Sothea Has
email: sothea.has@lpsm.paris

Consensual Aggregation on Random Projected High-dimensional Features for Regression

Keywords: Consensual aggregation, random projection, regression. 2010 Mathematics Subject Classification: 62G08, 62J99, 62P30

In this paper, we present a study of a kernel-based consensual aggregation on randomly projected high-dimensional features of predictions for regression. The aggregation scheme is composed of two steps: the high-dimensional features of predictions, given by a large number of regression estimators, are randomly projected into a smaller subspace using Johnson-Lindenstrauss Lemma in the first step, and a kernel-based consensual aggregation is implemented on the projected features in the second step. We theoretically show that the performance of the aggregation scheme is close to the performance of the aggregation implemented on the original high-dimensional features, with high probability. Moreover, we numerically illustrate that the aggregation scheme upholds its performance on very large and highly correlated features of predictions given by different types of machines. The aggregation scheme allows us to flexibly merge a large number of redundant machines, plainly constructed without model selection or cross-validation. The efficiency of the proposed method is illustrated through several experiments evaluated on different types of synthetic and real datasets.

Introduction

In supervised machine learning problems, one aims at predicting values of any quantities of interest using the corresponding input information. When the quantity of interest or response takes continuous values (which is the focus of this paper), the task is called regression. On the other hand, it is called classification if the response takes values in any finite sets (few unique values).

Nowadays, several machine learning models are invented, can be easily implemented and used in any supervised prediction problems. Those methods aim at approximating the relationship between inputs and the corresponding outputs by minimizing some empirical criterion, which is a function of the training data. Hence, the performances of those predictive models strongly depend on the data fed to them. In practice, we may try to implement different types of models according to the context of the problems, and the one with strong generalization capability would be selected. However, selecting the best method may require a lot of efforts and consideration. Therefore, another approach is to automatically combine those candidate predictors in a flexible way, in a sense that the performance of the combination biases towards the best basic estimators.

Up to now, many combining estimation methods have been introduced, for instance, ensemble learning methods which combines an homogeneous type (trees) of predictors such as Random Forest [START_REF] Friedman | Bagging predictors[END_REF]) and Boosting [START_REF] Friedman | Greedy function approximation: A gradient boosting machine[END_REF]). Moreover, some other methods allowing to combine a bunch of different types of individual estimators using some convex combination are also introduced, for example, in [START_REF] Catoni | Statistical Learning Theory and Stochastic Optimization[END_REF], Juditsky and [START_REF] Nemirovski | Topics in Non-Parametric Statistics[END_REF], [START_REF] Nemirovski | Topics in Non-Parametric Statistics[END_REF], [START_REF] Yang | Combining different procedures for adaptive regression[END_REF][START_REF] Yang | Adaptive regression by mixing[END_REF], [START_REF] Yang | Aggregating regression procedures to improve performance[END_REF], [START_REF] Györfi | A Distribution-Free Theory of Nonparametric Regression[END_REF], [START_REF] Wegkamp | Model selection in nonparametric regression[END_REF], [START_REF] Audibert | Aggregated estimators and empirical complexity for least square regression[END_REF], [START_REF] Bunea | Aggregation and sparsity via 1 -penalized least squares[END_REF]Bunea et al. ( , 2007a,b),b), and [START_REF] Dalalyan | Aggregation by exponential weighting, sharp PAC-Bayesian bounds and sparsity[END_REF]. There are also a group of combining strategies that aggregate different instance estimators based on features of predictions given by the basic estimators such as stack generalization of [START_REF] Wolpert | Stacked generalization[END_REF] and stacked regression by [START_REF] Breiman | Stacked regression[END_REF]. Last but not least, some combining estimation methods aggregating different types of individual estimators based on consensus level of predictions given by the instances, which is the central idea of this chapter, are also introduced by [START_REF] Mojirsheibani | Combined classifiers via disretization[END_REF][START_REF] Mojirsheibani | A kernel-based combined classification rule[END_REF] and [START_REF] Mojirsheibani | An asymptotically optimal kernel combined classifier[END_REF] for classification problems, by [START_REF] Biau | COBRA: a combined regression strategy[END_REF] and [START_REF] Has | A Kernel-based Consensual Aggregation for Regression[END_REF] for regression problems, and for both frameworks by [START_REF] Fischer | Aggregation using input-output trade-off[END_REF], where in this last method the combination also takes into account the input part. The consistency result of each consensual aggregation method is provided under different assumptions, and is also confirmed through several numerical simulations.

This study focuses on a high-dimensional setting of combining estimation strategy for regressions by [START_REF] Has | A Kernel-based Consensual Aggregation for Regression[END_REF]. The method is an extension to a regular kernel-based framework of a combining strategy by [START_REF] Biau | COBRA: a combined regression strategy[END_REF], which is a regression configuration of combining classifiers by [START_REF] Mojirsheibani | Combined classifiers via disretization[END_REF]. More precisely, let r(x) = (r 1 (x), ..., r M (x)) denote the prediction vector of x ∈ R d , given by the M basic regression estimators r 1 , ..., r M , and suppose that n iid couples of supervised training data (X 1 , Y 1 ), ..., (X n , Y n ) are observed. Moreover, let . denote the Euclidean norm on R M , thus the prediction at any point x ∈ R d of the combining strategy by [START_REF] Has | A Kernel-based Consensual Aggregation for Regression[END_REF] is defined by

g n (r(x)) = n i=1 Y i K h ( r(x) -r(X i ) ) n j=1 K h ( r(x) -r(X j ) ) (1)
for some regular kernel function K with K h (x) = K(x/h) for some smoothing parameter h > 0, and the convention of 0/0 = 0. Note that COBRA method of [START_REF] Biau | COBRA: a combined regression strategy[END_REF] corresponds to naive kernel K(x) = M j=1 1 {|x j |<ε} for some window parameter ε > 0 to be tuned. It is theoretically shown that the combining strategy asymptotically outperforms the best individual estimator in L 2 sense. Moreover, the implementation of the classical method is available in COBRA library of R software (see [START_REF] Guedj | COBRA: Nonlinear Aggregation of Predictors[END_REF]), and a slightly different setting of its kernel-based configuration is available in Python library called pycobra (see [START_REF] Guedj | Pycobra: A python toolbox for ensemble learning and visualisation[END_REF].

