Justifying Six Sigma Projects in Manufacturing Management

Transcription

Justifying Six Sigma Projects in
Manufacturing Management
Tapan P. Bagchi1
Narsee Monjee Institute of Management Studies, Shirpur
Abstract
This paper develops and illustrates a case in
to examine and quantify such shortfalls—many
manufacturing management, using the instance of
being preventable by reduction of quality variance
justifying quality improvement of ball bearings—a
and/or part variety. Statistical and numerical
common precision product whose correct
models have been used. Thus, targeting beyond
manufacture and assembly greatly affects their
scrap and rework, this paper invokes modeling
efficiency, utility and life. Mass-produced at high
methods to quantify such not-so-visible constraints
speed, bearings extend a fertile domain for
that limit productivity and profits of high-volume
benefiting from QA apparatus including Gage R&R,
high-speed processes.
ISO standards, sampling, and SPC to Six Sigma
DMAIC (Pyzdek 2000). However, when large
Keywords: Precision Manufacturing, Variance
investments are involved, it becomes imperative
Reduction, Hidden Costs of Poor Quality, Numerical
that besides the obvious, the hidden costs of quality
Modeling, Monte Carlo Simulation.
be located and sized. This paper provides methods
ISSN: 0971-1023
NMIMS Management Review
Volume: April - May 2012
1
Dr. Tapan Bagchi is the corresponding author who can be reached at [email protected]
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exceptional domain in which quality assurance
Introduction
Managing precision manufacturing of specialized
products at their highest achievable performance
level is anything but trivial, but management
frequently finds itself unable to justify the large
investment entailed in superior technologies
required to do so. We illustrate a procedure for this
by using a real case—a firm's pursuit to upgrade the
quality of automotive ball bearings (Figure 1) that it
produced. Mounted on skateboards, passenger
vehicles, machine tools and even a space shuttle's
engine, bearings have been a major mechanical
innovation that reduces surface to surface contact
between moving surfaces, thereby reducing friction
and saving motive energy requirement and its
wasteful loss. Traced to drawings by Leonardo da
Vinci around 1500, bearings today help the
“bearing” of load typically between a shaft and a
rotating surface. Bearings are mass-produced by
manual to fully automated machining and assembly.
Their precise manufacture greatly affects their
efficiency, utility and life. Bearings, as contrasted
with appliances, toys, furniture, etc., also are an
methods from Gage R&R, ISO standards, SPC
(Montgomery 2005) and sampling to Six Sigma
DMAIC (Pyzdek 2000; Evans 2005) can impact
business.
A mid-size bearings manufacturer gave this writer
an extraordinary opportunity to observe first hand
the bearing production process, freely interact with
the expert staff manning the machines and work
stations and vary process parameters in
experiments to observe their effect on product
quality. This company had already trained its staff in
TQM
tools and TPM methods. However, no
measurable impact from these on either the bottom
line or top line could be discerned by management,
as is often the case. Therefore, a rigorous and
advanced method that could elevate profits and
customer satisfaction was sought. Six Sigma
appeared to promise such breakthrough—but, the
gains from it could not be projected beforehand.
This paper describes the modeling methodologies
that led to successfully justifying state-of-the-art
technology interventions in this company.
Figure 1 The Components of a Ball Bearing
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Justifying Six Sigma Projects
To scope the quality improvement project Cpk was
Motivation of the current study was to help the
assessed at quality-bottlenecked process steps. The
manufacturer find economic justification for
plant ran Orthogonal array experiments (Taguchi
possible major technology intervention that could
and Clausing 1990) to locate process factors
cut tangible and intangible COQ (cost of poor
speculated to affect quality by the plant. Thus, key
quality) (Gitlow et. al. 2005, Gryna et. al. 2008) and
quality deviations in need of attention could be
delays and raise profits by reducing production of
identified, but no firm basis could be cited to
marginal quality bearings. With improved quality,
motivate impacting them. However, such studies led
the company could possibly sell to premium bearing
to re-statement of the project's charter, which
markets.
became “predict variability of the final bearing
assembly based on information available on part
This paper is organized as follows. The next section
variability.” Key parts in question here were the
of this paper outlines the relevant aspects of bearing
inner and outer rings of the bearing and the rolling
parts manufacture and assembly, and then states
balls (Figure 1).
the problem of immediate focus—low yield
(proportion of acceptable production) of quality
Deductive variance prediction from parts to whole
bearings, resulting from parts with high
proved too complex as it led to queuing or inventory
dimensional variability (σ ). The manufacturer
type models (Bhat 2008) involving random
variables discretised (rounded down) from real
numbers. General forms of such models (see (1) and
(2) later in this paper) have not yet been solved
theoretically. Consequently, the process—the
assembly of complete bearings from parts
separately manufactured by grinding/honing
machines with significant variability in them—was
first numerically modeled and then studied by
Monte Carlo simulation. The objective was to
quantify the relationship of high variability (σ) in
manufactured ring sizes (outer and inner) and the
variety of bearing balls needed to complete the
2
wanted to be competitive in both quality and
profitability. Subsequently, we portray a key
operational bottleneck that the plant faced—the
challenge of selecting balls of correct size to match a
random pair of outer and inner rings produced by
track grinding. Next, we provide a statistical
perspective of bearing assembly since all machining
operations are subject to random variation yielding
rings with considerable variance in their
dimensions. Then, we show the steps to numerically
determine the dependence of distinct ball size
requirements on ring grinding variance, and then
relate this to yield.
assembly. Till this point, “experience” had guided
the creation of the large assortment of ball sizes that
the plant used. Producing a wide assortment of ball
sizes with frequent machine set up changes (a
hidden cost) was a burden for the plant. But
management could find no sound method to answer
why this practice should be changed. They were
“committed to deliver high performance bearings to
customers”, so the issue remained stuck there.
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Subsequently, a simulation procedure is developed
to predict process yield within stated precision
given specified randomness of outer and inner ring
sizes. Typical questions that management will
confront that could be successfully tackled by such
simulation are presented next. Results of a number
of designed simulation experiments indicate that a
rising variety is required in distinct ball sizes as
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grinding variance (σ ) goes up (Figure 4). Next we
(CTQ) dimensions remain within 0 and 1.2 micron
illustrate one such use of simulation to determine
and surface finish is acceptable.
2
the distinct categories of “standard” ball sizes
required in high yield assembly given Cpk ratings of
The Production of Balls
track grinding. Subsequently, we use assembly costs
Balls are the most critical engineering component of
and visible COQ (scrap and rework) to help project
bearings as they directly “bear” the load while
the justifiable capital expenditure in technology
providing minimal resistance to movement. Each
that could improve grinding precision (i.e., reduce
ball must be precision-machined and polished, and
σ ). The paper ends with a summary of conclusions
together balls cost typically about 40% of a
that management should expect to see in such a
bearing's manufacturing cost. Ball manufacturing
involves the following steps (The Manufacture of
study.
