DesignDirector Version 1.0(E)

Transcription

1 Statistical Design Support System DesignDirector Version 1.0(E) User s Guide NHK Spring Co.,Ltd. Copyright NHK Spring Co.,Ltd All Rights Reserved.

2 Copyright DesignDirector is registered trademarks of NHK Spring Co., Ltd. Microsoft Windows, Windows 95, Windows 98 Windows NT, Excel Word are registered trademarks of the Microsoft Corporation NHK Spring Co., Ltd. All rights reserved. DesignDirector Software and all accompanying documentation are copyrighted with all rights reserved by NHK Spring Co., Ltd. The Software and the documentation may not be licensed or distributed except by NHK Spring Co., Ltd. and its designated suppliers. Unauthorized copying, duplicating, selling, licensing, or otherwise distributing the Software or documentation is a violation of the Copyright Law.

3 Contents Contents 1.About DesignDirector Features of DesignDirector Operation Flow and Functions 3 2.Operating Instructions Investigation of the Problem Setting Factors and Interactions Setting the Factors Setting Interactions Setting the Number of Error Columns Operation for Sample Problem Orthogonal Array Changing Assignments Assigning Mixed Numbers of Levels to the Orthogonal Array Operation for Sample Problem Inputting the Response Values Prescription Table Obtaining the Response Values Inputting the Response Values Operation for Sample Problem ANOVA (Analysis of Variance) Pooling Operation Displaying the Supplementary Tables Supplementary Graph Comparison Graph of Response Values and Estimated Values Reanalysis Sensitivity Analysis Sensitivity Values, Sensitivity Equations Sensitivity Graph 53

4 Contents 2.8 Evaluation of Dispersion Factor Dispersion Dispersion Equation Response Dispersion Component Table Dispersion Graph Optimization Calculation Setting the Initial Values User Equations Setting the Objective Function and Behavioral Constraint Function Optimization Calculation Operations for the Sample Problem Theory of Statistical Design Support System Introduction Statistical Design Support System Flow of Statistical Design Support System Effectness Analysis Design of Experiments Structural Analysis Analysis of Variance and Estimation Expression Sensitivity Analysis Re-analysis Evaluation of Dispersion Optimum Calculation Optimum Calculation of Continuous Variables Optimum Calculation for Discrete Variables Application Analysis of Effectness Design of Experiments Finite Element Analysis Analysis of Variance Estimation Expression 91

5 Contents Sensitivity Analysis Evaluation of Dispersion Optimum Calculation Conclusion References 95

6 About DesignDirector 1.About DesignDirector In structural analysis, the work normally takes the form of a repeated cycle. The strength and rigidity of a given shape are analyzed, and if the design targets are not met, the values of the design factors are changed and the structural analysis is carried out again. And when there are multiple design targets, as when the design must reduce the weight of a product at the same time that it satisfies strength and rigidity requirements, the amount of trial and error work increases even more. One effect of carrying out structural design by this method is that both the worthiness of the design and the length of the design period depend to a large extent on the talents of the designer. And where there is dispersion in the design factors, which is accompanied by variation in the responses being analyzed, it becomes critical to have a stable, robust design that ensures satisfactory responses even as the design factors vary. Statistical design support systems [1][2] have been developed to create optimum designs that are not influenced by the designer's experience and to respond to the demand for more efficient and precise design work. DesignDirector embodies such a statistical design support system in the form of software. Recently in the structural analysis field, many designers are dealing with the problem of non-linear phenomena. Almost all the current optimum design systems use a method where structural analysis and sensitivity analysis are combined in the optimization calculation loop. As a result, these systems bring with them certain problems. The optimization of non-linear phenomena requires a huge number of calculations, and a corresponding amount of processing time, which decreases efficiency. And the systems are limited in regard to problems they can deal with. However, in DesignDirector, the structural analysis and optimization functions are independent of one another. At the same time, because DesignDirector can use a response surface equation derived by ANOVA (analysis of variance) for the objective function or the constraint function, it can calculate a practical optimization even for non-linear problems. Also, if the dispersion evaluation is conducted ahead of time, the dispersion can be taken into account when optimization is carried out. Because the statistical design support system is based on the experimental design method, it can do a quantitative evaluation of the factors that influence responses in fields where the - 1 -

7 About DesignDirector experimental design method has conventionally been used, such as chemistry, pharmacology, and agriculture. It can also easily evaluate the optimum combination of factors. In this way, DesignDirector can use experimental data as well as data obtained from simulations, so it is practical for use in a wide range of fields and can do design comprehensively and efficiently. 1.1 Features of DesignDirector In the various everyday problems of design work, numerous design factors act independently or in concert to influence the target responses (Fig. 1.1). Gaining an understanding of what factors act to what extent upon the target responses is a critical matter for the advancement of research and development. The DesignDirector system not only clarifies what factors influence a target response, but also makes it possible to express a target response as a function of the design factors (response surface equation), which can then be used to obtain useful information for design. Target response surface equation = F (Design factor 1, Design factor 2,...) Design factor 2 Design factor 3 Design factor 1 Design factor 4 Target Response Fig. 1.1 Relationship of Target Response and Design Factors (1) Because the system, based on the experimental design method, derives the target response surface equation (response surface) as a function of the design variables, and then uses the equation in optimization by the mathematical programming method, it makes it practical to carry out optimization even when there is non-linearity in the target responses. (2) Because the system can perform sensitivity analysis, dispersion evaluation, and reanalysis based on the target response surface equation obtained by the experimental - 2 -

8 About DesignDirector design method, it can obtain quantitative information that is useful for design and manufacturing. (3) In order to obtain the target response surface equation, the system must use the experimental design method to obtain the response values from the combinations of design factors, but DesignDirector can use either experimentally or analytically derived data for this purpose. (4) Analytical tools such as FEM software are not incorporated into DesignDirector itself. Indeed, the approach is quite the opposite. No matter what simulation software is used, as long as the user can obtain the response values with the software he/she has, DesignDirector can use the data. It is exceptionally flexible in this regard. (5) DesignDirector uses the experimental design method, but even users who are not familiar with the experimental design method can use it easily with its automatic functions such as assignment to orthogonal arrays, and pooling operations in ANOVA. 1.2 Operation Flow and Functions The operation flow in DesignDirector is shown in Fig The operations up to the point where ANOVA is performed using the experimental design method are essential operations. The remaining operations, sensitivity analysis, dispersion evaluation, reanalysis, and optimization calculation, may be performed as required. Design of experiments Sensitivity analysis Select design factors and level values Arrange in orthogonal array Analysis of variance (Effectiveness analysis) Structural analyses or experiments Select response values Reanalysis Evaluation of dispersion Optimization Select objective function and conditions of constraints Generate response surface equations Optimum calculation Fig. 1.2 Operation Flow (1) Experimental design method (analysis of influence) By carrying out experiments and analyses based on the experimental design method, - 3 -

9 About DesignDirector and by performing ANOVA for the response values obtained from the experiments and analyses, it is possible to clarify what factors influence the response values. Because ANOVA is performed by decomposing factor into orders, it can go so far as to make clear what order of a factor influences the response value. Also, taking into account the factors and orders that influence the response, the system builds the response surface equation of polynomials. Using these polynomials when performing sensitivity analysis and subsequent analyses, the system thereby makes it possible to obtain the quantitative information that is necessary for design. In cases where there are multiple responses, the influence of factors can be analyzed simultaneously for them. The system can also output supplementary tables and graphs, as well as graphs comparing the response values with the estimated values obtained from the response surface equation. Before analyzing the influence of factors, it is necessary first to study the responses of the problem one wishes to investigate, as well as the factors that are thought to influence the responses and the interaction between the factors. After the factors have been assigned to an orthogonal array, the experiments or analyses may be conducted in accordance with the conditions of the experimental design method to obtain the response values. (2) Sensitivity analysis A partial derivation of the response surface equation with respect to the factors yields the response sensitivity equation. The sensitivity that is determined by this sensitivity equation expresses the amount of change in the response per unit change in the factors. This provides a quantitative understanding of the influence of the factors on the response value. The system can output the sensitivity equation for factors with respect to the response, as well as sensitivity values both numerically and in graph form. (3) Dispersion evaluation By applying the primary approximation/secondary moment method to the response surface equation, it is possible to create an equation for the dispersion of the response values that arises from the dispersion of a given factor. It is then possible to use the standard deviation of the response and the coefficient of variation derived from this - 4 -

10 About DesignDirector equation to evaluate the influence of dispersion in the factor on dispersion in the response. The system can output the dispersion equation, and if there is no dispersion in other factors, it can output a dispersion component equation as well. It can also output the dispersion values both numerically and in graph form. (4) Reanalysis By using the response surface equation obtained by the experimental design method, it is easily possible to infer the value of the response when the value of a factor changes, without having to carry out experiments or simulations all over again. In cases where there are multiple responses, estimates of response values can be obtained simultaneously for multiple responses. (5) Optimization calculation By setting the response surface equation as the objective function or constraint function of the optimization problem, it is possible to perform the optimization calculation very efficiently. It is also possible to set user-defined functions as the objective function or constraint function. The optimization calculation method uses mathematical optimization for continuous variables and a round-robin calculation method for all combinations of discrete variables. The sequence of screens displayed in DesignDirector and the corresponding operating procedures are shown in Fig

11 About DesignDirector Investigation of the Problem - Identifying the factors - Determining the number of levels - Determining the level value for each factor - Investigating interactions between factors : Main steps : Sub steps Factors and Interactions Number of Error Columns Setting in the cases of failing an automatic assignment Orthogonal Array - Carrying out experiments or analyses based on an orthogonal array - Getting response values Response Values Prescription Table Sort Supplementary Graph Supplementary Table ANOVA Table Pooling Criteria Dispersion Component Table Evaluation of Dispersion Comparison Graph Dispersion Graph Dispersion Equation Reanalysis Edit Interactive Factor Values Edit Lower/Upper Limits Sensitivity Sensitivity Graph Carrying out these according to the problem. - Set an objective function - Set constraints Optimization User Equations Edit Interactive Factor Values Fig. 1.3 Sequence of Screens in DesignDirector - 6 -

12 2.1 Investigation of the Problem 2. Operating Instructions 2.1 Investigation of the Problem (1) Determining the response values - Decide what responses you want to evaluate. - Even if there are multiple responses, DesignDirector can evaluate them simultaneously. (2) Identifying the factors - Identify the factors that are thought to influence the responses you want to evaluate. (3) Determining the number of levels - Assume that with respect to its influence on the response, a given factor has either a first-order influence (the response can be expressed by a first-order equation of the factor) or a second-order influence (the response can be expressed by a second-order equation of the factor). The minimum number of levels must be no less than the order of influence plus one. - For example, when a factor is thought to exert first-order influence on the response (or is assumed to exert first-order influence), the number of levels must be set to two or more. If the factor is thought to exert second-order influence, the number of levels must be set to three or more. - The precision of the response surface equation improves as the number of levels increases, but the number of times the experiment (analysis) must be conducted also increases. If the experiment (analysis) can be carried out easily, this is not a problem. However, when conducting experiment (analysis) just once requires an inordinate number of man-hours, the decision of how many levels to set must take into account both the desired precision and the man-hours required for the experiment. (4) Determining the level value for each factor - Set upper and lower limits for the level value based on the range of values each factor can actually have. (5) Investigating interactions between factors - Investigate whether or not interactions between factors must be taken into account. If interactions must be taken into account, investigate what interactions between what factors must be considered. It sometimes happens that taking interactions into account increases the number of times the experiment (analysis) must be conducted, so the decision of whether or not to consider interactions must take into account both the - 7 -

