Authors: YehLiang Hsu, TzuChi Liu, MingSho Hsu, Francis Thibault (20011021);
recommended: YehLiang Hsu (20011022).
Note: This paper is revised and presented at 「中國機械工程學會第十八屆學術研討會」.
The original paper was presented at The Sixth International Conference on CSCW
Design, July 1214, 2001, London,
Ontario, Canada.
. .
A fuzzy optimization algorithm
for blow moulding process
Abstract
This paper demonstrates a fuzzy optimization algorithm for
determining the optimal gap openings of the programming points in the blow
moulding process. Traditional numerical optimization algorithms treat the
optimization problem as pure mathematical problems. Valuable engineering
knowledge is not utilized in the optimization process. The idea of the fuzzy
optimization algorithm is that, instead of using purely numerical information
to get the new design point in the next iteration, engineering knowledge and
human supervision process can be modeled in the optimization algorithm using
fuzzy rules. The fuzzy optimization algorithm developed in this paper works
well on the blow moulding examples.
Keywords: fuzzy optimization, blow
moulding.
1. Introduction
Blow moulding is the forming of a hollow part by “blowing” a
mouldcavityshaped parison that is made by thermoplastic molten tube [Lee,
1990]. The thickness of parison determines the thickness of the blown hollow
part, which is controlled by the gap opening space between the die and the
mandrel. Figure 1(a) illustrates
parison programming, which manipulates the gap opening
of die at programming points to control the thickness of parison. In order to
obtain uniform thickness distribution of the hollow part as shown in Figure
1(b), the thickness of programmed parison must be nonuniform. For example, in Figure
1(b), the parison thickness for the greatest expansion area must be thicker
than those of the other areas.
(a) (b)
Figure 1. Illustration of parison programming
BlowSim is a finite element software package designed to simulate
the extrusion blow moulding, injection stretch blow moulding, and thermoforming
processes. It is developed by the Industrial Material Institute (IMI) of
National Research Council (NRC), Canada. Figure 2 shows the
thickness distribution of a sample problem simulated by BlowSim when the gap opening
is set at 75% at all time.
Figure 2. The sample blow moulding problemn
In the blow
moulding process, it is desirable to obtain a final part of constant thickness.
It is therefore an optimization problem on how to control the gap openings to
minimize the deviation of the thickness of the final part from the target
thickness. For the sample problem shown in Figure 2, the objective is to
minimize the average deviation of the thickness of all nodes from the target
thickness:
min. _{} (1)
where T_{t} is the
target thickness, T_{i }^{j} is the thickness at node j
in the ith iteration, and n is the total number of nodes. We are
able to extract the thickness of all nodes from the simulation results of BlowSim
and plug them into Eq.(1) to get the objective function value. The gapopening
rates at discrete time points are the design variables. Obviously, the
thickness at node j in the ith iteration T_{i }^{j}
are functions of the design variables, namely, the gap opening rates.
We can certainly use a gradient type numerical optimization
algorithm to solve for the optimal opening rates. On the other hand,
manufacturing engineers usually adjust the gap opening rates empirically:
reduce the gap opening rate if the final part is too thick, and vice versa.
The purpose of this paper is to illustrate how the concept of “fuzzy
optimization algorithms” to closes the gap between numerical optimization
algorithms and engineering experience, using the optimization of the blow
moulding process as an example. This paper first explains the concept of fuzzy
optimization algorithms. Then in our blow moulding optimization problem,
engineering experiences for adjusting the gap opening rates are modeled as
fuzzy rules which determines the search direction of a line search algorithm.
The optimization results are presented, and the advantages of this approach are
discussed.
2. The concept of “fuzzy
optimization algorithms”
The optimization process can be viewed as a closedloop control
system. Figure 3 shows a general block diagram of an automatic control system.
An error detector compares a signal obtained through feedback elements, which
is a function of the output response, with the reference input. Any difference
between these two signals constitutes an error or actuating signal, which
actuates the control elements. The control elements in turn alter the
conditions in the plant (controlled member) in such a manner as to reduce the
original error.
Figure 3. General block diagram of an automatic
control system
Figure 4 shows a general block diagram for an optimization process.
Comparing with Figure 3, an optimization model in an optimization process is
analogous to the plant in a control system; an optimization algorithm is
analogous to the controllers. Initial parameters are input to the optimization
algorithm, which in turn generates a trial design point according to its search
rules. The optimization model is then evaluated at this trial design point, and
the information such as objective and constraint function values and
sensitivity is fed back to the termination test. If the termination test fails,
the optimization algorithm is triggered again to generate the next design
point, using the numerical information from previous iterations.
Figure 4. General block diagram of a design
optimization process
Traditional numerical optimization algorithms are analogous to
direct digital controllers. The algorithms are usually “crisply” designed for
well defined mathematical models. Their numerical rules for generating the next
design point are exact and definite, and they can usually be proved to have
nice converging behavior when applying to well defined mathematical models.