Until now, the study of high-dimensional case of the described consensual aggregation method has not been considered yet. Therefore, this study aims at filling this gap by considering exponential kernel-based consensual aggregation for regression on high-dimensional features of predictions. In other words, we are interested in combining a large number of basic machines, which might be obtained by varying the hyperparameters of any types of predictive models, or from mixtures of different types of models. Moreover, these basic machines can be constructed without any model selection or cross-validation techniques. One can simply see this aggregation scheme as a method to merge the candidate models into one final prediction that is asymptotically optimal with respect to all the basic machines.

However, working in high-dimensional spaces often brings along some difficulties such as highly computational cost and curse of dimensionality, which refers to the situation where Euclidean distance loses its meaning. In this study, these problems are handled using dimensional reduction technique based on Johnson and Lindenstrauss Lemma (J-L). Johnson and Lindenstrauss showed that for any δ > 0 given, one can embed a given finite set of high-dimensional vectors of Euclidean spaces into a lower-dimensional sub-space, preserving the pairwise Euclidean distances between data points up to an error δ, with high probability (see, for example, [START_REF] Johnson | Extensions of lipschitz maps into a hilbert space[END_REF] and [START_REF] Johnson | Extensions of lipschitz maps into banach spaces[END_REF]). This result has become a very powerful technique of dimensional reduction aiming at preserving pairwise Euclidean distances between data points (Frankl andMaehara (1988, 1990) and [START_REF] Dasgupta | An elementary proof of a theorem of johnson and lindenstrauss[END_REF]). J-L method is suitable for our setting not only because of the pairwise-distance preserving property, but also because of its computational efficiency. The implementation of this method is as simple as simulating M independent random vectors (rows of projection matrix), and performing a matrix multiplication. Dimensional reduction based on J-L technique has also been applied in several machine learning studies, for instance, in image processing and text analysis by [START_REF] Bingham | Random projection in dimensionality reduction: Applications to image and text data[END_REF], in Lipschitz embeddings of graphs into normed spaces by [START_REF] Frankl | The johnson-lindenstrauss lemma and the sphericity of some graphs[END_REF], in approximating nearest-neighbor in high-dimensional spaces by [START_REF] Kleinberg | Two algorithms for nearest-neighbor search in high dimensions[END_REF] and [START_REF] Indyk | Approximate nearest neighbors: Towards removing the curse of dimensionality[END_REF], in linear regression framework by [START_REF] Maillard | Linear regression with random projections[END_REF], and also in unsupervised clustering in Hilbert spaces by Biau et al. (2008a).

In this work, we propose an aggregation scheme on random projected features of high-dimensional predictions. The scheme is composed of two steps. First, we randomly embed the original features of predictions of dimension M (large) into a lower subspace of dimension m (m < M ) using dimensional reduction based on J-L Lemma. Then, the consensual aggregation (1) is implemented on the projected features of predictions in the second step. We aim in this study to provide a probability bound of the difference between the classical consensual aggregation and the aggregation implemented on projected features of predictions. We also numerically illustrate the performance of the full aggregation scheme on several simulated and real-world datasets.

This chapter is organized in the following manner. Section 2 details the construction of the proposed aggregation scheme. Section 3 provides the theoretical performance of the method. Section 4 illustrates performance of the method through several numerical experiments evaluated on different types of datasets. Lastly, the proofs of the theoretical results stated in this paper are collected in Section 6.

The aggregation method

Notation

Assume that (X, Y ) is an R d × R-valued generic random variable, and that we have at hand a training dataset containing iid copies of (X, Y ):

D n = {(X 1 , Y 1 ), (X 2 , Y 2 ), ..., (X n , Y n )}.
We assume moreover that M basic regression estimators or machines r 1 , r 2 , ..., r M , are constructed independently of D n (otherwise, a simple splitting technique can be used as described, for example, in [START_REF] Biau | COBRA: a combined regression strategy[END_REF] and [START_REF] Has | A Kernel-based Consensual Aggregation for Regression[END_REF]). These basic machines can be any regression estimators of the same type (with different parameters), or constructed based on completely different theories. We only require that they can predict the training data and any new data points since the aggregation is done based only on those predictions.

To alleviate notation, when the context is clear, all Euclidean norms will be denoted by . without mentioning the dimension of the space. Moreover, this paper deals with exponential kernel, K(t) = exp(-t α /σ), for some σ > 0 and α ≥ 0, which has numerically been shown to be the most outstanding one so far in the previous studies. Moreover, let µ denote the distribution of X with respect to Lebesgue measure, and the regression function is denoted by η

(x) = E[Y |X = x].

Random projection: Johnson-Lindenstrauss Lemma

In the sequel, the prediction matrix of the training data is denoted by

r(X ) =      r 1 (X 1 ) r 2 (X 1 ) . . . r M (X 1 ) r 1 (X 2 ) r 2 (X 2 ) . . . r M (X 2 ) . . . . . . . . . . . . r 1 (X n ) r 2 (X n ) . . . r M (X n )      n×M . (2)
For any positive integer m < M , let G = (G ij ) 1≤i≤M,1≤j≤m be a random projection matrix where the entries G ij are iid centered Gaussian random variables with variance 1/m, for all i = 1, 2, ..., M and j = 1, 2, ..., m. Embedding the predicted features (2) into a subspace of dimension m via J-L random projection is simply done by multiplying the matrix of original features r(X ) by a random projection matrix G i.e.,

r(X ) = r(X ) × G =    r 1 (X 1 ) . . . r M (X 1 ) . . . . . . . . . r 1 (X n ) . . . r M (X n )    ×    G 11 . . . G 1m . . . . . . . . . G M 1 . . . G M m    =      r1 (X 1 ) r2 (X 1 ) . . . rm (X 1 ) r1 (X 2 ) r2 (X 2 ) . . . rm (X 2 ) . . . . . . . . . . . . r1 (X n ) r2 (X n ) . . . rm (X n )      n×m .
The ith row-vector of r(X ) is the vector of embedded features evaluated at

X i , denoted by r(X i ) = (r 1 (X i ), r2 (X i ), ..., rm (X i )) for i = 1, 2, ..., n.
It is easy to check that given the original features r(X i ) and r(X j ), the Euclidean distance between its projection r(X i ) -r(X j ) , is equal to the Euclidean distance between the original pair r(X i )-r(X j ) in expectation with respect to G. More precisely, since G ij are centered and iid, one has

