c Heldermann Verlag ISSN 0940-5151
Economic Quality Control Vol 17 (2002), No. 1, 23 – 38
Process Targeting for Optimal Capability when the Product is Subject to Degradation Alan Veevers and Ross Sparks
Abstract: A performance capability measure is proposed that is useful in a wide class of production processes. Traditional quality improvement methodologies concentrate on getting the process right in order to meet the capability requirements of the product immediately after production, as measured for a key quality characteristic by one of the popular capability indexes. This is not the best strategy when the characteristic degrades over time in use. The implications of this type of degradation on targeting the production process are investigated and illustrated for five typical degradation models. A method is given which enables the optimum settings for the target values of the process mean and standard deviation to be chosen, given a particular model for product degradation. Commonly used philosophies which advocate targeting at the centre of the specification range are shown to be sub-optimal. A useful consequence of better targeting practice is that warranty periods can be extended without increasing the costs associated with supporting warranty claims. Keywords: Production process; product capability; performance capability; degradation model; optimal targeting; warranty period.
1
Introduction
Production processes are usually designed so that their product satisfies a set of specifications. In the age of Quality Improvement, the specifications for key quality characteristics are regarded as being supplied by the customer. The customer may be an animate object such as a person, or an inanimate object such as the next process to which the product in question is an input. Following the Shewhart paradigm, a process is stable and in control with respect to those quality characteristics if measurements of them on samples of product items taken over a suitably long period deviate from a constant mean only because of common-cause sources of variation. The capability of a stable process (i.e. its ability to consistently produce product within specification) is measured, for a particular quality characteristic, by standard capability indexes such as Cp , Cpk and Cpm (see, for example, Kotz and Johnson (1993), Kotz and Lovelace (1998)). The emphasis here is on the process rather than the product. One problem that has received a considerable amount of attention is to find the best target value for the mean of the process so that the producer’s profit is maximized. An early formulation and solution to this problem was given by Hunter and Kartha (1977), updated
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Alan Veevers and Ross Sparks
by Nelson (1978). This area of research is identified with the so-called “optimal filling problem” where the product is typically a packet of food or a bottle of liquid. There is a lower, and sometimes an upper, specification limit on the quality characteristic, which is usually a weight or volume, and the filling process is subject to stable common-cause variation. An objective function is set up from costs such as those associated with over-fill, waste, prime product, sub-standard product and other factors, and then the target mean is found which optimizes this function. An up-to-date discussion of this research is given in Misiorek and Barnett (2000). The problem addressed in the present paper is related in the sense that it is concerned with optimal targeting of the mean (and standard deviation) of a product quality characteristic, but different in that it optimizes over the behaviour of the product whilst in use. When items produced by a highly capable process go into service, the best that can be said of them is that they meet the specifications in the “as new” state. From the actual users’ point of view this is a desirable property. However, it is also desirable that the item continues to meet specifications for a length of time in service sufficient to satisfy the customer. Clearly, if degradation takes place, the item will eventually cease to meet specifications and may then be repaired or scrapped. An important implication of degradation to the manufacturer concerns the warranty period for the item. If it is set too long, a large warranty cost will occur; if it is set too short, market share will be lost to competitors offering longer warranty periods. In view of the above, it is evident that product performance is not adequately measured by quoting Cp , Cpk or Cpm values at the end of production. A description of how degradation affects the key characteristics of the product over time is also required. This knowledge can be used to advantage to determine better values for the production process settings so as to maintain its capability whilst maximizing the length of time the product remains in specification during its service life. If this can be achieved it represents a “win-win” situation for the producer and the customer. An obvious benefit is that, by optimal targeting instead of simply aiming for the centre of the specification range, a warranty period could be extended without increasing the cost to the manufacturer. Alternatively, the warranty period could be kept the same, resulting in a significant saving on warranty costs. This article demonstrates how some realistic degradation models can be used to determine optimal target values for the mean and standard deviation of a quality characteristic at the point of manufacture. Issues relating to degradation estimation are briefly mentioned, but are not addressed in detail.