Ball Bearings 2009):
Ball Bearing Manufacture
Ball bearing production is now generic—used by
industry worldwide. Some steps may be automated
while others are kept manual. Many steps are
augmented by automated inspection and SPC. All
ball bearings comprise the outer ring, the inner ring,
and the rolling balls along with some support parts
(Figure 1). Each of these parts is a precision product
made from special steel and it must be produced,
tested and then assembled correctly in order to
enable the completed bearing to perform at location
as expected. Ring grinding also called track grinding
comprises a sequence as follows (Ball Bearing 2009;
The Manufacture of a Ball bearing 2009):
• Cold or hot forming operation using steel wire or
rods by a heading machine. This leaves a ring of
metal (called flash) around the ball.
• Removal of flash by rolling between grooved rill
plates, giving each ball a very hard surface
greatly needed for its load bearing capacity.
Process settings include pressure and spinning
speed while squeezing by rilling hardens the
ball.
• Heat treatment.
• Setting ball grinder and grinding the ball to its
specified dimension.
• Lapping to render a perfectly smooth shiny
surface, without removing any more material.
• Turning of raw material—steel tubes and
bars—into raw rings (a step that is often
Bearing Assembly
outsourced).
Outer and inner ring pairs and the corresponding
• Heat treatment of raw rings.
correct size balls are then selected. Rings are
• Precision face grinding.
manually or automatically deformed lightly to
• Precision outer diameter (OD) and inner
insert the balls into the tracks between the rings.
diameter (ID) grinding to produce tracks in rings
Retainer rings and lubricants may be added. Each
on which the balls will roll.
final bearing assembly is 100% tested for clearance
• Final honing to create surface finish.
and noise.
Ring width and track dia are the control targets in
Critical in final bearing assembly is the selection of
track grinding. At each step, sampled inspection is
balls that will result in the specified radial clearance
done to ensure that the final critical-to-quality
between the balls and rings (see Section 2). Note
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that the wider is the variation in dimensions of the
Process Variability and its Hidden
bearing's inner and outer rings, the larger will be the
Impact on Productivity and Costs
number of and variety in the size (diameter) of the
high-precision hardened steel balls required to
complete the ball selection step. To adapt to ring dia
variation (imprecise grinding or high σ), industry
produces balls of several different “standard”
sizes—incrementing in 1 or 2 micron steps in
diameter. Such availability of balls of different sizes
helps the plant reach the desired radial clearance in
the maximum proportion (measured as p) of
bearings assembled, even with ring size variability.
p measures the yield (= fraction of on-spec bearings
automatically assembled from the total outer/inner
ring pairs produced) of the assembly line. Ring pairs
for which a matching ball size cannot be
automatically found reduce p. Such rings are
separated and assembled manually by using presorted matching rings. Rings that cannot be
manually matched are scraped.
A typical precision bearing costs about USD 10 to
make. The subject plant made 30 million
bearings/year and was incurring an internal failure
financial loss of about 1% annually in scrap and
rework, very significant and substantial, and visible
to management. Additionally, poor quality caused
intangible losses. For instance, the outsourced
vendor put “extra” steel on the raw turned rings
(and charged for it) that were machined off to
produce the feed to the precision grinding process.
Such non-value added machining reduced the
plant's productive capacity. Besides, management
remained curious whether improved machining
precision (high Cpk at grinding or low σ) could
reduce the variety in the size of components (here
balls) that must be produced and stocked to provide
the desired clearance and “custom” matching
during assembly (explained in Section 2). This
“variety” of required balls—each kind custom-
Manual ring-ball matching is slow (<1/10th of the
made—was a significant, but an unknown
hourly yield of automated assembly) and a costly
component of a bearing's production cost.
operation. Note, also that each “standard” ball size
Management speculated that the existing poor
must be separately produced, requiring extra setups.
grinding precision (high σ) caused this variety and
Thus, COQ considerations will urge one to lower the
lost production capacity due to frequent set up
2
variation σ (or σ) of the track grinding operation.
changes on ball machines. This could perhaps be
Intuitively, one feels that lower the σ, smaller will be
optimized by a study of ball production which
the needed number of (“standard”) balls of distinct
currently accounts fo 40% of total production cost.
diameters for final assembly. Hence, lower overall
On the other hand, to find technology benchmarks,
cost of bearing production. These considerations
the plant had checked the output of outer rings on
led us to develop the quantitative relationship
two different track grinders, one 25 years old and
between ring (track) grinding variability (σ) and the
the other new. Cpk differed by 0.69 to 2.02 between
variety in “standard” ball sizes needed to keep yield
the two, showing a realizable possibility for
(p) high. This methodology is described Section 4
reducing σ provided the monetary incentive for such
onward.
technology upgradation could be quantified.
However, as noted, installation of all new machines
was to be a large investment that implored
quantification (monetizing) of the incentives. This
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task the plant found difficult.
Difference = (ID of OR) – (OD of IR) = RC + 2 Ball dia
when RC > 0
(2)
Thus, one could intuit that wide dimensional
variation (σ) in bearing parts—inner rings, balls,
The notations used here are ID for inner track dia of
and outer rings—led to production of many rings
the outer ring (OR) and OD for the outer track dia of
and balls that could not be automatically assembled
the inner ring (IR). Ball dias are stepped from a
to make finished bearings while maintaining the
smallest practical size (s) in discrete units of 1 or 2
desired RC (radial clearance, see Section 2) and
micron. Hence, for a feasible assembly.
other quality characteristics. One approach to raise
hourly yield (the proportion of correct assemblies
Ball Dia = [Difference - RC]/2
(3)
from all parts produced) would be to reduce all
variances. A partial solution to this quandary would
Industry has found it expedient to manufacture
be the use of “standard” sized large assortment balls
balls used in bearings to “standard” dimensions in
as most bearing manufacturers currently do. The
specified steps, like shoes, and not in continuous
convenient though expensive way would be to sort
dimensions (SKF Bearings Handbook 2009). So the
all parts produced and then find matches that meet
selection is made by rounding down to the nearest
CTQs including RC. Yet another approach to reduce
size standard ball to the calculated “Ball Dia”
the cost of poor quality (visible and hidden) would
determined by (3) for each IR/OR pair being
be to seek optimal variance reduction considering at
assembled. In this plant, “standard balls” are made
the minimum all measureable costs in ring grinding,
in one micron steps.
ring matching (pairing) and then the selection of
balls from the resulting smaller assortment of
Note that ID and OD are subject to grinding
bearing parts.
