Statistical design of experiments, while popular in many disciplines,
originated with agricultural applications in the 1920's. This paper
discusses the fundamental ideas behind various branches of design of experiments.
This includes treatment allocation, multiple comparisons, blocking, combinatorics,
response surface methodology, and the following types of designs: fractional
factorials, ANOVA-type, optimal, bayesian, Taguchi, mixture, uniform, supersaturated,
and other related designs. Relationships among them will be described.
Through these relationships, it is hoped that we can generated future research
ideas on design of experiments. This seems to be particularly important
for the information technology era we are currently in and that can be
expected to continue for decades.
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This is a survey article on known results about analytic solutions and
some numerical solutions of optimal designs for various regression models
for experiments with mixtures. The regression models include polynomial
models, models containing homogeneous functions, models containing inverse
terms and ratios, log contrast models, models with process variables, and
models containing the amount of mixture. Optimality criteria considered
include D-, A-, e- F_{p}-
and I_{l }- optimalities.
Uniform design and uniform optimal design for mixture components, and efficiencies
of the {q,2} simplex-centroid design are briefly discussed.
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While dealing with the problem of synthesis on process system engineering(PSE),
we often meet multi-peak and singularity, even fail to get optimum. In
this paper, we introduce sequential algorithm for optimization into the
synthesis of PSE, and we do some improvement to this algorithm by the technique
of genetic algorithm(GA). Finally, we test the enhanced sequential
algorithm for optimization by some test functions form GA and from standard
synthesis problem in PSE .
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The weighted regression analysis is listed in paper, and the reciprocal
of trial variance is taken as the weight. The theoretical optimal
coefficient of determination of data and the degree of lack of fit on the
regression function could be estimated. An example is detailed and
its problems in SAS software are pointed out. Author does not think
it is good that using average value of the repeated trial results analyze
data and the stepwise procedure of regression analysis select variables
if size of variables is more than sample size.
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Most classical experimental designs request duplicates of experiment
points. Obviously, repeated experiments increase the budget and time
of the experiments, but give in general better estimation of parameters
in the underlying model. Is it necessary to have duplicates in experimental
design? In this note we give several examples to show the usefulness
of repeated experiments and to discuss when one can do experiment without
duplicated.
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The spread of the uniform design method is an important task after the
invention of uniform designed method. Through studying and summarizing
the spread practice since 1994, this paper describes how construct framework
for the spread of the uniform design method and amplify service function
of the spread system. Organized and planned to proceed technology transfer
and diffusion. Call up science and technology researcher and community
to apply this method in our country all trades and demesnes for quick Formation
scale and benefit. Acceleration the uniform design method to conversion
reality productive forces.
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When uniform design (UD) is used to solve an optimization problem, a
set of uniform distributed points in the test area is produced based on
the Number Theorem ( NTM). Usually the number of points is very large,
for example, to do an experiment for a problem with S factors and Q levels,
S! possible selections are produced and Q^{S} records are written
into the database. Knowledge discovery in database (KDD) and data mining
(DM) is another technique for optimization, it discovers the rich knowledge
from the experiment data, so it is better to combine these two methods
in solving optimization problems. In this paper a case study is presented
by using the joined method. In this study, the joined method is applied
to the problem of material mix in designing the deep-water leakage-proof
cofferdam in the Three Gorge Project. The UD method is used to do 10 experiments
and the data from the experiments are used to train a neural network, then
two mix results are obtained. It has been verified that the result is satisfactory.
This shows the ability of NTM-based UD method to simplify optimization
problems and extract useful information. At the same time, if the UD is
not combined with Artificial Neural Network (ANN), then the solution can
only be selected among the data groups 6, 7 and 8, which is worse than
the obtained one and cannot meet the requirement in respect to1995's standard.
In this case study, by using ANN, 900 data groups are "mined out" from
the 10 data "mineral deposit", so two satisfactory solutions are "discovered".
These solutions produce the mix for the flexible material that meets the
mechanics requirement under the water of 100 meters. This shows important
role played by the technique of KDD & DM in UD applications.
In Fang, Wang 1996, [3] it is pointed out that in optimization NTM
makes great challenge to other methods including simulated annealing method
(SAM). Thus, it gives rise to a question that can we use UD to conduct
the ANN to make knowledge discovery?