E G [ r(X i ) -r(X j ) 2 |r(X i ), r(X j )] = m p=1 E G [(r p (X i ) -rp (X j )) 2 |r(X i ), r(X j )] = m p=1 E G M k=1 (r k (X i ) -r k (X j ))G kp 2 |r(X i ), r(X j ) = m p=1 M k=1 (r k (X i ) -r k (X j )) 2 E G [G 2 kp |r(X i ), r(X j )] (E G [G kp ] = 0) = m p=1 M k=1 (r k (X i ) -r k (X j )) 2 /m (E G [G 2 kp ] = 1/m) = M k=1 (r k (X i ) -r k (X j )) 2 = r(X i ) -r(X j ) 2 ,
where E G denotes the expectation with respect to G. Moreover, as the pth coordinate of vector r(X i ) -r(X j ) is given by

(r(X i ) -r(X j )) p = rp (X i ) -rp (X j ) = M k=1 (r k (X i ) -r k (X j ))G kp ,
and one has

(r(X i ) -r(X j )) p ∼ N (0, r(X i ) -r(X j ) 2 /m), for all p = 1, 2, ..., m. Therefore, m r(X i ) -r(X j ) 2 r(X i ) -r(X j ) 2 ∼ χ 2 (m).
Then, the gap between the original and projected features can by controlled using concentration inequalities, for example, by applying Chernoff bound for χ 2 (m) distribution (see [START_REF] Chernoff | Chernoff Bound[END_REF]), for any rows r(X i ) and r(X j ) of r(X ), and for any δ > 0, one has

P G r(X i ) -r(X j ) 2 r(X i ) -r(X j ) 2 -1 > δ ≤ e m[-δ+ln(1+δ)]/2 (3)
and

P G r(X i ) -r(X j ) 2 r(X i ) -r(X j ) 2 -1 < -δ ≤ e m[δ+ln(1-δ)]/2 , ( 4
)
where P G denotes the probability under the law of G. The union bound of inequalities ( 3) and ( 4), together with the following inequalities

ln(1 + δ) ≤ δ -δ 2 2 + δ 3 3 ln(1 -δ) ≤ -δ -δ 2 2 -δ 3 3 , (5)
for any δ ∈ (0, 1), yields the following proposition.

Proposition 1 (Johnson-Lindenstrauss) Let S n = {z j ∈ R M : j = 1, 2, ..., n} denote a subset containing n points of R M and z 0 ∈ R M fixed. Moreover, let z0 and zj denote the projected point of z 0 and z j respectively into R m using random projection described above. Thus, for any δ ∈ (0, 1), with probability at least 1 -2n exp(-m(δ 2 /2 -δ 3 /3)/2), one has:

z0 -zj 2 z 0 -z j 2 -1 ≤ δ, for all z j ∈ S n .

Aggregation on random projected features

We are now in a position to formally describe our aggregation strategy on random projected features of high-dimensional predictions. We first embed the original M -dimensional features of predictions r(X ) using J-L random projection, simply by multiplying r(X ) by a random projection matrix G to obtain the projected features r(X ). Then, the aggregation method ( 1) is implemented on the projected features r(X ) in the last step. More precisely, the prediction of any point x ∈ R d is defined by

g n (r(x)) = n i=1 Y i K h ( (r(x) -r(X i ) ) n j=1 K h ( r(x) -r(X j ) ) . ( 6
)
Note that for any x ∈ R d one has r(x) ∈ R m and the Euclidean norm used in ( 6) is defined on R m while the one used in ( 1) is defined on R M .

Theoretical performance

In the sequel, we assume that dimension M of the predicted features is large. Moreover, the consensual aggregation method implemented on the original M -dimensional features of predictions (respectively m-dimensional projection features) is called full (respectively projected) aggregation method.

We are now in a position to state the main theoretical result regarding the difference between the full and projected aggregation methods. More precisely, for any ε > 0, we are interested in controlling the following probability:

P g n (r(X)) -g n (r(X))| > ε (7)
where g n (r(.)) and g n (r(.)) are the two aggregation methods defined respectively in (1) and ( 6). The key difference between the two methods is the features of predictions used for the aggregation, therefore the proof relies on the theoretical result of J-L Lemma. The control of this probability is given in the following theorem.

Theorem 1 Assume that all the machines r 1 , r 2 , ..., r M and the response variable Y are bounded almost surely by R 0 , thus for any h, ε > 0, n ≥ 1, and for any δ ∈ (0, 1), with the choice of m satisfying:

m ≥ C 1 log[2/(1 -n √ 1 -δ)] h 2α ε 2 , with C 1 = 3(2 + α) 2 (2R 0 ) 2(1+α) /σ 2 ,
one has:

P |g n (r(X)) -g n (r(X))| > ε ≤ δ.
The probability of Theorem 1 is computed under the laws of X, the training data D n = {(X i , Y i ) n i=1 } and the random projection matrix G. It can be viewed as the loss of aggregation method when projecting the features of predictions into smaller subspace of dimension m. Note that in this result, the constant C 1 depends on R 0 , which is in practice can be scaled to be, for example, less then 1. Therefore, the constant C 1 ≈ 12 for Gaussian kernel, and the lower bound of m is roughly of order:

O log(2n/δ) ε 2 h 2α ,
for large n and small δ.

Numerical simulation

This section is devoted to numerical experiments carried out on several simulated and real datasets to illustrate the performance of the proposed method.

The basic regression machines considered in this section are of five different types:

• kNN: k-nearest neighbors for regression (R package FNN, see Li ( 2019)).

• Elas: lasso and elastic-net regularized generalized linear models (R package glmbet, see Jerome et al. ( 2021))

• Bag: bagging tree for regression (R package ipred, see Andrea et al. ( 2021)).

• RF: regression random forest (R package randomForest, see Liaw and Wiener (2002a)).

• Boost: gradient boosting (R package gbm, see [START_REF] Brandon | gbm: Generalized boosted regression models[END_REF]).

To produce high-dimensional features of predictions, we construct the basic machines of each type using various options of the corresponding parameters as described below:

• 200 values of k ∈ {2, 3, ..., 201} for kNN.

