Justifying Six Sigma Projects in Manufacturing · PDF file74 Justifying Six Sigma Projects 75...

72ISSN: 0971-1023NMIMS Management ReviewVolume: April - May 2012

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Mr. Mihir Ajgaonkar is a doctoral student and a research scholar at Birla Institute of Technology, Mesra,

where Dr. Utpal Baul is a professor in the Department of Management with research and teaching

interests in business to business marketing, human resource and industrial management, and

organizational development and theory. Dr. S.M. Phadke is a management consultant and organizational

psychologist.


Justifying Six Sigma Projects inManufacturing Management

1Tapan P. BagchiNarsee Monjee Institute of Management Studies, Shirpur

Abstract

This paper develops and illustrates a case in

manufacturing management, using the instance of

justifying quality improvement of ball bearings—a

common precision product whose correct

manufacture and assembly greatly affects their

efficiency, utility and life. Mass-produced at high

speed, bearings extend a fertile domain for

benefiting from QA apparatus including Gage R&R,

ISO standards, sampling, and SPC to Six Sigma

DMAIC (Pyzdek 2000). However, when large

investments are involved, it becomes imperative

that besides the obvious, the hidden costs of quality

be located and sized. This paper provides methods

to examine and quantify such shortfalls—many

being preventable by reduction of quality variance

and/or part variety. Statistical and numerical

models have been used. Thus, targeting beyond

scrap and rework, this paper invokes modeling

methods to quantify such not-so-visible constraints

that limit productivity and profits of high-volume

high-speed processes.

Keywords: Precision Manufacturing, Variance

Reduction, Hidden Costs of Poor Quality, Numerical

Modeling, Monte Carlo Simulation.

1Dr. Tapan Bagchi is the corresponding author who can be reached at [email protected] Between Organizational Citizenship Behavior and Job Characteristics

Justifying Six Sigma Projects 7574ISSN: 0971-1023NMIMS Management ReviewVolume: April - May 2012

ISSN: 0971-1023NMIMS Management ReviewVolume: April - May 2012

Justifying Six Sigma Projects

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

exceptional domain in which quality assurance

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

To scope the quality improvement project Cpk was

assessed at quality-bottlenecked process steps. The

plant ran Orthogonal array experiments (Taguchi

and Clausing 1990) to locate process factors

speculated to affect quality by the plant. Thus, key

quality deviations in need of attention could be

identified, but no firm basis could be cited to

motivate impacting them. However, such studies led

to re-statement of the project's charter, which

became “predict variability of the final bearing

assembly based on information available on part

variability.” Key parts in question here were the

inner and outer rings of the bearing and the rolling

balls (Figure 1).

Deductive variance prediction from parts to whole

proved too complex as it led to queuing or inventory

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

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.

Motivation of the current study was to help the

manufacturer find economic justification for

possible major technology intervention that could

cut tangible and intangible COQ (cost of poor

quality) (Gitlow et. al. 2005, Gryna et. al. 2008) and

delays and raise profits by reducing production of

marginal quality bearings. With improved quality,

the company could possibly sell to premium bearing

markets.

This paper is organized as follows. The next section

of this paper outlines the relevant aspects of bearing

parts manufacture and assembly, and then states

the problem of immediate focus—low yield

(proportion of acceptable production) of quality

bearings, resulting from parts with high

2dimensional variability (σ ). The manufacturer

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.

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




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

exceptional domain in which quality assurance

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

To scope the quality improvement project Cpk was

assessed at quality-bottlenecked process steps. The

plant ran Orthogonal array experiments (Taguchi

and Clausing 1990) to locate process factors

speculated to affect quality by the plant. Thus, key

quality deviations in need of attention could be

identified, but no firm basis could be cited to

motivate impacting them. However, such studies led

to re-statement of the project's charter, which

became “predict variability of the final bearing

assembly based on information available on part

variability.” Key parts in question here were the

inner and outer rings of the bearing and the rolling

balls (Figure 1).

Deductive variance prediction from parts to whole

proved too complex as it led to queuing or inventory

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

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.

Motivation of the current study was to help the

manufacturer find economic justification for

possible major technology intervention that could

cut tangible and intangible COQ (cost of poor

quality) (Gitlow et. al. 2005, Gryna et. al. 2008) and

delays and raise profits by reducing production of

marginal quality bearings. With improved quality,

the company could possibly sell to premium bearing

markets.

This paper is organized as follows. The next section

of this paper outlines the relevant aspects of bearing

parts manufacture and assembly, and then states

the problem of immediate focus—low yield

(proportion of acceptable production) of quality

bearings, resulting from parts with high

2dimensional variability (σ ). The manufacturer

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.

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

2grinding variance (σ ) goes up (Figure 4). Next we

illustrate one such use of simulation to determine

the distinct categories of “standard” ball sizes

required in high yield assembly given C ratings of pk

track grinding. Subsequently, we use assembly costs

and visible COQ (scrap and rework) to help project

the justifiable capital expenditure in technology

that could improve grinding precision (i.e., reduce

σ). The paper ends with a summary of conclusions

that management should expect to see in such a

study.

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):

• Turning of raw material—steel tubes and

bars—into raw rings (a step that is often

outsourced).

• Heat treatment of raw rings.

• Precision face grinding.

• Precision outer diameter (OD) and inner

diameter (ID) grinding to produce tracks in rings

on which the balls will roll.

• Final honing to create surface finish.

Ring width and track dia are the control targets in

track grinding. At each step, sampled inspection is

done to ensure that the final critical-to-quality

(CTQ) dimensions remain within 0 and 1.2 micron

and surface finish is acceptable.

The Production of Balls

Balls are the most critical engineering component of

bearings as they directly “bear” the load while

providing minimal resistance to movement. Each

ball must be precision-machined and polished, and

together balls cost typically about 40% of a

bearing's manufacturing cost. Ball manufacturing

involves the following steps (The Manufacture of

Ball Bearings 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.

Bearing Assembly

Outer and inner ring pairs and the corresponding

correct size balls are then selected. Rings are

manually or automatically deformed lightly to

insert the balls into the tracks between the rings.

Retainer rings and lubricants may be added. Each

final bearing assembly is 100% tested for clearance

and noise.

Critical in final bearing assembly is the selection of

balls that will result in the specified radial clearance

between the balls and rings (see Section 2). Note

that the wider is the variation in dimensions of the

bearing's inner and outer rings, the larger will be the

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 pre-

sorted matching rings. Rings that cannot be

manually matched are scraped.




thManual ring-ball matching is slow (<1/10 of the

hourly yield of automated assembly) and a costly

operation. Note, also that each “standard” ball size

must be separately produced, requiring extra setups.

Thus, COQ considerations will urge one to lower the 2variation σ (or of the track grinding operation.

Intuitively, one feels that lower the σ, smaller will be

the needed number of (“standard”) balls of distinct

diameters for final assembly. Hence, lower overall

cost of bearing production. These considerations

led us to develop the quantitative relationship

between ring (track) grinding variability (σ) and the

variety in “standard” ball sizes needed to keep yield

(p) high. This methodology is described Section 4

onward.

σ)

Process Variability and its Hidden

Impact on Productivity and Costs

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 C at grinding or low ) could pk

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-

made—was a significant, but an unknown

component of a bearing's production cost.

Management speculated that the existing poor

grinding precision (high σ) caused this variety and

lost production capacity due to frequent set up

changes on ball machines. This could perhaps be

optimized by a study of ball production which

currently accounts fo 40% of total production cost.

On the other hand, to find technology benchmarks,

the plant had checked the output of outer rings on

two different track grinders, one 25 years old and

the other new. C differed by 0.69 to 2.02 between pk

the two, showing a realizable possibility for

reducing σ provided the monetary incentive for such

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

σ

2grinding variance (σ ) goes up (Figure 4). Next we

illustrate one such use of simulation to determine

the distinct categories of “standard” ball sizes

required in high yield assembly given C ratings of pk

track grinding. Subsequently, we use assembly costs

and visible COQ (scrap and rework) to help project

the justifiable capital expenditure in technology

that could improve grinding precision (i.e., reduce

σ). The paper ends with a summary of conclusions

that management should expect to see in such a

study.

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):

• Turning of raw material—steel tubes and

bars—into raw rings (a step that is often

outsourced).

• Heat treatment of raw rings.

• Precision face grinding.

• Precision outer diameter (OD) and inner

diameter (ID) grinding to produce tracks in rings

on which the balls will roll.

• Final honing to create surface finish.

Ring width and track dia are the control targets in

track grinding. At each step, sampled inspection is

done to ensure that the final critical-to-quality

(CTQ) dimensions remain within 0 and 1.2 micron

and surface finish is acceptable.

The Production of Balls

Balls are the most critical engineering component of

bearings as they directly “bear” the load while

providing minimal resistance to movement. Each

ball must be precision-machined and polished, and

together balls cost typically about 40% of a

bearing's manufacturing cost. Ball manufacturing

involves the following steps (The Manufacture of

Ball Bearings 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.

Bearing Assembly

Outer and inner ring pairs and the corresponding

correct size balls are then selected. Rings are

manually or automatically deformed lightly to

insert the balls into the tracks between the rings.

Retainer rings and lubricants may be added. Each

final bearing assembly is 100% tested for clearance

and noise.

Critical in final bearing assembly is the selection of

balls that will result in the specified radial clearance

between the balls and rings (see Section 2). Note

that the wider is the variation in dimensions of the

bearing's inner and outer rings, the larger will be the

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 pre-

sorted matching rings. Rings that cannot be

manually matched are scraped.




thManual ring-ball matching is slow (<1/10 of the

hourly yield of automated assembly) and a costly

operation. Note, also that each “standard” ball size

must be separately produced, requiring extra setups.

Thus, COQ considerations will urge one to lower the 2variation σ (or of the track grinding operation.

