Conference Matrices for Optimizing and Applications

Researchers use their accumulated knowledge as a base to create and embody new functions, improve the efficiency of existing mechanisms, and reduce the side effects of existing technologies. There is also the problem of global warming. A researcher tries to find a feasible combination of conditions from the conditions taken up. In this case, he wants to catch the overall optimum condition by partial experiments. It is an experimental design method. As a new experimental design method, we will introduce a method of applying the design matrices to the conference C matrices and analyzing it as a coefficient graph by regression analysis of the observed values. This chapter presents experimental procedures and analysis methods using actual cases. In addition, the method for adjusting the research goals from the analysis results is also presented.

Conference matrices were first used in Belevitch’s (Belevitch, Electric Communication 26:231–244, 1950) telephone network diagnostics. Hadamard matrices and mixed orthogonal arrays are vertical matrices. However, the conference matrices is　the square orthogonal matrices. In principle, we will explain the generation mechanism of the conference matrices that can be constructed every two rows.

Regression analysis is performed on the experimental observation data according to the conference matrices. Practical researchers have few opportunities to learn regression analysis. Therefore, in this chapter, we will explain the analysis process and statistical implications of a series of regression analyses using case studies.

There are times when it is not possible to obtain the experimental results of research as a continuous quantity. It is expressed as the passing rate, which classifies the products into acceptable and unacceptable products, and the non-defective product ratio, which classifies the non-defective products into non-defective products. In addition, there are often missing samples, conditions that cannot be synthesized, and samples that exceed the measurement range. Here, we show (1) classification value/rank data (Mori, T.: Application and mathematics of the Taguchi method. Chapter 13.4: Analysis of classification value and ranking data, p. 264, Mori Consulting Office (2005)), (2) data (Mori, T.: Pocket Guidebook, Processing method when data cannot be obtained (Processing of missing data), pp. 26–27. Mori Consulting Office (2009)), (3) missing values, and (4) functional window analysis method.

Using the conference matrices reduces the number of experiments to 1/3 to 1/2 of the mixed orthogonal array. About 90% of the research subjects are quality improvement and process improvement such as new product development, productivity improvement, elimination of pollutants, and defective product countermeasures by changing conditions. This chapter explains quality improvement research with the case studies related to 90%.

Hadamard orthogonal tables correspond to interactions in linear graphs. But it only represents part of the interaction. Mixed orthogonal arrays are highly confounding to the main effects of interactions. These experimental design methods capture the main effects, but cannot grasp the interaction. About 62% of the optimal conditions for the mixed system orthogonal array are below the best experimental no. value. This chapter proposes a method for selecting optimal conditions that considers both interactions and main effects. We compared the no. best value condition (a) and the best level condition (b) of the factor effect in the orthogonal array experiment and designed it assuming that the difference level factor was strongly related to the interaction and the common level factor to the main effect. The method is described in detail as [a, b] analysis.

We replace the conventional mixed orthogonal matrices with a conference matrices of square orthogonal matrices and perform [a,b] analysis. Difference-level factors are assumed to be strongly related to interactions, and common-level factors to main effects. As a model example, we report optimization research on the pact motors.

We replace the conventional mixed orthogonal table with the square orthogonal conference matrix and design it by [a,b] analysis. By comparison, we assume that difference-level factors are strongly related to interactions, and common-level factors to main effects. As a model example, we report optimization research on the bonding of aluminum plates and double-sided tape.

The purpose of this report was to chemically treat dye-colored wastewater and find the conditions for sedimentation and separation in a short period of time through beaker experiments rather than using actual equipment shows how to obtain.

