A neuro-data envelopment analysis approach for optimization of uncorrelated multiple response problems with smaller the better type controllable factors
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Abstract
In this paper, a new method is proposed to optimize a multi-response optimization problem based on the Taguchi method for the processes where controllable factors are the smaller-the-better (STB)-type variables and the analyzer desires to find an optimal solution with smaller amount of controllable factors. In such processes, the overall output quality of the product should be maximized while the usage of the process inputs, the controllable factors, should be minimized. Since all possible combinations of factors’ levels, are not considered in the Taguchi method, the response values of the possible unpracticed treatments are estimated using the artificial neural network (ANN). The neural network is tuned by the central composite design (CCD) and the genetic algorithm (GA). Then data envelopment analysis (DEA) is applied for determining the efficiency of each treatment. Although the important issue for implementation of DEA is its philosophy, which is maximization of outputs versus minimization of inputs, this important issue has been neglected in previous similar studies in multi-response problems. Finally, the most efficient treatment is determined using the maximin weight model approach. The performance of the proposed method is verified in a plastic molding process. Moreover a sensitivity analysis has been done by an efficiency estimator neural network. The results show efficiency of the proposed approach.
Keywords
Multiple response optimization Artificial neural networks Data envelopment analysis Smaller-the-better-type controllable factorsIntroduction
Today’s competitive environment impels companies to improve the quality of their products proactively, so the design of experiments (DOEs) can be one of the most efficient methods for this purpose. The experimental design helps us find the effects of the controllable and nuisance factors on one or more responses. Finding the combination of controllable factor levels, namely treatment, which leads to the most appropriate process outputs is one of the common challenges in quality engineering researches.
The Taguchi method is a common strategy in the robust design and involves designing experiments with the use of orthogonal arrays for finding the treatment which optimizes a given performance measure, typically a signal-to-noise (SN) ratio.
The Taguchi method was initially proposed for single response problems (Taguchi and Chowdhury 2000; Maghsoodloo et al. 2004; Robinson et al. 2004; Zang et al. 2005). However, in real industrial problems, there are more than one responses and their simultaneous optimization is of the interest.
Multiple-response optimization (MRO) problems have been studied by several researchers. Existing methods in this field are classified into three basic categories by Ortiz et al. (2004). The performance of each category depends on the complexity of the problem. In the first category, overlying contour plots of each response is applied for finding the space where each response satisfies its specification limits. Myers and Montgomery (2002) expressed that this method is applicable only in the situations where the numbers of controllable factors are very few. In the second category, the most important response for the decision maker (DM) is used as the main objective and the rest of responses are considered as constraints. However the approaches in this category do not conform to the basic idea of multiple-response optimization which is the simultaneous optimization of all responses.
The third category, which contains a major proportion of studies in MRO, consists of three main steps to find the optimal treatment. In the first step, a model for describing the relation between controllable factors and responses is built. In the second step, an aggregation approach of responses is applied. Finally, in the third step, an optimization method is used for optimizing the single response which is obtained from the last step. In the mentioned categories, Ordinary Least Square (OLS) is one of the common methods for building the model of relation between controllable factors and responses (first step). Although in some researches such as Bashiri and Moslemi (2013), robust optimization methods are applied; one of the major shortcomings of the studies in this category is that when the mean square error (MSE) of the regression model is a high value, the ability of the model to describe the relationship of the response variables and the controllable factors would be poor (Erzurumlu and Oktem 2007). For overcoming this problem, ANN can be used as a proper substitute method for response estimation. Some authors have compared response surface and regression models with ANN in model building and the preciseness of ANN has been verified in their results (Tsao 2008; Desai et al. 2008; Namvar-Asl et al. 2008; Gauri and Pal 2010).
Furthermore, Niaki and Hoseinzade (2013) used ANN for forecasting S&P indices where the experimental design was used for tuning the parameters of the neural network.
