From: Subject: Article - "Process Tolerancing: A Solution to the Dilemma of Worst-Case Versus Statistical Tolerancing" Date: Mon, 1 Dec 2008 15:07:39 +0100 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C953C6.8D903CC0" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.5579 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C953C6.8D903CC0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www.variation.com/techlib/ta-2full.html Article - "Process Tolerancing: A Solution to the = Dilemma of Worst-Case Versus Statistical Tolerancing"

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PROCESS = TOLERANCING:
A SOLUTION TO=20 THE DILEMMA OF
WORST-CASE VERSUS STATISTICAL = TOLERANCING

Dr. Wayne A. Taylor

When solving tolerancing problems, one must choose between = worst-case=20 tolerancing and statistical tolerancing. Both of these methods have = their pros=20 and cons. If worst-case tolerancing is used, all tolerances must be = specified=20 as worst-case tolerances. If statistical tolerancing is used, all = tolerances=20 must be specified as statistical tolerances. In reality, the behavior = of=20 certain inputs is best described using worst-case tolerances while the = behavior of other inputs is best described by statistical tolerances. = Still=20 other inputs are not adequately represented by either type of = tolerance. This=20 article reviews the two current methods of tolerancing along with = their pros=20 and cons. It then introduces a new method of tolerancing, called = process=20 tolerancing, which solves many of the problems associated with the = current=20 approaches. Process tolerancing represents a unified approach to = tolerancing=20 that encompasses both of the previous approaches. Using process = tolerancing,=20 worst-case tolerances and statistical tolerances can be combined into = the same=20 analysis. The flexibility provided by process tolerancing results in = solutions=20 to such common problems as multiple components from the same lot and=20 off-center processes.

KEY WORDS: Statistical Tolerance, Worst-Case Tolerance, Process = Tolerance,=20 Variation Transmission Analysis, Propagation of Error, and Six Sigma=20 Quality.

INTRODUCTION

Tolerance analysis is used during product design. For example, = consider the=20 design of a pump. Flow rate is a key characteristic that must be = controlled. It=20 is called an output variable. Flow rate is affected by such factors as = diameter=20 of the piston, stroke length, motor speed and viscosity of the solution. = These=20 factors are called input variables. The goal of tolerance analysis is to = predict=20 the behavior of the output variable based on an understanding of how the = inputs=20 behave. The behavior of the inputs can then be adjusted to cause the = output to=20 behave as desired.

Tolerance analysis is equally important to process development. For = example,=20 consider the installation of a new heat seal machine. Seal strength is = the=20 output variable. Input variables include die temperature, seal time, and = pressure. Using tolerance analysis, a process window can be determined = for the=20 inputs that ensures that seal strength stays within its specification=20 limits.

Two methods of tolerancing currently exist: worst-case and = statistical=20 tolerancing. Both have their pros and cons. These two methods will be = reviewed=20 and compared. A new method of tolerancing, called process tolerancing, = will then=20 be introduced. Process tolerancing represent a unified approach to = tolerancing=20 that encompasses both the previous approaches.

The greatest benefit of process tolerancing is that it allows both = worst-case=20 and statistical tolerances to be combined in the same analysis. Some = inputs are=20 best described by worst-case tolerances. For example, consider solution=20 viscosity for the pump. Different customers will use the pump with = different=20 solutions. Each customer wants the correct flow rate. As a result, a = worst-case=20 tolerance best describes the behavior of solution viscosity. Some inputs = are=20 best described by statistical tolerances. For example, consider die = temperature=20 on the heat seal machine. Die temperature is controlled automatically = using=20 close-loop feedback. This ensures it remains at the specified target. = Its=20 behavior is best described using a statistical tolerance. The purpose of = a=20 tolerance analysis is to predict the behavior of the output based on = knowledge=20 of how the inputs behave. Key to its success is the ability to = accurately=20 describe the behavior of the inputs. Using process tolerancing, you can = select=20 the type of tolerance that best describes the behavior of each = input.

