Statistical analysis of a study on a press tool’s life-length

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Statistical analysis of a study on a press tool’s

life-length

Takako Westlind

U.U.D.M. Project Report 2005:13

Examensarbete i matematisk statistik, 10 poäng Handledare: Lennart Wärnborg, Sandvik Coromant, Gimo

Examinator: Lennart Norell December 2005

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Preface

This is a project report in mathematical statistics of the Natural Scientist Program at Uppsala University. This is based on my study of life length of a press tool at Sandvik Coromant, Gimo during the period from February to April in 2005.

This study consists of two parts. One is a part of collecting data including measurements which was the most time-consuming part. The other one is a part of analysis of the data collected. Through this study at the company, I learned that theory does not always work perfectly in reality. Roughly speaking, as sometimes said by statisticians “All models are wrong, but some of them are useful, namely models that detect the

underlying phenomena.” It is the aim for statisticians to build such a model that bridges the gap between theory and reality.

I wish to thank Mr. Lennart Wärnborg, Mr. Stefan Söderblom and their colleagues at Sandvik Coromant, Gimo for their encouragement and support during the period this study had been going on. I also thank Dr. Lennart Norell, my supervisor and

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1 Introduction

……… 1

2 Data

materials ……….

2 2.1 Definition of wear (Slitage)

2.2 Conceivable factors of wear

2.3 Collecting IC data

2.4 Measurement of clearance

2.5 Calculation of clearance

3 Statistical

methods

……….. 5

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1 Introduction

At various places in the report, there are special footnote references for Sandvik Coromant, Gimo.

Sandvik Coromant in Gimo is a part of the Sandvik Tooling Group and is the world leading manufacturer of cutting tools for the metal working industry. The tools are used for turning, milling and drilling. In metal cutting, the metal is removed by a small “cutting edge” of a cemented carbide insert in contact with the work piece. My work in Gimo was to study the tool used at the very start of this process making this insert, especially the wear on the part of the tool creating this “cutting edge”.

The making of an insert starts by pressing cemented carbide powder in a press tool.

The tool consists of a die, containing a cavity. Powder is injected into this cavity and compacted in between a top punch and a bottom punch. When a tool is new, there is an initial clearance between the die and the punch. The powder pressed is very abrasive and will make this clearance increase when the tool is used. Also the cavity in the die will increase in size. Both these facts have negative effects on the product’s quality. It is therefore important to be able to predict the

condition of a press tool. In this report, a model predicting the tool wear is presented with focus on a certain group of tools, whose shape is called “negative”.

The company’s vision is to be chosen as “the number one productivity partner” by its customers. To accomplish this, Sandvik works close to its customers, and putting great effort into research and development. The main purpose of my study at Sandvik Coromant, Gimo was to find a

statistical model that estimates a press tool’s life-length so that the company can keep serving high quality products for its customers.

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2 Data materials

2.1 Definition of wear (Slitage)

Wear of the press tool is divided into two parts, i.e. wear of the punch and wear of the die. These two components have a clearance between each other in the pressing position. It is the change in this clearance that is defined as wear. The initial clearance is determined when the press tool is made.

Wear = Present clearance – Initial clearance

2.2 Conceivable factors of wear

Conceivable factors of the wear given from the company were the following:

1) Powder grade (Pulversort)

2) Initial clearance between punch and die (Ursprungligt spel) 3) Press machine number where the press tool will be used at (Press nr)

4) Number of pressed blanks (Pressat antal =PRANT)

5) Size of product (Storlek)

A ` blank´ is a semi-finished sintered insert of hard metal consisting of a mixture of different powder grades, which is pressed and formed by press tools (die and punches). This will later be ground, edge-treated and coated.

Except for 2), the data was obtained from the company’s data base IC. Measurement of factor 2) will be explained in section 2.4. Among the five factors, 1), 2) and 4) were factors that the company suspected to be important for the wear.

2.3 Collecting IC data

Files from 22 years of production (1984 – 2005) were obtained from Sandvik’s database, IC. Because of limited time resources, 60 press tools having a certain form, called “negative”, were selected to be observed. Among the data of these 60 tools, 7 were excluded from final data, because their numbers of pressed blanks were not appropriate to use for the analysis. Detailed information will be referred to section 3.1.2.