In MRO problems, different multi-criteria decision making methods have been used to determine the optimum treatment (for a review see Amiri et al. (2012)). Data envelopment analysis (DEA) is one of these techniques which have been used in several researches. Caporaletti et al. (1999) proposed a pure input DEA model for the nominal-the-best (NTB)-type responses using ${\left({\overline{y}}_{\mathit{ij}}-{y}_{i}\right)}^{2}$ and ${S}_{\mathit{ij}}^{2}$ as input variable. In this study, just the experimented treatments have been evaluated. Liao and Chen (2002) proposed an input-oriented basic DEA ratio known as the Charnes, Cooper and Rhodes (CCR) model introduced by Charnes et al. (1978) that uses the normalized mean responses as input variables when the responses are the NTB or the smaller-the-better (STB) type; also, Goel et al. (2007) proposed a new method in multiple-response optimization using the Pareto optimal solution.
In the model of Liao and Chen (2002), when the responses are LTB type, the normalized mean responses are considered as the output variable. Herein again, only the real experimented treatments and their corresponding responses are considered. Liao (Goel et al. 2007) also used a back propagation (BP) neural network (trained with the data of the actual treatments) to estimate the SN ratio of responses for all treatments and then efficient treatments are determined by the CCR DEA model, considering normalized SN ratio as outputs.
In Liao (Goel et al. 2007), the same DEA model is used but all possible treatments are estimated using a BP neural network. In their proposed approach, the most treatment is not selected.
Gutierrez and Lozano (2010) used a similar approach to find the efficient treatment and then sieve the most efficient among the efficient ones.
In the mentioned literature of the studies which have used DEA for the determination of the optimum treatment, only the response variables have been focused on, while the main philosophy of DEA is maximization of the overall process outputs versus the minimization of the total consumed inputs.
Many real world processes can be exampled where the controllable factors are STB type. For instance, in a plastic molding process, one of the factors is the barrel temperature, of which the less value is more preferred. The higher temperature impels more electricity consumption and equivalently more costs. By considering other similar STB controllable factors as inputs and the SN ratio of response variables as the outputs of the process, the main philosophy of DEA would be realized. The final results of DEA for such a problem would assure us that the most appropriate quality of the process is obtained by the least consumption of input variables.
In this paper, a new approach is developed based on the neural network and data envelopment analysis where controllable factors are the STB type and the responses are uncorrelated. Besides, it is. In this method, the ANN is used to estimate the response values for unpracticed treatments in a way that the multiple responses are obtained simultaneously. In this paper, the used neural network is tuned by the method proposed by Bashiri and Geranmayeh (2011). In their method, the Central Composite Design (CCD) and the Genetic Algorithm (GA) have been used to tune and determine the optimum parameters of the neural network, specifically the number of the layers and the number of neurons in each layer.
After tuning the ANN, the unpracticed treatments with smaller intervals of factor levels are generated and their corresponding responses are estimated by the ANN.
In the next step, by using DEA, the efficiency value of each treatment is computed. Finally, the most efficient treatment is determined using the maximin weight model approach.
The remainder of the article is organized as follows: in the next section, the steps of the proposed approach are described. In section 3, the proposed method is implemented in a real world case study and the steps of the proposed method are explained thoroughly. Finally, section 4 summarizes and draws the conclusion.
Proposed approach
Experimentation phase
In the first step of any design of experiment problems, the responses and their descriptions such as their types should be determined. Also, the controllable factors and their corresponding levels should be specified according to the technical knowledge of the process and the execution limitations. By knowing the required information of responses and the controllable factors, a Taguchi design is chosen and conducted and the response data are collected. The outputs of this step are the SN ratio values computed for each response in each treatment. Note that the responses are assumed to be uncorrelated.
Tune-effective parameters of ANN for estimation of responses
For obtaining desirable results by using Artificial Neural Networks, tuning the parameters of ANN seems to be necessary. For example, the number of layers and the number of neurons in each layer are effective parameters in the performance of neural networks. Some authors have used the design of experiments (DOE) for determining the best combination of effective parameters of ANN. Khaw et al. (1995) used the Taguchi method as well as two simulated data collections to determine the effective parameters of ANN, which caused to increase the velocity and convergence of the Back Propagation (BP) algorithm. Also, other similar researches have been proposed in this field, such as the studies of Kim and Yum (2004), Sukthomya and Tannock (2005), Tortum et al. (2007), Packianather et al. (2000), and Peterson et al. (1995). Bashiri and Geranmayeh (2011) proposed a method for tuning the parameters of the artificial neural network based on CCD and genetic algorithm. Because of the accuracy and generalization capability of this approach, in this study, we applied this method for tuning the parameters of the neural network with some necessary changes in determining the neural network’s performance criteria. The condition of the problem determines the proper performance criteria of the neural network.
