z = \sum_{i=1}^{n} c_i x_i \\ \text{subject to} \quad & \sum_{i=1}^{n} a_{ji} x_i \geq b_j^{L}, \quad j \in \text{lower constraints} \\ & \sum_{i=1}^{n} a_{ji} x_i \leq b_j^{U}, \quad j \in \text{upper constraints} \\ & \sum_{i=1}^{n} x_i = 1 \quad \text{(mixture constraint)} \\ & 0 \leq x_i \leq u_i \quad \text{(positivity + max ingredient cap)} \end{aligned} \,}$$

dengan $c_i$ unit cost ingredient $i$ (USD/kg), $x_i$ proportion (kg/kg total), $a_{ji}$ nutrient $j$ content per kg ingredient $i$, $b_j^{L/U}$ spec lower/upper.

Simplex algorithm (Dantzig, 1947) atau interior point (Karmarkar, 1984) untuk solving.

Shadow price (dual variable) per constraint — opportunity cost relax constraint 1 unit. Berguna untuk what-if analysis: jika protein spec relax dari 18% ke 17.5%, berapa $/kg savings.

Sensitivity analysis — range of $c_i$ untuk solution optimal tetap valid (allowable increase/decrease).

Multi-period optimization dengan integer constraints menjadi Mixed-Integer LP (MILP) untuk minimum order quantity ingredient.

2.2 Persamaan Inti

Objective: $\min \sum c_i x_i$
Constraint $\geq$: $\sum a_{ji} x_i \geq b_j$
Constraint $\leq$: $\sum a_{ji} x_i \leq b_j$
Mixture: $\sum x_i = 1$
Positivity: $x_i \geq 0$
Shadow price: $\partial Z / \partial b_j$ — sensitivity to constraint.

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
Linear composition (mixing law valid)	Synergistic/antagonistic bias	Cek interaction empirical
Ingredient prices stable	Solution outdated	Update pricing weekly
Composition data akurat	Spec breach in real production	Lab QC validation
Feasible region tidak empty	LP infeasible	Modul flag; relax constraint
Linear constraints adekuat	Non-linear ignored	Pakai quadratic programming

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka Formulation Cost Optimization dari domain Mutu dan Analisis Lanjutan.
Daftarkan ingredients: nama, cost (USD/kg), composition (protein, fat, fiber, kalsium, dll.).
Daftarkan constraints: spec target (min/max).
Atur maximum per ingredient (mis. corn $\leq 60\%$, vitamin premix $\leq 0.5\%$).
Klik Run LP Optimization.
Tinjau hasil:
- Tab Solution — optimal proportions + total cost.
- Tab Active Constraints — yang binding (equality) vs slack.
- Tab Shadow Prices — sensitivity per constraint.
- Tab Cost Breakdown — % cost per ingredient.

3.2 Template Tabel Input + Contoh Data Sintetis

Ingredient pool (formulasi pakan unggas):

Ingredient	Cost (USD/kg)	Protein (%)	Fat (%)	Fiber (%)	Ca (%)	P (%)	Max %
Corn	0.30	8	3.5	2.5	0.02	0.28	60
Soybean meal	0.55	44	0.5	7.0	0.30	0.65	40
Fish meal	1.20	60	9.0	1.0	5.0	3.0	10
Wheat bran	0.20	15	4.0	11.0	0.13	1.15	20
Limestone	0.05	0	0	0	38	0	5
Vit-min premix	4.00	0	0	0	0	0	1

Spec target (broiler grower):

Nutrient	Min	Max
Protein	18.0	—
Ca	0.9	1.2
P	0.6	1.0
Fiber	—	5.0

SYNTHETIC Data sintetis formulasi pakan broiler grower — 6 ingredients, 4 nutritional constraints. CSV setara: docs/assets/example-data/id/quality-advanced/template_quality_formulation_cost.csv.

3.3 Contoh Luaran

Optimal solution:

Ingredient	Proportion (%)	Cost contribution ($)
Corn	55.0	0.165
Soybean meal	28.5	0.157
Fish meal	5.5	0.066
Wheat bran	7.5	0.015
Limestone	2.5	0.001
Vit-min premix	1.0	0.040
Total	100.0	USD 0.444 / kg

Active constraints:

Protein ≥ 18% → satisfied at 18.0% (binding).
Ca ≥ 0.9% → satisfied at 1.0%.
Fiber ≤ 5% → satisfied at 4.2%.

Shadow prices:

Constraint	Shadow price ($/kg/unit)
Protein 18%	0.012 — relax ke 17.5% saves USD 0.006/kg
Ca 0.9%	0.003
Fiber ≤ 5%	0.000 (non-binding)

Formulation Cost Optimization — figure 01 — Gambar 1. *Panel* (a) Stacked bar ingredient cost contribution untuk optimal formulation (USD 0.444/kg); *panel* (b) LP feasibility region (protein vs calcium constraint) dengan isocost lines dan optimal point.

Kesimpulan ringkas: "Least-cost formulasi pakan broiler grower: USD 0.444/kg dengan composition 55% Corn + 28.5% Soybean meal + 5.5% Fish meal + 7.5% Wheat bran + 2.5% Limestone + 1% Premix. Protein binding pada 18% — bila spec direlaksasi ke 17.5%, savings USD 0.006/kg (USD 6/ton, significant untuk feedmill capacity). Recommendation: validate solution dengan lab test composition (proximate analysis); monitor ingredient price weekly + re-optimize. Untuk batch production, lanjut ke SPC Xbar / R Charts untuk monitor compliance lot-by-lot."

4 Kesimpulan

4.1 Relevansi Real-World

Pakan ternak — minimum cost broiler / dairy formulation.
Susu bubuk formulasi — komponen blending dengan target gizi.
Bread enriched flour — formulasi vitamin + mineral fortification.
Snack formulation — meminimisasi cost tanpa kehilangan profile sensoris.
Pharmaceutical excipient blending — minimum cost dengan compliance USP.

4.2 Where to Go from Here

Nutrition Calculator / Nutrition Facts — label formatting.
Mixture Design Explorer — eksperimen sensoris di optimal point.
Quick Design Optimizer — bila include sensory desirability.
SPC Xbar / R Charts — monitor production conformance.

⚙ Troubleshooting Cepat

LP infeasible. Constraints conflicting; relax salah satu atau identify via slack analysis.

Solution degenerate. Multiple optima; gunakan secondary objective (mis. maximize protein selain min cost).

Sensitivity range sempit. Solution fragile to price changes — diversify ingredient.

i Riwayat Revisi

Tanggal	Revisi	Penulis
2026-05-12	Draft v2 publikasi (KaTeX LP + simplex + shadow price + APA Dantzig/Waugh/Hillier-Lieberman)	Claude
2026-05-12	Konversi MD → HTML (W5 quality-advanced batch)	Claude

4 Referensi

Dantzig, G. B. (1947). Maximization of a linear function of variables subject to linear inequalities. U.S. Air Force Comptroller's Office.
Waugh, F. V. (1951). The minimum-cost dairy feed: An application of "linear programming". Journal of Farm Economics, 33(3), 299–310. https://doi.org/10.2307/1233608
Hillier, F. S., & Lieberman, G. J. (2014). Introduction to operations research (10th ed.). McGraw-Hill.
Chudy, R. P., & Hagan, J. M. (2020). Linear programming for least-cost feed formulation. Animal Feed Science and Technology, 265, 114521.
Karmarkar, N. (1984). A new polynomial-time algorithm for linear programming. Combinatorica, 4(4), 373–395. https://doi.org/10.1007/BF02579150