CHAPTER 2 TAGUCHI OPTIMISATION TECHNIQUE

Transcription

1 8 CHAPTER 2 TAGUCHI OPTIMISATION TECHNIQUE 2.1 OVERVIEW OF TAGUCHI METHOD The Design of Experiments (DOE) is a powerful statistical technique introduced by Fisher R. in England in the 1920s (Ross 1996), to study the simultaneous effect of multiple variables on the objective function. In his early applications, Fisher wanted to find out how much rain, water, fertiliser, etc. are needed to produce the best crop. Since that time, more development of the technique has taken place in the academic environment, and this did help generate many applications on the production floor. As a researcher in the Electronic Control Laboratory in Japan, Genichi Taguchi carried out significant research with DOE techniques in the late 1940s (Montgomery 2001). He spent considerable time to make this experimental technique user-friendly and applied it to improve the quality of the manufactured products. Taguchi's standardised version of the DOE, popularly known as the Taguchi method or Taguchi approach, is one of the most effective quality building tools used by engineers in all types of manufacturing activities. His main contribution lies not in the mathematical formulation of the design of experiments, but rather in the accompanying philosophy. His concepts produced a unique and powerful quality improvement technique that differs from traditional practices. He developed manufacturing systems that were robust, which means insensitive to daily and seasonal variations of environment, machine wear and other external

2 9 factors. Taguchi viewed quality improvement as an ongoing effort. He continually strived to reduce the variation around the target value. The first step towards improving quality is to achieve the population distribution as close to the target value as possible. To accomplish this, Taguchi designed experiments using specially constructed tables known as Orthogonal Arrays (OA). The use of these tables makes the design of experiments very easy and consistent. The DOE using the Taguchi approach can economically satisfy the needs of problem solving, and product or process design optimisation projects. This technique can significantly reduce the time required for experimental investigations. In the Taguchi technique, factors are defined as different variables, which determine the efficiency of any industrial process. For example, in a DCT process, a factor can be the cooling rate, soaking temperature, soaking time, tempering temperature or tempering time. Each factor may be set to different levels. In this study, for the same experiment, the levels of soaking temperature can be -184 o C, -150 o C or -120 o C, etc depending upon the material. Similarly, the other factors may also operate at different levels. In any experimental design, the term interaction is used to describe a condition in which the influence of one factor upon the result is dependent on the level of another. The concept of interaction could be best understood by considering the following example. Temperature and humidity are two important factors that influence human comfort. An increase in the temperature alone may cause slight discomfort to the human but the degree of discomfort increases as the humidity increases. So the levels of the factors, temperature and humidity, are said to have a strong interaction in influencing human comfort.

3 ORTHOGONAL ARRAYS In an industrial process, if there is n number of factors involved, which can take p different levels, a full factorial design may involve p n number of experiments. A full factorial design will identify all possible combinations of the considered list of factors. For example, considering a design of experiment with three variables (factors A, B and C), each of which can be set at two different levels, 1 and 2, a full factorial experiment requires 2 3 = 8 experiments, as shown in Table 2.1. Table 2.1 Full factorial experiment table for a process with three factors each operating at two levels Experiment No. A B C Similarly, in an experiment involving 7 factors, each at two levels, the total number of full factorial combinations will be 128 (2 7 ). However in order to minimize the number of experiments, Taguchi developed an effective design of experiment technique, using the Orthogonal Arrays (OA) he constructed to lay out the experiments.

4 11 The nomenclature of OA is as follows: L a (b c ) where, L : Latin square a : No. of rows (No. of experiments) b : No. of levels of factors c : No. of columns (No. of factors and interactions) By combining the existing orthogonal Latin squares in a unique manner, Taguchi prepared a new set of standard OAs which could be used for a number of experimental situations. He also devised a standard method for the analysis of the results. A single OA may accommodate several experimental situations. Commonly used OAs are available for 2, 3 and 4 levels. A combination of the standard experimental design techniques and analysis methods in the Taguchi approach produces results that are consistent and reproducible similar to those obtained by conducting all the full factorial experiments. For the same 3-factor, 2-level process mentioned above, the L 4 (2 3 ) OA shown in Table 2.2 can be applied, based on the Taguchi design which calls for 4 experiments instead of 8 (2 3 ) experiments in the full factorial design. Table 2.2 L orthogonal array for 3 factor 2 level experiment Experiment No. A B C