In the past 20 years, regression model is widely used to analyze the
data produced by UD experiments. In this way, less than one tenth of KDD
& DM is used. We use UD to conduct DELPHI polling, with four factors
considered and six exports selected , 24 tables of U_{6} (6^{4})
can be produced. By using the method proposed by the author of this paper
in [9], a database be called UD database with 6^{4} = 1296 records
is produced. By using the criteria of extenics and based on the practical
applications, we can select a set of solutions for consideration.
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The paper analyses the factors effecting compressive strength.
According to the experiment extent decided by the factors the research
is carried out by improving uniform design of experiments with mixtures
and a perfect technique for increasing compressive strength of magnesia
brick is found.
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Evaluation of terrain suitability for navigation is a difficult problem
encountered in the development of a terrain-aided navigation system. An
evaluation method based on the uniform design principle is proposed. It
provides a total solution for the problem.
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Da Chengqi Tang, composed of Rhubarb (X_{1}), Mirabilite (X_{2}),
Immature bitter orange (X_{3}) and Magnolia bark (X_{4}),
is one of purgative prescriptions. The compatible properties of this
recipe was explored by experimental researches with uniform design.
The results show that X_{1}, X_{2} and X_{4} act
as principal drug to expel large intestine of rat ( Y = 11.1537 +
2.7750X_{1} + 1.7493X_{2}^{2} + 1.4814X_{4}
- 0.3036X_{1}X_{2} - 0.0814X_{3} ), that
X_{1}, X_{4} and X_{3} possess the effect of purging
stool but X_{2} inversely (Y = 4.5291 + 0.3763X_{1}
- 0.5357X_{2 }+ 0.1986X_{3 }+ 0.2458X_{4} ), that
X_{2} is used to increase the water inside intestine softening
hard mass ( Y = 0.8723 + 0.3510X_{2} - 0.0394X_{2}^{2}
+ 0.0115X_{4 }+ 0.0046X_{3} ) and to increase the mice
numbers of diarrhea (Y= - 0.270 + 3.8985X_{2 }+ 0.0234X_{3 }-
0.3558X_{2}^{2 }- 0.1531X_{1}). These results
match with those of orthogonal design. A new result was found that
X_{2} has a optimal dose which was not found by orthogonal design.
This experiment shows that uniform design is availability in analyzing
the compatible properties of TCM prescription.
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The researches of Artificial Neural Networks (ANN) have achieved remarkably
wide application and development since its revival in the 1980's. But the
characteristic feature -- the abilities of strong learning, strong adaptation,
large scale parallel processing, have not been shown obviously. The learning
capacity and generalization capacity of ANN are not satisfactory. Yet,
its adaptive abilities is weak and it is difficulty to construct networks.
As a result, the two major problem in ANN engineering application are researching
the optimization design of the topology structure of networks and efficient
learning algorithm.
The structure design of ANN mainly depended on experience and repeated
experiments, there was not a systematically method to determine the satisfactory
networks structure all along. A kind of heuristic construction algorithm
or prune algorithm to design networks automatically has been proposed during
recent years, but this algorithm is too complicated to be applied. In this
paper, uniform design is used to optimize NN structure design, achieving
a good effect.
The method of ANN structure optimization design based on uniform design
is introduced in this paper, making use of NN detecting pulse compressed
signal as example. Three layers structure of ANN is adopted, which consists
of input layer, hidden layer, one node in output layer. The 13 digits Barker
code signal is taken for input signal, and input layer is acted as classifier.
The output value will approach to 1 on condition that receiving code sequential
signals is pulse compressed signals. As optimizing design, Backpropagation
NN (BP) and Radius Base Function (RBF) NN are adopted respectively, and
the number of units in hidden layer, the learning step and the learning
rate are chose as three factors for design, and learning error as the performance
index.
The conclusion is derived through the experiment in 7 levels of 3 factors:
both BP NN and RBF NN should employ 52 hidden units and take 20 as learning
rate, 28 as learning step in order to achieve the best effect, that is
to say, the time is the shortest and the error is the smallest .
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The regression analysis should be used in the data analysis of the uniform
design, however, the multi-attribution determination method (MADM) only
was used in the data analysis of the uniform design for rubber formula
experiment. In view of this situation, the author had analysis the same
data with regression analysis method of the computer aided research (CAR)
system programmed by the author. The procedure and MADM was pointed, it
is impossible for other method to replace the regression analysis in the
data analysis of uniform design.
