Modified variable return to scale back-propagation neural network

Engineering Optimization

ISSN: 0305-215X (Print) 1029-0273 (Online) Journal homepage: http://www.tandfonline.com/loi/geno20

Modified variable return to scale back-propagation neural network robust parameter optimization procedure for multi-quality processes Sahand Daneshvar & Kehinde Adewale Adesina To cite this article: Sahand Daneshvar & Kehinde Adewale Adesina (2018): Modified variable return to scale back-propagation neural network robust parameter optimization procedure for multiquality processes, Engineering Optimization, DOI: 10.1080/0305215X.2018.1524463 To link to this article: https://doi.org/10.1080/0305215X.2018.1524463

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ENGINEERING OPTIMIZATION https://doi.org/10.1080/0305215X.2018.1524463

Modified variable return to scale back-propagation neural network robust parameter optimization procedure for multi-quality processes Sahand Daneshvar and Kehinde Adewale Adesina Industrial Engineering Department, Eastern Mediterranean University, Famagusta, Turkey ABSTRACT

ARTICLE HISTORY

Selecting the optimum process parameter level setting for multi-quality processes is cumbersome. Previous methods were plagued by complex computational search, unrealistic assumptions, ignoring the interrelationship between responses and failure to select optimum process parameter level settings. The methods of variable return to scale (VRS) backpropagation neural network (BPNN) previously adopted were limited by the use of weak models, poor discriminatory tendency and an inability to select the optimum parameter level setting. This study applied a modified VRS–adequate BPNN topology model in the robust parameter procedure to solve this problem. Here, standard VRS models are allowed to self-assess, leading to partitioning. The upper bound of the free variable of the VRS model is restricted and the VRS penalization coefficient is adopted to determine the optimum process parameter level setting. The effectiveness of the proposed model measured by the total anticipated improvement yielded the highest total improvement over the existing methods.

Received 5 December 2017 Accepted 29 August 2018 KEYWORDS

Modified VRS; VRS penalization coefficient; parameter level setting; robust parameter optimization; VRS discrimination

1. Introduction Traditional on-line methods of quality assurance are based primarily on inspecting products as they are discharged from the production line and rejecting those that fail to meet the specified acceptance range. However, no amount of inspection can improve the product’s quality attributes and that quality must be built into the product right from inception (Taguchi, Chowdhury, and Wu 2005). Off-line quality control procedures, where efforts have been focused on the reduction of the effects of variations, have been used to complement quality monitoring. The Taguchi robust method has been widely used to improve quality through the reduction of the effect of uncontrollable factors (noise factors) on the quality response at the process and product design stages. However, one of the major problems of the Taguchi method is its inability to effectively and efficiently optimize processes with multi-quality responses (Al-Refaie and Al-Tahat 2011). Several attempts have been made to solve this problem, but ended up complicating it (Al-Refaie 2011; Liao and Chen 2002). The method of assigning weight has been plagued with the difficulty of how to describe and evaluate the weights of responses in a real case. The regression method further complicates the computational process owing to its failure to establish vividly the correlations among the responses. This is evident from the larger mean square error (MSE). Liao (2004) reported that principal component analysis (PCA) could not trade off to select feasible solutions when more than one eigenvalue CONTACT Kehinde Adewale Adesina

[email protected]

Supplemental data for this article can be accessed at https://doi.org/10.1080/0305215X.2018.1524463 © 2018 Informa UK Limited, trading as Taylor & Francis Group

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S. DANESHVAR AND K. A. ADESINA