However, in engineering optimization problems, we seldom have well
defined mathematical models. In the blow moulding example, we know that the
thickness at node j in the ith iteration T_{i }^{j}
are functions of the design variables, namely, the gap opening rates. But these
functions often do not have exact algebraic forms in terms of the design
variables, and they can only be evaluated through experiments or computer
simulations, which are expensive and imprecise in nature. Usually sensitivity
calculation is also done through imprecise finite difference methods. Very
often the cost of the number of function evaluations required to meet the
“crisp” definition of the numerical algorithms is simply too high to be
affordable.
On the other hand, when we apply numerical optimization algorithms
on an engineering problem, we treat the engineering problem as a pure mathematical
problem. Engineering heuristics are totally ignored in the numerical
optimization algorithms. This motivates the idea that, in addition to crisp
numerical rules, the human supervision process should also be modeled in an
optimization algorithm using fuzzy rules; the “controllers” in the optimization
process may as well be fuzzy controllers!
This concept was proposed by Hsu et al [1995]. Mulkay and Rao [1998]
also presented the same idea. In both work, fuzzy heuristics are used to control
the parameters of the optimization algorithm to improve its performance. In the
following section, fuzzy heuristics are used to generate the new design point
of the next iteration.
3. The optimization algorithm
used in “BlowOp”
“BlowOp” is the optimization module of BlowSim used to obtain a
uniform thickness distribution and minimize part weight. In BlowOp, the parison
thickness, which are directly affected by gap opening rates, are used as design
variables. The optimization procedure is given by the following equation:
_{} (2)
where _{}and _{} are parison
thickness of each programming point at iteration i+1 and iteration i.
They are related to gap opening rates and extrusion time. _{} is the thickness
of the part at iteration i. _{} is target
thickness of the final part, which has uniform thickness distribution. a_{u} is a userdefined proportional gain; while a_{p }is a gain defined by the inflation model:
a_{p} = A_{i}/T_{i}. (3)
This optimization procedure is a line search type of algorithm,
which is commonly used in various numerical optimization algorithms. The
standard line search algorithm will be [Vanderplaats, 1984]:
_{} (4)
where x is the vector of design variables, s is the
search direction, and a is the step length.
In standard line search algorithms, step length a is decided
after several trials according to certain numerical rules. In the procedure
shown in equation (2), a takes a fixed value 0.5.
In numerical optimization algorithms, search directions are defined by
numerical information, for example, gradient or Hessian of the objective
function. However, in the procedure shown in equation (2), the search direction
is determined by a_{p} = A_{i}/T_{i},
and (T_{i} T_{s}).
Figure 5 shows the iteration history of this optimization procedure,
when applied on the sample problem shown in Figure 2. The gap opening is set at
75% at all time in the first iteration. The target thickness T_{s}
is 2mm. In the first iteration,
the maximum part thickness is 6.804mm,
the minimum part thickness is 1.394mm,
and the value of the objective function (Eq. (1)) is 2.474. After 10
iterations, the objective function value reduces from 2.474 to 0.953.
Figure 5. Iteration history of BlowOp
The search algorithm in BlowOp actually uses engineering heuristics
in a crisp format. The change in parison thickness between iterations is
determined by a_{p}(T_{i}T_{s}).
The term a_{p} is affected by the geometry of
the bottle. The part thickness becomes smaller if the radius of the bottle is
large. Therefore a_{p} decreases as the radius
increases. The term (T_{i}T_{s}) represents the
simple rule mentioned earlier: reduce the gap opening rate if the final part is
too thick, and vice versa.
While the search procedure in Eq. (2) makes perfect sense to
engineers, there is no theoretical justification to why the change in parison
thickness between iterations is determined by a_{p} “multiplied” by (T_{i}T_{s}). If a 3^{rd}
or 4^{th} factors are found, it will be hard to find a proper “crisp”
formula to combine these 3 or 4 terms together.
4. The fuzzy optimization
algorithm
The engineering huristics used in the optimization procedure discussed
in the previous section are
(1) If the thickness of a certain node is larger than the target
thickness, then reduce the respective opening rate.
(2) If the thickness of a certain node is smaller than the target
thickness, then increase the respective opening rate.
(3) If the radius at a certain node is large, the change of opening
rate is large.
(4) If the radius at a certain node is small, the change of opening
rate is small.
Engineering rules (1) and (2) are first used to try out the fuzzy
optimization idea on the sample problem shown in Figure 2. Eq.(4) can be
written into the following scalar form for our optimization problem:
_{} (4)
where the gapopening rates at 7 discrete time points as the design
variables: _{}, _{}, _{}, _{}, _{}, _{}, _{}. In our fuzzy optimization algorithm, the change of opening
rate S_{j} is to be determined by fuzzy rules. The input
variables to the fuzzy rules are thickness T_{i} at the 7
discrete time points
Five levels are defined to describe the linguistic variables: PB(Positive
Big), PS(Positive Small), ZE(Zero), NS(Negative Small), and NB(Negative Big).
Standard membership functions shown in Table 1 and Figure 6 are used. Five
fuzzy rules are established:
R1:
If thickness is PB then S_{j} is NB
R2:
If thickness is PS then S_{j} is NS.
R3:
If thickness is ZE then S_{j} is ZE.
R4:
If thickness is NS then S_{j} is PS.
R5:
If thickness is NB then S_{j} is PB.
Table 1. Membership function

