Factor Screening Experiments Research Articles

Advanced Methods for Constructing 𝑼𝑬(𝑺 𝟐 ) Optimal Supersaturated Designs in Factor Screening Experiments

Advanced Methods for Constructing 𝑼𝑬(𝑺 𝟐 ) Optimal Supersaturated Designs in Factor Screening Experiments

Construction of optimal supersaturated designs via generalized Hadamard matrices

A supersaturated design (SSD) is a factorial design whose run size is not enough for estimating all the main effects. Such designs have received much recent interest because of their potential in factor screening experiments. This paper first shows that the design obtained by the Kronecker sum of a balanced design and a generalized Hadamard matrix (i.e., a matrix with both itself and its transpose being difference matrices) has some nice properties. Based on these findings, some new methods for constructing -optimal SSDs via generalized Hadamard matrices are developed. Meanwhile, the non-orthogonality of the proposed designs is well controlled by the source designs. In addition, some generalized Hadamard matrices with nice properties are constructed for obtaining -optimal SSDs. The proposed methods are easy to implement and many new SSDs can then be constructed.

Multi-level and mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) constitute an important class of fractional factorial designs that could be extremely useful in factor screening experiments. Most of the existing studies have focused on balanced designs. This paper provides a new lower bound for the \(E(f_{NOD})\)-optimality measure of SSDs with general run sizes. This bound is a generalization of existing bounds since it is applicable to both balanced and unbalanced designs. Optimal multi and mixed-level, balanced and nearly balanced SSDs are constructed by applying a k-circulant type methodology. Necessary and sufficient conditions are introduced for the generator vectors, in order to pre-ensure the optimality of the constructed k-circulant SSDs. The provided lower bounds were used to measure the efficiency of the generated designs. The presented methodology leads to a number of new families of improved SSDs, providing tools for directly constructing optimal or nearly-optimal k-circulant designs by just checking the corresponding generator vector.

Formula omitted]- and [formula omitted]-optimal supersaturated designs

Supersaturated designs are useful for factor screening experiments under the factor sparsity assumption that only a small number of factors are active. The popular E(s2)-criterion for choosing two-level supersaturated designs minimizes the sum of squares of theentries of the information matrix over the designs in which the two levels of each factor appear equal number of times. Jones and Majumdar (2014) proposed the UE(s2)-criterion which is essentially the same as the E(s2)-criterion except that the requirement of factor-level-balance is dropped. Removing this constraint makes UE(s2)-optimal designs easy to construct, but it also produces many UE(s2)-optimal designs with diverse performances. It is necessary to choose better designs from them, especially those with good lower-dimensional projection properties. While E(s2)-optimal designs tend to have better lower-dimensional projections than arbitrary UE(s2)-optimal designs, they are usually very difficult to construct. We propose a secondary criterion and provide simple and systematic constructions of superior UE(s2)-optimal designs having good projection properties. We also derive conditions under which E(s2)-optimal designs are UE(s2)-optimal as well and identify several families of designs that are optimal under both criteria.

Factor screening experiments using fractional factorial split plot designs and regression analysis in developing a top-down nanomanufacturing system for recycling of welding rod residuals

A promising technology for integration of top-down nanomanufacturing is equal channel angular pressing, a process transforming metallic materials into nanostructured or ultra-fine grained materials with significantly enhanced performance characteristics. To bridge the gap between process potential and actual manufacturing output a novel prototype system identified as indexing equal channel angular pressing (IX-ECAP) was developed to capitalize on sustainable engineering opportunities of transforming spent or scrap engineering elements into key engineering commodities by recycling 4043 aluminium alloy welding rod residuals. A resolution III fractional factorial split-plot experiment assessed significance of predictors on the response, microhardness, with multiple linear regression used for model development. Five process parameters involving pressing temperature, number of passes, pressing speed, back pressure, and vibration were studied. Microhardness conversions allowed theoretical determination of grain sizes employing Hall-Petch relationships. IX-ECAP offered a viable solution whereby processing of discrete variable length work pieces proved very successful.

