Determining a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality
Consider an assembly which includes components of type A and B. Within type A there are n varieties A,, A2,..., A„ with fractions a,, a2,..., a„ (a, +a2 + ... + a„ = l) and within type В there are also n different varieties Bb B2,..„ B„ with fractions bx, b2,..., b„ (bx +b2 +... + b„ = l). A component of type A and a component of type
В are randomly purchased without any knowledge of their varieties. An assembly is of high reliability only if each variety A, from type A is combined with exactly one particular variety from type B.
Without restricting generality (the indices of the varieties from types A and В can always be reassigned appropriately), it can be assumed that a high reliability assembly is only present if variety A is paired with variety Bu variety A2 with B2,..., variety A„ with variety B„.
Such a case is commonly present if, for example, in an assembly, the two types of components A and В are in contact and the varieties are ‘normal’ (variety 1), ‘hard’ (variety 2) and ‘soft’ (variety 3) according to the surface hardness of the components. A high-reliability assembly exists if the contacting surfaces are both with normal hardness, both hard or both soft. The rest of the combinations are associated with fast wear of one of the contacting components, which compromises the durability of the assembly.
Suppose that the purchase of A and В component is from two separate batches, one of the batches containing А-components only and the other batch containing й-components only. The fractions of normal-hardness components, hard components and soft components in the А-batch are denoted by aua2,a3, while the corresponding fractions in the й-batch are denoted by bi,b2,b}, correspondingly. It is known that in each batch, the fraction of components with normal hardness is always the largest, followed by the fractions of hard components and soft components: a, >a2> a}, b >b2> by.
If the exact fractions of the separate varieties in each batch are unknown, the question of interest is the lower bound of the probability that from four assemblies at least one will be a high-reliability assembly.
The answer reduces to deriving a lower-bound estimate for the probability of a high-reliability assembly.
Since the probability R of a high-reliability assembly is given by
and, in addition, the constraints a, +a2 +a3 = 1, bt +b2 +b2 = 1 also hold, the lower bound can be conveniently estimated by using the Chebyshev sum inequality:
Since ai + a2 + a3 = 1 and b{ + b2 + b} = 1, the right side equals 1/3 and, as a result, the probability of a durable assembly is at least 0.33.
Consequently, the probability that a single assembly will be not be of high reliability never exceeds 0.66, irrespective of the actual proportions aua2, a3 and bhb2, b} of the different varieties.
Since the probability of an inferior assembly does not exceed 2/3, the probability that from four assemblies, at least one will be a high-reliability assembly is at least
1 — (2/3)4 =0.8. This is a lower bound of the probability that from four randomly selected assemblies at least one high-reliability assembly will be present, irrespective of the actual proportions ah a2, a} and bb bb b3 of the varieties in the batches.
Deriving a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality and Jensen Inequality
The example in the previous section can be continued by considering another application featuring an assembly which includes components of n varieties with unknown proportions xh x2,..., x„ (x, +x2 + ... + x„ = I). Two components are selected randomly and installed in a device. An assembly is fully functional only if both components are of the same variety. Such a case is present in selecting electronic components built into a sensitive circuit, which can only work reliably if the selected components are of the same variety.
The question of interest is to obtain an upper bound of the probability that the components will be of different variety and the assembly will not be fully functional.
To determine an upper bound of the risk of a faulty assembly, the initial step is to list all component configurations which lead to a faulty assembly. A faulty assembly is present only if there are at least two components of different variety. If two components are selected (n varieties), the probability of selecting two components of different variety is given by Pr(faulty assembly) = pf = X,2x,x/. This is a sum of the probabilities of all possible mutually exclusive events: A,y - the first component is of /-variety and the second component is of/-variety and AM - the first component is of/-variety and the second component is of /-variety (Pr(A;/) = x,x,; Pr(A^) = xyxf; joi). The probability pr that the two selected components will be of the same variety (the probability of a functional assembly) is given by p, = Y.'Uxh
An assembly can either be functional or faulty (complementary events), therefore pr + pt = 1 or
Consequently, for the probability of a faulty assembly, the next relationship holds:
An upper bound p/>max for the probability p, = ’Tdi
Pf, max — 1 Pr, minThe Chebyshev inequality provides the opportunity to segment an initial complex expression into simpler expressions. The segmentation capability offered by the Chebyshev inequality will be used to evaluate a lower bound />,- min of X,"=i x,2.
Without loss of generality, it can be assumed that x, < x2 <... < x„. The conditions of the Chebyshev sum inequality are fulfilled and the following inequality holds:
Substituting X| +x2 + ... + x„ = 1 in (9.24) gives the lower bound of
For example, for three varieties, from (9.25) the following lower bound for the probability of a fully-functional assembly is obtained:
The upper bound for the probability of a faulty assembly is
The probability of a faulty assembly does not exceed 0.66 (p,,max = 2/3) irrespective of the fractions xbx2 and x} characterising the different varieties.
For four different varieties (n = 4), Xtix? -1/4 the upper bound for the probability of a faulty assembly is
The probability of a faulty assembly never exceeds 0.75 (/;/лШ1Х = 3/4) irrespective of the fractions x,,x2, x3 and x4 characterising the different varieties.
These results have been confirmed by a Monte Carlo simulation. The simulation algorithm is very simple and details have been omitted. Thus, for three varieties with different proportions adding up to unity, the probability of selecting two different varieties (faulty assembly) never exceeded 0.6667. For four available varieties with proportions adding up to unity, the probability of selecting two different varieties never exceeded 0.75. These bounds are also obtained from the right parts of inequalities (9.26) and (9.27).
Now, suppose that for a functional assembly, three components must be selected from the same variety. To estimate the lower bound of X"=ix,3, again, without loss of generality, it can be assumed that x, < x2 <... < x„. The conditions of the Chebyshev inequality are fulfilled and
Since xl + ... + x2n > 1 In, the substitution in (9.28) gives
or
The lower bound provided by inequality (9.29) can now be used for obtaining an upper bound of the probability that there will be at least two components of different variety, which will cause a faulty assembly. Consequently, the upper bound /;,max for the probability of a faulty assembly is given by
These results have been confirmed by a Monte Carlo simulation. Thus, for three varieties, with proportions adding up to unity and three selected components, the probability that there will be at least two components of different variety never exceeded 0.8889. For four varieties with proportions adding up to unity and three selected components, the empirical probability of having at least two components of different variety never exceeded 0.937.
These results can be generalised and confirmed by invoking the Jensen’s inequality for convex functions.
For any convex function /(x), the Jensen inequality states that
where w,- (i = 1,are numbers (weights) that satisfy 0 < w, < 1 and w, + w2 +... + = 1.
Let u'i = w2=... = w„ = 1 In. The function f(x) = xm is a convex function because fx) = m(m -1) > 0 for m > 2. Therefore, according to the Jensen inequality
with equality attained only for x, =x2 =... = x„. Substituting in (9.32) x, +x2+...+ x„ = 1 gives the lower bound for the probability of a fully-functional assembly
The upper bound for the probability of faulty assembly is
Substituting in (9.34) n = 3, m = 2 produces the upper bound given by inequality (9.26); substituting in (9.34) n = 4, m = 2 produces the upper bound (9.27) and finally, substituting in (9.34) n = 4, m = 3 produces the upper bounds for the probability of a faulty assembly determined by the Monte Carlo simulation.
