起源故事 · 田口玄一与二次损失函数
Origin Story · Genichi Taguchi and the Quadratic Loss Function
1970 年代,日本工程师 田口玄一(Genichi Taguchi)反对一件事:把"还在规格内"全部视为零损失。
他认为只要偏离目标 T,就有看不见的成本(客户体验下降、装配返工、售后投诉)。他用 二次损失函数 L(x) = k(x−T)² 量化这种损失——
偏一寸亏一寸的平方。1988 年 Chan / Cheng / Spiring 三人把这个观点封装成公式 Cpm,把 σ² 和 (μ−T)² 加在根号里得到"等效 σ",
再走 Cp 的老路。从此 Six Sigma 高阶报告必带 Cpm。
In the 1970s, Japanese engineer Genichi Taguchi rejected a comfortable assumption: that "still inside spec" meant zero loss.
He argued that the moment a part drifts away from target T, hidden costs are already piling up — degraded customer experience, downstream rework, warranty claims.
He quantified this with the quadratic loss function L(x) = k(x − T)²: every step away from target costs you the square of that step.
In 1988, Chan, Cheng, and Spiring packaged the idea into Cpm — fold σ² and (μ − T)² into one root, call it the effective σ, and run Cp's formula on it.
Ever since, serious Six Sigma reports carry Cpm alongside Cpk.
1 偏靶不是免费的:钟形 + 田口损失漏斗 Off-Target Isn't Free: Bell Curve + Taguchi Loss Funnel
μ = T 时损失为零
Cpm =
(USL − LSL)
÷
[6 × √(σ² + (μ−T)²)]
=
—
2 Cp / Cpk / Cpm 三柱:递进的诚实度 Cp / Cpk / Cpm Bars: Honesty in Three Steps
拖 T 滑块 → Cp 完全不变(只看分散),Cpk 微动(偏心相对 LSL/USL 中点),Cpm 大幅反应(直接惩罚偏靶)。 Drag the T slider → Cp doesn't budge (it only sees spread); Cpk barely moves (off-center is measured from the LSL/USL midpoint); Cpm swings hard — it penalizes off-target directly.
3 现实里的 Cpm Cpm in the Real World
客户图纸标 T=±0:精密机械、光学透镜、半导体线宽,客户明确"目标值就是 T",Cpm 必报。
Drawing calls out T = ±0: precision machining, optical lenses, semiconductor linewidth — the customer explicitly names the target. Cpm is non-optional in the report.
注塑成品厚度:USL/LSL=2.05/1.95 mm,T=2.00。实测 μ=2.02 时 Cpk 还说"挺好",Cpm 已经掉下来——客户已感到偏厚。
Injection-molded part thickness: USL/LSL = 2.05 / 1.95 mm, T = 2.00. Measured μ = 2.02 — Cpk still says "looks fine", but Cpm has already dropped. The customer is already noticing the parts feel thick.
药片含量:T=100 mg ±2。μ=99 在规格内 Cpk 合格,但 Cpm 揭示偏靶——药效已偏。
Tablet potency: T = 100 mg ±2. μ = 99 is in-spec and Cpk passes, but Cpm exposes the off-target bias — the dose is already drifting.
Master Black Belt 报告:Six Sigma 高级项目(DFSS、产品设计放行)的能力页几乎都用 Cpm 而非 Cpk。
Master Black Belt reports: high-end Six Sigma projects — DFSS, design-for-release — almost always quote Cpm rather than Cpk on the capability page.
一句话In One Line
三层递进 Cp ≥ Cpk ≥ Cpm:能力 → 现实 → 客户感受。
· Cp 答:"理论上规格够不够宽"
· Cpk 答:"会不会做出废品"
· Cpm 答:"客户感受好不好"
Cpk 高 + Cpm 低 = 没废品但客户在抱怨偏靶。田口的洞察是:质量损失从离开 T 的那一刻就开始,不是越过规格才开始。 Three honest layers: Cp ≥ Cpk ≥ Cpm — capability → reality → customer feel.
· Cp answers: "In theory, is the spec wide enough?"
· Cpk answers: "Will we ship scrap?"
· Cpm answers: "Will the customer be happy?"
High Cpk plus low Cpm means no scrap, but customers are complaining about off-target parts. Taguchi's insight: quality loss starts the moment you leave T — not when you cross the spec line.
· Cp 答:"理论上规格够不够宽"
· Cpk 答:"会不会做出废品"
· Cpm 答:"客户感受好不好"
Cpk 高 + Cpm 低 = 没废品但客户在抱怨偏靶。田口的洞察是:质量损失从离开 T 的那一刻就开始,不是越过规格才开始。 Three honest layers: Cp ≥ Cpk ≥ Cpm — capability → reality → customer feel.
· Cp answers: "In theory, is the spec wide enough?"
· Cpk answers: "Will we ship scrap?"
· Cpm answers: "Will the customer be happy?"
High Cpk plus low Cpm means no scrap, but customers are complaining about off-target parts. Taguchi's insight: quality loss starts the moment you leave T — not when you cross the spec line.
常见误用Common Mistakes
T 没定就算 Cpm,默认取 (USL+LSL)/2。Cpm 必须有客户或工程师明示的 T;没 T 时退回 Cpk,别造一个伪 T。
Computing Cpm without a defined T, defaulting to (USL + LSL) / 2. Cpm requires an explicit T from the customer or engineer. No T? Fall back to Cpk — don't invent a phantom target.
Cpm 当 Cpk 用,互相替换报。两者本质不同:Cpk 用 μ 相对最近规格边距离,Cpm 用 √(σ²+(μ−T)²) 同时惩罚偏靶。语义不通用。
Substituting Cpm for Cpk in reports. They mean different things: Cpk measures μ's distance to the nearest spec limit; Cpm uses √(σ² + (μ−T)²) to penalize off-target. They are not interchangeable.
认为"Cpm 必须小于 Cpk"。当 μ=T 居中时 Cpm = Cp(满分);偏靶后 Cpm 才下降。Cpm 永远 ≤ Cp,但与 Cpk 的大小取决于偏心方向。
Assuming "Cpm is always below Cpk". When μ = T (on-center), Cpm = Cp — a perfect score. Cpm only drops as you drift off-target. Cpm ≤ Cp always, but versus Cpk the order depends on which side of center you are.