Optimal process sequence identification and optimal process tolerance assignment in computer-aided process planning

Computers in Industry 17 (1991) 19-32 Elsevier

19

Applications

Optimal process sequence identification and optimal process tolerance assignment in computer-aided process planning Z. Dong and W. Hu Department of Mechanical Engineering, University of Victoria, Victoria, B.C., Canada

An approach for generating the optimal process tolerances and for evaluating alternative process sequences in process planning is presented. The formulations for evaluating production costs of a process sequence and its production operations are established. Several new production cost-tolerance models for production operations, which can significantly improve the modelling accuracy of empirical data of typical production processes, are introduced. Constrained nonlinear optimization is applied to identify the optimal process sequence and to determine the optimal process tolerances with least production costs. The method can improve present CAPP methods by introducing quantitative analysis. It can be combined with the knowledge-based generative CAPP approach to automatically generate the optimal sequence of production operations and the optimal process tolerances for a given design feature. An example is used to illustrate the method. Keywords: Computer-aided process planning (CAPP), Optimi-

zation, Cost-tolerance model, Optimal process tolerance assignment.

1. Introduction Process planning is concerned with determining the sequence of production operations for making a given mechanical part. As a traditional, but critical, c o m p o n e n t of design and manufacturing, process planning has being a u t o m a t e d to meet the needs for higher p r o d u c t i o n efficiencies and lower production costs. Considerable work has been done on computer-aided process planning ( C A P P ) [1-10]. It has also b e e n r e c o g n i z e d that

computer-aided process planning can serve as a crucial link to the a u t o m a t e d modulus of computer-aided design ( C A D ) and computer-aided m a n u f a c t u r i n g (CAM). I m p r o v e m e n t s in C A P P can essentially change the art of mechanical design and m a n u f a c t u r i n g [3-10]. Researches on the new generative C A P P focus on the methods for creating the process plan by synthesizing a given part with the built-in m a n u facturing knowledge. The knowledge-based system or expert system a p p r o a c h is an effective tool for developing a generative C A P P system [4-10]. Heuristics for process planning and mathematical models (or machinability data) for cutting parameter selection are included in a comprehensive knowledge for process generation. A generative C A P P system, acting as an experienced production engineer, creates the process plan based on its knowledge of available p r o d u c t i o n operations and their capabilities, p r o d u c t i o n codes, tooling and machining parameters, as well as constraints imposed in part manufacturing. These studies have significantly contributed to our understanding of computer-aided process planning. However, the knowledge-based a p p r o a c h is qualitative in nature. Quantitative studies in process planning are currently limited to the computerized tolerance charting that provides a systematic m e t h o d for assigning consistent process dimensions and tolerances [11], and the optimal cutting parameter determination [2]. They b o t h emerge at the later

0166-3615/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved

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Computers in lndust W

stage of process plan generation when the process sequence has been determined. The quantitative analysis aspect of process sequence generation has rarely been addressed. Since no quantitative comparison is made among the competitive candidate production sequences, and among the different process tolerance combinations of each production sequence, an optimal process plan cannot be expected. This work introduces a new quantitative approach for generating the optimal process tolerance and for evaluating alternative process sequences. It can be combined with the present knowledge-based system approach to considerably improve the methodology of generative computeraided process planning.

2. Process sequence generation in CAPP A major function of process planning is process sequence generation. A process sequence is a selected group of production operations in a specified order. These production operations first present a mechanical feature in the raw material form, and gradually change it to the designed specifications of dimension, tolerance and surface finish. Each production operation improves the feature to an intermediate process tolerance. A typical process sequence for producing a rotational surface is illustrated in Fig. 1. The sequence is composed of four production operations: (a) Zuomin Dong is an assistant professor of Mechanical Engineering at the University of Victoria. His research interests include automated tolerancing in CAD, generation of the optimal production sequence in CAPP, concurrent design and modeling, and intelligent machining of sculptured parts.

d Weiping Hu is a research assistant in the Department of Mechanical Engineering at the University of Victoria. Mr. Hu is currently working for a Master of Applied Science Degree. His research focuses on generation of the optimal production sequence in CAPP, combining numerical analysis and a knowledge-based system.