A cyclic construction of saturated and supersaturated designs

Supersaturated design (SSD) has received much interest because of its potential in factor screening experiments. Inspired by the construction of cyclic orthogonal designs with prime power run numbers in Plackett and Burman (1946), a method for constructing χ2(D) optimal multi-level SSDs with cyclic structure is explored in this paper. Some newly constructed designs are tabulated for practical use. Meanwhile, some of the resulting designs are shown to be also optimal under the maxχ2 criterion. In addition, we show that a sufficient condition for a balanced mixed-level saturated design to be orthogonal is that the weighted coincidence number between any two distinct rows of this design is a constant. An easier proof for the orthogonality of the Plackett and Burman designs with prime power run numbers is also presented.

Construction of supersaturated design with large number of factors by the complementary design method

Supersaturated designs (SSDs) have been widely used in factor screening experiments. The present paper aims to prove that the maximal balanced designs are a kind of special optimal SSDs under the E(fNOD) criterion. We also propose a new method, called the complementary design method, for constructing E(fNOD) optimal SSDs. The basic principle of this method is that for any existing E(fNOD) optimal SSD whose E(fNOD) value reaches its lower bound, its complementary design in the corresponding maximal balanced design is also E(fNOD) optimal. This method applies to both symmetrical and asymmetrical (mixed-level) cases. It provides a convenient and efficient way to construct many new designs with relatively large numbers of factors. Some newly constructed designs are given as examples.

New Statistic to Increase Correctness in Simulation Factor Screening Using Frequency Domain Method

Simulation models often include a large number of input factors, many of them may be unimportant to the output; justifying the use of factor screening experiments to eliminate unimportant input factors from consideration in later stages of analysis. With a large number of factors, the challenge is designing experiments so that total number of runs and consequently the required time and cost decrease while achieving a satisfactory detection rate. This article employs frequency domain method (FDM) which is applicable in discrete-event simulation models to propose a new statistic defined as the ratio of estimated signal spectrum to maximum estimated noise spectrum. The proposed method not only has the FDM advantages compared to classic screening approaches but also helps to reduce the error of associated with distinguishing important effects from unimportant ones. Furthermore, as an alternative to the existing statistics, it is shown that not only the proposed statistic does not deteriorate the power of the screening test but in some instances it helps to improve it.

Crashworthiness optimization design of honeycomb sandwich panel based on factor screening

This paper presents a crashworthiness robust optimization design of aluminum honeycomb sandwich panel with regular hexagonal core cell based on explicit finite element analysis. First, the crashworthiness of honeycomb sandwich panel and the bare honeycomb core are compared with each other in the aspects of specific energy absorption and peak crushing force. The comparative results show that honeycomb sandwich panel can absorb more energy than the bare honeycomb core, though the peak crushing force of honeycomb sandwich panel is larger. Then the factor screening experiments are carried out to find out variables that have significant effects on the crashworthiness of honeycomb sandwich panel. The ultimate purpose of factor screening experiments is to decrease the computational expense by reducing the number of design variables. Finally, the dual response surface method and the crossed array design are employed to formulate the complex robust optimization design problem. The regression expressions of specific energy absorption and peak crushing force are defined as the objective and constraint function respectively in the robust design of crashworthiness optimization.

On construction of optimal mixed-level supersaturated designs

Supersaturated design (SSD) has received much recent interest because of its potential in factor screening experiments. In this paper, we provide equivalent conditions for two columns to be fully aliased and consequently propose methods for constructing $E(f_{\mathrm{NOD}})$- and $\chi^2$-optimal mixed-level SSDs without fully aliased columns, via equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs have been obtained. Furthermore, it is proved that the nonorthogonality between columns of the resulting design is well controlled by the source designs. A rather complete list of newly generated optimal mixed-level SSDs are tabulated for practical use.

