A systematic approach to preliminary polymer process development: modeling and design

Materials

& Design.

Vol. 17, NO. 5, pp. 235-244, 1996 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0261-3069196 $15.00 + 0.00

Pll: 6026%3069(97)00010-1

A systematic approach to pre~imina~ polymer pro&s development: modeling and design Li Chen, School U.S.A.

Kuang

Ku, Karthik

of Mechanical

Received

16 December

Ramani

Engineering,

and S.S. Rao

Purdue

1996; accepted

University,

25 January

West Lafayette,

Indiana

47907,

1997

A general and complete methodology is presented to facilitate systematic modeling and design of polymer processes during the early development period. To capture and handle the subjective type of uncertainty, embedded in the preliminary process development, fuzzy theories are used as a basis to model and design the process in the presence of ambiguity and vagueness. Physical membership functions are developed for mapping the relation between process variables and the associated fuzzy uncertainties. Based on the qualitative results generated using our previously proposed “linguistic based preliminary design method,” the process modeling can be followed even in the absence of any process governing equations. The modeling is carried out by establishing an appropriate fuzzy reasoning system which provides a specific functional mapping that relates input process variables to one (or more than one) output performance parameter(s). A reduced yet feasible domain is generated by our qualitative design scheme to constrain the process variables. Now, any optimization routine can then be employed to search for a proper process design. We demonstrate the effectiveness of the proposed methodology by its application to a typical compression molding process. 0 1997 Elsevier Science Ltd.

Keywords:

modeling and design; polymer process; fuzzy uncertainty;

Introduction

Although polymer processes have received close attention for many years, process design research issues still remain open. For savings in time and cost during the early process development period, a systematic guide to the description and selection of proper process ‘conditions’ and ‘parameters’* is required. The successful design of a polymer process to achieve a desired material performance depends significantly on the selection of the process conditions. Thus, the quality control over the entire process can be fulfilled indirectly through the process design. Traditional modeling and design of a polymer process require intense analytical or numerical work to reveal the associated characteristics before optimizing the process. Furthermore, profound knowledge of governing equation(s) for a specific process is also presumed and required. This traditional procedure has been implemented for many years; however, it is often accompanied by uncertainties resulting from ~sumption(s) or approx~a~on(s) due to s~p~ified process modeling. The unce~~nties may confine the capability and applicability of a real-world process development/realization. On the other hand, due to the complex and uncontrollable nature of the process, weakness of the equation-modeling-based approaches is apparent. In the real world, detailed equations or formulas may not capture key processing physics. The availability and validity of the analytical/numerical modeling methods depend on the complexity inherent in each process. Clearly, conventional“For convenience, simpiy as (processing

we may refer to process conditions or design) parameters or variables.

and parameters

preliminary

engineering

type modeling and design routine has disadvantages. The exactness, correctness or completeness present in the equation-based formulation (or statement) for early process development are difficult to achieve. In many applications, there exist considerable uncertainty and insufficient information to describe the problem. Typically, the subjective type of unce~n~, known as ‘fuzziness,’ dominates the p~~~n~ phase of process ~velopment. ’ During the early process development period, a process engineer is not very clear about the ‘exact’ parameter values or conditions to be used. Often, the engineer initiates the process development mainly in terms of his prior ‘imperfect’ knowledge or past ‘limited’ experience. However, this becomes extremely difficult in the presence of a large number of unknown process variables. In essence, this is true at the early stage of any engineering process development. As an example, consider the processing of ‘ultra-high molecular weight polyethylene’ (UHMWP E) by compression molding. Several factors affect the properties of the resulting part, such as the grade of the polymer, particle-size ~s~bution, pressure and tern~mt~ cycle during processing, cooling rate, atmosphere during processing, and eventual sterilization and aging of the part. All these variables add to the uncertainty or fuzziness to the exact definition of an ideal model for the process. For instance, it has been demonstrated that the physical and mechanical proprties were dependent on the initial polymer morphology. Compression-molded parts showed much lower wear rates as compared to parts machined from an extruded stock.3 Such variables add to the uncertainty of the problem at hand. For this reason, the quditative design approaches have been Materials

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L. Chen et al.: A systematic

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presented to indicate an alternative way to rigorously treat the initial process design.lP4 Linguistic variables have been applied to facilitate the description and selection of process conditions and parameters at the preliminary process stage.’ With the aid of qualitative design results, a reduced yet feasible design space is readily generated. Within the resulting solution domain, a set of physical process parameters involving their magnitudes or conditions can be further searched and finally identified for the preliminary process development. In this work, a generalized complete methodology is developed to facilitate the systematic ‘modeling and design’ of polymer processes at the early development stage. After introducing fundamentals related to the present work, we first outline our previous work on how to narrow down the range of fuzzy-type uncertainty inherent in the early process development period. The proposed methodology is then presented in detail for ‘preliminary’ process modeling and design in the presence of ambiguity and vagueness. The case study of a typical compression molding process is finally considered to illustrate the computational aspects of the proposed methodology.