• The coefficients of elastic-net model are defined by

β = argmin β { Y -βX 2 2 + λ[α β 1 + (1 -α) β 2 2 ]},
where α is the trade-off parameter between L 1 and L 2 penalty, and λ is the penalty parameter. In this case, 5 × 100 = 500 values of the couple (α, λ) ∈ {0, 0.25, 0.5, 0.75, 1} × {0.00005, ..., 1} are considered. Note that α = 0 (respectively α = 1) corresponds to Ridge (respectively Lasso) regression.

• 100 values of ntree ∈ {18, 21, ..., 315} for the three remaining tree-based methods: Bag, RF and Boost.

Remark 1 With the choices of parameters of each model, one may expect the features of predictions to be very highly correlated or redundant. For example, many values of parameter k of kNN, and ntree of Bag and RF are not very interesting in a normal setting, however, in our context, it is quite interesting to see the performance of the aggregation method in such a large highly correlated features. This is interesting in a sense that, without model selection or cross-validation technique, the aggregation method can merge the features of predictions in a robust way.

Therefore, the features of predictions are of dimension 1000. The performance of any regression estimator f is measured using the following root mean square error (RMSE) evaluated on an independent testing dataset:

RMSE(f ) = 1 n test ntest i=1 (f (x i ) -y i ) 2
where n test denotes the number of testing sample.

Simulated datasets

In this part, we consider 5 simulated models of size n where the d-dimensional input data is uniformly distributed on

[-1, 1] d , denoted by X ∼ U ([-1, 1] d ).
The five simulated models are defined as follows:

Model 1 : n = 600, d = 10, Y = X 2 1 -X 2 3 + 3X 4 exp(-X 5 ) -X 3 7 exp(-X 8 X 9 + X 5 X 10 ) + N (0, 1). Model 2 : n = 800, d = 30, Y = 5 j=1 [3X 3 2j exp(X 30-j -X 2j+1 ) -2X 3 2j-1 exp(X 2j -X 30-3j )] + N (0, 1). Model 3 : n = 800, d = 50, Y = 1 -X 2 1 + 2X 3 X 4 1.1 + X 5 -2 1 + 5 j=1 1 + X 5+j 2 -X 45+j exp(-X 10 + X 20 -X 30 ) + N (0, 1). Model 4 : n = 800, d = 100, Y = (X 2 1 -X 2 2 )(1 -exp(-X 5 X 7 )) + 3X 3 exp(- 10 j=1 X 10j ) + N (0, 1). Model 5 : n = 800, d = 100, Y = 1 + sin(X 1 + X 2 ) 1 -sin(X 1 X 2 ) - 10 j=1 2 j + 1 2 j -1 X 5j X 10j X j + N (0, 1).
In each simulation, we randomly split the simulated data into 80% and 20% training and testing set respectively. Then, the training data is split further into two parts of sizes n 1 and n 2 such that n 1 = n train /2 = n train -n 2 . The first part of the training data of size n 1 is used to construct the 1000 machines yielding predictions of the remaining parts. On top of that, to study the impact of the projected dimension m, the matrix of original features of predictions r(X ) is embedded into two groups of subspaces. The first group corresponds to the case of m ∈ {100, 200, ..., 900}, and the second group consists of much smaller values of m ∈ {2, 3, ..., 9}, associated with different random projection matrices G. Then, the kernel-based consensual aggregation method of equation ( 6) is implemented. Moreover, the aggregation on the original features defined in equation ( 1) is also computed and used to compare with all the projected cases.

The average RMSE and the associated standard error (into bracket) over 30 independent runs of each model are reported in Table 1 below. For the sake of readability, only the best performance of each type of the five basic machines is reported, followed by the performance of all the aggregation methods. In this table, the first block consists of five columns (2nd to 6th), corresponding to the performances of the best cases of the five basic machines (kNN, Elas, Bag, RF and Boost), and the second block contains 9 columns (two rows in each column) corresponding to the results of the aggregation method with different values of m. The column's names of this block are of the form m 1 /m 2 , where m 1 and m 2 are the dimensions of the projected subspaces reported in the first and second row respectively (except for the last column 900/Comb_Full). More precisely, the first row of this block contains the results of the projected aggregation methods with m ∈ {100, 200, ....900}, and the second row contains the performances of the methods with m = 2, 3, ..., 9, plus the full aggregation method, which is the aggregation on the original predicted features of dimension M = 1000 (the second row of the last column). In each case, the best performance of each block is written in boldfaced.

We observe in Table 1 that Boost shows the best performance comparing to other basic machines in the first block. In the second blocks, we see that the performances of all aggregation methods are quite similar which confirms the theoretical result stated in Theorem 1. Moreover, the performances of the aggregations bias towards, sometimes even outperform, the best method of the first block. We can also see that the full aggregation method (second row of the last column) performs really well despite being implemented on a very large redundant set of machines. And more interestingly, the performances of all the proposed methods are preserved in much lower dimensional spaces (second rows of the second block). In addition to that, Figure 2 below provides the computational efficiency of the method implemented using a computational machine with the following characteristics:

• Processor: 2x AMD Opteron 6174, 12C, 2.2GHz, 12x512K L2/12M L3 Cache, 80W ACP, DDR3-1333MHz.

• Memory: 64GB Memory for 2 CPUs, DDR3, 1333MHz.

Remark 2 Note that in all simulations, smoothing parameter h is estimated using gradient descent algorithm discussed in [START_REF] Has | A Kernel-based Consensual Aggregation for Regression[END_REF]. In all cases, the same learning rate is used, that is why on some datasets, the algorithm struggles around the optimal values of parameter, leading to slower computational times (Model 4 and Model 5 of Figure 2). In real situation, this can be improved by choosing more suitable values of parameter in the optimization method for any given datasets. With approximately the same accuracy, the proposed methods are at least 3 times faster than the full aggregation.

Real datasets

We consider in this section two public datasets (available and easily accessible on the internet) and two private energy datasets. The first dataset called Abalone (available at Dua and Graff (2017a)) contains 4177 rows and 9 columns of measurements of abalones observed in Tasmania, Australia. We are interested in predicting the age of each abalone through the number of rings (Rings) using its physical characteristics such as gender, size, weight, etc. The second dataset, named Boston, is available in MASS library of R software (see [START_REF] Brian | Mass: Support functions and datasets for venables and ripley's mass[END_REF]), comprises of 14 columns corresponding to median house prices (medv) and other variables of 506 suburbs in Boston such as per capita crime rate (crim), average number of rooms per dwelling (rm), pupil-teacher ratio by town (ptratio), nitrogen oxides concentration (ox), etc.