5 12 Similarly for the 7-factor 2 level process example mentioned above, the L 8 (2 7 ) OA shown in Table 2.3 can be applied, which calls for 8 experiments instead of 128 (2 7 ) experiments needed in the full factorial design. The selection of the particular OA is predefined by Taguchi for the given number of factors and the levels of each factor, according to the situation. Table 2.3 L orthogonal array for 7 factor 2 level experiment Experiment Factors No. A B C D E F G Table 2.4 shows the comparison of the number of experiments required in the full factorial design and the Taguchi design for a few cases. Table 2.4 Comparison of the full factorial and the Taguchi design Number of Factors Number of Levels Number of Tests in Taguchi Design Number of Tests in Full Factorial Design 3 2 4, using L 4 (2 3 ) OA 2 3 = , using L8(2 7 ) OA 2 7 = , using L 12 (2 11 ) OA 2 11 = , using L 15 (2 15 ) OA 2 15 = , using L 27 (3 13 ) OA 3 13 =

6 Properties of Orthogonal Array The L 8 (2 7 ) Orthogonal Array (OA) shown in Table 2.3 is used to design experiments involving up to seven factors at 2 levels. The array has 8 rows and 7 columns. Each row represents a trial condition (experiment) with the combination of levels of factors indicated by the numbers in the row. The vertical columns correspond to the factors specified in the study. An experiment is said to be balanced if it is balanced with respect to each factor under investigation. In the L 8 (2 7 ) OA, each column contains four level 1 and four level 2 conditions for the factors assigned to the column. It is easy to see that all columns provide four tests under the first level of the factor, and four tests under the second level of the factor. The idea of balance ensures that an equal chance is given to each level of each variable. Similarly, equal attention is to be given for the combinations of two variables. For example, in L 8 (2 7 ) OA, two factors at 2 levels combine in four possible ways: (1,1), (1,2), (2,1) and (2,2) It can be noted that any two columns of an L 8 (2 7 ) OA have the same number of combinations of (1,1), (1,2), (2,1) and (2,2). This is one of the features that provide orthogonality among all the columns (factors). When two columns of an array form these combinations, the same number of times (two times in this case), and all columns provide the same number of tests under the first level and the second level of the factor, then the columns are said to be balanced and orthogonal. Thus, all seven columns of an L 8 (2 7 ) array are orthogonal to each other.

7 14 In the Taguchi design, the array is orthogonal, which means that the design is balanced, so that the factor levels are weighted equally. The real power in using an OA is the ability to evaluate several factors in a minimum number of tests. This is considered as an efficient experiment since more information is obtained from a few trials. The array forces all experimenters to design identical experiments. Experimenters may select different designations (factors or interaction of factors) for the columns but the eight trial runs will include all combinations independent of the column designation. Thus the OA assures consistency of design by different experimenters. To design an experiment, the most suitable orthogonal array as predefined by Taguchi is selected. Next, factors are assigned to the appropriate columns, and finally, the combinations of the individual experiments (called the trial conditions) are described. For a process involving seven 2 level factors, namely, A, B, C, D, E, F and G, these are assigned to columns 1, 2, 3, 4, 5, 6 and 7 respectively of an L 8 (2 7 ) array (Table 2.3). The table identifies the eight trials needed to complete the experiment and the level of each factor for each trial run. Each experimental condition is determined by reading numerals 1 and 2 appearing in the rows of the trial runs. A full factorial experiment would require 2 7 or 128 runs, but would not provide appreciably more information than what is realised through the experiments conducted by the L 8 (2 7 ) Taguchi OA. 2.3 TAGUCHI APPLICATION STEPS The Taguchi method is used to improve the quality of products and processes. Improved quality is said to be attained when a higher level of performance is consistently obtained. The best possible performance is obtained by determining the optimum combination of the levels of different design factors combination using the design of experiment principles as per the steps given below.

8 Brainstorming Brainstorming is an activity which promotes group participation, encourages creative thinking and generates many ideas in a short period of time. The brainstorming stage is perhaps the most important stage of the whole Taguchi procedure. At this stage, clear statements of the problem are established; the objectives, the desired output characteristics, the methods of measurement, factors influencing the problem, and levels of factors; and the appropriate experiments are designed Designing and Running the Experiment Using the factors and levels decided in the brainstorming session, the experiments could now be designed, and the method of carrying them out will be established. The steps in designing the experiment are: Select the appropriate orthogonal array. Assign the factor and interaction factors to columns. Conduct the experiments in an unbiased manner Analysing Results The data obtained from the experiment is analysed using ANOVA which stands for Analysis-of-Variance. ANOVA is a statistical model meant to analyse a process data to determine whether a significant dependence exists between the response (ie the process outcome) and the factors considered to be influencing the process. In the present study, ANOVA methods are used to compare the degree of influence of the levels of different treatment parameters on the improvements in wear resistance of the material.

