### Determination of Tolerances

Taguchi's loss function methodology can be used to determine the tolerance for a parameter y used to control any quality characteristic.

For a characteristic which is best on target, the loss is proportional to the square of the deviation from the target. Thus we can write the loss function as L(y) = k(y-T) ^{2}, where k needs to be determined.

Suppose the consumer has a functional requirement of the product to be T+/-d_{0}, where T is the target specification and d_{0} his allowable tolerance. Let CL be the consumer's loss when the product is just beyond his requirement of T+/-d_{0}. Equating this loss to the loss function, CL = k(T+d_{0}-T)^{2}. Or k = CL/d_{0}^{2}. Therefore L(y) = (CL/d_{0}^{2}) (y-T)^{2}.

Let the product specification be T+/-d, where d is half the tolerance interval to be permitted to the producer. Let PC be the cost to the producer to rework or scrap the product if it is just beyond the specification of T+/-d, at the factory itself. This, in general, will be lower than CL, since the producer does not have to incur expenses like transportation to obtain the replacement. Again, if we substitute this loss into the loss function we get PL = (CL/d_{0}^{2}) (T+d-T)^{2}. Solving for d, we obtain d = d_{0}*SQRT(PL/CL).

For example, suppose if a pane of glass turns out to be the wrong size at the installation site the consumer's loss CL is Rs1500. Let the consumer's allowable tolerance d_{0} be 3mm and the producer's cost of replacement PL be Rs300. Then the tolerance for the producer d = d_{0}*SQRT(PL/CL) = 3*SQRT(300/1500) = 1.34mm.

**Reference:**

Taguchi, Genichi, "Introduction to Quality Engineering - Designing Quality into Products and Processes", Asian Productivity Organization, Tokyo, 1986, pp.17-19.

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