greater than 1 was obtained. Furthermore, PCA is based on the strict assumption that the residual errors are randomly multivariate normally distributed (Al-Refaie 2011). Salmasnia, Kazemzadeh, and Tabrizi (2012) further improved PCA by incorporating the desirability function and adaptive neuro-fuzzy inference system (ANFIS), an artificial intelligence tool, but this also failed to account for the significant responses during the optimization. Gomes et al. (2013) proposed an improved PCA modelled by weighted multivariate MSE, integrated with PCA and response surface methodology (RSM), and an uncorrelated weighted object function that required additional optimization algorithms was achieved. PCA, genetic algorithm (GA), desirability function, grey relational analysis, exponential desirability function, simulated annealing and multiple adaptive neuro-fuzzy inference systems have been used for robust optimization, but are too complicated to be applied by many decision makers (Chang 2008; Chang and Chen 2011; Noorossana, Davanloo, and Saghaei 2009; Salmasnia, Kazemzadeh, and Tabrizi 2012; Sibalija et al. 2011). Furthermore, most of these methods assume that the variance between responses is constant throughout, thereby disregarding the dispersal effect. Rivero and Garcia (2001) concluded that parametric procedures could not adequately deal with uncertainties or variations within the system. Studies conducted by Arvidsson and Gremyr (2008) and Rao et al. (2008) confirmed that the Taguchi robust parameter is adequately disposed to render the control factors insensitive to the noise factors. Further studies based on RSM techniques (Aggarwal et al. 2007; Benyounis and Olabi 2008; Robinson, Borror, and Myers 2004; Balestrassi et al. 2009; Myers et al. 2004) stressed that although some design of experiments (DoE) are highly applicable, Taguchi robust parameter design is precisely time efficient and effective, and could greatly improve the quality and reliability of the product at low cost. Approaches such as the central composite design of RSM (see Montgomery 2009 for an explanation of this concept), as applied by Brito et al. (2014), used the MSE and concluded that it offered reduced sensitivity to the effect of process variability. In addition, work by Luzano and Gutierrez (2010) and Wu and Chyu (2002) adapted the signal-to-noise ratio (SN) as a function of the MSE. Therefore, Taguchi robust design is principally aimed at making the response of the process insensitive to the effects of variations, while DoE and RSM are based on eliminating these noise factor effects by striking a compromise within the process. These noise factors causing deviations from the quality target could be of external, internal and unit-to-unit dimensions. External noise factors are due to variations in the conditions of use, internal noise factors are due to production variations and unit-to-unit factors are a result of variations in the time of use. Arvidsson and Gremyr (2008) reported that Taguchi robust design can deal with all these categories of noise factors simultaneously, unlike DoE techniques which can deal only with the unit-to-unit type. Rocha et al. (2016) and Al-Refaie (2012) explained that a constant return to scale (CRS) Charnes, Cooper and Rhodes (CCR) model assumes that all appraised decision-making units (DMUs) are at the optimal production scale level (optimal state of CRS), with its technical efficiency encapsulating some components of the scale efficiency. Ma et al. (2014) discussed that the variable return to scale (VRS) model accounts for the component of scale efficiency, thereby making it easy for processes to be examined in the regions of increasing, constant and decreasing return to scale. The weaknesses of these classical models are: (1) their inability to offer scale (pure) technical efficiency; (2) that the efficiency score at input orientation is the same as the efficiency score at output orientation; and (3) their tendency to assign misleading scores to inefficient units. Previous efforts were only geared towards removing the inability of the standard CCR model to produce scale (pure) technical efficiency, but weaknesses (2) and (3) were not thoroughly dealt with. Adler, Friedman, and Sinuany-Stern (2002) conducted a thorough review and application of some of the proposed ranking methods, and concluded that none of them could be prescribed as an adequate solution to fully rank the DMUs in the data envelopment analysis (DEA) approach. This research used the VRS model because scale problems had been solved. Thus, the VRS model is a veritable basis for partitioning and provides adequate leverage for the DMUs to self-assess, paving the way for the estimation of the restriction of the free variable. This mean that a restriction is placed on the free variable instead of on the weights of input

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and output variables, as was previously done. The study attempts to solve this problem by imposing VRS partitioning within a modified VRS model. Cross-efficiency, aggressive formulation, benevolent formulation (BF), DEA games, linear discriminant analysis and discriminant analysis of ratios (Al-Refaie and Al-Tahat 2011; Adler, Friedman, and Sinuany-Stern 2002), which have been proposed to completely rank the DMUs, are inadequate in many ways. The integrated CCR–back-propagation neural network (BPNN) robust design procedure proposed by Liao (2004), involving censored data, found only the significant and non-significant controllable factors and could not select the optimum factor level setting that will maximize the multiple-response objective. Luzano and Gutierrez (2010) attempted to solve problems by incorporating BPNN with DEA to determine the mean responses of the parameter level settings with a three-step approach (see e.g. Luzano and Gutierrez 2010 on this concept). The advantage of the technique was that it achieved the selection of the optimum factors level. However, the identified drawback of the VRS model as identified in weakness (3) (see above) was not resolved, the adequacy of the BPNN topology was not tested, and the upper bound of the free variable uo of the VRS was free to assume any value between −∞ and +∞. These shortcomings are capable of making the outcome of the model vague. Salmasnia, Bastan, and Moeini (2012) proposed an application of GA with a selected adequate BPNN topology for training the model before prediction in the robust design. However, the selection was based only on the value of the MSE of testing and training data, and the prediction was made from the normalized signal-to-noise ratio (NSN). An adequate topology selection is based on the determination of the appropriate number of neurons in the hidden layer. This search has been thoroughly carried out by different methods: trial and error, heuristic search, exhaustive search, pruning and constructive algorithm, and GA search (Stathalkis 2009). The present study anticipates a holistic selection of the topology through MSE and the coefficient of determination or regression coefficient R2 of the training using the trial-and-error method. Discrimination among the DMUs is enhanced through the imposition of VRS partitioning, and adequate BPNN topology is incorporated for predicting the responses in any experiment with censored, missing and incomplete experimental data, or whenever responses are required beyond the data obtained experimentally (see Liao 2004 for examples of censored and missing data, and experiments with incomplete data). This study proposes the fractional factorial number of the orthogonal array (OA) as the number of neurons that should be located in the hidden layer of the BPNN. In the proposed model, assumptions are drastically reduced, the DMUs are allowed to self-assess to produce the input and output weights, computations are simplified and the procedure is completely non-parametric. This article is structured as follows. Section 2 presents the theoretical consideration of models for robust design, standard VRS, conditions for modified VRS, VRS penalization coefficient and BPNN selection. Section 3 presents the proposed model and numerical applications to illustrate the viability of the proposed procedure. Section 4 presents conclusions.