Some inputs are not well represented by either worst-case or = statistical=20 tolerances. For example, inputs from processes that are off-center or = lack=20 statistical control. Process tolerancing offers much greater flexibility = than=20 the current two methods and can be used to better represent such = situations.=20 Because of this added flexibility, process tolerancing provides = solutions to=20 such common problems as multiple components from the same lot and = off-center=20 processes.

WORST-CASE = TOLERANCES

Worst-case tolerances are specified in the form 1.00 =B1 0.03". The tolerance half-width of 0.03" = represents the=20 maximum amount that a "good" unit can deviate from target. This = worst-case=20 tolerance can also be represented in interval form as (0.97", 1.03"). A = unit=20 falling outside the range of 0.97" and 1.03" fails to meet the tolerance = and is=20 called a nonconformance or defect. With worst-case tolerances, units of = product=20 can fall anywhere in the tolerance interval. All units could be 0.97", = all units=20 could be 1.03", or the units could be uniformly spread across the = interval.=20 Nonsymmetrical tolerances can also be specified, but are ignored here = because=20 they can be easily translated into equivalent symmetrical = tolerances.

If worst-case tolerances are specified for the inputs, a worst-case = tolerance=20 can be calculated for the output. Let the output be represented by = y and=20 the inputs by x1, =85, xn. Suppose y = =3D=20 f(x1, =85, xn) and that the inputs have = worst-case=20 tolerances txi =B1 D xi. Then the = resulting=20 worst-case tolerance for y, denoted ty =B1 D = y, is:

3D"Equation (Eq. 1)

3D"Equation (Eq. 2)

3D"Equation (Eq. 3)

3D"Equation (Eq. 4)

The terms ly and uy represent the = minimum=20 and maximum values of y over the worst-case tolerances for the = inputs.=20 The worst-case tolerance for y is the interval = (ly,=20 uy). The target for y, ty, is = the=20 midpoint of this interval while Dy is=20 half the width of the interval. The following example demonstrates the = use of=20 these equations.

Example: Suppose that four identical components are stacked on = top of=20 each other and that we are interested in controlling the overall height. = Overall=20 height is denoted by h. Specification limits for overall height = are from=20 3.9" to 4.1". The height of the four components are denoted by=20 d1, =85, d4. Suppose these = components have a=20 tolerance of 1.00 =B1 0.03". Then:

3D"Equation (Eq. 5)

3D"Equation (Eq. 6)

3D"Equation (Eq. 7)

3D"Equation (Eq. 8)

3D"Equation (Eq. 9)

The resulting worst-case tolerance for h is 4.00 =B1 0.12" or (3.88", 4.12"). This tolerance is = shown in Figure=20 1 along with the specification limits. Worst-case tolerancing, predicts = that the=20 specification limits will not be meet and that the tolerance for = component=20 height must be tightened.

3D"Figure

Figure 1: Worst-case = tolerance for=20 overall height
4-component stack example

When the output is the sum of the inputs as in the above example, the = worst-case tolerance equations simplify to:

3D"Equation (Eq. 10)

3D"Equation (Eq. 11)

Equation 11 is frequently cited in textbooks on tolerancing. However, = when=20 more complex equations are involved, Equations 1-4 must be used = instead.

STATISTICAL = TOLERANCES

Statistical tolerances are determined by a target, = tx, and=20 maximum standard deviation, sx. In=20 this article they are denoted by tx =A3=20 sx. They are more commonly = denoted by=20 tx =B1 D=20 x . When using the symbol,=20 it is most common to have D = x =3D=20 3sx. However, this = relationship is=20 not formalized in the standards and the six sigma approach uses D x =3D 6sx. Throughout this article we will = assume=20 that D x =3D 3sx. Using this interpretation, both = 1.00 =A3 0.01" and 1.00 =B1 = 0.03" denote the statistical tolerance with target = tx =3D=20 1.00" and maximum standard deviation sx =3D 0.01".

The worst-case tolerance 1.00 =B1 0.03" = and the=20 statistical tolerance 1.00 =B1 0.03" both assume that units of product fall in the interval = (0.97",=20 1.03"). However, the statistical tolerance is more restrictive. The = statistical=20 tolerance further requires that the units of product have an average of = 1.00"=20 and a standard deviation no greater than 0.01". Statistical tolerancing = assumes=20 centered processes. It further assumes that the inputs vary independent = of each=20 other, i.e., are not correlated. Figure 2 shows a production batch that = meets=20 the worst-case tolerance 1.00 =B1 0.03" but = fails to meet=20 the assumptions of the corresponding statistical tolerance 1.00 =B1 0.03" .