Basic theories Fuzzy theories5P6are extended in this work to seize fuzzytype uncertainties inherent in the early process development phase. It is noted that most computational quantities in the present work are nothing but fuzzy numbers (rather than real numbers). To bridge the present methodology to our previous work, fundamental theories are highlighted and summarized in this section. Relational matrix A relational matrix is an input matrix whose elements are used to designate the engineer’s intuition and experience concerning the influence of each process variable on a particular output performance parameter. The relational matrix depicts a particular qualitative mapping relation between the individual input variables and the output performance parameter. The domain of each process variable is decomposed into different divisions (or levels) linguistically, such as low, medium and high. These levels enable the engineer to relate the ‘degree of influence’ at respective levels of the process variables to the specific performance objective. The degree of influence is defined by ‘linguistic variables’ which are quantified by fuzzy numbers. The domain of an output performance parameter is categorized into several decompositions, such as very bad, bad, okay, good and very good to indicate various levels of degree of influence. In the present work, triangular-type fuzzy numTable Process

1

Relational

matrix

Material Low 2 2 4 3 3 2 3 3

x2 x3

X4 5 X6 X7 x8

Legend:

236

Clystallinity Medium 4 3 3 2 3 3 3

1 = very bad; 2 = bad, 3 = ok; = good;

Materials

process

development

bers are employed to represent the linguistic quantities. As an example, Table I illustrates the relational matrix for a compression molding process. Fuzzy design metric In the absence of any governing equation during the preliminary process development, fuzzy design metric is defined as a ‘relative’ measure to evaluate an output performance parameter in the presence of vagueness. Thus, the functional mapping relation between input process parameters and a particular output performance characteristic can be quantified. Consider the$h process design alternative with regards to multiple process variables: the corresponding fuzzy design metric, pjl can be expressed as - iv

(1)

Pj = >: rVwV i=l

where WVand rii indicate the normalized fuzzy weight and degree of influence for the ith process variable, respectively. ’ The normalized fuzzy weight is characterized by a fuzzy number through the construction of ‘fuzzy-positive reciprocal matrix’ .7S8The weight is used not only to screen the order of importance among the various process parameters, but also to discern the degree of interaction induced by likely coupled parameters. Thus, the ‘degree of importance’ of each process parameter can be identified for a desired output performance parameter. Under the specified performance objective, the degree of influence is determined as a fuzzy quantity for each (input) process variable. For computational details, the reader is referred to references 1S8,9for a complete discussion. Linguistic design of experiments (LDOE) The method of ‘linguistic design of experiments’ (LDOE) is essentially a qualitative-based preliminary design procedure, developed as an extension of the Taguchi method.’ The Taguchi robust philosophy is well-known for off-line product quality controls through its designed-in performance.“Figure 1 shows the schematic flow chart of LDOE. A three-stage procedure involving the linguistic fuzzification, inference and defuzzification constitutes the framework for LDOE. In the present work, the LDOE will be applied to reduce original fuzzy uncertainties associated with process variables. At the first ‘linguistic fuzzification’ stage, an input interface is established. Three linguistic quantities, specified as low, medium and high, are considered as input quantities for each process variable. Five linguistic quantities, coupled with appropriate weights associated with process variables,

’

variable

Xl

polymer

& Design Volume

LOW

High 3 3 2 2 3 3 2 3

4 2 5 5 2 4 3 5

and 5 = very good

17 Number

5 1996

properties/performance Tensile strength MedilUIl 3 4 4 5 3 4 4

High

Low

3 4 3 3 4 5 3 4

2 2 2 2 5 3 3 2

Tensile modulus Medium 4 4 4 2 5 3 3

High 3 2 3 3 2 2 2 3

L. Chen et al.: A systematic

o--

approach

to pre~imi~~~

polymer

process development

Step 1

Construct relational matrices

LinguZktic Ft&j%ation

Step 2

routine to determin metric for each

Linguistic Inference

Step 3 Linguistic Defuzzifikation

Figure 1 Lin~istic-bid

qualitative design method

are specified as very bad, bad, okay, good and very good for a desired output performance parameter. These five quantities are also referred to as the degree of influence that reflect the effects of various process variables on a specific output performance parameter. Under a specified performance objective, various fuzzy-positive reciprocal matrices are set up at this stage for each input process variable. Meanwhile, the relational matrix is constructed, through

which the mapping from the input variab’tes to the particuhu output performance parameter can be identified qualitatively. At the second ‘linguistic interference’ stage, inte~ediate computations are carried out. Under the specific material performance objective, the normalized fuzzy weights are determined for each process variable. The Taguchi-robustdesign format is utilized to evaluate the fuzzy design metric Materials

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L. Chen et al.: A systemafic