2
Problem Definition and Notation
Consider a single quality characteristic, Y , of a product. In a stable process a suitable model for the variation in Y can often be a Normal distribution with mean µ and standard deviation σ. Suppose this to be the case, and define upper and lower specification limits, U SL and LSL, respectively, for Y .
Process Targeting for Optimal Capability when the Product is Subject to Degradation
25
It is important at this point to clearly distinguish between degradation in the production process and degradation of the product in service. The former occurs when the machinery involved in the production process becomes progressively less able to perform its desired function, for example when tool-wear occurs. The latter, however, is a property of the finished product and refers to its ability to perform in the desired manner during its lifetime. There is an extensive body of literature on degradation models, set in the context of estimating the lifetime of a product using measurements on characteristics of the product whose degradation will ultimately lead to its failure. Robinson and Crowder (2000) give a review of growth-curve models before proposing their own Bayesian method of analysis. Our concern is not with the development of degradation models, rather, it is with the optimal targeting of the process when degradation is known to occur for that product. Current 6-sigma philosophy, for example, is geared up to designing the process, with respect to the quality characteristic Y , to have a sufficiently small standard deviation and to have µ targeted at the centre of the specification range for Y directly after production. Like most process improvement philosophies, the emphasis of this one is on the production process and not directly on the performance of the product over time in use. The 6-sigma methodology has a built-in tolerance of ±1.5σ around the centre of the specification range to allow for the inability to accurately target µ at the centre point. This feature also ensures that, even if µ drifts around its target or degradation takes place in the process, as long as µ stays within ±1.5σ of the centre point the 6-sigma aims will still be achieved. In addition, this property goes some way towards allowing for degradation of the product whilst in service. However, targeting µ at the centre of the specification range is clearly not the best strategy for maximizing performance capability when, for example, Y decreases at a linear rate due to degradation of the product in service. To be specific, consider a time of usage from new, T , which may be thought of as the warranty period. The question of interest is where should µ be targeted to get the best average conformance to specification for the product during a time T in use from new? This issue is addressed by deciding what values of µ and σ should be aimed for, in the finished product, to achieve at least a given proportion of items within specification after usage time T . An example is provided to illustrate the methodology.
3
Illustrations of Degradation Models
It is convenient to introduce a time suffix on Y , µ and σ at this point. Let Yt be the value of the quality characteristic at time t after going into service. Hence, Y0 represents the value immediately after production and µ0 and σ 0 are the distribution mean and standard deviation, respectively, to be targeted at suitable values. Let the probability density of Yt at time t be ft (y) = f (yt | µt , σ t )
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and the formal model be Yt = µt + t , where t is an error term having mean zero and standard deviation σ t . Degradation models, 1 to 5 below, which can be represented by simple functions describing how µt and/or σ t change with time will be used for illustrative purposes. The interval from t = 0 to t = T is taken to be the period of interest. Figures 1 to 5 show specific instances of the five models using snapshots at t = 0, T /2 and T . 1. No degradation, µt = µ0 and σ t = σ 0 : Here the starting distribution and the distribution for the product throughout the period (0, T ) are the same. It is evident that the value of the capability index directly after manufacture provides all the information required for specifying product performance, because the product does not degrade.
Figure 1: No degradation.
2. Linear degradation in mean only, µt = µ0 + ct with c a constant, σ t = σ 0 : Here the deterioration rate remains constant during the period of interest. The sign of the rate constant c determines the direction of degradation, thus, a positive value for c indicates that degradation increases the value of the quality characteristic whilst a negative value for c indicates that degradation decreases the value.
Figure 2: Linear degradation in mean only, c < 0.
Process Targeting for Optimal Capability when the Product is Subject to Degradation
27
3. Degradation in variance only, µt = µ0 and σ 2t = (1 + d2 t2 )σ 20 , d constant: Here the mean remains constant but the standard deviation increases with t (the variance increasing by an amount proportional to t2 ), depicting a situation where the quality characteristic becomes more variable as time in use passes.
Figure 3: Degradation in variance only. 4. Degradation in mean and variance, µt = µ0 + ct and σ 2t = (1 + d2 t2 )σ 20 , c and d constants: This model, which includes models 1, 2 and 3 as special cases, allows for increasing variability as the mean performance degrades.