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The processing of addition glucose in Glutamic acid fermentation were
optimized by using uniform design method. The yield of Glutamic acid
increased by 0.79 g/dl, the transformation rate from glucose into Glutamic
acid increased by 2.0%.
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"Uniform design" and optimizing technology are used in the research
of additive component optimization and hence desire results are got quickly.
When use the technology in emulsifying component optimization, 5 different
compounds are pre-selected as base elements of the promoter for enhancing
the emulsifier and U_{10}^{*}(10^{8}) form (applying
form when S=5) is used designing experiment. Adding amount of every element
is used as X_{i }and dewatering experiment results as Y.
The data are treated with optimization software and the experiential model
is built. According to the direction of the model, new promoter components
are selected and used in the emulsifying experiments. New results are added
to the data and new experiential model is built. After several cycles,
desire promoter is made up and the desire delusion results are got.
The promoter is blended with two samples of polyethoxy polypropoxy
emulsifier (product name SP169) taking from different producers and the
mixtures are added to the crude oil emulsion. Compare with the demulsifier
without promoter, the dewatering ability increases obviously in the experiments
for two different Liaohe crude. When using promoter 15% in the demulsifying
components (15% promoter and 85% demulsifier), the water contain in the
demulsified crude can be reduced from 6.30% to 2.41% and 2.38% to 2.04%
for two SP169 samples. The promoter cost is much lower than the demulsifier
used in the experiments.
When use the technology in toluene emulsion project, 4 factors are
examined and the U_{7} form is used for designing experiment. In
the same way as in the crude demulsion project, after several cycles, stable
toluene-water emulsion is got quickly. Because many different toluene-water
emulsion components can meet the demand, the stable area chart is made
out and the effect of different emulsifying agent on emulsion stability
is clear.
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The test is an important link for scientific research. It can offer
a large number of evidence and examine if the result of theory analysis
is right. The method of test affects seriously its results. A simple and
definite method is adopted usually.
The concrete filled steel square tube is a composite material that
the concrete is placed into empty steel tube. It has high compressive strength.
It is suitable to be used as the columns of buildings. This kind of structure
has many advantages. For instance, the weight of a building is only about
50 percent of reinforced concrete structure. The size of a column is merely
30 percent of a reinforced concrete column suffering the same number of
outer load. The construction time is shortened considerably. The amount
of steel required to build equals to that of reinforced concrete structure;
it is about 50 percent of that of steel structure. Thus, the building costs
are reduced on the same condition. The new type of structure has a very
good future of development. So, the performance of every kind of some structural
pieces is needed to investigate next.
The experiment plan is made using the Uniform Design on concrete filled
steel square tubular columns under bias compressive loading. The number
of specimens is minimized compared to the Orthogonal Design. The data are
more reasonable. A large amount of labor, materials and money is saved.
The equation to calculate the bearing capacity of the columns is deduced.
It compensates the disadvantages of the equation of the ordinary concrete
filled steel tubular columns under bias compressive loading. The discount
coefficient on the basis of short axial columns is given by regression
analysis. It is the function of the ratio between length and radius and
eccentricity. According to the equation of the ordinary concrete filled
steel tubular columns, if the columns have different ratios between length
and radius and the same eccentricity, they have the same bearing capacity.
There is something wrong in theory with this result. It is proved in this
paper. The columns with different ratios between length and radius and
the same eccentricity have different bearing capacity. The calculation
results with this equation are very accurate compared with the experiment
results.
Through the analysis as above, the Uniform Design method can be used
in structural experiments and it is very convenient.
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This article provides a new high efficiency, time-saving access to the
process parameter establishment for the welding of new materials and new
constructions. It applies Number Theoretic Methods to the design of Slit
Type cracking test of welded joint between two different base metals, and
Linear Regression Analysis to the processing of test result. A Mathematics
Model is established based on the test to describe the relationship between
the cracking rate of welded joint and its several effective factors. Process
parameters of welding of different base metals are also given out from
the test. By using this testing method, the amount and cost of test are
reduced dramatically.
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Concerns with comparisons between Orthogonal Designs (LDs) and Uniform
Designs (UDs) from many angles, such that Star L_{2}-discrepancy,
Symmetric L_{2}-discrepancy, Centered L_{2}-discrepancy,
Unanchored L_{2}-discrepancy, D-optimality and A-optimality. The
results show that UD has lower discrepancy especially when the number of
experiments (n) equal to 8, 12, 16 (level, q = 2 in LD) and
27, and UD needs less experiments in the case that LD and UD have a close
discrepancy. On the other hand, under the seven models in this article,
LD gives better A-optimality and D-optimality in linear models or when
n is equal to 16 (q = 4 in LD) and 25.