forging, (b) rough turning, (c) semi-finish turning, and (d) finish grinding. A usual process sequence consists of at least two of the following possible production operations: (i) blank part shape forming, (ii) rough machining, (iii) semi-finish machining I, (iv) semi-finish machining II, (v) finish machining. and, (vi) forming by special equipment. Among these operations, the blank part shape forming generally includes casting or forging. Ready formed bar materials can also be used. The choice of raw material forms is usually based upon design considerations. On the other hand, there may be more than one design-independent alternative for the other production operations. The selection of production operations is based upon the available machine tools and their working conditions, the cutting tools used, the part material and dimensions. An additional production operation may be added to the sequence. A production operation may be replaced by other operations, or dropped from the process sequence. For the mentioned rotational surface manufacturing example, the semi-finish machining of turning could be replaced by grinding, and the finish machining of grinding could be replaced by high-accuracy turning. These variations on the previously selected production operations have introduced two more alternative process sequences. Since multiple candidates exist for a machined feature, a quantitative measure to evaluate different alternatives and to identify the optimal process sequence, with least production cost, becomes necessary. The production costs associated with a process sequence are determined by the number of production operations of the sequence, the production operations selected, and the way that the process tolerances are distributed among different production operations. The influence of process tolerance distributions on production costs can be illustrated by the tolerance distribution between the blank part making and the later machining operations. A part, with relatively tight blank tolerance and consequently high blank making cost, requires less effort in its subsequent machining due to less required tolerance improvements, and this results in relatively lower machining costs. On the other hand, a part, with relatively loose blank tolerance and low blank making cost, needs more effort in later machining due to more required tolerance improvements, and leads to rela-

Computers in Industry

Z. Dong, 14( Hu / Optimal process sequence identification

-7[

(a)

(b)

2_

(C)

21

A

(d)

Fig. 1. Typical process sequence for producing rotational surface: (a) forging, (b) rough turning, (c) semi-finish turning, and (d) finish grinding.

tively higher machining costs. The appropriate distribution of process tolerances in a process sequence is critical to achieving low production costs. The tolerance charting technique used in traditional process planning provides a convenient tool for decomposing a complex process dimension and tolerance generation problem [12,13]. It helps a production engineer identify the relations among process tolerance and design tolerances. In current practice, several key process tolerances have to be assigned by the production engineer based on his experiences and judgements. Other process tolerances can then be calculated, based on the specified key tolerances and the identified process tolerance relations. The tolerance charting method produces consistent process tolerances, thus assuring that the process tolerances are able to be produced and that the design tolerances are satisfied. With the tolerance charting method, a good process tolerance combination for a given process sequence can be introduced by an experienced production engineer; but an optimal process tolerance combination with least production cost cannot be expected. In addition, it cannot provide any quantitive measure to the production cost of a process sequence, and cannot lead to the optimal sequence selection. In order to identify the optimal process sequence and to generate the optimal process tolerance distribution for that sequence, quantitative evaluations of alternative process sequences and of different process tolerance combinations, based on production costs, need to be introduced. The work focuses on the formulation of production costs for a process sequence, under a given process tolerance combination. New models for produc-

tion cost-tolerance data are introduced to support the formulation. Constrained nonlinear optimization is applied to identify the optimal process sequence, and to determine the optimal process tolerances of the sequence.

3. Production cost-tolerance models for a process sequence and its production operations

3.1. Empirical production cost-tolerance relations Through a process sequence, a mechanical feature is modified from its raw material form to the designed shape and accuracy. The mechanical tolerance and surface finish of a part are continually improved by the selected production operations. It has been recognized that a tighter tolerance leads to higher production cost. Experiments

I 300o

i i

g,

I !

25 0 o

20.0~

,v

70.05

0.1

0.2

0.3

0.4

True Position Tolerance, + ~ ( nun ) Fig. 2. Production cost versus true position tolerance.

0.5

22

Applications

Computers in Industry

have been carried out to obtain the empirical cost-tolerance data for production cost control and for optimal tolerance design. Trucks compiled various empirical data curves for different production operations and processes [14]. One of these curves, the empirical relation of production cost vs. the true position tolerance in hole making, is illustrated in Fig. 2, where, 8 denotes a tolerance that describes the hole location accuracy, and c(8) denotes the production cost to produce a hole to the location accuracy of 8. The curve covers production operations of drilling, jig boring and machining using special equipment. It represents a hole-making process sequence, which includes the production operations of rough machining, semifinish machining, finish machining, and forming by special equipment. For each production operation, such as drilling or boring, the tolerance that it produces and the machining cost present a similar curve over a smaller valid tolerance region. The production cost-tolerance curve of a process sequence is actually the embracing curve for the cost-tolerance curves of its component machining operations. This is illustrated in Fig. 3. This observation of the authors is based on the empirical production data of different production operations.

3.2. Production cost-tolerance model for a production operation The tolerance (or accuracy level) that a production operation produces, and the machining cost

[

[

l

I

~oool

g~

_~ooo

"d

'L

C(62) J•C •

C( 8~ )

-

_"-,,c,,,,,,_

/

I

[

I

01

02

0.3

0.4

0

True Position Tolerance, +~d( m m ) Fig. 3. Cost-tolerance curves for a process sequence and its production operations.

