# Identifying the Source Whose Removal Causes the Largest Reduction of the Worst-Case Variation

The upper-bound variance inequality can be used to remove the most critical sources of variation, which achieves the most significant reduction in the upper bound V_{max} of the variance in inequality (9.2).

Suppose that the average weight (in grams) of material used for fabrication of a particular item from four manufacturing units is =5039, *p _{2}* =5043,

*p*з = 5045, = 5044 and

*p*= 5046, and variances

_{5}*V*= 120,

*V*= 140,

_{2}*V*= 98,

_{}}*Vi* =150, V5 = 122. Substituting these values in Algorithm 9.1 yields V_{max} = 178.58, obtained at sampling source one with probability *p _{Umax}* =0.41 and source four, with probability p

_{4},

_{max}=0.59.

Removing one of the two sources of variation (one or four) will yield the maximum drop in the upper bound of the variance. The direct check by running Algorithm

9.1 two more times shows that removing source one achieves the largest drop of the upper-bound variance, from V_{max} = 178.58 to V|'_{lax} = 154.7. The variance upper bound of Кшх = 154.7 attained after the removal of source one is attained from sampling the new source one and source three. The procedure is repeated if more sources of variation need to be removed.

Note that removing any other source of variation different from sources one and four has no effect on the upper bound of the variance. Thus, removing source two with mean *p _{2}* = 5043 and variance

*V*140 has no effect on the upper bound of the variance. It is still equal to V

_{2}=_{max}= 178.58.

If sources of variation can be removed from the production process, the same approach can be applied to decrease the variation of any other parameter characterising components coming from multiple sources. For example, this approach can be applied to decrease the variation of the positioning distance of a part, done by several positioning devices. Identifying the positioning device which causes the largest drop in the variance upper bound and removing it from the operational cycle achieves the maximum reduction in the maximum possible variability of the positioning distance.

It needs to be pointed out that while the mean of a parameter can easily be corrected by a readjustment, reducing the variance of the parameter is not easy. Reducing the variance of a process usually requires fundamental technological changes associated with substantial investment. Again, an example can be given with positioning of identical components at a certain distance by different positioning devices. While the mean positioning distance of each device can be easily readjusted to a target value by simply adding/subtracting an appropriate value, the variance cannot be reduced in such a simple way.

After the new upper bound for the variance is found, the design is checked whether it is capable of accommodating this worst-case variation without a substantial loss of functionality.

# Increasing the Robustness of Electronic Products by Using the Variance Upper-Bound Inequality

Resistors are part of many electronic devices. In order for the devices to deliver their specified function, the variation in the resistances of the resistors should not exceed a critical value. Thus, if a pair of resistors are used as *current dividers* or *voltage dividers*, variations in the resistances of the separate resistors change the parameters of the circuit (Floyd and Buchla, 2014). It is important to design the circuit so that it can accommodate the worst possible variation of the resistances without a substantial loss of functionality.

Electronic components come from different suppliers. Suppose that resistors are sourced from four suppliers *(M =* 4) with unknown shares, characterised by means

and standard deviations

of the supplied resistors. The variances of the supplied resistors are obtained from *Ц* =of, *i* = 1,...,4 or И = 82.81; V_{2} = 139.24; V, = 163.84; V_{4} = 112.36, correspondingly.

Since the mixing proportions of the suppliers are unknown, substituting these values as input data in Algorithm 9.1 yields F_{max} = 167.14 or <7_{max} = 12.9£T This value is obtained from taking the first supplier with a mixing proportion /?,,_{max} = 0.165 and the third supplier with a mixing proportion p_{3},_{max} = 0.835.

A smaller variance is obtained for different mixing proportions. For example, for mixing proportions p, = 0.25, *p _{2}* =0.10,

*p*0.35, p

_{2}=_{4}= 0.30, X,

^{4}=i

*Pi*= 1, the obtained variance from equation (9.1) is

*V =*143.9. However, the design must be capable of operating under the worst possible variance of V

_{max}= 167.14 obtained from mixing only the first and the third supplier.

# Determining Tight Bounds for the Fraction of Items with a Particular Property

Suppose that /;, < *p _{2} <... < p*

_{m}, are the fractions of items with a particular property

*X*in

*m*batches. The exact number of components in the separate batches is unknown. If all batches are combined, a question of interest is the bounds of the fraction

*p*of components with the property

*X*in the combined batch.

To answer this question, the fractions of items with property *X* are presented as*:pi = щ/щ, p _{2} = a_{2}/n_{2},... ,p„, = ajn_{m},* where

*a,*is the number of items with property

*X*in the ith batch and

*n*is the total number of items in the ith batch.

_{x}It can be shown by using mathematical induction that

Consider, the trivial case where *m =* 2. For two batches, for the percentage of faulty components *p* in the combined batch, the following inequality holds:

Indeed, from /?, < *p _{2},* which is the same as by multiplying both sides with

the positive value n,*n _{2},* it follows that

*a*<

_{{}n_{2}*a*Adding a

_{2}n_{t}._{:}n, to both sides of this inequality results in

Factoring *cii* from the left-hand side and *щ* from the right-hand side results in
Dividing both sides by the positive value *п _{л}(п*,

*+n*results in

_{2})

The part *p* = < * _{Pl}* = |l from the inequalities in (9.5) is established in a similar

fashion.

Now assume for a particular *k>* 2, where p, = *афц < p _{2}= a_{2}/n_{2} <... < p_{k} = a_{k}/n_{k}, *that the inequalities

hold.

From the left inequality, by multiplying both sides with the positive value И|(и, *+n _{2} +... + n_{k}),* it follows that

Without loss of generality, suppose that there are *к* + 1 batches of components and * P* = «i/«i ^

**Pi = a**

_{2}

**/n**

_{2}

**<... < p**_{k}=*a*

_{k}/n_{k}

**< p**_{k+l}= a_{k}Jn_{M}.From /?, < *p _{k+l},* which is the same as

*^*< ^j-, by multiplying both sides with the positive value

*n*it follows that

_{t}n_{M},*a*Adding a,(n,

_{t}n_{M}< a_{k+l}n_{t}.*+n*to the left side of this inequality and n,(a,

_{2}+ ... + n_{k})

**+a**

_{2}

*+ ... +*

*to the right side and considering the inductive assumption (9.8) results in*

**a**_{k})

which is equivalent to

Considering the proved trivial case *к = 2* (see inequality 9.5), according to the principle of the mathematical induction, the inequality

holds for any *m> k.*

The first part of the inequalities in (9.4) has been proved. In a similar fashion, the second part of the inequalities can also be proved.

The tight lower and upper bounds for the fraction of items with the property *X, *for items pooled from several sources, are thus obtained without knowledge related to the actual number of items in the batches coming from the individual sources.