The files of the 22 years’ production were collected from IC and sorted by each press tool’s number. The overlapped observations were excluded from final data. Files of the year 1982 and 1983 were not available. Of the 53 press tools, 4 were produced in 1982 and 1983.

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The data from IC Number of different items IC term

Order number 5311 (ORDER NR)

Press tool’s identity no. 53 (PRVERKT)

Press machine’s identity no. 79 (PRESS NR)

Powder grade’s identity no. 68 (SORT)

Size of blanks 2 (STORLEK)

Number of pressed blanks (PRANT)

The data for the 53 press tools was not sufficient for a detailed model with individual parameters for all 79 press machines and all 68 powder grades. Due to a large number of parameters, a reduction was necessary. This was performed by grouping of press machines and powder grades. The press machines were divided into 2 groups 1), type1 = mechanical and type2 = hydraulic, because it was impossible to identify the press machine used for each order (despite of the existence of the IC data). The 68 powder grades were grouped into 21 groups according to similarity of chemical contents. The first group is called g1 and it consists of 20 unknown powder grades, which is so called “grades under development (utvecklingssorter)”. This group consists of powder grades which are not able to be allocated in any other powder grade group.

By this grouping it became possible to estimate 26 parameters for additive effects; 2 for types of press machines, 21 for groups of powder grade, 2 for sizes and 1 for initial clearance.

2.4 Measurement of clearance

A press tool consists of a die and a number of punches. The top and bottom punches’ present dimensions were measured using hand measuring tools. Present wear of dies was measured by the company’s coordinate measuring machine, CMM. The original dimensions of every single press tool are stored. Each tool has its own documentation where the initial clearance could be found. There were different numbers of punches for a press tool. Measurements were done from 3 to 7 punches per tool. In a negative tool, a punch can be used either as a bottom or a top punch. It is impossible to identify which punch has been used for each observation (order no.) of IC data. A mean value of the punches was, therefore, registered for respective press tool. All of these punches have similar forms, negative and parallelogram-shaped and they are of size 12 or 16. Dies were collected from the storage and brought to the measuring room for measurement by a CMM. Five dies a week were measured. All dies were measured at the clip line position.

2.5 Calculation of clearance

By definition, the clearance between punch and die is the sum of the punches under-dimension and die’s over-dimension. [See Figure 2 on the next page.]

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To define the wear let D0 be the initial dimension of the die, possibly adjusted by 0.002 mm.

The present dimension of the die is denoted D1. The initial dimension of the punches was

assumed to be equal to P0 and the present average of the punch dimensions is denoted by P1.

The wear is thus calculated as

D1 – D0 – (P1 – P0) = (D1 – P1 ) – (D0 – P0 )

where D1 – D0 > 0 is the over-dimension of the die and P1 – P0 < 0 is the under-dimension of the

punches.

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3 Statistical methods

The following statistical packages were available for the analysis: Statistica 6.0 2), Simca-P and Modde at Sandvik Coromant, Gimo

SAS 8.0 and Minitab 14at Uppsala University

3.1 Data preparation and preliminary analysis

To prepare the estimation of the parameters, the number of pressed blanks was summed for each press type and for each powder grade group for respective press tool, using Excel, “pivot table”. An example of a press tool is as follows:

A part of pivot table:

Press tool no. Press type Total

12938-00 1 118644

2 515

12938-00 Total 119159

Press tool

Powder

grade group Total

12938-00 1 144 3 6060 5 9733 6 22268 7 5184 8 35712 9 1874 12 19242 13 17806 19 1136 12938-00 Total 119159

A part of the table of data

Press tool’s No Press type1 Press type2 Sum (press type) g1 g2 g3 g4 g5 g6 12938-00 118644 515 119159 144 0 6060 0 9733 22268 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16 5184 35712 1874 0 0 19242 17806 0 0 0 g17 g18 g19 g20 g21 sum(group) Size Initial clearance Present clearance slitage 0 0 1136 0 0 119159 12 0.0020 0.0110 0.0090

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Preliminary analysis was done by using SAS, Simca P, and Minitab.

