Is Treatment Adjustment in SBIBD Necessary

An arrangement of $v$ varieties or treatments in $b$ blocks of size $k, (k < v),$ is known as a balanced incomplete block design if every variety occurs in $r$ blocks and any two varieties occur together in $\\lambda$ blocks. These parameters obviously satisfy the equations \\begin{equation*}\\tag{1} bk = vr\\end{equation*}\\begin{equation*}\\tag{2} \\lambda(v - 1) = r(k - 1)\\end{equation*} Fisher [1] has also proved that the inequality \\begin{equation*}\\tag{3} b \\geq v, \\quad r \\geq k\\end{equation*} must hold. If $v, b, r, k$ and $\\lambda$ are positive integers satisfying (1), (2) and (3), then a balanced incomplete block design with these parameters possibly exists, but the actual existence of a combinatorial solution is not ensured. These conditions are thus necessary but not sufficient for the existence of a design. Fisher and Yates in their tables [2] have listed all designs with $r \\leq 10$ and given combinatorial solutions, where known. A balanced incomplete block design in which $b = v$, and hence $r = k$ is called a symmetrical balanced incomplete block design. The impossibility of the symmetrical designs with parameters $v = b = 22, r = k = 7, \\lambda = 2$ and $v = b = 29, r = k = 8, \\lambda = 2$ was first demonstrated by Hussain [3], [4] essentially by the method of enumeration. The object of the present note is to give an alternative simple proof of the impossibility of these designs and to show that the only unknown remaining symmetrical design in Fisher and Yates' tables, viz. $v = b = 46, r = k = 10, \\lambda = 2,$ is definitely impossible. Symmetrical designs with $\\lambda \\leq 5, r, k \\leq 20$, which are impossible combinatorially, are also listed.

Several construction methods have been introduced to build the elements of balanced incomplete block designs (BIBDs) for specific parameters, with different techniques suggested for testing their existence. There is still no general technique to determine the existence of the designs that have been realized. In this study the efficiencies of some given automorphic symmetric balanced incomplete block designs (AUSBIBD) formed by sum construction method have been presented in details alongside existence and non-existence of such designs. The process involved the application of sum construction method to give new designs of parameters D (v, b, λ1+λ2) and an application of Bruck Ryser Chwola theorem extensively. A test design constructed using the described method was found to be existing with an efficiency of 76.38%. Sum constructed designs are more applicable in agricultural fields as witness in case of KEFRI-Kenya.

Motivated by the field experimental designs in agriculture, the theory of block designs has been applied to several areas such as statistics, combinatorics, communication networks, distributed systems, cryptography, etc. An explicit formula and its fast computational algorithm for a class of symmetric balanced incomplete block designs are presented. Based on the formula and the careful investigation of the modulus multiplication table, the algorithm is developed. The computational costs of the algorithm is superior to those of the conventional ones.

The design and analysis of an efficient load balancing algorithm employing the symmetric balanced incomplete block design

Abstract This paper illustrates, through a special case, an effect of non-independent numerators on F-test's in an analysis of variance. Symmetrical balanced incomplete block designs are considered with "blocks" a random effect and no interaction. The joint density of the adjusted sums of squares is determined under the null hypothesis of no treatment effect. Given the result of testing for the block effect, the conditional probability of a Type I error when testing the null hypothesis concerning the treatment effect is evaluated for a number of designs.

The Taguchi and analysis of variance (ANOVA) methods were applied to investigate the effects of the structural and operational parameters on the heat transfer performance of a vertical sinter cooling packed bed. The analysed parameters were the gas flow rate, the air inlet temperature, the sinter inlet temperature, the cooling bed diameter and the cooling bed height, all of which contain three levels. The purpose was to improve the heat transfer performance of a vertical sinter cooling bed. A numerical analysis model was established to assess the heat transfer performance with respect to the varying parameters and their different levels. This mathematical model was validated by using data from practical industrial processes. The Taguchi method for the L27 (35) orthogonal design experiment was selected to evaluate the impacts of the design parameters on the heat transfer performance and to acquire the optimum combination of parameters. The analysis of variance was applied to assess the impact weights and the order of significance of the design parameters. The results show that the sinter inlet temperature and cooling bed diameter have great influences and impact the exergy of the wasted heat recovery by 61.65% and 23.31%, respectively. However, the gas flow rate and the air inlet temperature have small effects on the response. Furthermore, the air and sinter inlet temperatures have the most significant impacts on the efficiency of heat transfer by 68.83% and 23.31%, respectively. The optimal parameter combination (A1B1C3D3E3) was obtained, and the optimal results were validated by confirmation tests.