3D"Figure

Figure 2: Production batch = satisfying=20 the requirements of a
worst-case tolerance but not a statistical=20 tolerance

If statistical tolerances are specified for the inputs, a statistical = tolerance can be calculated for the output. This amounts to determining = the=20 average and standard deviation of the output. Simulations can always be = used to=20 predict these two values. However, there also exist a variety of methods = for=20 deriving approximate and sometimes exact equations for the average and = standard=20 deviation. Details can be found in Taylor (1991), Cox (1986) and Evans = (1975).=20 These approaches have a variety of names including statistical tolerance = analysis, propagation of errors and variation transmission analysis.

Returning to the case where y =3D f(x1, =85,=20 xn), equations can be derived for the average and = standard=20 deviation of y in terms of the average and standard deviation of = the=20 xi=92s. These will be denoted m=20 y (m x1, =85, m=20 xn, s=20 x1, =85, s xn) and=20 s y (m=20 x1, =85, m xn, s x1, =85,=20 s xn)=20 respectively. The statistical tolerance for y can then be = calculated as=20 follows:

3D"Equation (Eq. 12)

3D"Equation (Eq. 13)

When the output is the sum of the inputs, the statistical tolerancing = equations simplify to:

3D"Equation (Eq. 14)

3D"Equation (Eq. 15)

Multiplying both sides of Equation 15 by 3 gives:

3D"Equation (Eq. 16)

Equation 16 is frequently cited in textbooks on tolerancing. It still = holds=20 even if we assume D x =3D = 6sx. When more complex equations are = involved,=20 Equations 12 and 13 should be used instead.

Example (continued): Returning to the example of four = components=20 stacked on top of each other, assume each component has a statistical = tolerance=20 of 1.00 =A3 0.01". Then the statistical = tolerance for=20 overall height is:

3D"Equation (Eq. 17)

3D"Equation (Eq. 18)

3D"Equation (Eq. 19)

The resulting statistical tolerance for h is 4.00 =A3 0.02" or (3.94", 4.06"). This tolerance is = shown in Figure=20 3 along with the specification limits. Statistical tolerancing predicts = that the=20 specification limits will be easily met. No change is required to the = tolerance=20 for component height.

3D"Figure

Figure 3: Statistical = tolerance for=20 overall height
4-component stack example

COMPARISON OF THE TWO = APPROACHES TO=20 TOLERANCING

In the 4 component stack example, worst case tolerancing results in a = tolerance for overall height of 4.00 =B1 = 0.12" while=20 statistical tolerancing results in a tolerance of 4.00 =B1 0.06" . The worst-case tolerance is double the width of the = statistical=20 tolerance. Statistical tolerancing predicts that the specification = limits will=20 be met and that the tolerance for component height does not need to be=20 tightened. Worst-case tolerancing predicts that the specification limits = will=20 not be met and that the tolerance for component height must be = tightened. The=20 two methods of tolerancing produce different results and that can lead = to=20 opposite conclusions. Selecting which method of tolerancing to use is an = important decision. This section examines the pros and cons of each=20 approach.

Worst-case tolerancing is the safer approach. If the inputs are = within their=20 respective tolerances, the output is guaranteed to be within its = worst-case=20 tolerance. This is especially important for products like heart values = or=20 critical components on airplanes. However, this guarantee comes at a = high cost.=20 Worst-case tolerancing guards against a worst-case scenario that is = highly=20 unlikely if not impossible in most situations. Worst-case tolerancing = guards=20 against all inputs simultaneously being at the same extreme. In the = 4-component=20 stack example, this would require that all four components be either = 0.97" or=20 1.03". For this to occur repeatedly, would require the entire batch of=20 components to be at one of the two extremes. This could only occur if = the=20 process producing the components had zero variation.