13 2.1 Investigation of the Problem desired precision and the man-hours required for the experiment. This investigation of the problem is part of the preparation for using DesignDirector. This prior investigation is critical for making practical use of DesignDirector. If some factors that influence the response are missing, or if the influence on the response of interactions between the factors is not considered, the reliability of the results obtained from DesignDirector (the degree of influence of the factors, the response surface equation, sensitivity, etc.) will be impaired accordingly. Sample Problem The operation of DesignDirector will be explained using a sample problem in which the response value is determined by three factors, x 1, x 2, and x 3. Normally, DesignDirector uses analysis of the degree of influence to determine the degree of influence on the response of the factors and to derive the response surface equation. However, in this sample problem, the response has already been set ahead of time, expressed in equation (1) as a polynomial function of the factors. Let us now investigate how the response surface equation that DesignDirector obtains from analysis of the degree of influence might compare to the pre-set equation (1). Response function = 1+x 1 +x 2 +x 3 +x 2 1 +x x 3 +x 1 x 2 +x 1 x 2 2 +x 2 1 x 2 +x x 2 +x 2 x 3 +x 2 x 2 3 +x 2 2 x 3 +x x 3 +x 3 x 1 +x 3 x 2 1 +x 2 3 x 1 +x x (1) (1) Factors: The three individual factors, x 1, x 2, and x 3 shall be given the factor names x1, x2, and x3. (2) Number of levels: Because the function expressing the response value contains second-order terms for the factors, the number of levels shall be set to 3. (3) Level values: Level values shall be set as shown in Table

14 2.1 Investigation of the Problem Table 2.1 Level Values for Each Factor Factor names Level 1 Level 2 Level 3 x x x (4) Interaction between factors: The response function above contains terms such as x 1 x 2, the product of factor x 1 and factor x 2. This means that there is interaction between factor x 1 and factor x 2. Therefore, in the case of this sample problem, the interactions between all the factors must be taken into account. (5) Response value: The response value is actually obtained by experiment or analysis. But in this sample problem, the response value corresponds to the right side of equation (1), so the response value can be determined by substituting the level values of the factors for the variables in equation (1) Interaction A simple explanation of interaction is provided here. For a detailed explanation, please refer to the special materials [3][4] on the experimental design method. Assume that we have three factors, x1, x2, and x3, and that response Y is expressed by equation (2) as a function of the three factors. Y= 5 + 2*x1 + 3*x2 - x3 - x1*x (2) Also, the level values for factors x1, x2, and x3 are set as shown in Table 2.2 [ Relationship of factor x1 and factor x2] If the value of factor x3 is set to 3, the midpoint between the values of the two levels, then equation (2) can be recast in the form of equation (3) to determine the relationship between factor x1 and factor x2. Table 2.2 Level Values Factors Level 1 Level 2 x1 1 5 x2 1 5 x3 1 5 Y= 2+ 2*x1 + 3*x2 - x1*x (3) Using equation (3), if x1 and x2 are assigned the values shown for level 1 and level 2 in Table 2.2, then the values for response Y will be as shown in Table 2.3. Fig. 2.1 shows a graph of the results. The effect of factor x1 differs according to the value of factor x2, so in this case we can say that there is interaction between factor x1 and factor x2. When - 9 -

15 2.1 Investigation of the Problem interaction is present, the two lines in the graph will not be parallel, as shown in Fig In this case, the interaction is due to the term x1 * x2 in equation (3). [ Relationship of factor x1 and factor x3] If the value of factor x2 is set to 3, the midpoint between the values of the two levels, then equation (2) can be recast in the form of equation (4) to determine the relationship between factor x1 and factor x3. Y= 14 - x1 - x (4) Using equation (4), if x1 and x3 are assigned the values shown for level 1 and level 2 in Table 2.2, then the values for response Y will be as shown in Table 2.4. Fig. 2.2 shows a graph of the results. The effect of factor x1 is the same, regardless of the value of factor x3, so in this case we can say that there is no interaction between factor x1 and factor x3. When interaction is not present, the two lines in the graph will be parallel, as shown in Fig [ Relationship of factor x2 and factor x3] If the value of factor x1 is set to 3, the midpoint between the values of the two levels, as in the previous examples, then the results of the relationship between factor x2 and factor x3 will be as shown in Table 2.5. There is no interaction between factor x2 and factor x3. Table 2.3 x1 and x2 Table 2.4 x1 and x3 Table 2.5 x2 and x2 x2 x3 x x x x Response Y Level 1 x1 x2 Level 1 x2 Level 2 Fig. 2.1 Relationship of x1 and x2 Level 2 Response Y Leve 1 x1 x3 Level 1 x3 Level 2 Level 2 Fig. 2.2 Relationship of x1 and x3-10 -

16 2.2 Setting Factors and Interactions 2.2 Setting Factors and Interactions When the program is started, the "Factors and Interactions" screen is displayed. [ Screen configuration ] The screen is configured into two tabs, one called "Factors" and the other called "Interactions between Factors". Fig. 2.3 "Factors" Tab Screen Fig. 2.4 "Interactions between Factors" Tab Screen

17 2.2 Setting Factors and Interactions [ Child screen icons ] (1) (2) Fig. 2.5 "Factors and Interactions" Screen Icons (1) Forward to 'Orthogonal Array': Takes the user to the "Orthogonal Array" screen. (2) Number of Error Columns: Sets the number of error columns in the orthogonal array for use when the factors and interactions are automatically assigned to the array. [ Items for input, selection, and setting ] (a) "Factors" tab screen - Input the factor names and the upper and lower limits of the level values. (b) "Interactions between Factors" tab screen - Set the interactions between the factors. The default setting is that interactions will not be taken into account. (c) "Levels" combo box - Select the number of levels for the factors Setting the Factors (1) Select the number of levels for the factors from the "Levels" combo box. (2) In the "Factors" tab screen, enter in the "Name" column the names of the factors that were identified during the investigation of the problem as influencing the response. Enter the upper and lower limits of the level values for each factor in the "Upper" and "Lower" columns respectively. If the upper and lower limit values are input, the system will automatically calculate the level values for level 1, level 2, etc., in such a way that the level values will be set at equally spaced points from the lower limit value to the upper limit value. Then the calculated level values will be displayed. (3) If interactions between factors will not be taken into account, operation is complete for this screen. Click on the icon and proceed to the "Orthogonal Array" screen. (4) If interactions between factors will be taken into account, click on the "Interactions between Factors" tab and set the interactions following the procedures described in section 2.2.2, "Setting Interactions". (5) The system is designed to function properly even if the user leaves blank rows between factors when inputting the factor names and upper and lower limit values. However, it is recommended that the information be input in order, starting with the first row

18 2.2 Setting Factors and Interactions (6) The system can not automatically assign mixed numbers of levels to an orthogonal array. Therefore, if you want to use an orthogonal array in which different numbers of levels are mixed together, go to the "Orthogonal Array" screen as described in section 2.3, select an orthogonal array, and assign the factors to the array manually. When this is done, DesignDirector will ignore the number of levels set in the "Levels" combo box and instead set the number of levels automatically Setting Interactions The default setting is that no interactions between factors will be taken into account, so if you want to take interactions into account, make the settings described below. (1) Click on the "Interactions between Factors" tab and the screen for setting the interactions will be displayed. (2) If an interaction between specific factors is to be taken into account, click on the corresponding cell on the screen. The word "TRUE" will be displayed in the cell, and the interaction between the specified factors will be taken into account. (3) If all interactions between factors are to be taken into account, click on the "Select All" button. The word "TRUE" will be displayed in all of the cells, and all interactions between factors will be taken into account. (4) To deselect interactions that have already been set, click on the cell corresponding to the interaction you wish to deselect. The word "TRUE" will disappear from the cell, indicating that the interaction has been deselected. (5) To deselect all interactions that have already been set, click on the "Deselect All" button. The word "TRUE" will disappear from all cells that were set, indicating that all settings have been deselected. (6) Any interactions that are set will be reflected automatically in the "Interactions" column of the "Factors" tab screen, as shown in Fig For example, in Fig. 2.3, the numbers 2 and 3 are displayed in the "Interactions" column for factor x1. This means that for factor x1, interactions have been set for factor x2 listed in row 2 and for factor x3 listed in row 3. (7) When all factor settings and interaction settings have been completed, click on the icon and proceed to the "Orthogonal Array" screen

19 2.2 Setting Factors and Interactions (8) At the point where you proceed to the "Orthogonal Array" screen, if the number of factors or the number of combinations of interactions is too large, the system may not be able to assign them even to the largest orthogonal arrays available for each level. In that event, the message box shown in Fig. 2.6 will be displayed. Try the assignment function again after reducing the number of factors and/or interactions. (9) If the number of factors or the number of combinations of interactions is large, the system may not be able to assign them automatically to the orthogonal arrays. In that event, the message box shown in Fig. 2.6 will be displayed. Proceed to the "Orthogonal Array" screen and assign the factors and interactions to the arrays manually. Fig. 2.6 "Cannot Assign" Message Box The system can not process interactions when mixed numbers of levels are used. Therefore, when you want to take interactions between factors into account, use an orthogonal array in which all factors have the same number of levels Setting the Number of Error Columns When you finish setting the factors and interactions as shown in Fig. 2.3 and Fig. 2.4 and proceed to the "Orthogonal Array" screen by clicking on the icon, DesignDirector automatically selects an appropriate orthogonal array based on the data you entered. At the same time, it automatically assigns the factors and interactions to the columns of the orthogonal array. In this case, the user can set how many error columns to use in the orthogonal array by

20 2.2 Setting Factors and Interactions using the "Minimum Error Columns" function before proceeding to the "Orthogonal Array" screen. A different orthogonal array may be selected automatically depending on the value to which the number of error columns is set. If the number of error columns is set to a large value, a large orthogonal array will be selected. The system is programmed so that the default setting will be a minimum of one error column. Fig. 2.7 "Minimum Error Columns" Dialog Box (1) If you click on the icon on either of the screens shown in Fig. 2.3 or Fig. 2.4, the "Minimum Error Columns" dialog box (Fig. 2.7) will be displayed. (2) Select one of the option buttons in the box labeled "Minimum error columns at automatic assignment" to set the desired number of error columns. (3) If you select the option "This value or more", enter the desired minimum number of error columns as a positive integer in the text box to the right of that option. (4) When you have completed the setting, click on the "OK" button