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process

for each design alternative formalized through the orthogonal-based combinations of in~vidual parameters at various levels. To accommodate the Taguchi’s criterion of ‘largerthe-better’ (LB) or ‘smiler-me-better’ (SB), the fuzzy design metric in eqn. (1) is modified as

(3), as follows

LB : Pjk = -10 log,,

or

devebpmenf

(2’ SB : pmultj = - 10 log,, { EwObjkPjk,‘)

or SB : pjk = -10

where Nabj denotes the number of objectives or criteria and pjk denotes the design metric ~o~s~n~g to the jth experiment or design alternative and krh objective. Thus, the proper setting for each design parameter can be determined qualitatively using the same procedure stated earlier,

log,,

where the subs~~pts, i, j, k, denote the ith process parameter, thejth Taguchi experiment or design alternative, and the &h design objective (i.e. output performance parameter)* respectively, In eqn. (2) or (3), rjk is a fuzzy (or linguistic) value indicating the degree of influence; wiRis a normalized fuzzy weight associated with rik. The value of rik is identified by referencing the index of the Taguchi orthogonal array and then mapping it to a ~o~es~n~ng level of influence indicated in the relational matrix. At the final ‘linguistic defuzzification’ stage, an output interface is genemted. The fuzzy results obtained earlier are defuzzified at each level of the process parameters. This defuzzification procedure is fulfilled by finding a geometrical center (centroid, x’, for the output fuzzy value at each level of the pmeters (Rggre 2). Subseque~~y~ the final design solution is selected among the respective three levels (i.e. low, medium and high) for each process parameter. The qualitative solution is the level that corresponds to the maximum centroid value. In the case of multicriteria qualitative design, the methodology presented earlier is equally applicable where multiple (output) desired material performance parameters are targeted sim~~eously. Note that the ~zy-~sitive reciprocal matrix is set up for each single material ~~o~~~e @&), rather than for each process parameter, to determine the no~~i~ed fuzzy weights (w,e k). The fuzzy multicriteria design metric Qmutti) is formulated in terms of individual design metrics o;)jk, indicated in eqns. (2) and

Fuzzy reasoning system

Fuzzy reasoning utilizes a human-like language rather than differential or difference equations to characterize a system’s behavior. A typical fuzzy reasoning system, as shown in Figwe 3,” is essentially the one that establishes and transforms a mapping of fuzzy sets indicating input conditions to fuzzy sets indicating output performance characteristic. The input to the fuzzy reasoning system is given as precise or crisp values. The step of fuzziflcation interprets the input, that contains ambiguous or vague information, as per its associated ‘degree of fulfilment’ @OF). The DOF is the membership level to which a fuzzy value is held true, Basically, the reasoning process evaluates a collection of fuzzy ‘if-then’ rules in parallel. The “if-then’ rule base provides the information that functionally relates the input entities to the output responses through prescribed fuzzy relations using linguistic descriptions. Each rule encodes one piece of loosely-defined knowledge in the following pattern: if x ~input} is Ai then y (ou~ut) is Bi

f@ which generates an association between the fuzzy sets (or values) Ai and Bi. The resulting fuzzy answer is produced after fusing all the results generated from each rule (fired independent of each other). This overall fuzzy result is

1 E lSz 0.75 CE 3 0.5 E 3 g 0.25 E 0 -14

-12

-10

-8

-6

-4

3?a3zyDesign Metric (X) Figure

2

238

Materials

Defmzifkation

using centroid

methI

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5 1996

c5)

-2

0

L. Chen et al.: A systematic

:

:

approach

to preliminary

polymer

process

development

:

Figure 3 Fuzzy reasoning system ‘I

finally interpreted as a representative crisp value through the procedure of defuzzification, such as finding the centroid of the resultant fuzzy value. For completeness, computational details can be found in references.“3’2 In the absence of detailed governing equations available during the early process development, the fuzzy redoing system will be used in this work as a mathematical means for approximating continuous real-valued functions.

Proposed methodology The methodology proposed for sys~mati~aliy modeling and designing polymer processes in the early development period involves the following major steps. The linguis~c-based qualitative design method is applied fast to narrow down the original fuzzy uncertainties associated with the process variables. Using the results generated from the previous step, an app~p~ate fuzzy reasoning system is then constructed for modeling the process using the ‘if-then’ linguistic rules. The fuzzy reasoning system is viewed as producing a continuous real-valued function, which functions similar to conventionally developed governing equations. Constrained in the reduced domain (generated in the first step) for various process variables, the process design is ftily reached by properly searching for a precise or crisp set of process variables via the maximization of the required performance objective. The following sections will address the modeling and design related issues in more detail.

for simplicity. At the early stage of process development, triangular membership functions are usually sufficient. In the current work, three fuzzy sets (or numbers) designating Iow, me&m and high are assigned to each input process variable, as shown in Figwe 4. These fuzzy components consti~te a continuous universe of discourse for the process variables. It can be seen that three a priati values corresponding to the membership grade of one are selected in Figure 4. To offset likely uneven effects induced by, for instance, a nonline~ty inherent in physical data acquisition, the following formulation is developed to define the membership function (see Figure 4)