3.1.1 Preliminary analysis by Simca P

Principal component analysis (PCA) is useful for a preliminary analysis when there are many variables. The basic idea of the methods is to describe the variations of a set of multivariate data in terms of a set of uncorrelated variables, each of which is a particular linear

combination of the original variables.

The graph of wear against the covariates showed that the number of pressed blanks (PRANT) has a positive effect, i.e. the more number of pressed blanks, the more wear, which is quite reasonable. It also showed the initial clearance between punch and die (Urspr.spel) has a negative effect, which means that the larger clearance the smaller wear. The effect of the powder grade had a small variation.3)

3.1.2 Plot of wear against covariates

First of all, wear against number of pressed blanks, type 1, type 2, initial clearance, and size were plotted using Minitab. Plot of wear vs. number of pressed blanks (= type1 and type2) showed a clear linear tendency except for 7 press tools. These tools formed their own linear relation with another slope. 4) (See Appendix Minitab plot)

These outliers were investigated by the company5). It was found that the original dies of these 7 press tools had been discarded, and new dies were set without changing the number of pressed blanks in IC data to 0 at the new starting point. These 7 press tools were removed from the analysis, and thus data from 53 press tools was used.

3.1.3 Preliminary analysis

It might be asked if the number of tool settings, instead of the number of pressed blanks has a large effect on wear. In the same way as number of pressed blanks were summed for each 53 tools and put into a table, a new table of number of tool settings (antal monteringar, omgångar) was made using Excel. Then wear was plotted against respective group whose data is based on number of tool settings. These plots did not either indicate any clear linearity between wear and powder grade groups. Then the analysis would be continued as before, with data based on number of pressed blanks (PRANT).

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3.2 General linear models

In a general linear model, GLM, the observed value of the dependent variable yi for observation

number i (i=1, 2, …n) is modelled as a linear function of p independent variables x1 , x2 , …, xp as

yi = β0+ β1 xi,1 + … + βp xi,p+ ei , i = 1, 2, … , n

or in matrix terms, y = X β + e, where

y = (y

1

, y

2

, ... y

n

)'

,

X = , ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ p n n p p x x x x x x , 1 , , 2 1 , 2 , 1 1 , 1 . . . 1 . . . . . . . . . . . . 1 . . . 1

β = (β

0

,

β

1

,

…

β

p

)'

, and e = (e1, e2, …en)' , ei ~ N(0, σ2) 6) .

Estimation of parameters in general linear models is often done using the method of least squares. For normal theory models this is equivalent to maximum likelihood estimation. The parameters are estimated with those values for which the sum of the squared residuals

∑

i i

e2

is minimal. In matrix terms, this sum of squares is

e'e = (y – Xβ)' (y – Xβ) .

Minimizing this expression with respect to the parameters in β gives the normal equations, X' X β = X' y .

If the matrix X' X is non-singular, this yields, as estimators of the parameter of the model, = (X' X)

βˆ -1_{X' y .}

If the inverse of X' X does not exist, we can still find a solution, although the solution may not be unique. We can use so called generalized inverses and find a solution as

= (X' X)

β~

X' y .

Alternatively we can restrict the number of parameters in the model by introducing constraints that lead to a non-singular X' X. 7)

If a class variable at q levels is included, it can be written β1 xi,1 + … + βq xi,q where exactly

one of the dummy variables xi,1 , … , xi,q equals 1 and the others are 0. To simplify notation,

the effect is usually written αj , j=1,…, q, and a corresponding declaration is done in the

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3.3 Model building: GLM approach

Typical tests:

1. All p independent variables considered together explain a significant amount of the variation in wear?

yi = β0 + β1 xi,1 +β2 xi,2 + … + βp xi,p+ ei , i = 1, 2, … , 53

Overall F-test of H0 : β1 = β2 = ….= βp= 0 against H1 : At least one βk ≠ 0 , k = 1, 2, … p. The null

hypothesis means that none of the p independent variables is needed.

2. Does xksignificantly improve the prediction of wear?

Partial F- test of H0 : βk = 0 against H1 : βk ≠ 0. This test can be generalised to a group of two or

more independent variables. In the case with one variable, the t-test can be used as an alternative.

