Taguchi Design Experiment - Research Paper Example

Taguchi Design Experiment [Student Name] [Institutional Affiliation] [Date] Introduction Taguchi method entails the reduction of process variations through the design of experiment processes that are robust. The primary objective of the method is to allow production of high quality products at a cost that is low to the manufacturers. This method was brought about by Genichi Taguchi who came up with a method for design of experiments that was aimed at investigating how various process parameters impact on the variance and mean characteristics for process performance. Many factors are put into consideration when making a product and mostly when coming up with a new product. These factors include inputs required, outputs expected for the process of production and all other aspects which constitute the process of production. The Taguchi method is basically a structured and well calculated approach for determining the best combination production factors in coming up with the best achievable product or service based on a design of experiments(DOE). DOE is a method adopted to systematically establish relationships between all factors affecting production and come up with the best relations in factors of production which will give optimum output (Condra, 23). According to Taguchi methods, products should be designed to be defect free and maintain high quality standards. This technique enables designers to simultaneously determine the individual and interactive effects of the various factors affecting outputs. It also gives a full understanding of interactions between design components. Design of experiments methodology mainly makes use of the uncontrollable and controllable factors for the input factors, blocking, responses, replication, hypothesis, testing and interaction. The techniques for DOE allow the designers to simultaneously make a determination of both the interactive and individual effects for the many factors that would impact on the results for a given design. What is more, DOE offers a lot of insight into the interactions between the elements of design and thus useful in modifying a standard design into robust designs. In fact, DOE assist in pinpointing some of the critical areas in the design and which would impact of the yields of the design. Such an advantage helps the designers to mitigate some of the problems and come up with designs that are more robust than the initial production process designs (Condra, 29) Controllable and Uncontrollable Production Factors to Consider Controllable factors of production refers to the factors that can be determined when one is carrying out a production experiment and they are quantity of inputs and quality of inputs while uncontrollable production factors refers to the factors that cannot be determined by the one carrying out an experiment and they are mainly the natural occurrences, for example the weather. Response or output measures are carried out to measure the deviation of the produced product from the expectations of the producer. The hypothesis testing uses statistical methods to give options/possibilities of outcomes while blocking does away with the possibility of getting any unwanted outcomes. Disturbances/interruptions refer to the occurrences which take place in the process of production and bring a design performance deviation from the expected outcome. Taguchi method divides disturbances into three categories; External disturbances: The external disturbances refer to the environmental variations which occur in the location where the product is applied. Internal disturbances Internal disturbances are the actual tear and wear inside a specific product unit. Disturbances in the production process also take into consideration the deviation of a produce from the expected outcome. Steps for Achieving a Robust Taguchi Design A three step method is used to achieve a robust design in Taguchi experiment. The three steps involved include: i. Concept design ii. Parameter design iii. Tolerance design Concept design Concept design is the process of examining the competition a product is facing from any present competitive technologies in production. It generally assists a producer in choosing production methods which are best suited to counter competition by analyzing choices of technology and choices in production design. A control prototype is designed which can be produced to meet customers’ needs irrespective of the disturbances available. Parameter design Taguchi method is mainly related with the parameter design whereby selection of control factors and their optimal levels is done with an objective to make the design robust. Control factors refer to those processes which production management can influence in a certain way. This includes the technical producers used and their level of expertise in the field. The optimal parameter levels are determined