variation. Hence, as RC increases due to bearing
design requirements, the random “Difference” (ID of
Ball Selection Process for Bearing Assembly
OR) – (OD of IR) in (2) for many IR / OR pairs will
Sorting randomly produced rings and balls to find
increase, and hence the variety required in standard
matches that will successfully fit is slow and effort-
ball sizes for the different randomly picked outer /
(cost-) intensive, even if the task is automated.
inner ring pairs. Conversely, if the “Difference” in (2)
Nevertheless, ball selection is a critical practice in
were small, a smaller number of standard ball
bearing assembly worldwide due to the
choices will be required. Of course, if extra effort
considerable dimensional variation of machined
was made to sort all IRs and ORs before assembly so
parts—inner and outer rings. Such selection aims at
that the pairs would result in final radial clearances
achieving the CTQ target radial clearance (RC) that
close to the target RC, perhaps only one or two
must meet the engineering spec of each assembled
standard-sized balls will be required.
bearing. RC (using X+ to represent the maximum of X
or 0) is given by the formula:
RC (Radial Clearance) = [(Inner Dia of OR - Outer Dia
of IR)+ – 2 Ball Dia]+ (1)
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Such sorting of rings before assembly, unless done
(using equation (3)) held in stock. The ball
completely and cheaply, is not justifiable as it will
selected should be such that the final assembly
likely require such matching to be automated, with a
should result in the target RC between the balls
great deal of rejections and recycling of rings since
and the ring tracks.
IR/OR dimensions vary randomly, before a matched
pair is passed to the ball insertion workstation.
• Assemble the bearing by pushing the balls into
the tracks.
Selecting a “standard sized” ball from an assortment
• Conduct visual and dimensional checks and
of balls, therefore, is the preferred option
performance tests (e.g., noise at full speed) on
worldwide in bearings assembly and there are
each final assembly.
specialists who manufacture such automated
bearing assembly machines. Swiss bearing makers
• Reject bearings that are unacceptable. Accept
others for further processing.
use this procedure routinely.
With wide variation within (ring-to-ring) the inner
Ball selection logic is as follows. Generally, it is
ring and also within the outer ring dimensions
desired that the output produced by a
produced, a relatively large number of trials are
manufacturing process should fall within the
required in Step 3 above to find the best matching of
specified range, fixed by tolerance or “spec”.
balls for the inner and outer ring. But, as such ring
Furthermore, the larger the spec range the greater
grinding variation (σ) decreases, within a few trials
will be the permitted variation in the output that is
the best fitting balls—due to lower dimensional
acceptable. Since a bearing comprises the assembly
variability of rings—may be found. This is why
of the inner ring, outer ring and balls, each produced
leading bearing producers are moving towards
with some variation, to deliver a “quality” bearing,
raising Cp/Cpk of ring manufacture. Such action
the manufacturer has to find the best match of an
reduces output variations and hence the average
inner ring, an outer ring and a ball size such that the
“Difference” in (2). The result is that then fewer
balls fit correctly (with a clearance) within those
“standard” ball sizes will be required to assemble
inner and outer rings. The bearing will then possess
the bearings while one would still deliver the
the desired target RC to allow the balls to roll
targeted finished bearing performance.
between the rings and have good life. Therefore, ball
selection is implemented in the following steps:
So, as Cp/Cpk or the “Sigma” metric (Pyzdek 2000) of
the ring grinding process goes up, it reduces not
• Measure and dimensionally sort all finished
inner and outer rings.
only the process cycle time that includes matching,
but also production of defective bearings (possible
• Produce and sort an estimated required
marginal misfits) and rejection of rings in bearing
assortment (distinct sizes) of standard balls and
assembly. With low Cp/Cpk many inner/outer ring
keep them in stock. This step is guided by the
pairs randomly picked will not match at all, creating
shop's experience with the quantities of
scrap and raising the cost of poor quality.
unmatched rings generated at assembly.
• Find matching ring pairs that will lead to on-spec
This condition raises a question for the bearing
assembly using the rings chosen and a ball size
manufacturer: What should be the relative
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precision with which the rings should be
A Statistical Perspective of Variations in
manufactured? In this paper, we outline procedures
Bearing Assembly
to help relate process variability (σ) in inner and
outer ring machining to the variety required in ball
sizes to make high performance bearings.
Hence, rather than bear only on inductive or
intuitive reasoning as alluded to in Section 2, we
sought a stronger case for variance reduction based
on analytical reasoning. It was clear that with
incentives thus made visible, one would adopt a
data-driven and fact-based rather than intuitive
quality improvement stance.
Assume that inner rings are produced with a
nominal outer track mean dimension μI and
standard deviation σI. Similarly, assume that outer
rings are produced with inner track mean μO and
standard deviation σO. Let the size of a randomly
picked inner ring be I and that of an outer ring be O.
Since rings are independently produced, there is no
relationship between random dimensions I and O.
However, during assembly, a “feasible” bearing can
be assembled using I and O only when:
Prima facie, as noted above, grinding Cpk estimates
O - I – 2 B – RC ≥ 0 with RC > 0
obtained at the start of the project hinted at a
as specified by engineering
(4)
significant opportunity to reduce cost of set up
changes as well as rework and the production of
Here, RC is the designed (targeted) radial clearance
unacceptable rings. Management already intuited
and B is the diameter of the balls to be placed
that if ring size variation could be reduced, fewer
between the outer and inner rings. In (4), RC is an
standard ball sizes would do the job, a lot of set up
engineering constant (> 0), dictated by bearing life
hours could be reduced, and the manual assembly
considerations. B is the size of the (identical) balls
done with inner and outer rings rejected by the
selected to be placed in the bearing to make the
automatic assembly machine could perhaps even be
assembly possible. As noted in Section 2, B has to be
eliminated. However, quantitative estimates of such
carefully chosen for each I and O pair so as not to let
incentives were unavailable to them. Due to the
the final radial clearance of the assembled bearing
processes being random, the tools to help tackle this
deviate too much from the design RC value.
situation could either be the exact theoretical
Otherwise, the fit will be too tight or loose, affecting
modeling of the assembly process incorporating the
the bearing's installed performance and life.
rounding-to-the-lower-dimension practice to pick
balls or a numerical approach or Monte Carlo
As described in Section 2, in order to make the
simulation. In this study, each of these methods was
process workable, industry produces “standard”
explored.
precision balls of various sizes and keeps those in
stock. But, these sizes (like shoe sizes in a shoe
store) do not vary continuously. One produces balls
only at certain “stepped” sizes, usually in steps of 1
(or 2) micron starting with the smallest ball. It is not
difficult to see from (4) that the higher are track
dimension variances σI2 and σO2+, the wider will be
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the probable difference between the OR/IR rings
intuitively as said in Section 1 and elsewhere. A
pairs randomly picked for assembly and the larger
small point to be noted is that the starting ball size s
will be the number of differently sized balls
required will be smaller when track grinding
required to complete the assembly. In fact, if the
variability (σ) is high. (This is shown later in Figure
distributions of the outer and inner ring sizes
4 with RC = 20 and σ = 1 vs. cases when σ is higher.)
overlap due to high variance, many candidate pairs
will be rejected during the automatic assembly step
Recall next that O and I are dimensions of two
that picks one outer and one inner ring and checks
rings—one outer and one inner—that we randomly
their dimensions for a feasible assembly.
picked with the hope of matching them successfully
by fitting them with the appropriately sized balls.