For training ANNs, the data are divided into three subsets: training, validation, and testing sets. In this study, root mean square error (RMSE) of the test and validation data are considered as ANN’s performance criteria in estimation of responses in the multiple response optimization problem. In this step, the optimum number of hidden layers and the number of neurons in hidden layers are obtained.
Train ANN using tuned parameters and estimate the unpracticed treatments
In the previous step, optimal parameters of ANN for this problem are obtained. So, the neural network is ready to be trained and estimate the response values of unpracticed treatments.
After the training phase, neural network builds a model and can estimate other treatments which are not experimented. Since the trained neural network’s response estimation is not affected by the number of factor levels, new levels are defined between the initial factor levels. This procedure improves the accuracy of the solution.
In mentioned studies, the neural network is used to estimate the SN ratio or the mean square deviation (MSD) of responses. But in the cases where a nuisance factor exists, the effect of the nuisance factor is neglected and the nuisance factor is treated as a replicate. In this study, for solving this problem, ANN is applied for estimating responses, not the SN ratios or MSDs of responses. As Chang (Ozcelik and Erzurumlu 2006) applied ANN in dynamic multiple response experiments, factors and nuisance factors are considered as inputs for training of the neural network.
Evaluate the efficiency of all treatments using DEA
DEA is used to compute the relative efficiency of a group of competing decision-making units (DMUs), while there are several inputs and outputs for each DMU (Tbanassoulis 2001). The relative efficiency is the ratio of the weighted sum of outputs to the weighted sum of inputs.
where u_{ r } is the weight of output r, v_{ i } is the weight of input i, y_{ rj } is the value of output r from DMU j, and x_{ ij } is the value of input i from DMU j and where DMU_{0} is the DMU under study.
In this paper, each treatment is considered as a DMU, each controllable factor is considered as an input variable and finally each response variable is considered as an output.
For using DEA in MRO problems, input variables should be STB type and response variables should be LTB type; however this condition does not necessarily hold in all processes. For this reason, before applying DEA, it should be checked that the controllable factors are STB type variables and the responses are LTB-type ones.
At the end of this step, an efficiency value is computed for each treatment which represents that how well the input variables, controllable factors, have been minimized and how well the output variables, SN ratios of responses, have been maximized. The efficiency value is a measure ranging from 0 to 1 and sometimes more than one treatment would have the efficiency equal to 1.
Use maximin weight model to choose among the efficient treatments
If there would be k efficient units, by solving maximin weight model represented in Equation 7 for each efficient unit, a group of maximin weights, ${w}_{i1}^{\ast},{w}_{i2}^{\ast},\dots {w}_{\mathit{ik}}^{\ast},$ are obtained and the DMU with the largest value of w is considered as the most efficient treatment.
Illustrative example
Response definitions for the injection molding process example
Response | Description | Specification limit ( mm) | Type |
---|---|---|---|
Y _{1} | The size of the upper side | 483.5 + 0.3 | NTB |
Y _{2} | The size of the lower side | 483.6 + 0.3 | NTB |
Factors and their levels for the injection molding process example
Row | Factors | Levels | Type |
---|---|---|---|
A | Injection pressure (percent of machine max pressure) | 40, 50 | STB |
B | Injection speed (percent of machine max speed) | 55, 60, 65 | STB |
C | Holding Pressure1 (percent of machine max pressure) | 40, 45, 50 | STB |
D | Holding Pressure1 (percent of machine max pressure) | 75, 80, 85 | STB |
E | Holding pressure Time (seconds) | 6, 8, 10 | STB |
N | Nuisance factors, injection machines | 550A, 550B | STB |
According to the information given in Table 2, the L_{18} Taguchi design is selected for design of experiments. Since two similar machines can be used for production of the mentioned part, two nuisance factors are considered.