Suppose instead that the process producing the components had a = standard=20 deviation of 0.005". As shown in Figure 4, the process average must stay = between=20 0.985" and 1.015". Otherwise some components would fall outside the = tolerance=20 limits. The two normal curves in Figure 4 represent the worst-case = conditions=20 for component height when the standard deviation is 0.005".

3D"Figure

Figure 4: Worst-case = conditions for=20 component height when s =3D=20 0.005"

When component height has an average of 0.985" and a standard = deviation is=20 0.005", overall height will have an average of 3.94" and a standard = deviation of=20 0.0025". Likewise when component height has an average of 1.015" and a = standard=20 deviation is 0.005", overall height will have an average of 4.06" and a = standard=20 deviation of 0.0025". These two sets of conditions represent the = worst-case=20 scenarios for overall height. They are shown in Figure 5. As a result of = the=20 variation, so long as the components meet their tolerance, overall = height will=20 easily meet the specification limits.

3D"Figure

Figure 5: Resulting = worst-case=20 conditions for overall height when s =3D 0.005"

Worst-case tolerancing generally overestimates the range of the = outputs. This=20 can result in the unnecessary tightening of tolerances for the inputs, = which=20 drives up costs. Worst-case tolerancing is the safer but more costly of = the two=20 approaches.

Statistical tolerancing results in narrower tolerances for the = outputs. In=20 the 4-component stack example, statistical tolerancing resulted in a = tolerance=20 for overall height that was half as wide as that obtained using = worst-case=20 tolerancing. As a result, the tolerances of the inputs do not have to be = tightened as frequently or by as much. Statistical tolerancing is the = less=20 costly of the two approaches. When applicable, we would like to use = statistical=20 tolerancing. However, statistical tolerancing assumes centered = processes.=20 Violation of this assumption can result in defective product.

To see how, let us return to the 4-component stack example. Figure 6 = shows=20 how component height is suppose to behave when a statistical tolerance = of 1.0=20 =B1 0.03" is specified (m =3D 1.0", s =3D 0.01"). It also shows the corresponding = behavior of=20 overall height (m =3D 4.0", s =3D=20 0.02").

3D"Figure

Figure 6: How component = height and=20 overall height are suppose to behave

Consider what would happen if the component height actually behaved = as in=20 Figure 7 (m =3D 0.976", s =3D=20 0.002"). All the component heights are within the tolerance. However, = the=20 process is off-center. Figure 7 also shows the resulting behavior of = overall=20 height (m =3D 3.904", s =3D=20 0.004"). As a result of the components being off-center, the overall = height of=20 some units exceeds the specification limits and all the units exceed the = statistical tolerance of 3.94" to 4.06". Statistical tolerancing, while = less=20 costly, can underestimate the range of the outputs.

3D"Figure

Figure 7: How component = height and=20 overall height actually behave

To counter the tendency of statistical tolerances to underestimate = the range=20 of the outputs, Bender (1962) recommends multiplying statistical = tolerances by a=20 factor of 1.5. For the 4-component stack example, this results in a = tolerance of=20 4.0 =B1 0.09" or (3.91", 4.09"). The = resulting tolerance=20 is half way between the statistical and worst-case tolerances. It is = probably a=20 more accurate estimate of the actual behavior of overall height. = However, the=20 factor 1.5 is somewhat arbitrary and certainly not appropriate for every = problem.

Another way of adjusting for the tendency of statistical tolerances = to=20 underestimate the range of the outputs is to estimate separately the = amount that=20 the average varies around the target and then calculate the total = variation of=20 the input as follows:

3D"Equation (Eq. 20)

Benderizing tolerances amounts to assuming that the process average = has a=20 standard deviation equal to 112% of that of the within lot standard=20 deviation.

These two approaches are only appropriate when the process varies = around the=20 target over longer periods of time. Neither approach properly accounts = for a=20 process that is consistently run off-target. For example, sheeting = thickness=20 might purposely be run closely to the lower specification limit in order = to=20 reduce material usage.