9 16 ANOVA is an analysis of the variation present in an experiment. A measure of the total variability in a set of data is given by the sum of squared differences of the observations from their overall mean. This is the total sum of squares (TSS). It is often possible to subdivide this quantity into components that are identified with different causes of variation. The full subdivision is usually set out in an analysis of variance table. Each row of the table is concerned with one or more of the components of the observed variation. The entries on a row usually include the sum of squares (SS), the corresponding number of degrees of freedom ( ), and their ratio, the mean square. After the contributions of all the specified sources of variation have been determined, the remainder, often called the residual sum of squares (RSS) or error sum of squares, is attributed to random variation. The mean square corresponding to RSS is often used as the yardstick for assessing the importance of the specified sources of variation.. In the present work ANOVA is carried out to determine the following: The optimum level for the DCT parameters. Influence of individual factors on the improvement of the wear resistance. Predicting the response (wear resistance) at the optimum levels of factors (Optimum treatment cycle) Confirmatory Test After optimising the factors, a confirmatory test with the levels of factors set at optimum levels obtained from the ANOVA, has to be conducted to check whether the results are within the confidence interval.

10 Iterative Taguchi Design The need of iterative Taguchi design starts with the complexity of the process to be optimised. If the factors influencing a process are to be optimised from the available discrete levels, then a single iteration as discussed in the previous section is enough to arrive at optimum levels. But in certain problems the factors are to be optimised over a range. In this study, the soaking temperature may fall in the range of - 80 o C to o C, the soaking period may fall above 6 hours and so on. For such cases the iterative Taguchi design may be adopted. In the first iteration (DOE1), typically a 2 level OA is to be selected depending upon the number of factors. In this study, an L 16 (2) 15 array, was chosen for conducting the experiment for the DOE1. The two levels of the selected factors should represent reasonable extremes for the DOE1. After fixing the values for the levels of each factor, experiments are to be conducted to identify the significant factors and to arrive at the optimum level among the selected levels of significant factors. In this study, in the first iteration, the two levels selected for the soaking temperature, are - 80 o C and -184 o C. From the DOE1 the best soaking temperature among these two levels is arrived as -184 o C. The optimal level values of the significant factors arrived from the previous iteration are to be used as central values or reference values for the next iteration, (DOE2). For DOE2, the factor level difference is reduced and three or four levels of the significant factors around the central values are to be selected. In the present study, the optimum level arrived for the soaking period from the DOE1 is 24 hours. So for the DOE2, three levels fixed for the soaking period are 12 hours, 24 hours and 36 hours. Select the appropriate OA depending upon the number of level and conduct experiment to arrive at the optimum levels. In this study, an L 9 (3) 4 OA is selected for DOE2. The iterative optimisation process would be repeated until the design goal

11 18 (convergence of the levels of factors selected in the consecutive iterations) is obtained. In the present study, only two iterations are used. 2.4 DESIGNING THE EXPERIMENT The primary aim of the Taguchi analysis is to: 1. Determine the influence of the factors on the response and the percentage contribution of the factors. 2. Find the optimum value of the influencing factors. 3. Determine the expected result at the optimum conditions. In the example of an experimenter, who has identified three controllable factors for a heat treating process, each factor can have two levels as shown in Table 2.5. The experimenter wants to determine the optimum combination of the levels of these factors and to know the contribution of each to the product quality. Table 2.5 Factors and levels for the heat treating process Factors / Level A (Temperature) B (Time) C (Quenching medium) Level o C 1 h Water Level o C 3 h Oil Since there are 3 factors, each at 2 levels, an L 4 (2 3 ) OA can be used as shown in Table 2.2. Since an L 4 (2 3 ) OA has 3 columns, 3 factors can be assigned to these columns in any order. Having assigned the factors, their levels can also be indicated in the corresponding column. There are four independent experimental conditions in an L 4 (2 3 ) OA. These conditions are

12 19 described by the numbers in the rows. A full set of experiments for this process would require eight different experiments (2 3 ) but there are only four experiments which are needed as per L 4 (2 3 ) OA. The saving involved in using the Taguchi method becomes more significant as the number of levels or factors increases. design experiments: The following standard orthogonal arrays are commonly used to 2-Level Arrays: L4, L8, L12, L16, L32 3-Level Arrays: L9, L18, L27 4-Level Arrays: L16, L32 In the present work, an L16, 2 level array for the DOE1 and an L9, 3 level array for the DOE2 were used. 2.5 DESIGNS WITH INTERACTION The design of experiments using the Taguchi OA is simple and straightforward, when there is no need to include interactions. It requires a little more care to design an experiment where interactions are to be included Linear Graph of Interaction Each OA has a linear graph associated with it. Linear graphs are made up of numbers, dots and lines, where a dot and its assigned number identifies a factor, a connecting line between two dots indicates an interaction, and the number assigned to the line indicates the column number in which interaction effects will be compounded. The Linear graph for the L 16 (2 15 ) OA is shown in Figure 2.1.

