Simplex Design Research Articles

Statistical designs for the optimization of the nitrogen-phosphorus gas chromatographic detector response

Statistically designed and evaluated factorial experiments were used to approach optimum operating conditions for the rubidium bead type of nitrogen--phosphorus gas chromatographic detector to detect small quantities of nitrogen compounds in a hydrocarbon matrix, and to study the effects of the operating variables. Simplex designs were used to define the optimum set of conditions. The test variables were hydrogen pressure, air pressure, and bead heating current. Only two 2/sup 3/ factorial designs and ten simplex vertices were needed to accomplish these purposes. Detector responses for 12 nitrogen and 12 hydrocarbon compounds were compared with both the nitrogen--phosphorus and flame ionization detectors. The average value of the ratio of nitrogen to hydrocarbon response with the nitrogen-sensitive detector was about 1000:1. Chromatography of several fractions from a chemical extractive fractionation of a coal liquefaction product oil demonstrated the discrimation capability of the nitrogen--phosphorus detector at optimum conditions. 4 figures, 4 tables.

Comparison of Simplex Designs for Quadratic Mixture Models

Comparison of Simplex Designs for Quadratic Mixture Models

Comparison of Simplex Designs for Quadratic Mixture Models

Designs for quadratic regression are considered when the possible values of the controlable variable are mixtures x = (x 1, x 2, …, x q + 1) of nonnegative components x i with Σ q + 1 1 x i = 1. The designs that are optimum with respect to the D-, A-, and E-optimality criteria are compared in their performance relative to these and other criteria. Computational routines for obtaining these designs are developed, and the geometry of optimum structures is discussed. Except when q = 2, the A-optimum design is supported by the vertices and midpoints of edges of the simplex, as is the case for the previously known D-optimum design. Although the E-optimum design requires more observation points, it is more robust in its efficiency, under variation of criterion: but all three designs perform reasonably well in this sense.

Simplex optimization of carbon electrodes for the hydrogen—oxygen membrane fuel cell

Progress in, and the results of studies on the preparation of optimum carbon electrodes with a platinum catalyst are presented. The electrodes are of the bilayer type. Polyethylene has been used as the binding agent. The simplex design method of Spendley, Hext and Himsworth has been applied. The temperature, pressure and time of heating and finally the content of polyethylene and carbon in both layers of the electrode have been optimized.

Simplex optimization of carbon electrodes for the hydrogen—oxygen membrane fuel cell

Progress in, and the results of studies on the preparation of optimum carbon electrodes with a platinum catalyst are presented. The electrodes are of the bilayer type. Polyethylene has been used as the binding agent. The simplex design method of Spendley, Hext and Himsworth has been applied. The temperature, pressure and time of heating and finally the content of polyethylene and carbon in both layers of the electrode have been optimized.

Application of modified Rosenbrock's method for optimization of nutrient media used in microorganism culturing.

The Rosenbrock's procedure has been modified for optimization of nutrient medium composition and has been found to be less tedious than the Box-Wilson method, especially for larger numbers of optimized parameters. Its merits are particularly obvious with multiparameter optimization where the gradient method, so far the only one employed in microbiology from a variety of optimization methods (e.g., refs, 9 and 10), becomes impractical because of the excessive number of experiments required. The method suggested is also more stable during optimization than the gradient methods which are very sensitive to the selection of steps in the direction of the gradient and may thus easily shoot out of the optimized region. It is also anticipated that other direct search methods, particularly simplex design, may be easily adapted for optimization of medium composition. It is obvious that direct search methods may find an application in process improvement in antibiotic and related industries.

Simplex Optimization of Reaction Yields

The sequential simplex algorithm, an efficient optimization strategy, rapidly improved reaction yield as a function of time and temperature in the synthesis of pi-C(5)H(5)Mo(CO)(2)CSN(CH(3))(2). The work demonstrates the feasibility and efficiency of the simplex design and suggests its application and usefulness in other syntheses.

Use of simplex experimental designs in detergent formulation

AbstractResearch in the field of detergent formulation is deeply involved in the optimization of numerous detergent systems. The use of experimental designs in looking for optimal formulations can save a large amount of laboratory time. This paper describes a coherent strategy of use for a special class of simplex designs adapted from Scheffé's work. The technique applies to multicomponent systems, as they are generally met in detergent formulation. Proper designs provide polynomial equations and, in addition, convenient graphical representations that enable the chemist to predict responses for a wide range of mixtures. Calculations are easy. A computer may help for routine application of these designs, but is not strictly required. The method is illustrated by a problem in the solubilization of light duty liquid detergents and at the same time provides arguments for better use of statistics in formulation. Finally, this simplex method is positioned among some other statistical techniques dealing with mixture designs.