21 2.2 Setting Factors and Interactions Hints for setting the number of error columns - Setting the number of error columns to one-fourth or one-third of the total columns is considered preferable, but if the number of error columns is set to a large number, a large orthogonal array may be selected. As the orthogonal array becomes larger, the number of times that the experiment must be conducted increases, so consider how easy or difficult the experiment is to conduct when you set the number of error columns. - DesignDirector will perform the necessary calculations even if there are no error columns in the orthogonal array, but it is recommended that the number of error columns be set to no less than one Operation for Sample Problem (1) In the "Levels" combo box on the "Factors" tab screen, select "3 levels". (2) Input the factor names and the upper and lower limit values on the "Factors" tab screen. In the case of our sample problem, there are three factors, so enter the data in rows 1 through 3. - In row 1, enter "x1" in the "Name" column, "1" in the "Lower" column, and "3" in the "Upper" column. - In row 2, enter "x2" in the "Name" column, "1" in the "Lower" column, and "3" in the "Upper" column. - In row 3, enter "x3" in the "Name" column, "1" in the "Lower" column, and "3" in the "Upper" column. (3) On the "Interactions between Factors" tab screen, click on the "Select All" button, so that all interactions between factors will be taken into account. (4) The default setting of a minimum of one error column will be used, so it is not necessary to use the "Minimum Error Columns" dialog box

22 2.3 Orthogonal Array 2.3 Orthogonal Array At the point when the "Orthogonal Array" screen is displayed, what is displayed on the screen are the results of a process in which DesignDirector selects an appropriate orthogonal array for the factor data and interaction data that the user has set and then automatically assigns the factors and interactions to the columns of the orthogonal array. Level values are also automatically assigned to the columns to which factors have been assigned. All remaining columns to which factors and interactions have not been assigned are treated as error columns, and the word "Error" is displayed in the column heading on the screen. [ Screen configuration ] The screen is configured as an "Orthogonal Array" sheet. Fig. 2.8 "Orthogonal Array" Screen

23 2.3 Orthogonal Array [ Child screen icons ] (1) (2) (3) (4) (5) Fig. 2.9 "Orthogonal Array" Screen Icons (1) Back to 'Factors and Interactions': Returns the user to the "Factors and Interactions" screen. (2) Forward to 'Response Values': Takes the user to the "Response Values" screen. (3) Clear Assignments: Clears all factor assignments from the orthogonal array. (4) Automatic Assignments: Automatically assigns factors to the orthogonal array. (5) Hint: Provides hints for using the "Orthogonal Array" screen. [ Items for input, selection, and setting ] (a) If the selected orthogonal array and the assignments of the factors to the orthogonal array are both good, input is not necessary on this screen. (b) If you wish to change the orthogonal array or the assignments, assign the factors as described in section 2.3.1, "Changing Assignments". (c) When the settings have been completed, click on the "Response Values" screen. icon and proceed to the Based on the data input in "Factors and Interactions" (section 2.2), the system automatically assigns values to an appropriate orthogonal array, but if the number of factors and the number of combinations of interactions is large, the system may not be able to conduct automatic assignments properly. In that event, assign the factors and interactions to the orthogonal array manually, using the procedures described in section 2.3.1, "Changing Assignments" Changing Assignments (1) If you click on the icon on the "Orthogonal Array" screen shown in Fig. 2.8, all factor assignments will be cleared. At this time, the "Clear All Assignments" confirmation message box (Fig. 2.10) will be displayed. Click on the "OK" button to clear the assignments

24 2.3 Orthogonal Array Fig "Clear All Assignments" Confirmation Message Box (2) If you click on the "Factors" cell of the column for which you want to make an assignment, the pull-down "Factors" combo box shown in Fig will be displayed below the cell. Select a factor you want to assign by clicking on it. Fig Click on "Factors" Cell to Display "Factors" Combo Box (3) Assign the factors to the orthogonal array by repeating step (2) as necessary. If interactions were set on the "Interactions between Factors" screen shown in Fig. 2.4, then when two factors for which an interaction has been set are assigned, the interaction will be assigned automatically to the appropriate columns in the orthogonal array. When all assignments have been completed, the "Assignments Complete" message box (Fig. 2.12) will be displayed. Fig "Assignments Complete" Message Box

25 2.3 Orthogonal Array (4) If you wish to change the orthogonal array, select an appropriate orthogonal array from the "Orthogonal Arrays" combo box. At this time, the "Clear All Assignments" confirmation message box shown in Fig will be displayed. If you click on the "OK" button, all assignments will be cleared, and the display will change to the orthogonal array that you have selected. Follow the procedures described in steps (2) and (3) above to assign the factors to the columns of the orthogonal array. If you have selected an orthogonal array that is too small for the number of factors and interactions, you will not be able to assign all the factors. In that event, the "Change Orthogonal Array" confirmation message box (Fig. 2.13) will be displayed. Select a larger orthogonal array. Fig "Change Orthogonal Array" Confirmation Message Box (5) (If, instead of clearing all the assignments by clicking on the icon as described in step (1) above, you click on the "Factors" cell in a column to which a factor has been assigned, the "Factors" combo box will be displayed. If you select "Clear" from the combo box, all assignments related to that factor will be cleared. After repeating this procedure for as many factors as you wish to change, you can assign the factors with the procedures described in steps (2) and (3) above. (6) If you click on the icon, the system will automatically assign the factors to the columns of the orthogonal array. (7) For a detailed explanation of assignments to the orthogonal array, please refer to the special materials [3][4] on the experimental design method

26 2.3 Orthogonal Array Assigning Mixed Numbers of Levels to the Orthogonal Array DesignDirector provides the following four types of orthogonal arrays for handling mixed numbers of levels: L12(2 11 ), L18( ), L32( ), L50( ) With L12, interactions cannot be handled, because the interaction components in two of the columns become mixed into the remaining nine columns a little at a time. Because of this, it has no mixing of the number of levels, but in DesignDirector, it is included in the orthogonal arrays for handling mixed numbers of levels. In the L18, L32, and L50 orthogonal arrays, assign a factor with two levels to column 1. From column 2 on, assign the factors with 3 levels in L18, the factors with 4 levels in L32, and the factors with 5 levels in L50. (1) Select an appropriate orthogonal array for handling mixed numbers of levels from the "Orthogonal Arrays" combo box. The message box shown in Fig and the "Clear All Assignments" confirmation message box shown in Fig will be displayed. If everything is all right, click on the "OK" button in each box. Fig Message Box Displayed when Orthogonal Array for Mixed Numbers of Levels Is Selected (2) Follow the procedures described in steps (2) and (3) of section 2.3.1, "Changing Assignments", to assign the factors to the columns in the orthogonal array. When the factors are assigned, level values will be assigned to the orthogonal array automatically Operation for Sample Problem (1) The three-level orthogonal array L27 that was selected automatically for the sample program and the assignment of factors x1, x2, and x3 to columns 1, 2, and

27 2.3 Orthogonal Array respectively are fine as they stand, so no changes need to be made on this screen, and we will confirm the assignments. Note that the interactions between factors x1 and x2 have been assigned automatically to columns 3 and 4, the interactions between factors x1 and x3 have been assigned to columns 6 and 7, and the interactions between factors x2 and x3 have been assigned to columns 8 and

28 2.4 Inputting the Response Values 2.4 Inputting the Response Values Input the response values on this screen. You can also create a prescription table. [ Screen configuration ] The screen is configured as a "Response Values" sheet. Fig, 2.15 "Response Values" Screen [ Child screen icons ] (1) (2) (3) (4) (5) (6) (7) (8) Fig "Response Values" Screen Icons (1)Back to 'Orthogonal Array': Returns the user to the "Orthogonal Array" screen. (2)Forward to 'ANOVA Table': Takes the user to the "ANOVA Table" screen. (3)Add Input Sheet: Adds response values input sheet. (4)Delete Input Sheet: Deletes selected response values input sheet. (5)Increase Replication: Adds response values input columns. (6)Decrease Replication: Deletes selected response values input columns. (7)Prescription Table: Displays the prescription table. (8)Hint: Provides hints for using the "Response Values" screen

29 2.4 Inputting the Response Values [ Items for input and selection ] (a) If it is necessary to randomize the experiments, click on the icon to display the prescription table. If it is not necessary to randomize the experiments, the "Response Values" screen shown in Fig can be used as the prescription table. (b) If response values were obtained in accordance with the prescription table, input the response values that were obtained. (c) When you have finished inputting the response values, click on the icon to proceed to the "ANOVA Table" screen Prescription Table In order to input the response values, it is necessary first to obtain the response values by carrying out the experiments or analyses based on the experimental conditions recorded in the orthogonal array. When the experiments are conducted to obtain the response values, rather than conducting them in the order shown in the orthogonal array, it is preferable to randomize the order of the experiments so that experimental error will not be introduced. If you click on the icon on the "Response Values" screen (Fig. 2.15) to display the "Prescription Table" screen (Fig. 2.17), what will be displayed are the results of the randomizing of the order of the experiments. When the response values are obtained by means of analysis, experimental error is not introduced, so the prescription table is not really necessary. The response values may be obtained in accordance with the conditions in the orthogonal array (Fig. 2.15)

30 2.4 Inputting the Response Values [ Screen configuration ] The screen is configured as a "Prescription Table" sheet with a "Replication" spin button. [ Child screen icons ] (1) (2) (3) Fig "Prescription Table" Screen Fig "Prescription Table" Screen Icons (1)Close: Close the "Prescription Table" screen. (2)Clear Notes: Clears the "Note" column of the prescription table. (3) Sort: Switch between sorting the experiments in order by experiment number or in randomized order. [ Items for input and selection ] (a) Set the number of replications and enter the data in the "Note" column, if necessary. (b) If it is necessary to sort the prescription table by experiment number, sort it by using the icon. (c) When operation has been completed, click on the Table" screen. icon to close the "Prescription

31 2.4 Inputting the Response Values [ Operation method ] (1) On the "Response Values" screen (Fig. 2.15), click on the icon to display the "Prescription Table" screen. When the screen is displayed, the order of the experiments will have been randomized. (2) There may be occasions when the same experiment is carried out several times in order to reduce the influence of measurement errors on the results. If the experiments will be carried out more than once, use the "Replications" spin button to set the number of times the experiments will be conducted. The maximum number of replications that can be set is 20. (3) The contents of the "Note" column have no bearing on the analysis that the system performs. Use the "Note" column to record notes, etc., when the experiment is carried out. Not entering anything in the "Note" column will not in any way interfere with the subsequent analysis. Click on the icon to clear the "Note" column. (4) If you click on the icon, a message box saying, "Clear prescription table when it is closed?" will be displayed. If you select the "NO" button, the contents of the randomized prescription table will be preserved, and the next time you open the prescription table, the same contents can be displayed. If you select the "YES" button, the next time you open the prescription table, the contents will be newly randomized. In other words, if the randomized sequence that has been set is good, select "NO", and if you want to conduct randomization over again, select "YES". Fig "Confirmation Screen" Message Box (5) Obtain the response values by means of experiments or analyses based on the prescription table. Note that the work of obtaining the response values is not a part of DesignDirector. (6) Once you have obtained the response values, input them on the "Response Values" screen (Fig. 2.15)

32 2.4 Inputting the Response Values Sorting the experiments If you click on the icon, the "Sort" dialog box will be displayed, and you can sort the prescription table. Fig "Sort" Dialog Box (1) If you select the "Standard Order" option and click on the "OK" button, the prescription table will be displayed, sorted in order by the experiment number. This order is not randomized, but follows the order in the orthogonal array. When the response values are obtained by analysis, there is no need to randomize the order of the experiments (analyses), so it is more convenient to use the prescription table based on the orthogonal array. The content on the prescription table is the same as on the "Response Values" screen (Fig. 2.15), but you can use this table to record notes with respect to specific experiment numbers. (2) If you select the "Run Order" option and click on the "OK" button, the prescription table in experiment number order will revert to the randomized order