10

fpI =

and p) i

1=

log,a(l +a$

aij 2 0

log10 i+

l&j < 0

- p* 2

77ifh)- qp 2=

2

1

Physical membership functions

To rn~pula~ fuzziness present in the early process development, physical membership functions are established to relate individual process variables to their associated fuzzy uncertainties. A membership function characterizes a specific fuzzy set (or number) to indicate the extent of a singleton element belongs to the set. If no other information is available, triangular-type membership functions are used

0 $0

rii(hf

Figure 4 Fuzzy quantities corresponding to low, medium and high

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objective (PJ. The normalized fuzzy weight is evaluate? in advance using the algorithm introduced in reference . These three fuzzy components, namely, A,, rik and wjk are used to construct a fuzzy ‘if-then’ rule as follows

8 P 0.75 Y E g

polymer

Single criterion:

0.5

if Xi is Ai then Pk is rikwik

(10)

5 d E

or

025 -

Multiple criteria 0

N&j 0

Figure

5

Fuzzy

0.25

(linguistic)

0.5

0.75

output PeIfomlance values

for output

1

if Xi iS Au then PO k3

(11)

z(rikWik)W&jk k=l

performance

where index j = 1, 2, 3 corresponds to individual levels (i.e. linguistic or fuzzy values) low, medium and high, respectively; a0 represents an a priori value for the ith process variable (xi) that corresponds to the membership grade of one at the jth level; and A1 and A2 are shape parameters used to define the level medium. Obviously, the following inequality is always held true: ail 5 ui2 5 Ui3. Note that p(qi) is set to one in the case of qi 5 #) or 77’2 $‘). With further process development, the membership ftnctions can be constructed based upon collected statistical data.13 Another domain considered for output performance parameters is composed of five decompositions indicating very bad, bud, okay, good and very good, as shown in Figure 5. For expedience, the universe of discourse for these fuzzy components are defined over the interval O-l.

where the multiplication term in the consequent part indicates a fuzzy design metric, corresponding to the ith singleton input Xi, to evaluate the kth (or overall) output peIfOmEUEe parameter Pk (Or PO) in the absence Of the process governing equations. Since the fuzzy quantity low or high overlaps with the quantity medium, as indicated in Figure 4, care should be taken in constructing the ‘if-then’ rules associated with these two quantities. Under the circumstance where xi is ZOW (Ail) or Xi is high (A&, GUI additional ‘if-then’ rule corresponding to ‘xi is medium’ (Ai2) needs to be incorporated into the rule base simultaneously. This implementation will be instantiated in Section 4. The same procedure of construction of the ‘if-then’ rules is repeated for all the process variables from 1 to N. The cumulative rules form the whole rule base for the fuzzy reasoning system under a specific performance objective, say Pk (k = 1, 2, . . . , Nobj). This reasoning system provides a pahXd~ functional mapping, say Fk, that relates the input variables (Xi, i = 1, 2, . . . , N) to the output performance parameter Pk. Thus, we have a goveming-equationlike functional form

Modeling procedure: a combined LDOE routine

(xl, x2,

with i= 1, 2, . ..) N;

j = 1, 2, 3

Before modeling a polymer process, as stated earlier, the LDOE should be applied first to narrow down initial uncertainties or ranges of the process variables. The resulting uncertainty range corresponding to an appropriate process variable is characterized by one of the fuzzy (or linguistic) quantities among low, medium and high. Our modeling procedure starts with these qualitative results to establish a corresponding ‘if-then’ rule base (in terms of the qualitative results generated earlier). This rule base constitutes a fuzzy reasoning system in which a precise or crisp value of a specified output performance can be generated at each time with respect to input values of the process variables. It has been shown that the fuzzy reasoning systems can be efficiently used as an alternate computational tool to approximate continuous real-valued functions with certain accuracy.‘1*14 The inside computational efforts do not require any analytical or numerical governing equations. The detailed procedure for process modeling is presented below. For the ith process variable xi, consider a fuzzy quantity A, indicating the qualitative result generated by LDOE in the previous procedure. This fuzzy quantity can be either low (j = l), medium (j = 2) or high (j = 3). Using this linguistic-based qualitative information, another linguistic quantity (rik), coupled with its associated normalized fuzzy weight (w&T can be tracked and screened out in the corresponding predefined relational matrix under the kth performance 240

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. . . , XN)

: f:

(12)

Pk

or F&l,

x2,

.-.,

XN)

(13)

=pk

In this work, the Mamdani inference method l2 is modified by finding an appropriate centroid value for each rule individually and then fusing them together through the use of logical ‘OR’. The centroid-style method is applied to defuzzify the overall fuzzy result, with a likely arbitrary shape, integrated by all individual results resulting from each rule, in the form ;

k= 1, 2, ... ,Nobj

(14)

where pk is a precise or crisp value indicating the overall defuzzfied result corresponding to the kth output performance parameter; p4 indicates the defuzzified result (i.e. centroid value) with respect to the qth fuzzy rule; sp indicates an appropriate geometric area during the defuzzlfication at the qth fuzzy rule; and M is the total number of ‘if-then’ rules. This concludes the modeling work for the polymer process. Design scheme: un optimization-oriented

routine

The polymer process design problem can eventually be formulated as an appropriate optimization-like problem.