THE = DILEMMA

Statistical tolerancing is less costly. We would like to use it when=20 appropriate. If statistical tolerancing is selected, statistical = tolerances must=20 be specified for all inputs. However, statistical tolerances are not = appropriate=20 for all inputs. To be safe, if some inputs are best represented by = worst-case=20 tolerances, worst-case tolerancing should be used instead. But if = worst-case=20 tolerancing is selected, worst-case tolerances must be specified for all = inputs,=20 even those meeting the assumptions of statistical tolerancing. This = dives up=20 cost. What is needed is a method of combining the two types of = tolerances into a=20 single analysis. Process tolerancing, the topic of the next section will = allow=20 us to do just this.

PROCESS = TOLERANCING

Process tolerancing is a new approach to tolerancing that is = implemented in=20 Taylor (1997). A process tolerance consists of a minimum average, = maximum=20 average and maximum standard deviation. For example, it might be = specified that=20 the average for component height be between 0.99" and 1.01" and that the = standard deviation be no more than 0.0067". This process tolerance is = shown in=20 Figure 8. The two normal curves represent the minimum and maximum = average with=20 the maximum standard deviation.

3D"Figure

Figure 8: Process tolerance = for=20 component height

Process tolerances will be denoted by tx =B1 D = x =A3 s = x. The term=20 tx =B1 D=20 x denotes an interval containing the average. The = minimum=20 average is tx - D = x=20 and the maximum average is tx + D=20 x. The maximum standard deviation is s x. For example, the process = tolerance=20 shown in Figure 8 is denoted by 1.0 =B1 0.01 = =A3 0.0067". Process tolerances ensure that = individual units=20 all fall in the range (tx-D = x-3s x,=20 tx+D = x+3s x). As shown in Figure 8, the = process=20 tolerance 1.0 =B1 0.01 =A3 0.0067"=20 ensures individual components fall in the interval (0.97", 1.03").

Process tolerances differ from worst-case tolerances in that = worst-case=20 tolerances represent requirements for individual components while = process=20 tolerances represent requirements for the process producing the = components.=20 Hence its name. Process tolerances are more similar to statistical = tolerances.=20 However, process tolerances do not require that the process remain = centered.=20 Instead, process tolerances specify an operating window that the process = average=20 must remain within.

If process tolerances are specified for the inputs, a process = tolerance can=20 be calculated for the output. This requires equations for the average = and=20 standard deviation. Returning to the case where y =3D = f(x1,=20 =85, xn), let these two equations be represented by m y ( m=20 x1, =85, m xn, s x1, =85,=20 s xn)=20 and s y (m=20 x1, =85, m xn, s x1, =85,=20 s xn).=20 These two equations can be obtained using the same methods used to = calculate=20 statistical tolerances. Further let the process tolerance for = xi be=20 denoted txi =B1 = D xi =A3=20 s xi. Then the = resulting=20 process tolerance for y, denoted ty =B1=20 D y =A3 s y, is:

3D"Equation (Eq. 21)

3D"Equation (Eq. 22)

3D"Equation (Eq. 23)

3D"Equation (Eq. 24)

3D"Equation (Eq. 25)

If the inputs satisfy their respective process tolerances, then = y is=20 guaranteed to satisfy the process tolerance given by Equations = 21-25.

When the output is the sum of the inputs, the process tolerancing = equations=20 simplify to:

3D"Equation (Eq. 26)

3D"Equation (Eq. 27)

3D"Equation (Eq. 28)

Equation 27 is the same as Equation 11 for worst-case tolerances. = Equation 28=20 is the same as Equation 15 for statistical tolerances. A process = tolerance=20 represents a worst-case tolerance for the average and a statistical = tolerance=20 for the variation around the average. When more complex equations are = involved,=20 Equations 21 through 25 should be used instead.

Example (continued): Returning to the example of four = components=20 stacked on top of each other, assume each component has a process = tolerance of=20 1.00 =B1 0.01 =A3 = 0.00667". This=20 tolerance was shown in Figure 8. It is roughly equivalent to the = worst-case=20 tolerance 1.00 =B1 0.03" and the statistical = tolerance=20 1.00 =A3 0.01". All three tolerances result = in units in=20 the range (0.97", 1.03"). The resulting process tolerance for overall = height is=20 calculated as follows:

3D"Equation (Eq. 29)

3D"Equation (Eq. 30)

3D"Equation (Eq. 31)

The resulting process tolerance for h is 4.00 =B1 0.04 =A3 0.01334". = This tolerance is=20 shown in Figure 9 along with the specification limits. Individual units = of=20 product are expected to fall in the interval (3.92", 4.08"). Process = tolerancing=20 predicts that the specification limits will be met. No change is = required to the=20 tolerance for component height.