Figure 4: Degradation in mean and variance, c < 0. 5. Non-linear degradation in mean only, µt = µ0 − a 1 − e−bt , a, b > 0 constants, σ t = σ 0 : In this case, the mean drops more rapidly at first then less rapidly later, as time in use increases.
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Figure 5: Non-linear degradation in mean only. These models cover instances where, (i) the mean remains constant or degradation of the mean is either monotonically increasing or decreasing over the time period (0, T ), and (ii) the variance remains constant or degradation of the variance is monotonically increasing over the time period (0, T ). These examples accord well with practical experience of degradation mechanisms, so are not particularly restrictive. The first instance includes the common situations of material wear and gradual fall-off in performance; the second simply excludes cases (difficult to find in practice) where performance becomes less variable as time passes. Other models can obviously be postulated, for example, ones for which the variance is made proportional to the mean, and other non-linear functions could be used in model 5. It is not the purpose here to suggest and fit degradation models, but to develop a method of optimally targeting the production process given that a particular degradation model is appropriate.
4
Product Performance Capability over a Period (0, T )
Conceptually, it is necessary to think of a large population of product items going into service at t = 0 and then following them through until t = T . In reality, those items going out of specification during that time would not reach time T (they would be repaired or scrapped) but this does not invalidate the methodology based on probabilities calculated from the above concept. Consider such a population of items at t = 0 with quality characteristic Y modelled by a Normal distribution with mean µ0 and standard deviation σ 0 . Imagine trimming
Process Targeting for Optimal Capability when the Product is Subject to Degradation
29
off any portion of the distribution that is outside specifications at this stage. Now, as the distribution moves forward in time, with its mean and standard deviation changing smoothly according to their respective degradation models, imagine any further parts of the distribution which stray over the specification limits to be trimmed off as well. It is helpful to place one restriction on the practical situations covered here, namely that an item which is out of specification on the high side cannot subsequently go out of specification on the low side, or vice-versa, during the period (0, T ). Excluding situations where the entire population of items is culled before time T , there will be, at time T , a portion of the distribution left within the specification limits (not necessarily spanning the distance between them). Define a measure of performance capability to be the proportion of items continually meeting specifications throughout the period t = 0 to t = T . This is given by p = 1 − (pU + pL )
(1)
where LSL pL = sup ft (y)dy 0≤t≤T
and
(2)
−∞
∞
pU = sup
0≤t≤T
ft (y)dy
(3)
U SL
Essentially, pU is the maximum proportion of items that would be seen outside specification on the high side, if culling did not take place, as t increases from 0 to T , and pL is the corresponding quantity on the low side. Hence, the proportion conforming, p, is a capability index that ranges from 0 to 1, with 1 meaning perfect capability and 0 representing the pathological case where the last of the surviving items goes out of specification just at time T . This performance capability index may also be quoted as a percentage, in which case 100p is the appropriate value. In a stable production process with mean targeted at µ0 and with standard deviation σ 0 , the expected production capability might be calculated to be Cpk = 1.33, say. This is traditionally interpreted as a highly capable process with only 32 ppm expected to be outside the specification limit nearest to µ0 . However, this comfortable position is shattered if, after a time T in use with degradation taking place, the performance capability is assessed to be only, say, 80%. Clearly, the performance capability needs to be kept above an acceptable critical value, such as 99.9%, by choosing the values of µ0 and σ 0 that will achieve this. This will ensure a suitably satisfactory Cpk as a by-product. It is intuitively obvious that making σ 0 arbitrarily small will make p arbitrarily close to 1. Hence, optimal performance capability must be defined in terms of choosing the best µ0 for a given σ 0 . The value of σ 0 needs to be chosen by specifying the minimum value tolerable for p, e.g. plim = 0.999 representing a 99.9% capability, and then solving the appropriate equation for σ 0 . The following examples will make this clear.