6

T

I

6~j n

62

8~

8m~~

Fig. 4. Cost tolerance relation for a production operation.

to achieve this tolerance through the operation, have the relation shown in Fig. 4. This kind of empirical cost-tolerance curves of production operations, including turning, milling, grinding, casting, etc., can be found in [14]. The curve represents the feasible tolerance range that the operation covers and the corresponding relative production costs. The production cost reflects the required manufacturing efforts for changing a mechanical feature from the raw material state with considerably larger errors (very loose tolerance) to the tolerance corresponding to the cost. A tighter tolerance can be achieved with higher production cost, when high-quality producing and inefficient cutting parameter settings, such as small cutting depth, slow feed rate, and fine turning of the machine tool are adopted. The infinite initial tolerance can be considered to correspond to zero cost. A mechanical feature with small original errors (tight blank part tolerance) requires less manufacturing effort in machining, and leads to low production costs. As illustrated in Fig. 4, the relative production cost for improving the feature tolerance from 8~ to 82 can be calculated by: AG,_~= = c ( a = ) - c ( a , ) ,

i

oo

C(8)

(1)

where 81 and 82 represent the tolerances of a mechanical feature before and after the production operation, c(81) and c(82) are the corresponding relative production costs for changing the tolerance of a feature from an infinite tolerance (very large error) to the tolerances of 8~ and 82, and AC~,_8= is the relative production costs to improve the tolerance of a feature from 81 to 82.

Z. Done, W. Hu / Optimal process sequence identification

Computers in Industry

C( 8'

\C~r (8)

23

A general formulation of the production cost for a process sequence with p production operations is:

,

C(30 . . . . . 3p)sequ.... p p

= E a c , = E [ c , ( 3 , ) - c,(3,_,)], i=1

83

Gz

&

Go

Fig. 5. C o s t - t o l e r a n c e model for'a process sequence.

In the present process tolerance assignment, a recommended process tolerance 3 is suggested by the manufacturing handbooks [15] for a given machining operation. The recommended tolerance represents a point on the cost-tolerance curve. Since this fixed tolerance was generated for a general case without considering the interactions among different production operations, optimal process tolerances for a process sequence cannot be obtained. 3.3. Production cost-tolerance model for a process sequence

(3)

i=1

where, 30 is the tolerance of the blank part, and 3p is the tolerance accomplished by the production process, it is equal to (or smaller than) the designed tolerance 3 of the feature. In practice, a part is often unloaded from a machine tool after one production operation and loaded onto another machine tool to proceed to the next production operation. The set-up error for the ith production operation, esetup.i, is introduced. When a part is reloaded, the errors that the production operation has to handle include the error of the part, which has been left over from the previous production operation, and the error that has been introduced in the re-loading. In other words, the existing tolerance for the i th production operation is 3'i--I = 3i 1 + esetup,i, rather than 3;_ 1- The general formulation of the production cost for a process sequence with p production operations then becomes:

C(30 . . . . . 3p)se q. . . . . From .the above analysis, the process for changing a mechanical feature from its raw material state with considerably larger error, to the finished part state with the designed tolerance, can be modeled as shown in Fig. 5, provided the process s~quence consists of three production operations, the rough machining, the semi-finish machining, and the finish machining. The production cost of this process sequence can be calculated by: Cseq . . . . . = ACr + ACsf "F A C f

:

cr(3o)] + [Csf(3 + [cf(33) - cf(32) ],

)- csf (31)] (2)

where 30 is the tolerance of the blank part, 33 is the tolerance accomplished by the production process, cr(3 ), c~f(3) and cf(3) are the production cost-tolerance models for the rough machining, the semi-finish machining, and the finish machining, and AC, represents the tolerance improvement from the corresponding production operation.

P

P

= 2 AC/= y' [c/(3,)- c/(3/_,)], i=1

/=1

3'i -- 1 = 3i_ 1 +

esetup,i •

(4)

The value of esetupj is process-dependent as being discussed. Values of esetup, i for various machining operations could be obtained by experiments. If a part is not reloaded before the ith production operation, esetup, i becomes zero. In the introduced model, the production cost for a process sequence, Cseq...... is determined by the design tolerance of the feature, 3 = 3p, the blank part tolerance, 80, and the process tolerances, 31 . . . . . 3p_ b which represent the switching points from one production operation to another. In searching for the best switching tolerance between adjacent production operations, it has been found that the cost, Cseq...... depends upon the relative magnitudes of derivatives of the production cost-tolerance curves over the overlapped tolerance region, and independent to the relative

24

Computers in Industry

Applications

operation is less than the derivative for the preceding production operation; the introduced production cost, AC = AC, + AC,_ a, will be minimum if 8~..... is used as the switching tolerance, i.e. 3" = ~

c(a) Cl-i

i AI1LIX

I

C1-1

F

5'= 8,,~,

.8

&

(a) C'(5 ),-, ~ C'(6),

C(5)

{

O-1

ACt-1

/

t

I

3"= 5 ,-,,m,.

c'(8),_ 1 < c'(8),

if8°<6<6,

......

c'(8),

1=c'(8),

if8=8

c'(8),

,>c'(6),

if, ,.mi, < 8 < 8

°, °.