Summary of experimental results for injection molding process example
Run | L_{18} | Y _{ 1 } | Y _{ 2 } | ||||||
---|---|---|---|---|---|---|---|---|---|
A | B | C | D | E | N _{ 1 } | N _{ 2 } | N _{ 1 } | N _{ 2 } | |
1 | 40 | 55 | 40 | 75 | 6 | 483.36 | 483.32 | 483.96 | 483.98 |
2 | 40 | 55 | 45 | 80 | 8 | 483.34 | 483.38 | 483.74 | 484.06 |
17 | 50 | 65 | 45 | 75 | 10 | 483.4 | 483.42 | 483.84 | 484.14 |
18 | 50 | 65 | 50 | 80 | 6 | 483.38 | 483.32 | 483.88 | 484.06 |
Tune-effective parameters of ANN for estimation of responses
Algorithm and its parameters considered for training the neural network
Training algorithm | Maximum number of epochs to train | Performance goal | Maximum validation failures | Minimum performance gradient | Initial mu | mu decrease factor | mu increase factor | Maximum mu |
---|---|---|---|---|---|---|---|---|
Levenberg-Marquardt back propagation | 100 | 0 | 5 | 1e-10 | 0.001 | 0.1 | 10 | 1e10 |
Parameters and their levels studied in experiments based on CCD design in ANN parameter tuning
Factors | Cube points | Central point | Axial points | ||
---|---|---|---|---|---|
Low | High | Low | High | ||
The number of neuron in first layer | 4 | 8 | 6 | 2 | 10 |
The number of neuron in second layer | 1 | 3 | 2 | 0 | 4 |
Optimum values of effective parameters in performance of ANN
Parameter | Optimum value |
---|---|
The number of neurons in first layer | 5 |
The number of neurons in second layer | 3 |
RMSE of test data | 14.32 |
RMSE of validation data | 14.70 |
Train the ANN using tuned parameters and estimate unpracticed treatments
In this step, data are divided into testing, training, and validation data subsets as mentioned in the previous section, and finally, the training of ANN is conducted using the tuned parameters obtained from the previous step.
Factors and their new defined levels for the injection molding process example
Row | Factors | Levels |
---|---|---|
A | Injection pressure (percent of machine max pressure) | 40, 45, 50 |
B | Injection speed (percent of machine max speed) | 55, 57.5, 60, 62.5, 65 |
C | Holding pressure 1 (percent of machine max pressure) | 40, 42.5, 45, 47.5, 50 |
D | Holding pressure 1 (percent of machine max pressure) | 75, 77.5, 80, 82.5, 85 |
E | Holding pressure time (seconds) | 6, 7, 8, 9, 10 |
N | Nuisance factors, injection machines | 550A, 550B |
Estimated values by neural networks for actual experiments for the injection molding process example
Combination | L_{18} | Y _{ 1 } | Y _{ 2 } | ||||||
---|---|---|---|---|---|---|---|---|---|
A | B | C | D | E | N _{ 1 } | N _{ 2 } | N _{ 1 } | N _{ 2 } | |
C1 | 40 | 55 | 40 | 75 | 6 | 483.36 | 483.34 | 483.95 | 484.02 |
C2 | 40 | 55 | 40 | 75 | 7 | 483.35 | 483.35 | 483.9 | 484.05 |
C1874 | 50 | 65 | 50 | 85 | 9 | 483.32 | 483.33 | 483.78 | 484.13 |
C1875 | 50 | 65 | 50 | 85 | 10 | 483.32 | 483.35 | 483.79 | 484.1 |
Evaluate the efficiency of all treatments using DEA
In this step, we assume each of the estimated treatments reported in Table 8 as a certain DMU. The SN ratios of estimated responses are computed and then normalized in each treatment using Equations 3 and 5. In this step, evaluation of the efficiency for each treatment can be conducted by solving Equation 1.
Matlab software (MathWorks, Inc., Natick, MA, USA) is used to solve 1,875 linear models to determine the efficiency of each treatment. Finally, the efficiency value for 12 of them is obtained as equal to 1. In the next step, the most efficient treatment would be selected among these 12 treatments.