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5
Targeting Strategies for Optimal Capability
The general method will be illustrated by finding the optimal targeting strategy for each of the five degradation models introduced in Section 3. Although models 1, 2 and 3 are special cases of model 4, it is illuminating to consider them separately, starting with the simplest case and progressing to the general case. Attention is confined to applications for which the cost to the manufacturer of an item being out of specification at any point in the time period of interest is a constant, or at least approximately so. Applications for which this does not hold require a modified approach not covered in this article. Let Φ(·) and φ(·) represent the cumulative distribution function and the probability density function, respectively, of the Standard Normal distribution. 5.1
No degradation, µt = µ0 and σ t = σ 0
For fixed σ 0 > 0, it is obvious from symmetry considerations that the optimum value for . Nonetheless, it is illuminating to establish the theory using this µ0 is µopt = U SL+LSL 2 simple case. Evidently ∞ pU =
f0 (y)dy
LSL and pL = f0 (y)dy ∞
U SL
so
(4)
LSL − µ0 U SL − µ0 +Φ p=1− 1−Φ σ0 σ0 dp LSL − µ0 U SL − µ0 1 φ −φ = dµ0 σ0 σ0 σ0
(5) (6)
The derivative is zero when −
e
(U SL−µopt )2 2σ 2 0
−
=e
(LSL−µopt )2 2σ 2 0
(7)
that is, when (U SL − µopt )2 = (LSL − µopt )2
(8)
and, taking the appropriate sign in the square root, U SL − µopt = −(LSL − µopt )
(9)
giving, as anticipated, U SL + LSL µopt = (10) 2 This shows that optimizing with respect to µ0 is independent of the value of the fixed σ 0 (= 0). The largest acceptable (optimal) standard deviation, σ opt , which just achieves the performance capability p, can be found by solving for σ 0 in the above equation for p, after substituting µ0 = µopt and p = plim . Doing this gives U SL − LSL −1 (11) plim = 2Φ 2σ opt
Process Targeting for Optimal Capability when the Product is Subject to Degradation
31
hence, σ opt =
U SL − LSL 2Φ−1 1+p2lim
(12)
Targeting µ0 at the centre of the specification range and reducing variation, if necessary, so that σ 0 ≤ σ opt will ensure that p ≥ plim . For example, for plim = 0.999, the largest acceptable standard deviation is found by dividing the specification range by 6.581, requiring a Cpk at production of at least 1.097. For plim = 0.9999, the corresponding divisor is 7.781 and Cpk at production would need to be at least 1.297. 5.2
Linear degradation in mean only, µt = µ0 + ct with c a constant, σ t = σ 0
If c = 0, this reduces to model 1 above. If c < 0 the gradual decrease in µt means that ∞ pU =
f0 (y)dy
LSL and pL = fT (y)dy
U SL
so
(13)
−∞
U SL − µ0 LSL − µ0 − cT p=1− 1−Φ +Φ σ0 σ0 dp LSL − µ0 − cT U SL − µ0 1 φ −φ = dµ0 σ0 σ0 σ0
(14) (15)
The derivative is zero when (U SL − µopt )2 = (LSL − µopt − cT )2
(16)
and, taking the appropriate sign in the square root, U SL − µopt = −(LSL − µopt − cT )
(17)
giving, µopt =
U SL + LSL − cT 2
(18)
This means that µ0 must be targeted at a distance |cT2 | above the centre of the specification range. The linear decrease in mean results in µT being the same distance below the centre of the specification range at time T . The same result for µopt is obtained for c > 0, which can be seen by a symmetry argument or by carrying through the calculation. As before, the choice of µopt is independent of σ 0 , so σ opt can be found by solving for σ 0 in the above equation for p, as done for model 1. This time we obtain U SL − LSL ± cT −1 (19) plim = 2Φ 2σ opt where the plus sign is taken in the ambiguity when c is negative, and the minus sign when c is positive. Hence,
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Alan Veevers and Ross Sparks
σ opt =
U SL − LSL − |cT | 2Φ−1 1+p2lim
(20)