As shown in Figure 6 (c), the derivatives of the cost-tolerance curves for the two adjacent production operations are identical at 6 = 6 o, and gradually change from c ' ( 8 ) i 1 < c'(6)i to c ' ( 6 ) , 1 > c ' ( 6 ) i as 8 decreases. For this case, the introduced production cost, AC = AC, + AC,_ t, will be minimum if 6 ° is used as the switching tolerance, i.e. 8* = 6 °

AC

8

"

(b) Smooth-stiff case, where c ' ( 8 ) , _ 1 <~c'(8)i. As shown in Fig. 6(b), the derivative of the cost-tolerance curve for the present production operation is greater than the derivative for the preceding production operation; the introduced production cost, AC = AC, + AC, ~, will be minimum if 6~ ~.mm is used as the switching tolerance, i.e. 8" = 8i 1.ram" (C) Uncertainty case, where:

&

(b).C ' (5),-t <- C '(5 ),

C(5)

4. Models for production c o s t - t o l e r a n c e relations -r ACt.~ I

ACE

5

8'= 5 o

&

(c) C ' ( & ) , - , : C'(&)~ Fig. 6. Adjacent production operations and best switching tolerance.

4.1. Existing production cost-tolerance models In formulating the optimal tolerance design problem, different production cost-tolerance models were introduced to represent the empirical production cost-tolerance relations, such as the one shown in Fig. 2. The existing models include: • Exponential model (Exponent) by Speckhart [16], c ( 8 ) = a o e .,8.

• Reciprocal squared Spotts [17], positions of the cost-tolerance curves. There are three meaningful cases. They are illustrated in Fig. 6 and discussed below ( c ' ( 6 ) i = ] d c ( 8 ) J d 6 1): (a) Stiff-smooth case, where c' ( 6)~_ t >>.c' ( 8) r As shown in Fig. 6(a), the derivative of the cost-tolerance curve for the present production

c ( 8 ) = t o / 8 2.

(5)

model

(R-Squared)

by (6)

• Reciprocal powers model (R-Power) by Sutherland and Roth [18], c( 8 ) = ao 6 "'.

(7)

Computers in Industry

Z. Dong, W. Hu / Optimal process sequence identification

120

A P P r

i

25

[

/ Turning 100

~

I-grind

0 X

/

i m

a t e r e

R-grind

80 F-mill H-posi 60

1 a t i V

D-cast

l-cast

40

e

Dieter

\ c o s t

\

20

0 L

0.1

0

0.2

0.4

0.3 Tolerance

0.5

-0.6

(mm)

Fig. 7. Empirical production cost-tolerance curves.

• Reciprocal powers and exponential hybrid model ( R P - E Hybrid) by Michael and Siddall [191, c ( 8 ) = ao8 "' e .,a. • Reciprocal model Greenwood [20],

(S) (Recip.)

c ( 8 ) = ao/8.

by

Chase

and 4.2. New production cost-tolerance models

(9)

• Modified exponential model (M-Exponent) by Dong and Soom [21-23], c ( a ) = a 0 e-"'(a-":) + a 3

relations. However, it has been recognized that updated and more complete production costtolerance data, and better mathematical models to represent the actuality of these data, are necessary [21,22,25].

(10)

8min -~< 8 ~ 8 . . . .

where [~min,~max] is the valid tolerance range representing the physical capability of a production process. • Discrete model (Discrete) by Lee and Woo [24]. The above existing production cost-tolerance models were used in different optimal tolerance design formulations. They can provide quantitative measures of the production cost-tolerance

Five new production cost-tolerance models that better describe the empirical production c o s t Table 1 Empirical production cost-tolerance curves Empirical curve

Production process

Dieter D-cast I-cast H-posi F-mill Turning R-grind I-grind

General cost-tolerance relation by Dieter Die casting Investment casting True position of hole producing Face milling Turning on lathe Rotary surface grinding Internal grinding

Applications

26

Computers in Industry

tolerance relations are introduced in the work. These new models are: • Combined reciprocal powers and exponential function (Combined RP-E), c(8)

= a o + al 8 - ' ~ + a 3

• Combined linear (Combined L - E L c(8)=a

0+a18+a

and

e -"48.

(11)

exponential

function

2 e ,,8.

(12)

ponential function models, are introduced by adding a reciprocal powers term or a linear term to the original exponential model to improve the approximation of the existing models at the trouble-prone tight tolerance zone. The polynomial functions of cubic, fourth and fifth orders are introduced to find the best representations of an empirical data curve.