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A mathematical model is proposed and applied to develop the polymer-modified
asphalt products with UST software. A suitable factor-level form is selected
to arrange experiments and the model is set up by regression method using
experimental data. The results of verification tests indicate that all
the expected results can be achieved by this model.
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This report had represented adaptation of uniform design method for
arranging design plan in the research of oxidation reaction of the wax
of by-product PE (polyethylene). To achieve the optimum of process by the
way of statistical optimization, and attain the goal of optimization of
developing experiment and results of forecast optimum. Moreover,
the author put forward application for fuzzy conception, and method of
uniform design to realize fuzzy simulate optimization of process, and thus
attain the goal of analyze and transcend of patent.
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UNIFORM DESIGN METHOD illustrated in this article is the result of special
need of China at the time, China's emphasis on basic theory research, Professor
Wang Yuan and Fang Kai-Tai have high research level in this field, and
cooperation of the scientists and engineer of CASC. Adopting uniform design
method has such advantages: promote tactical and technical level; achieve
best technological process and prescription; improve products performance;
achieve better products quality and reliability; high value/cost ratio
and make excellent economical and social benefit.
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The paper probes into the relation between test design and economic
benefit, and specially gives a detailed discussion on the relations among
even design, opportunity cost, product life cycle and investment risks.
Based on these relations and marketing theory, the paper advances and proves
a mathematical model of economic benefit evaluation for uniform design:
(1) p(t)
= R(t)-[C^{1}(t)
+ C^{2}(t)]
= P(t)Q(t) - [C_{1}^{1}(Q)
+ C_{2}^{1}(Q) + C_{3}^{1}(t)
+ C_{1}^{2}(Q) + C_{2}^{2}(Q)
+ C_{3}^{2}(t)]
Herein, t in Formula (1) refers to the time for the test,
and the length of t distinguishes different test design approaches:
the shortest time is uniform design, next is orthogonal design, the longest
is overall test; p(t)
in Formula (1) indicates economic benefits (profits) with different
test designs; id., R(t)
and [C^{1}(t)
+ C^{2}(t)]
denotes the corresponding revenues and costs respectively (C^{1}(t)
-- accounting cost, C^{2}(t)
-- opportunity cost); P(t), Q(t), C_{1}^{1}(Q),
C_{2}^{1}(Q), C_{3}^{1}(t),
C_{1}^{2}(Q), C_{2}^{2}(Q),
C_{3}^{2}(t) are the related price, sales
quantity, production accounting cost, sales accounting cost, R&D accounting
cost, production opportunity cost, sales opportunity cost, and R&D
opportunity cost respectively.
Besides, with five assumptions, the paper puts forward 3 models of
economic benefit evaluation for uniform design.
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The instinct features of foam lightweight material are light weight
and cellular structure. The raw materials of the foam material are cement,
sand, foam and others. As the foam existing, every composition of
the raw material shows complex nonlinear relationship to the mechanics
index of the lightweight material. In order to find out the optimum
mix proportion, the experimental design is conducted by application of
uniform design theory. Since the experimental data are too scattered
to analyze by using regression method, neural networks model is adopted
for analyzing experimental result. Back propagation (BP) algorithms neural
networks has very strong capacity of mapping nonlinearity, the artificial
neural networks model is trained by given sample of input and output, so
that it can research a simulate relationship of the input and the output.
The trained network model can forecast the mechanics index of the foam
material in any mix proportion, which have not been experimented.
So, the optimum mix proportion for the engineering request can be found
by means of the trained neural networks model. In a word, this paper
verifies that uniform design theory is reasonable for experimental design
of foam lightweight material, and the neural networks model is practical
and efficient for tackling experimental data with uniform design.
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The cultivation experiment with four factors and four levels was carried
out in oilseed rape (cv. high-oil 605) using the uniform design. Results
showed that the suitable sowing date was 28 September and the seedling
age was 47 days. The population of 135000 plants/hm^{2} and the
nitrogen of 37500kg standard fertilizer/hm^{2} were recommended
in the present experiment. Similar results were obtained from the calculated
values of the uniform design and the observed values of the direct measurement.