3. Are some β– parameters equal? F- test of H0 : β1 = … = βq , q<p

As mentioned before, number of pressed blanks was summed for each press type and for each powder grade group for respective press tool to estimate the parameters β0 , β1, … βp by GLM

function. Contrasts of powder grade groups were tested. Table of abbreviations is as follows;

Covariates In Swedish Name in SAS program

Size of blanks 1-2 Storlek storlek

Press type1-2 Prsstyp typ1, typ2

Powder grade group1-21 Pulversortgrupp g1 – g21

Initial clearance (i-clearance) Ursprungligt spel urspr_spel Cumulative pressing pressure Kumulativ presstyrka cpp

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3.3.1 Model inclusive size of blanks

Model 1

A model with size, type1(size), type2(size), group and i-clearance(size) was analysed. yi,j = β0+ αj+ β1,j xi,j,1 + β2,j xi,j,2 + β3 xi,j,3 + ... +β23 xi,j,23 + β24,j xi,j,24 + ei,j

j= 1, 2, i = 1, …, nj, n1 = 42 n2 =11 αj = effect of size, j=1,2 ( 1 = size12, 2 = size16)

β1,j xi,j,1 = effect of pressing in type1, size j β2,j xi,j,2 = effect of pressing in type2, size j

β3 xi,j,3, ... , β23 xi,j,23 = effect of pressing in powder grade group1, …, group 21

β24,j xi,j,24 = effect of initial clearance, size j

Size as class variable

The following tests were done: 1) Equality of size effect, H0 : α1 = α2

2) Equality of type1 effects between sizes H0 : β1,1 = β1,2

3) Equality of type2 effects between sizes H0 : β2,1 = β2,2

4) Equality of i-clearance effects between sizes H0 : β24,1 = β24,2

5) Equality of powder grade groups 8) H0 : β3 = β4= ….= β23

These tests were done by the following SAS program. SAS program (The contrast statement is shortened here.)

proc glm data=verkt;

class storlek;

model slitage = storlek typ1(storlek) typ2(storlek) g1 -g21 urspr_spel(storlek) / ss1 e;

estimate 'storlek' storlek 1 -1;

estimate 'typ1(storlek)' typ1(storlek)1 -1;

estimate 'typ2(storlek)' typ2(storlek)1 -1;

estimate 'urspr_spel(storlek)' urspr_spel(storlek) 1 -1;

contrast 'grupper'g1 1 g2 -1, g2 1 g3 -1, ... g20 1 g21 -1;

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A part of the output

The GLM Procedure

Dependent Variable: slitage slitage Sum of

Source DF Squares Mean Square F Value Pr > F Model 27 0.00045665 0.00001691 8.05 <.0001 Error 25 0.00005254 0.00000210

Corrected Total 52 0.00050918

R-Square Coeff Var Root MSE slitage Mean 0.896816 22.24472 0.001450 0.006517 Contrast DF Contrast SS Mean Square F Value Pr > F grupper 20 0.00007312 0.00000366 1.74 0.0948 Parameter Estimate Pr > |t| size 0.00149809 0.6287 type 1(size) 0.00000001 0.7505 type 2(size) -0.00000084 0.0592 i-clearance(size) -0.31628072 0.6661

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3.3.2 Model exclusive size of blanks

Model 2

A model with type1, type2, group and i-clearance was analysed.

yi = β0 + β1 xi,1 + β2 xi,2 + β3 xi,3 + ... +β23 xi,23 + β24 xi,24 +ei, i = 1, 2, … , 53

β1 xi,1 = effect of pressing in type1

β2 xi,2 = effect of pressing in type2

β3 xi,3, …, β23 xi,23 = effect of pressing in powder grade group 1, …, 21 β24 xi,24 = effect of initial clearance

The points of a special interest were:

1) Test of equality between type1 and type2: H0 : β1 = β2

2) Test of equality of powder grade groups: H0 : β3 = β4= ….= β23

3) Estimates of separate regression coefficients for type 1 and type 2 as mean coefficients over all powder grade groups.

4) Estimates of separate regression coefficients for each powder grade group as mean coefficients over both press machine types.

5) Plots of ordinary and standardised residuals against size, i-clearance, type 1 and type 2 in order to check model assumptions.