33 2.4 Inputting the Response Values Obtaining the Response Values Obtain the response values by carrying out experiments or analyses based on the prescription table displayed on the "Response Values" screen (Fig. 2.15) or the "Prescription Table" screen (Fig. 2.17). In the sample problem, the L27 orthogonal array is used, and the replications have been set to 1, so the experiments will be carried out 27 times. Because the response has been set as a polynomial function of the factors, the response values can be obtained by substituting the level values of the factors for the variables in the function. The response name for these values is then set to "No error". In order to explain the operation of DesignDirector, we will establish two additional types of responses for which errors have been intentionally introduced. These responses are named "Error 1" and Error 2". So all together, our sample problem has three types of responses. The three response values for each experiment number are shown in Table 2.6. Table 2.6 Response Values for Sample Problem Response values Response values Experiment number No error Error 1 Error 2 Experiment number No error Error 1 Error

34 2.4 Inputting the Response Values Inputting the Response Values On the "Response Values" screen (Fig. 2.15), input the response values that were obtained by analysis or experiment in accordance with the prescription table. (1) The default number of responses is 1, and the name of the sheet tab is "Response 1". (2) If you double-click on the sheet tab, the "Sheet Name" dialog box (Fig. 2.21) will be displayed. Change the name from "Response 1" to some appropriate name. A name of up to 10 characters may be input. Fig "Sheet Name" Dialog Box (3) Enter the response values for each experiment number in the "1th" column. (4) If there are multiple responses, add input sheets by clicking on the icon. The "Response Name" input dialog box will be displayed. Input the response name. The default response name is "Response i". After the default response name has been set, the procedure in step (2) above may be used to change the response name. Fig "Response Name" Input Dialog Box (5) Input the response values on the added input sheet. (6) Repeat steps (4) and (5) for as many responses as you have. (7) In the operations described above, response values are input each time that a new input sheet is added, but it is perfectly all right to click on the icon repeatedly to add as many input sheets as you have responses before you actually input any data. Click on

35 2.4 Inputting the Response Values the tab of the desired input sheet to make it the active sheet, and input the response values. (8) To delete an input sheet, first click on the tab of the sheet you want to delete to make it the active sheet. Then click on the icon. (9) When you have finished inputting the response values, click on the icon to proceed to the "ANOVA Table" screen. When the experiments will be replicated (1) When two or more replications of the experiments are carried out, click on the icon. Each time you click on the icon, a new column (called "2th", "3th", etc.) is added to the sheet so that you can input the response values. Input the response values in the appropriate cells. (2) If you want to delete unneeded replication columns, first select the columns you want to delete by clicking on their headings, then click on the icon to delete the selected columns. Fig Screen with 2th Response Value Input Column Added by [INCREASE] Icon

36 2.4 Inputting the Response Values Operation for Sample Problem (1) Double-click on the tab for the "Response 1" sheet to display the "Sheet Name" dialog box (Fig. 2.21). Change the name from "Response 1" to "No error". (2) In the "1th" column, input the response values for "No error" (19, 39, 69, 469) in rows 1 to 27. (3) Click on the icon to add another input sheet. Using the procedures described in step (1) above, change the name of the sheet from "Response 2" to "Error 1". (4) Using the procedures described in step (2) above, input the response values for "Error 1" (20, 41, 65, 462) in the "1th" column. (5) Use the same procedures to add another input sheet, change its name to "Error 2", and input the response values. [ Screen showing added input sheets, response names, and response values for the sample problem ] Fig Example of Input for Sample Problem

37 2.5 ANOVA (Analysis of Variance) 2.5 ANOVA (Analysis of Variance) The ANOVA table and the response surface equations are displayed on the "ANOVA Table" screen. When the "ANOVA Table" screen is displayed, the ANOVA results and response surface equations for all of the responses are automatically calculated and displayed. If there are multiple responses, you can view the results for any one response by clicking on the sheet tab on which the response name is displayed. [ Screen configuration ] The screen is configured of the ANOVA table and a text box that displays the response surface equation. The system automatically creates as many separate sheets as there are responses. Fig "ANOVA Table" Screen In the sample problem, the value we are using for the response value for "No error" was obtained by substituting the level values of the factors for the variables in the established polynomial function, so no errors occur in ANOVA. As a result, the variance value of the

38 2.5 ANOVA (Analysis of Variance) error in the ANOVA table is "0", which means that the variance ratio F0 cannot be derived for each factor and its component orders. Therefore, the word "Nothing" is displayed in the columns for the variance ratio F0 and significant level. In this case, the marks indicating a significant level of 5% or higher (*) or a significant level of 1% or higher (**) are not displayed, but because these results are significant, pooling is not carried out. Also, the equation that is displayed in the lower portion of the screen is a function that is consistent with the function of the response values. [ Child screen icons ] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) Fig "ANOVA Table" Screen Icons (1)Back to 'Response Values': Returns the user to the "Response Values" screen. (2)Forward to 'Optimization': Takes the user to the "Optimization" screen. (3)Reanalysis: Carries out reanalysis. (4)Sensitivity: Carries out sensitivity analysis. (5)Dispersion: Carries out dispersion analysis. (6)Supplementary Table: Displays the supplementary tables. (7)Supplementary Graph: Displays graphs of the supplementary tables. (8) Comparison Graph: Displays a graph comparing the response values with the estimated values. (9)Pooling: Carries out pooling for specified factors one at a time. (10)Auto Pooling (one by one): Carries out automatic pooling one factor at a time. (11)Auto Pooling: Carries out automatic pooling. (12)Reset Pooling: Resets all pooling. (13)Pooling Criteria: Sets the pooling criteria. (14)Hint: Provides hints for using the "ANOVA Table" screen. [ Items for input and setting ] (a) Input is not required on this screen. (b) If necessary, carry out pooling for the ANOVA table for each response. (c) Click on the icons described above to carry out a variety of analyses based on the response surface equation. (d) Click on the icon to proceed to the "Optimization" screen. Items displayed in the ANOVA table (1) Order - In order to perform a detailed analysis in ANOVA, DesignDirector orthogonally

39 2.5 ANOVA (Analysis of Variance) decomposes the influence of the factors on the response into each component of the polynomial and evaluates the results. As a result, the influence of a factor on the response is broken down into the factor's component orders (first-order, second-order, etc.) and expressed accordingly. (2) Freedom (f) - Because the system performs orthogonal decomposition, the degree of freedom is "1" for every component order of all the factors. (3) Sum of Squares (S) - The system displays the variation value (sum of squares) for each component order of the factors. (4) Variance (V) - The variance is the value calculated by dividing the sum of squares (S) by the freedom (f) and expresses the sum of squares per unit freedom. - Because the system orthogonally decomposes a factor into the component orders, the sum of squares and the variance have the same values. (5) Variance Ratio (F0) - The variance ratio (F0) expresses the ratio of variance V A (the variance of the component orders of factor A) to the variance V e (the variance of the error). - F0= V A / V e (6) Significant Level - Among the factors and component orders that have been determined to influence the response (a significant difference is found), those with a significant level of 5% or more are marked with an asterisk (*), while those with a significant level of 1% or more are marked with two asterisks (**). - The significant level represents the maximum probability that the results of the F test are mistaken. - The variation in factors that do not show significant F test results (factors not marked with asterisks) can be judged to be within the range of error. Such factors are subject to pooling. (7) Net Variation (S ) - Because the variation for a given factor includes not only variation in the factor but variation in the errors as well, it is necessary to exclude the portion of the variation that is attributable to the errors in order to evaluate the true effect of the factor. The

40 2.5 ANOVA (Analysis of Variance) resulting value is called the net variation. (8) Contribution Ratio - The contribution ratio is the ratio of the net variation to the total variation, and it represents the magnitude of the influence of the factor. However, the contribution ratio is influenced by the level range for the factor (the upper and lower limit range for the factor). That is to say, if the level range is great, the contribution ratio will be large, and if the level range is narrow, the contribution ratio will be small. Therefore, in order to evaluate the quantitative influence of the factor on the response, it is necessary to carry out sensitivity analysis, as explained in section 2.7. (9) F value - When the F test is carried out on the component orders of factors, the values used as the threshold are F(0.05) and F(0.01), which correspond to the 5% point and the 1% point of the F distribution. - The values F(0.05) and F(0.01) are displayed on the bottom line of the ANOVA table. (10) Equation - The response surface equation is derived using Chebyshev orthogonal polynomials. These orthogonal polynomials have the characteristic that at the same time that the low-order term is given priority, all orders of terms are mutually independent. Because of this, even if the equation is arbitrarily truncated, it still indicates the best regression equation for the orders that remain. - However, because Chebyshev orthogonal polynomials give priority to the low-order term, component orders that are lower than the highest order that has an effect can not be ignored even if they have no effect themselves. Therefore, in such cases, the orders can not be pooled just because they have no effect, and the low-order component must be left as it is, without being pooled. Because DesignDirector derives the response surface equation in conjunction with pooling, care must be taken when carrying out pooling. - For example, if a 3-level orthogonal array is used, the factors will have second-order components when orthogonal decomposition is performed. If the second-order component of the factor is then judged to be "significant", and the first-order component is judged to be "not significant", do not conduct pooling of the first-order component just because it is not significant. Pooling in this case would lower the precision of the response surface equation

41 2.5 ANOVA (Analysis of Variance) Pooling Operation Generally, evaluation of ANOVA is performed by comparing the value of the variance ratio F0 for each factor and component order against the values at the 5% significance point and the 1% significance point of the F distribution and then testing for a significant difference. At this stage, among the factors and component orders that have been judged to have no significant difference, if there are factors and component orders that have particularly small values for the variance ratio F0, then the variation and freedom values for those factors and orders are added to the variation and freedom values for the error, and the error variance is derived again. This new error variance is then used to derive the values for the variance ratio F0 for those factors and component orders, and the significant difference test is carried out again. By repeating this procedure until it eliminates all factors and component orders that have a value for the variance ratio F0 that is smaller than the value at the 5% significance point of the F distribution, it is possible to obtain the final ANOVA results. This procedure, which means that any effect of a factor that is no greater than the effect of error is itself regarded as the effect of error, is called pooling. DesignDirector is programmed with three pooling methods. In automatic pooling, the threshold value for pooling is set to the value at the 5% significance point of the F distribution, but it is possible to change the threshold value. (A) Pooling Carry out pooling sequentially for factors and component orders in the ANOVA table that are not marked with an asterisk (*) in the "Significant Level" column for having a significance level of 5% or more or two asterisks (**) for having a significance level of 1% or more, starting with the factor and component order having the smallest value in the "Variance Ratio (F0)" column. (1) On the "ANOVA Table" screen shown in Fig. 2.25, click on the row heading of the factor and component to be pooled, then click on the icon to pool the selected factor and component. After pooling, the value displayed for the variance, etc., of the selected factor and component will be "0". The response surface equation displayed in the lower portion of the screen will also be changed in conjunction with the pooling