Then, the goal is to predict the median house prices of those suburbs using all quantitative characteristics. The third dataset (Air) considered in this section is a private dataset containing six columns corresponding to Air temperature, Input Pressure, Output Pressure, Flow, Water Temperature and Power Consumption, along with 2 026 rows of hourly observations of these measurements of an air compressor machine provided by [START_REF] Cadet | Monitoring energy performance of compressors with an innovative auto-adaptive approach[END_REF]. The goal is to predict the power consumption of this machine using the five remaining explanatory variables. The last dataset (Turbine) is provided by the wind energy company Maïa Eolis. It contains 8 721 observations of seven variables representing 10-minute measurements of Electrical power, Wind speed, Wind direction, Temperature, Variance of wind speed and Variance of wind direction measured from a wind turbine of the company (see [START_REF] Fischer | Statistical learning for wind power: A modeling and stability study towards forecasting[END_REF]). In this case, we aim at predicting the electrical power produced by the turbine using the remaining six measurements as explanatory variables.

The performances obtained from 30 independent runs, computed using the same computer mentioned in the previous section, are provided in Table 2 below. We observe that the performances of the aggregation methods approach, and sometimes outperform the best estimator on all datasets. Moreover, all the aggregation methods perform equally well in each case regardless of the size of projected dimension. In addition, the performances (the best and the worst cases) of all machines and the aggregation methods are summarized in boxplots of Figure 3

Conclusion

This chapter fills the gap by studying high-dimensional case of consensual aggregation for regression. The aggregation scheme is composed of two steps: high-dimensional features of predictions are first random projected into a smaller space using Johnson-Lindenstrauss method, then the exponential aggregation method is implemented on the projected features. First, we theoretically show that the performance of the projected and full aggregation methods are close, with high probability. Then, we numerically illustrate that the full aggregation method upholds its performance on very large redundant features given by different types of predictors. Together, this indicates the robustness of the method in a sense that, one can plainly construct several types of predictive models with different values of parameters in parallel, then flexibly aggregate them directly without any model validation step. All these results are confirmed through several numerical experiments carried out on different types of simulated and real datasets. On top of that, in term of computational speed, the proposed method is often much faster (from 3 to 20 times) compared to the full aggregation method according the optimization process (learning rate, for instance).

Proofs

Proof of proposition 1

Under the assumption of the proposition, using the results of (3), ( 4) and ( 5), the union bound probability implies for any δ ∈ (0, 1): ≤ ne -m(δ 2 /2-δ 3 /3)/2 + ne -m(δ 2 /2+δ 3 /3)/2 ≤ 2ne -m(δ 2 /2-δ 3 /3)/2 .

P ∃z j ∈ S n : z0 -zj 2 z 0 -z j 2 -1 > δ = P ∃z j ∈ S n : z0 -zj 2 z 0 -z j 2 -1 > δ + P ∃z j ∈ S n : z0 -zj 2 z 0 -z j 2 -1 < -δ ≤ n j=1 P z0 -zj 2 z 0 -z j 2 -1 > δ +
We conclude the proof using the complementary probability, P z0 -zj 2 z 0 -z j 2 -1 ≤ δ, ∀z j ∈ S n ≥ 1 -2ne -m(δ 2 /2-δ 3 /3)/2 .

Proof of Theorem 1

For the sake of readability, for any j = 1, 2, ..., n, let

• K j h = K h ( r(X) -r(X j ) ). • Kj h = K h ( r(X) -r(X j ) ).

20

For any x ∈ R d and for any h > 0,

|g n (r(X)) -g n (r(X)

)| = n i=1 Y i K i h n j=1 K j h - n i=1 Y i Ki h n j=1 Kj h = n i=1 Y i K i h n j=1 K j h - n i=1 Y i Ki h n j=1 K j h + n i=1 Y i Ki h n j=1 K j h - n i=1 Y i Ki h n j=1 Kj h ≤ R 0 n i=1 |K i h -Ki h | n j=1 K j h + R 0 n j=1 Kj h | n i=1 Ki h -n i=1 K i h | n j=1 K i h n j=1 Kj h ≤ R 0 n i=1 |K i h -Ki h | j=1 K j h + R 0 n i=1 | Ki h -K i h | n j=1 K j h = 2R 0 n i=1 |K i h -Ki h | n j=1 K h = 2R 0 n i=1 K i h |1 -Ki h /K i h | n j=1 K j h ≤ 2R 0 max 1≤i≤n 1 - Ki h K i h .
Therefore, for any ε > 0, one has:

P |g n (r(X)) -g n (r(X))| > ε ≤ P 2R 0 max 1≤i≤n 1 - K h ( r(X) -r(X i ) ) K h ( r(X) -r(X i ) ) > ε = 1 -P 2R 0 max 1≤i≤n 1 - K h ( r(X) -r(X i ) ) K h ( r(X) -r(X i ) ) ≤ ε .
One can compute the last probability using independency of (X i ) n i=1 and Fubini's theorem as follow And if one take ε = 2R 0 exp δ 0 (1 + α/2)(2R 0 /h) α /σ -1 , thus

δ 0 = σ ln(1 + ε/(2R 0 )) (1 + α/2)(2R 0 ) α h α = C 0 σεh α (1 + α/2)(2R 0 ) 1+α ,
where the constant C 0 ≈ 1 for small ε > 0, and will be ignored. Therefore, for any x, v ∈ R d , and using the fact that for any δ 0 ∈ (0, 1) : δ 2 0 /2-δ 3 0 /3 ≥ δ 2 0 /6, one has

P G 2R 0 1 - K h ( (r(x) -r(v))G ) K h ( r(x) -r(v) ) ≤ ε ≥ 1 -2 exp - m(δ 2 0 /2 -δ 3 0 /3) 2 ≥ 1 -2 exp - mδ 2 0 12 ≥ 1 -2 exp - m(σh α ε) 2 3(2 + α) 2 (2R 0 ) 2(α+1) = 1 -2 exp - mh 2α ε 2 C 1 ,
where the constant C 1 = 3(2 + α) 2 (2R 0 ) 2(α+1) > 0. Therefore, one has