Intuitively, one feels that lower the σ, smaller will be

the needed number of (“standard”) balls of distinct

diameters for final assembly. Hence, lower overall

cost of bearing production. These considerations

led us to develop the quantitative relationship

between ring (track) grinding variability (σ) and the

variety in “standard” ball sizes needed to keep yield

(p) high. This methodology is described Section 4

onward.

σ)

Process Variability and its Hidden

Impact on Productivity and Costs

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 C at grinding or low ) could pk

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-

made—was a significant, but an unknown

component of a bearing's production cost.

Management speculated that the existing poor

grinding precision (high σ) caused this variety and

lost production capacity due to frequent set up

changes on ball machines. This could perhaps be

optimized by a study of ball production which

currently accounts fo 40% of total production cost.

On the other hand, to find technology benchmarks,

the plant had checked the output of outer rings on

two different track grinders, one 25 years old and

the other new. C differed by 0.69 to 2.02 between pk

the two, showing a realizable possibility for

reducing σ provided the monetary incentive for such

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

σ




task the plant found difficult.

Thus, one could intuit that wide dimensional

variation (σ) in bearing parts—inner rings, balls,

and outer rings—led to production of many rings

and balls that could not be automatically assembled

to make finished bearings while maintaining the

desired RC (radial clearance, see Section 2) and

other quality characteristics. One approach to raise

hourly yield (the proportion of correct assemblies

from all parts produced) would be to reduce all

variances. A partial solution to this quandary would

be the use of “standard” sized large assortment balls

as most bearing manufacturers currently do. The

convenient though expensive way would be to sort

all parts produced and then find matches that meet

CTQs including RC. Yet another approach to reduce

the cost of poor quality (visible and hidden) would

be to seek optimal variance reduction considering at

the minimum all measureable costs in ring grinding,

ring matching (pairing) and then the selection of

balls from the resulting smaller assortment of

bearing parts.

Ball Selection Process for Bearing Assembly

Sorting randomly produced rings and balls to find

matches that will successfully fit is slow and effort-

(cost-) intensive, even if the task is automated.

Nevertheless, ball selection is a critical practice in

bearing assembly worldwide due to the

considerable dimensional variation of machined

parts—inner and outer rings. Such selection aims at

achieving the CTQ target radial clearance (RC) that

must meet the engineering spec of each assembled +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)

Difference = (ID of OR) – (OD of IR) = RC + 2 Ball dia

when RC > 0 (2)

The notations used here are ID for inner track dia of

the outer ring (OR) and OD for the outer track dia of

the inner ring (IR). Ball dias are stepped from a

smallest practical size (s) in discrete units of 1 or 2

micron. Hence, for a feasible assembly.

Ball Dia = [Difference - RC]/2 (3)

Industry has found it expedient to manufacture

balls used in bearings to “standard” dimensions in

specified steps, like shoes, and not in continuous

dimensions (SKF Bearings Handbook 2009). So the

selection is made by rounding down to the nearest

size standard ball to the calculated “Ball Dia”

determined by (3) for each IR/OR pair being

assembled. In this plant, “standard balls” are made

in one micron steps.

Note that ID and OD are subject to grinding

variation. Hence, as RC increases due to bearing

design requirements, the random “Difference” (ID of

OR) – (OD of IR) in (2) for many IR / OR pairs will

increase, and hence the variety required in standard

ball sizes for the different randomly picked outer /

inner ring pairs. Conversely, if the “Difference” in (2)

were small, a smaller number of standard ball

choices will be required. Of course, if extra effort

was made to sort all IRs and ORs before assembly so

that the pairs would result in final radial clearances

close to the target RC, perhaps only one or two

standard-sized balls will be required.

Such sorting of rings before assembly, unless done

completely and cheaply, is not justifiable as it will

likely require such matching to be automated, with a

great deal of rejections and recycling of rings since

IR/OR dimensions vary randomly, before a matched

pair is passed to the ball insertion workstation.

Selecting a “standard sized” ball from an assortment

of balls, therefore, is the preferred option

worldwide in bearings assembly and there are

specialists who manufacture such automated

bearing assembly machines. Swiss bearing makers

use this procedure routinely.

Ball selection logic is as follows. Generally, it is

desired that the output produced by a

manufacturing process should fall within the

specified range, fixed by tolerance or “spec”.

Furthermore, the larger the spec range the greater

will be the permitted variation in the output that is

acceptable. Since a bearing comprises the assembly

of the inner ring, outer ring and balls, each produced

with some variation, to deliver a “quality” bearing,

the manufacturer has to find the best match of an

inner ring, an outer ring and a ball size such that the

balls fit correctly (with a clearance) within those

inner and outer rings. The bearing will then possess

the desired target RC to allow the balls to roll

between the rings and have good life. Therefore, ball

selection is implemented in the following steps:

• Measure and dimensionally sort all finished

inner and outer rings.

• Produce and sort an estimated required

assortment (distinct sizes) of standard balls and

keep them in stock. This step is guided by the

shop's experience with the quantities of

unmatched rings generated at assembly.

• Find matching ring pairs that will lead to on-spec

assembly using the rings chosen and a ball size

(using equation (3)) held in stock. The ball

selected should be such that the final assembly

should result in the target RC between the balls

and the ring tracks.

• Assemble the bearing by pushing the balls into

the tracks.

• Conduct visual and dimensional checks and

performance tests (e.g., noise at full speed) on

each final assembly.

• Reject bearings that are unacceptable. Accept

others for further processing.

With wide variation within (ring-to-ring) the inner

ring and also within the outer ring dimensions

produced, a relatively large number of trials are

required in Step 3 above to find the best matching of

balls for the inner and outer ring. But, as such ring

grinding variation (σ) decreases, within a few trials

the best fitting balls—due to lower dimensional

variability of rings—may be found. This is why

leading bearing producers are moving towards

raising C /C of ring manufacture. Such action p pk

reduces output variations and hence the average

“Difference” in (2). The result is that then fewer

“standard” ball sizes will be required to assemble

the bearings while one would still deliver the

targeted finished bearing performance.

So, as C /C or the “Sigma” metric (Pyzdek 2000) of p pk

the ring grinding process goes up, it reduces not

only the process cycle time that includes matching,

but also production of defective bearings (possible

marginal misfits) and rejection of rings in bearing

assembly. With low Cp/Cpk many inner/outer ring

pairs randomly picked will not match at all, creating

scrap and raising the cost of poor quality.

This condition raises a question for the bearing

manufacturer: What should be the relative




task the plant found difficult.

Thus, one could intuit that wide dimensional

variation (σ) in bearing parts—inner rings, balls,

and outer rings—led to production of many rings

and balls that could not be automatically assembled

to make finished bearings while maintaining the

desired RC (radial clearance, see Section 2) and

other quality characteristics. One approach to raise

hourly yield (the proportion of correct assemblies

from all parts produced) would be to reduce all

variances. A partial solution to this quandary would

be the use of “standard” sized large assortment balls

as most bearing manufacturers currently do. The

convenient though expensive way would be to sort

all parts produced and then find matches that meet

CTQs including RC. Yet another approach to reduce

the cost of poor quality (visible and hidden) would

be to seek optimal variance reduction considering at

the minimum all measureable costs in ring grinding,

ring matching (pairing) and then the selection of

balls from the resulting smaller assortment of

bearing parts.

Ball Selection Process for Bearing Assembly

Sorting randomly produced rings and balls to find

matches that will successfully fit is slow and effort-

(cost-) intensive, even if the task is automated.

Nevertheless, ball selection is a critical practice in

bearing assembly worldwide due to the

considerable dimensional variation of machined

parts—inner and outer rings. Such selection aims at

achieving the CTQ target radial clearance (RC) that

must meet the engineering spec of each assembled +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)

Difference = (ID of OR) – (OD of IR) = RC + 2 Ball dia

when RC > 0 (2)

The notations used here are ID for inner track dia of

the outer ring (OR) and OD for the outer track dia of

the inner ring (IR). Ball dias are stepped from a

smallest practical size (s) in discrete units of 1 or 2

micron. Hence, for a feasible assembly.

Ball Dia = [Difference - RC]/2 (3)

Industry has found it expedient to manufacture

balls used in bearings to “standard” dimensions in

specified steps, like shoes, and not in continuous

dimensions (SKF Bearings Handbook 2009). So the

selection is made by rounding down to the nearest

size standard ball to the calculated “Ball Dia”

determined by (3) for each IR/OR pair being

assembled. In this plant, “standard balls” are made

in one micron steps.

Note that ID and OD are subject to grinding

variation. Hence, as RC increases due to bearing

design requirements, the random “Difference” (ID of

OR) – (OD of IR) in (2) for many IR / OR pairs will

increase, and hence the variety required in standard

ball sizes for the different randomly picked outer /

inner ring pairs. Conversely, if the “Difference” in (2)

were small, a smaller number of standard ball

choices will be required. Of course, if extra effort

was made to sort all IRs and ORs before assembly so

that the pairs would result in final radial clearances

close to the target RC, perhaps only one or two

standard-sized balls will be required.

Such sorting of rings before assembly, unless done

completely and cheaply, is not justifiable as it will

likely require such matching to be automated, with a

great deal of rejections and recycling of rings since

IR/OR dimensions vary randomly, before a matched

pair is passed to the ball insertion workstation.

Selecting a “standard sized” ball from an assortment

of balls, therefore, is the preferred option

worldwide in bearings assembly and there are

specialists who manufacture such automated

bearing assembly machines. Swiss bearing makers

use this procedure routinely.

Ball selection logic is as follows. Generally, it is

desired that the output produced by a

manufacturing process should fall within the

specified range, fixed by tolerance or “spec”.