4.3. Model comparisons

• Cubic polynomial (Cubic-P), c(8)

= a 0 + a18 + a2 ~2 + a3 ~3.

(13)

• Fourth-order polynomial (4th-P),

c(8)=ao+a18+a282+a383+a484.

(14)

• Fifth-order polynomial (5th-P), c ( 8 ) = a 0 + al8 + a26 2 +

a383 + 1484 +

1585.

(15) The combined reciprocal powers and exponential function, and the combined linear and ex-

Previous work on optimal tolerance design introduced a production cost-tolerance model, or compared various existing models based on one particular empirical curve. Studies on model applicabilities and modelling errors are limited to a few isolated data curves. A systematic study of inherent modelling errors for the existing and new models, to the empirical production cost tolerance relations of typical production processes, including hole producing (true position), face milling, turning, rotational surface grinding, internal

Table 2 Model parameters a o. a 1..... of the new models Mathe-

Empirical production cost-tolerance curve

matical model

Dieter

D-cast

I-cast

H-posi

Combined 23.22 81.002 74.41 30.92 reciprocal 6021 7516 4645 7824 powers and 3.71 80.44 -5.13<10 4 9.144 exponential 0.5 × 10 3 - 0 . 1 × 10 4 0.59 0.008 function 1.18 0.153 0.947 7.68 Combined linear and exponential function

23.21 5.96;,<103 0.1×10 7 1.33

76.7 1.11 ×104 4.1×10 21 16.6

69.5 4.42× 103 0.0153 2.9

29.4 9.77×103 2.3×10 14 11.34

F-mill

Turning

443.4 533.09 11490 52832 69.08 -2.5×10 4 0.71 0.39 -0.1 × 10 4 16.21

R-grind 38.55 49357 -2.5×10 0.69 -0.1 ×10

I-grind 51.41 45965 5 - 2 . 5 ×10 0.005 ~ 55.8

4

390.5 1.02× 104 0.965 0.1 × 10 5

591.3 4.84 × 104 5.2×10 5~1 34.84

32.7 4.45 × 104 28.84 1.06

69.16 2.05 × 104 0.0 31.93

Cubic polynomial

22.95 -1.023<103 1.71 × 1 0 4 -9.643<104

75.7 4.09×103 8.18 × 104 5.55 × 105

71.3 -2.783<103 4.18×104 -2.31 × 105

34.88 -1.49×103 2.86 × 104 -1.8×105

322.1 l -2.41×104 6.19× 105 - 5.46× 106

270.9 -1.61×104 2.82× 105 -1.41 ×106

33.2 1.27×104 1.84× 106 9.64×107

106.7 -2.19×104 3.81 × 106 --2.45 × 10 ~

Fourthorder polynomial

23.9 1.21×103 2.67× 104 2.76×105 1.08×106

87.3 -6.57×103 2.22× 105 -3.41×106 1.91 × 107

79.4 -4.01×103 9.79×104 -1.19×106 5.59×106

38.78 -2.26×103 6.87× 104 -9.35×105 4.63 ×106

280.7 -l.5×104 1.17×104 1.23×107 1.83 ×10 s

317.01 -2.7×104 9.03× 105 -1.08×107 4.17X 107

39.27 1.88×104 3.95×106 -3.993<10 ~ 1.54X10 I°

104.2 -1.42×104 -3.093<106 2.13X109 -2.71 3<10n~

23.82 97.16 1.19×103 - 9 . 4 4 × 1 0 3 2.513<104 4.62x 105 -2.22×10 ~ -1.16×107 3.62 × 105 1.41 × 108 3.51×10 ~' - 6 . 5 3<10 s

86.54 5.42×103 1.89×105 -3.75×106 3.77 × 107 -1.52×108

42.07 -3.13×103 1.363<105 -3.07×106 3.37 × 107 1.42 × 10 ~

11.08 5.33×104 - 6 . 8 4 × 106 3.26×10 ~ - 6.98 × 109 5.62 × 10 m

365.7 -5.083<104 2.71X 106 - 5 . 8 9 × 107 5.54×108 - 1.86×10 '~

43.22 -2.38×104 6.33× 106 - 9 . 3 3 × 10 ~ 7.21 × 10 m 2.3×1012

104.4 -1.5×104 2.09× 106 1.57× 109 -1.35<101~ -1.23×1013

Fifth-

order polynomial

Computers in Industry

Z. Dong, W. Hu / Optimal process sequence identification

35

0

27

(a)

30

/ J Reciprocal Powers

25

'~,

20

~~,

.0

E o

~~ . RP-E Hybrid

Exponential

~1 ~ : L . . ~ " "*~'"

c,,

<

~~

~

0.1

Dieter's Empirical Curve J

~

~

0.2

J Reciprocal Squared .

.