P 2R 0 max 1≤i≤n 1 - K h ( r(X) -r(X i ) ) K h ( r(X) -r(X i ) ) ≤ ε ≥ 1 -2 exp - mh 2α ε 2 C 1 n .
And this implies P |g n (r(X)) -g n (r(X))| > ε

≤ P 2R 0 max 1≤i≤n 1 - K h ( r(X) -r(X i ) ) K h ( r(X) -r(X i ) ) > ε ≤ 1 -1 -2 exp - mh 2α ε 2 3R 2 1 n .
Thus, for any δ ∈ (0, 1),

1 -1 -2 exp - mh 2α ε 2 3R 2 1 n ≤ δ ⇔ m ≥ C 1 log[2/(1 -n √ 1 -δ)] h 2α ε 2 .
Moreover, for any large n, one has (1 -n √ 1 -δ) ≈ -log(1 -δ)/n, which implies that the lower bound of m is approximately

C 1 log[-2n/ log(1 -δ)] h 2α ε 2 .
Moreover, for small δ, the order of this bound is roughly O log(2n/δ) h 2α ε 2 .

Figure 1 :

 1 Figure 1: Boxplots of average RMSEs computed on simulated datasets. From left to right, the first ten boxplots are the best and the worst performance of kNN, Elas, Bag, RF and Boost machines respectively. The last eighteen boxplots represent the performances of the aggregation methods Combm with m = 2, 3, ..., 9, 100, 200, ..., 900 and Comb_Full respectively. The full aggregation performs well on 1000 dimensional predicted features (very highly correlated). Moreover, the performances of the aggregation scheme on much lower dimensional subspaces are almost preserved compared to the full aggregation with slightly larger variances.

Figure 2 :

 2 Figure 2: Running times of all the combining methods on simulated datasets. With approximately the same accuracy, the proposed methods are at least 3 times faster than the full aggregation.

Figure 3 :

 3 Figure 3: Boxplots of average RMSEs computed on real-life datasets.

Figure 4 :

 4 Figure 4: Running times of the combining methods on real-life datasets.

Table 1 :

 1 Average RMSEs on all simulated datasets.

	2.031	(0.127)	2.044	(0.131)	15.604	(3.564)	15.745	(3.609)	0.953	(0.042)	0.954	(0.041)
	2.038 2.035 2.264 2.028 2.037 2.040 2.145 2.041	3.454 2.518 (0.126) (0.135) (0.855) (0.130) (0.141) (0.132) (0.582) (0.140)	(0.768) (0.584) (0.382) (0.526) (0.333) 2.116 2.124 2.173 2.060 2.072 2.070 2.082 2.060	(0.150) (0.181) (0.619) (0.160) (0.163) (0.146) (0.166) (0.152)	15.672 15.677 15.610 15.785 15.573 15.822 15.814 15.741	18.752 18.350 17.844 18.706 17.708 (3.566) (3.488) (3.528) (3.532) (3.536) (3.449) (3.753) (3.539)	(4.847) (4.626) (4.497) (4.409) (4.632) 16.962 16.823 16.914 16.210 16.362 16.142 16.150 16.092	(4.993) (5.049) (4.912) (4.889) (4.723) (4.874) (4.963) (4.976)	0.955 0.955 0.953 0.956 0.955 0.955 0.956 0.956	1.417 1.169 1.021 1.076 1.031 (0.039) (0.043) (0.040) (0.042) (0.040) (0.044) (0.040) (0.041)	(0.114) (0.086) (0.046) (0.068) (0.045) 0.950 0.951 0.948 0.942 0.956 0.953 0.950 0.953	(0.054) (0.057) (0.067) (0.048) (0.057) (0.050) (0.050) (0.050)
					4			5

  below. Finally, Figure4illustrates time efficiency of the proposed methods.

	Basic machines Aggregation method Combm	kNN Elas Bag RF Boost 100/2 200/3 300/4 400/5 500/6 600/7 700/8 800/9 900/Comb_Full	2.135 2.105 2.114 2.113 2.113 2.115 2.112 2.114 2.113	2.052 2.092 2.174 2.213 2.106 (0.051) (0.046) (0.051) (0.047) (0.048) (0.045) (0.049) (0.044) (0.047)	(0.061) (0.055) (0.060) (0.052) (0.055) 2.198 2.165 2.143 2.144 2.156 2.138 2.149 2.152 2.114	(0.155) (0.095) (0.066) (0.061) (0.067) (0.067) (0.078) (0.063) (0.044)	3.048 3.039 3.073 3.041 3.055 3.043 3.049 3.049 3.051	6.855 5.039 4.410 3.574 3.811 (0.351) (0.348) (0.378) (0.376) (0.373) (0.369) (0.372) (0.352) (0.383)	(0.547) (0.576) (0.468) (0.402) (0.437) 4.033 3.431 3.436 3.459 3.227 3.344 3.198 3.293 3.044	(1.099) (0.724) (0.672) (0.596) (0.737) (0.631) (0.555) (0.679) (0.362)	136.424 136.535 136.532 136.487 135.961 136.424 136.108 136.509 136.075	291.435 177.581 341.514 210.910 153.538 (3.178) (4.276) (4.535) (4.122) (3.704) (4.383) (4.580) (4.237) (4.507)	(9.084) (4.763) (16.110) (15.899) (5.868) 169.592 151.757 148.344 146.905 144.371 143.118 142.619 143.028 136.828	(20.127) (9.602) (5.556) (7.005) (6.294) (4.599) (4.572) (5.743) (3.616)	39.850 36.968 36.671 36.694 36.602 36.675 36.568 36.643 36.635 36.622	39.348 67.978 68.110 35.932 (1.127) (1.146) (1.099) (1.148) (1.184) (1.092) (1.034) (1.123) (1.125)	(1.119) (2.505) (1.498) (1.038) (0.976) 38.916 37.843 37.390 37.183 36.970 36.542 36.673 36.490 36.465	(2.363) (1.201) (1.228) (1.244) (1.035) (0.745) (0.759) (0.880) (1.117)
	Model			Abalone			Boston			Air		Turbine

Table 2 :

 2 Average RMSEs of real-life datasets.

The last bound of the above inequality is obtained by Jensen's inequality. Next, for any x, v ∈ R d , given all the basic machines (r k ) M k=1 , Johnson-Lindenstrauss Lemma implies that for any δ 0 ∈ (0, 1), with probability at least 1 -2e -m(δ 2 0 /2-δ 3 0 /3)/2 , one has:

where the last inequality above is obtained using the following inequality:

1 -(1 -δ 0 ) α ≤ δ 0 (1 + α), ∀δ 0 ∈ (0, 1), ∀α > 0.