42 2.5 ANOVA (Analysis of Variance) (2) If you click on the row heading of a factor and component that have been pooled, then click on the icon, the pooling will be reset, and the screen will display the values shown before pooling. (3) Repeat step (1) above as many times as necessary. (B) Auto Pooling (one by one) Factors and components for which the default value in the "Variance Ratio (F0)" column is less than F(0.05) (i.e., their significance level is less than 5%) are automatically detected and pooled. In DesignDirector, auto pooling is carried out sequentially for non-significant components (components having a variance ratio F0 that is less than the pooling threshold value), beginning with the component having the lowest value for the variance ratio F0. However, with regard to main factors, a set of conditions under which pooling can not be done has been established. Factor components and interaction components that satisfy these conditions are excluded from auto pooling. The conditions are as follows: (a) If there is a significant orthogonal decomposition component for a main factor, any lower-order components of that factor are excluded from pooling. Ex. 1: If the fourth-order component of main factor X1 is significant, auto pooling is not carried out for the first- through third-order components of main factor X1, even if those components are not significant. Ex. 2: If the second-order component of main factor X1 is significant, auto pooling is not carried out for the first-order component of main factor X1, even if it is not significant. (b) If there is a significant orthogonal decomposition component in an interaction, the orders of any main factor in the interaction that are components of a lower order than the significant component of the interaction are excluded from auto pooling. Ex. 1: If the X1^fourth-order * X2^fourth-order component of the interaction is significant, pooling is not carried out for the first- through fourth-order components of main factor X1 or for the first- through fourth-order components of main factor X2. Ex. 2: If the X1^fourth-order * X2^first-order component of the interaction is significant, pooling is not carried out for the first- through fourth-order components of main factor X1 or for the first-order component of main factor X

43 2.5 ANOVA (Analysis of Variance) When pooling is performed in the one by one mode, a confirmation message box is displayed. (1) Click on the icon on the "ANOVA Table" screen (Fig. 2.25). (2) A confirmation dialog box (Fig. 2.27) is displayed with the message, "Factors/interactions in the following case will be pooled into the error. 'Variance Ratio (F0) <= F(0.05)." Click on the "OK" button to carry out auto pooling. Fig "Start Auto Pooling" Message Box (3) A confirmation dialog box (Fig. 2.28) is displayed with a message asking if you want to select and pool factors and component orders in sequence, beginning with the factor having the lowest value for the variance ratio F0. Click on the "YES" button to carry out pooling. The equation displayed in the lower portion of the screen will be changed automatically in conjunction with the pooling. Fig "Auto Pooling (one by one) Confirmation" Message Box (4) Repeat step (3) until a confirmation dialog box (Fig. 2.29) is displayed indicating that pooling has been completed for all factors with a variance ratio of F(0.05) or less. Click on the "OK" button

44 2.5 ANOVA (Analysis of Variance) Fig "Pooling Complete" Message Box (C) Auto Pooling In this form of auto pooling, confirmation dialog boxes are not displayed, and the default is that pooling will be carried out until it has been completed for all factors with a variance ratio of F(0.05) or less. (1) Click on the icon on the "ANOVA Table" screen (Fig. 2.25). (2) A confirmation dialog box (Fig. 2.30) is displayed with the message, "Factors/interactions in the following case will be pooled into the error. 'Variance Ratio (F0) <= F(0.05)." Click on the "OK" button to carry out auto pooling. Fig "Start Auto Pooling" Message Box (3) When pooling has been completed for all factors with a variance ratio of F(0.05) or less, a confirmation dialog box (Fig. 2.29) is displayed. Click on the "OK" button. (D) Setting Pooling Criteria In auto pooling, it is possible to set the threshold value for pooling. The default threshold value for pooling is set to F(0.05). (1) Click on the icon on the "ANOVA Table" screen (Fig. 2.25) to display the "Pooling Criteria" dialog box (Fig. 2.31)

45 2.5 ANOVA (Analysis of Variance) Fig "Pooling Criteria" Dialog Box (2) Chose the desired threshold value from among the three types of auto pooling threshold values that are displayed in the "Pooling Criteria" dialog box. - Variance Ratio (F0) <= F(0.05) (default) - Variance Ratio (F0) <= F(0.01) - Variance Ratio (F0) <= This value : If you select this option, input a positive number in the text box on the right for the value of the variance ratio F0. (3)When the setting has been completed, click on the "OK" button. (E) Reset Pooling Click on the icon on the "ANOVA Table" screen (Fig. 2.25) to clear all pooling that has been carried out and revert to the state before pooling was performed

46 2.5 ANOVA (Analysis of Variance) Displaying the Supplementary Tables The "Supplementary Table" screen displays the supplementary tables for the main factors and interactions. The total response values and mean response values for each level of each factor are displayed. If there are no interactions, the user can understand the influence of the factors on the response from looking at the supplementary table. If there are interactions, sensitivity analysis must be carried out for a quantitative evaluation. [ Screen configuration ] The "Supplementary Table" screen is configured of two sheets, one for the main factors and the other for interactions. Fig "Supplementary Table" Screen [ Child screen icons ] (1) (2) Fig "Supplementary Table" Screen Icons (1) Close: Closes the "Supplementary Table" screen

47 2.5 ANOVA (Analysis of Variance) (2)Supplementary Graph: Displays a graph of the supplementary table. [ Items for input and setting ] (a) Input is not required on this screen. (b) Click on the icon to close the "Supplementary Table" screen. [ Operations on the "Supplementary Table" screen ] (1) On the "ANOVA Table" screen (Fig. 2.25), click on the sheet tab of the response for which you want to display the supplementary tables, then click on the icon to display the supplementary tables for the selected response. In this procedure, if multiple responses are selected, the corresponding supplementary tables will be displayed simultaneously. (2) If you want to view the supplementary table data for a given factor in the form of a graph, click on the row heading of the factor, then click on the icon, and the supplementary graph for that factor will be displayed. Supplementary graphs for multiple factors can be displayed simultaneously. (3) Click on the icon to close the "Supplementary Table" screen Supplementary Graph The "Supplementary Graph" screen displays a graph of the mean values of the response for each level of the factor. [ Screen configuration ] The "Supplementary Graph" screen for a main factor is configured of a plot area and a title area. The "Supplementary Graph" screen for an interaction is configured of a plot area, a title area, and an area for the legend

48 2.5 ANOVA (Analysis of Variance) Title area Plot area Fig "Supplementary Graph (Main Factor)" Screen Title area Area for the legend Plot area Fig "Supplementary Graph (Interaction)" Screen

49 2.5 ANOVA (Analysis of Variance) [ Child screen icons ] (1) (2) Fig "Supplementary Graph" Screen Icons (1) Close: Closes the "Supplementary Graph" screen. (2) Exchange the X axis factor: Changes the factor of the X axis between two interactive factors. This icon is disabled on the "Supplementary Graph" screen for a main factor. [ Operations on the "Supplementary Graph" screen ] (1) On the "ANOVA Table" screen (Fig. 2.25), click on the row of the factor for which you want to display the supplementary graph, then click on the icon to display the supplementary graph for the selected factor. Clicking on a factor row causes a supplementary graph to be displayed for the main factor. Clicking on an interaction row causes a supplementary graph to be displayed for the selected interaction. (2) Alternatively, on the "Supplementary Table" screen (Fig. 2.32), click on a row heading of factor to select the factor, then click on the icon, and the supplementary graph for that factor will be displayed. Clicking on an interaction row causes a supplementary graph to be displayed for the selected interaction. (3) In the case of a supplementary graph for an interaction, each time you click on the icon, the factor of the X axis changes. (4) Interval estimates of 95% and 90% can be displayed by using the "Interval Estimate" combo box. - The width of the reliability interval for the difference between the mean value with a reliability of 95% and the mean value of the point estimates is t(f e,0.95)sqrt((v e /n)) - The width of the reliability interval for the difference between the mean value with a reliability of 90% and the mean value of the point estimates is t(f e,0.90)sqrt((v e /n)) Where F e : Degree of freedom of error, V e : Variance of error, n: Number of replication (5) The least significant difference can be displayed in the "lsd. line" combo box. The least significant difference is calculated by the formula lsd = t(f e, 0.95)SQRT((2V e /n)), and the lsd. line can be displayed for each level. If the difference between the mean values of two adjacent levels is less than the least significant difference, the system handles the situation as if there were no difference between the levels

50 2.5 ANOVA (Analysis of Variance) (6) Click on the icon to close the "Supplementary Graph" screen. [ Viewing the supplementary graph ] If there are no interactions, the user can understand the trend of the influence of a main factor on the response from looking at the supplementary graph. If the supplementary graph for a main factor rises from left to right, it indicates that the response value can be increased by increasing the level value of the factor. The supplementary graph for the interaction between factors x1 and x2 of the sample problem is shown in Fig The graph shows the mean response values for the level value of factor x1 with respect to the three level values for factor x2. The fact that the three lines in the graph are not parallel indicates that interaction exists between factor x1 and factor x2. Since the lines rise from left to right, the response value can be increased by increasing the level value of factor x1. However, in cases where interaction exists, sensitivity analysis must be carried out for a quantitative evaluation

51 2.5 ANOVA (Analysis of Variance) Comparison Graph of Response Values and Estimated Values The "Comparison Graph" screen makes it possible to see the correspondence between the response values obtained by analysis or experiment and the estimated values obtained from the response surface equation, with respect to each experimental condition, by displaying a graph that compares the response values and the estimated values. [ Screen configuration ] The "Comparison Graph" screen is configured of a plot area, a title area, and an area for the legend. Title area Plot Area Area for the legend Fig "Comparison Graph" Screen Comparing Response Values and Estimated Values

52 2.5 ANOVA (Analysis of Variance) [ Child screen icons ] (1) (2) Fig "Comparison Graph" Screen Icons (1) Close: Closes the "Comparison Graph" screen. (2) Show Equation: Displays the equation for the graph on the screen. [ Operations on the "Comparison Graph" screen ] (1) Click on the icon on the "ANOVA Table" screen (Fig. 2.25) to display the "Comparison Graph" screen. (2) Click on the icon to display the equation on the screen. Click on the icon again to cancel the equation display. (3) Click on the icon to close the "Comparison Graph" screen

53 2.6 Reanalysis 2.6 Reanalysis The "Reanalysis" screen calculates the estimated response values relative to each factor value, using the equation for the responses that were obtained. [ Screen configuration ] The "Reanalysis" screen consists of a reanalysis table. [ Child screen icons ] (1) (2) (3) Fig "Reanalysis" Screen Fig "Reanalysis" Screen Icons (1)Back to 'ANOVA Table': Returns the user to the "ANOVA Table" screen. (2)Add a Reanalysis Row: Adds a reanalysis row to the table. (3)Delete Reanalysis Row(s): Deletes a reanalysis row from the table. [ Operations on the "Reanalysis" screen ] (1) For each experiment number, the factor values and the corresponding response values are displayed, along with the estimated response values obtained from the response surface

54 2.6 Reanalysis equation. (2) Click on the icon to add a reanalysis row to the table. As each factor value is input into the cells, the estimated values obtained from the response surface equation are displayed. (3) The response surface equation ensures precision by using the upper and lower limit range values for the factor that were input on the "Factors and Interactions" screen as explained in section 2.2. That is to say, the basic rule for inputting factor values for reanalysis is that the values should be within the specified range. However, because occasions may arise when the user wants to estimate response values for factor values that lie outside the specified range, even if that means that the estimate will be correspondingly less precise, the system does permit the user to input factor values that are outside the specified range. However, when the user inputs factor values that are outside the specified range, the system displays a message box urging caution. In addition to the "Reanalysis" screen, the same message box is displayed on the "Sensitivity" and "Evaluation of Dispersion" screens when a factor value that is outside the specified range is input, because both sensitivity analysis and dispersion evaluation are based on the response surface equation. Fig "The value is out of the factor range" Message Box (4) Repeat step (2) for as many factors as you have to input. (5) If there are two or more responses and a factor value is input on one response sheet, the estimated values for the other responses are also calculated and displayed on the respective response sheets. (6) To delete an added reanalysis row, click on the row heading to select the row, then click on the icon to delete it. (7) Click on the icon to return to the "ANOVA Table" screen (Fig. 2.25)