0.3

.

Reciprocal

0.4

0.5

0.6

Tolerance (mm) 25 (b) 20 o0 (D

E o

15

10

Combined RP-E* Combined L-E* Cubic-P* 4tt~:P:

Dieter's Empirical Curve

Q.

<

0

0

ot

0:2

o'3

04

os

0.6

Tolerance (mm)

Fig. 8. Comparisons of existing and new models.

grinding, die casting and investment casting, is conducted. The empirical production cost-tolerance data of these production processes, which have been compiled by Trucks [14], are used. The data curves used in the study are listed in Table 1

and plotted in Fig. 7. Dieter presented an empirical curve representing the general cost-tolerance relation [26]. This curve is included in the study, to compare the results with the previous analysis of existing cost-tolerance models done by

28

Applications

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Wu et al. [25]. Twenty data points, 3i, gi ( i = 1 ..... 20), were picked up from each of the eight production cost-tolerance curves. The model parameters of the existing and the new models were determined using the least-square approximations to these data points on the empirical curve. The calculated model parameters, a 0, a 1, a 2. . . . . for new models are listed in Table 2. The modelling errors of all existing and new models are tabulated in Table 3 to compare the inherent modelling accuracies of these models. The errors are normalized to the widely used exponential model. A new model is marked by an asterisk. The Dieter's empirical curve is used as the test production data curve. The curves that are created, using the existing models and the new models, are shown in Fig. 8. The new models showed significant modelling accuracy improvements over the existing models. The modelling errors of the best new model, the fifth-order polynomial, over the whole valid tolerance region, is only 2.3 percent of the modelling errors of the current best model, the reciprocal powers and exponential hybrid model, and only 1.8 percent of the commonly used exponential model. This indicates 97.7-98.2 percent error deductions. Even the modelling error of the worst new model, the combined reciprocal powers and exponential function, is only 18.7 percent of the reciprocal powers and exponential hybrid model, and 14.7 percent of the exponential model, meaning 81.3 85.3 percent error deductions. Even the new models have more parameters and more complex forms: a test in the work showed their influences to computation time are minor.

5. Feasible process sequence g e n e r a t i o n - - a n overview

The method for generating all feasible production sequences, using an A N D / O R tree representation and search, is beyond the scope of this publication. Instead, the method is explained and illustrated below using an example. In order to create all feasible production sequences for a design feature, frequently used production processes, discussed by Trucks [14], are clustered into feature-oriented groups, including the blank part preparation group, the rotational surface group, the flat surface group, the hole making group, the hole location group, etc. Each group consists of various process-oriented cost-tolerance models for rough machining, semi-finish machining and finish machining. A part can be machined using different combinations of the processes in the group. For the rotational surface group, the involved production processes include: sand casting, die casting, rough turning, semi-finish turning, finish turning, semi-finish grinding, finish grinding and horn. The p r o d u c t i o n cost tolerance models of these processes are illustrated in Fig. 9. This group of production processes, if modelled using the A N D / O R tree, can be represented as shown in Fig. 10. The tolerance, 3, of the rotational surface can be improved gradually using machining operations of blank part forming, rough machining, semi-finish machining and finish machining. Feasible production sequences for a given tolerance 3 can be generated using a search on the A N D / O R tree representation of relevant produc-

Table 3 T h e relative e r r o r s of existing a n d n e w m o d e l s Mathematical

C o s t - t o l e r a n c e curves

models

Dieter

D-cast

l-cast

H-posi

F-mill

Turning

R-grind

l-grind

errors, e

Exponent R-squared R-Power RP E Hybrid Recip. Combined RP-E * Combined L-E * Cubic-P * 4th-P * 5th-P *

1 75.91 6.310 0.860 15.04 0.037 0.034 0.122 0.011 0.010

1 15.31 19.63 0.350 1.880 0.486 0.117 0.630 0.210 0.055

1 21.26 33.07 1.180 2.490 0.150 0.119 0.150 0.042 0.017

1 14.94 0.026 0.013 4.120 0.060 0.020 0.110 0.038 0.009

1 0.370 81.27 0.100 1.370 0.060 0.035 0.021 0.018 0.007

1 0.730 0.099 0.230 0.094 0.024 0.038 0.590 0.370 0.230

1 0.043 2.470 1.530 0.120 0.003 0.0016 0.0008 0.0004 0.0002

1 1301 1.080 0.210 555.3 0.040 0.580 0.029 0.0108 0.0106

1 178.6 17.99 0.559 72.55 0.107 0.118 0.206 0.087 0.042

* N e w models.

Average

Computers in Industry r

4OC 35C

Z. Dong, W. Hu / Optimal process sequence identification i

I

i

i

Horn /

FinishTurning ExternalGrinding

25(

Semi-FinishTurning

201

R o u g h Turning

, 150 ,0 100

.