L. Chen et at.: A systematic This ~sfo~ed design problem can be stated in the following mathematical form: Find a process design vector X

x= [x,,x2, *.., XJ which maximizes Single criterion: Pk =F&x);

to preliminary

approach

k= 1, 2, .*

Or

multiple criteria:

polymer

process development

signify tow, edict and high, respectively. The relational matrix was also constructed to characterize the type of effects for each process parameter on the desired material properties (see Table I). The qualitative results, obtained from the LDOE, identified the search range for each process parameter. The quali~tive results will be used in the next two steps in identifying physical parameter values for the individual process variables. In this case study, the eight process parameters and four desired ptput) material performance objectives are listed below. Process ~ara~eter~s~~ 1. platen temperature (“F),

07)

subject to xj E Dj;

i=l,

2, . . . . N

(18)

Di = {Au 1 j = 1, 2, 3)

(19)

or in more detail Ai1 = low [Gl, aiZ]; [air + 1, ~2 -l- 21; Ai2 = medium = high [ai:!, ai ; Ai

(20)

where & denotes the solution domain, found earlier by LDOE, for the ith process variable (xi). Thus, any available standard optimization technique can be directly employed to solve this design problem for a proper set of physical process parameter values. The resulting solution represents a preacne process design for a particular polymer process. The following section will demonstrate the computational details of the proposed approach.

Case study: compression molding of UHMWPE The concepts and solution procedure in the proposed methodology will be demonstrated in this section through the study of the compression molding of UHMWPE. The solution procedure can be categorized into three major steps. They are: (a) ~ce~in~ reduction through the linguisticbased qualitative design approach, (b) establis~ent of an appropriate fuzzy reasoning system for process modeling, and (c) identification of precise or crisp values for various processparametersbyusingavailabledesignoptimizationroutine. ~~litati~e design: an ~ncerta~n~-reduced procedure In the previous work,’ the qualitative preliminary design method (i.e. LDOE) was used to determine the proper parameter settings for the compression molding of U~PE. The linguistic-bred qualimtive results are tabulated in Tabbe 2, where the parameter levels 1, 2, and 3 Table 2 Linguistic-based Process

qualitative

p~amete~s)

Cxystallinity Tensile strength Tensile modulus Multi-criterion Legend:

4. r~~stallization cooling rate (“F/m&r), 5. melt pressure (MPa), 6. recrystallization pressure (MPa), 7. melt soak (min), and 8. recrystallization soak (min). Desired material perfonnunce:

with

Di =

2. heating rate (“F/r&r), 3. melt cooling rate (“FAnin),

parameter

level:

1 = hv,

design

results

1. maximize tensile strength (MPa), 2. maximize crystallinity (a), 3. maximize tensile modulus (MPa), and 4. multicriteria design: maximize tensile strength, crystal-

limty and tensile modulus ‘simultaneously.’

Process yodeling: a ale-base-established procedure In this case study, the fuzzy reasoning system is applied to determine proper physical parameters value corresponding to each linguistic parameter level. As mentioned in previous sections, the fuzzy reasoning system categorizes and describes the system’s behavior by compiling a list of linguistic (or fuzzy) ‘if-then’ statements. These ‘if-then’ statements (or fuzzy rules) are used to describe the mapping relations of the system by relating input parameter values (Xl I at “‘, xH) to an output performance objective (Pk). By assigning a design vector (X) containing precise input parameter values, a physical output performance value or integrated design metric (Pk) can be calculated through the process of inference mapping, defuzzification, and fusing. Inference mapping is established by dete~ng the ‘degree of fulfilment’ (DOF) while de~zzi~cation is achieved by the centroid-based method. Fusing through logical ‘OR’ is then applied to integrate the design metrics of each fuzzy rule as the final output of the system corresponding to a given X. This procedure is then combined with a nonlinear opti~zation subroutine to determine the final proper parameter values by maximizing an objective function that reflects the desired output material performance Pk. The final results represent the preliminary process design. A general form of the fuzzy rule ‘if-then’ statement is written in the form of eqn. (10) or (11). More than one ‘if-

’