3D"Figure

Figure 9: Process tolerance = for overall=20 height
4-component stack example

The results of the three different approaches to tolerance analysis = are=20 summarized in Figure 10. Process tolerancing predicts the overall = heights will=20 be in the interval (3.92", 4.08"). This falls between the worst-case = tolerance=20 of (3.88", 4.12") and the statistical tolerance of (3.94", 4.06").

3D"Figure

Figure 10: Comparison of = three=20 approaches to tolerance analysis
4-component stack = example

A UNIFIED APPROACH TO=20 TOLERANCING

Process tolerancing represent a unified approach to tolerancing that=20 encompasses both the previous approaches. The process tolerance=20 tx =B1 D=20 x =A3 0 is in fact a = worst-case=20 tolerance. In this case, Equations 21-25 simplify to Equations 1-4. The = process=20 tolerance tx =B1 0 =A3 s = x is in fact a=20 statistical tolerance. In this case, Equations 21-25 simplify to = Equations=20 12-13. Worst-case tolerancing guards against the worst-case conditions = which can=20 only occur in the unlikely event of zero variation (s=20 x =3D 0). Statistical tolerances make the restrictive = assumption=20 of centered processes (D x = =3D 0). In=20 one sense statistical tolerances represent the best-case conditions. = Neither of=20 these two extreme cases adequately describes all inputs. Process = tolerancing=20 provides a middle ground. The purpose of a tolerance analysis is to = predict the=20 behavior of the output based on knowledge of how the inputs behave. Key = to its=20 success is the ability to accurately describe the behavior of the = inputs.=20 Process tolerancing, provides the needed flexibility to accurately = describe the=20 behavior of the inputs and to therefore accurately predict the behavior = of the=20 outputs.

Even if you never specify a process tolerance, you can still benefit = from=20 process tolerancing. Process tolerancing allows both worst-case and = statistical=20 tolerances to be combined in the same analysis. Some inputs are best = described=20 by worst-case tolerances. Other inputs are better described by = statistical=20 tolerances. However, if worst-case tolerancing is used, all tolerances = must be=20 specified as worst-case tolerances and if statistical tolerancing is = used, all=20 tolerances must be specified as statistical tolerances. Using process=20 tolerancing, you can select the type of tolerance for each input that = best=20 describes its behavior. The decision you must make changes from "Which = type of=20 tolerancing should I use?" to "Which type of tolerance best describes = the=20 behavior of each input?" The following seven examples illustrate how = tolerances=20 might be specified for different circumstances.

Example 1 =96 Usage Temperature (environmental input): The = flow rate of=20 a pump is affected by usage temperature. The pump is intended for use = over a=20 temperature range of 40=B0 F to 120=B0 F. A worst-case tolerance of 80=B0 F=20 =B1 40=B0 F best = describes the=20 behavior of usage temperature since individual pumps could be run = continuously=20 at each of the temperature extremes. In general, inputs representing = usage and=20 environmental conditions as well as other inputs outside the control of = the=20 designer should be specified as worst-case tolerances.

Example 2 =96 Die Temperature (automatically controlled = input): Die=20 temperature affects the seal strength of a heat seal machine. The heat = sealer=20 continuously monitors die temperature and automatically adjusts it. This = ensures=20 that die temperature is centered on the target. The behavior of die = temperature=20 is best described using a statistical tolerance. Suppose the target is = 150=B0 F and that the standard deviation of die = temperature is=20 5=B0 F. Then the statistical tolerance = 150=B0 F =A3 5=B0 F=20 could be specified. In general, inputs with automatic controls to keep = them=20 centered should be specified as statistical tolerances.