Vivek Vijayakumar

Fabio Sgarbossa

Tools for Evaluating Human Factor Aspects in Production and Logistics System

Keywords: Human factors, Production and Logistics System, System Settings, Process, Technology

Even though the advancement of technology has a great influence on the production and logistics system (PLS), companies are more dependent on manual human work because of their cognitive ability and flexibility. Inversely, many decision support models in PLS have neglected the characteristics of human workers which could degrade their working conditions. Therefore, this paper would assist the managers in a production and logistics system to evaluate the current workload on their workers. This study would suggest different tools for evaluating the human factor aspects of operations management. Initially, the paper introduces the three aspects of human factors such as physical, mental and psychosocial. Thereafter, tools for evaluating these aspects are presented. The tools are NMQ, NASA TLX, SWAT, and JCQ. Then, this paper summarizes the contribution of tools towards the production and logistics system by classifying it into three segments such as process, system settings, and technology.

Introduction

Production and logistics systems are in a thrust to sustain their operations process with globalization and the challenging market. Thus, researchers are active in developing decision support system that improves the production and logistics system (PLS). Also, operations processes necessitate a huge volume of manual work, such as material handling and assembly even though automation provides a great opportunity. This is because humans are more flexible with their blend of motor and cognitive skills. However, decision support models for production and logistics have neglect the characteristics of human workers, which resulted in unrealistic planning outcomes that could harm the workers employed in the system [START_REF] Grosse | Incorporating human factors into decision support models for production and logistics: current state of research[END_REF]. Thus, it is important to understand the human worker characteristics when planning and designing a system, the worker's characteristics could be classified under three aspects such as mental, physical, and psychosocial [START_REF] Neumann | Human factors: spanning the gap between OM and HRM[END_REF]. Analyzing the characteristics of human workers would help operations managers to properly plan and take decisions at the strategic level to improve the design of products, processes, and workstations. Also in recent research papers, they explain the importance of considering human factors in production and logistics system [START_REF] Sgarbossa | Human factors in production and logistics systems of the future[END_REF]. However, there still exist a gap in literature that shows how to evaluate HF in production and logistics system. Therefore, purpose of this paper is to address different tools that would assist the managers to evaluate HF aspects in production and logistics system. The paper is being built on integrative review methodology [START_REF] Snyder | Literature review as a research methodology: An overview and guidelines[END_REF], by examining the literature and thereby synthesizing the results into a model elaborating the relationship between the tools and PLS.

The rest of the paper is structured as follows. The next section introduces the three aspects of HF. Section 3 describes the tools for evaluating the workload on operators. Section 4 explains the contribution of tools towards PLS. Finally, the paper winds up with a conclusion.

Three aspects of HF

According to IEA council (2000), human factors are defined as " Ergonomics (or human factors) is the scientific discipline concerned with the understanding of interactions among humans and other elements of a system, and the profession that applies theory, principles, data, and methods to design in order to optimize human well-being and overall system performance". Therefore, HF could be explained as the relationship between humans and the system. Therefore, there is a great need to understand different aspects of humans, thus they are classified into three aspects, which are explained below, Mental Aspect: or the Cognitive Aspect is defined as the cognitive ability of humans to make decisions and solve problems based on their attention, learning and forgetting capabilities. For example, assembly workers require excessive cognitive demand due to the high amount of information, system complexities and variants of components required in assembling [START_REF] Brolin | Experimental study of cognitive aspects affecting human performance in manual assembly[END_REF]. It is seen that with the increase in product variation could increase the mental load to the assembly operators [START_REF] Lim | Strategies in performing a manual assembly task[END_REF].

Physical Aspect: The physical wellbeing of humans to perform a task. The increased level of physical fatigue is studied under the physical aspect. Fatigue could be general body fatigue or fatigue to a particular muscle of a worker engaged in manual work. For instance, workers who are engaged in manual material handling experience physical fatigue and discomfort which account for a high risk of developing musculoskeletal disorders (MSDs). Thus, operators are encouraged to take a break in between to reduce the fatigue accumulation on them [START_REF] Rose | Fatigue and Recovery During and After Static Loading[END_REF].

Psychosocial Aspect: is defined as the individual's demand at work concerned with his or her ability to control the activities and utilize their skills. It is seen that when workers demand high psychological factor with low control on their work would lead the workers to high stress and ill health [START_REF] De Jonge | Job strain, effortreward imbalance and employee well-being: a large-scale cross-sectional sudy[END_REF]. Factors like high autonomy, high control and a high degree of variety could improve employee's motivation [START_REF] Demerouti | Job characteristics, flow, and performance: The moderating role of conscientiousness[END_REF]. In today's working life these factors contribute to encourage better health at work. Therefore, to develop a better healthy and working environment, there is a need to consider these aspects of human factors from management point of view when designing and planning new processes in a production and logistics system.

Tools for evaluating workload on operators

For evaluating the workload on the operators engaged in manual workload from a system level, questionnaires had proved to be one of the best methods to collect the necessary data from workers [START_REF] Malchaire | Musculoskeletal complaints, functional capacity, personality and psychosocial factors[END_REF]. The questionnaire could be self-administered by the worker, which eliminates the chances of observer bias. Also, Questionnaires would reduce the cost of collecting data, by avoiding the practicality of scheduling and follow up of scrutiny [START_REF] Offenbächer | Validation of a German version of the'Disabilities of Arm, Shoulder and Hand 'questionnaire (DASH-G)[END_REF]. Moreover, questionnaires are the finest subjective assessment technique that could assist managers to evaluate the workload and its effects on labors with long term perspective.

Thus, the questionnaires were selected based on literature search. Initially, the keywords and syntax for the search of tools were defined based on the three HF aspects. Thereafter, the paper with the most potential and relevant questionnaires under each human factor aspects are extracted out. Finally it showed up to be, NMQ [START_REF] Kuorinka | Standardised Nordic questionnaires for the analysis of musculoskeletal symptoms[END_REF] under physical aspect; SWAT and NASA TLX [START_REF] Rubio | Evaluation of Subjective Mental Workload: A Comparison of SWAT, NASA-TLX, andWorkload Profile Methods[END_REF] under mental aspect and JCQ [START_REF] Karasek | The new work organization, conductive production and work quality policy. Labor market changes and job insecurity: A challenge for social welfare and health promotion[END_REF] under psychosocial aspect were selected. When NMQ, NASA TLX and SWAT evaluates the workload, JCQ is the tools which evaluates the outcomes from the workload.