2. Theoretical considerations 2.1. Robust parameter design procedure The three quality response estimations for the robust design, according to the SN expression given by Li and Zhu (2017), are given in Equations (1)–(3), respectively, for larger-the-better (LTB), smallerthe-better (STB) and nominal-the-better (NTB). The normalized signal-to-noise ratios (NSNij ) are estimated using Equations (4), (5) and (6), respectively (Raza, Ahmad, and Kamal 2014):  n  1 1 SN = −10 log (1) n i=1 yij2 

1 2 y SN = −10 log n i=1 ij n

 (2)

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S. DANESHVAR AND K. A. ADESINA

 SN = −10 log

NSNij =

y¯ ij2



s2ij

(3)

NSNij =

Yij − min(Yij i = i = 1, 2, . . . , n) max(Yij i = i = 1, 2, . . . , n) − min(Yij i = 1, 2, . . . , n)

(4)

NSNij =

min(Yij i = 1, 2, . . . , n) − Yij max(Yij i = i = 1, 2, . . . , n) − min(Yij i = 1, 2, . . . , n)

(5)

|Yij − Target| − min(|Yij − Target|, i = 1, 2, . . . , n) max(|Yij − Target|, i = 1, 2, . . . , n) − min(|Yij − Target|, i = 1, 2, . . . , n)

(6)

where n is the number of replications and yij is the observed data, for variance s2ij determined for input i for j DMUs. 2.2. Consideration of DEA models To solve weakness (2) identified in Section 1, standard VRS is carried out to determine the efficiency scores (θ) at both input and output orientations, as applied by Daneshvar, Izibirak, and Javadi (2014), expressed as: where yro is the output of the DMU under investigation, xij is the input data to DMU j, yrj is the output data to DMU j, input weight vi , output weigh ur , uo is the upper bound of free variable of the optimal solution, m is the total number of input data, and s is the total number of output data Max s.t.

s  r=1 m  i=1 s 

ur yro + uo vi xio = 1 ur yrj −

r=1

m 

vi xij + uo ≤ 0

(7)

i=1

ur ≥ 0 r = 1, . . . s vi ≥ 0 i = 1, . . . m uo free where yro is the output of the DMU under investigation, xij is the input data to DMU j, yrj is the output data to DMU j, vi is the input weight, ur is the output weight, uo is the upper bound of the free variable of the optimal solution, m is the total number of input data, and s is the total number of output data. Similarly, at output orientation the efficiency score (η) is obtained by: Min s.t.

s  r=1 m  i=1 s  r=1

vi xio + vo ur yro = 1 ur yrj −

m  i=1

vi xij + uo ≤ 0

ur ≥ 0 r = 1, . . . s vi ≥ 0 i = 1, . . . m uo free

(8)

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2.3. Conditions for modified VRS Daneshvar, Izibirak, and Javadi (2014) and Daneshvar (2009) provided insights into how the partitioning of the VRS model can result in efficient points (EPs) or strong efficient points (SEPs) when θ = 1 and η = 1 (or vice versa), weak efficient points (WEPs) when θ = 1 and η < 1 (or vice versa), and inefficient points when θ < 1 and η < 1. It has been proven that the global optimal solution u∗o ∗ +∗ +∗ must satisfy the inequality that u−∗ o ≤ uo ≤ uo . For instance, if at input orientation uo = 1 then −∗ ∗ the inequality becomes uo ≤ uo ≤ 1; thus, there exists an intersection between WEP and EP. Therefore, if it is possible to restrict the free variable u∗o such that it is strictly less than 1, then it is possible to completely partition the frontier into EPs and WEPs. As applied by Adesina and Daneshvar (2018), for the output and input weights ut and v t , respectively, the upper and lower limits, u− and u+ , of the free variable are obtained as follows. u+ o is obtained by Max uo s.t. ut yo + uo = 1 v t xo = 1 ut yo − v t xo ≤ 0 ut ≥ 0 vt ≥ 0 uo free

(9)

In the same manner, u− o is obtained by Min uo s.t. ut yo + uo = 1 v t xo = 1 ut yo − v t xo ≤ 0 ut ≥ 0 vt ≥ 0 uo free

(10)

The restriction is estimated by + ε = max{u− o |u0 = 1

for efficient DMUs}

(11)

The modified VRS model is expressed as Max ut yo + uo s.t. v t xo = 1 ut yo − ut xo , i = 1 ut ≥ 0 r = 1, . . . s v t ≥ 0 i = 1, . . . m uo ≤ ε

(12)

2.4. DEA penalization coefficient conditions To select the optimum parameter setting from the efficient settings obtained in Section 2.3, the second VRS model is adapted after Gutiérrez and Lozano’s (2010) penalization coefficient WJ , where J is the

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S. DANESHVAR AND K. A. ADESINA

index of factor level combination for DMUJ , given as Max s.t.

WJ m  i=1 m 

ui NSDij ≥ 1

∇j = J

ui NSDiJ = 1

i=1 ui ≥

WJ WJ ≥ 0

(13)

∇i

This step indicates the efficacy of each efficient parameter setting to produce its equivalent maximum lower bound, which is a kind of penalty that can be imposed on the responses and thus replaces the step of setting a trade-off between the variables. 2.5. BPNN model selection At this juncture, it should be noted that BPNN in the model is intended for prediction purposes only. Neural fitting (nftool) feed-forward back-propagation with the Levenberg–Marquart algorithm ® is employed, in MATLAB release 2014a. The hidden layer is transformed by the sigmoid function while the output layer uses a linear fit function. Adequate topology is selected based on the MSE and the coefficient of determination or regression coefficient R2 of the training and cross-validation.