levels


2

1

0

1

2

NB

1.0000

0.0235

0.0000

0.0000

0.0000

NS

0.0235

1.0000

0.0235

0.0000

0.0000

ZE

0.0000

0.0235

1.0000

0.0235

0.0000

PS

0.0000

0.0000

0.0235

1.0000

0.0235

PB

0.0000

0.0000

0.0000

0.0235

1.0000

Figure 6. Membership functions
Table 2 is the quantization table for the input and out put
variables. This table is derived from engineering experience to the blow
moulding process and the optimization process. Step size a in Eq.(4)
is fixed at 1.0. Figure 5 shows the nice converging behavior of this
optimization procedure, when applied on the sample problem shown in Figure 2.
The gap opening is set at 75% at all time in the first iteration. Figure 7
compares the iteration history with that of BlowOp. From
Figure 7, it appears that BlowOp and the fuzzy optimization algorithm has
similar performance in this case. After 10 iterations, the objective function
value reduces from 2.474 to 0.834.
Table 2. Quantization table
Thickness

fuzzy input


Fuzzy reasoning

_{}

Target thickness + 2*Max(Thickness target thickness)/3

2


2

1_{}

Target thickness + Max(Thickness target
thickness)/3

1


1

[1_{}]/2

Target thickness

0


0

0

Target thickness + Min(Thickness target
thickness)/3

1


1

[_{}/2]

Target thickness + 2*Min(Thickness target
thickness)/3

2


2

_{}

Figure 7. Iteration history of the fuzzy
optimization algorithm with one fuzzy input
5. Adding a new set of fuzzy
rules
The fuzzy optimization algorithm discussed in the previous section
considers only the engineering heuristics related to the final part thickness. We
now include the other engineering heuristics related to the radius of the die into
the fuzzy optimization algorithm:
R6:
If the Radius is PB, then S_{j} is PB.
R7:
If the Radius is PS, then S_{j} is PS.
R8:
If the Radius is ZE, then S_{j} is ZE.
R9:
If the Radius is NS, then S_{j} is NS.
R10:
If the Radius is NB, then S_{j} is NB
The five levels and their membership functions are defined in Section
4. Table 4 is the quantization table for the input and out put variables. Note
that right now there are two fuzzy inputs.
Figure 8 compares the iteration history of the fuzzy optimization
algorithm with two fuzzy inputs with that of BlowOp. The
fuzzy optimization algorithm has better performance in this case. It converges
faster, and the final objective function value is lower. After 10 iterations,
the objective function value reduces from 2.474 to 0.811.
Figure 8. Iteration history of the fuzzy
optimization algorithm with two fuzzy inputs
Ideally, the objective function should converge to zero, which
indicates constant thickness through out the whole part. Figure 8 also shows
the iteration history when we increase number of programming points from 7 to 31,
using the same fuzzy optimization algorithm. With this increase in resolution,
the value of the objective function further drops to 0.515.
Table 4. Quantization table with engineering rules
Thickness

Fuzzy input

Target thickness + 2*(Max(Thicknesstarget
thickness))/3

2

Target thickness + Max(Thicknesstarget
thickness)/3

1

Target thickness

0

Target thickness + Min(Thicknesstarget
thickness)/3

1

Target thickness + 2*(Min(Thicknesstarget
thickness))/3

2

Radius

Fuzzy input

Mean(Radius) +(2*(Max(Radius)mean(Radius))/3)

2

Mean(Radius) +(Max(Radius)mean(Radius))/3

1

mean(Radius)