Furthermore, the larger the spec range the greater

will be the permitted variation in the output that is

acceptable. Since a bearing comprises the assembly

of the inner ring, outer ring and balls, each produced

with some variation, to deliver a “quality” bearing,

the manufacturer has to find the best match of an

inner ring, an outer ring and a ball size such that the

balls fit correctly (with a clearance) within those

inner and outer rings. The bearing will then possess

the desired target RC to allow the balls to roll

between the rings and have good life. Therefore, ball

selection is implemented in the following steps:

• Measure and dimensionally sort all finished

inner and outer rings.

• Produce and sort an estimated required

assortment (distinct sizes) of standard balls and

keep them in stock. This step is guided by the

shop's experience with the quantities of

unmatched rings generated at assembly.

• Find matching ring pairs that will lead to on-spec

assembly using the rings chosen and a ball size

(using equation (3)) held in stock. The ball

selected should be such that the final assembly

should result in the target RC between the balls

and the ring tracks.

• Assemble the bearing by pushing the balls into

the tracks.

• Conduct visual and dimensional checks and

performance tests (e.g., noise at full speed) on

each final assembly.

• Reject bearings that are unacceptable. Accept

others for further processing.

With wide variation within (ring-to-ring) the inner

ring and also within the outer ring dimensions

produced, a relatively large number of trials are

required in Step 3 above to find the best matching of

balls for the inner and outer ring. But, as such ring

grinding variation (σ) decreases, within a few trials

the best fitting balls—due to lower dimensional

variability of rings—may be found. This is why

leading bearing producers are moving towards

raising C /C of ring manufacture. Such action p pk

reduces output variations and hence the average

“Difference” in (2). The result is that then fewer

“standard” ball sizes will be required to assemble

the bearings while one would still deliver the

targeted finished bearing performance.

So, as C /C or the “Sigma” metric (Pyzdek 2000) of p pk

the ring grinding process goes up, it reduces not

only the process cycle time that includes matching,

but also production of defective bearings (possible

marginal misfits) and rejection of rings in bearing

assembly. With low Cp/Cpk many inner/outer ring

pairs randomly picked will not match at all, creating

scrap and raising the cost of poor quality.

This condition raises a question for the bearing

manufacturer: What should be the relative

precision with which the rings should be

manufactured? In this paper, we outline procedures

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.

Prima facie, as noted above, grinding C estimates pk

obtained at the start of the project hinted at a

significant opportunity to reduce cost of set up

changes as well as rework and the production of

unacceptable rings. Management already intuited

that if ring size variation could be reduced, fewer

standard ball sizes would do the job, a lot of set up

hours could be reduced, and the manual assembly

done with inner and outer rings rejected by the

automatic assembly machine could perhaps even be

eliminated. However, quantitative estimates of such

incentives were unavailable to them. Due to the

processes being random, the tools to help tackle this

situation could either be the exact theoretical

modeling of the assembly process incorporating the

rounding-to-the-lower-dimension practice to pick

balls or a numerical approach or Monte Carlo

simulation. In this study, each of these methods was

explored.




A Statistical Perspective of Variations in

Bearing Assembly

Assume that inner rings are produced with a

nominal outer track mean dimension μ and I

standard deviation σ . Similarly, assume that outer I

rings are produced with inner track mean μ and O

standard deviation σ . Let the size of a randomly O

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:

O - I – 2 B – RC ≥ 0 with RC > 0

as specified by engineering (4)

Here, RC is the designed (targeted) radial clearance

and B is the diameter of the balls to be placed

between the outer and inner rings. In (4), RC is an

engineering constant (> 0), dictated by bearing life

considerations. B is the size of the (identical) balls

selected to be placed in the bearing to make the

assembly possible. As noted in Section 2, B has to be

carefully chosen for each I and O pair so as not to let

the final radial clearance of the assembled bearing

deviate too much from the design RC value.

Otherwise, the fit will be too tight or loose, affecting

the bearing's installed performance and life.

As described in Section 2, in order to make the

process workable, industry produces “standard”

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 2 2+dimension variances σ and σ , the wider will be I O

the probable difference between the OR/IR rings

pairs randomly picked for assembly and the larger

will be the number of differently sized balls

required to complete the assembly. In fact, if the

distributions of the outer and inner ring sizes

overlap due to high variance, many candidate pairs

will be rejected during the automatic assembly step

that picks one outer and one inner ring and checks

their dimensions for a feasible assembly.

For illustration, let the balls be made precisely in

steps of 2 micron, starting with the smallest ball of

size s. If balls are sized successively, they will have

diameters s, s + 2, s + 4, s + 6, …, s + 2(k – 1), … Then, if

the ball with size s + 2(k – 1) gets matched, given I, O

and RC (> 0), we shall have

O – I – RC = 2(s + 2(k – 1)) (5)

Equation (5) leads to

k - 1 = [(O – I – RC)/2 – s]/2

This leads to

k = 1 + [(O – I – RC)/2 – s]/2 (6)

This expression simplifies to

k = 1 + (O – I)/4 – RC/4 – s/2 (7)

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

intuitively as said in Section 1 and elsewhere. A

small point to be noted is that the starting ball size s

required will be smaller when track grinding

variability (σ) is high. (This is shown later in Figure

4 with RC = 20 and σ = 1 vs. cases when σ is higher.)

Recall next that O and I are dimensions of two

rings—one outer and one inner—that we randomly

picked with the hope of matching them successfully

by fitting them with the appropriately sized balls.

Frequently, σ for the grinding process—inner or

outer—is nearly the same. However, in the general 2case, let the variance of outer ring track dia O be σ 0

2and that for the inner ring track dia I be σ . Then, the I

2 2larger are process variances σ and σ , the wider the 0 I

possible random difference or gap (O – I) is likely to

be. And, if the distributions of O and I overlap, the

higher will be the proportion of ring production that

cannot lead to successful bearing assembly when O

and I due to their high variability would not leave

much clearance between them. In fact, random

variables O and I being independent (the rings are

separately produced), the variance of the random

variable (O – I) is the sum of the variances of the two

random variables O and I. Hence,

2Variance (O – I) = σ (8)O-I

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 2 standard balls and RC, one should reduce both σO

2 2= σ + σ O I

precision with which the rings should be

manufactured? In this paper, we outline procedures

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.

Prima facie, as noted above, grinding C estimates pk

obtained at the start of the project hinted at a

significant opportunity to reduce cost of set up

changes as well as rework and the production of

unacceptable rings. Management already intuited

that if ring size variation could be reduced, fewer

standard ball sizes would do the job, a lot of set up

hours could be reduced, and the manual assembly

done with inner and outer rings rejected by the

automatic assembly machine could perhaps even be

eliminated. However, quantitative estimates of such

incentives were unavailable to them. Due to the

processes being random, the tools to help tackle this

situation could either be the exact theoretical

modeling of the assembly process incorporating the

rounding-to-the-lower-dimension practice to pick

balls or a numerical approach or Monte Carlo

simulation. In this study, each of these methods was

explored.




A Statistical Perspective of Variations in

Bearing Assembly

Assume that inner rings are produced with a

nominal outer track mean dimension μ and I

standard deviation σ . Similarly, assume that outer I

rings are produced with inner track mean μ and O

standard deviation σ . Let the size of a randomly O

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:

O - I – 2 B – RC ≥ 0 with RC > 0

as specified by engineering (4)

Here, RC is the designed (targeted) radial clearance

and B is the diameter of the balls to be placed

between the outer and inner rings. In (4), RC is an

engineering constant (> 0), dictated by bearing life

considerations. B is the size of the (identical) balls

selected to be placed in the bearing to make the

assembly possible. As noted in Section 2, B has to be

carefully chosen for each I and O pair so as not to let

the final radial clearance of the assembled bearing

deviate too much from the design RC value.

Otherwise, the fit will be too tight or loose, affecting

the bearing's installed performance and life.

As described in Section 2, in order to make the

process workable, industry produces “standard”

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 2 2+dimension variances σ and σ , the wider will be I O

the probable difference between the OR/IR rings

pairs randomly picked for assembly and the larger

will be the number of differently sized balls

required to complete the assembly. In fact, if the

distributions of the outer and inner ring sizes

overlap due to high variance, many candidate pairs

will be rejected during the automatic assembly step

that picks one outer and one inner ring and checks

their dimensions for a feasible assembly.

For illustration, let the balls be made precisely in

steps of 2 micron, starting with the smallest ball of

size s. If balls are sized successively, they will have

diameters s, s + 2, s + 4, s + 6, …, s + 2(k – 1), … Then, if

the ball with size s + 2(k – 1) gets matched, given I, O

and RC (> 0), we shall have

O – I – RC = 2(s + 2(k – 1)) (5)

Equation (5) leads to

k - 1 = [(O – I – RC)/2 – s]/2

This leads to

k = 1 + [(O – I – RC)/2 – s]/2 (6)

This expression simplifies to

k = 1 + (O – I)/4 – RC/4 – s/2 (7)

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

intuitively as said in Section 1 and elsewhere. A

small point to be noted is that the starting ball size s

required will be smaller when track grinding

variability (σ) is high. (This is shown later in Figure

4 with RC = 20 and σ = 1 vs. cases when σ is higher.)

Recall next that O and I are dimensions of two

rings—one outer and one inner—that we randomly

picked with the hope of matching them successfully

by fitting them with the appropriately sized balls.

Frequently, σ for the grinding process—inner or

outer—is nearly the same. However, in the general 2case, let the variance of outer ring track dia O be σ 0

2and that for the inner ring track dia I be σ . Then, the I

2 2larger are process variances σ and σ , the wider the 0 I

possible random difference or gap (O – I) is likely to

be. And, if the distributions of O and I overlap, the

higher will be the proportion of ring production that

cannot lead to successful bearing assembly when O

and I due to their high variability would not leave

much clearance between them. In fact, random

variables O and I being independent (the rings are

separately produced), the variance of the random

variable (O – I) is the sum of the variances of the two

random variables O and I. Hence,

2Variance (O – I) = σ (8)O-I

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 2 standard balls and RC, one should reduce both σO

2 2= σ + σ O I




and . Next, when and are large, many more

outer and inner rings will have a wider gap between

them, leading to a larger k, or a wider variety of

differently-sized balls to be stocked for the

automatic (or even manual) assembly.