XI

x2

x3

x4

x5

%

x7

xs

3 1 3 1

2 3 1 3

1 1 2 1

I 1 2 1

3 2 1 2

2 3 2 3

1 2 2 2

2 1 2 2

5 1996

241

2 = me&run

and 3 = high

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then’ statement is required to describe the relationship between a parameter level and its output performance objective due to overlapping of linguistic parameter value ranges. For example, if the linguistic design method suggests the process parameter level to be low (level = l), one cannot be certain that the physical input parameter value will only fall within the low region because there is an intersection region between the levels of low and medium. The effects of intersection are compensated, since an input parameter value with a higher corresponding DOF in the low region will naturally have a lower corresponding DOF in the medium region. The same argument can also be applied to the case where the parameter levels are set at medium or high. In the current study, parameter 1, xi (i.e. platen temperature), can only hold two input values (low and high). Therefore, its parameter range will not contain any intersection or overlapping region. In the case where the performance objective ‘crystallinity’ is considered, the proper level for process parameter 2 was determined to be medium. Therefore, three ‘if-then’ statements (low, medium and high) are used to describe the effects for parameter 2, as described in eqns. (23)-(25). The r21 value is determined from the relational matrix. For x2 = low, the r21 value corresponds to very good, as shown in eqn. (23). The same procedure can be applied to parameter 6 and 8 since their proper levels are also medium. For parameters 3,4 and 7, the proper parameter levels were set to be low. Thus, only two ‘if-then’ statements are needed to categorize this parameters, namely, low and medium. For parameter 5, the effects can be categorized by medium and high. The detailed ‘if-then’ rule base for the performance objective ‘crystallinity’ is constructed and given as follows RI : if x1 is low, then Pr is rrtwrr

process

development

Rg :if x4 is medium, then PI is r~rw4~ (r4r = ok) OR (29) RIO : if x5 is medium, then PI is rst ~51 (~,t = bad) OR (30) R11:if x5 is high, then PI is rstw51 (~1 =ok) OR (31) R12:if X6 is low, then PI is r6rw61 (&1 =bad) OR

(32)

R13:if X6 is medium, then PI is r6rws1

(&5r = ok) OR (33) R14: if X6 is high, then PI is rstw61 (r61 = ok) OR (34) Rt5:if x7 is low, then PI is r7rw7t (r7r = ok) OR (35) R16 : if x7 is medium, then PI is r7rw7t R17:if x8 is low, then PI is rstwgl

(r71 = ok) OR

(rst = ok) OR

(36) (37)

R18 :if Xg is medium, then PI is rstwst Rtg :if xs is high, then PI is rstwsl

(rst = ok) OR (38) (~1 =ok) OR (39)

where Wik denotes the normalized fuzzy weight for the ith process parameter (i = 1, 2, . . . , 8) under consideration of the ti performance objective k = 1,2, and 3, i.e. 1 = crystallinity, 2 = tensile strength and 3 = tensile modulus). The normalized fuzzy weights in this case study were pre-determined, as tabulated in Table 3, using the methods discussed in reference ‘. Using the results from the linguistic based design and relational matrix, the other ‘if-then’ rule bases can be constructed similarly for the remaining performance objectives.

(rrt =very good) OR (21)

R2: if x1 is high, then Pr is rt2wtt

Quantitative design: an optimization-based procedure Based upon the ‘if-then’ rule base generated in the previous procedure, the output performance objective Pk can be evaluated through the integration of each linguistic rule statement. This is accomplished by inferring a design vector X with physical (i.e. precise) input values to the overall fused performance value fi using the modified centroid method. This relationship has been presented in eqns. (12 and 13). As a result, an optimization routine can then be easily implemented by using eqn. (13) as the objective function. Table 4 From the linguistic design, the uncertainty range for each parameter has been reduced, and the resulting range was indicated by its proper level.’ By applying eqns. (7)-(9), as well as eqn. (20), the physical bounds for each parameter can be identified. These results are summarized in Table 5.

(rtr =very good) OR (22)

R3 : if x2 is low, then Pr is r21w21 (~1 =bad) OR

(23)

Rq: if x2 is medium, then Pt is r2twzt R5 :if x2 is high, then Pt is ~1~21

(~1 = good) OR (24) (r21 = ok) OR (25)

Rs:if x3 is low, then Pr is tstwst

(rst = good) OR (26)

R7 :if x3 is medium, then PI is r3rwst R8 :if x4 is low, then PI is r4tw4t

Table Process

3

Normalized

fuzzy

weights

(w,)

(r31 = ok) OR

(27) (28)

(r41 = ok) OR

for each process

parameter Normalized

variable Left 0.0342 0.0208 0.2572 0.0319 0.0938 0.1586 0.8665 0.1811

Xl x2 x3

X4 X5 X6 x7

XS Note:

Each normalized

242

Materials

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weight

crystallinity Mode 0.0495 0.0286 0.3575 0.0479 0.1366 0.2336 0.1290 0.2438 is characterized