SAS program

model slitage =typ1 typ2 g1 - g21 urspr_spel /e ss1;

estimate 'typ1-typ2' typ1 1 typ2 -1;

contrast 'grupper'g1 1 g2 -1, g2 1 g3 -1, ... g20 1 g21 -1;

estimate 'typ1:medel' intercept 21 typ1 21 g1 1 g2 1 ... g21 1 /divisor=21;

estimate 'typ2:medel' intercept 21 typ2 21 g1 1 g2 1 ... g21 1 /divisor=21;

estimate 'grupp1:medel' intercept 1 typ1 0.5 typ2 0.5 g1 1;

estimate 'grupp2:medel' intercept 1 typ1 0.5 typ2 0.5 g2 1; ...

estimate 'grupp21:medel' intercept 1 typ1 0.5 typ2 0.5 g21 1;

output out=new p=yhat r=e rstudent=rstudent;

proc print;

var litas ge yhat e rstudent;

proc plot;

plot e*(yhat storlek urspr_spel typ1 typ2)/vref=0;

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Sum of

R-Square Coeff Var Root MSE slitage Mean 0.879719 22.29930 0.001453 0.006517

Contrast DF Contrast SS Mean Square F Value Pr > F grupper 20 0.00007567 0.00000378 1.79 0.0743

Standard

Parameter Estimate Error t Value Pr > |t| typ1-typ2 0.00000000 0.00000003 0.14 0.8870 typ1:medel 0.00546775 0.00069570 7.86 <.0001 typ2:medel 0.00546774 0.00069570 7.86 <.0001 grupp1:medel 0.00547125 0.00069595 7.86 <.0001 grupp2:medel 0.00547248 0.00069574 7.87 <.0001 grupp3:medel 0.00547128 0.00069594 7.86 <.0001 grupp4:medel 0.00547128 0.00069595 7.86 <.0001 grupp5:medel 0.00547128 0.00069596 7.86 <.0001 grupp6:medel 0.00547129 0.00069595 7.86 <.0001 grupp7:medel 0.00547120 0.00069597 7.86 <.0001 grupp8:medel 0.00547135 0.00069594 7.86 <.0001 grupp9:medel 0.00547128 0.00069595 7.86 <.0001 grupp10:medel 0.00547122 0.00069594 7.86 <.0001 grupp11:medel 0.00545238 0.00069493 7.85 <.0001 grupp12:medel 0.00547127 0.00069595 7.86 <.0001 grupp13:medel 0.00547129 0.00069595 7.86 <.0001 grupp14:medel 0.00546825 0.00069680 7.85 <.0001 grupp15:medel 0.00547129 0.00069595 7.86 <.0001 grupp16:medel 0.00547885 0.00069653 7.87 <.0001 grupp17:medel 0.00541046 0.00069160 7.82 <.0001 grupp18:medel 0.00547118 0.00069594 7.86 <.0001 grupp19:medel 0.00547142 0.00069602 7.86 <.0001 grupp20:medel 0.00547121 0.00069590 7.86 <.0001 grupp21:medel 0.00547116 0.00069596 7.86 <.0001

The X’X matrix was found to be singular due to linear dependency11) of the covariates, and a generalized inverse was used to solve the normal equations. (See Appendix SAS output pages 11 - 14)

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As mentioned before, wear against g1, g2, …, g21 were plotted by Excel (section 3.1.3), and none of these plots did not indicate any special linear tendency. The contrast test showed no significant differences between these 21 groups (P-value=0.0743) at 5 % level. Mean value of respective group did not either indicate any clear difference among groups, although group17 showed a slight difference from other groups.

The plots of residuals against size, i-clearance, type1 and type2 indicated that the errors were independent and normally distributed with a constant variance. (See Appendix SAS output pages 16 -20)

3.3.3 A new variable and final data for analysis

Analysis of the data by SAS showed that the five factors 1) – 5) in section 2.2 did not

sufficiently explain the wear of press tools, and they were not enough to build a proper model. It is therefore necessary to find a new variable which contains factors, powder grade and number of pressed blanks. This new variable was named cumulative pressing pressure (Kumulativ presstyrka). It is a summation of the mean pressing pressure times the number of pressed blanks of each powder grade.