3. The proposed modified VRS-BPNN robust parameter framework 3.1. Construction of the modified VRS robust parameter procedure The framework of the procedure is depicted in Figure 1. The description and illustration of the phases involved in the method are presented in the following subsections. 3.2. Numerical demonstration of the proposed method 3.2.1. Optimization of a hard disc drive Procedure A Steps 1–3: The quality of a hard disc drive was investigated, with four responses: 50% pulse width (PW), peak shift (PS), overwrite (OW) and high-frequency amplitude (HFA). PW and PS are STBtype responses, while OW and HFA are LTB-type responses. Five controllable process factors used for the investigation involve (A) disc writability, (B) magnetization width, (C) gap length, (D) coercivity of media, and (E) rotational speed. The input and response data of the hard disc as given by AlRefaie and Al-Tahat (2011) and Al-Refaie (2012) are presented in Supplementary Table S1. In total, 18 parameter level combinations were achieved; therefore, the L18 OA is adopted and 18 DMUs are defined. Procedure B Steps 1–3: The step is not required since all the data have been obtained from the experiment and no further data beyond those obtained by the experiment are anticipated. Procedure C Step 1: Estimate the SN of responses by applying Equation (3) to PW and PS, and Equation (2) to OW, treated as STB, and HFA. For DMU 1, SN, PW = −10 ∗ log(64.75) ∧ 2 = −36.22479 SN, PS = −10 ∗ log(11.45) ∧ 2 = −21.176109

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Figure 1. Proposed modified variable return to scale (VRS)–back-propagation neural network (BPNN) robust parameter framework for solving multiple-response experiments. DEA = data envelopment analysis; DMU = decision-making unit; SN = signal-tonoise ratio; NSN = normalized signal-to-noise ratio; NN = neural network; EP = efficient point; SEP = strong efficient point; WEP = weak efficient point.

SN, OW = −10 ∗ log(31.15) ∧ 2 = 29.86916 SN, HFA = −10 ∗ log(1/272.15) ∧ 2 = 48.6961 Step 2: Estimate the NSN of the SNs by applying Equations (4) and (5) appropriately: NSN, PW =

−33.73272 − (−36.22479) = 0.4800 −33.73272 − (−38.92413)

NSN, PS =

20.00 − (−21.176109) = 0.1906 −20.00 − (−26.17128)

NSN, OW =

29.86916 − 25.48315 = 0.5771 33.08353 − 25.48315

NSN, HFA =

48.6961 − 47.00108 = 0.3074 52.51447 − 47.00108

Procedure D Step 1: Solve the standard VRS model of Equation (7) using MaxDEA version 6.0 software. Step 2: Solve the standard VRS model of Equation (8) using MaxDEA version 6.0 software. The input and output weights obtained are shown in Supplementary Table S2. Step 3: Extract the efficiency scores obtained from Steps 1 (θ) and 2 (η). If a DMU is efficient at both orientations, then it is an EP/SEP; otherwise it is a WEP. − Step 4: Solving Equations (9) and (10), u+ o and uo are estimated using LINGO 6.18 as: The remaining DMUs are calculated in the same manner and the results are presented in Supplementary Table S3. Equation (10) is used to determine the upper bound restriction , obtained as 0.3957. The modified VRS model (Equation 12) is solved to determine the efficiency of the DMUs (Supplementary Figure S1).

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Robust parameter Design factors (input variables)

NSN

SN

Modified VRS model

DMU

A

B

C

D

E

PW50

PS

OW

HFA

PW50

PS

OW

HFA

Score (θ )

Score (η)

Partitioning

Modified

Penalization coefficient

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2

1 2 3 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1

0.4800 0.4891 0.5196 0.2236 0.2667 1.0009 0.4826 0.4916 0.5184 0.0000 0.7470 0.7415 0.0000 0.7481 0.7415 0.2236 0.2667 1.0000

0.1906 0.2914 0.4886 0.0000 0.0952 0.8620 0.1277 0.1844 0.5613 0.1844 0.7551 0.8076 0.1083 0.6214 0.8734 0.3141 0.3419 1.0000

0.5771 0.6788 0.7345 0.6256 0.7281 0.8398 0.5567 0.6955 1.0000 0.0091 0.6465 0.6922 0.2815 0.5071 0.8236 0.0000 0.7135 0.7967

0.3074 0.5957 0.7817 0.5215 0.7067 0.0000 0.3158 0.5643 0.4491 1.0000 0.3370 0.6092 0.9942 0.4432 0.2273 0.7523 0.7486 0.2209

−36.22 −36.27 −36.43 −34.89 −35.12 −38.93 −36.24 −36.28 −36.42 −33.73 −37.61 −37.58 −33.73 −37.62 −37.58 −34.89 −35.12 −38.92

−21.18 −21.80 −23.02 −20.00 −20.59 −25.32 −20.79 −21.14 −23.46 −21.14 −24.66 −24.98 −20.67 −23.83 −25.39 −21.94 −22.11 −26.17

