Quality Engineering & Lean Manufacturing
Taguchi Quadratic Quality Loss Function Formula
Replace binary goalpost tolerance thinking with Genichi Taguchi’s quadratic loss L(y) = (A_0 / Delta_0^2)*(y - m)^2 and expected batch loss k * (sigma^2 + (mu - m)^2).
Why parts barely inside USL/LSL "goalposts" still cost real warranty and field-failure money!
Traditional inspection treats a shaft at m + 0.99*Delta_0 ("barely inside USL") as 0-loss perfection and m + 1.01*Delta_0 as 100% scrap cost A_0! Genichi Taguchi revolutionized quality engineering by modeling societal/warranty dissatisfaction around Nominal Target m via the Quadratic Quality Loss Function L(y) = k * (y - m)^2, calibrated where customer functional failure / scrap cost A_0 occurs at tolerance half-width Delta_0: Loss Coefficient k = A_0 / (Delta_0^2)! Across a production batch with Process Mean mu and Process SD sigma, taking expectations E[(Y - m)^2] = Var(Y) + (E[Y] - m)^2 yields Taguchi’s exact Expected Loss per Unit E[L] = k * (sigma^2 + (mu - m)^2) — decomposing quality cost cleanly into Variance Loss k * sigma^2 plus Off-Center Bias Loss k * (mu - m)^2!
Verify on Nominal_Target m = 50.00 mm (B2), Tolerance_Half_Width Delta_0 = 0.50 mm (B3), Rework/Warranty_Cost_At_Limit A_0 = 25.00 $/unit (B4), Observed_Batch_Mean mu = 50.20 mm (B5), and Batch_SD sigma = 0.15 mm (B6): Loss Coefficient is k = 25.00 / (0.50^2) = 100.00 $/mm^2! Variance Loss per unit is 100 * (0.15^2) = 2.25 $/unit, Off-Target Bias Loss is 100 * (0.20^2) = 4.00 $/unit, and Total Expected Quality Loss is 2.25 + 4.00 = 6.25 $/unit (even though mu + 2*sigma = 50.50 barely touches USL — centering the machine on m = 50.00 saves $4.00/unit instantly at zero capital cost!)!
| Taguchi Quality Metric (m=50, Delta_0=0.50, A_0=$25, mu=50.20, sigma=0.15) | Formula | Verified Output |
|---|---|---|
| Taguchi Loss Coefficient (k) | A_0 / (Delta_0^2) = 25 / 0.25 | 100.00 $ / mm^2 |
| Dispersion / Variance Loss per Unit | k * sigma^2 = 100 * 0.0225 | 2.25 $ / unit |
| Off-Target Centering Loss per Unit | k * (mu - m)^2 = 100 * 0.0400 | 4.00 $ / unit (64% of Loss!) |
| Expected Batch Quality Loss E[L] | k * (sigma^2 + (mu - m)^2) | 6.25 $ / unit |
Taguchi Loss Constant k = A_0 / Delta_0^2 Formula
Validate strictly positive tolerance half-width Delta_0 > 0 and failure cost A_0 > 0.
=IF(OR(B3 <= 0, B4 <= 0), "Invalid Taguchi Input", ROUND(B4 / (B3 ^ 2), 4))Expected Per-Unit Batch Loss E[L] = k * (sigma^2 + (mu - m)^2) Formula
Centering Process_Mean mu onto Nominal_Target m eliminates (mu - m)^2 immediately without needing tighter machining variance sigma^2!
=LET(m_tg, B2, dl_tl, B3, a_sc, B4, mu_pr, B5, sg_pr, B6, IF(OR(dl_tl <= 0, a_sc <= 0, sg_pr < 0), "Invalid Taguchi Input", LET(k_ls, a_sc / (dl_tl ^ 2), ROUND(k_ls * ((sg_pr ^ 2) + ((mu_pr - m_tg) ^ 2)), 2))))Try the example
Use the original values or change them to test your understanding.