55 2.7 Sensitivity Analysis 2.7 Sensitivity Analysis The "Sensitivity" screen displays the sensitivity values and sensitivity equations for the factors with respect to the response. A sensitivity equation is obtained for each factor by taking a partial derivative of the response surface equation for the factor, and it expresses the amount of change in the response per unit change in the factor. It can therefore be used to evaluate quantitatively the amount of change that will occur in the response if the factor is changed Sensitivity Values, Sensitivity Equations [ Screen configuration ] The lower portion of the "Sensitivity" screen displays a list of the variables and the corresponding factors, as well as sensitivity equations. The upper portion of the screen displays the sensitivity values obtained from the sensitivity equations. In the table of sensitivity values, the sensitivity of each factor with respect to the response is shown in the corresponding row. For example, in Fig. 2.42, the sensitivity of factor x2 with respect to the response "No Error" is shown in the "Sensitivity" column of the second row of the table of sensitivity values, the row labeled "x2". In this case, the sensitivity is "65", which means that if the value of factor x2 changes by 1, the value of the response "No Error" will change by 65. In this example, the system computes the sensitivity values by setting the values of all of the factors, x1 through x3, to 2 in the sensitivity equations. The table of sensitivity values and the sensitivity equations are displayed for one response at a time. A separate tab sheet is created for each response, with the response name displayed on the tab

56 2.7 Sensitivity Analysis Fig "Sensitivity" Screen [ Child screen icons ] (1) (2) (3) Fig "Sensitivity" Screen Icons (1) Back to 'ANOVA Table': Returns the user to the "ANOVA Table" screen. (2) Midpoint of Factor Value: Calculates the sensitivity value using the midpoint value for the factor. (3) Sensitivity Graph: Displays the sensitivity graph. [ Items for input and setting ] (a) The sensitivity value changes with the value of the factor. The default is that the sensitivity value with respect to the response is calculated for each factor using the midpoint value for the factor. However, if you want to obtain the sensitivity value using a different factor value, input the desired factor value in the appropriate cell. The sensitivity value will automatically be calculated and displayed. (b) Click on the icon to return to the "ANOVA Table" screen

57 2.7 Sensitivity Analysis [ Operations on the "Sensitivity" screen ] (1) On the "ANOVA Table" screen (Fig. 2.25), click on the icon to display the "Sensitivity" screen. (2) The sensitivity values that correspond to the midpoint values of the factors are displayed by default. If you want to obtain the sensitivity values using different factor values, input the desired factor values in the appropriate cells. Basically, the factor values to be input should be within the range of the upper and lower limits for the factor. Therefore, when the user inputs factor values that are outside the specified range, a message box (Fig. 2.41) will be displayed to urge caution. (3) If you want to display the sensitivity value that corresponds to the midpoint value of a given factor, select the desired factor row by clicking on the heading, then click on the icon to calculate and display the sensitivity value. (4) If you want to display the sensitivity values in graph form for a given factor, select the desired factor row by clicking on the heading, then click on the icon to display a sensitivity graph based on the sensitivity equation. (5) Click on the icon to return to the "ANOVA Table" screen (Fig. 2.25)

58 2.7 Sensitivity Analysis Sensitivity Graph The "Sensitivity Graph" screen displays a graph of the sensitivity of a selected factor for each of its level values. Multiple sensitivity graphs can be displayed simultaneously. [ Screen configuration ] The "Sensitivity Graph" screen is configured of a plot area to display the graph itself, a title area to display the response name, and a notes area to display the values for other factors. Title area Notes area Plot area [ Child screen icons ] (1) (2) Fig "Sensitivity Graph" Screen Fig "Sensitivity Graph" Screen Icons (1) CLOSE: Closes the "Sensitivity Graph" screen. (2) Edit Constants: Allows input of values for other factors

59 2.7 Sensitivity Analysis [ Operations on the "Sensitivity Graph" screen ] (1) On the "Sensitivity" screen (Fig. 2.42), click on the row heading of the factor for which you want to display the sensitivity graph, then click on the icon to display a graph of the sensitivity of the selected factor with respect to the response for each of the factor's level values. (2) When there are interactions between the selected factor and the other factors, the other factors are present in the sensitivity equations. So, in order to calculate the sensitivity values for the selected factor, it is necessary to input the values of the other factors into the table of sensitivity values. For example, to calculate the sensitivity value for factor x1, the values of factors x2 and x3 are required. The system default is that the calculation uses the midpoint values of the other factors to calculate the sensitivity and display the graph. (3) If you want to set values for the other factors other than the values shown in the table of sensitivity values, click on the icon to display the "Edit Interactive Factor Values" dialog box. Fig "Edit Interactive Factor Values" Dialog Box (4) Input factor values into the cells of the "Edit Interactive Factor Values" dialog box as necessary. Alternatively, select a cell for input, then move the slider to set the factor value that you want to use for calculating sensitivity. The slider can be used to set each factor to any value between its upper and lower limits. When the factor value is input directly into the cell, there are no restrictions on the value that is input, but the precision of the sensitivity equation that is derived from the response surface equation can be guaranteed only when the factor value lies between the upper and lower limits

60 2.7 Sensitivity Analysis For that reason, if a factor value is input that is outside the specified range, the message box shown in Fig is displayed as a warning. (5) If you click on the "Restore Defaults" button, the factor values will revert to the values initially displayed in the "Edit Interactive Factor Values" dialog box. (6) If you want to display multiple graphs, repeat step (1) above as necessary. (7) Click on the icon to close the "Sensitivity Graph" screen

61 2.8 Evaluation of Dispersion 2.8 Evaluation of Dispersion The system uses the first approximation/second moment method to analyze the influence of dispersion in the factors on dispersion in the response. The dispersion in the response is calculated using the dispersion in the factors, the coefficients of correlation between factors, and the sensitivity equations. The system can give the factor dispersion in two ways, as the coefficient of variation and as the standard deviation (tolerance). If the factor dispersion value is set to the standard deviation, then factor dispersion will be constant, regardless of the level values for the factor. By contrast, the factor dispersion can be made to vary with the level value of the factor by setting the factor dispersion value to the coefficient of variation, which is used as a measure of relative dispersion. For example, assume that the factor in question is the thickness of a sheet, and that there are three levels for the factor, 10 mm, 15 mm, and 20 mm. If the dispersion in sheet thickness is set to the standard deviation of 0.1 mm, then the dispersion in sheet thickness will be a constant 0.1 mm for all three levels, from 10 mm to 20 mm. On the other hand, if the dispersion in sheet thickness is set to the value of the coefficient of variation, 1% for example, then the dispersion in sheet thickness will be 0.1 mm when the thickness is 10 mm, 0.15 mm when the thickness is 15 mm, and 0.2 mm when the thickness is 20 mm Factor Dispersion [ Screen configuration ] The "Evaluation of Dispersion" screen is configured of two tab screens, one called "Factor Dispersion" and the other called "Correlation Coefficient"

62 2.8 Evaluation of Dispersion Fig "Evaluation of Dispersion" Screen Fig "Correlation Coefficient" Screen [ Items for input and setting ] (a)"factor Dispersion" tab - Input factor dispersion values. (b)"correlation Coefficient" tab - Click on the icon to enable the "Correlation Coefficient" tab. - Input the coefficient of correlation between factors as necessary. If no input is made, only autocorrelation will be considered (i.e., the coefficient of correlation with other factors is set to zero). The default is that only autocorrelation will be considered. (c) When the input step above has been completed, proceed to the "Dispersion Equation"

63 2.8 Evaluation of Dispersion screen or the "Dispersion Component Table" screen. (d) Click on the icon to return to the "ANOVA Table" screen. [ Child screen icons ] (1) (2) (3) (4) (5) (6) Fig "Evaluation of Dispersion" Screen Icons (1)Back to 'ANOVA Table': Returns the user to the "ANOVA Table" screen. (2)Estimated equation of dispersion: Displays the "Dispersion Equation" screen. (3)Dispersion Component Table: Displays the "Dispersion Component Table" screen. (4) Variation Coefficient: Allows the user to set the factor dispersion value to the coefficient of variation. (5) Standard Deviation: Allows the user to set the factor dispersion value to the standard deviation. (6)Correlation between factors: Enables the "Correlation Coefficient" sheet tab. [ Operations on the "Evaluation of Dispersion" screen ] (1) On the "ANOVA Table" screen (Fig. 2.25), click on the icon to display the "Evaluation of Dispersion" screen. (2) Input the coefficient of variation (%) or the standard deviation (tolerance) as the value for the factor dispersion. (3) If you input the coefficient of variation, click on the icon. The column label in the table will be displayed as "Variation Coefficient (%)". (4) If you input the standard deviation (tolerance), click on the icon. The column label in the table will be displayed as "Standard Deviation (tolerance)". (5) If there is factor dispersion for which input has not yet been made, a message box will be displayed with the message, "Some factor dispersions are empty. All empty values will be filled with '0'." (6) If the coefficient of correlation between factors is to be considered, click on the icon. The "Correlation Coefficient" tab will be enabled to allow values to be input. (7) Because the system default is that only autocorrelation will be considered, the default value of the correlation coefficient is zero, indicating no correlation between the factors. If you want to input the coefficient of correlation between two factors, input the value of the coefficient in the corresponding cell. (8) Click on the icon to display the "Dispersion Equation" screen (Fig. 2.50)

64 2.8 Evaluation of Dispersion (9) Click on the icon to display the "Dispersion Component Table" screen (Fig. 2.52). (10) Click on the icon to return to the "ANOVA Table" screen (Fig. 2.25) Dispersion Equation [ Screen configuration ] The "Dispersion Equation" screen is configured as a text box that displays the dispersion equation for the response and the dispersion component equation for each individual factor, which is obtained by setting the dispersion values for all other factors to zero. Fig "Dispersion Equation" Screen [ Child screen icons ] (1) Fig "Dispersion Equation" Screen Icons (1)Close: Closes the "Dispersion Equation" screen. [ Operations on the "Dispersion Equation" screen ] (1) After inputting the factor dispersion values on the "Evaluation of Dispersion" screen (Fig. 2.47) (or if necessary, after inputting the coefficients of correlation between the factors on the "Correlation Coefficient" screen (Fig. 2.48)), click on the icon to

65 2.8 Evaluation of Dispersion display the "Dispersion Equation" screen. (2) The "Dispersion Equation" screen displays the variance in the response and the components attributed to each factor in the variance in the response (i.e. the variance in the response when there is only one factor dispersion). (3) The standard deviation equation for the response is the square root of the dispersion equation (variance equation). (4) If there are multiple response values, the screen will display a tab for each response, and the user can obtain the dispersion equation for each response by clicking on the desired tab. (5) Click on the icon to close the "Dispersion Equation" screen Response Dispersion Component Table [ Screen configuration ] The "Dispersion Component Table" screen is configured of tab screens labeled "Dispersion of Response (SD)", "Estimated Value of Response", and "Variation Coefficient of Response (%)". Fig "Dispersion Component Table" Screen [ Child screen icons ] (1) (2) (3) Fig "Dispersion Component Table" Screen Icons (1)Close: Closes the "Dispersion Component Table" screen. (2) Midpoint of Factor Value: Sets the factor value in the dispersion component table to the