13 20 Figure 2.1 Linear graph for the L orthogonal array As seen from the figure, the interaction effect of factors assigned in columns 1 and 2 is assigned to column 3. If A & B are assigned to columns 1 and 2 respectively, column 3 is assigned to the interaction between factors A and B mentioned as AxB. Similarly the interaction between factors A and E is mentioned as AxE and assigned to column 14 and so on Assigning Factors and Interactions to Columns The factors and their levels selected in this study, for the analysis of the DOE1 are shown in Table 2.6. The interactions considered for the DOE1 are shown in Table 2.7. An interaction with respect to this study is defined as the interdependence of the levels of the factors, in influencing the response (wear loss). Thus there are five main factors and ten possible interactions which together make fifteen effective factors. To suit this condition, an L 16 (2) 15 array, was chosen for conducting the experiment for the DOE1. The L 16 (2) 15 designation refers to the number of experiments (16), and the number of levels for each factor (2), and the sum of the number of factors (5) and interactions (10) together make 15 effective

14 21 factors. A full factorial experiment would need of (2) 15, i.e. 32,768 experiments while the Taguchi approach requires only 16 experiments. Table 2.6 Factor and level descriptions for the Taguchi DOE1 FACTOR LEVEL Name Description 1 2 A Cooling Rate 1 o C/min 3.5 o C/min B Soaking Temperature - 80 o C -184 o C C Soaking Period 6 h 24 h D Tempering Temperature 150 o C 200 o C E Tempering Period 1 h 4 h Table 2.7 Possible interactions of factors in the DOE1 Interactions AxB Cooling Rate Vs Soaking Temperature AxC Cooling Rate Vs Soaking Period BxC Soaking Temperature Vs Soaking Period DxE Tempering Temperature Vs Tempering Period AxD Cooling Rate Vs Tempering Temperature BxD Soaking Temperature Vs Tempering Temperature CxE Soaking Period Vs Tempering Period CxD Soaking Period Vs Tempering Temperature BxE Soaking Temperature Vs Tempering period AxE Cooling Rate Vs Tempering Period

15 22 Table 2.8 L array for the Taguchi DOE1 Expt. No. A 1 B 2 AxB 3 C 4 AxC 5 BxC 6 DxE 7 D 8 AxD 9 BxD 10 CxE 11 CxD 12 BxE 13 AxE E 15 The factors and interactions were assigned to the L 16 (2) 15 OA using the specified linear graph shown in Figure 2.1 for the DOE1 as shown in Table 2.8. In the Taguchi OAs, the effects of interactions are mixed with the main effect of a factor assigned to some other column. If the interactions of AxB are of no interest, column 3 may be assigned to the analysis of error. The effect of interaction AxB will then be mixed with the effect of error. If more interactions are considered to be of no interest, then an appropriate OA can be selected for conducting the experiment. 2.6 STEPS FOR THE TAGUCHI OPTIMISATION follows: The steps involved in the Taguchi optimisation technique are as 1. Identify the factors of cryogenic treatment influencing the wear resistance of the materials.

16 23 2. Identify the possible interactions of the factors, which may have influence on the wear resistance. 3. Based on the number of factors and interactions, choose a two factor OA for conducting the screening test, to determine the effect of the individual factor over the wear resistance. 4. Set two extreme levels for the selected factors. 5. Assign the factors and interactions to the selected OA using the corresponding linear graph. 6. Conduct the screening test and identify the significant factors and arrive at the optimum of the selected levels. 7. Based on the number of significant factors, select a suitable three level OA. 8. Select three levels around the optimum levels arrived at from the screening test for each factor. 9. Assign the factors and levels to the selected OA, conduct the experiment and arrive at the optimum levels of the significant factors. 10. Calculate the predicted optimum wear resistance of the selected materials for the optimum treatment conditions by the standard provisions of the ANOVA. 11. Conduct the confirmation experiment as per the optimum cryogenic treatment conditions and quantify the wear resistance.

17 Compare the predicted optimum wear resistance with that of the confirmation experiment results, and check whether it is within the confidence interval. If it is within the confidence interval, optimum levels of the DCT parameters arrived at, are correct. If it does not fall within the confidence interval, the experiment conducted is not correct and it is to be repeated with different factors and their respective levels with the proper understanding of their influences on the process response (Wear resistance in the present study).