A METHODOLOGY TO REDUCE THE COUNTER GEAR SUBASSEMBLY REWORKS

Transcription

1 Journal of Engineering Science and Technology Special Issue on ICMTEA 2013 Conference, December (2014) School of Engineering, Taylor s University A METHODOLOGY TO REDUCE THE COUNTER GEAR SUBASSEMBLY REWORKS J. BOBY* SQC and OR Unit, Indian Statistical Institute, Bangalore, India * boby@isibang.ac.in Abstract The counter gear subassembly is an integral part of an actuator. The actuators are used to regulate the functions of the control valves. A company manufacturing actuators is facing the severe problem of counter gear subassembly reworks. The past data showed that around 58% of the counter gear assemblies are reworked monthly. Hence this study is undertaken to solve to the counter gear subassembly rework problem. The discussions with technical and design experts revealed that four critical dimensions are majorly causing the rework, namely gear 8 teeth thickness (x 1 ), sleeve with chamfered gear length (x 2 ), gear with chamfer (x 3 ) and gear 8 teeth and drive milling depth (x 4 ). Hence it is decided to study the effect of the aforementioned factors using design of experiments. Two levels are chosen for each of the four factors. Two interactions namely x 1 x 2 and x 3 x 4 are also included in the study. For each treatment combination, twenty counter gears are assembled and the number of assemblies reworked is counted. Since the response variable is binary, a model is developed for proportion of reworks using generalized linear model approach with response variable distribution as binomial and link function as logit function. The optimum combination of factor levels which would reduce the percentage of the reworks below 10% is identified. The results are validated by assembling another 50 counter gears with optimum factor level combination. The study resulted in reducing the rework from 58 % to 8%. The study also demonstrated a methodology to develop designs with minimum number experiments by augmenting the traditional fractional factorial designs with additional runs. This will help in separating out the effects of confounded interactions in traditional fractional factorial designs. The study also demonstrated an approach for analysing the design of experiments with binary response variable. Keywords: Actuator, Counter gear subassembly, Design of experiments, Binary response variable, Generalized linear model, Link function, Optimization. 46

2 A Methodology to Reduce the Counter Gear Subassembly Reworks 47 Nomenclatures exp GLM MLE x y i Exponential function Generalized Linear Model Maximum Likelihood Estimation Vector of factors Response of i th experiment Greek Symbols α Significance level β Vector of model coefficients ε Random error component Π i Probability that y i = 1 1. Introduction A counter gear subassembly is an integral part of an actuator. The actuators are used to control the opening and closing of valves [1]. The control valve needs to be opened and closed according to the user requirements. This is accomplished by translating the user input into a torque value and ensuring that the actuator trips at this specified torque [2]. The components, especially the counter gear subassembly, affect the functioning of actuators. Many times, the counter gear subassemblies need to be reworked to ensure proper functioning of actuator. An actuator manufacturing company is facing the severe problem of counter gear subassembly reworks. The past data on monthly percentage reworks is given in Fig Nov-12 Dec-12 Jan-13 Feb-13 Mar-13 Apr-13 Fig. 1. Month Wise % Rework of Counter Gear Subassembly. Fig.1 shows that on an average 58% of counter gear assemblies are reworked monthly. The approximate cost of counter gear subassembly rework has been Indian Ruppees 1,92,000 annually. Hence this research study is undertaken to develop a methodology to reduce the % rework of counter gear subassemblies to at least 10%.

