CHAPTER 4 WEIGHT-BASED DESIRABILITY METHOD TO SOLVE MULTI-RESPONSE PROBLEMS

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1 72 CHAPTER 4 WEIGHT-BASED DESIRABILITY METHOD TO SOLVE MULTI-RESPONSE PROBLEMS 4.1 INTRODUCTION Optimizing the quality of a product is widespread in the industry. Products have to be manufactured such that they best fit some quality properties. Varying the product settings leads to different product qualities and the aim of the manufacturer is to find the factor settings that simultaneously optimize the quality properties. The classical approach to solve such optimization problem is based on response surface methodology. First, a designed experiment is used to collect data and to adjust models capturing the relationship between the responses of interest and the factor settings. Those fitted models can then predict the quality properties for any design point of the experimental domain. Secondly, a desirability index is built to combine the predicted properties into a value belonging to the [0; 1] interval. This index provides a ranking of possible factor settings in the solutions space and the optimum can be found by an adequate optimization algorithm. But model predictions are suited with error; so are the desirability index and the optimal solution found. In practice, in the related literature and design of experiment software, this error is neglected.

2 DESIRABILITY APPROACH The desirability function approach is one of the most widely used methods in industry for dealing with the optimization of multiple response problems. It is based on the idea that the quality of a product that has multiple quality characteristics is completely unacceptable if one of the characteristics lies outside the desired limits. This method assigns a score to a set of responses and chooses factor settings that maximize that score. Desirability function approach was first proposed by Harrington (1965) and then revised by Derringer and Suich (1980). The multi-response optimization problem has been addressed by several authors by using different expressions of the desirability function. The method finds operating conditions x l... x k that provide the "most desirable" response values. For each response Y 1, a desirability function d i assigns numbers between 0 and 1 to the possible values of Y 1, with d i =0 representing a completely undesirable value of x 1 x k, and d i =1 representing a completely desirable or ideal response value. The individual desirability is then combined using an aggregation criterion such as a geometric mean, which gives the overall desirability (D). Different forms of the desirability function are proposed according to the nature of the optimization problem: minimization, maximization or tracking a target. For each case, a different expression is proposed and shown in Figure 4.1. The desirability of the response increases as it becomes closer to its target value T i. It reaches the maximum value of 1 only if the response value is equal to the target T. The overall desirability is then given by the desirability index which is the geometric mean of the individual desirability. It is noticed that the definition of the desirability function does not depend on any distribution assumption. In what follows the nonconformity ratio is considered as a response variable and the property that the desirability value

3 74 increases when the response becomes closer to its target is used to make the higher the better rule hold for any type of distribution and for any type of specification limit. There are many statistical techniques for solving multiple response problems like overlaying the contour plot for each response, constrained optimization problems and desirability approach. The desirability method is recommended due to its simplicity, availability in the software and provides flexibility in weighting and giving importance for individual response. Solving such multiple response optimization problems using this technique involves using a technique for combining multiple responses into a dimensionless measure of performance called the overall desirability function. The desirability approach involves transforming each estimated response, Y i, into a unit less utility bounded by 0 < d i < 1, where a higher d i value indicates that response value Y i is more desirable, if d i =0; this means a completely undesired response. The individual desirability function for the various responses under consideration is then incorporated into a single function which gives the overall assessment of the desirability of the combined response. Then, the solution x is found that maximizes the geometric mean of the individual response desirability. There are three categories of response, and the desirability function can be defined as follows. The combined desirability can be calculated by using the equation D = [d 1 w1 d 2 w2 d 3 w3 d 4 w4 ] / Σw i (4.1) where D is the overall desirability index and w i represents weight assigned to each response. Here weights are taken from Eigen values of responses.

4 DESIRABILITY FUNCTIONS AND DESIRABILITY INDEX The concept of desirability was introduced by Harrington (1965) to provide a solution to multi-response optimization problems. It allows to balance the optimized properties Y i s against one another, taking into account their target value, their relative importance and their scale. Harrington proceeds in two steps First, each response Y i is transformed to the same scale using a desirability function, denoted by d j, such that d i (Y i )[0,1]. If d i (Y i )=0, the product is not at all acceptable according to the specifications of the i th property and if d i (Y i ) =1, the product fullfils them perfectly. The most well-known desirability functions are those of Harrington (1965) j based on the exponential function of a linear transformation of the y i s and those of Derringer and Suich (1980) based on power of a liner transformation of the Yi. Gibb et al. (2001) and Govaerts et al (2005) proposed smoother and differentiable desirability functions based on the logit function normal density and normal distribution functions. These three types of desirability functions are presented in Table 4.1 for the cases where the response must be maximized, minimized or reach a target value. The target value T and the parameters a and b have to be adjusted according to the cumulative distribution function of the standard normal.