Example 3 =96 Motor Speed (manually controlled input): The = flow rate of=20 a pump is affected by motor speed. The motors are purchased from a = supplier. A=20 capability study performed on the supplier=92s process indicates that = their=20 process is in control with a standard deviation of 0.1 rpm. The = statistical=20 tolerance 18.0 rpm =A3 0.1 rpm might be = considered.=20 However, there is a concern that the capability study, run during the = production=20 of a single lot, is not representative of the full range of variation = that will=20 be experience over more extended periods of time. One possibility is to=20 Benderize the tolerance by multiplying it by 1.5. This results in the=20 statistical tolerance 18.0 rpm =A3 0.15 rpm. = Another=20 approach is to allow an operating window for the average. Suppose that = an control chart is used to keep the process centered. If the = process=20 shifts off-target by 1.5s , the first point = following=20 the shift has a 70% chance of falling outside the control limits. This = control=20 chart is capable of keeping the process within 1.5s of=20 target. A process tolerance of 18 =B1 0.15 = =A3 0.1 rpm could be specified. This allows a =B1 1.5s operating = window around the=20 target. In general, manually controlled processes are best represented = by=20 process tolerances specifying the operating window that can be = maintained.=20 Stable processes and processes controlled using control charts typically = hold a=20 =B1 1.5s operating = window.

Example 4 =96 Line voltage (unstable process): The line = voltage effects=20 the seal strength of a sonic sealing machine. Voltage fluctuates between = 120=20 =B1 10 volts. It cycles up and down. It can = run for long=20 periods of time in the upper end or the lower end of the range. This = input is=20 best described using the worst-case tolerance 120 =B1 10=20 volt. In general, unstable processes should be handled using worst-case = or=20 process tolerances. If process tolerances are used, the minimum and = maximum=20 averages should represent the extremes seen over extended periods of = time.

Example 5 =96 Sheeting Thickness (consistently off-target = input):=20 Sheeting thickness affects seal strength of a heat sealer. The = requirements have=20 been specified as 15 =B1 1 mil (thousandth of = an inch).=20 Suppose the sheeting supplier purposely runs below the target in order = to reduce=20 material costs. They average 14.6 mils with a standard deviation of 0.2 = mils.=20 Off-center processes violate the assumption of statistical tolerancing.=20 Statistical tolerances should not be used is this case. The process = tolerance 15=20 =B1 0.4 =A3 0.2 = mil accounts for=20 the off-center average. In general, only process or worst-case = tolerances should=20 be used for consistently off-target processes.

Example 6 =96 4-Component Stack (multiple identical = components): In the=20 4-component stack problem, the 4-components are identical. As a result, = finished=20 units consist of four components coming from the same supplier lot. A = capability=20 study performed on the supplier=92s process indicates that their process = is in=20 control with a standard deviation of 0.00667". Using historical data it = is=20 estimated that the process average varies from lot to lot with a = standard=20 deviation of 0.0075". The total standard deviation is then = ". However, specifying a statistical tolerance = of 1.0=20 =A3 0.01" and performing a tolerance analysis = can produce=20 misleading results. When the lot average is low, all the components tend = to be=20 low. When the lot average is high, all the components tend to be high. = This=20 creates a correlation between the inputs violating the assumption of=20 independence. Statistical tolerances should not be used is this case. A = process=20 tolerance of
1.0 =B1 0.015 =A3=20 0.00667" could be used instead. It accounts for the effect of below and = above=20 average lots. In general, only process and worst-case tolerances should = be used=20 for products containing multiple identical components or correlated = inputs.

Example 7 =96 Spring Force (deterioration over time): The = closure force=20 of a car door hinge is affected by spring force. The springs are = purchased from=20 a supplier. The target is 10 lbs. A capability study performed on the = supplier=92s=20 process indicates that their process is in control with a standard = deviation of=20 0.2 lbs. A process tolerance of 10 =B1 0.3 = =A3 0.2 lb. could be used, allowing a =B1 1.5s operating = window around the=20 target. This represents the manufacturing variation. However, in this = case the=20 springs are also prone to deterioration over time. Life testing of the = springs=20 indicates that they deteriorate by 1 lb. over a 5 year period. = Deterioration is=20 best represented by a worst-case tolerance since the closure force must = be=20 within specifications at both the beginning and the end of the 5 year = period.=20 Deterioration is best represented by the worst-case tolerance 10 +0 =961 = lb.=20 Combining these two tolerances results in 10 + 0.3 =961.3 =A3 0.2 lb. In general, deterioration of an input = should be=20 represented as a worst-case tolerance. The deterioration effect should = then be=20 combined with the manufacturing variation to obtain the final = tolerance.