To summarize, this section has found why reduction

of variability (as urged for instance by the advocates

of Six Sigma) may raise yield, reduce the variety of

balls and reduce reprocessing rejected rings

manually. This will reduce COQ. In the next section,

the impact of high machining variance is

numerically assessed. We see that as σ rises, so I - O

does the variety required of “standard” balls.

Numerical Assessment of Impact of Variance

Reduction on Ball Variety k

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:

I = diameter of a randomly produced inner ring. x

O = diameter of a randomly produced outer ring. x

RC = desired radial clearance after assembly.

B = ball size used to assemble the ball bearing. x

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 + th1) ball in this sequence of “standard” balls will be of

dia (s + i) micron.

In bearing manufacturing almost nothing is thrown

away since scrap is visible. It is collected and re-

processed, but at additional cost. In bearing

assembly, attempt is made to find a part that will fit

2 2 2σ σ σI O Isatisfactorily with another part. If part size

variability is too high, however, this process slows

down the generally high speed production rate and

introduces hidden costs of poor quality. When outer

and inner rings are mass produced, to find a

matched (I , O ) pair, a relationship is used to guide x x

ball selection that guarantees the required radial

clearance RC. This relationship, shown below,

determines the correct ball size for the (I , O ) ring x x

pair.

O – I = RC + 2 B = RC + 2(s + i)vx x x

The correct ball size B is given by x

Let the largest size standard ball available be of

diameter (s + k). Our attempt here will be to link (O – x

I ) to k. We expect that when the tracks of the outer x

and inner rings are ground, there will be some

variability. Intuitively, we feel that larger the 2 2resulting track diameter variances σ and/or σ , Ox Ix

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 (O – I ) to lead x x

to a correct bearing assembly may be determined.

Such an assembly will result in the desired radial

clearance RC to assure good life and other

performance criteria (SKF Bearings Handbook

2009). These limits are given by

This gives

and

This produces the two “not a feasible assembly”

lower and upper limit conditions as

and

(9)

(10)

We will now attempt to find the distribution of (O – x

I ), the random gap between the outer and inner ring x

track diameters in which the balls will sit. Let the

probability density functions of two independent

random variables X and Y be f(x) and g(y). Then the

distribution of U (= X – Y) will be convoluted (Rice

2007, p. 97).

Determining P[U u] for arbitrary distributions f(x)

and g(y) is difficult. However, we can determine P[U

u] for some common distributions assumed for X

and Y as follows.

In the current application, samples of over 300

inner ring sizes were found to be normally 2distributed; hence, Ix ~ N[μ , σ ] (Sharma 2009). I I

Similarly, the machined outer ring track dia were 2also normal; thus Ox ~ N[μ , σ ]. Such observations O O

are commonplace in mechanical metal removal by

grinding (Gigo 2005).

When X and Y are independent and normally 2 2distributed such that X ~ N[μ σ ] and Y ~ N[μ ], x, x y, y

2 2then U is distributed as N[μ -μ σ + σ ]. Therefore, x y, x y

if F (u) represents the cdf of U, then the probability U

that a randomly picked inner and outer ring will

lead to a feasible bearing assembly is

P[RC + 2s U RC+2(s + k)] = F [RC+2(s + k)]-F [RC + 2s]U U

σ

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

Φ(u)-the cdf of the standard normal distribution-is

a monotonic function of u. Next, note that in (11),

engineering considerations fix the quantities RC, s,

μ and μ . Therefore, if we wish to have 99% of the O I

machined inner and outer rings correctly

assembled by picking two equal-sized balls from the

size range [s, s + k], the larger is , the larger

will be the value of k, the variety in size of balls

required to produce a feasible bearing assembly.

2 2Note further that σ and σ , respectively, represent O I

the variances of the track diameters of the machined

outer and inner rings. Hence, higher the variability

in track grinding, larger will be k, the variety in

different sized balls required-the assortment




and . Next, when and are large, many more

outer and inner rings will have a wider gap between

them, leading to a larger k, or a wider variety of

differently-sized balls to be stocked for the

automatic (or even manual) assembly.

To summarize, this section has found why reduction

of variability (as urged for instance by the advocates

of Six Sigma) may raise yield, reduce the variety of

balls and reduce reprocessing rejected rings

manually. This will reduce COQ. In the next section,

the impact of high machining variance is

numerically assessed. We see that as σ rises, so I - O

does the variety required of “standard” balls.

Numerical Assessment of Impact of Variance

Reduction on Ball Variety k

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:

I = diameter of a randomly produced inner ring. x

O = diameter of a randomly produced outer ring. x

RC = desired radial clearance after assembly.

B = ball size used to assemble the ball bearing. x

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 + th1) ball in this sequence of “standard” balls will be of

dia (s + i) micron.

In bearing manufacturing almost nothing is thrown

away since scrap is visible. It is collected and re-

processed, but at additional cost. In bearing

assembly, attempt is made to find a part that will fit

2 2 2σ σ σI O Isatisfactorily with another part. If part size

variability is too high, however, this process slows

down the generally high speed production rate and

introduces hidden costs of poor quality. When outer

and inner rings are mass produced, to find a

matched (I , O ) pair, a relationship is used to guide x x

ball selection that guarantees the required radial

clearance RC. This relationship, shown below,

determines the correct ball size for the (I , O ) ring x x

pair.

O – I = RC + 2 B = RC + 2(s + i)vx x x

The correct ball size B is given by x

Let the largest size standard ball available be of

diameter (s + k). Our attempt here will be to link (O – x

I ) to k. We expect that when the tracks of the outer x

and inner rings are ground, there will be some

variability. Intuitively, we feel that larger the 2 2resulting track diameter variances σ and/or σ , Ox Ix

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 (O – I ) to lead x x

to a correct bearing assembly may be determined.

Such an assembly will result in the desired radial

clearance RC to assure good life and other

performance criteria (SKF Bearings Handbook

2009). These limits are given by

This gives

and

This produces the two “not a feasible assembly”

lower and upper limit conditions as

and

(9)

(10)

We will now attempt to find the distribution of (O – x

I ), the random gap between the outer and inner ring x

track diameters in which the balls will sit. Let the

probability density functions of two independent

random variables X and Y be f(x) and g(y). Then the

distribution of U (= X – Y) will be convoluted (Rice

2007, p. 97).

Determining P[U u] for arbitrary distributions f(x)

and g(y) is difficult. However, we can determine P[U

u] for some common distributions assumed for X

and Y as follows.

In the current application, samples of over 300

inner ring sizes were found to be normally 2distributed; hence, Ix ~ N[μ , σ ] (Sharma 2009). I I

Similarly, the machined outer ring track dia were 2also normal; thus Ox ~ N[μ , σ ]. Such observations O O

are commonplace in mechanical metal removal by

grinding (Gigo 2005).

When X and Y are independent and normally 2 2distributed such that X ~ N[μ σ ] and Y ~ N[μ ], x, x y, y

2 2then U is distributed as N[μ -μ σ + σ ]. Therefore, x y, x y

if F (u) represents the cdf of U, then the probability U

that a randomly picked inner and outer ring will

lead to a feasible bearing assembly is

P[RC + 2s U RC+2(s + k)] = F [RC+2(s + k)]-F [RC + 2s]U U

σ

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

Φ(u)-the cdf of the standard normal distribution-is

a monotonic function of u. Next, note that in (11),

engineering considerations fix the quantities RC, s,

μ and μ . Therefore, if we wish to have 99% of the O I

machined inner and outer rings correctly

assembled by picking two equal-sized balls from the

size range [s, s + k], the larger is , the larger

will be the value of k, the variety in size of balls

required to produce a feasible bearing assembly.

2 2Note further that σ and σ , respectively, represent O I

the variances of the track diameters of the machined

outer and inner rings. Hence, higher the variability

in track grinding, larger will be k, the variety in

different sized balls required-the assortment




starting with the smallest dia s to the largest size s +

k. Figure 2 shows the numerically determined

dependency between grinding sigma (standard

deviation σ) and ball variety shown in ball size

increments in micron. For typical grinding

operations performed on outer and inner rings, 2 2machining variances σ and σ on machines of O I

comparable condit ion were stat ist ical ly

indistinguishable, represented here by σ.

Table 1 displays the numerically estimated fraction

of correctly assembled ball bearings with assumed 2 2track dia grinding variances σ and σ . We used O I

here numeric values of Φ(.), the standard normal

cdf. Observe the impact of increasing minimum k on

yield (p). Desirable yields exceeding 99% with σ= 3 2were obtained with k values near 10. For lower σ O

2and σ one would require a smaller k or fewer sizes I

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 σscenario # sizes of balls required assembled bearings

1 1μ 1 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%

Outer ring σInner ring k = minimum distinct p = % yield of correctly

μ

This analysis, done numerically, reasserts the

fundamental credence—larger the ring machining 2 2variability (here σ and σ ), larger will be the O I

variety balls needed to ensure correct assembly.

From cost of quality standpoint, each additional

“standard” sized ball requires its own independent

set up and production run with its own settings on

rilling and other machines. Each time such a set up is

changed, it reduces the ball plant's throughput,

hence, its productivity. Variety or frequent changes

also reduce scale (larger lot size) advantages. These

considerations will prompt one to seek production

methods or strategies requiring fewer assortments

of balls. All this is consistent with the spirit of Six ®Sigma , which champions variability reduction

(Pyzdek 2000; Evans and Lindsay 2005).