For illustration, let the balls be made precisely in
Frequently, σ for the grinding process—inner or
steps of 2 micron, starting with the smallest ball of
outer—is nearly the same. However, in the general
size s. If balls are sized successively, they will have
case, let the variance of outer ring track dia O be σ02
diameters s, s + 2, s + 4, s + 6, …, s + 2(k – 1), … Then, if
and that for the inner ring track dia I be σI2. Then, the
the ball with size s + 2(k – 1) gets matched, given I, O
larger are process variances σ02 and σI2, the wider the
and RC (> 0), we shall have
possible random difference or gap (O – I) is likely to
be. And, if the distributions of O and I overlap, the
O – I – RC = 2(s + 2(k – 1))
(5)
Equation (5) leads to
cannot lead to successful bearing assembly when O
k - 1 = [(O – I – RC)/2 – s]/2
and I due to their high variability would not leave
This leads to
k = 1 + [(O – I – RC)/2 – s]/2
much clearance between them. In fact, random
(6)
This expression simplifies to
k = 1 + (O – I)/4 – RC/4 – s/2
higher will be the proportion of ring production that
variables O and I being independent (the rings are
separately produced), the variance of the random
(7)
variable (O – I) is the sum of the variances of the two
random variables O and I. Hence,
Equation (7) indicates some important things. First,
since RC (radial clearance) and s (the smallest size
standard ball available for assembly based on
engineering considerations) are constants, the
number k is directly dependent on the difference (O
– I), the dimensional difference between the outer
and inner ring track diameters that we are trying to
assemble into an acceptable and properly
functioning ball bearing. Equation (7) suggests that
the wider is the difference between O and I, the
larger will k be, indicating the number of differently
sized balls that we must have on hand must then
also be large in order to complete bearing assembly
with outer and inner rings produced with wide
variability. This deduction confirms what is held
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2
2
Variance (O – I) = σO-I = σO + σI
2
(8)
Sometimes—since the outer and inner rings are
independently produced and picked for assembly
randomly—we may even have picked two rings
when O < I! Those two rings must then be put aside
for manual matching from a bin of assorted rings in
stock.
What, therefore, is the message? The first is that in
order to reduce the fraction of outer and inner rings
rejected by the automatic assembly machine
because (O – I) does not leave enough room for two
standard balls and RC, one should reduce both σO2
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and σI . Next, when σO and σI are large, many more
satisfactorily with another part. If part size
outer and inner rings will have a wider gap between
variability is too high, however, this process slows
them, leading to a larger k, or a wider variety of
down the generally high speed production rate and
differently-sized balls to be stocked for the
introduces hidden costs of poor quality. When outer
automatic (or even manual) assembly.
and inner rings are mass produced, to find a
2
2
2
matched (Ix, Ox) pair, a relationship is used to guide
To summarize, this section has found why reduction
ball selection that guarantees the required radial
of variability (as urged for instance by the advocates
clearance RC. This relationship, shown below,
of Six Sigma) may raise yield, reduce the variety of
determines the correct ball size for the (Ix, Ox) ring
balls and reduce reprocessing rejected rings
pair.
manually. This will reduce COQ. In the next section,
the impact of high machining variance is
Ox – Ix = RC + 2 Bx = RC + 2(s + i)v
numerically assessed. We see that as σI - O rises, so
The correct ball size Bx is given by
does the variety required of “standard” balls.
Numerical Assessment of Impact of Variance
Let the largest size standard ball available be of
Reduction on Ball Variety k
diameter (s + k). Our attempt here will be to link (Ox –
Is the cry for variance reduction (Evans and Lindsay
2005) only hard sell? What if track grinding
variability (σ) in ring production was reduced by
half? How would that impact yield p at automatic
assembly? In this section, we numerically analyze
this issue to quantify this impact. We first define
some variables as:
Ix = diameter of a randomly produced inner ring.
Ox = diameter of a randomly produced outer ring.
RC = desired radial clearance after assembly.
Bx = ball size used to assemble the ball bearing.
Ix) to k. We expect that when the tracks of the outer
and inner rings are ground, there will be some
variability. Intuitively, we feel that larger the
resulting track diameter variances σOx2 and/or σIx2,
larger will be k or the variety in the sizes of
“standard” balls required for correct assembly. This
relationship can be numerically established as
follows:
Given k, the limits on the range or gap (Ox – Ix) to lead
to a correct bearing assembly may be determined.
Such an assembly will result in the desired radial
Also, let s micron be the smallest size standard ball
available, balls being made in sizes stepped by one
unit (1 micron), starting from size s. Thus, the (i +
1)th ball in this sequence of “standard” balls will be of
clearance RC to assure good life and other
performance criteria (SKF Bearings Handbook
2009). These limits are given by
dia (s + i) micron.
This gives
In bearing manufacturing almost nothing is thrown
away since scrap is visible. It is collected and reprocessed, but at additional cost. In bearing
and
assembly, attempt is made to find a part that will fit
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This produces the two “not a feasible assembly”
When X and Y are independent and normally
lower and upper limit conditions as
distributed such that X ~ N[μx, σx2] and Y ~ N[μy, σy2],
(9)
then U is distributed as N[μx -μ y, σx2 + σy2]. Therefore,
if FU(u) represents the cdf of U, then the probability
and
that a randomly picked inner and outer ring will
(10)
lead to a feasible bearing assembly is
P[RC + 2s U RC+2(s + k)] = FU[RC+2(s + k)]-FU[RC + 2s]
We will now attempt to find the distribution of (Ox –
Ix), the random gap between the outer and inner ring
track diameters in which the balls will sit. Let the
probability density functions of two independent
In the current application, samples of over 300
random variables X and Y be f(x) and g(y). Then the
inner ring sizes were found to be normally
distribution of U (= X – Y) will be convoluted (Rice
distributed; hence, Ix ~ N[μI, σI2] (Sharma 2009).
2007, p. 97).
Similarly, the machined outer ring track dia were
also normal; thus Ox ~ N[μO, σO2]. Such observations
Determining P[U u] for arbitrary distributions f(x)
are commonplace in mechanical metal removal by
and g(y) is difficult. However, we can determine P[U
grinding (Gigo 2005).
u] for some common distributions assumed for X
and Y as follows.