0

2*mean(Radius)/3

1

mean(Radius)/3

2

Fuzzy reasoning

_{}

2

_{}

1

_{}

0

0

1

_{}

2

_{}




Figure 9. Variations in thickness distribution
through the iterations
Figure 9 shows the variations in thickness distribution through the
iterations. It appears that the thickness distribution is close to constant in
the final part from iteration 10. Figure 10 compares the profiles of optimal
gap openings of programming points obtained from BlowOp, the fuzzy optimization
algorithm with one fuzzy input, the fuzzy optimization algorithm with two fuzzy
inputs, and the case with 31 programming points.
Figure 10. The profiles of optimal gap opening of
programming points
6. Automobile fluid reservoir
example
We then apply the same fuzzy optimization algorithm described in the
previous sections to an automobile fluid reservoir example, again with one
fuzzy input and two fuzzy inputs. As shown in Figure 11, the shape of the
reservoir is very complicated and unsymmetrical. The quantization table and the
membership functions are the same as defined in Section 4 and in Section 5. The
target thickness is 5 mm.
Figure 11 is the final thickness distribution after 10 iterations
using the fuzzy optimization with one fuzzy input. Figure 12 compares the
iteration history of the fuzzy optimization algorithm with that of BlowOp. Judging from Figure 12, BlowOp and the fuzzy
optimization algorithm have similar performance in this case. After 10
iterations, the objective function value of BlowOp decreases from 1.4773 to
1.2338, and the objective function value of fuzzy optimization algorithm with
one fuzzy input decreases to 1.2934. Because the shape of the reservoir is unsymmetrical,
at one horizontal cross section near the bottom of the reservoir, the thickness
may vary from 3.8643 mm to 4.1446 mm. Therefore it may
not be possible to further reduce the objective function value using only gap
openings as design variables.
Figure 11. Variations in thickness distribution
through the iterations
Figure 12. Iteration history of the fuzzy
optimization algorithm and BlowOp
The “radius” of a cross section of the reservoir is hard to define.
We use “mean radius” instead in our section fuzzy input. But the fuzzy
optimization algorithm with two fuzzy inputs works poorly in this case. As
shown in Figure 12, the objective function value increases from 1.4773 to
1.6538, which indicates that we may need more experiences in the blow moulding
of complex geometry model to construct better fuzzy rules.
7. The bottle case with internal
pressure and top load
Back to the bottle case, but now we have two types of loading, one
is internal pressure and the other is top load, as illustrated in Figure 13. A
performance optimization [Thibault, 2001] to minimize the weight of the bottle subject
to stress constraints has been performed, and the result obtained is a
nonuniform part thickness distribution as shown in Figure 14.
Figure 13 Two different types of loading were applied:
(left) internal pressure and (right) a top loading .
We used the same
fuzzy optimization algorithm described in previous sections to obtain the
optimal gap openings that will achieve this nonuniform part thickness. Figure
14 compares the thickness distribution obtained by the optimal gap openings and
the target nonuniform thickness distribution after 30 iterations.
Figure 14 Compare the
thickness with Fuzzy optimization and target
8. Conclusions and discussion
This paper demonstrates how to develop a fuzzy optimization
algorithm for determining the optimal gap openings of the programming points in
the blow moulding process. Traditional numerical optimization algorithms treat
the optimization problem as pure mathematical problems. Valuable engineering
knowledge is not utilized in the optimization process. The idea of developing a
fuzzy optimization algorithm is that, instead of using purely numerical
information to get the new design point in the next iteration, engineering
knowledge and human supervision process can be modeled in the optimization
algorithm using fuzzy rules. The fuzzy optimization algorithm developed in this
paper works well on our blow moulding examples.
In the blow moulding examples, one major advantage of the fuzzy
optimization algorithm over traditional numerical optimization algorithms is
that the first order gradient information, which is often expensive or
unavailable in real engineering problems, is not required. Another advantage is
its flexibility. As shown in the paper, it is easy to expand the fuzzy
optimization algorithm if new rules and/or new fuzzy inputs are considered.
9. Acknowledgement
This research is supported by the joint project between National
Science Council, Taiwan, and
Industrial Material Institute, National Research Council, Canada, NSC
892212E155019. This support is gratefully acknowledged.
10 References
Lee, N. C., 1990, “Plastic Blow Molding Handbook,” Van Nostrand Reinhold, New
York.
Hsu, Y. L., Lin, Y. F., Guo, Y. S., “A Fuzzy Sequential Linear
Programming Algorithm for Engineering Design Optimization,” 1995 Design
Engineering Technical Conferences, Volume 1, pp. 455462, ASME.
Thibault, F., Gauvin, C., Laroche, D., and DiRaddo, R., 2001,
“Development of an MDO Software Environment for the Blow Moulding Process”, The Sixth International Conference on CSCW in Design
July 1214, 2001, London, Ontario,
Canada.
Vanderplaats, G.. N., 1984, Numerical Optimization Techniques For
Engineering Design, McGrawHill.