Solving Factorio Quality

This explains how the Space Age expansion’s five-tier Quality system in Factorio can be modeled and planned using linear algebra. Items and recipes now come in five quality tiers and quality modules give a probabilistic chance q of bumping output tiers; multiple module slots stack to raise q (up to ~24.8% with four slots). Representing a recipe’s mapping from ingredient-tier probabilities to product-tier probabilities as a 5×5 transition matrix T_quality(q) makes multi-step chains trivial via matrix multiplication. A zero-quality chance yields the identity matrix; successive processes (mining, smelting, assembly) are T_quality matrices multiplied in sequence. An explicit numeric example (mining 7.5%, smelting 5%, crafting 10%) gives a final pipe distribution of roughly 79.09% normal, 17.43% uncommon, 2.97% rare, 0.45% epic, 0.07% legendary.

The piece then applies the same formalism to quality strategies that use the recycler (which returns 25% of ingredients) to “wash” items until they reach a target tier. Filtering and recycling are encoded as matrices (keep/extract filters and L = T_filterKeep · 1/4 · T_quality(q)), and the full extraction distribution is the geometric series fresh_unit · (∑i≥0 L^i) · T_filterExtract. This reduces the looping, probabilistic upcycling problem to finite linear-algebra operations, enabling exact planning of ingredient consumption, expected yields, and tradeoffs between opportunistic crafting and repeated recycling. An online calculator implements these computations.