The Guarino Manufacturing Effectiveness Metric: Overall Equipment Effectiveness and the Lean Six Sigma Quantitative Core as Complete Dimensionless Bases

Abstract

Overall equipment effectiveness is taught as an industrial convention: three factors, availability times performance times quality, adopted because decades of practice found them useful. This paper proves that it is not a convention. Over the minimal variable set of discrete manufacturing, and with piece count admitted as a base dimension alongside time, the Buckingham π theorem forces exactly three independent dimensionless groups, and those three groups are exactly the availability, performance, and quality factors. The two formulas every OEE practitioner learns, the factored form A × P × Q and the collapsed form τG/T_p, are revealed as the factored and telescoped presentations of one and the same π monomial. Nakajima wrote down the correct answer in 1988. The theorem that makes it inevitable is proved here. The move that makes this work is dimensional rather than statistical. Counts of a specified entity are treated as quantities of a kind rather than as bare numbers, exactly as the International System treats amount of substance for the mole. With the unit dimension U in the basis, ideal cycle time carries T U⁻¹, throughput carries U T⁻¹, and count ratios become true π groups instead of informal percentages. Without U the count variables are dimensionally invisible, the matrix loses a row, and the machinery of the theorem cannot see the structure it is about to reveal. Nothing metaphysical is claimed for U; the instrumental reading, that U is the bookkeeping dimension of a specified countable product, carries every result in the paper. Two completeness theorems. Theorem 1 establishes that over the base dimensions T and U the minimal set {T_p, T_r, τ, N, G} admits exactly three independent groups and that the OEE triple is a complete basis, so every dimensionless quantity constructible from those five variables is a product of powers of availability, performance, and quality. Theorem 2 establishes a completeness result of exactly parallel shape for the statistical layer: over the single dimension of a quality characteristic, the set {Δ, σ, δ} admits exactly two groups, and the capability pair C_p = Δ/(6σ) and the centering index k = 2δ/Δ is a complete basis. The demonstrated capability index C_pk = C_p × (1 − k), the sigma level Z = 3C_pk, rolled throughput yield, and a closed form for normalized Taguchi loss all follow as members of that basis. The extension ladder. Each variable appended to the minimal set introduces no new base dimension until energy enters, so each purchases exactly one further independent group, and at every rung the new group already has a name on the factory floor. Total calendar time gives TEEP. Takt time gives takt coverage. Mean time between failures and mean time to repair give the reliability burden and with it the classical inherent availability as an exact identity, together with a ledger that resolves the opaque quantity 1 − Π_A, the availability loss that OEE reports but cannot explain, into named and separately actionable channels. Changeover time gives the SMED setup burden. Energy gives an efficiency group that only the rank form of the theorem can find, since a naive count of base dimensions predicts three groups and would wrongly conclude that no independent energy group exists. The flow variables give the Factory Physics WIP efficiency together with a closed identity for process cycle efficiency, PCE = Π_F × u, which tells a diagnosing engineer whether poor flow is an inventory problem or a starvation problem. The bridge law. The two layers join at exactly one point. For a process in statistical control with normally distributed output, the capability pair determines a ceiling on the quality group, Π_Q* = Φ(3C_p(1 − k)) + Φ(3C_p(1 + k)) − 1. Determination runs one way: capability sets a ceiling that operations can fail to reach but cannot exceed, and the gap between the ceiling and the observed quality group is itself diagnostic, because an in-control capable process that nonetheless scraps parts is losing them to special causes and startup transients rather than to inherent spread. Consequences for the canon. The Six Big Losses of total productive maintenance and the eight wastes of Lean close one to one against the group registry, each located in the group whose degradation it names. Lean and Six Sigma are re-read as dimensional analysis performed on two different variable sets, the flow variables and the spread variables, by communities that did not know they were doing it, which is offered as an explanation of why their merger into a single methodology worked in practice. The normalized Taguchi loss acquires the closed form ΛQ = (9C_p²)⁻¹ + k², which separates the spread term from the centering term additively and repairs a known defect of Taguchi practice, whose signal-to-noise ratios take logarithms of dimensional quantities. The composite and what it changes. A plant-level composite, Ψ_Plant = Ψ_OEE × Γ_Demand × Γ_Energy × Γ_Flow, becomes an additive loss ledger under logarithms. Because the composite is a product, its log-elasticity with respect to every factor is exactly unity, so no factor carries more marginal leverage than another and improvement priority is set entirely by headroom, which is precisely what the ledger measures. A worked injection-molding line scored over one operating week returns Ψ_OEE = 0.680, a respectable value squarely inside the typical industrial band, while the composite returns Ψ_Plant = 0.0638. The ledger says why: flow contributes seventy-six percent of the log deficit. A single-lever intervention table follows. Buying uptime, the move a manager reading only the OEE number would make, returns 1.12 fold. Cutting work in process to its critical level multiplies the composite eightfold, from 0.064 to 0.510, without touching availability, performance, or quality at all. The intervention that feels natural is not the intervention that pays, and the framework tells them apart before a dollar is spent. Scope and limitations, stated plainly. The bridge law assumes statistical control and normality; for non-normal characteristics the ceiling must be computed from the fitted distribution. Reliability enters through a renewal approximation. The minimal set is single-product, and clamping conventions are declared wherever imposed, with the raw unclamped groups always reported alongside. The worked example is synthetic. Its inputs are constructed to be consistent with published benchmark values rather than drawn from a single instrumented plant, so it demonstrates the machinery and its diagnostic reading but is not an empirical validation; full single-plant instantiation with direct measurement is the subject of the next paper in this program. The paper positions itself explicitly against five bordering literatures, the OEE literature, process-physics dimensional analysis, the dimensional analysis tradition inside operations management, data envelopment analysis and index numbers, and Factory Physics, and states its priority claim as scoped to that survey, with correction from the community invited.