An algorithmic foldover procedure for nearly orthogonal arrays with projection

Nearly orthogonal arrays are a class of experimental designs that are useful for accommodating multiple mixed-level factors and small run sizes for factor screening experiments. Several algorithmic approaches have been used to construct such arrays using D, A, ave(S2), and most interestingly, projection properties to evaluate and classify candidate arrays. Nearly orthogonal arrays constructed to contain good projection properties make it possible to estimate higher order interactions for subsets of design columns but can introduce significant correlation as to make some factor estimates terribly imprecise. This paper proposes a method to exploit the projection properties of these nearly orthogonal arrays while guarding against correlated factor effects. The usefulness of this foldover technique is demonstrated by a live-virtualconstructive simulation experiment.

An algorithm for constructing mixed-level [formula omitted]-circulant supersaturated designs

Supersaturated design (SSD) has received much interest because of its potential in factor screening experiments. Recently, Chen and Liu [Chen, J., Liu, M.Q., 2008a. Optimal mixed-level k -circulant supersaturated designs. J. Statist. Plann. Inference 138, 4151–4157] explored the construction of E ( f NOD ) -optimal mixed-level SSDs using k -cyclic generators. As an extension, this paper further studies how to construct optimal mixed-level k -circulant SSDs under the χ 2 ( D ) criterion. A sufficient condition under which the generator vector produces a χ 2 ( D ) -optimal SSD is obtained, and some other properties of the resulting SSDs are investigated. Note that, the construction of optimal mixed-level k -circulant SSDs under the χ 2 ( D ) criterion is much more complicated than that under E ( f NOD ) . Hence, an algorithm for finding generators of (nearly) χ 2 ( D ) -optimal designs is provided, and many (nearly) χ 2 ( D ) -optimal mixed-level SSDs are then constructed.

Optimal mixed-level supersaturated design with general number of runs

Supersaturated design (SSD) has received much interest because of its potential in factor screening experiments. Most studies focused mainly on balanced designs. This paper considers SSDs with general run sizes. The lower bounds of the χ 2 proposed by [Yamada, S., Lin, D.K.J., 1999. Three-level supersaturated designs. Statist. Probab. Lett. 45, 31–39] and E ( f NOD ) proposed by [Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003b. Optimal mixed-level supersaturated design. Metrika 58, 279–291] are obtained along with the sufficient and necessary conditions for achieving them, respectively. Some construction methods for optimal mixed-level SSDs with general run sizes are proposed. Many new designs are also tabulated for practical uses.

Construction of multi-level supersaturated design via Kronecker product

Supersaturated design is a kind of fractional factorial design in which the number of runs is not enough to estimate all the main effects. It has recently received much interest because of its potential in factor screening experiments. In this paper, we propose a method for constructing multi-level supersaturated design via Kronecker product. The resulting design has the good properties such as: its χ 2 value can be computed exactly and the upper bound on the dependency among all design columns is under control. Illustrating examples followed by some detailed discussions are presented.

Projection properties of some orthogonal arrays

In factor screening experiments, one generally starts with a large pool of potentially important factors. However, often only a few of these are really active. Under this assumption of effect sparsity, while choosing a design for factor screening, it is important to consider projections of the design on to smaller subsets of factors and examine whether the projected designs allow estimability of some interactions along with the main effects. While the projectivity properties of symmetric 2-level and a few 3-level fractional factorial designs represented by orthogonal arrays have been studied in the literature, similar studies in respect of asymmetric or, mixed level factorials seems to be lacking. In this paper, we initiate work in this direction by providing designs with good projectivity properties for asymmetric factorials of the type t × 2 m based on orthogonal arrays. We also note that the results of Cheng (1995) regarding the projectivity of symmetric two-symbol orthogonal arrays do not necessarily extend to arrays with more than two symbols.