The specification range is effectively reduced by the absolute amount of degradation, |cT |, before division by the factor involving plim yields σ opt . As an example, a realistic value for c might be such that |cT | = 0.1(U SL − LSL), in which case the optimum is , where D = 7.312 for plim = 0.999 and D = 8.646 for plim = 0.9999. The σ opt = U SL−LSL D corresponding minimum production Cpk values required are 1.097 and 1.297, respectively, which are the same as for model 1 above. The reason for this is that the smaller standard deviation compensates for the off-centre target for the mean. The practical significance of this example is that when linear degradation of the mean is expected, with standard deviation remaining constant, it is best to target µ0 such that the same proportion of items is out of specification at the time of production as will be the case after time T of use. This may seem surprising at first sight, but it is clear from the above reasoning that any other apparently sensible choice for µ0 will lead to a greater proportion of items being out of specification over the period (0, T ). 5.3
Degradation in variance only, µt = µ0 and σ 2t = (1 + d2 t2 )σ 20 , d constant
In this example the mean remains constant but the standard deviation increases, so the maximum proportions outside specifications will occur, for both tails, at time T . Hence, ∞ pU =
fT (y)dy U SL
LSL and pL = fT (y)dy
(21)
−∞
giving an expression for p, to be optimized, like that for model 1 above, replacing σ 0 with σ T . It follows, as is obvious from a symmetry argument, that U SL + LSL µopt = (22) 2 Substituting appropriately in the equation for p gives U SL − LSL √ −1 (23) plim = 2Φ 2σ opt 1 + d2 T 2 hence, U SL − LSL σ opt = √ 2 1 + d2 T 2 Φ−1 1+p2lim
(24)
Here, the standard deviation at t = 0, σ opt , is sufficiently small so that by the time t = T the degraded (increased) standard deviation is such that the plim criterion is just , met. For example, if d is such that 1 + d2 T 2 = 1.12 , it follows that σ opt = U SL−LSL D where D = 7.239 for plim = 0.999 and D = 8.559 for plim = 0.9999. The corresponding minimum production Cpk values required are 1.207 and 1.427.
Process Targeting for Optimal Capability when the Product is Subject to Degradation
5.4
33
Degradation in mean and variance, µt = µ0 + ct and σ 2t = (1 + d2 t2 )σ 20 , c and d constants
First take c < 0, from which it follows that pL will take its maximum value at time T , since the mean is falling and the variance is increasing with time. However, the maximum of pU may occur either at t = 0 or t = T , depending on whether or not the decrease in it due to the falling mean is faster or slower, respectively, than the increase in it due to the expanding variance, as time passes. Now, for c < 0, U SL − µ0 − ct √ (25) pU = sup 1 − Φ σ 0 1 + d2 t2 0≤t≤T which will take its maximum value at t = 0 or t = T according to whether or not √ (U SL − µ0 ) 1 + d2 T 2 (26) U SL − µ0 − cT is less than or greater than unity, respectively. After some manipulation, we find that for c < 0 the maximum proportion of items out of specification on the high side will occur at time T if 2 cT 1 2 1− −1 (27) d > 2 T U SL − µ0 If the inequality is reversed, then the maximum proportion of items out of specification on the high side will occur at time t = 0. 5.4.1
Case 1
Continuing with c < 0 and the inequality for d2 holding, we have LSL ∞ pU = fT (y)dy and pL = fT (y)dy U SL
hence
(28)
−∞
LSL − µ0 − cT U SL − µ0 − cT +Φ p=1− 1−Φ σT σT dp LSL − µ0 − cT U SL − µ0 − cT 1 φ −φ = dµ0 σT σT σT
(29) (30)
The derivative is zero when (U SL − µopt − cT )2 = (LSL − µopt − cT )2
(31)
and, taking the appropriate sign in the square root, U SL − µopt − cT = −(LSL − µopt − cT )
(32)
giving, µopt =
U SL + LSL − 2cT 2
(33)
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Alan Veevers and Ross Sparks
This is clearly sensible, since, with µ0 = µopt it follows that µT = been degradation in the mean for a time T .