& Design Volume

Right

Left

0.0709 0.0398 0.5015 0.0704 0.2004 0.3464 0.1727 0.3350

0.0233 0.0451 0.0572 0.0711 0.2406 0.1870 0.0151 0.0396

as a linear triangular

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fuzzy

5 1996

fuzzy weight Tensile strength Mode 0.035 1 0.0701 0.0856 0.1063 0.3564 0.2616 0.0229 0.0618

number

[lef,

mo&

Tensile Right

Left

0.0499 0.1028 0.1202 0.1487 0.4928 0.5815 0.0309 0.0897

0.0542 0.0163 0.7417 0.0462 0.2658 0.2024 0.0435 0.0781

right]

modulus Mode

Right

0.0777 0.0202 0.1043 0.0634 0.3447 0.2563 0.0258 0.1074

0.1113 0.0273 0.1412 0.0623 0.4611 0.3312 0.0338 0.1506

L. Chen et al.: A systematic Table

4

Normalized

Performance

fuzzy

weights

Mode

Right

0.0513 0.4683 0.2129

0.0618 0.6527 0.2856

0.0819 0.8999 0.4080

Note: Each normalii fuzzy weight fuzzy number [left, mode, right].

is characterized

PO = 5 k=l

(40)

3.6 5 x4 _<9.0

(41)

15.6 5 x5 5 23.4

(42)

47.66 5 x6 < 82.59

(43)

x7

< 30.0

Bounding

Results

Using the above procedure, an unconstrained optimization routine (NCONF) from IMSL was used to identify the set of physical parameter values by maximizing the output performance value Pk (k = 1, 2, 3). The final design results corresponding to each material performance desired are outlined in Table 6. From Table 6, it can be observed that the values of the final design metric for each material performance did not increase by a large margin. In some cases, the parameter values were only changed by a slight amount depending on the initial design inputs selected. Nevertheless, these results represent a proper set of physical parameter values for the preliminary process design. This design outcome may guarantee a certain level of quality of the product based on the designer’s initial intuition. Values in the optimization process are based upon a systematic methodology. Since a bound for each parameter level has been specified, the engineer will have enough guidance in determining the input values that are not only reasonable, but also consistent

(44)

Process variable

Input

values

for process

a p&n’

x3 x4

x5 X6 X7

%3 Note:

Parameter

Table

6

Process

Level ai

2.4 3.6 3.6 7.8 38.9 0.0 0.0

x2

variables

values

ai1 low

Xl

6.2 9.0 9.0 15.6 58.4 30.0 30.0

bounds

Preliminary

for crystallinity design

s.

are highlighted

7.0 5.0 4.0 20.0 60.0 10.0 35.0

x4

X5 x6 x7

X8 parameter

Pk

Upper low

low 2.4 3.6 3.6 7.8 38.9 0.0 0.0

6.2 9.0 9.0 15.6 58.4 30.0 30.0

2 (medium)

Lower 3.92 5.73 5.73 11.04 47.66 5.39 5.39

Level

3 (high) Lower

Upper 7.52 15.61 15.61 19.11 82.59 42.43 42.43

Upper high

high 6.2 9.0 9.0 15.6 58.4 30.0 30.0

9.1 27.0 27.0 23.4 116.8 60.0 60.0

for illustration

0.1587

Desired material performance/properties Tensile strength (k = 2) Tensile Initial Final Initial

(k = 1) Final

high

X1 x3

Level

results Crystallinity Initial

Objective

9.1 27.0 27.0 23.4 116.8 60.0 60.0

1 (low) Lower

ai high

variable

x2

(46)

(‘ikwik)wobjk

w&j

(45) Thus the process design problem is finally transformed to a corresponding equivalent optimization design problem, which can be formulated exactly in the same way as indicated in eqns. (15)-(18). Any available optimization technique can be employed to find the solution to this design problem. It is noted that the same solution procedure can be followed to solve the remaining design problems with their individual performance objectives. 5

development

where k denotes the normalized fuzzy weights for the kth single performance objective . The procedure involving the establishment of fuzzy rules, identification of DOF and determination of ‘clipped’ design metric by inference mapping, fusing, and defuzzification of the final objective parameter is equally applicable to this multi-objective process design problem.

5.39 5 xs 5 42.43

Table

process

as a linear trianguhu

3.6 5 x3 5 9.0

I

polymer

In the multicriteria design case, the singleton performance parameter Pk corresponding to each fuzzy rule is replaced by simultaneously considering the three different objective parameters as

For illustration, the physical bounds for process parameter 2, x2 (i.e. heating rate), will be tracked for the output performance ‘crystallinity’. From Table 2, parameter 2 has been assigned a proper parameter level of 2 (medium). In Table 5, the bounds for parameter 2 at level 2 are given as 3.92 5 x2 5 7.52. Therefore, when the optimization routine is applied to maximize the objective value Pk, the values 3.92 and 7.52 are used as the lower and upper bounds, respectively, for the process variable 2. Using the same procedure, the bounding values for the other process variables can be determined similarly as follows