On the other extreme, if production machinery is set

up arbitrarily, a really large assortment of balls will

be needed by the assembler. The “rejects” from such

an automated assembly will require a wider

support to hand-match rings that the assembly

machine cannot accept within its normal

setting—again raising COQ, hence, adding to

production cost. Thus, a trade-off appears possible

between high C ring grinders (with small variance) pk

on one hand and producing and stocking a larger

variety of “standard” balls with sizes varying from s

to (s + k) on the other. As shown, if are

reduced, k (the number of distinct ball sizes) will be

smaller. Figure 2 displays this relationship. When

the relevant costs are available, an optimization can

be attempted—best done before capital is invested

in plant machinery and technologies. We note that

the automatic selection of “fitting” parts is a

common technique implemented in thousands of

production systems worldwide. As reconfirmed

here for bearing production, grinding variance

reduction will lead to the handling of fewer sizes of

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 μ and μ , and grinding standard O I

deviations σ and σ . Prevailing dimensions and O I

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

2 2σ and σ O I

effort, in probing the effect of operational variations

and different scenarios for which a numerical model

involving unusual distributions without probability

tables may not be easy to build.

Monte Carlo Simulation of Ball Bearing

Assembly

Figure 3 is a screen shot of the Monte Carlo model ®built in Excel for bearing assembly simulation. The

top portion of the worksheet shows where first the

process parameters (nominal dimensions μ and μ , O I

and standard deviations σ and σ ) are entered. O I

Parameters s and k indicate ball sizes. In this model,

s is specified by the analyst while incremental ball

sizes are automatically determined by the model by

finding the correct k using (7) above. At the

beginning, distribution of dia variations was

established as normal—common in grinding (Gijo

2005).




starting with the smallest dia s to the largest size s +

k. Figure 2 shows the numerically determined

dependency between grinding sigma (standard

deviation σ) and ball variety shown in ball size

increments in micron. For typical grinding

operations performed on outer and inner rings, 2 2machining variances σ and σ on machines of O I

comparable condit ion were stat ist ical ly

indistinguishable, represented here by σ.

Table 1 displays the numerically estimated fraction

of correctly assembled ball bearings with assumed 2 2track dia grinding variances σ and σ . We used O I

here numeric values of Φ(.), the standard normal

cdf. Observe the impact of increasing minimum k on

yield (p). Desirable yields exceeding 99% with σ= 3 2were obtained with k values near 10. For lower σ O

2and σ one would require a smaller k or fewer sizes I

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 σscenario # sizes of balls required assembled bearings

1 1μ 1 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%

Outer ring σInner ring k = minimum distinct p = % yield of correctly

μ

This analysis, done numerically, reasserts the

fundamental credence—larger the ring machining 2 2variability (here σ and σ ), larger will be the O I

variety balls needed to ensure correct assembly.

From cost of quality standpoint, each additional

“standard” sized ball requires its own independent

set up and production run with its own settings on

rilling and other machines. Each time such a set up is

changed, it reduces the ball plant's throughput,

hence, its productivity. Variety or frequent changes

also reduce scale (larger lot size) advantages. These

considerations will prompt one to seek production

methods or strategies requiring fewer assortments

of balls. All this is consistent with the spirit of Six ®Sigma , which champions variability reduction

(Pyzdek 2000; Evans and Lindsay 2005).

On the other extreme, if production machinery is set

up arbitrarily, a really large assortment of balls will

be needed by the assembler. The “rejects” from such

an automated assembly will require a wider

support to hand-match rings that the assembly

machine cannot accept within its normal

setting—again raising COQ, hence, adding to

production cost. Thus, a trade-off appears possible

between high C ring grinders (with small variance) pk

on one hand and producing and stocking a larger

variety of “standard” balls with sizes varying from s

to (s + k) on the other. As shown, if are

reduced, k (the number of distinct ball sizes) will be

smaller. Figure 2 displays this relationship. When

the relevant costs are available, an optimization can

be attempted—best done before capital is invested

in plant machinery and technologies. We note that

the automatic selection of “fitting” parts is a

common technique implemented in thousands of

production systems worldwide. As reconfirmed

here for bearing production, grinding variance

reduction will lead to the handling of fewer sizes of

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 μ and μ , and grinding standard O I

deviations σ and σ . Prevailing dimensions and O I

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

2 2σ and σ O I

effort, in probing the effect of operational variations

and different scenarios for which a numerical model

involving unusual distributions without probability

tables may not be easy to build.

Monte Carlo Simulation of Ball Bearing

Assembly

Figure 3 is a screen shot of the Monte Carlo model ®built in Excel for bearing assembly simulation. The

top portion of the worksheet shows where first the

process parameters (nominal dimensions μ and μ , O I

and standard deviations σ and σ ) are entered. O I

Parameters s and k indicate ball sizes. In this model,

s is specified by the analyst while incremental ball

sizes are automatically determined by the model by

finding the correct k using (7) above. At the

beginning, distribution of dia variations was

established as normal—common in grinding (Gijo

2005).




Figure 3 The automated bearing assembly simulator

In this simulation of automated bearing assembly,

ring sizes O and I are randomly found, presently

normally distributed dimensions based on μ and μ , O I

and σ and σ , respectively, as entered in the green O I

zone of the worksheet. Random samples are drawn ®by Excel using the NORMINV() function. Each row

from Row 26 downward uses one simulated outer

ring (O value) and one simulated inner ring

dimension (I value). Column headings (Column F

onward) implement the relationship (7) in the steps

of computation. The simulation progresses as

follows:

Identify the sequence of processing steps,

control and noise factors, any information on

randomness and its nature (the associated

distributions). Determine the relationships

•

among the variables (e.g., relationships of the

type (7), (9), (10), etc.). The variables presently

involved were outer and inner ring track dia, ball

size, and the ball variety k.

• Set up the experimental framework—the

variables to be manipulated (here k), the

constants (the smallest ball size s, and radial

clearance RC) and the random sampling

mechanism from specified distributions (O and

I).

• Once the model is coded and set up (here the ®Excel model of Figure 3), conduct pilot runs to

estimate the variance of the response (here p or

yield) and then the required sample size (length

of the simulation run).

•

complete output analysis.

• Replicate runs with identical seeds to reduce

variance of the yield (p) estimated (Law and

Kelton 2000).

Results of one round of simulated assembly with

inputs from worksheet cells C26, D26, E26, etc.,

appear in Column K—“Micron Size of Balls

Required.” Note that in practice calculated ball sizes

are rounded down to a whole number standard size

for conservative (slightly larger) clearance resulting

in the assembled bearing. If a feasible size of balls is

found, i ts s ize is noted. Otherwise the

corresponding row (i.e., the simulated O and I pair

put up for assembly) reports “ball unavailable”.

Simulate for the desired sample; collect data and The total count of “ball unavailable” indicates ring

size mismatches that could not be automatically

assembled into a bearing using the k standard

assorted ball sizes provided. This estimates p and

the manual work needed to complete the job. The

simulation model was validated using physical

inner/outer ring production lots of size 300 each at

the prevailing grinding variability (σ) level (5

micron) and a target RC of 20 micron. Hand

assembly produced about 10% mismatches (too

small or too large rings) that could not deliver the

target RC using 12 different standard sized balls at

hand (cf. Table 2). For given σ and σ for grinding, a O I

sample size of 1,000 simulated assemblies provided

a conservative 2-digit precision of yield estimate,

sufficient to illustrate the utility of simulation.

Table 2 Numerical estimation of yield p (fraction of bearings automatically assembled)using the standard normal cdf




Figure 3 The automated bearing assembly simulator

In this simulation of automated bearing assembly,

ring sizes O and I are randomly found, presently

normally distributed dimensions based on μ and μ , O I

and σ and σ , respectively, as entered in the green O I

zone of the worksheet. Random samples are drawn ®by Excel using the NORMINV() function. Each row

from Row 26 downward uses one simulated outer

ring (O value) and one simulated inner ring

dimension (I value). Column headings (Column F

onward) implement the relationship (7) in the steps

of computation. The simulation progresses as

follows:

Identify the sequence of processing steps,

control and noise factors, any information on

randomness and its nature (the associated

distributions). Determine the relationships

•

among the variables (e.g., relationships of the

type (7), (9), (10), etc.). The variables presently

involved were outer and inner ring track dia, ball

size, and the ball variety k.

• Set up the experimental framework—the

variables to be manipulated (here k), the

constants (the smallest ball size s, and radial

clearance RC) and the random sampling

mechanism from specified distributions (O and

I).

• Once the model is coded and set up (here the ®Excel model of Figure 3), conduct pilot runs to

estimate the variance of the response (here p or

yield) and then the required sample size (length

of the simulation run).

•

complete output analysis.

• Replicate runs with identical seeds to reduce

variance of the yield (p) estimated (Law and

Kelton 2000).

Results of one round of simulated assembly with

inputs from worksheet cells C26, D26, E26, etc.,

appear in Column K—“Micron Size of Balls

Required.” Note that in practice calculated ball sizes

are rounded down to a whole number standard size

for conservative (slightly larger) clearance resulting

in the assembled bearing. If a feasible size of balls is

found, i ts s ize is noted. Otherwise the

corresponding row (i.e., the simulated O and I pair

put up for assembly) reports “ball unavailable”.

Simulate for the desired sample; collect data and The total count of “ball unavailable” indicates ring

size mismatches that could not be automatically

assembled into a bearing using the k standard

assorted ball sizes provided. This estimates p and

the manual work needed to complete the job. The

simulation model was validated using physical

inner/outer ring production lots of size 300 each at

the prevailing grinding variability (σ) level (5

micron) and a target RC of 20 micron. Hand

assembly produced about 10% mismatches (too

small or too large rings) that could not deliver the

target RC using 12 different standard sized balls at

hand (cf. Table 2). For given σ and σ for grinding, a O I

sample size of 1,000 simulated assemblies provided

a conservative 2-digit precision of yield estimate,

sufficient to illustrate the utility of simulation.