U SL+LSL 2
after there has
Optimizing for σ 0 proceeds by substituting for plim , µ0 and σ T in the equation for p above and solving for σ opt . After some simplification, we obtain U SL − LSL √ −1 (34) plim = 2Φ 2σ opt 1 + d2 T 2 hence, U SL − LSL σ opt = √ 2 1 + d2 T 2 Φ−1 1+p2lim
(35)
This result is the same as for model 3, which is not surprising since it is a consequence , as shown earlier in this example. Taking the specific of the fact that µT = U SL+LSL 2 case where d2 T 2 = 0.75, i.e. σ T = 1.323σ 0 , and cT = −0.1(U SL − LSL), we note that , where D = 8.706 for the inequality for d2 is satisfied. It follows that σ opt = U SL−LSL D plim = 0.999 and D = 10.294 for plim = 0.9999. The corresponding minimum production Cpk values required are 1.161 and 1.372. 5.4.2
Case 2
Keeping c < 0 but reversing the inequality for d2 , we have LSL ∞ pU = f0 (y)dy and pL = fT (y)dy −∞
U SL
hence
LSL − µ0 − cT U SL − µ0 +Φ p=1− 1−Φ σ0 σT The condition for a maximum is that LSL − µopt − cT U SL − µopt 1 √ =√ φ φ σ0 σ 0 1 + d2 T 2 1 + d2 T 2 from which µopt opt is found as a root of the quadratic equation √ 2 2 2 2 2 2 2 2 2 d T X + 2RX − R − 2σ 0 1 + d T ln 1+d T =0
(36)
(37)
(38)
(39)
where X = U SL − µopt and R = U SL − LSL + cT The solution does not have a simple algebraic form, so is not given here. Optimizing for σ 0 for chosen values of plim can be done, but no simple formula exists. As an example, take d2 T 2 = 0.25, i.e. σ T = 1.118σ 0 , and cT = −0.1(U SL − LSL), U SL = 22, LSL = 20 and σ 0 = 0.3. We now find that the inequality for d2 is not satisfied, which is the required condition for this case. Solving the above quadratic equation gives
Process Targeting for Optimal Capability when the Product is Subject to Degradation
35
µopt = 21.144, to three decimal places. It follows that σ opt = 0.258 for plim = 0.999 and σ opt = 0.218 for plim = 0.9999. The corresponding minimum production Cpk values are then calculated to be 1.106 and 1.309. It should be noted that optimizing with respect to µ0 is not independent of σ 0 as has been the case in the previous examples. However, the degree of dependence is slight as illustrated by the fact that taking σ 0 = 0.218 instead of 0.3 in the quadratic equation, gives µopt = 21.147, which differs from the previous value by a practically insignificant amount. This very small effect can hardly be seen over a wider range of values of both µ0 and σ 0 by examining the next two figures. Figure 6 is a contour plot of plim over a range of values of µ0 and σ 0 (for model 2 above) where µopt = 21.1 for any values of plim and σ 0 . Figure 7 is the corresponding plot for the present case where optimizing with respect to µ0 is not independent of σ 0 , yet the departure of the optimal ridge line from a vertical line (µopt = 21.144 is shown) is hardly noticeable.
Figure 6: Contours of plim over a range of values of µ0 and σ 0 for model 2, where µopt is independent of σ 0 .
Figure 7: Contours of plim over a range of values of µ0 and σ 0 for case 2 of model 4, where µopt is not independent of σ 0 .
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Alan Veevers and Ross Sparks
Two other cases occur for model 4, arising when c > 0. They do not need to be covered here, however, as the arguments used in cases 1 and 2 apply mutatis mutandis. Non-linear degradation in mean only, µt = µ0 − a 1 − e−bt , a, b > 0 constants, σ t = σ 0
5.5
Here the mean reduces from µ0 whilst the standard deviation remains constant, so the maximum proportion of items out of specification on the high side will be at t = 0 and on the low side will be at t = T . Hence LSL ∞ f0 (y)dy and pL = fT (y)dy (40) pU = −∞
U SL
therefore
p=1− 1−Φ
U SL − µ0 σ0
+Φ
LSL − µ0 + a(1 − e−bT ) σ0
Following the usual procedure for maximizing in the appropriate range, we obtain U SL − µopt = −LSL + µopt − a 1 − e−bT
(41)
(42)
giving,
U SL + LSL + a 1 − e−bT µopt = 2 Optimizing for σ 0 in the usual way leads to U SL − LSL − a 1 − e−bT −1 plim = 2Φ 2σ opt hence, σ opt
U SL − LSL − a 1 − e−bT = 2Φ−1 1+p2lim
(43)
(44)
(45)
Taking a = 0.5, bT = 2.0, U SL = 22 and LSL = 20 provides a realistic example. In this case µopt = 21.216 and σ opt = 0.238 for plim = 0.999 reducing to σ opt = 0.201 for plim = 0.9999. The corresponding minimum production Cpk values required are 1.097 and 1.297, respectively, which are the same as for models 1 and 2 above. This is not surprising, since the standard deviation remains constant and µopt is as much above the centre of the specification range as µT is below it. The fact that the trajectory of µt over time T is curved rather than a straight line, as in models 1 and 2, does not affect the result.