An example of the calculation is as follows:

Press tool Powder grade no. Mean pressing pressure Number of pressed blanks

09047-00 234 111.3 576

553 115.0 62524

….. … …

565 89.9 120

Cumulative pressing pressure of press tool “09047-00”:

111.3*576 + 115.0*62524 + ….. + 89.9*120 = 7302306.8

After dividing the new variable by 109, it was added as cpp to the table in section 3.1.

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3.3.4 Model with size, i-clearance(size), cpp(size)

Model 3

A model with size, i-clearance(size) and cpp(size) was analysed.

yij = β0+ αj+ β1,j xi,j,1 + β2,j xi,j,2 + ei,j j= 1, 2, i = 1, … , nj, n1 = 42, n2 =11 αj = effect of size, j=1,2 ( 1 = size12, 2 = size16)

β1,j xi,j,1 = effect of initial clearance, size j

β2,j xi,j,2 = effect of cumulative pressing pressure, size j

The following parameter differences were estimated. 1) Overall size effects: α1 - α2

2) Size effect on i-clearance: β1,1 - β1,2

3) Size effect on cpp: β2,1 - β2,2

Residuals were plotted against size, i-clearance, and cpp in order to check model assumptions. SAS program

class storlek;

model slitage = storlek urspr_spel(storlek) cpp(storlek) / ss1 e;

estimate 'storlek' storlek 1 -1;

estimate 'urspr_spel(storlek)' urspr_spel(storlek) 1 -1;

estimate 'cpp(storlek)' cpp(storlek) 1 -1;

output out=new p=yhat r=e rstudent=rstudent;

proc print;

var slitage yhat e rstudent;

proc plot;

plot e*(yhat storlek urspr_spel cpp )/vref=0;

run;

A part of output

Sum of

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Parameter Estimate Pr > |t| size difference -0.00091380 0.5485 i-clearance between sizes 0.33503274 0.3539 cpp between sizes 0.01213780 0.7730

The R2 was 0.759 and none of the estimates were significantly different from 0. Hence, none of the hypotheses H0: α1 = α2, H0: β1,1 = β1,2 and H0: β2,1 = β2,2 can be rejected.

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3.3.5 Model with i-clearance and cpp

Model 4

A model with two factors; initial clearance (urspr_ spel) and cumulative press pressure (ccp) was analysed.

y

i

= β

0

+ β

1

x

i,1

+ β

2

x

i,2

+ e

i

,

i = 1, 2, … , 53

β1 xi,1 = effect of initial clearance

β2 xi,2 = effect of cumulative pressing pressure

Prediction limits for the individual value of each observation were produced.

The leverage hi, i=1, 2,…, 53 was displayed and checked. Leverage is a measure of the

importance of the ith observation in determining the model parameters. Leverage values are such that 0 h≤ i≤ 1. It is recommended scrutinizing any observation for which

n p h_i > 2( +1) 0.113 53 ) 1 2 ( 2 = + = .

Details can be found in “Applied regression analysis and other multivariable methods” by Kleinbaum, David G. and others.

Residuals against fitted value, cpp and i-clearance were plotted. SAS program

model slitage= urspr_spel cpp /solution p cli e;

output out=newcpp p=yhat r=e rstudent=rstudent h=hii;

proc print;

var slitage yhat e rstudent hii;

proc plot;

plot e*(yhat urspr_spel cpp)/vref=0;

run;

Sum of

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Parameter Estimate Pr > |t| Intercept 0.0060024230 <.0001 i-clearance -.6224352093 0.0001 ccp 0.1582801461 <.0001

The result indicated that all parameters are highly significant (P-value≤ 0. 0001) at 5% level. The R2of this model was 0.753.

Observation no.50 gave the highest leverage value of 0.159, and observation no.25 showed the next highest value of 0.130, both above the recommended limit value, 0.113. (See Appendix p.35) As can be seen in the plot in the Appendix p.36, none of these two values with the largest values of h_i indicate any deviance from the model.

The observed, predicted, and residual values for each observation are also displayed in the output. The most extreme studentized residual of - 2.28 was obtained for observation no.4, considered as an observation from a t-distribution, this value does not indicate any extreme outlier value. The first four observations correspond to tools manufactured before 1984 and thus, all use of them is not documented. However, their studentized residuals were 1.52, 1.40, 1.51 and -2.28, and their yˆ_ivalues were not extreme12). Hence, there is no indication that these four should be excluded from the analysis. Separate analysis without those tools gave

essentially the same result.