Table 2 Numerical estimation of yield p (fraction of bearings automatically assembled)using the standard normal cdf




Experiments could be now run by treating process

or machining variances σ and σ as experimental O I

variables with nominal sizes μ and μ and O I

engineering parameters RC, s and k held at specified

levels. Plots of the output data will graphically

indicate the effect of variances σ and σ on the O I

process output p (yield)—the fraction of rings

submitted for assembly that could be finished into

complete bearings. This exercise showed how the

effect of comptemplated process changes may be

quantitatively estimated. Simulation is a well-

known method to use here (Law and Kelton 2000).

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-so-

visible cost in bearing production is the

requirement of large variety of balls. Each of these

ball sizes must be precision-made to exact specs th(10 or better in micron). Manual sorting and

assembly using matched outer and inner rings and

balls is the traditional fallback. But this results in

errors and the consequent unwanted variation in

bearing characteristics. Some operations are,

therefore, automated. The common one in bearing

production is assembly.

Still, many questions remain about the estimation of

the visible and hidden COQ and the economic

optimality of technology interventions. Some of

these questions may be probed by Monte Carlo

simulation. Examples are:

• 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 C /C at different p pk

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

Given C of Ring Grinding Machinespk

Earlier, we had hinted that low C /C will lead to p pk

extra manual work, larger variety required in ball

sizes, as well as possible degradation of

performance of bearings that are near-marginal.

Low C /C , i.e., high natural variability or process p pk

will also lead to extra manual work (re-work) to find

matching ring pairs that an automated assembly

machine would reject. Such matters are intuitively

known to most bearing manufacturers. They believe

that if variation in track grinding, for instance, is

reduced, fewer varieties in the standardized sizes of

balls would be required, considerable waste could

be reduced and the manual assembly operation

done using inner and outer rings rejected by the

automatic assembly could even be eliminated.

However, the theoretical derivation of the link

between grinding variance and the extra ball sizes

required (see Figure 2) is non trivial.

Using numerical or analytical (where feasible)

models or Monte Carlo simulation, the assembly

process may be studied to relate the statistical

variance of track grinding to ball size variety. While

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 σ and/or σ ) goes down. O I

Figure 4 displays the distribution of ball sizes

required determined by simulation for certain pre-

stated grinding variability (σ and σ ). Other O I

scenarios may be similarly evaluated.

Some inferences may be easily drawn from Table 3

and Figure 4. As ring grinding variability

represented here by σ and σ improves (i.e., O I

standard deviations σ and σ reduce in value), O I

bearing production yield improves. This implies

drop of rework and possible stoppage of scrapping

unmatched rings.

Figure 4 displays another important effect of

variance reduction. As grinding σ rises so does the

variety of balls required to assure the correct

bearing assembly. Engineering considerations

dictate that balls must fit into the grooves as well as

leave the required clearance RC within the bearing.

Hence, given a ball size, wider the ring size

variability σ, larger will be the number of trials with

different rings needed to complete the assembly.

The quantitative relationship is not difficult to infer

here. When costs of automated and manual

assembly are known and so is the extra cost of

producing an extra variety of ball, one may work out

the trade off to determine the optimum grinding

machine capability or σ or the corresponding C . pk

The appropriateness of technology upgradation

may thus be found.




Experiments could be now run by treating process

or machining variances σ and σ as experimental O I

variables with nominal sizes μ and μ and O I

engineering parameters RC, s and k held at specified

levels. Plots of the output data will graphically

indicate the effect of variances σ and σ on the O I

process output p (yield)—the fraction of rings

submitted for assembly that could be finished into

complete bearings. This exercise showed how the

effect of comptemplated process changes may be

quantitatively estimated. Simulation is a well-

known method to use here (Law and Kelton 2000).

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-so-

visible cost in bearing production is the

requirement of large variety of balls. Each of these

ball sizes must be precision-made to exact specs th(10 or better in micron). Manual sorting and

assembly using matched outer and inner rings and

balls is the traditional fallback. But this results in

errors and the consequent unwanted variation in

bearing characteristics. Some operations are,

therefore, automated. The common one in bearing

production is assembly.

Still, many questions remain about the estimation of

the visible and hidden COQ and the economic

optimality of technology interventions. Some of

these questions may be probed by Monte Carlo

simulation. Examples are:

• 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 C /C at different p pk

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

Given C of Ring Grinding Machinespk

Earlier, we had hinted that low C /C will lead to p pk

extra manual work, larger variety required in ball

sizes, as well as possible degradation of

performance of bearings that are near-marginal.

Low C /C , i.e., high natural variability or process p pk

will also lead to extra manual work (re-work) to find

matching ring pairs that an automated assembly

machine would reject. Such matters are intuitively

known to most bearing manufacturers. They believe

that if variation in track grinding, for instance, is

reduced, fewer varieties in the standardized sizes of

balls would be required, considerable waste could

be reduced and the manual assembly operation

done using inner and outer rings rejected by the

automatic assembly could even be eliminated.

However, the theoretical derivation of the link

between grinding variance and the extra ball sizes

required (see Figure 2) is non trivial.

Using numerical or analytical (where feasible)

models or Monte Carlo simulation, the assembly

process may be studied to relate the statistical

variance of track grinding to ball size variety. While

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 σ and/or σ ) goes down. O I

Figure 4 displays the distribution of ball sizes

required determined by simulation for certain pre-

stated grinding variability (σ and σ ). Other O I

scenarios may be similarly evaluated.

Some inferences may be easily drawn from Table 3

and Figure 4. As ring grinding variability

represented here by σ and σ improves (i.e., O I

standard deviations σ and σ reduce in value), O I

bearing production yield improves. This implies

drop of rework and possible stoppage of scrapping

unmatched rings.

Figure 4 displays another important effect of

variance reduction. As grinding σ rises so does the

variety of balls required to assure the correct

bearing assembly. Engineering considerations

dictate that balls must fit into the grooves as well as

leave the required clearance RC within the bearing.

Hence, given a ball size, wider the ring size

variability σ, larger will be the number of trials with

different rings needed to complete the assembly.

The quantitative relationship is not difficult to infer

here. When costs of automated and manual

assembly are known and so is the extra cost of

producing an extra variety of ball, one may work out

the trade off to determine the optimum grinding

machine capability or σ or the corresponding C . pk

The appropriateness of technology upgradation

may thus be found.




Figure 4 Ball size distributions for different grinding σ

RC = 20, σ = 1 RC = 20, σ = 2

RC = 20, σ = 3 RC = 20, σ = 4

RC = 20, σ = 5 RC = 20, σ = 6

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

C /C at grinding. In fact, simulation may be extended backward where raw rings are received form p pk

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

Grinding p Distinct ballvariability with 11 balls sizes required(std dev) for 100% yield

σ=1 100% 7

σ=2 99% 10

σ=3 96% 13

σ=4 90% 19

σ=5 83% >21

σ=6 74% >22



σ=1 100% 6

σ=2 100% 10

σ=3 99% 15

σ=4 94% 17

σ=5 89% >20

σ=6 79% >22

For new business development, to enter superior

bearing markets, the required grinding σ hence

machining capabilities (C ) may be similarly pk

determined. In fact, due to variability, a part of

current production may already qualify to be sold as

high precision bearings. Many other uses may be

made of the methods and tools illustrated here. The

greatest of all is that such methods produce

quantified information. This will enable

management select best intervention options on the

basis of reliable estimates of the gains such as

production yield improvement. Found this way, the

financial returns become significantly more certain

and measurable.




Figure 4 Ball size distributions for different grinding σ

RC = 20, σ = 1 RC = 20, σ = 2

RC = 20, σ = 3 RC = 20, σ = 4

RC = 20, σ = 5 RC = 20, σ = 6

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

C /C at grinding. In fact, simulation may be extended backward where raw rings are received form p pk

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



σ=1 100% 7

σ=2 99% 10

σ=3 96% 13

σ=4 90% 19

σ=5 83% >21

σ=6 74% >22



σ=1 100% 6

σ=2 100% 10

σ=3 99% 15

σ=4 94% 17

σ=5 89% >20

σ=6 79% >22

For new business development, to enter superior

bearing markets, the required grinding σ hence

machining capabilities (C ) may be similarly pk

determined. In fact, due to variability, a part of

current production may already qualify to be sold as

high precision bearings. Many other uses may be

made of the methods and tools illustrated here. The

greatest of all is that such methods produce

quantified information. This will enable

management select best intervention options on the

basis of reliable estimates of the gains such as

production yield improvement. Found this way, the

financial returns become significantly more certain

and measurable.




Outline of a Procedure to Economically Justify

Variance Reduction

Balls are what make the bearing bear the load and

ensure friction-free movement for the life of the

b e a r i n g . B a l l s a re , t h e re fo re , c a re f u l ly

manufactured, starting with thick steel wire, cold

heading, cut into pieces and then smashed between

two steel dies (The Manufacture of a Ball bearing

2009). Then, the flash is removed and balls are heat-

treated to make them very hard, then tempered to

make them tough. Finishing requires grinding

between grinding wheels and then lapped with very

fine abrasive slurry to polish them for several hours

to reach correct dia and mirror-like finish (Figure

5). Each ball type takes 6 to 10 hours to finish. A

medium precision ball cannot be out of round more

than 25 millionth of an inch while high speed

precision bearing balls are allowed only five-

millionth of an inch roundness variation. Therefore,

to change ball size, set up has to be changed with

much care and effort and the process must be

stabilized before production begins. As shown

above, as ball dia variety increases, so does the

required number of set ups, each set up reducing

production capacity.