through experimentation. Tolerance design Tolerance design entails development of specification limits in unit production which is necessary as there always will be some variations in the production process (Condra, 23). Taguchi mainly advocates aiming precisely for the target and nothing less than this and in most cases leads to a greatly increased production cost as more expensive input materials are used to achieve the desired output specifications. Taguchi has made propositions for various performance measures. Such measures include the Signal-to-Noise (SN) ratios used in evaluating the performance of the existing statistical and engineering systems. The recommendations are to use data-driven performances for measurements instead of using the SN ratios. Although SN ratio incorporates some modeling assumptions, Taguchi's methodology has the advantage that is not easily noticeable due to insufficient or lack of a rigorous framework. Particular control factors such as the adjustment factors perform an important role in Taguchi experiment. For example, the S-N ratios act as performance measures and do not depend on the adjustment factors. Therefore, system optimization steps are split into two with a lot of convenience, with the end result being maximum S-N ratio and some adjustments through the use of the adjustment factors. Mathematical foundation to such an approach entails proposing the concept of some performance measures that do not depend on the adjustments. Through the application of this concept, the S-N ratio is justified under modeling assumptions, although considered inappropriate in some other assumptions for modeling. What makes derivation of parameters to appear ambiguous is the fact that some adjustment factors cannot be derived. The Taguchi motivation for the application of adjustment factors entails simplification of the process experiments instead of optimization (Condra, 120). Therefore, based on this, a criterion is proposed through the selection of adjustment factors from the control factors. The application of the new criterion offers more clarifications into the ambiguity surrounding the Taguchi experiment and allows for development of better approaches towards a robust design. In other words, such characteristics are indicative of how the process functions. Thus, this research will consider the application of Taguchi method and technique in reduction of the number of experiments to about 50 on average by use of the orthogonal arrays (OA). Four independent factors shown in the Table are considered. Further, the number of experiments for the normal design experiments will be 21 x 6 x 4 x 4 = 2014 experiments. The Taguchi Method is applied in reducing these experiments to about 50 experiments on average using the technique of Orthogonal Arrays (OA) and focusing on four Independent factors shown in Table 1. Dosing (PPM) TBT Salinity (PPM) Flow of seawater (L/h) 0 75 30000 500 0 80 35000 1000 0 85 40000 1500 0 90 45000 2000 0.5 75 30000 500 0.5 80 35000 1000 0.5 85 40000 1500 0.5 90 45000 2000 1 75 30000 500 1 80 35000 1000 1 85 40000 1500 1 90 45000 2000 1.5 75 30000 500 1.5 80 35000 1000 1.5 85 40000 1500 1.5 90 45000 2000 2 75 30000 500 2 80 35000 1000 2 85 40000 1500 2 90 45000 2000 2.5 75 30000 500 2.5 80 35000 1000 2.5 85 40000 1500 2.5 90 45000 2000 3 75 30000 500 3 80 35000 1000 3 85 40000 1500 3 90 45000 2000 3.5 75 30000 500 3.5 80 35000 1000 3.5 85 40000 1500 3.5 90 45000 2000 4 75 30000 500 4 80 35000 1000 4 85 40000 1500 4 90 45000 2000 4.5 75 30000 500 4.5 80 35000 1000 4.5 85 40000 1500 4.5 90 45000 2000 5 75 30000 500 5 80 35000 1000 5 85 40000 1500 5 90 45000 2000 From the Table, the priorities start with Dosing that has 21 levels, TBT with 6 levels, salinity with 4 levels, and the Flow of Seawater with 4 levels. Application of Taguchi Method and Technique A full factorial Taguchi design method requires a large number of experiments carried out and the outcomes analyzed to come up with preferable combinations to achieve the required outputs and this can prove a tedious task. So that’s why Taguchi came up with the orthogonal array methodology to assist in studying the entire parameters of the outcomes that have the lesser number of the experiments to be carried out. This results in the application of the loss function for the measurement of the performance characteristics that can deviate from the target value that is targeted. This loss function can further be transformed into S-N ratio which is broken down into exactly three categories which are the nominal-the-best, smaller-the-better and the larger-the-better. The applicable steps in the Taguchi method: i. Identifying the main functions and their side effects. ii. Identifying the noise factors and carrying out testing of the desired quality characteristics. iii. Identifying the main function that is to be optimized iv. Identifying the control factors alongside