29.87 30.64 31.07 30.24 31.02 31.87 29.71 30.77 33.08 25.55 30.40 30.74 27.62 29.34 31.74 25.48 30.91 31.54

48.70 50.29 51.31 49.88 50.90 47.00 48.74 50.11 49.48 52.51 48.86 50.36 52.48 49.44 48.25 51.15 51.13 48.22

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9902 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.5551 0.4720 0.3961 0.7296 0.5566 0.3700 0.6744 0.3954 0.8379 0.4023 0.3508 0.7225 0.4311 0.3751 0.7752 0.4829 0.3314

Note: DMU = decision-making unit; A = direction of gear hobbing; B = number of passes; C = source of hob; D = feed; E = speed; F = job run-out; NSN = normalized signal-to-noise ratio; SN = signal-to-noise ratio; PW50 = 50% pulse width; PS = peak shift; OW = overwrite; HFA = high-frequency amplitude; VRS = variable return to scale; EP = efficient point; SEP = strong efficient point.

S. DANESHVAR AND K. A. ADESINA

Table 1. Efficiency scores for the standard orientations, modified Banker, Charnes and Cooper (BCC) model and penalization coefficients for the hard disc case study.

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Figure 2. Average signal-to-noise ratio (SN) response for factor level setting for the hard disc drive.

Procedure E Equation (13) is solved for penalization coefficient WJ , for only the efficient DMUs obtained in procedure D above. For DMU 1, W 1 is estimated to be 0.5551 (Supplementary Figure S2). Table 1 shows the results of SN, NSN, θ, η, partitioning, modified VRS efficiency scores and the penalization coefficient obtained by all 18 DMUs at both orientations as EPs/SEPs. The proposed method criticizes and discriminates among the DMUs by correcting the problems associated with the standard model. It is able to reveal those inefficient DMUs that have been returned as efficient by the standard VRS model. With this method, the efficiency of DMU 8 changes, indicating that it either is a WEP or is comparable with the WEP. This DMU 8 is discarded at this point since WEPs and their companions cannot yield an optimum output. Furthermore, DMU 8 is not within the convex combination of the process design factors and did not show any possibility that a virtual output could be formed from this combination. The VRS penalization coefficient estimation yielded the highest score of 0.8379 for DMU 10 which is optimal on the OA; similarly, in Figure 2, factor level combination DMU 10 has the highest average SN response value; hence, DMU 10 with A2 B1 C1 D3 E3 is selected as the optimum factor level combination for the hard disc process. The same DMU was obtained by the BF method of Al-Refaie and Al-Tahat (2011). To confirm the efficacy of the proposed method over other previously applied methods, the anticipated improvement (e.g. Liao and Chan 2002) of the proposed method is obtained and compared with those of BF, data envelopment analysis ranking (DEAR) and PCA. The initial conditions for the significant design factors have been reported previously (e.g. Su and Tong 1997). Therefore, for the optimal DMU 10, PW = −33.73 − (−36.28) = 2.55 PS = −25.14 − (−21.48) = 0.34 OW = −25.55 − (−31.51) = 2.96HFA = 52.51 − (50.47) = 2.04 Total anticipated improvement = 2.55 + 0.34 + 2.96 + 2.04 = 10.890 The total anticipated improvement for the proposed method is 10.890; for the BF technique it was 10.681, PCA 2.61 and DEAR 3.35 (Table 2), clearly confirming that the proposed method outperformed the previous methods. 3.2.2. Gear hobbing operation As previously illustrated with BF (Al-Refaie and Al-Tahat 2011), Supplementary Table S4 presents the input six controllable factors: direction of gear hobbing (A), number of passes (B), source of hob (C), feed (D), speed (E) and job run-out (F); with four quality responses: left profile (LP) error, right

10 S. DANESHVAR AND K. A. ADESINA

Table 2. Summary of the anticipated improvement of previous methods and the proposed method for the hard disc case study. SN of the optimal combination obtained (2)

Response

Initial condition (1)

PCA (Su and Tong 1997)

PW −36.28 −33.74 PS −21.48 −19.37 OW −31.51 27.71 HFA 50.47 52.23 Total improvement anticipated

Anticipated improvement (2) – (1)

DEAR (Liao and Chen 2002)

Benevolent formulation (Al-Refaie and Al-Tahat (2011)

Proposed method (modified VRS-BPNN robust parameter)

PCA (Su and Tong 1997)

DEAR (Liao and Chen 2002)

Benevolent formulation (Al-Refaie and Al-Tahat (2011)

Proposed method (modified VRS BPNN-robust parameter)

−33.74 −19.17 28.97 51.51

−33.73 −21.05 −25.67 52.95

−33.73 −21.14 25.55 52.51

2.54 2.11 −3.80 1.76 2.61

2.54 2.31 −2.54 1.04 3.35

2.54 0.44 5.22 2.48 10.68

2.55 0.34 5.96 2.04 10.89

Note: PCA = principal component analysis; DEAR = data envelopment analysis ranking; VRS = variable return to scale; BPNN = back-propagation neural network; PW = pulse width; PS = peak shift; OW = overwrite; HFA = high-frequency amplitude.