SIX SIGMA = QUALITY

There is an important link between six sigma quality and process = tolerancing.=20 Six sigma quality, a term introduced at Motorola, requires that a = measurable=20 characteristic such as flow rate or seal strength have no more the 3.4 = defects=20 per million units produced. This is in turn translated into the = requirements=20 that the average and standard deviation meet the conditions specified by = the=20 process tolerance ty =B1 = D y =A3 s y where:

3D"Equation (Eq. 32)

3D"Equation (Eq. 33)

3D"Equation (Eq. 34)

USL is the upper specification limit and LSL is the lower = specification=20 limit. This process tolerance is shown in Figure 11.

3D"Figure

Figure 11: Six Sigma Process = Tolerance

For the process tolerance in Figure 11, the worst-case scenario is = when the=20 average is 1.5s off-target resulting in an = average that=20 is 4.5s from the nearest specification limit. = Based on=20 the normal distribution, when the process average is 4.5s from the specification limit, the defect rate is = 3.4=20 defects per million. This is the definition of six sigma quality. The = name comes=20 from the fact that six sigma quality requires specifications that are = =B1 6s wide.

Since the requirements for six sigma quality are specified in the = form of a=20 process tolerance, it is only natural that process tolerancing be used = to ensure=20 that these tolerances are achieved. Process tolerancing provides an = alternative=20 to the combined dynamic and static mean tolerance analysis procedure = described=20 in Harry and Lawson (1992).

SUMMARY

Worst-case tolerancing tends to overestimate the variation of the = output.=20 Cost suffers when the variation of the output is overestimated. = Statistical=20 tolerancing, tends to underestimate the output variation. Quality = suffers when=20 the variation of the output is underestimated. Using process = tolerancing, it is=20 possible to accurately predict the behavior of the output, providing the = desired=20 quality at a lower cost.

Process tolerancing provides a unified approach to tolerancing with=20 encompasses both worst-case and statistical tolerancing. It allows = worst-case=20 tolerances to be used for some inputs and statistical tolerance are used = for=20 others. No longer does one have to use the same type of tolerance for = all=20 inputs. The purpose of a tolerance analysis is to predict the behavior = of the=20 output based on knowledge of how the inputs behave. Key to its success = is the=20 ability to accurately describe the behavior of the inputs. Using process = tolerancing, you can describe behaviors for inputs not adequately = described by=20 either worst-case or statistical tolerances. This provides solutions to = commonly=20 occurring problems such as off-center processes and multiple identical=20 components.

REFERENCE = LIST

Bender, A. (1962). Benderizing Tolerances - A Simple Practical = Probability=20 Method of Handling Tolerances for Limit-Stack-Ups. Graphic = Science=20 Dec.

Cox, N. D. (1986). How to Perform Statistical Tolerance = Analysis.=20 Milwaukee: ASQC Quality Press.

Evans, D. H. (1975). Statistical Tolerancing: The State of the Art.=20 Journal of Quality Technology 7 (1): 1-12.

Harry, M. J. and Lawson J. R. (1992). Six Sigma Producibility = Analysis and=20 Process Characterization. New York: Addison-Wesley.

Taylor, W. A. (1991). Optimization and = Variation=20 Reduction in Quality. Libertyville, IL: Taylor Enterprises, = Inc.

Taylor, W. A. (1997). VarTran User = Manual.=20 Libertyville, IL: Taylor Enterprises, Inc.

 

Presented at Fall Technical Conference,=20 1995

Send mail to info@variation.com with=20 questions or comments about this web site.
Copyright =A9 1997-2005 = Taylor=20 Enterprises, Inc.
Last modified: January 24, 2000

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