their apparent levels. v. Selecting of the appropriate orthogonal array and constructing a matrix. vi. Performing the matrix experiment. vii. Data analysis and examination and prediction of the optimum control factor levels & its performance. viii. Conducting the verification experiment. Experimental design According to the given table above, the control factors are flow of sea water, TBT and the salinity given while the noise factor is the dosing. S/N Ratio S-N ratio is crucial in Taguchi design method. The factors controlled by designers (the design parameters) and the factors that are not controlled by the designers (noise factors) have to be considered due to their influence on the quality of a product. In this case, Signal to Noise, denoted as S/N ratio is applied in the analysis to take care of both the variability of the experimental results and the mean. This ratio relies on the process quality or product characteristics which are supposed to be optimized. Orthogonal arrays can be viewed as plans of multifactor experiments where the factors and the columns correspond, and the entries of the columns also correspond to the test level of the factors and rows correspond to test runs. Assumptions Underlying the Taguchi Method A fractional factorial plan is responsible for the uncorrelated estimation of every factorial impact and will include the underlying linear model that assumes that all other effects are zero is called an orthogonal plan. Fractional factorial plans based on orthogonal arrays, regardless of the level of fractionation, are fundamentally referred to as the orthogonal plans. This is the primary reason for the popularity of fractional factorials based on orthogonal arrays. S-N Ratio S/n ratio =n=-10 Where n=dosage and y=flow of sea water The main function in our experiment is to apply Taguchi method to reduce experiments using orthogonal arrays. In accordance to the OA table given above, this experiments should be conducted with their levels and factors as stated in the Table. The experimental layout alongside the selected values of the factors is illustrated in the Table and were conducted 5 times ( leading to a total of 45 experiments) that account for variations which may take place as a result of the noise factors. The salinity (Ra), in this case, was measured through the use of the agent for testing salinity and shown in the Table below. This table indicates the different salinity values for the different experiments. Control factors Experiment Number Salinity Sea water flow TBT 1 20000 500 50 2 23500 700 55 3 22000 1200 60 4 27000 800 65 5 12000 800 70 6 13000 1300 75 7 25000 1000 80 Measured values of experiment reduction factor Computing s/n ratio s/n ratio =n=-10 s/n ratio =10=-10 =17.97020 The above formula shows how we compute the s/n ratio(dB) for the rest of the experiments. Experiment no s/n ratio 1 17.97020 2 19.25680 3 20.43210 4 21.08740 5 22.23430 6 19.78540 7 20.43980 8 19.657100 9 20.44830 The S-N ratio for individual control factors is computed as shown in the Table below. Ss1= (η₁+η₂+η3), Ss2=(η4+η5+η6) & Ss3=(η7+η8+η9) Sf1=(η₁+η4+η7), Sf2=(η2+η5+η8) & Sf3=(η3+η6+η9) St1=(η₁+η5+η9), St2=(η2+η6+η7), and St3=(η3+η4+η8) While selecting the values of η₁, η2, η3, among others and for calculating Ss1, Ss2 & Ss3, this table is useful where ηk is the S-N ratio corresponding to experiment k. The average S-N ratio corresponding to cutting speed at level 1 is S s1/3 and the average S-N ratio corresponding to cutting speed at level 2 is S s2/3. What is more, average S-N ratio corresponding to the cutting speed at level 3 is S s3/3 j. The values for Sfj and Stj are computed for dosing and sea water flow(lh) and S-N ratio can be computed for other factors. The factor levels that correspond to the highest S-N ratio were chosen to optimize the condition. Degree or Levels of Freedom The degree of freedom when it comes to grand total for the sum of squares can be equated to the overall number of rows that are within the design of the matrix. Thus, this level of freedom can be attributed to the average of one of the mean square. On the other hand, the level of freedom about the sum of squares can be equated to the overall number of the rows in the design matrix. The level of freedom for an error can be equated to the level of freedom for the overall sum of the squares less the total of the degrees of freedom for different factors. In the present case-study, the degrees of freedom for the error will be zero. Hence an approximate estimate of the error sum of squares can be obtained by pooling the total of the squares that correspond to the factors which have the minimum mean square. As a rule of thumb, the sum of squares corresponding to the bottom half of the factors (as defined by lower mean square) is applied in estimation of the error sum of the squared. In the current scenario, the factors C and D are used for estimation of