5 76 Table 4.1 Different types of desirability functions Maximum Minimum Target Value Y T Harrington (1965) exp (-exp(-a-by)) 1-exp(-exp(-a-bY) exp b Derringer and Suich (1980) 0 if Y<a s Y a b a if a<y<b 1 if Y>b 1 if Y>b s a Y a b if <Y<a 0 if Y<a n 0 if Y<a a a 2 2 s if a 1 <Y<T Y if T < Y<a 2 T 2 0 if Y>a 2 Le Bailly and Govaerts (2005) ( Y a b Y a 1 b Y a 1 Y a 2 (1 ( )) b1 b2 4.4 OPTIMIZATION STEPS USING DESIRABILITY FUNCTION METHOD Determining the Optimal Parameter Combination Step 1: Calculate S/N Ratio for the corresponding responses using the following formula The Nominal-the-Best (NTB) When the value of y equals T (the target value), the desirability value equals 1; if the departure of y exceeds a particular range from the target, the desirability value equals 0, and such a situation represents the worst case. The desirability function of the-nominal-the-best scenario can be written as:

6 77 Yi Li Ti Li a1, Li y Ti di(y 1) Yi Ui Ti Ui a 2, Ti y Ui 0, Y < Li, or, y>ui (4.2) The Larger-the-Better (LTB) When the value of y exceeds a particular criterion value, which can be viewed as the required value, the desirability value equals 1; if the value of y is less than a particular criterion value, which is unacceptable, the desirability value equals 0. The desirability function of the-larger-the-best scenario can be written as, 0, yˆ i Li a 2 Yi Ui di(y ), Li yˆ U Ti Ui 1, yˆ i Ui 1 i i (4.3) The Smaller-the-Better (STB) When the value of y is less than a particular criterion value, the desirability value equals 1; if the value of y exceeds a particular criterion value, the desirability value equals 0. The desirability function of the-smaller-the-best can be written as, 0, yˆ i Li a 2 Yi Ui di(y ), Li yˆ U Ti Ui 1, yˆ i Ui 1 i i (4.4)

7 78 where L i is i th lower limit and U i is i th upper limit, T i is the i th target of the response and y i is the i th response and a 1, a 2 represent the weight of the response. Step 2: The individual desirability index value has been calculated using the formula as di= [(Yi-a)/ (b-a)] r Here the value of r = 0.5. formula as, Step 3: The overall desirability (D) has been computed using the value. D = [ d i.w i ]/ w i. Here, the value of w i is taken based on Eigen Step 4: Then the mean desirability index can be calculated and also optimal combinations will be chosen. The weight based DM can be calculated as, WBDM = D 1ij * weight 1 +D 2ij * weight 2 (4.5) Step 5: ANOVA can be determined from the desirability index value in order to find the significance. 4.5 DETERMINING THE OPTIMAL COMBINATION FOR WEDM OPERATIONS BY USING EXISTING DESIRABILITY INDEX METHOD The normalized values are taken from Table 3.4, and then the individual desirability index value has been calculated using the formula d i = [(Yi-a)/ (b-a)] r and given in Table 4.2. The overall desirability (D) has been computed using the formula D = [ d i.w i ]/ w i. Here, the value of w i is arbitrarily taken as 0.5. The mean desirability value for each factor has been worked out using the L 18 OA and the results are given in Table 4.3 and the factor effects are shown in Figure 4.1. From the main effects the optimal

8 79 combination by the existing method using desirability index is A 1 B 1 C 1 D 3 E 2 F 2 G 1. Table 4.2 Desirability index and overall desirability value for Case-1 Exp. No. Normalized S/N Ratio d 1 d 2 d 3 D Table 4.3 Mean desirability value for Case-1 Parameters Levels Max-Min A B C D E F G

9 80 Mean Desirability Index Levels A B C D E F G Figure 4.1 Factor effects on desirability index for Case-1 Table 4.4 Results of the pooled ANOVA on desirability for Case-1 Factors SS Dof MS F % Contribution C D E Error Total The result of the pooled ANOVA given in Table 4.4 indicates that all factors have almost equal contribution towards affecting the multiple quality characteristics.