3 48 J. Boby The remaining part of this paper is organized as follows: the details on how the study is designed is given in section 2. The data collection and analysis are provided in section 3. The section 4 discusses the identification of optimum combination. The validation of the results is given in section 5. The paper ends with the conclusion in section Designing the Study The analysis of the past counter gear subassembly rejections and discussions with the technical professionals of the organization revealed that the rejections are mainly caused by four component dimensions namely gear 8 teeth thickness (x 1 ), sleeve with chamfered gear length(x 2 ), gear with chamfer (x 3 ) and gear 8 teeth and drive gear milling depth (x 4 ). Hence it is decided to conduct statistically designed experiments [3, 4] to study the effect of the aforementioned dimensions on the counter gear subassembly reworks. The design of experiments helps to evaluate the effect of many factors on the process output or response. The design of experiments is extensively used in industry for problem solving and process optimization [5-13]. The four critical dimensions are taken as factors for the experiments. Two levels are chosen for each factor. The organization does only design, subassembly, final assembly and testing at its facility. The components are manufactured by its subcontractors. The design department suggested that the levels should be within the approved design specification of the components. Therefore levels of the factors are chosen within the tolerance specified by the design team. The technical team also suspected the interaction between factors gear 8 teeth thickness and sleeve with chamfered gear length (x 1 x 2 ) and gear with chamfer and gear 8 teeth and drive gear milling depth (x 3 x 4 ).The factors with levels chosen for the experiment is given in Table 1. Table 1. Factors with the Levels. Factor Name Factor Levels Code Gear 8 teeth thickness x mm 5.3 mm Sleeve with chamfered gear length x 2 Minimum Maximum Gear with chamfer x mm 1.6 mm Gear 8 teeth and drive gear milling depth x 4 Short Full The minimum number of experiments required for studying four factors each at two levels and two interactions are only six. So the nearest design is an eight run fractional factorial design [14] with defining relation x = (1) 4 x1x2 x3 But fractional factorial design with defining relation (1) is not suitable for this study as x 1 x 2 interaction will be aliased with x 3 x 4 interaction. The aliased structure of the design is x = = (2) 3 x4 x3. x1x2 x3 x1x2

4 A Methodology to Reduce the Counter Gear Subassembly Reworks 49 The full factorial design requires sixteen runs, which is much higher than the minimum required runs. Taguchi s approach also requires sixteen run L 16 orthogonal array design [15]. Otherwise run fractional factorial in stage one and if the analysis shows that aliased interactions are significant, then to de alias the interactions, run the remaining eight runs in stage two. If the interactions are significant then this will eventually result in 2 4 full factorial experiments. Another option is to run additional experiments in stage two to separate out the x 1 x 2 interaction effect from that of x 3 x 4 interaction. In fact to de alias x 1 x 2 interaction from x 3 x 4 interaction effect, only one additional run is required. The additional run required is x 1 = +1, x 2 = +1, x 3 = +1 and x 4 = -1 so that x 1 x 2 = +1 and x 3 x 4 = -1. But executing this additional run as stage a two experiment may have misleading results if the time between the two stages of experimentation has a significant effect [16]. Hence a new design is developed by augmenting the traditional fractional factorial design with an additional run to de alias x 1 x 2 and x 3 x 4 interactions. It is also decided to run all the nine experiments in one stage. The modified design is given in Table 2. The 9 th run is the additional one added to design to separate out the effect of x 1 x 2 interaction from that of x 3 x 4 interaction. The elements in x 1 x 2 and x 3 x 4 columns of Table 2 are exactly same except for the 9 th experiment. Table 2. The Experimental Design. Experiment number x 1 x 2 x 1 x 2 x 3 x 4 x 3 x Before starting the experiment, the experimental design is validated by the technical professionals of the company. Some of the production engineers pointed out the experiments 1 and 2 would result in high rework and hence need not be run. But without experiment 1 and 2, there would be only seven data points. Hence the estimation of four main effects and two interaction effects would be difficult due rank deficiency problem. Finally the engineers agreed to conduct all the nine experiments as given in Table Data Collection and Analysis The experiments are carried out as per the design. For each run twenty counter gear subassemblies are made and counted the number of subassemblies to be reworked. The data collected through experimentation is given in Table 3.