the error sum of squares and together they account for four degrees of freedom and their sum of squares is 400. Level TBT Salinity Sea water flow Sum() Avg s-n Sum() Avg s-n Sum() Avg s-n 1 71.8808 23.9602 236742.3 78914.43 4366.6 1455.53 2 62.534 17.642 183543.2 61543.63 5432.4 1810.8 3 76.54 25.510 262542..4 87514.133 6542.74 2173.34 4 72.54 24.1620 324577.2 115321.34 7221.54 2464.2 5 64.21 16.452 26144.542 25317.21 3363.63 1576.31 6 52.421 17.540 33643.30 1122.120 4242.20 2121.1 7 67.432 22.430 46322.40 15152.30 3232.60 1616.24 The major inferences from the table above are given in this section. When making reverences to the sum of the squares within the Table, dosing accounts for the largest contribution for the overall sum of the squares (2450/3800) x 100 = 64.5%]. The factor TBT forms the second largest contribution by 25% of the total sum of squares and the factors such as salinity and sea water flow together make only 10.5% of the contribution. If the contribution of a given factor to the overall sum of squares is larger, then the ability of that factor is larger enough to influence n. Further, larger F-values result in larger factor effects when compared to the error variance or error mean square. Total sum of squares = (13) 291i2i)dB( 3800)m(=−ηΣ= Sum of squares due to factor A = [(number of experiments at level A1) × (mA1-m)2 ] + [(number of experiments at level A2) × (mA2-m)2 ] + (14) [(number of experiments at level A3) × (mA3-m)2] = [3 × (-20+41.67)2] + [3 × (-45+41.67)2] + [3 × (-60+41.67)2] = 2450 (dB)2. The total of the squares due to the factors B, C and D is found to be 950, 350 and 50 respectively. Full factorial analysis is made of different experiments that take into consideration all the likely combinations for both the factors and their respective levels. Therefore, as far as these experiments are concerned, the three factors, that is: salinity, TBT, and sea water flow of dosing are considered at 3 different levels. The design of experiments that Taguchi proposed involves the application of orthogonal arrays in organizing of the parameters which impact on the process and the process levels where such parameters would be varied. As an alternative to testing of the possible combinations such as in factorial designs, Taguchi’s method undertakes tests by combination of pairs, thus allowing for the collection of the required data for determination of factors that highly impact on the quality of a product through the minimum number of experiments, and in the process saving on both resources and time. Therefore, the method works well when we have the number of variables falling between 3 – 50, few variables make significant contribution and when there are few interactions between those variables. The use of orthogonal arrays as special standard for experimental designs needs a little number of experimental trials in order to discover some of the principle factors that impact on the output. Prior to selection of the orthogonal arrays, the lowest number of the experiments to be carried out is fixed by applying the formula bellow. NTaguchi = 1 + NV(L-1) Where NTaguchi = Total Number of experiments NV = Total number of the parameters considered And L = Total number of levels In this Case, NV = 2014 L = 21 NTaguchi = 1 + 2014 (35-1) = 68475 In this case, the variables are assigned to the columns as per the principle of the orthogonal arrays and the last column is kept dummy and no row is left out. The selection of experiments is done according to the level of combinations and after the orthogonal array has been chosen. The noting of the output or performance parameter is done during each experimental run. Even though three level designs help in understanding the non linear impact of the parameters for the process , the number of experiments can increase a lot in response to increase of the process parameters. For example, the number of experiments involved in three level designs with three, four and five factors is twenty seven (33=27), eighty one (34=81) and two hundred and forty three (35=243), respectively. The principle of central composite rotatable design (CCD) reduces the total number of experiments without a loss of generality [2]. This is widely used as it can allow a second order multiple regression model to work as a function of the independent parameters of the process. The principle of central composite rotatable design includes 2 numbers of factorial experiments to estimate the linear and the interaction effects of the independent variables on the responses, where f is the number of factors or independent process variables. In addition, a number (nC) of repetitions [nC> f] are made at the center point of the design matrix to calculate the model independent estimate of the noise variance and 2f number of axial runs are used to facilitate the incorporation of the quadratic terms into the model. The term rotatable indicates that