In this study, the effect of various parameters on the power coefficient (CP) performance of a vertical axis wind turbine (VAWT) was optimized using the Taguchi method. The optimization was conducted with the Taguchi method, employing three control factors: blade-spoke connection (BSC), turbine blade pitch angle (β), and turbine blade pitch direction (φ). Subsequently, the analysis of variance (ANOVA) method as employed to determine the contribution ratio of each control factor. Regression analysis (RA) was applied to develop an empirical equation predicting the CP of the VAWT, incorporating the control factors. The results indicated an optimal parameter configuration of BSC = 0.5c, β = 2°, and φ = (-), which maximizes the system performance. The performance of the optimal model was observed to exceed that of the conventional VAWT (B1) by 5.82%. Using the ANOVA method, the contribution of parameters on the CP performance of the VAWT was ranked as follows: φ > BSC > β. The φ parameter has the most significant effect at 82.07%, whereas the β parameter exhibits the least effect of 1.17%. Moreover, the predictive accuracy of the developed regression model was validated, yielding R2 values of 0.9221 for the training data and 0.9908 for the test data.

This research is about the roundness characteristics of the turning process parameters on St 42 material using the Taguchi and ANOVA (Analysis of Variance) methods. A cylindrical product requires a good roundness value to ensure its quality. One of the factors that determine the quality of these products is the machining process parameters. In this study, optimization of the turning process parameters using the Taguchi and ANOVA methods was carried out on the roundness characteristics of the workpiece. The workpiece used is Steel St 42. The lathe used is the Krisbow Precision Gear Head Machine 410X1000MM. After data analysis using the Taguchi method, the parameters with the optimum roundness value of 7.385 µm were 510 rpm spindle rotation, 1mm depth of cut, and coolant opening valve. The results of the ANOVA analysis show the percent contribution of each parameter, respectively, as follows: spindle rotation 26.97%, depth of cut 24.82%, and coolant 38.63%.

Some new construction methods of the optimum chemical balance weighing designs and pairwise efficiency and variance balanced designs are proposed, which are based on the incidence matrices of the known symmetric balanced incomplete block designs. Also the conditions under which the constructed chemical balance weighing designs become A-optimal are also been given.

Cloud computing is one of the recent emerging technologies. Heavy data sharing among multiple users is an open issue. Data sharing in cloud computing enables multiple participants to freely share the group data, which improves the efficiency of work in cooperative environments and has widespread potential applications. However, how to ensure the security of data sharing within a group and how to efficiently share the outsourced data in a group manner are formidable challenges. In this paper we focused on security issues with the help of key agreement protocols to perform efficient data sharing in cloud environment. We proposed a novel block design based key agreement method in support of symmetric balanced incomplete block design (SBIBD). The main objective of this method is to supports multiple participants, which can flexibly extend the number of participants in a cloud environment according to the structure of the block design. Based on the proposed group data sharing model, we present general formulas for generating the common conference key K for multiple participants. In addition, the fault tolerance property of our protocol enables the group data sharing in cloud computing to withstand different key attacks, which is similar to Yi’s protocol.

This paper aims to present the analysis of a part manufactured in three shifts, which has a specific characteristic dimension, using DFSS (Design for Six Sigma) ANOVA (Analysis of Variance) method. In every shift, the significant characteristic, “SC”, dimension should be produced within the given tolerance. The question that arises is: “Does the shift have any influence on the “SC” dimension realization?” By using the one way ANOVA method, one can observe the variation between the means of each of the three shifts. Afterwards, specific action can be undertaken to adjust, if necessary, the differences between the shifts.

: It is shown that among binary and connected block designs with and without equal block sizes and equal replications only a symmetric balanced incomplete block design produces a completely symmetric information matrix for the treatment effects whenever the number of blocks is equal to the number of treatments and the number of experimental units is a multiple of the number of treatments. This result contains a known result as a special case.

Symmetric balanced incomplete block designs (SBIBD) and Hadamard matrices

All the symmetric balanced incomplete block (SBIB) designs have been characterized and a new generalized expression on parameters of SBIB designs has been obtained. The parameter b has been formulated in a different way which is denoted by b i , i = 1, 2, 3, associating with the types of the SBIB design D i . The parameters of all the designs obtained through this representation have been tabulated while corresponding them with the suitable formulae for the number of blocks b i and the expression S i for the convenience of practical users for constructional methods of certain designs, which is the main theme of this paper.

A Machine learning approach to predict re-key in wireless sensor network using symmetric balanced incomplete block designs