6
Example
Consider a 12-volt battery that is used to power a high-water-level indicator in a sewer system. These batteries are custom made for the hostile environment in which they
Process Targeting for Optimal Capability when the Product is Subject to Degradation
37
operate and are meant to perform within specification for at least a year without human intervention. The specification for this type of battery is that its on-demand voltage be 12 ± 1 volts. Degradation of the available voltage takes place over time, and may be modelled by example 2 or 5 above. Traditional manufacturing practice is to target µ0 at 12 volts and to have a production standard deviation not greater than 0.3 volts. If this were to be achieved, the minimum production capability could be stated to be Cpk = 1.11. Using model 2 with a typical degradation loss of 0.5 volts per year over the first year (c = −0.5 and T = 1), the theory presented above gives the optimal target for µ0 to be µopt = 12.25 volts. For a value of σ 0 = 0.3, the performance capability is calculated as p = 98.76%. This represents a useful increase over the value p = 95.18% which would be the case if the current value of 12 volts continued to be used as the target for the mean. If the manufacturer wanted to achieve a performance capability of at least 99.5%, say, then the production standard deviation would need to be reduced from 0.30 to be not greater than σ opt = 0.267. If it were not possible to reduce the standard deviation beyond 0.30 but the manufacturer still wanted to achieve a given performance capability, then the remaining option is to reduce the value of T . For p = 99.5%, the new value of T would have to be T = 0.632 (i.e. 230 days) and the target for the mean becomes µopt = 12.16 volts. If T is a warranty period, then this is the calculation needed to determine a new, in this case smaller, warranty period to satisfy requirements.
7
Concluding Remarks
The performance capability measure, p, proposed here is useful in situations where the cost incurred by a non-conforming item is the same (or approximately so) wherever it goes out of specification in the time period (0, T ). As well as being interested in the quality of the product as new, manufacturers and customers are interested in whether or not it remains within specification over a reasonable period of usage time. The manufacturer usually has the warranty period in mind, though the customer would be thinking of a much longer period. In any event, both would want an acceptable conformance level during the warranty period. This paper seeks to focus attention away from simply aiming for a particular capability at the end of production, which measures product performance at the start of usage, onto measuring capability over its warranty period. A method is given which enables the optimum settings for the target values of the process mean and standard deviation to be chosen, given a particular model for product degradation. Optimizing in this way can make significant savings on warranty costs whilst satisfying both producer and consumer.
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References [1] Hunter, W.G. and Kartha, C.P. (1977). Determining the most profitable target value for a production process. Journal of Quality Technology, 9, No.4, 176-181. [2] Kotz, S. and Johnson, N.L. (1993). Process Capability Indices. London: Chapman and Hall. [3] Kotz, S, and Lovelace, C.R. (1998). Process Capability Indices in Theory and Practice. London: Arnold. [4] Misiorek, V.I., and Barnett, N.S. (2000). Mean selection for a filling process with implications to ’Weights and Measures’ requirements. Journal of Quality Technology, 32, No.2, 111-121. [5] Nelson, L.S. (1978). Best target value for a production process. Journal of Quality Technology, 10, No.2, 88-89. [6] Robinson, M.E., and Crowder, M.J. (2000). Bayesian methods for a growth-curve degradation model with repeated measures. Lifetime Data Analysis, 6, 357-374.
Alan Veevers CSIRO Mathematical and Information Sciences Private Bag 10, Clayton South MDC Vic 3169, Australia
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