Simplex design of the experiment for developing optimum conditions of analysis of thiphen and diprophen by two-side differential spectrophotometry

Simplex design of the experiment for developing optimum conditions of analysis of thiphen and diprophen by two-side differential spectrophotometry

Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation

A technique for empirical optimisation is presented in which a sequence of experimental designs each in the form of a regular or irregular simplex is used, each simplex having all vertices but one in common with the preceding simplex, and being completed by one new point. Reasons for the choice of design are outlined, and a formal procedure given. The performance of the technique in the presence and absence of error is studied and it is shown (a) that in the presence of error the rate of advance is inversely proportional to the error standard deviation, so that replication of observations is not beneficial, and (b) that the “efficiency” of the technique appears to increase in direct proportion to the number of factors investigated. It is also noted that, since the direction of movement from each simplex is dependent solely on the ranking of the observations, the technique may be used even in circumstances when a response cannot be quantitatively assessed. Attention is drawn to the ease with which second-order designs having the minimum number of experimental points may be derived from a regular simplex, and a fitting procedure which avoids a direct matrix inversion is suggested. In a brief appendix one or two new rotatable designs derivable from a simplex are noted.

Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation

A technique for empirical optimisation is presented in which a sequence of experimental designs each in the form of a regular or irregular simplex is used, each simplex having all vertices but one in common with the preceding simplex, and being completed by one new point. Reasons for the choice of design are outlined, and a formal procedure given. The performance of the technique in the presence and absence of error is studied and it is shown (a) that in the presence of error the rate of advance is inversely proportional to the error standard deviation, so that replication of observations is not beneficial, and (b) that the “efficiency” of the technique appears to increase in direct proportion to the number of factors investigated. It is also noted that, since the direction of movement from each simplex is dependent solely on the ranking of the observations, the technique may be used even in circumstances when a response cannot be quantitatively assessed. Attention is drawn to the ease with which second-o...

Optimum Designs in Regression Problems, II

Extending the results of Kiefer and Wolfowitz [10], [11], methods are obtained for characterizing and computing optimum regression designs in various settings, and examples are given where $D$-optimum designs are computed. In Section 1 we introduce the main definitions and notation which will be used in the paper, and discuss briefly the roles of invariance, randomization, number of points at which observations are taken, and nonlinearity of the model, in our results. In Section 2 we prove the main theoretical results. We are concerned with the estimation of $s$ out of the $k$ parameters, extending an approach developed in [10] and [11] in the case $s = k$. There is no direct way of ascertaining whether or not a given design $\xi^{\ast}$ is $D$-optimum for (minimizes the generalized variance of the best linear estimators of) the $s$ chosen parameters, and T+eorems 1 and 2 provide algorithms for determining whether or not a given $\xi^\ast$ is $D$-optimum. If all $k$ parameters are estimable under $\xi^\ast$, we can use (2.7) to decide whether $\xi^\ast$ is $D$-optimum, while if not all $k$ parameters are estimable we must use the somewhat more complicated condition (2.17) (of which part (a) or (b) is necessary for optimality, while (a), (c), or (d) is sufficient). An addition to Theorem 2 near the end of Section 3 provides assistance in using (2.17) (b). Theorem 3 of Section 2 characterizes the set of information matrices of the $D$-optimum designs. In Section 3 we give a geometric interpretation of the results of Section 2, and compare the present approach with that of [10]. In the case $s = k$, the present approach reduces to that of Section 5 of [10] and of [11]. When $1 < s < k$, we obtain an algorithm which differs from that of Section 4 of [10] and which appears to be computationally easier to use. When $s = 1$, the results of the present paper are shown to reduce to those of Section 2 of [10]; in particular, we obtain the game-theoretic results without using the game-theoretic machinery of [10]. In Section 4 we determine $D$-optimum designs for the problems of quadratic regression on a $q$-cube and polynomial regression on a real interval with $1 < s < k$. Part II of the paper is devoted entirely to the determination of $D$-optimum designs for various problems in the setting of simplex designs considered by Scheffe [12]. Various unsolved problems are mentioned throughout the paper. Further examples will be published elsewhere.

Use of the Simplex Design in the Study of Joint Action of Related Hormones

In certain toxicological studies the term joint action has come to take a special meaning. If members of a group of related compounds all cause death of an organism when administered separately the simultaneous action of these substances is called their joint action. Bliss (1939) first discussed the analysis of data obtained in this manner. From this time the problem has been examined in terms of tolerance distribution theory and developed in relation to probit analysis (Finney, 1952). Plackett and Hewlett (1951) have extended the tolerance distribution theory to different theoretical forms of joint action and developed a set of mathematical models, each of which is based on many assumptions and is very difficult to fit to experimental data. Fisher (1954) has shown that parameters of the binomial distribution may be estimated without tolerance distribution assumptions. The aim of the present paper is to show that the study of joint action by means of an appropriate experimental design-the simplex designallows ready interpretation of experimental data with no reference to a joint tolerance distribution, and no further assumptions than normally required in quantal analysis. The method is also appropriate without modification to the study of joint action of substances eliciting a graded response simply by applying the standard estimation procedures. Examples will be drawn from the study of the action of oestrogens on the vagina of the ovariectomized mouse. The quantal response in this case is cornification of the vaginal epithelium.