66 2.8 Evaluation of Dispersion midpoint value for the factor. (3)Dispersion Graph: Displays the dispersion graph. [ Operations on the "Dispersion Component Table" screen ] (1) On either the "Evaluation of Dispersion" screen (Fig. 2.47) or the "Correlation Coefficient" screen (Fig. 2.48), click on the icon to display the "Dispersion Component Table" screen. (2) The "dispersion component" is a value that indicates how much of the dispersion in the response can be attributed to dispersion in an individual factor (when there is no dispersion in the other factors). For example, the standard deviation obtained from the variance equation shown under the heading "[Dispersion attributed to x1]" on the "Dispersion Equation" screen (Fig. 2.50) is displayed for factor x1 in the first row of the "Dispersion Component Table" screen (Fig. 2.52). (3) The user can display the "Dispersion of Response (SD)", "Estimated Value of Response", and "Variation Coefficient of Response (%)" for each factor level value by clicking on the respective sheet tabs. (4) The dispersion in the response (standard deviation) is the square root of the value obtained from the dispersion component equation (variance equation) shown on the "Dispersion Equation" screen (Fig. 2.50), and it is calculated separately for each level of each factor. If there are other factors present in the dispersion equation, the dispersion that is calculated will vary according to the values of those factors. The default is that the midpoint values for each factor are used to calculate the dispersion in the response. (5) If you want to change the values of the other factors to something other than the midpoint values, input the desired values directly into the corresponding cells. If you input a numerical value that is outside the range defined by the upper and lower limit values for the factor, the warning message box shown in Fig will be displayed to urge caution. (6) To revert the value of a factor to the midpoint value, select the factor by clicking on the row heading, then click on the icon. (7) The estimated value of the response is the value calculated by the response surface equation that is displayed on the "ANOVA Table" screen (Fig. 2.25). (8) The variation coefficient of the response is calculated by dividing the response standard

67 2.8 Evaluation of Dispersion deviation by the estimated value of the response and expressing it as a percentage. (9) If you want to display a dispersion graph for a specific factor, click on the row heading for that factor to select it, then click on the icon. A graph of the factor's influence on dispersion in the response will be displayed. (10) Click on the icon to close the "Dispersion Component Table" screen Dispersion Graph The "Dispersion Graph" Screen displays a graph of the response dispersion when there is dispersion in only one factor. [ Screen configuration ] The "Dispersion Graph" screen is configured of three tab screens, called "Response Dispersion", "Standard Deflection", and "Variation Coefficient" respectively. Each tab screen is configured of a plot area to display the graph itself, a title area to display the response name, a notes area to display the values for other factors, and an area for the legend. Title area Notes area Area for the legend Plot area Fig "Dispersion Graph" Screen

68 2.8 Evaluation of Dispersion [ Child screen icons ] (1) (2) Fig "Dispersion Graph" Screen Icons (1)Close: Closes the "Dispersion Graph" screen. (2)Edit Constants: Allows input of values for other factors. [ Operations on the "Dispersion Graph" screen ] (1) On the "Dispersion Component Table" screen shown in Fig. 2.52, select the factor for which you want to display a dispersion graph by clicking on the corresponding row heading. Next, click on the icon to display the dispersion graph. The graph shows the response dispersion when there is dispersion only in the selected factor (and no dispersion in the other factors). (2) The user can view the "Response Dispersion", "Standard Deflection", and "Variation Coefficient" graphs by clicking on the corresponding tabs. (3) The "Response Dispersion" graph displays the estimated values for the response, plus or minus the standard deviation. (4) Graphs of the standard deviation and the coefficient of variation for the response value are displayed on the "Standard Deflection" and "Variation Coefficient" tab screens, respectively. (5) When there are interactions between the selected factor and the other factors, the other factors are present in the dispersion component equations, so in order to calculate the response dispersion values for the selected factor, it is necessary to input the values of the other factors. The system default is to use the midpoint values of the other factors. If you want to set the values for the other factors to something other than the default values, input the desired values on the "Dispersion Component Table" screen shown in Fig (6) If you click on the icon on the "Dispersion Graph" screen (Fig. 2.54), the "Edit Interactive Factor Values" dialog box (Fig. 2.56) will be displayed. If you want to change the values of the other factors, input the desired values directly into the appropriate cells, or move the slider to set the factor values that you want to use to calculate the dispersion. The slider can be used to set each factor to any value between its upper and lower limits. When the factor value is input directly into the cell,

69 2.8 Evaluation of Dispersion there are no restrictions on the value that is input, but if a factor value is input that is outside the specified range (upper and lower limit range), the warning message box shown in Fig is displayed to urge caution. (7) If you want to display multiple graphs, repeat step (1) above as necessary. (8) Click on the icon to close the "Dispersion Graph" screen. Fig "Edit Interactive Factor Values" Dialog Box

70 2.9 Optimization Calculation 2.9 Optimization Calculation The Optimization screen carries out the optimization calculation to figure the minimum or maximum values for the objective function based on the given initial values and constraints, and finds the factor value when the optimum value is obtained. (1) Optimization method - The system can carry out optimization in two different ways, by optimizing the factors as continuous variables, or by optimizing the factors as discrete variables. - For optimization of continuous variables, the system uses the successive second approximation method. For optimization of discrete variables, the system uses the round-robin calculation method for all possible combinations. - Optimization of the factors as discrete variables is performed only when the values of the factors can be nothing but discrete values, for example, when the values of factors X 1, X 2, and X 3 are defined as 1.0, 1.2, and 1.4 respectively. - The user can choose between using continuous variables or discrete variables by clicking on the and icons respectively on the "Optimization" screen (Fig. 2.57). Note that it is not possible to set individual factors as continuous or discrete variables. (2) Initial values - The system uses the successive second approximation method for optimization of continuous variables, but the overall optimum value is not always obtained by this method. In some cases, only a limited optimum value in the vicinity of the initial value is obtained. That is to say, if the initial value is changed, a different optimum value may be obtained. Therefore, the calculation is repeated several times using different initial values, and the minimum or maximum value among all of the optimum values thus obtained is taken to be the overall optimum value. - It is possible to change the number of initial values for each individual factor. In the optimization of continuous variables, convergence calculations are carried out repeatedly, and the amount of time required to perform the calculations depends on the length of the objective function and on the number and length of the behavioral constraint functions. It is therefore recommended that the user set the number of initial values for every factor to "1" and perform one round of calculation in order to check the calculation time before increasing the number of initial values. (3) Constraints

71 2.9 Optimization Calculation - The system sets constraints in the form of two types of constraint functions, side constraint functions and behavioral constraint functions. Side constraint functions prescribe the constraints based on the upper and lower limit values for the factors. Behavioral constraint functions are set when optimization is carried out using constraints other than the side constraint functions. The response surface equation or user equations can be used for the behavioral constraint functions. - The precision of the response surface equation that is obtained from ANOVA is guaranteed when the factor values are between their upper and lower limit values. Conversely, when the factor values lie outside the range specified by the upper and lower limit values, the precision of the response surface equation becomes worse. Therefore, when the optimization calculation is carried out with the response surface equation being used for the objective function or the behavioral constraint functions, the system performs the calculation under the constraint that the factor values must lie within the range specified by the upper and lower limit values. This condition is set by the side constraint functions. - The side constraint functions are set automatically using the upper and lower limit values. However, because the user may want to perform the optimization calculation using a broader range of values for the side constraint functions, even though the precision of the response surface equation will be sacrificed to some extent, the system makes it possible to change the upper and lower limit values for the factors. - The system is capable of handling behavioral constraint functions that are either equality constraint functions (function = value) or an inequality constraint functions (function >= value, function <= value). - When the response surface equation is used for the behavioral constraint functions, the dispersion in the response can be taken into account if the dispersion equation has been calculated. (4) Objective function - The number of objective functions that the system can handle is "1". - The objective function can be set using either the response surface equation that was derived by ANOVA or a user equation. - The user can switch between minimization or maximization of the objective function

72 2.9 Optimization Calculation - When the response surface equation is used for the objective function, the dispersion in the response can be taken into account if the dispersion equation has been calculated. (5) User equations - The user may independently define an equation other than the response surface equation that was derived by ANOVA to use for the objective function or behavioral constraint function. - The user can define a function using the factors as variables on the "User Equations" screen (Fig. 2.62). [ Screen configuration ] The "Optimization" screen is configured of four tab screens, called "Continuous Variables", "Discrete Variables", "Constraints/Objective Function", and "Optimum Calculation" respectively. However, because the user can switch between the "Continuous Variables" tab screen and the "Discrete Variables" tab screen by using the icons, only three tab screens are displayed. Fig "Optimization" Screen

73 2.9 Optimization Calculation [ Child screen icons ] (1) (2) (3) (4) (5) (6) Fig "Optimization" Screen Icons (1)Back: Returns the user to the "ANOVA Table" screen. (2)User Equations: Allows the user to define user functions. (3) Continuous Variables: Click on this icon to carry out optimization using continuous variables. (4)Discrete Variables: Click on this icon to carry out optimization using discrete variables. (5)Run an Optimal Calculation: Carries out optimization. (6)Stop the Optimal Calculation: Stops optimization. [ Items for input and setting ] (a) "Continuous Variables" tab screen - Select the method of setting the initial values. - Input the number of initial values for each factor. (b) "Discrete Variables" tab screen - Input the number of discrete values for each factor. - Input the discrete values for each factor. (c) "Constraints/Objective Function" tab screen - Select an objective function and the direction of optimization for the objective function. - If optimization is to be carried out taking into account the dispersion in the objective function, input a multiple of the standard deviation. - Select the behavioral constraint functions and conditions, and input the constraint values. - If optimization is to be carried out taking into account the dispersion in the behavioral constraint functions, input a multiple of the standard deviation. (d)after inputting the data as described above, carry out the optimization calculation. The results will be displayed on the "Optimum Calculation" tab screen. (e)click on the icon to return to the "ANOVA Table" screen shown in Fig

74 2.9 Optimization Calculation Setting the Initial Values (A) Setting the initial values when the factors are continuous variables Fig Setting Initial Values as Continuous Variables (1) Click on the icon on the "ANOVA Table" screen (Fig. 2.25) to display the "Optimization" screen. (2) Select the method of setting the initial values. - If you select "Equally spaced points in the range of upper-lower", the initial values will automatically be set at equally spaced points from the lower limit value to the upper limit value. - If you select "Any points in the range of upper-lower", you can set the initial values as you wish. To do so, input the desired values directly into the cells. (3) If you want to change the number of initial values for each factor, input a numerical value in the "Number of Values" cell. The number of columns for inputting initial values will automatically be increased or decreased as necessary to match the number of initial values. (4) Repeat the initial values setting procedure for each individual factor. For example, if the number of initial values for factor x1 is two, the number of initial values for factor x2 is two, and the number of initial values for factor x3 is three, the total number of combinations of initial values is twelve. Therefore, optimization will be carried out with these twelve combinations of initial values as the departure points. (5) If you want to change the side constraint functions even though the precision of the response surface equation will drop accordingly, you can change the values of the