Figure 2 Effect of ring dia variability on the variety of ball sizes required for high yield bearing assembly
Expression (11) is highly informative. Note first that
will be the value of k, the variety in size of balls
Φ(u)-the cdf of the standard normal distribution-is
required to produce a feasible bearing assembly.
a monotonic function of u. Next, note that in (11),
engineering considerations fix the quantities RC, s,
Note further that σO2 and σI2, respectively, represent
μO and μI. Therefore, if we wish to have 99% of the
the variances of the track diameters of the machined
machined inner and outer rings correctly
outer and inner rings. Hence, higher the variability
assembled by picking two equal-sized balls from the
in track grinding, larger will be k, the variety in
size range [s, s + k], the larger is
different sized balls required-the assortment
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, the larger
83
starting with the smallest dia s to the largest size s +
Table 1 displays the numerically estimated fraction
k. Figure 2 shows the numerically determined
of correctly assembled ball bearings with assumed
dependency between grinding sigma (standard
track dia grinding variances σO2 and σI2 . We used
deviation σ) and ball variety shown in ball size
here numeric values of Φ(.), the standard normal
increments in micron. For typical grinding
cdf. Observe the impact of increasing minimum k on
operations performed on outer and inner rings,
yield (p). Desirable yields exceeding 99% with σ= 3
machining variances σO2 and σI2 on machines of
were obtained with k values near 10. For lower σO2
c o m p a ra b l e c o n d i t i o n we re s t a t i s t i c a l ly
and σI2 one would require a smaller k or fewer sizes
indistinguishable, represented here by σ.
of balls to complete the assembly without rejecting
rings. When k is set at 0, i.e., when only balls of
nominal size are available, very few ring pairs are
expected to match; hence, almost no bearings could
be assembled.
Table 1 Impact of rising track dia variability (σ) on k (minimum variety of balls required
for 100% yield in bearing assembly)
Process
scenario #
σOuter ring
σInner ring
k = minimum distinct
sizes of balls required
p = % yield of correctly
assembled bearings
1
1μ
1μ
3
100%
2
2μ
2μ
7
100%
3
3μ
3μ
10
100%
4
4μ
4μ
14
100%
5
5μ
5μ
17
100%
6
6μ
6μ
21
100%
7
1μ
5μ
12
100%
8
5μ
1μ
12
100%
This analysis, done numerically, reasserts the
also reduce scale (larger lot size) advantages. These
fundamental credence—larger the ring machining
considerations will prompt one to seek production
2
2
variability (here σO and σI ), larger will be the
methods or strategies requiring fewer assortments
variety balls needed to ensure correct assembly.
of balls. All this is consistent with the spirit of Six
From cost of quality standpoint, each additional
Sigma®, which champions variability reduction
“standard” sized ball requires its own independent
(Pyzdek 2000; Evans and Lindsay 2005).
set up and production run with its own settings on
rilling and other machines. Each time such a set up is
On the other extreme, if production machinery is set
changed, it reduces the ball plant's throughput,
up arbitrarily, a really large assortment of balls will
hence, its productivity. Variety or frequent changes
be needed by the assembler. The “rejects” from such
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ISSN: 0971-1023
an automated assembly will require a wider
effort, in probing the effect of operational variations
support to hand-match rings that the assembly
and different scenarios for which a numerical model
machine cannot accept within its normal
involving unusual distributions without probability
setting—again raising COQ, hence, adding to
tables may not be easy to build.
production cost. Thus, a trade-off appears possible
between high Cpk ring grinders (with small variance)
Monte Carlo Simulation of Ball Bearing
on one hand and producing and stocking a larger
Assembly
variety of “standard” balls with sizes varying from s
Figure 3 is a screen shot of the Monte Carlo model
2
2
to (s + k) on the other. As shown, if σO and σI are
built in Excel® for bearing assembly simulation. The
reduced, k (the number of distinct ball sizes) will be
top portion of the worksheet shows where first the
smaller. Figure 2 displays this relationship. When
process parameters (nominal dimensions μO and μI,
the relevant costs are available, an optimization can
and standard deviations σO and σI) are entered.
be attempted—best done before capital is invested
Parameters s and k indicate ball sizes. In this model,
in plant machinery and technologies. We note that
s is specified by the analyst while incremental ball
the automatic selection of “fitting” parts is a
sizes are automatically determined by the model by
common technique implemented in thousands of
finding the correct k using (7) above. At the
production systems worldwide. As reconfirmed
beginning, distribution of dia variations was
here for bearing production, grinding variance
established as normal—common in grinding (Gijo
reduction will lead to the handling of fewer sizes of
2005).
balls getting assembled into finished bearings,
lowering the cost of poor quality. Such cost
reduction explored during planning could possibly
achieve optimum technology configuration in the
plant.
In the following section, we describe a Monte Carlo
method to project the fraction of ring mismatches
occurring during assembly, if the analyst specifies
the ball sizes available, radial clearance desired, the
nominal ring sizes μO and μI, and grinding standard
deviations σO and σI. Prevailing dimensions and
standard deviations may be obtained from the
relevant X Bar-R control charts maintained on the
shop floor. We envisage two reasons for attempting
to study the bearing assembly process by Monte
Carlo simulation. The first is the flexibility that it
affords in respect to the distributions of the part
dimensions. The second is the flexibility simulation
extends, albeit at the cost of higher computational
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85
Figure 3 The automated bearing assembly simulator
In this simulation of automated bearing assembly,
among the variables (e.g., relationships of the
ring sizes O and I are randomly found, presently
type (7), (9), (10), etc.). The variables presently
normally distributed dimensions based on μO and μI,
involved were outer and inner ring track dia, ball
and σO and σI, respectively, as entered in the green
size, and the ball variety k.
zone of the worksheet. Random samples are drawn
by Excel® using the NORMINV() function. Each row
• Set up the experimental framework—the
from Row 26 downward uses one simulated outer
variables to be manipulated (here k), the
ring (O value) and one simulated inner ring
constants (the smallest ball size s, and radial
dimension (I value). Column headings (Column F
clearance RC) and the random sampling
onward) implement the relationship (7) in the steps
mechanism from specified distributions (O and
of computation. The simulation progresses as
I).
follows:
• Once the model is coded and set up (here the
• Identify the sequence of processing steps,
Excel® model of Figure 3), conduct pilot runs to
control and noise factors, any information on
estimate the variance of the response (here p or
randomness and its nature (the associated
yield) and then the required sample size (length
distributions). Determine the relationships
of the simulation run).
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ISSN: 0971-1023
• Simulate for the desired sample; collect data and
complete output analysis.