Table 3. Efficiency scores for the standard orientations, modified Banker, Charnes and Cooper (BCC) model and the penalization coefficient for the gear hobbing case study. Robust parameter Design factors (input variables)

NSN

SN

Modified VRS model

DMU

A

BC

D

E

F

LP

RP

LH

RH

LP

RP

LH

RH

Score (θ )

Score (η)

Partitioning

Modified

Penalization coefficient

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2

1 2 3 2 3 1 1 2 3 3 1 2 3 1 2 2 3 1

0.1249 0.6707 0.4180 0.5218 0.6195 0.0000 0.5733 0.9001 1.0000 0.3257 0.4233 0.0251 0.5733 0.8112 0.1780 0.6639 0.6246 0.7098

0.2929 0.3304 0.1665 0.7261 0.5724 0.2871 0.0000 0.4122 0.0445 0.6941 1.0000 0.4933 0.3735 0.3692 0.4079 0.9594 0.7316 0.0045

0.4034 0.0000 0.5012 0.6773 1.0000 0.1061 0.5520 0.5847 0.6165 0.2836 0.4357 0.2760 0.0556 0.2321 0.2836 0.3527 0.2986 0.4799

0.4968 0.3581 0.7582 0.8690 0.9933 0.4757 1.0000 0.5649 0.9940 0.6364 0.1580 0.0000 0.1810 0.3088 0.4059 0.2040 0.3646 0.6459

−37.21 −37.58 −37.41 −37.48 −37.54 −37.13 −37.51 −37.73 −37.80 −37.35 −37.41 −37.14 −37.51 −37.67 −37.25 −37.57 −37.55 −37.60

−37.38 −37.41 −37.28 −37.73 −37.61 −37.38 −37.14 −37.48 −37.18 −37.71 −37.96 −37.54 −37.45 −37.44 −37.47 −37.92 −37.74 −37.15

−33.51 −30.22 −34.31 −35.74 −38.38 −31.08 −34.72 −34.99 −35.25 −32.53 −33.77 −32.47 −30.67 −32.11 −32.53 −33.10 −32.66 −34.13

−32.65 −31.84 −34.17 −34.81 −35.53 −32.53 −35.57 −33.04 −35.54 −33.46 −30.68 −29.76 −30.81 −31.56 −32.12 −30.95 −31.88 −33.52

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8429

1.0000 1.0000 1.0000 1.0000 1.0000 0.9803 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.5319 1.0000 1.0000 0.8504

EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP WEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP WEP EP, SEP EP, SEP Inefficient

1.0000 1.0000 1.0000 1.0000 1.0000 0.9978 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.94921 1.0000 1.0000 0.8429

0.7587 0.7357 0.5423 0.3579 0.3139 0.4705 0.4062 0.3766 0.5155 0.4957 1.2588 0.845 0.5809

Note: VRS = variable return to scale; DMU = decision-making unit; A = direction of gear hobbing; B = number of passes; C = source of hob; D = feed; E = speed; F = job run-out; NSN = normalized signal-to-noise ratio; SN = signal-to-noise ratio; LP = left profile; RP = right profile; LH = left helix; RH = right helix; EP = efficient point; SEP = strong efficient point; WEP = weak efficient point.

ENGINEERING OPTIMIZATION

0.4587 0.4952

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Figure 3. Average signal-to-noise ratio (SN) response for factor level setting for the gear hobbing operation.

profile (RP) error, left helix (LH) error and right helix (RH) error. Factors A, B and C are at two levels each; factors D, E and F are at three levels each. Factors B and C are assigned to the same column as BC to make up a modified L18 OA for this case study. The modified OA can, therefore, hold three-level factors in its first three column and three-level factors in the next three columns (see e.g. Jeyapaul, Shahabudeen, and Krishnaiah 2006 on this concept of modified OA for this particular case study). Similarly, the upper bound restriction ε according to the experimental data is obtained as 0.8915 (Supplementary Table S5). At the input orientation, Table 3 reveals that DMUs 1–17 are efficient while DMU 18 is inefficient. At the output orientation, DMUs 6, 15 and 18 are inefficient while the others remain on the frontier. On partitioning, DMUs 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16 and 17 are EPs and SEPs while DMUs 6, 15 and 18 are WEPs. On the application of the proposed modified VRS model, DMUs 6, 15, and 18 are returned as inefficient. Therefore, DMUs 6 and 15 are strictly WEPs, and since DMU 18 is inefficient at both orientations and on the proposed modified VRS, then it is strictly inefficient and cannot be said to be in the same possible production set and within the convex combination of the controllable factors for this gear hobbing operation. The highest penalization coefficient score of 1.2588 is obtained for DMU 12 and, coupled with Figure 3, DMU 12 with A2 B1 C1 D3 E2 F2 is selected as the optimum factor level combination. Similarly, the BF approach obtained the same; the total anticipated improvement (Table 4) for the proposed model is 11.8224, GA (see e.g. Jeyapaul, Shahabudeen, and Krishnaiah 2006 on this selection) was 4.1498 and BF was 11.2506, clearly showing that the proposed model outperformed BF and GA. 3.2.3. Apple dehydration operation Di Scala et al. (2013) demonstrated an artificial neural network–GA to predict the quality characteristics of apples in a convective dehydration process involving hot air flow at three different temperatures, 40°C, 60°C and 80°C, and at three air flow rates, 0.5 m/s, 1 m/s and 1.5 m/s. The quality characteristics examined are total phenolic content (TPC), surface colour (SC) and water holding capacity (WHC), with correlation coefficients (r2 ) of 0.987, 0.990 and 0.994 respectively. This case study is used to demonstrate how BPNN can be tested, trained and validated for the prediction of the responses beyond the experimental input variables. This reveals that a hidden layer containing nine neurons would give the lowest MSE with the highest regression coefficient r2 . The adequate topology adopted as shown in Supplementary Table S6 has the lowest MSE and the highest regression coefficient R value of 0.000338 and 0.999, respectively, for the training; this is supported by the cross-validation, with an R value of 0.998 and MSE of 10.306. This well-trained topology can be used to estimate, within acceptable error limits, the multi-quality response of any process parameter settings beyond those experimentally tested (Table 5). From the obtained results, the fractional factorial number of