75 2.9 Optimization Calculation upper and lower limits. - Select the factor for which you want to change the upper and lower limit values by clicking on the corresponding row heading. - If you right click on the sheet, a pop-up menu will appear. Select "Edit Upper/Lower Limits" from the menu. - The "Edit Upper/Lower Limits" dialog box (Fig. 2.60) will be displayed. Input the desired values for the upper and lower limits. Fig "Edit Upper/Lower Limits" Dialog Box (B) Setting the initial values when the factors are discrete variables Fig Setting Initial Values as Discrete Variables (1) Click on the icon to display the "Discrete Variables" tab screen

76 2.9 Optimization Calculation (2) By default, the "Number of Values" cell for each factor is set to the number of levels for that factor, and the "Discrete Value" cells are set to the level values. If you want to change the number of discrete values for any factor, input a numerical value in the "Number of Values" cell for that factor. The number of columns for inputting discrete values will automatically be increased or decreased as necessary to match the number of discrete values. Input the discrete values. (3) Repeat the discrete values setting procedure for each individual factor. (4) For example, if the number of discrete values for factor x1 is two, the number of discrete values for factor x2 is three, and the number of discrete values for factor x3 is three, the total number of combinations of discrete values is eighteen. Therefore, the constraints and the objective function will be calculated for these eighteen combinations of discrete values. The combination that satisfies the constraints and indicates minimum or maximum objective function is chosen from among the results to be the optimum value. (5) If you want to change the side constraints even though the precision of the response surface equation will drop accordingly, change the values of the upper and lower limits. The method of changing the values is the same as for continuous variables User Equations If the user wants to use functions which he/she defines for the behavioral constraint functions or the objective function, the "User Equations" dialog box can be used to define and register those functions. [ Screen configuration ] The screen for the "User Equations" dialog box is configured of a table called "Variables", a text box called "Name" for inputting the equation name, a text box called "Equation" for inputting the equation itself, and a list box called "User Equations"

77 2.9 Optimization Calculation Fig "User Equations" Dialog Box [ Operations on the "User Equations" dialog box ] (1) Click on the icon on the "Optimization" screen to display the "User Equations" dialog box (Fig. 2.62). (2) Input the name of the function you want to define in the "Name" text box. (3) Input the function you want to define in the "Equation" text box, using the variable names for the factors that are displayed in the "Variables" table. Be careful to start each variable name with an "x" (either upper or lower case letters may be used), so that the variable names will be x1, x2, and so on. (4) If you click on the "Register" button, the user equation will be registered and the function name will be displayed in the "User Equations" list box. (5) Click on the "Finish" button to close the "User Equations" dialog box. (6) It is possible to edit or delete the registered user equations. If you click on the function name in the "User Equations" list box, the function name and the equation will be displayed in the "Name" text box and the "Equation" text box respectively. - If you change the name in the "Name" text box and click on the "Register" button, the function will be registered as a new function. - If you change the function in the "Equation" text box without changing the name in

78 2.9 Optimization Calculation the "Name" text box, and then click on the "Register" button, a message box will be displayed with the message, "Replace existing user equation?" Click on the "OK" button to replace the existing equation and register the new one. - To delete a registered user equation, select the equation by clicking on the function name in the "User Equations" list box, then click on the "Delete" button. - If you click on the "New" button, the contents of the "Name" text box and the "Equation" text box will be cleared. List of functions that can be used for user equations - Computational operators: +, -, *, /, ^, (, ) - ABS: Absolute value of a numerical value ABS( ) - COS: Cosine of a specified angle (rad) COS( ) - COSH: Hyperbolic cosine of a specified numerical value COSH( ) - EXP: Power to which a specified numerical value having e as a base is raised EXP( ) - LN: Natural logarithm of a numerical value (logarithm having e as a base) LN( ) - LOG10: Common logarithm of a numerical value (logarithm having 10 as a base) LOG10( ) - PI: Circular constant PI( ) - SIN: Sine of a specified angle (rad) SIN( ) - SINH: Hyperbolic sine of a specified numerical value SINH( ) - SQRT: Square root of a specified numerical value SQRT( ) - TAN: Tangent of a specified angle (rad) TAN( ) - TANH: Hyperbolic tangent of a specified numerical value TANH( ) The circular constant PI does not have an argument, but when using it, please be sure to add parentheses after it, so that it reads "PI( )"

79 2.9 Optimization Calculation Setting the Objective Function and Behavioral Constraint Function Click on the "Constraints/Objective Function" tab on the "Optimization" screen to set the objective function and the behavioral constraint function. (A) Setting the objective function (1) Click on the "Equations Name" combo box to display a list of the names of the response surface equation and the user equations. Select the desired equation from the combo box. The user equation names that will be displayed are those that have been defined as described in section (2) Select the direction of optimization of the objective function by clicking on either the "Minimize" button or the "Maximize" button. (B) Setting the behavioral constraint function (1) Click on the "Constraints" cell to display a list of the names of the response surface equation and the user equations. Select the equation you want to use as a constraint from among the names on the list. (2) Click on the "Condition" cell to display the computational operators "=", "<=", and ">=". Select the desired operator. (3) Input the constraint value in the "Constraint Value" cell. Fig Setting the Behavioral Constraint Function and Objective Function (C) Setting the objective function and behavioral constraint function when dispersion is

80 2.9 Optimization Calculation taken into account If the dispersion equation has been calculated, the optimization calculation for the objective function and the behavioral constraint function can be performed taking into account the dispersion in the response. If the dispersion equation has not been calculated, the "Multiple of Standard Deviation" text box and the "Multiple of Standard Deviation" column of the "Constraints" table will be disabled. (a) Setting the dispersion for the objective function (1) If the response surface equation is selected as the objective function, the "Multiple of Standard Deviation" text box will be enabled. If you want to take dispersion into account, input the multiple of standard deviation. (2) If the optimization direction is set to "Minimize", the objective function will be set as follows: Response surface equation + Multiple of standard deviation Response standard deviation equation (3) If the optimization direction is set to "Maximize", the objective function will be set as follows: Response surface equation - Multiple of standard deviation Response standard deviation equation (b) Setting the dispersion for the behavioral constraint function (1) If the response surface equation is selected as the constraint, the "Multiple of Standard Deviation" column of the "Constraints" table will be enabled. If you want to take dispersion into account, input the multiple of standard deviation. (2) If the condition is set to "=", the behavioral constraint function will be set as follows: Response surface equation + Multiple of standard deviation * Response standard deviation equation (3) If the condition is set to "<=", the behavioral constraint function will be set as follows: Response surface equation + Multiple of standard deviation * Response standard deviation equation (4) If the condition is set to ">=", the behavioral constraint function will be set as follows: Response surface equation - Multiple of standard deviation Response standard deviation equation

81 2.9 Optimization Calculation Fig Setting the Behavioral Constraint Function and Objective Function when Dispersion Is Taken into Account Optimization Calculation When you have finished making the settings for the initial values, the objective function, and the behavioral constraint function, the optimization calculation can be carried out

82 2.9 Optimization Calculation (A) Optimization calculation for continuous variables Fig Optimization Results for Continuous Variables (1) Click on the icon to start the optimization calculation. The status of the calculation will be displayed in the panel at the upper right of the screen. (2) The user can see the relationship of the factors and variables, the constraints and their numbers, and the objective function by referring to the "Factors", "Constraints", and "Objective Function" boxes, respectively. (3) When the calculation is completed, the results will be displayed in the table in the lower part of the screen. (a) Convergence: "OK" means that the calculation results converged. The cell will display "err" if the calculation results did not converge using the initial values. (b) Optimum Value: The objective function derived from the calculation is displayed. This represents the optimum value if the "Convergence" cell displays "OK". (c) X1opt, etc.: These cells display the factor values when the optimum value is obtained. (d) X1ini, etc.: These cells display the initial factor values at the start of the

83 2.9 Optimization Calculation optimization calculation. (e) Constraint 1, etc.: These cells display the values for the left side of each constraint function listed in the "Constraints" list box. (4) Because the successive second approximation method does not always obtain the overall optimum value, it is necessary to perform the optimization calculation for several different sets of initial values. In this case, the minimum or maximum optimum value for which the convergence status is "OK" will be redisplayed in the first row. If the same optimum value is obtained for more than one set of initial values, the optimum value with the lowest sequence number will be displayed. If all the calculations have a convergence status of "err", the word "Nothing" will be displayed in the optimum value cells. (5) Click on the icon while the calculation is running to stop the calculation. (B) Optimization calculation for discrete variables Click on the icon to start the optimization calculation. The results will be displayed on the "Optimum Calculation" tab sheet. (1) The display of the calculation results (Fig. 2.66) will differ from that for optimization of continuous variables. The results will be displayed in columns for the objective function values, the discrete factor values ("X1ini", etc.), and the constraint function values ("Constraint 1", etc.). (2) The objective function values and constraint function values calculated for each combination of discrete values are displayed in the corresponding rows. (3) In the case of discrete variables, the system does not perform optimization using the mathematical programming method, but rather calculates the objective function and the constraint function for each discrete value of each factor. Therefore, if a value for the left side of each equation in the "Constraints" list box is not zero for equality constraint functions and not a positive value for inequality constraint functions, the constraint conditions have not been satisfied. (4) The minimum or maximum objective function value that satisfies the conditions stated in item (3) above is displayed as the optimum value in the first row. If there are no results that satisfy the constraint conditions, the word "Nothing" will be displayed in the objective function column

84 2.9 Optimization Calculation Fig Optimization Results for Discrete Variables Operations for the Sample Problem (1) Optimization method Carry out optimization for continuous variables. (2) Setting initial values In the case of the sample problem, the calculation time for one initial value is short, so perform the calculation for two initial values for each factor. The initial values will be set automatically by selecting "Equally spaced points in the range of upper-lower". Also, the six individual constraint functions listed below will automatically be set as the side constraint functions, based on the upper and lower limit values. X 1 1 >= 0, 3 X 1 >= 0 (1 <= X 1 <= 3) X 2 1 >= 0, 3 X 2 >= 0 (1 <= X 2 <= 3) X 3 1 >= 0, 3 X 3 >= 0 (1 <= X 3 <= 3)

85 2.9 Optimization Calculation Fig Setting Initial Values (3) Setting the user equation In the "User Equations" dialog box (Fig. 2.62), define the function X 1 + X 2 + X 3 for use as a behavioral constraint function and name it "Equality Behavioral Constraint Function". (4) Setting the objective function On the "Constraints/Objective Function" tab screen (Fig. 2.63), set the "No Error" response as the objective function, and select "Minimize" for the direction of optimization. The response dispersion will not be taken into account. (5) Setting the behavioral constraint function On the "Constraints/Objective Function" tab screen (Fig. 2.63), set the equality condition for the registered user equation "Equality Behavioral Constraint Function" to "=5". (6) Optimization calculation The results of the optimization calculation carried out using the settings made in steps (1) through (5) above will be displayed on the "Optimum Calculation" tab screen as shown in Fig When x1 = 1, x2 = 1, and x3 = 3, the minimum value for the "No Error" response is 69. Note that there are other combinations of x1, x2, and x3 that yield the minimum value, but in