The plot of residuals against i-clearance and cpp, see Appendix SAS output pages 36-38, indicated independence, normal distribution with a constant variance (variance

homoscedasticity). Thus, the assumption of GLM was satisfied.

3.4 Model comparison

The ordinary and adjusted R2 13) of the models are as follows:

R2 R2adj

Model 1- size, type1(size), type2(size),group and i-clearance(size) 0.897 0.786 Model 2- type1, type2, group and i-clearance 0.880 0.785 Model 3- size, i-clearance(size) and cpp(size) 0.759 0.733

Model 4- i-clearance and cpp 0.753 0.743

As the R2 valuesshow, model with more variables is of course more descriptive than a simpler model. However, it can be more difficult to interpret. A simpler model can therefore be preferable. As shown above, the R2adj are almost the same for all four models, and this

motivates the simplest model 4 as the best fitting model. The tests of model 4 indicated also this model as the final model.

3.5 Multi-collinear variables

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be careful so that covariates are not related to each other. Variance inflation factor (VIF) is often used to measure collinearity in a multiple regression analysis. The larger value of VIF we have, the harder to identify the effect of a single variable. A rule of thumb is to be concerned about the variable when VIF is larger than 10.0.

To check multi-collinearity, a model with initial clearance, cumulative pressing pressure, and number of pressed blanks (Press antal =PRANT) was tested. Because of strong relationship between cumulative press pressure and number of pressed blanks, the number of pressed blanks was not significant. The VIF for cumulative pressing pressure (cpp) was 20.5, and for number of pressed blanks (PRANT) was 20.4. This is a clear indication for multi-collinearity and only one of the variables is meaningful to include in the model. (See Appendix SAS output p.39-40)

3.6 A final model

According to the analyses above, a model with cumulative pressing pressure and initial clearance (i-clearance) was chosen as a final model.

4 Results

Wear of these 53 negative parallelogram-shaped punches and dies depends mainly on initial clearance and cumulative pressing pressure. The second variable is a summation of mean press pressure for respective powder grade times number of pressed blanks.

The final model is:

y

i

= β

0

+β

1

x

i,1

+ β

2

x

i,2

+ e

i , i = 1, 2, …, 53

where x1 is initial clearance and x2 is cumulative pressing pressure for each powder grade. For

a detailed description of x1 and x2, see sections 2.5 and 3.3.3. 14)

Standard

Parameter Estimate Error Pr > |t| β0 0.00600 0.00063 <.0001

β1 -0.62244 0.14849 0.0001

β2 0.15828 0.01411 <.0001

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5 Discussion

This result contains some uncertainties due to the following reasons:

- The basic data contains only orders from 1984 to spring of 2005. The data for 4 press tools that were produced before 1984 are, therefore, partly not accessible. However, study of these tools did not indicate any unnatural wear.

- Mean value of upper punches’ IC for a press tool was used for the calculation of current initial clearance, because it is impossible to identify which upper punch is used for each of the 5311 orders.

- The part of the initial clearance contributed by the punches is assumed to be zero, because the data did not exist.

- As mentioned before, we have 79 press machines and 68 powder grades. When these were grouped into 2 types of press machines and 21 powder grades groups, the effect of each press machine and each powder grade are not detectable any more.

- Study was done only for tools of a certain shape.

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References

Kleinbaum, David G. and others. (1998). Applied regression analysis and other multivariable methods, Third edition. Duxbury Press, Pacific Grove, CA

Everitt, B. S. and Dunn G. (2001). Applied multivariate data analysis, Second edition. Arnold, London

Johnson, R. A. and Kam-Wah Tsui. (1998). Statistical reasoning and methods, Wiley, New York

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Appendix

Minitab plot of wear vs. number of pressed blanks

Number of pressed blanks

We a r 1600000 1400000 1200000 1000000 800000 600000 400000 200000 0 0,014 0,012 0,010 0,008 0,006 0,004 0,002 0,000

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Statistical analysis of a study on a press tool’s life-length