The total count of “ball unavailable” indicates ring
size mismatches that could not be automatically
assembled into a bearing using the k standard
• Replicate runs with identical seeds to reduce
assorted ball sizes provided. This estimates p and
variance of the yield (p) estimated (Law and
the manual work needed to complete the job. The
Kelton 2000).
simulation model was validated using physical
inner/outer ring production lots of size 300 each at
Results of one round of simulated assembly with
the prevailing grinding variability (σ) level (5
inputs from worksheet cells C26, D26, E26, etc.,
micron) and a target RC of 20 micron. Hand
appear in Column K—“Micron Size of Balls
assembly produced about 10% mismatches (too
Required.” Note that in practice calculated ball sizes
small or too large rings) that could not deliver the
are rounded down to a whole number standard size
target RC using 12 different standard sized balls at
for conservative (slightly larger) clearance resulting
hand (cf. Table 2). For given σO and σI for grinding, a
in the assembled bearing. If a feasible size of balls is
sample size of 1,000 simulated assemblies provided
found, its size is noted. Otherwise the
a conservative 2-digit precision of yield estimate,
corresponding row (i.e., the simulated O and I pair
sufficient to illustrate the utility of simulation.
put up for assembly) reports “ball unavailable”.
Table 2 Numerical estimation of yield p (fraction of bearings automatically assembled)
using the standard normal cdf
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87
Experiments could be now run by treating process
ball sizes must be precision-made to exact specs
or machining variances σO and σI as experimental
(10th or better in micron). Manual sorting and
variables with nominal sizes μO and μI and
assembly using matched outer and inner rings and
engineering parameters RC, s and k held at specified
balls is the traditional fallback. But this results in
levels. Plots of the output data will graphically
errors and the consequent unwanted variation in
indicate the effect of variances σO and σI on the
bearing characteristics. Some operations are,
process output p (yield)—the fraction of rings
therefore, automated. The common one in bearing
submitted for assembly that could be finished into
production is assembly.
complete bearings. This exercise showed how the
Still, many questions remain about the estimation of
effect of comptemplated process changes may be
the visible and hidden COQ and the economic
quantitatively estimated. Simulation is a well-
optimality of technology interventions. Some of
known method to use here (Law and Kelton 2000).
these questions may be probed by Monte Carlo
simulation. Examples are:
A Strategic Application of the Assembly
Simulation Model Created
A plant typically confronts questions for which
quantitative answers are not easily found to guide
strategic changes such as adopting new technology.
Answers are often sought based on the tacit
(experiential and intuitive) knowledge of senior
management, and the machinists and quality
control staff with practical hands-on experience.
However, industry now generally appreciates that
reduction in variability of parts and the final
product or service should be a key target for an
enterprise. For ball bearings, part size variability
affects the final radial clearance achieved, which
affects it life, performance and production cost.
Managers also want consistency and, therefore,
look to locate the “problem” stages in the shop
processes that lead to high variability of output. As
found here, high variance of dimensions, for
instance, raises rework and scrap, and thus chokes
throughput. Importantly, it raises hidden costs. Due
to the inescapable variability in track grinding and
the stringency demanded in reaching the target
clearance in each bearing assembled, a key not-sovisible cost in bearing production is the
requirement of large variety of balls. Each of these
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• Quantitative projection of the impact of high
machining variability on the variety of standard
balls (k) needed to complete the final assembly at
high yield—p, the proportion of acceptable
bearing assemblies produced.
• Effect of Radial Clearance (RC) on utilization of
balls.
• Combined impact of high variability of inner and
outer rings on yield.
• Incentives for tightening machining tolerances at
key process steps, and at input—the raw ring
grinding stage.
• Quantification of the extent of manual re-work
given specified levels of Cp/Cpk at different
processing steps.
• Cost optimization of the complete bearings
manufacturing process based on quantified
estimates of scrap, rework and capital cost of
work centers.
• Business development—help determine plant
capabilities required to move into making
superior quality ball bearings used in machine
tools, aerospace and similar applications.
In the following paragraphs we examine one such
question.
Estimating the Variety Required in Ball Sizes
Some inferences may be easily drawn from Table 3
Given Cpk of Ring Grinding Machines
and Figure 4. As ring grinding variability
Earlier, we had hinted that low Cp/Cpk will lead to
represented here by σO and σI improves (i.e.,
extra manual work, larger variety required in ball
standard deviations σO and σI reduce in value),
sizes, as well as possible degradation of
bearing production yield improves. This implies
performance of bearings that are near-marginal.
drop of rework and possible stoppage of scrapping
Low Cp/Cpk, i.e., high natural variability or process  
unmatched rings.
will also lead to extra manual work (re-work) to find
matching ring pairs that an automated assembly
Figure 4 displays another important effect of
machine would reject. Such matters are intuitively
variance reduction. As grinding σ rises so does the
known to most bearing manufacturers. They believe
variety of balls required to assure the correct
that if variation in track grinding, for instance, is
bearing assembly. Engineering considerations
reduced, fewer varieties in the standardized sizes of
dictate that balls must fit into the grooves as well as
balls would be required, considerable waste could
leave the required clearance RC within the bearing.
be reduced and the manual assembly operation
Hence, given a ball size, wider the ring size
done using inner and outer rings rejected by the
variability σ, larger will be the number of trials with
automatic assembly could even be eliminated.
different rings needed to complete the assembly.
However, the theoretical derivation of the link
The quantitative relationship is not difficult to infer
between grinding variance and the extra ball sizes
here. When costs of automated and manual
required (see Figure 2) is non trivial.
assembly are known and so is the extra cost of
producing an extra variety of ball, one may work out
Using numerical or analytical (where feasible)
the trade off to determine the optimum grinding
models or Monte Carlo simulation, the assembly
machine capability or σ or the corresponding Cpk.
process may be studied to relate the statistical
The appropriateness of technology upgradation
variance of track grinding to ball size variety. While
may thus be found.
this study is attempted, one can restrict the answers
such that the designed (target) radial clearance is
always maintained. This relationship, determined
by the Monte Carlo simulator is shown in Table 3
and Figure 3. Ring (OR and IR) lot size was 1,000 for
each simulation. The inference that can be
immediately drawn is that yield (% of bearings
correctly assembled in one pass) goes up as ring
grinding σ (representing σO and/or σI) goes down.
Figure 4 displays the distribution of ball sizes
required determined by simulation for certain prestated grinding variability (σO and σI). Other
scenarios may be similarly evaluated.