A simulation-based evaluation of the cost of cycle time reduction in Agere Systems wafer fabrication facility—a case study

With the vast amount of capital invested in the semiconductor industry and the high level of competitiveness, semiconductor manufacturers are continuously searching for new methods to gain more market share and increase their profits. New technologies are continuously in demand, and are pushing the semiconductor manufacturers to seek better market position through constant improvement in their processes and products. Agere Systems is one of the leading companies in the microelectronics industry. This paper presents a comprehensive framework for strategic capacity expansion of production equipment at Agere Systems wafer fabrication facility (fab). We integrate simulation modeling, design of experiments, statistical analysis, and economic justification tools to aid in this highly complex decision-making process. We compliment the framework with a particular implementation. The procedure involved studying the factors in the fab that significantly impact the cycle times of the four main technologies that are currently in process and will stay in demand for at least the next few years. Moreover, the goal was to quantify the economic impact of reducing cycle times. To accomplish the objectives of this research, a valid simulation model of the production line of the fab was built. The production equipment (tool sets) that significantly influence production cycle time were identified through factor screening experiments. Based on these factors, several scenarios involving the acquisition of additional tools, aimed at cutting down cycle times, were identified and the operating characteristics curves were constructed. These characteristics curves were used to relate cycle time to production volume capabilities. Finally, an economic analysis was conducted to evaluate the return on investment in additional tools. The outcome of the case study was the identification of bottleneck facilities in the fab that have a significant impact on the cycle time of the four technologies. In the economic analysis, the cost of adding machines to these facilities was compared with the benefits resulting from the increase in production capacities.

Some theory and the construction of mixed-level supersaturated designs

Supersaturated design is a form of fractional factorial design and has recently received much interest because of its potential in factor screening experiments. This paper mainly concerns the existing criteria for mixed-level designs, i.e. the χ 2( D) criterion (for design D) proposed by Yamada and Matsui (J. Statist. Plann. Inference 104 (2002) 459), the E( f NOD) criterion proposed by Fang et al. (Metrika 58 (2003b) 279) and the minimum moment aberration (MMA) proposed by Xu (Statist. Sinica 13 (2003) 691). A lower bound of χ 2( D) is obtained along with the sufficient and necessary condition for achieving it. The connections between χ 2( D) and other criteria are discussed. Especially, it is shown that χ 2( D) and the second power moment K 2( D) are in fact equivalent to each other and hence the two criteria, χ 2( D) and MMA, share the same optimal supersaturated designs, which can be constructed from saturated orthogonal arrays and other supersaturated designs.

Construction of optimal supersaturated designs by the packing method

A supersaturated design is essentially a factorial design with the equal occurrence of levels property and no fully aliased factors in which the number of main effects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently unrelated kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersaturated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties.

A flexible simulation tool for manufacturing-cell design, II: response surface analysis and case study

We present a two-phase approach to design and analysis of manufacturing cells based on simulated experimentation and response surface methodology using a general manufacturing-cell simulation model. The first phase involves factor-screening simulation experiments to identify design and operational factors that have a significant effect on cell performance as measured by a comprehensive annual cost function. In the second phase of experimentation, we construct simulation (response surface) meta-models to describe the relationship between the significant cell design and operational factors (the controllable input parameters) and the resulting simulation-based estimate of expected annual cell cost (the output response). We use canonical and ridge analyses of the estimated response surface to estimate the levels of the quantitative input factors that minimize the cell's expected annual cost. We apply this methodology to an assembly cell for printed circuit boards. Compared to the current cell operating policy, the simulation metamodel-based estimate of the optimum operating policy is predicted to yield average annual savings of approximately $425 000, which is a 20% reduction in annual cost. In a companion paper, we detail the structure and operation of the manufacturing-cell simulation model.

A General Method of Constructing E(s2)-Optimal Supersaturated Designs

Summary There has been much recent interest in supersaturated designs and their application in factor screening experiments. Supersaturated designs have mainly been constructed by using the E(s2)-optimality criterion originally proposed by Booth and Cox in 1962. However, until now E(s2)-optimal designs have only been established with certainty for n experimental runs when the number of factors m is a multiple of n-1, and in adjacent cases where m=q( n-1) +r (|r|⩽2, q an integer). A method of constructing E(s2)-optimal designs is presented which allows a reasonably complete solution to be found for various numbers of runs n including n,=8 12, 16, 20, 24, 32, 40, 48, 64.