As indicated in earlier sections, in bearing assembly,

the critical process characteristic is the standard

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,

reduction in does not come free, it frequently

requires a step jump in grinding technology as it is

affected by tools, grinding speed, dressing, material

being ground, operator skills, coolant temperature,

spindle vibration, etc. Thus, the benefits of reducing

must be economically more than the cost of variance

reduction. We outline this analysis as follows:

Let the automatic assembly cost of a bearing be a.

Assume that the % yield of the automatic assembly

operation is p. Therefore, the proportion of manual

rework required to complete assembling all inner

and outer rings manufactured is (1 - p). Let the

rework cost be ca or ca, with the assumption that

the factor c ≥ 1. Therefore, the total assembly cost of

a bearing will be (Gryna et al 2008, Chapter 2)

Total assembly cost = a (fraction automatically

assembled) + ca (fraction manually assembled)

= a p + ca (1 - p)

= ca - p(c-1)a (12)

Now, as seen in the sections above, p is a function of

σ and k. Figure 6 illustrates this relationship for an

example in which radial clearance RC has been

assumed to be 20 micron and inner and outer mean

track diameters are as shown in Table 3.

Figure 6 Dependency of yield of automated assembly on σ and ball variety k

Some observations from Figure 6 are straight-

forward. For a given variety (k) of balls available, as

σ (i.e., track grinding variability) increases, p falls.

Also, to increase p at a given grinding variability (σ)

level, k must be raised, i.e., a wider variety in ball

sizes must be available to increase the yield of

automatic assembly. From data generated by

numerical modeling, it is possible to empirically

relate p to process parameters and k. Therefore,

when this relationship is known, one can estimate

the total assembly cost using (12) given any values

of and k. Using cost accounting methods such as

Activity Based Costing the relevant costs may be

estimated. For illustration we used the numerical

framework used in producing Table 3 to develop an

empirical model relating p to σ and k as follows. Such

empirical models (Gujarati and Sangeetha 2007)

are helpful when the direct theoretical derivation of

the required relationships is not possible. Thus,

following Jianxin and Tseng (1999) and Al-Omiri

and Drury (2007), 2p = 0.693689 + 0.088342 k – 0.10595 σ - 0.00413 k +

20.00232 σ + 0.005933 k (13)

2Model (13) has a R of 0.80. The other cost that one

needs to quantify is that of producing an additional

variety of balls. This cost is a “hidden” COQ

component of production cost and is determined,

among other things, by the length of the set up time.

It is incurred when the ball grinding machine is

reset to make a different size ball, assuming that the

plant is operating in a “sold out” market and can sell

all it can produce. This hidden cost has three

components: (a) the cost of resources expended on

physically changing the actual setup; (b) the loss

due to production lost during the switchover; and (c)

the extra cost of storing and managing the stock of

an extra variety of balls. Alternatively, if balls are

outsourced, it will lead to purchasing and stocking

an additional size ball. This information too is

quantifiable. Thus, given ring grinding variability

and ball variety k (ignoring material cost) we find

Total (ball + assembly cost) = ca - p(c-1)a + Cost of

making k different ball sizes (14)

where p is given by (13). Expression (14) is a

function of and k.

The only cost that is missing so far is the cost of

reducing or the track grinding variability. Grinding 2variance (σ ) is a function of technology and the




Outline of a Procedure to Economically Justify

Variance Reduction

Balls are what make the bearing bear the load and

ensure friction-free movement for the life of the

b e a r i n g . B a l l s a re , t h e re fo re , c a re f u l ly

manufactured, starting with thick steel wire, cold

heading, cut into pieces and then smashed between

two steel dies (The Manufacture of a Ball bearing

2009). Then, the flash is removed and balls are heat-

treated to make them very hard, then tempered to

make them tough. Finishing requires grinding

between grinding wheels and then lapped with very

fine abrasive slurry to polish them for several hours

to reach correct dia and mirror-like finish (Figure

5). Each ball type takes 6 to 10 hours to finish. A

medium precision ball cannot be out of round more

than 25 millionth of an inch while high speed

precision bearing balls are allowed only five-

millionth of an inch roundness variation. Therefore,

to change ball size, set up has to be changed with

much care and effort and the process must be

stabilized before production begins. As shown

above, as ball dia variety increases, so does the

required number of set ups, each set up reducing

production capacity.

As indicated in earlier sections, in bearing assembly,

the critical process characteristic is the standard

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,

reduction in does not come free, it frequently

requires a step jump in grinding technology as it is

affected by tools, grinding speed, dressing, material

being ground, operator skills, coolant temperature,

spindle vibration, etc. Thus, the benefits of reducing

must be economically more than the cost of variance

reduction. We outline this analysis as follows:

Let the automatic assembly cost of a bearing be a.

Assume that the % yield of the automatic assembly

operation is p. Therefore, the proportion of manual

rework required to complete assembling all inner

and outer rings manufactured is (1 - p). Let the

rework cost be ca or ca, with the assumption that

the factor c ≥ 1. Therefore, the total assembly cost of

a bearing will be (Gryna et al 2008, Chapter 2)

Total assembly cost = a (fraction automatically

assembled) + ca (fraction manually assembled)

= a p + ca (1 - p)

= ca - p(c-1)a (12)

Now, as seen in the sections above, p is a function of

σ and k. Figure 6 illustrates this relationship for an

example in which radial clearance RC has been

assumed to be 20 micron and inner and outer mean

track diameters are as shown in Table 3.

Figure 6 Dependency of yield of automated assembly on σ and ball variety k

Some observations from Figure 6 are straight-

forward. For a given variety (k) of balls available, as

σ (i.e., track grinding variability) increases, p falls.

Also, to increase p at a given grinding variability (σ)

level, k must be raised, i.e., a wider variety in ball

sizes must be available to increase the yield of

automatic assembly. From data generated by

numerical modeling, it is possible to empirically

relate p to process parameters and k. Therefore,

when this relationship is known, one can estimate

the total assembly cost using (12) given any values

of and k. Using cost accounting methods such as

Activity Based Costing the relevant costs may be

estimated. For illustration we used the numerical

framework used in producing Table 3 to develop an

empirical model relating p to σ and k as follows. Such

empirical models (Gujarati and Sangeetha 2007)

are helpful when the direct theoretical derivation of

the required relationships is not possible. Thus,

following Jianxin and Tseng (1999) and Al-Omiri

and Drury (2007), 2p = 0.693689 + 0.088342 k – 0.10595 σ - 0.00413 k +

20.00232 σ + 0.005933 k (13)

2Model (13) has a R of 0.80. The other cost that one

needs to quantify is that of producing an additional

variety of balls. This cost is a “hidden” COQ

component of production cost and is determined,

among other things, by the length of the set up time.

It is incurred when the ball grinding machine is

reset to make a different size ball, assuming that the

plant is operating in a “sold out” market and can sell

all it can produce. This hidden cost has three

components: (a) the cost of resources expended on

physically changing the actual setup; (b) the loss

due to production lost during the switchover; and (c)

the extra cost of storing and managing the stock of

an extra variety of balls. Alternatively, if balls are

outsourced, it will lead to purchasing and stocking

an additional size ball. This information too is

quantifiable. Thus, given ring grinding variability

and ball variety k (ignoring material cost) we find

Total (ball + assembly cost) = ca - p(c-1)a + Cost of

making k different ball sizes (14)

where p is given by (13). Expression (14) is a

function of and k.

The only cost that is missing so far is the cost of

reducing or the track grinding variability. Grinding 2variance (σ ) is a function of technology and the




tightness with which the grinding process is

controlled by operators (operator skill).

However, as outlined above, it is possible to quantify

the cost savings resulting from reducing σ for a

given volume of annual bearing production. It is

possible to estimate the NPV of accumulated yearly

savings and consequently how much investment

can be justified in ring grinding precision

improvement to benefit business. This can be

projected for a planned period of selling those

particular bearings. The startling surprise is that if a

plant produces 10 million bearings annually and

one is able to reduce cost of production by 2

cents/bearing, assuming that the business runs for

7 years, an NPV in excess of USD 1 million will

materialize at nominal interest rates. Such sums can

just i fy s ignif icant process improvement

interventions by DMAIC (Pyzdek 2000).

Furthermore, a reduction of σ will improve bearing

performance and thus create intangible benefits of

supplying superior quality bearings. Such benefits

will be additional and their value and returns will be

strategic (Pyzdek 2000; Evans and Lindsay 2005).

The actual costs at the plant in question cannot be

shown due to proprietary reasons. However,

subsequent to this study, based on the insights

gained and incentives estimated based on delays,

production lost due to set up changes in ball

manufacture and the potential to upgrade products,

the company decided that it now had sufficient basis ®to launch a full-blown Six Sigma project involving

multi-factor Orthogonal Array experiments to pin

down factors that could raise p, the company's first-

pass bearings yield.

Every plant manager aims to reduce defects.

However, few in the supervisory or engineering staff

are able to formally quantify the cost of poor quality

even where monetary values (losses) are suspected

to be large. The result is the perpetuation of status

quo, unless a new facility with superior

technologies is proposed and justified. In a

m a n u f a c t u r i n g s e t u p t h e “ t i p o f t h e

iceberg”—scraps, rework and warrantee

service—are generally visible and reported

monthly or yearly. But, the hidden factory that runs

alongside of the brick-and–mortar factory or other

hidden costs of poor quality (COQ) typically evade

estimation. Still, without the benefits monetized,

management will not be interested in large scale

interventions such as Six Sigma, even if the

methodology has worked wonders at scores of

organizations worldwide. This paper illustrates a

Conclusions

procedure to quantify hidden losses—here due to

the extra sets of bearing balls that must be

precision-manufactured, stocked and used to raise

yield in automated bearing assembly. The method

employs numerical as well as Monte Carlo models, ®all done in Excel , and it results in quantitative

estimates of yield, manual work required and the

variety required in ball bearing balls before any

commitment needs to be made to alter plan

equipment, workforce or facilities.