the variance of the model prediction would be the same at all points located equidistant from the center of the design matrix. The choice of the distance of the axial points (ζ) from the centre of the design is important to make a central composite design (CCD) rotatable. The value of ζ for rotatability of the design schemes is estimated as ζ = (2 f)1/4 Adjustment factors For nominal best characteristics, Taguchi s-n can be calculated using the formula: s-n=µ where; µ= mean = variance The quadratic loss function should be minimized by setting control factors necessary for maximization of the s-n and then apply other adjustment factors so as to effectively adjust the desired mean target. Adjustment factor stated means any factor that can be changed to only impact the value of µ and have no influence on the s-n value. The factors stated above can be calculated through an equation shown below: EfL (R ) g = Y(X ;N) R = response L(R ) = quality loss function The control factors are divided by a group of two factors (X;Y) where N stands for set of adjustment factors. The expectation is should be considered with respect to noise factors and how they are distributed. The steps involved include; 1. Minimization of PN(X) = Y(X;NX)) with respect to variable X Where; NX) =arg minN Y(X;N) and the solution is denoted by X 2. Adjustment of N to N(X) In cases where the multiplicative error for X depends on X itself, then minimizing PN (X) can be equivalent to the minimization of the S-N ration. When Y has an addictive error model where the error depends on the value of X alone, then to minimize PN (X) can be equivalent to minimizing variance of Y. The adjustment factors are selected from a given set of factors so as to come up with the design of the product that can be more executable. It is easy to change the adjustment factors in order to meet the changes required for the design requirements. Workings for this Experiment In order to determine which array to use, the following array selector can be used: In this study, L9 orthogonal array is to be used. The L9 array is as shown below: Dosing (PPM) TBT Salinity (PPM) Flow of seawater (L/h) 0 75 30000 500 0 80 35000 1000 0 85 40000 1500 0 90 45000 2000 0.5 75 30000 500 0.5 80 35000 1000 0.5 85 40000 1500 0.5 90 45000 2000 1 75 30000 500 Such a setting allows for the testing of only the four variables instead of running all of them. Considering where three trials are to be conducted, then the data collected will be as shown bellow, where the SN ratio for every experiment is computed and a response chart is created. After this, the parameters with the lowest and highest effects are determined. After computing the trial, mean and standard deviation for the above, the following results are obtained as shown in the table below. Std dev Average Trial 3 Trial 2 Trial 1 Flow of seawater (L/h) Salinity (PPM) TBT Dosing (PPM) Experiment 8.5 80.1 70.7 82.3 87.3 500 30000 75 0 #1 5.9 69.6 63.2 70.7 74.8 1000 35000 80 0 #2 5.8 52.4 45.7 54.9 56.5 1500 40000 85 0 #3 9.7 73.4 62.3 78.2 79.8 2000 45000 90 0 #4 12.7 69.6 54.9 76.5 77.3 500 30000 75 0.5 #5 3.0 86.5 83.2 87.3 89.0 1000 35000 80 0.5 #6 4.7 60.9 55.7 62.3 64.8 1500 40000 85 0.5 #7 5.9 93.2 87.3 93.2 99.0 2000 45000 90 0.5 #8 6.80 71.0 63.2 74.0 75.7 500 30000 75 1.0 #9 For the first case, SNi Trial 3 Trial 2 Trial 1 Flow of seawater (L/h) C Salinity (PPM) B TBT A Dosing (PPM) Experiment Number 19.5 70.7 82.3 87.3 1 1 1 1 1 21.5 63.2 70.7 74.8 2 2 2 1 2 19.1 45.7 54.9 56.5 3 3 3 1 3 17.6 62.3 78.2 79.8 3 2 1 2 4 14.8 54.9 76.5 77.3 1 3 2 2 5 29.3 83.2 87.3 89 2 1 3 2 6 22.3 55.7 62.3 64.8 2 1 1 3 7 24.0 87.3 93.2 99 3 2 2 3 8 20.4 63.2 74 75.7 1 1 3 3 9 The maximum range is given by: Thus, salinity has the highest effect whereas dosing has the lowest. Summary and Conclusions This research covers the Taguchi experiment in terms of its main contributions which are presented in detail to express how to reduce the number of experiments from 2014 to 50 by use of the orthogonal arrays. The contribution this design experiment makes can be summarized herein as shown: Orthogonal Arrays are used for simplification of the use of Design of Experiments (DOE) Robust Designs (Parameter & Tolerance Designs) are used for identification of optimum settings so as to lower the process variability and getting the mean target. The definition and the application of the S-N ratio combine both the mean and standard deviation to form one measure. These details about Taguchi’s contribution are reviewed in detail and alongside the accolades and the criticism of every contribution in the process of reducing the 2014 experiments to about 50. Quality loss function is introduced by Taguchi as one of the greatest contributions. In this function, Gauss’s quadratic loss is applied and it enables Taguchi to quantify deviations from the ideal targets as opposed to the traditional view where cost was incurred