0.0

to preliminary

Case of multiple objectives

objective

Left

objective

Crystallinity Tensile strength Tensile modulus

C) for each design

(w&j

approach

high 7.51 3.6 4.18 19.12 58.62 9.06 36.48 0.1679

low 9.0 4.0 4.5 16.0 100.0 30.0 15.0 0.2498

low 8.96 4.03 4.57 16.28 101.95 29.57 14.98 0.2503

Materials

modulus

(k = 3) Final

high 3.0 10.0 12.0 8.0 60.0 35.0 25.0 0.1958

Multicriteria Initial

high 3.26 9.02 15.5 7.83 58.27 42.2 29.9 0.2062

& Design Volume

low 8.0 5.0 5.0 9.0 100.0 30.0 30.0 0.1673

7 Number

5 1996

Final low 6.2 5.69 5.7 16.58 103.44 29.88 29.5 0.2

243

1. Chen et al.: A systematic

with his initial unsatisfactory engineer must reciprocal and

approach

to preliminary

intuition or past experience. If the results are and conflicting to a known fact, then the re-examine his intuition in constructing the relational matrices.’

Concl~~ng remarks A generalized methodology has been presented for modeling and designing polymer processes in the early process development period. The theory developed has been illustrated through a case study using compression molding of UHMWPE. The results have been presented along with the discussion. It has been noted that membership functions used in this work are linear, engulf-type. However, they result in discont~uities in input~u~ut mapping because of their geometric nature. It has been pointed out that the shape of membership functions has a strong influence on the behavior of a given reasoning system when the number of rules and number of inputs/outputs are the same.16 To smooth-out the input-output mapping, smoother membership functions such as Gaussian membership functions are recommended. Software packages currently being used, such as C-mold for injection molding, are more suitable at a later, detailed process design stage when the engineer has developed sufficient confidence in the particuhu process design, Furthermore, such software packages also have many embedded assumptions, such as being applicable to only shell type parts for injection molding simulation. Hence a critical gap between the use of advanced software and the preliminary stage of process modeling and design needs to be bridged. In order to enhance process development speeds at the time and cost consuming preli~n~ stage, the proposed methodology enables computers to facilitate the conceptual process development during the preliminary development stage. This work provides a basis to build up a new generation of design software for the early stage of process development. With the advancement of modern computer techniques, the advent of computer-integrated or automated modeling and design of polymer processes appears feasible.

244

Materials

& Design Volume

17 Number

5 1996

polymer

process

development

References 1 Chen, L., Ku, K. and Ramani, K., A linguistic based preliminary procedure for qualitative polymer process design. Materials & Design, 1996 (in press). 2 Zachariades, A. E. and Kanamoto, T., The effect of initial morphology on the mechanical properties of ulna-high molecular weight polyethylene. Polymer Eng. and Sci., 1986,26, 6583. 3 Barkston, A. B., Keating, E. M., Ranawat, C., Farris, P. M. and Ritta, M. A., Comparison of polyethylene wear in machined versus molded polyethylene. Clinical Orthopedics and Related Research, 1995, 37, 317. 4 Bash, C. and Dohnal, M., Qualitative reasoning with reference to polymers. Materials & Design, 1995, 16(2), 103-108. Zadeh, L. A.,. Fuzzy Sets.,Inform. and Cont~, 1%5,8(13), 338-3.53. Kaufrnann, A. and Gupta, M. M., ~n~d~tian ro Fury Ar~t~etic, Van Nostrand Reinhold, New York, 199 1. Ben-Arieh, D. and T~~~phyllou, E., Quantifying data for group technology with weighted fuzzy features. Int. Journal of Pmduction Research, 1992,30(6), 1285-1299. 8 Chang, P. T. and Lee, E. S., The estimation of normalized fuzzy weights. Computers Mathematical Application, 1995, 29(5), 2142. 9 Saaty, T. L., A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 1977, 15(3), 234-281. 10 Roy, R., A Primer on the Taguchi Method, Van Nostrand Reinbold, New York, 1990. 11 Kosko, B., Neural deform and Fuzzy Systems, Prentice Hall, Englewood Cliffs, 1992. I2 Tsoukalas, L. H. and Uhrig, L. H., Fuzzy and Neural Approaches in Engineering, John Wiley and Sons, New York, 1996. 13 Civanlar, M. R. and Trussell, H. J., Constructing membership functions using statistical data. Fuzzy Sets and Systems, 1986, 18, 1-13. 14 St&, T. and Hammell II, R. J., Interpretation, completion, and learning fuzzy rules. IEEE Trans. SystemsMan Cybernetics, 1994, 24(2), 332-342. 15 Ramani, K. and Parasnis, N. C., Process induced effects in compresssion molding of ultra-high molecular weight polyethylene. In ASTM Symposium on Characterization and Properties on UHMWPE, New Orleans, LA, 1996. 16 Lotfi, A. and Tsoi, A. C., Importance of Membership functions: A Comparative Study on Different Learning Methods for Fuzzy Inference Systems. In Pmt. 1st IEEE Co@ on Fuzzy System,IEEE, 1994, pp. 1791-1796.