~

/

SandCasting ,

.

29

(2) rough turning-semi-finish turning-finish grinding (t-t-g); (3) die casting-semi-finish turning-finish turning (dc-t-t); (4) die casting-semi-finish turning-finish grinding (dc-t-g); (5) sand casting-semi-finish turning-finish turning (sc-t-t); (6) sand casting-semi-finish turning-finish grinding (sc-t-g). Semi-finish grinding is not included due to lack of empirical data.

%

50!

0.0

01

02

03

04

0.5

Tolerance, + 6 ( mm ) Fig. 9. Cost-tolerance models for production processes relevant

to rotational surface machining.

tion processes. For the tolerance that is labeled in Fig. 10, all production operations except horn can be used. The feasible production sequences that are identified by the A N D / O R tree search include: (1) rough turning-semi-finish turning-finish turning (t-t-t);

6. Process sequence evaluation and process tolerance optimization

The production cost of a process sequence has been formulated previously. For a process sequence with p production processes, the sequence production cost is a p-term sum of the operation production costs that are needed to improve the accuracy of a feature from a given blank part tolerance, 80, to the design tolerance, 8p. Given the p production operations, the production cost of a process sequence is determined by the switching tolerances among adjacent operations, 81 . . . . . 8p_ 1. The optimal switching tolerances, or the process tolerances of a production sequence with least production costs, 8~*. . . . . 6p*_1, can be

Blank Part Preparation

5 Rough,

T 0 L E R A N C E

Semi-finish Machining

Finish Machining

Ultra-fine

Machining Fig. 10. Production operations for rotational surfaces.

Applications

30

Computers in Industry

determined by a constrained non-linear optimization:

discussed in Section 5. Each production sequence has three production operations. The cost function, C, of a sequence then becomes,

min C(8, . . . . . 8p ,) 8,

C(8,, 8 2 ) = [ g f ( 8 ) -

P

= min ~ [ c , ( 8 , ) - c ~ ( 8 f ,)], 8, 8[ 1 = 8i

(16)

gf(8~)]

-{-[ gsf ( 8 2 ) -

i= 1

gsf ( 8 [ ) ]

+ [ g r ( 8 , ) -- gr(80)].

I + esetup,i,

Using the optimization formulation proposed in Section 6, we can assign optimal process tolerances for each production sequence, and obtain the required production cost for each production sequence. The calculated results are listed in Table 4. Based on the calculation results, the optimal production sequence with least production cost is identified. The production sequence "turning turning-grinding" presents least production cost. Its optimal process tolerances, 80. . . . . 83 = 6, are listed in the Table. The manually planned production sequence, "turning turning-turning", with the recommended process tolerances is also listed at the bottom of Table 4 for comparison. The calculated results indicate that: (i) the optimal process tolerance assignment presents a 7.1 percent productions cost reduction for the identical production sequence, rough turning, semi-finish turning and finish turning; and, (ii) the optimal production sequence with optimal process tolerances leads to 47.6 percent production cost reduction over the manual planned approach. The advantages of the optimal production sequence identification and optimal process tolerance assignment are demonstrated. The optimization problem was solved using both the Constrained Variable Matrix Method (CVM) that applies the BFGS updating formulas and watch-dog strategy, and the penalty method. The CVM algorithm was implemented in the opti-

subject to: 8,.rain <~ 8i <~ 8i ..... .

The optimization also calculates the least overproduction cost of a process sequence, C * (81" . . . . . 8/9*- l)min- Different process sequences that consist of different combinations of production operations, serving as alternatives, can be compared using this measure to identify the best process sequence for a designed feature. These alternatives can be generated using the A N D / O R tree representation and search method briefly discussed in Section 5, or the knowledge-based system approach, which is used in current generative CAPP systems. all

7. An example The optimal machining sequence generation and optimal process tolerance assignment, for the surface of a rotational part, is used to illustrate the discussed method. The tolerance of the bar material for the part is 0.5 mm. It is used as the initial reference tolerance 80. The expected size tolerance of the rotational surface 8 is 0.013 mm. To achieve this accuracy, the rough turning, semifinish turning, finish turning or fine grinding are required. The blank part can be prepared by sand casting, die casting, or using the bar material. There are six alternative production sequences as

Table 4 The calculated relative production costs Production sequence

80

81

62

8

Cm*n

Relative C

Turn-turn-turn T u r n - t u r n grind Die c a s t - t u r n - t u r n Die cast- turn-grind Sand c a s t - t u r n - t u r n Sand cast-turn-grind Manual planned ( t - t - t )

0.500 0.500 0.500 0.500 0.500 0.500 0.500

0.076 0.076 0.076 0.076 0.127 0.127 0.218

0.064 0.064 0.064 0.064 0.064 0.064 0.076

0.013 0.013 0.013 0.013 0.013 0.013 0.013

206.9 116.8 237.8 147.7 312.4 222.3 -

0.929 0.524 1.000

Computers in Industry

mization program base library, OPB, which has been developed by Professor J. Zhou and his colleagues at the Huazhong University of Science and Technology, China. The penalty method was included in the program package for design optimization, OPP5, which was developed by Professor R. Mayne and his colleagues at the State University of New York at Buffalo.