The subject of assembling the tool box to equip the truck is to shorten the welding time and reduce the welding strain. Therefore, in order to replace the welding of steel plates with adhesives, we conducted research to optimize the bonding conditions. In addition to changing the adherent to stainless steel plate, which is often used for tool boxes, parameter design is used to optimize for the bonding conditions between stainless steel plates that suit the production site by taking into account the bonding pressure and pressurization time for layout 3 levels and 6 factors are taken up as control factors. Conventionally, the orthogonal table L18 is used, but the conference matrices C₆, which can reduce the number of experiments to 1/3, is adopted. The bonding area is the signal factor, the temperature deterioration is the noise factor, and the tensile shear load is the characteristic value. We were able to find bonding conditions that were superior to the best conditions in the first and second experiments in terms of tensile shear load and its variation.

We have developed a method of joining parts by applying mechanical crimping using rollers, aiming for further weight reduction and cost reduction as a processing method that replaces this bolt fastening. This technology satisfies torque resistance and pull-out load resistance by crimping a part of the crimped side part into the groove provided on the crimped side part by low load and short processing time. The conference matrices C₁₀ was used in research on the production of the processing machine conditions and part dimensions, responding to the application of this technology with the filling rate. When compare the conventional way L₃₆ and C₁₀, C₁₀ will reduce 1/3.6 for the experiment numbers, times, cost, and investment.

In recent years, there has been a demand for improved fuel efficiency and reduced CO₂ emissions by lowering the resistance of power mechanisms. In order to prevent global warming, we will analyze the resistance of the mechanical parts (Gear Box) provided for the power mechanism and explore the possibility of energy saving. A special paid program was used for the analysis, and 1 factor of 2-level and 5 factors of 3-level were analyzed. Normally L₁₈ with 18 experiments is applied, but conference matrices C₆ with 6 experiments and 1/3 was applied. We were able to analyze the causal relationship, and we were able to search for low resistance conditions for the current conditions. We are sure the conference matrices contributed SDGs.

At present, it is necessary to realize low cost as a concrete technical means to cope with global warming. From this point of view, the “wet solar cell” is expected to be a “next-generation power generation system” Among wet-type solar cells, it is that electric power can be obtained simply by exposing a semiconductor film of titanium oxide coated with pigment (dye) to the electrolyte. This dye-sensitized solar cell (dye-sensitized cell) can realize a low-cost battery system. Currently, the issues of dye-sensitized solar cell research are the optimization of high-efficiency electrode materials, the development of more inexpensive dyes, and the improvement of durability. This chapter introduces a case study applying the conference matrices.

When optimizing using an orthogonal array, it is desirable to consider the various relationships between factors and assign many factors. Two-level orthogonal array can be assigned many factors. Three-level orthogonal array has the advantage of obtaining intermediate information on the level. For this reason, mixed type orthogonal arrays L₁₈ (2¹3⁷), L₃₆ (2¹¹3¹³), etc. are still used today (Bose and Bush in Ann Mathe Stat 23:208–524, 1952). The response of these mixed type orthogonal arrays is logarithmically converted to the SN ratio and sensitivity for optimization. This way also is called Taguchi methods (Taguchi, 1984, 1988). Parameter design with a two-step procedure for predicting the optimum conditions is performed from this SN ratio and sensitivity with factor effect graph. However, this method has two problems (1) and (2).

The confirmation value (b) had been expected as the optimum condition with minimized variation. But, there are the problems that this confirmation value (b) is worse than the best value (a) of the SN ratio of the orthogonal array used for estimation will appear for 62% of cases (Mori, 2018, 2020). So, the prediction accuracy for the optimum conditions is poor. In order to improve these problems (1) and (2), this paper report will propose a new method to apply the conference matrices to the layout and the coefficient figure to the analysis to the row data. This report provides an easy-to-understand explanation that the conference matrices reduces the number of experiments and improves prediction accuracy using the coefficient of variation, especially for researchers. We are sure our proposed ways to reduce the experimental number and the period and cost almost to 1/3–1/2 with the higher accuracy for optimizing, so we will recommend as the specific ways to solve the subjects of the Sustainable Development Goals. Especially it will contribute to create the effective countermeasures to Global Warning that has been requested immediately to take the actions to reduce the increasing temperature.