Table 4. Summary of the anticipated improvement of previous methods and the proposed method for the gear hobbing case study. SN of the optimal combination obtained (2)

Response

Initial condition (1)

GA (Jeyapaul, Shahabudeen, and Krishnaiah 2006)

LP error −37.8581 RP error −37.4952 LH error −36.6009 RH error −35.7397 Total improvement anticipated

−37.4917 −37.4045 −34.4082 −34.2396

BF (Al-Refaie and Al-Tahat (2011) −37.3728 −37.7724 −31.9040 −29.9858

−37.1800 −37.4984 −31.4320 −30.3328

Anticipated improvement (2) – (1)

Proposed method (modified VRS-BPNN robust parameter)

GA (Jeyapaul, Shahabudeen, and Krishnaiah 2006)

−37.1400 −37.5400 −31.4320 −29.7600

0.3664 0.0907 2.1927 1.5001 4.1498

BF (Al-Refaie and Al-Tahat (2011) 0.4853 −0.2772 4.6968 5.7539 10.6588

0.6781 −0.0032 5.1688 5.4069 11.2506

Proposed method (modified VRS-BPNN robust parameter) 0.7181 −0.0443 5.1689 5.9797 11.8224

Note: SN = signal-to-noise ratio; GA = genetic algorithm; BF = benevolent formulation; VRS = variable return to scale; BPNN = back-propagation neural network; LP = left profile; RP = right profile; LH = left helix; RH = right helix. ENGINEERING OPTIMIZATION 13

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Table 5. Back-propagation neural network (BPNN) demonstration of predicted response. Experimental data Input variables Drying temperature 40.00 40.00 40.00 60.00 60.00 60.00 80.00 80.00 80.00

Response

BPNN predicted response

Air flow rate

SC

TPC

WHC

SC

TPC

WHC

0.50 1.00 1.50 0.50 1.00 1.50 0.50 1.00 1.50

35.71 29.10 37.17 28.15 28.89 33.88 45.15 18.89 27.04

31.93 27.39 27.34 40.15 39.31 33.62 38.70 44.82 44.12

45.50 56.30 50.94 51.66 53.74 44.12 48.28 45.46 47.80

35.71 ± 0.0014 30.012 ± 0.9190 37.16 ± 0.0073 28.16 ± 0.0093 28.92 ± 0.0283 33.88 ± 0.0016 45.15 ± 0.0034 18.96 ± 0.0450 53.448 ± 26.408

31.71 ± 5.7e-4 26.87 ± 0.5227 27.34 ± 0.0033 40.14 ± 0.0087 39.32 ± 0.0083 33.65 ± 0.029 38.72 ± 0.0174 44.81 ± 0.0100 50.23 ± 6.108

45.15 ± 3.3e-4 50.84 ± 5.4590 50.94 ± 5.4e-4 51.68 ± 0.0190 53.72 ± 0.0215 44.08 ± 0.0414 48.27 ± 0.0096 45.47 ± 0.006 49.24 ± 1.443

Note: SC = surface colour; TPC = total phenolic content; WHC = water holding capacity.

Figure 4. Average signal-to-noise ratio (SN) response for factor level setting for apple dehydration. A1 = temperature, 40°C; A2 = temperature, 60°C; A3 = temperature, 80°C; B1 = air flow rate, 0.5 m/s, B2 = air flow rate, 1 m/s; B3 = air flow rate, 1.5 m/s.