in situations where results do not concur with the consumer specifications or do not meet the design specifications. Taguchi also induces some change in the manufacturing sector in order to realize how to low the overall cost and reduce variability to targets defined by the customers rather than just meeting the customer’s specifications. Even though Orthogonal Array in Taguchi Experiment is not invented by Taguchi, he is accredited for making the size of the array bigger. Further, this research offers explanation as to how OAs are designed, their importance both in terms of their structure and design solutions, and offers examples for the construction and offers use of some of the most significant OA in Taguchi experiments. Reference Top of Form Condra, Lloyd W. Reliability Improvement with Design of Experiments. New York: Marcel Dekker, 2001. Print. Bottom of Form Read More

Response or output measures are carried out to measure the deviation of the produced product from the expectations of the producer. The hypothesis testing uses statistical methods to give options/possibilities of outcomes while blocking does away with the possibility of getting any unwanted outcomes. Disturbances/interruptions refer to the occurrences which take place in the process of production and bring a design performance deviation from the expected outcome. Taguchi method divides disturbances into three categories; External disturbances: The external disturbances refer to the environmental variations which occur in the location where the product is applied.

Internal disturbances Internal disturbances are the actual tear and wear inside a specific product unit. Disturbances in the production process also take into consideration the deviation of a produce from the expected outcome. Steps for Achieving a Robust Taguchi Design A three step method is used to achieve a robust design in Taguchi experiment. The three steps involved include: i. Concept design ii. Parameter design iii. Tolerance design Concept design Concept design is the process of examining the competition a product is facing from any present competitive technologies in production.

It generally assists a producer in choosing production methods which are best suited to counter competition by analyzing choices of technology and choices in production design. A control prototype is designed which can be produced to meet customers’ needs irrespective of the disturbances available. Parameter design Taguchi method is mainly related with the parameter design whereby selection of control factors and their optimal levels is done with an objective to make the design robust. Control factors refer to those processes which production management can influence in a certain way.

This includes the technical producers used and their level of expertise in the field. The optimal parameter levels are determined through experimentation. Tolerance design Tolerance design entails development of specification limits in unit production which is necessary as there always will be some variations in the production process (Condra, 23). Taguchi mainly advocates aiming precisely for the target and nothing less than this and in most cases leads to a greatly increased production cost as more expensive input materials are used to achieve the desired output specifications.

Taguchi has made propositions for various performance measures. Such measures include the Signal-to-Noise (SN) ratios used in evaluating the performance of the existing statistical and engineering systems. The recommendations are to use data-driven performances for measurements instead of using the SN ratios. Although SN ratio incorporates some modeling assumptions, Taguchi's methodology has the advantage that is not easily noticeable due to insufficient or lack of a rigorous framework. Particular control factors such as the adjustment factors perform an important role in Taguchi experiment.

For example, the S-N ratios act as performance measures and do not depend on the adjustment factors. Therefore, system optimization steps are split into two with a lot of convenience, with the end result being maximum S-N ratio and some adjustments through the use of the adjustment factors. Mathematical foundation to such an approach entails proposing the concept of some performance measures that do not depend on the adjustments. Through the application of this concept, the S-N ratio is justified under modeling assumptions, although considered inappropriate in some other assumptions for modeling.

What makes derivation of parameters to appear ambiguous is the fact that some adjustment factors cannot be derived. The Taguchi motivation for the application of adjustment factors entails simplification of the process experiments instead of optimization (Condra, 120). Therefore, based on this, a criterion is proposed through the selection of adjustment factors from the control factors.

Read More