Therefore, there was a practical utilization of the uniform design in the
crop cultivation experiment with multiple factors and levels.
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This paper reports a study of the relationship between the driver-vehicle
closed-loop system characteristics and some vehicle parameters such as
vehicle velocity and the cornering stiffness of the front and rear wheels.
By means of the uniform design, the computer experiment for three factors,
vehicle velocity and the cornering stiffness of the front and rear wheels,
is arranged. The regression model of the closed-loop comprehensive index
with respect to the vehicle velocity and the cornering stiffness of the
front and rear wheels is established. The regression model is in the form
of quartic polynomial. Stepwise regression and the L-H algorithm, by which
a better regression subset can be generated, are used to choose the regression
variables. The testing results show that the regression model is highly
significant with the vehicle velocity and the cornering stiffness of the
front and rear wheels adopting their values over a large scale. The regression
model shows that there exist higher-order nonlinear relations among the
closed-loop comprehensive index, the vehicle velocity, and the cornering
stiffness of the front and rear wheels. The higher-order interactions among
the vehicle velocity, the cornering stiffness of the front and rear wheels
also contribute to the closed-loop comprehensive index.
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The method of uniform design with 3 factors and poly-levels is used
in this paper. The properties of the high fuction PRC material is researched.
The affecting rules of the pressure and quantity of fly ash and the fractal
dimension are investigated. The relevant material-PRC in superfine fly
ash is producted. The optimum solutions can be gotten so this paper is
important to develop the kind of materials further, and to meet the demands
of different engineering.
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With the development of social economy and the progress of civilization,
the important functions of S&T results transfer are being outstanding.
within this complicated transfer system, there are many actors relating
to science, economy, policies and rules. They act and affect each
other and restrain the transfer together. Therefore the method which
can be used to make these actors improve transfer's efficiency is very
critical and can not be ignored. After research it can be found that
as a test design method, Uniform Design is of great use value. It
can be used to provide consultation services during making decisions such
as transfer law's exploration, items selection, investment, technology
analysis and market forecast. Using Uniform Design to plan test can
reduce test cost, improve test efficiency, guide work according to test's
results and increase scientific level of decisions. Its application
can provide theoretical basis and methods of judge for decision-makers.
The research has some tries to use Uniform Design in S&T results
transfer through the discussion of basic principles, concepts concerned
and the analysis of transfer. Of course, it is theoretical try only.
There are a lot of actual questions to be solved. But all of these
may show that Uniform Design method can be used in many fields, which are
awaiting us to develop.
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The preparation conditions of the loading catalyst (cation exchange
resin as carrier) for esterification are studied by using Uniform Design.
The obtained regression equation shows well the metal salt quantity, loading
temperature, loading time and particle size of catalyst carrier effects
on the activity of catalyst for esterification. The loading catalyst prepared
by optimization experiment conditions has high activity. Using as the catalyst
of esterification reaction of ethyl alcohol and acetic acid, the percent
conversion of acetic ether is about 36.56% in one hour as the loading catalyst
quantity is 5% of acetic acid quantity in the reaction system. Compared
with non-loading cation exchange risen as catalyst, the conversion of acetic
ether rose 17.32 percent points in the same reaction conditions.
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Uniform design is superior to orthogonal design and optimality design in their table's usage. Uniform tables are flexible and need not specific assumptions. The outcomes of uniform design experiments are analyzed on various assumptions and modern statistical methods. If the response variables of the outcomes are dichotomous, for example, "yes" or "no", "qualified" or "disqualified", et la.. There are two stable states and a transition between them, which is not stable and strongly dependent on factor-level combinations. Such problems may be analyzed with aid of logistic model:
The uniform design is first proposed by the Chinese mathematicians,
and is highly valued and promoted internationally. The Northeast
Pharmaceutics Co. and the China Petrochemicals Corp. together have popularized
the technology of Uniform design and parametric optimization. Some
improvements have been obtained after using this technology by different
units, such as the Fushun Research Institute of Petroleum and Petrochemicals,
Yanshan Petrochemical Corp., Qilu Petrochemical Co., and Shanghai Research
Institute of Petroleum and Petrochemicals. After the related conference
organized by the China Petrochemicals Corp. is held in Beijing, the usage
of uniform design increases immediately and becomes more popular.
It also helps for the development in the field of Petrochemicals.
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