8. Conclusion Computer-aided process planning has been recognized as playing a crucial role in the integration of design and manufacturing. It is concerned with determining the sequence of individual production operations needed to make a given mechanical part. Previous work in CAPP has significantly improved the arts of process planning and automated this dominant module connecting design and manufacturing. However, current knowledge-based CAPP systems are qualitative in nature. The quantitative analysis for generating optimal process sequence is rarely addressed. This work introduces a new quantitative approach for evaluating alternative process sequences and for generating the optimal process tolerances. The optimal production process sequence selection, and the optimal production process tolerance generation for the selected sequence have been formulated as a nonlinear constrained optimization problem, based on least production costs. New production cost-tolerance models, including the combined reciprocal powers and exponential function, the combined linear and exponential function, and the cubic, fourth-order and fifth-order polynomials, to represent the production cost-tolerance relations, are introduced. The models have been evaluated using the empirical production cost-tolerance data of typical production processes, including hole producing, turning, milling, grinding, and casting. The introduced new models showed significant modelling accuracy improvements for the empirical production cost-tolerance data over all existing models. The example used to illustrate the method has shown significant production cost reduction introduced by the method. The method can be combined with the present knowledge-based system approach, to identify the optimal process sequence and optimal process

Z. Dong, W. Hu / Optimal process sequence identification

31

tolerances for producing a part. It can considerably improve the methodology of generative computer-aided process planning and lower production costs.

Acknowledgments The authors would like to thank Professor H.P. Wang for suggesting the discussions at the end of Section 3.3. The support of the optimization program library, OPB and OPP5, from Professor J. Zhou and Professor R. Mayne are gratefully acknowledged. The authors also acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada.

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Applications

[13] M.A. Curtis, Process Planning, Wiley, New York, 1988. [14] H.E. Trucks, in: H.B. Smith (ed.), Designing for Economical Production, Society of Manufacturing Engineers, Dearborn, MI, 1976. [15] Machinability Data Center, Machining Data Handbook, Metcut Research Associates Inc., Cincinnati, OH, 1980. [16] F.H. Speckhart, "Calculation of tolerance based on a minimum cost approach", J. Eng. Ind., Trans. ASME, Vol. 94, No. 2, 1972, pp. 447-453. [17] M.F. Spotts, "Allocation of tolerances to minimize cost of assembly," A S M E J . Eng. Ind., August 1973, pp. 762-764. [18] G.H. Sutherland and B. Roth, "Mechanism design: Accounting for manufacturing tolerances and costs in function generating problems," J. Eng. Ind., Trans. ASME, Vol. 98, 1975, pp. 283-286. [19] W. Michael and J.N. Siddall, "Optimization problem with tolerance assignment and full acceptance", J. Mech. Des., Trans. ASME, Vol. 103, 1981, pp. 842-848. [20] K.W. Chase and W.H. Greenwood, "Design issues in mechanical tolerance analysis", A S M E Manuf Rev., Vol. l, No. 1, March 1988, pp. 50-59.

Computers in lndustty [21] Z. Dong, and A. Soom, ~'Some applications of artificial intelligence techniques to automatic tolerance analysis and synthesis", in: D.T. Pham (ed.), Artificial Intelligence in Design, IFS/Springer-Verlag, Berlin, 1991, pp. 101-124. [22] Z. Dong and A. Soom, "Optimal tolerance design with automatic incorporation of manufacturing knowledge", Proe. 1989 l i E Integrated Systems Conf. and Society for Integrated Manufacturing Conf., November 12-15, 1989. pp. 409-414. [23] Z. Dong and A. Soom, "Automatic optimal tolerance design for related dimension chains", A S M E Manuf Rev., Vol. 3, No. 4, December 1990~ pp. 262-271. [24] W.J. Lee and T.C. Woo~ "'Optimum selection of discrete tolerances", J. Meeh. Transm., Autom. Des.. Trans. ASME~ Vol. 111, June 1989, pp. 243-251. [25] Z. Wu, W.H. Elmaraghy and H.A. Elmaraghy, "Evaluation of cost-tolerance algorithms for design tolerance analysis and synthesis", A S M E Manu. Rev., Vol. 1, No. 3. October 1988, pp. 168-179. [26] G.E. Dieter, Engineering Design." A Materials and Processing Approach, McGraw-Hill, New York, 1983.