10 DETERMINING THE OPTIMAL COMBINATION FOR WEDM OPERATIONS BY USING THE PROPOSED WEIGHT-BASED DESIRABILITY METHOD The normalized values are taken from Table 4.2 and then the individual desirability index (d) value has been calculated using the formula d i = [(Yi-a)/ (b-a)] r and given in Table 4.5. The overall desirability (D) has been computed using the formula D = [ d i.w i ]/ w i. Here, the value of w i is the Eigen value computed using Principal Component Analysis (PCA). By applying PCA on WBDM, the optimal weights are obtained. The optimal weights are , , and The Mean desirability value for each factor has been worked out using the L 18 OA and the results are given in Table 4.6. The factor effects are shown in Figure 4.2. Table 4.5 Normalized S/N values, individual desirability value and overall desirability values for Case-1 Exp. No Normalized S/N Ratio d 1 d 2 d 3 D

11 82 Table 4.6 Mean desirability index values for WBDM for Case-1 Factors Levels Max-Min A B C D E F G Mean Desirability Index A B C D E F G Levels Figure 4.2 Factor effects on desirability index for Case-1 Therefore from Table 4.5 and Figure 4.2, the optimal parameter combination is A 1 B 2 C 1 D 3 E 2 F 2 G 1. Table 4.7 shows that the controllable factors C, E and G contribute 23.63%, % and 17.25%, respectively.

12 83 Table 4.7 Results of the Pooled ANOVA on WBDM for Case-1 Factors SS Dof MS F % Contribution A B C D E G Error Total COMPARISON OF SOLUTIONS FOR WEDM CASE-1 The initial settings for the WEDM process were A 2 B 1 C 1 D 2 E 2 F 2 G 1. The optimal factor settings based on the selected techniques are A 2 B 1 C 1 D 3 E 2 F 2 G 1 (Grey), A 1 B 1 C 1 D 3 E 2 F 2 G 1 (DM) and the combinations for the proposed methodologies are A 1 B 2 C 1 D 3 E 2 F 2 G 1 (WBDM) and A 2 B 3 C 2 D 3 E 2 F 1 G 1 (WGBRA). To find the improvements under the optimum condition, S/N ratios for all the responses are determined using the additive model. The overall improvement percentage is calculated as the ratio between sum of the improvement values of all the responses and sum of S/N ratios of initial conditions of all responses. Table 4.8 presents the comparison of results. From Table 4.8, it is seen that the results from the weight-based Desirability method and weight-based GRA have shown improvements of % and 3.18 %, respectively, from the initial condition. It is also seen that the results obtained on the existing methods are compared with the solutions obtained

13 84 with the proposed methods. It is also to be noted that the techniques discussed in this research have used arbitrary method in assigning weights to the response variables. Whereas in the proposed weight-based desirability method and weight-based Grey relation analysis, the weights are scientifically derived using Principal Component Analysis (PCA). Table 4.8 Comparison of solutions for Wire EDM process parameters Responses Initial Grey Relation Analysis Desirability Method Desirability Method (WBDM) Weight-Based Grey Relation Analysis (WBGRA) MRR SF KERF Optimal Setting A 2 B 1 C 1 D 2 A 2 B 1 C 1 D 3 A 1 B 1 C 1 D 3 A 1 B 2 C 1 D 3 A 2 B 3 C 1 D 3 E 2 F 2 G 1 E 2 F 2 G 1 E 2 F 2 G 1 E 2 F 2 G 1 E 2 F 1 G 1 Improvements in SN ratios MRR SF KERF Overall Improvement (%) DISCUSSION This chapter has presented the use of desirability function for optimizing the multi-response problems in Taguchi method by comparing the existing method with weight-based desirability method. The method of averaging for obtaining the overall desirability value has been done scientifically; therefore, human judgment has been completely eliminated. Thus, the weight-based desirability method has improved the solution. Three

14 85 case studies are solved through this methodology and results are presented and discussed in chapter 5. The proposed procedure has the following merits: 1. The optimal solution can be easily obtained, since desirability can effectively resolve problems that have incomplete information. 2. It does not have any complicated mathematical theory or computation and can be easily understood by the engineers having limited knowledge of statistics.