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89
Figure 4 Ball size distributions for different grinding σ
90
RC = 20, σ = 1
RC = 20, σ = 2
RC = 20, σ = 3
RC = 20, σ = 4
RC = 20, σ = 5
RC = 20, σ = 6
ISSN: 0971-1023
Increasing RC will shift the outer ring dia outward if the same assortment of balls is to be used. Alternatively,
smaller balls will be more frequently required to maintain yield. Incentives for tightening machining
tolerances may also be similarly evaluated. The same may be done for any contemplated improvement in
Cp/Cpk at grinding. In fact, simulation may be extended backward where raw rings are received form
outsourced machine shops. This provides a basis to set incoming specs.
Figure 5 Finishing balls before inspection
(adopted from Reference The Manufacturing of a Ball Bearing)
Table 3 Yield (p) and ball sizes required as function of grinding variability
Radial Clearance = 15 micron
Radial Clearance = 20 micron
Grinding
variability
(std dev)
p
with 11 balls
Distinct ball
sizes required
for 100% yield
Grinding
variability
(std dev)
p
with 11 balls
Distinct ball
sizes required
for 100% yield
σ=1
100%
7
σ=1
100%
6
σ=2
99%
10
σ=2
100%
10
σ=3
96%
13
σ=3
99%
15
σ=4
90%
19
σ=4
94%
17
σ=5
83%
>21
σ=5
89%
>20
σ=6
74%
>22
σ=6
79%
>22
For new business development, to enter superior
quantified information. This will enable
bearing markets, the required grinding σ hence
management select best intervention options on the
machining capabilities (Cpk) may be similarly
basis of reliable estimates of the gains such as
determined. In fact, due to variability, a part of
production yield improvement. Found this way, the
current production may already qualify to be sold as
financial returns become significantly more certain
high precision bearings. Many other uses may be
and measurable.
made of the methods and tools illustrated here. The
greatest of all is that such methods produce
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91
Outline of a Procedure to Economically Justify
reduction in
does not come free, it frequently
Variance Reduction
requires a step jump in grinding technology as it is
Balls are what make the bearing bear the load and
affected by tools, grinding speed, dressing, material
ensure friction-free movement for the life of the
being ground, operator skills, coolant temperature,
b e a r i n g . B a l l s a re , t h e re f o re , c a re f u l ly
spindle vibration, etc. Thus, the benefits of reducing
manufactured, starting with thick steel wire, cold
must be economically more than the cost of variance
heading, cut into pieces and then smashed between
reduction. We outline this analysis as follows:
two steel dies (The Manufacture of a Ball bearing
2009). Then, the flash is removed and balls are heat-
Let the automatic assembly cost of a bearing be a.
treated to make them very hard, then tempered to
Assume that the % yield of the automatic assembly
make them tough. Finishing requires grinding
operation is p. Therefore, the proportion of manual
between grinding wheels and then lapped with very
rework required to complete assembling all inner
fine abrasive slurry to polish them for several hours
and outer rings manufactured is (1 - p). Let the
to reach correct dia and mirror-like finish (Figure
rework cost be ca or ca, with the assumption that
5). Each ball type takes 6 to 10 hours to finish. A
the factor c ≥ 1. Therefore, the total assembly cost of
medium precision ball cannot be out of round more
a bearing will be (Gryna et al 2008, Chapter 2)
than 25 millionth of an inch while high speed
precision bearing balls are allowed only five-
Total assembly cost = a (fraction automatically
millionth of an inch roundness variation. Therefore,
assembled) + ca (fraction manually assembled)
to change ball size, set up has to be changed with
much care and effort and the process must be
= a p + ca (1 - p)
stabilized before production begins. As shown
= ca - p(c-1)a
(12)
above, as ball dia variety increases, so does the
required number of set ups, each set up reducing
Now, as seen in the sections above, p is a function of
production capacity.
σ and k. Figure 6 illustrates this relationship for an
example in which radial clearance RC has been
As indicated in earlier sections, in bearing assembly,
assumed to be 20 micron and inner and outer mean
the critical process characteristic is the standard
track diameters are as shown in Table 3.
deviation (σ) of track dia produced by ring grinding.
This characteristic determines the yield of quality
bearings and the extent of rework or scrap
generated. And, lower the yield (p) in automatic
assembly, higher will be the share of manually
assembled bearings, both a slower and a more
costly process. As we saw, higher the , higher will be
the variety (k) required to complete the assembly
automatically (Table 3 and Figure 4). Generally
speaking, due to the additional ball manufacturing
set ups required, ball and hence bearing production
cost rises as the variety of ball size increases. But,
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ISSN: 0971-1023
Figure 6 Dependency of yield of automated assembly on σ and ball variety k
Some observations from Figure 6 are straight-
variety of balls. This cost is a “hidden” COQ
forward. For a given variety (k) of balls available, as
component of production cost and is determined,
σ (i.e., track grinding variability) increases, p falls.
among other things, by the length of the set up time.
Also, to increase p at a given grinding variability (σ)
It is incurred when the ball grinding machine is
level, k must be raised, i.e., a wider variety in ball
reset to make a different size ball, assuming that the
sizes must be available to increase the yield of
plant is operating in a “sold out” market and can sell
automatic assembly. From data generated by
all it can produce. This hidden cost has three
numerical modeling, it is possible to empirically
components: (a) the cost of resources expended on
relate p to process parameters and k. Therefore,
physically changing the actual setup; (b) the loss
when this relationship is known, one can estimate
due to production lost during the switchover; and (c)
the total assembly cost using (12) given any values
the extra cost of storing and managing the stock of
of and k. Using cost accounting methods such as
an extra variety of balls. Alternatively, if balls are
Activity Based Costing the relevant costs may be
outsourced, it will lead to purchasing and stocking
estimated. For illustration we used the numerical
an additional size ball. This information too is
framework used in producing Table 3 to develop an
quantifiable. Thus, given ring grinding variability
empirical model relating p to σ and k as follows. Such
and ball variety k (ignoring material cost) we find
empirical models (Gujarati and Sangeetha 2007)
are helpful when the direct theoretical derivation of
Total (ball + assembly cost) = ca - p(c-1)a + Cost of
the required relationships is not possible. Thus,
making k different ball sizes
(14)
following Jianxin and Tseng (1999) and Al-Omiri
and Drury (2007),
where p is given by (13). Expression (14) is a
2
p = 0.693689 + 0.088342 k – 0.10595 σ - 0.00413 k +
2
0.00232 σ + 0.005933 k
function of and k.
(13)
The only cost that is missing so far is the cost of
2
Model (13) has a R of 0.80. The other cost that one
reducing or the track grinding variability. Grinding
needs to quantify is that of producing an additional
variance (σ2) is a function of technology and the
ISSN: 0971-1023
93
tightness with which the grinding process is
interventions by DMAIC (Pyzdek 2000).
controlled by operators (operator skill).