the OA is suggested as the number of neurons in the hidden layer for solving this kind of problem in the robust parameter strategy. Nine parameter level settings are defined; therefore, L9 OA is adopted, with nine neurons in the hidden layer of the BPNN. The upper bound variable restriction is estimated from the input and output weights (Supplementary Table S7) to be 0.6013 (Supplementary Table S8). The partitioning reveals that DMU 6 is strictly inefficient while the others are efficient. The proposed method reveals a contrary result, that DMU 6 is efficient while DUMs 5, 8 and 9 are inefficient (Table 6). This means that standard VRS models would have discarded DMU 6 and accepted DUMs 5, 8 and 9, thereby misleading the quality engineer or production manager. DMUs 5, 8 and 9 would have been treated as members of the possible production set and included in the search for the optimum factor level setting. The proposed model has corrected the problem associated with the standard models. On the application of the VRS penalization and the response graph (Figure 4), DMU 8 with a temperature of 80°C and air flow rate of 0.5 m/s gives the highest penalization coefficient of 0.8905 and is selected as the optimal factor level setting. This implies that at these dehydration conditions of temperature and air flow rate, all the quality attributes will be favoured. This assertion agrees with the findings of Di Scala et al. (2013), where (1) the lowest surface colour change was noticed at 80°C at all flow rates (0.5, 1 and 1.5 m/s); (2) TPC degradation decreased with increasing temperature at the air flow rates of 0.5 and 1 m/s, with the lowest degradation of TPC observed at 80°C; and (3) WHC changed as the air temperature increased at a constant air flow rate. The anticipated improvement shows that the proposed model gives a higher total anticipated improvement value of 16.989, compared with 1.284 for GA (Table 7).

Table 6. Efficiency scores for standard orientations, modified Banker, Charnes and Cooper (BCC) model and penalization coefficient for apple dehydration. Robust parameter Design factors (input variables) DMU 1 2 3 4 5 6 7 8 9

NSN

SN

Modified VRS model

Air drying temperature

Air drying velocity

SC

TPC

WHC

SC

TPC

WHC

Score (θ )

Score (η)

Partitioning

Modified

Penalization

40.000 40.000 40.000 60.000 60.000 60.000 80.000 80.000 80.000

0.500 1.000 1.500 0.500 1.000 1.500 0.500 1.000 1.500

0.731 0.496 0.777 0.458 0.488 0.670 1.000 0.000 0.412

0.314 0.004 0.000 0.777 0.735 0.418 0.703 1.000 0.968

0.126 1.000 0.590 0.647 0.809 0.000 0.370 0.123 0.329

−31.056 −29.278 −31.403 −28.989 −29.215 −30.599 −33.093 −25.525 −28.640

30.084 28.752 28.736 32.074 31.890 30.532 31.754 33.029 32.893

33.160 35.010 34.141 34.263 34.606 32.893 33.675 33.153 33.589

1.000 1.000 1.000 1.000 1.000 0.743 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.0000 0.797 1.000 1.000 1.000

EP, SEP EP, SEP EP, SEP EP, SEP EP, SEP WEP EP, SEP EP, SEP EP, SEP

1.000 1.000 1.000 1.000 0.742 1.000 1.000 0.934 0.934

0.854 0.667 0.732 0.5314 0.482 0.891

ENGINEERING OPTIMIZATION

Note: VRS = variable return to scale; DMU = decision-making unit; NSN = normalized signal-to-noise ratio; SN = signal-to-noise ratio; SC = surface colour; TPC = total phenolic content; WHC = water holding capacity; EP = efficient point; SEP = strong efficient point; WEP = weak efficient point.

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Table 7. Summary of the anticipated improvement of previous and proposed method for apple dehydration. SN of the optimal combination obtained (2)

Response

Initial condition (1)

SC TPC WHC

38.651 −43.988 −34.255

GA (Di Scala et al. 2013)

Proposed method (modified VRS-BPNN robust parameter)

28.422 33.093 −32.475 −31.754 −34.255 −33.675 Total improvement anticipated

Anticipated improvement (2) – (1) GA (Di Scala et al. 2013)

Proposed method (modified VRS-BPNN robust parameter)

−10.229 11.513 0.000 1.284

−5.558 12.234 10.313 16.989

Note: SN = signal-to-noise ratio; GA = genetic algorithm; VRS = variable return to scale; BPNN = back-propagation neural network; SC = surface colour; TPC = total phenolic content; WHC = water holding capacity.

4. Conclusions This study proposes a modified VRS-BPNN robust optimization framework for optimizing multiquality response systems. Efficiency determination and optimization to select the optimum parameter settings are achieved in the most simplified, adequate and effective manner. The proposed model provides the largest anticipated improvement over all other previously used approaches (PCA, DEAR, GA and BF). Furthermore, the following advantages can be noted: • The suggestion of using the fractional factorial number of the OA as the number of neurons in the hidden layer of the BPNN proved to be adequate and it is capable of reducing errors and uncertainties. • Estimation of the restriction for the upper bound of the free variable of the VRS determined by self-evaluation within the DMUs replaces errors in setting the non-Archimedean infinitesimal, as proposed in some previous studies. • The partitioning and the modification provide an adequate selection of the optimum parameter setting through enhanced discrimination. • The approach does not require any initial information and assumptions and it is not based on cogent assumptions, in contrast to PCA, GA and other approaches. The discriminative tendency provides further insight into DMUs that are within the convex set of the factor level settings and those that should not be considered within the search, thereby making the computational search easier than in other reported methods. These attributes could be of interest to quality, process and product engineers, as well as project managers, operational managers and upper management, who could embrace the proposed procedure to optimize multi-quality response processes in the robust parameter design strategy. However, more studies are needed on the application of the suggested fractional factorial number of the OA, as the number of neurons in the hidden layer of the BPNN, and its predictive consistency, in order to generalize the viability of the BPNN in the proposed model. Similar studies using DEA competitive games and virtual DEA are also anticipated.

Disclosure statement No potential conflict of interest was reported by the authors.

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