Numerous other questions in strategizing

manufacturing could also be tackled by such

analytical procedures. Typical instances of these are

listed in Section 6. However, before one undertakes ®such a study to initiate Six Sigma DMAIC (see

Pyzdek 2000 or Evans 2005), all COQ (cost of

quality) components must be drilled into and

targeted for quantification. This would be the most

sensible way to initiate priority action using DMAIC,

done best at the “D” (Define) stage.

In this instance it was particularly gratifying to

quantitatively affirm the intuitive assertions of

plant management. Their reactions became

instrumental in delving deeper to locate tacit

opportunities that existed for raising profits.

®Excel models were deliberately used in this work to

serve as easy-to-use decision support tools usable

at the plant level, for many such decisions are often

made locally without the sophistication consultants ®typically engage in. Excel 's graphic and statistical

capabilities proved sufficient for the plant

management to comprehend the steps in the

analysis and see how the conclusions were reached.

This became a stepping stone to raise many related

questions about alternative ways to cut cost and

impact profit. One such initiative justified was the

upgradation of vendor management. Another was

to reset the settings in the automated facilities

toward the gradual removal of manual work. A

significant issue tossed up was the engineering

optimization of the target RC (radial clearance)

values for its close interplay with bearing

performance and the variety of balls required in

automatic assembly. This constituted the charter of ®a separate Six Sigma project.

References

• Al-Omiri M and Drury Colin (2007). A Survey of

facts influencing the Choice of Product Costing

Systems in UK Organisations, Management

Accounting Research, Vol 18(4), Dec, 399-424.

• Ball Bearing, http://www.madehow.com/

Volume-1/Ball-Bearing.html, accessed April 25,

2009.

• Bhat U Narayan (2008). An Introduction to

Queueing Theory: Modeling and Analysis in

Applications Series, Statistics for Industry and

Technology, Birkhouser.

• Evans J R and Lindsay W M (2005). An

Introduct ion to Six Sigma & Process

Improvement, Thomson .

• Gijo E R (2005). Improving Process Capability of

Manufacturing Process by Application of

Statistical Techniques, Quality Engineering, 17:

309-315.

• Gitlow H S, Oppenheim A J, Oppenheim R and rdLevine D M (2005). Quality Management, 3 ed.,

Tata McGraw-Hill.

•

Juran's Quality Planning and Analysis for thEnterprise Quality, 5 ed., Tata McGraw-Hill.

• Gujarati D N and Sangeetha (2007). Basic

Econometrics, Tata McGraw-Hill.Jianxin Jio and

Tseng Mitchel M (1999). A Pragmatic Approach

to Product Costing based on Standard Time

Estimates, International Journal of Operations

and Production Management, 735-755.

• Law A M and Kelton W D (2000). Simulation rdModeling and Analysis, 3 ed., McGraw-Hill.

• Montgomery D C (2005). Introduction to thStatistical Quality Control, 4 ed., Wiley.

• Rice J A (2007). Mathematical Statistics and rdData Analysis, 3 ed., Thomson

• Sharma Poonam (2009). Report on MBA Project,

Vinod Gupta School of Management, IIT

Kharagpur, India.

• SKF Bearings Handbook (2009). http:// www.

who-sells-it.com/r/skf-bearing-handbook.html,

accessed April 25, 2009.

Gryna F M, Chua R C H and Defeo J A (2008).




tightness with which the grinding process is

controlled by operators (operator skill).

However, as outlined above, it is possible to quantify

the cost savings resulting from reducing σ for a

given volume of annual bearing production. It is

possible to estimate the NPV of accumulated yearly

savings and consequently how much investment

can be justified in ring grinding precision

improvement to benefit business. This can be

projected for a planned period of selling those

particular bearings. The startling surprise is that if a

plant produces 10 million bearings annually and

one is able to reduce cost of production by 2

cents/bearing, assuming that the business runs for

7 years, an NPV in excess of USD 1 million will

materialize at nominal interest rates. Such sums can

just i fy s ignif icant process improvement

interventions by DMAIC (Pyzdek 2000).

Furthermore, a reduction of σ will improve bearing

performance and thus create intangible benefits of

supplying superior quality bearings. Such benefits

will be additional and their value and returns will be

strategic (Pyzdek 2000; Evans and Lindsay 2005).

The actual costs at the plant in question cannot be

shown due to proprietary reasons. However,

subsequent to this study, based on the insights

gained and incentives estimated based on delays,

production lost due to set up changes in ball

manufacture and the potential to upgrade products,

the company decided that it now had sufficient basis ®to launch a full-blown Six Sigma project involving

multi-factor Orthogonal Array experiments to pin

down factors that could raise p, the company's first-

pass bearings yield.

Every plant manager aims to reduce defects.

However, few in the supervisory or engineering staff

are able to formally quantify the cost of poor quality

even where monetary values (losses) are suspected

to be large. The result is the perpetuation of status

quo, unless a new facility with superior

technologies is proposed and justified. In a

m a n u f a c t u r i n g s e t u p t h e “ t i p o f t h e

iceberg”—scraps, rework and warrantee

service—are generally visible and reported

monthly or yearly. But, the hidden factory that runs

alongside of the brick-and–mortar factory or other

hidden costs of poor quality (COQ) typically evade

estimation. Still, without the benefits monetized,

management will not be interested in large scale

interventions such as Six Sigma, even if the

methodology has worked wonders at scores of

organizations worldwide. This paper illustrates a

Conclusions

procedure to quantify hidden losses—here due to

the extra sets of bearing balls that must be

precision-manufactured, stocked and used to raise

yield in automated bearing assembly. The method

employs numerical as well as Monte Carlo models, ®all done in Excel , and it results in quantitative

estimates of yield, manual work required and the

variety required in ball bearing balls before any

commitment needs to be made to alter plan

equipment, workforce or facilities.

Numerous other questions in strategizing

manufacturing could also be tackled by such

analytical procedures. Typical instances of these are

listed in Section 6. However, before one undertakes ®such a study to initiate Six Sigma DMAIC (see

Pyzdek 2000 or Evans 2005), all COQ (cost of

quality) components must be drilled into and

targeted for quantification. This would be the most

sensible way to initiate priority action using DMAIC,

done best at the “D” (Define) stage.

In this instance it was particularly gratifying to

quantitatively affirm the intuitive assertions of

plant management. Their reactions became

instrumental in delving deeper to locate tacit

opportunities that existed for raising profits.

®Excel models were deliberately used in this work to

serve as easy-to-use decision support tools usable

at the plant level, for many such decisions are often

made locally without the sophistication consultants ®typically engage in. Excel 's graphic and statistical

capabilities proved sufficient for the plant

management to comprehend the steps in the

analysis and see how the conclusions were reached.

This became a stepping stone to raise many related

questions about alternative ways to cut cost and

impact profit. One such initiative justified was the

upgradation of vendor management. Another was

to reset the settings in the automated facilities

toward the gradual removal of manual work. A

significant issue tossed up was the engineering

optimization of the target RC (radial clearance)

values for its close interplay with bearing

performance and the variety of balls required in

automatic assembly. This constituted the charter of ®a separate Six Sigma project.

References

• Al-Omiri M and Drury Colin (2007). A Survey of

facts influencing the Choice of Product Costing

Systems in UK Organisations, Management

Accounting Research, Vol 18(4), Dec, 399-424.

• Ball Bearing, http://www.madehow.com/

Volume-1/Ball-Bearing.html, accessed April 25,

2009.

• Bhat U Narayan (2008). An Introduction to

Queueing Theory: Modeling and Analysis in

Applications Series, Statistics for Industry and

Technology, Birkhouser.

• Evans J R and Lindsay W M (2005). An

Introduct ion to Six Sigma & Process

Improvement, Thomson .

• Gijo E R (2005). Improving Process Capability of

Manufacturing Process by Application of

Statistical Techniques, Quality Engineering, 17:

309-315.

• Gitlow H S, Oppenheim A J, Oppenheim R and rdLevine D M (2005). Quality Management, 3 ed.,

Tata McGraw-Hill.

•

Juran's Quality Planning and Analysis for thEnterprise Quality, 5 ed., Tata McGraw-Hill.

• Gujarati D N and Sangeetha (2007). Basic

Econometrics, Tata McGraw-Hill.Jianxin Jio and

Tseng Mitchel M (1999). A Pragmatic Approach

to Product Costing based on Standard Time

Estimates, International Journal of Operations

and Production Management, 735-755.

• Law A M and Kelton W D (2000). Simulation rdModeling and Analysis, 3 ed., McGraw-Hill.

• Montgomery D C (2005). Introduction to thStatistical Quality Control, 4 ed., Wiley.

• Rice J A (2007). Mathematical Statistics and rdData Analysis, 3 ed., Thomson

• Sharma Poonam (2009). Report on MBA Project,

Vinod Gupta School of Management, IIT

Kharagpur, India.

• SKF Bearings Handbook (2009). http:// www.

who-sells-it.com/r/skf-bearing-handbook.html,

accessed April 25, 2009.

Gryna F M, Chua R C H and Defeo J A (2008).



•

Quality, Harvard Business Review, January-

February.

• The Manufacturing of a Ball Bearing,

http://www.bearingsindustry.com/manufactu

ring.pdf, accessed April 25, 2009.

Taugchi G and Clausing Don (1990). Robust

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.

Acknowledgements

The author thanks Harsh Sachdev, Joydeep

Sengupta, Jyoti Mukherji and Tapan Mondal of Tata

Bearings. Together they provided a wealth of

practical shop floor knowledge and economic

insights into bearing manufacturing. Poonam

Sharma provided data collection assistance.

Dr. Gurumurthy Kalyanaram

Dean for Research, NMIMS University, Mumbai

Editor, NMIMS Management Review

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