Furthermore, a reduction of σ will improve bearing
performance and thus create intangible benefits of
However, as outlined above, it is possible to quantify
supplying superior quality bearings. Such benefits
the cost savings resulting from reducing σ for a
will be additional and their value and returns will be
given volume of annual bearing production. It is
strategic (Pyzdek 2000; Evans and Lindsay 2005).
possible to estimate the NPV of accumulated yearly
savings and consequently how much investment
The actual costs at the plant in question cannot be
can be justified in ring grinding precision
shown due to proprietary reasons. However,
improvement to benefit business. This can be
subsequent to this study, based on the insights
projected for a planned period of selling those
gained and incentives estimated based on delays,
particular bearings. The startling surprise is that if a
production lost due to set up changes in ball
plant produces 10 million bearings annually and
manufacture and the potential to upgrade products,
one is able to reduce cost of production by 2
the company decided that it now had sufficient basis
cents/bearing, assuming that the business runs for
to launch a full-blown Six Sigma® project involving
7 years, an NPV in excess of USD 1 million will
multi-factor Orthogonal Array experiments to pin
materialize at nominal interest rates. Such sums can
down factors that could raise p, the company's first-
j u st i fy si g n i fi c a n t p ro c e ss i m p rove m e n t
pass bearings yield.
Conclusions
Every plant manager aims to reduce defects.
procedure to quantify hidden losses—here due to
However, few in the supervisory or engineering staff
the extra sets of bearing balls that must be
are able to formally quantify the cost of poor quality
precision-manufactured, stocked and used to raise
even where monetary values (losses) are suspected
yield in automated bearing assembly. The method
to be large. The result is the perpetuation of status
employs numerical as well as Monte Carlo models,
quo, unless a new facility with superior
all done in Excel®, and it results in quantitative
technologies is proposed and justified. In a
estimates of yield, manual work required and the
manufacturing set up the “tip of the
variety required in ball bearing balls before any
iceberg”—scraps, rework and warrantee
commitment needs to be made to alter plan
service—are generally visible and reported
equipment, workforce or facilities.
monthly or yearly. But, the hidden factory that runs
alongside of the brick-and–mortar factory or other
Numerous other questions in strategizing
hidden costs of poor quality (COQ) typically evade
manufacturing could also be tackled by such
estimation. Still, without the benefits monetized,
analytical procedures. Typical instances of these are
management will not be interested in large scale
listed in Section 6. However, before one undertakes
interventions such as Six Sigma, even if the
such a study to initiate Six Sigma® DMAIC (see
methodology has worked wonders at scores of
Pyzdek 2000 or Evans 2005), all COQ (cost of
organizations worldwide. This paper illustrates a
quality) components must be drilled into and
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ISSN: 0971-1023
targeted for quantification. This would be the most
capabilities proved sufficient for the plant
sensible way to initiate priority action using DMAIC,
management to comprehend the steps in the
done best at the “D” (Define) stage.
analysis and see how the conclusions were reached.
This became a stepping stone to raise many related
In this instance it was particularly gratifying to
questions about alternative ways to cut cost and
quantitatively affirm the intuitive assertions of
impact profit. One such initiative justified was the
plant management. Their reactions became
upgradation of vendor management. Another was
instrumental in delving deeper to locate tacit
to reset the settings in the automated facilities
opportunities that existed for raising profits.
toward the gradual removal of manual work. A
significant issue tossed up was the engineering
®
Excel models were deliberately used in this work to
optimization of the target RC (radial clearance)
serve as easy-to-use decision support tools usable
values for its close interplay with bearing
at the plant level, for many such decisions are often
performance and the variety of balls required in
made locally without the sophistication consultants
automatic assembly. This constituted the charter of
®
typically engage in. Excel 's graphic and statistical
a separate Six Sigma® project.
References
•
Al-Omiri M and Drury Colin (2007). A Survey of
•
Juran's Quality Planning and Analysis for
Systems in UK Organisations, Management
Enterprise Quality, 5th ed., Tata McGraw-Hill.
•
Volume-1/Ball-Bearing.html, accessed April 25,
Tseng Mitchel M (1999). A Pragmatic Approach
2009.
to Product Costing based on Standard Time
Bhat U Narayan (2008). An Introduction to
Estimates, International Journal of Operations
Queueing Theory: Modeling and Analysis in
and Production Management, 735-755.
•
Law A M and Kelton W D (2000). Simulation
Modeling and Analysis, 3rd ed., McGraw-Hill.
Evans J R and Lindsay W M (2005).
An
•
Improvement, Thomson .
•
Rice J A (2007). Mathematical Statistics and
Data Analysis, 3rd ed., Thomson
•
Sharma Poonam (2009). Report on MBA Project,
Statistical Techniques, Quality Engineering, 17:
Vinod Gupta School of
309-315.
Kharagpur, India.
Gitlow H S, Oppenheim A J, Oppenheim R and
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Introduction to
Statistical Quality Control, 4 ed., Wiley.
Gijo E R (2005). Improving Process Capability of
Manufacturing Process by Application of
Montgomery D C (2005).
th
I n t ro d u c t i o n t o Si x Si g m a & P ro ce s s
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Basic
Econometrics, Tata McGraw-Hill.Jianxin Jio and
Technology, Birkhouser.
•
Gujarati D N and Sangeetha (2007).
Ball Bearing, http://www.madehow.com/
Applications Series, Statistics for Industry and
•
Gryna F M, Chua R C H and Defeo J A (2008).
facts influencing the Choice of Product Costing
Accounting Research, Vol 18(4), Dec, 399-424.
•
•
•
Management, IIT
SKF Bearings Handbook (2009). http:// www.
Levine D M (2005). Quality Management, 3 ed.,
who-sells-it.com/r/skf-bearing-handbook.html,
Tata McGraw-Hill.
accessed April 25, 2009.
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•
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Taugchi G and Clausing Don (1990). Robust
Acknowledgements
Quality, Harvard Business Review, January-
The author thanks Harsh Sachdev, Joydeep
February.
Sengupta, Jyoti Mukherji and Tapan Mondal of Tata
The Manufacturing of a Ball Bearing,
Bearings. Together they provided a wealth of
http://www.bearingsindustry.com/manufactu
practical shop floor knowledge and economic
ring.pdf, accessed April 25, 2009.
insights into bearing manufacturing. Poonam
Sharma provided data collection assistance.
Dr. Tapan Bagchi is the Director of Shirpur campus of NMIMS university. He was was recenctly
awarded Doctor of Science by IIT Kharagpur. His research interests have been in the area of quality
engineering and production. Dr. Bagchi has published prolifically. Prior to his current responsibility,
he has served in professorial and academic leadership capacities in IIT Kharagpur, NITIE and S.P. Jain
Institute of Management and Research.
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Justifying Six Sigma Projects in Manufacturing Management

Transcription

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