5 50 J. Boby Experiment number Table3. Experimental Data. x 1 x 2 x 3 x 4 Total assembled Number reworked In this study, the response is a binary variable. The response y i of each experimental unit can take only one of the two possible values namely accepted or reworked denoted by 0 or 1 with probabilities [17]. p( y = 1) = ; p( y = 0) = 1 (3) i i i i When the response variable is binary, generally the relation between the response and factors will be non-linear and most probably will be a monotonically increasing or decreasing s shaped model of the form exp( x' β ) p ( y) = + ε (4) 1+ exp( x' β ) where x is the vector of model factors, β is the vector of model coefficients and ε is the random error component. In traditional analysis of statistical experiments and empirical model building, the error ε is assumed to be normally distributed with mean zero and constant variance. But in Eq. (4), the error cannot possibly be normal and also error variance may not be constant [18]. Hence the generalized linear model (GLM) approach is used to build the model. In GLM the response variable distribution can be normal, binomial, poisson, exponential or gamma [19]. The parameters of the model can be estimated using maximum likelihood estimation (MLE) method. In this research, the parameters of the model is estimated using maximum likelihood estimation (MLE) method with response variable distribution as binomial and link function as logit function [20]. The details of the parameter estimation are given in Table 4. Table 4 shows that the p value is less than 0.05 for the factors namely gear 8 teeth thickness (x 1 ), sleeve with chamfered gear length (x 2 ), gear with chamfer (x 3 ) and gear 8 teeth and drive gear milling depth (x 4 ). Hence there is sufficient evidence to conclude that all the four factors have significant effect on the response or their coefficients are not zero at a α level of 5%. Similarly the interactions between gear 8 teeth thickness and sleeve with chamfered gear length (x 1 x 2 ) and gear with chamfer and gear 8 teeth and drive gear milling depth (x 3 x 4 ) are significant at an α level of 10% (p value < 0.1). Hence the estimated model is identified as

6 A Methodology to Reduce the Counter Gear Subassembly Reworks 51 exp( x x x x x 1 x x 3 x 4 ) p(y) = 1 + exp( x x x x x 1 x x 3 x 4 ) (5) Table 4. Model Parameter Estimation. Predictor Coefficients Std Error z P value Odds ratio Constant x x x 1 x x x x 3 x The estimated coefficient of a factor represents the change in the link function for each unit change in the factor, while all other factors are held constant. In other words, it represents the change in the ratio of the logarithm of the probability that the subassembly will be reworked to the probability that the subassembly will be accepted, when other factors are kept constant. Table 4 also shows that the odds ratio for factors and interactions are sufficiently larger than one. The odds ratio indicates that the odds that a counter gear subassembly will be reworked will increase by odds ratio times with a unit increase in the factor. For example the odds ratio for the factor gear 8 teeth thickness (x 1 ) is 4.83 indicating that the odds that a counter gear subassembly will be reworked will increase by 4.83 times with each unit increase in the gear 8 teeth thickness. The overall fit of the model is evaluated using log likelihood test. The log likelihood value with G statistic and p value are given in Table 5. Log likelihood Table 5. Log Likelihood. G statistic Degrees of freedom P value Table 5 shows that the p value = 0.00 < 0.05, rejecting the null hypothesis that the coefficients of all the factors and interactions in the model are equal to zero at α level of 5%. In other words there is sufficient evidence that coefficients of one or more factors or interactions are different from zero and the model is appropriate. To assess the fit of the model, three goodness of fit tests are also carried out namely Pearson chi square test, deviance based test and Hosmer-Lemeshow test. The goodness-of-fit statistics assess the fit of a logistic model against actual outcomes [21]. The output of the goodness of fit tests is given in Table 6. Table 6 shows the p value for all the three tests are greater than 0.05, indicating that model adequately fits the data. The table of observed and expected frequencies is given in Table 7. Table 7 shows that the expected frequencies are equal to observed frequencies for all the cases of reworked and accepted subassemblies, which once again confirm that the model adequately fits the data and

7 52 J. Boby also supports the goodness fit tests conclusions. The concordant, discordant and tied pairs of observed responses and predicted probabilities are given in Table 8. Table 7 showed that there are 180 cases out of which 86 subassemblies are reworked and the remaining 94 are accepted as good ones. So there are 86 x 94 = 8084 pairs with different response variables. Table 8 shows that out of 8084 pairs, 7231 (89.4%) are concordant pairs, 351 (4.30%) are discordant pairs and 502 (6.2%) are tied. Based on the model, a pair is concordant if a reworked subassembly has high probability of having rework, discordant if the opposite is true and tied if the probabilities are equal. The concordant pairs % of 89.4 shows that the model has good predictive capabilities. The rank correlation between observed responses and predicted probabilities are also computed and is given in Table 9. Table 6. Goodness of Fit Tests. Method Chi-square Degrees of freedom P value Pearson Deviance Hosmer-Lemeshow Table 7. Table of Observed and Expected Frequencies. Serial number Reworked Accepted Observed Expected Observed Expected Total Table 8. Concordant and Discordant Pairs. Pairs Number Percentage Concordant Discordant Ties Total Table 9. Rank Correlation Coefficients. Correlation coefficient Value Somers' D 0.85 Goodman-Kruskal Gamma 0.91 Kendall's Tau-a 0.43