Use maximin weight model to choose among the efficient parameter combinations
The maximin calculated weight for efficient combinations
Combination | A | B | C | D | E | w _{ j } |
---|---|---|---|---|---|---|
C100 | 40 | 55 | 47.5 | 85 | 10 | 0.000507 |
C530 | 40 | 65 | 42.5 | 75 | 10 | 0.000508 |
C636 | 45 | 55 | 40 | 80 | 6 | 0.000508 |
C675 | 45 | 55 | 42.5 | 85 | 10 | 0.000426 |
C748 | 45 | 55 | 50 | 85 | 8 | 0.000494 |
C1008 | 45 | 62.5 | 40 | 77.5 | 8 | 0.000502 |
C1201 | 45 | 65 | 47.5 | 75 | 6 | 0.000508 |
C1420 | 50 | 57.5 | 42.5 | 82.5 | 10 | 0.000477 |
C1505 | 50 | 60 | 40 | 75 | 10 | 0.000485 |
C1691 | 50 | 62.5 | 45 | 82.5 | 6 | 0.000077 |
C1768 | 50 | 65 | 40 | 82.5 | 8 | 0.000487 |
C1815 | 50 | 65 | 45 | 80 | 10 | 0.000505 |
Validating results
Run | SN ratio of the results | Relative efficient | w _{ j } | |
---|---|---|---|---|
SN_{1} | SN_{2} | |||
1 | 84.6542 | 90.6861 | 0.94 | |
2 | 84.6545 | 66.6024 | 0.63 | |
3 | 84.6549 | 70.6859 | 0.67 | |
4 | 76.6955 | 63.8360 | 0.00 | |
5 | 81.1332 | 67.7633 | 0.35 | |
6 | 81.1329 | 66.0764 | 0.33 | |
7 | 90.6753 | 71.6012 | 1.00 | 0.0316 |
8 | 84.6549 | 68.4083 | 0.57 | |
9 | 84.6552 | 73.7852 | 0.68 | |
10 | 78.6346 | 63.8382 | 0.15 | |
11 | 90.6757 | 69.8589 | 1.00 | 0.0297 |
12 | 81.1332 | 69.8579 | 0.37 | |
13 | 90.6746 | 70.6848 | 1.00 | 0.0256 |
14 | 84.6549 | 67.7630 | 0.57 | |
15 | 81.1332 | 66.0764 | 0.32 | |
16 | 84.6545 | 67.1635 | 0.57 | |
17 | 90.6760 | 67.1646 | 1.00 | 0.0000 |
18 | 81.1325 | 71.6012 | 0.39 | |
19 | 83.5709 | 106.2443 | 1.00 | 0.0348 |
Conclusions
In this study, a four-step approach was presented to find the optimal treatment in multiple-response optimization problems. After conducting a Taguchi-designed experiment and collecting the response data, by using a tuned neural network, the responses of unpracticed treatments were estimated considering new levels defined for the controllable factors. Each treatment was assumed to be as a DMU and the smaller-the-better-type controllable factors were assumed as the input variables, whereas the computed SN ratios of responses were assumed to be as the outputs for the DEA modeling. Then data envelopment analysis was applied to obtain the efficiency of each treatment. Finally, the maximin weight model was applied to find the most efficient treatment.
In the proposed method, for assisting the economic aspects of the process improvement, the type of controllable factors was accounted for, and using this approach, the overall output quality of the process was maximized while the usage of input variables was minimized. For validation of the study, the proposed method was applied in a plastic molding process as a real case and the results were compared and analyzed. Sensitivity analysis of the efficiency deviation, which was the useful tool for analyzing results, was presented by applying another neural network. The analysis showed that the proposed approach was a proper tool in discrete multiple response optimization especially for the STB controllable factors. As illustrated in this paper, the proposed approach was based on the response values and it did not consider the variations of responses; so, using other approaches which consider the variation can be future studies of STB-type controllable factors in MRO problems.
Notes
Acknowledgements
The authors would like to acknowledge the management of Iran office machine industry plastic factory for contributing in the case study.
Supplementary material
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