8 A Methodology to Reduce the Counter Gear Subassembly Reworks 53 Table 9 shows that all the three correlation coefficients namely Somers D, Goodman-Kruskal Gamma and Kendall's Tau-a are high indicating that the model has good predictive ability. Hence it is decided to use the estimated model given in Eq. (5) to optimize the factors or the component dimensions which would reduce the percentage of counter gear subassembly reworks. 4. Choice of Optimum Combination The estimated model given in Eq. (5) is used to identify the optimum values of the factors that would reduce the probability of counter gear subassembly rework. Since only four factors are used in this study each with two levels, the total number of factor level combinations is only sixteen. Using the model, the probability of rework is predicted for all the sixteen combinations. The sixteen combinations with predicted probability of rework is given in Table 10. Table10 shows that the predicted probability of rework is less than 0.1 (10%) for the combinations 10, 11, 14 and 15. The technical experts of the organization suggested that the combination 14 is easier and economical to implement than combinations 10, 11 and 15. Hence the combination 14 is identified as the optimum combination. The optimum combination of the factors is given in Table 11. Table 10. All Sixteen Combinations with Predicted Probability of Rework. Combination x 1 x 2 x 1 x 2 x 3 x 4 x 3 x 4 p(y) Serial number Table 11. Optimum Combination of Factors. Factor Optimum level 1 Gear 8 teeth thickness (x 1 ) 5.1 mm (-1) 2 Sleeve with chamfered gear length(x 2 ) Minimum(-1) 3 Gear with chamfer (x 3 ) 1.6 mm (1) 4 Gear 8 teeth and drive gear milling depth (x 4 ) Short (-1)

9 54 J. Boby 5. Validation of Results The results from the study are once again validated by assembling 5 batches of 10 each counter gear subassemblies with optimum combination of factors. The results of the pilot runs are given in Table 12. Table 12. Validation of Results. Serial number Number of subassemblies Number reworked Fraction reworked % reworked Total Table 12 shows that the validation runs resulted in overall 8% rework, which is very close to the predicted value of 7.28% by the model. Moreover the estimated annual cost for 8 % rework will be Indian Rupees 28,800 only, a saving of Indian Rupees 1,63,200 (1,92,000 28,800). Hence it is decided to use the optimum combination of factors for all future counter gear subassemblies. The combination mostly used in the past is x 1 (-1), x 2 (1), x 3 (1) and x 4 (1). Even though the aforementioned combination has been resulting in high rework, the assembly team are not able to reject the components as the dimensions are within the specification. With the experimental results, the assembly team is able to convince the design engineers to modify the component specifications. The company also helped the sub-contractors to enhance their production facility to meet the new requirements. 6. Conclusions This paper discusses a methodology to reduce the % reworks in counter gear subassemblies. An experiment is designed by taking four critical dimensions of the subassembly namely gear 8 teeth thickness, sleeve with chamfered gear length and gear with chamfer and gear 8 teeth and drive gear milling depth as factors. The interaction between factors namely gear 8 teeth thickness and sleeve with chamfered gear length and gear with chamfer and gear 8 teeth and drive gear milling depth are also studied. A new design, developed by augmenting fractional factorial design with an additional run to de alias the effects of the interactions, is used in the study. For each experimental run, twenty counter gear subassemblies are made and the number of subassemblies reworked is counted. Since the response variable is binary, a model is developed using generalized linear model (GLM) approach with response variable distribution as binomial and logit function as link function. Based on the model the optimum combination of factors which would reduce the % counter gear subassembly reworks below 10% is identified. The results of the study are once again validated by assembling 5 batches of 10 counter gear subassemblies at the optimum combination of factors. The validation runs also resulted in rework of 8% which is very close to the predicted value of 7.28% for optimum combination.

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