Nordic Musculoskeletal Questionnaire (NMQ)

NMQ is a standardized questionnaire for the analysis of musculoskeletal symptoms in a worker. They focus mainly on the symptoms that are associated with the work settings [START_REF] Dickinson | Questionnaire development: an examination of the Nordic Musculoskeletal Questionnaire[END_REF]. Also, the NMQ serves in decision making in occupational health practices [START_REF] Kuorinka | Standardised Nordic questionnaires for the analysis of musculoskeletal symptoms[END_REF]. NMQ consists of 28 questions which could be classified into two types of questionnaires, the general questionnaire and specific questions dealing on the low back, neck, and shoulder. General questions are structured based on "Do musculoskeletal troubles occur in a given population and if so, in what parts of the body are they localized?". The questions are structured based on 9 anatomical regions. The respondent has to reflect on the trouble caused in each anatomical region during the preceding 12 months. Thereafter, a special questionnaire focuses on the regions which are most common to musculoskeletal symptoms. Questions are based on the symptoms and the duration of the past time such as the previous 7 days, the last 12 months and entire life. An analysis is done based on the severity of the symptoms in terms of their effect on the work and during their leisure time and also in terms of duration of the symptoms during the past 1 year [START_REF] Kuorinka | Standardised Nordic questionnaires for the analysis of musculoskeletal symptoms[END_REF].

NASA TLX-Task Load Index

NASA TLX is a multidimensional rating procedure that provides an overall workload score based on the weight-average rating on six subscales. They are mental demands, physical demands, temporal demands, own performance, and frustration. In which the first three describe the demand based on the worker such as mental, physical and temporal and the rest three-dimension explains the interaction of the worker with the task such as effort frustration and performance [START_REF] Hart | Development of NASA-TLX (Task LoadIndex): Results of empirical and theoretical research[END_REF]. NASA TLX is a two-step evaluation procedure consisting of weights and ratings. Initially, the weight of each dimension provides data on the diagnostic information about the nature of the workload imposed by the task and each subscale is tallied in a range from 0 to 5. Where 5 corresponds to the most important factor. Thereafter, numerical ratings for each scale reflects on the magnitude of each dimension for a given task. Each scale is presented as a 12 cm line divided into 20 intervals. The overall workload score for each subject is computed by multiplying each rate by its weight given to that dimension by the subject [START_REF] Hart | NASA Task load Index (TLX)[END_REF].

SWAT-Subjective Workload Assessment Technique

SWAT is a subjective rating technique that uses three levels such as low, high and medium for each of three dimensions of time load, mental effort load and psychological stress load to assess workload [START_REF] Reid | The subjective workload assessment technique: A scaling procedure for measuring mental workload[END_REF] The sensitivity of the tool has been shown in a variety of task: Memory task, manual control tasks, display monitoring [START_REF] Rubio | Evaluation of Subjective Mental Workload: A Comparison of SWAT, NASA-TLX, andWorkload Profile Methods[END_REF].SWAT uses three distinct steps i.e. scale development, event scoring and calculating. Under the scale development, each operator sort 27 cards in a combination of three levels of each of three dimensions. Thereafter event scoring, in which actual rating of workload for each given task. Finally calculating, the three-dimensional rating is converted into numeric scores between 0 and 100 using the interval scale developed in the first step [START_REF] Reid | The subjective workload assessment technique: A scaling procedure for measuring mental workload[END_REF].

Job Content Questionnaire (JCQ)

JCQ is a tool which has been developed to assess the psychosocial characteristic of jobs. JCQ is associated with the domains of demand control, thereby classifying the worker under the following realms, such as active (high demand and high control), passive (low demand and high control), high strain (high demand and low control) and low strain (low demand and high control) [START_REF] Karasek | The Job Content Questionnaire (JCQ): an instrument for internationally comparative assessments of psychosocial job characteristics[END_REF]. JCQ could be used by the management to analyze the work quality of its workers. It allows the testing of new technologies, worker motivation and job satisfaction [START_REF] Karasek | The new work organization, conductive production and work quality policy. Labor market changes and job insecurity: A challenge for social welfare and health promotion[END_REF].JCQ contains 5 subscales. They are decision latitude, psychological demand, social support, physical demand, and job insecurity. These subscales all together have a length of 49 questions. Decision latitude describes skill discretion, decision authority, skill underutilization, workgroup decision authority, formal authority, and union influence. Psychological demands define psychological demands, role ambiguity, concentration, and mental work disruption. Social support describes socioemotional, instrumental and hostility support from supervisors as well as coworkers. Physical demands define general physical loading, isometric load, and aerobic load. Job insecurity describes general job insecurity and skill obsolescence.

Contribution of tools towards PLS

To analyze the involvement and usage of tools in production and logistics system from a managerial perspective, PLS at the system level is classified into three segments such as process, system settings, and technology. The process is determined as the combination of different tasks to achieve a goal. For example, the assembly process, order picking process, etc. The system refers to where and how the process should be performed. For instance, workstation layout and configuration in an assembly system; in order picking process, rack layout, order picking system such as parts to picker and picker to parts are the system settings. Technology provides a mainstay for the entire system and also responsible for proper communication between each task under the system. For example, the level of automation in an assembly process and the assistive technologies in an order picking process. Finally, the contribution of different tools in each segment are discussed below.

Fig. 1. Contribution of tools towards PLS

Process: HF aspects should be considered as a crucial integral part of the process mapping. While mapping a process, information regarding check points, critical decisions and feedback loop are only considered. However, integrating HF could improve an existing process to attain an enhanced workplace. Thus, there exists a necessity to have a generic view on HF aspects while mapping a process [START_REF] Neumann | Ergonomics action research II: A framework supporting the integration of HF into work system design[END_REF]. NASA TLX is the most appropriate tool while mapping a process for improvements concerned with workers. NASA TLX can measure the mental, physical and psychosocial demand. Besides, it is capable of measuring the effort, performance, and frustration, operators could face in a new process.

System Settings: Evaluating HF aspects in System settings could improve the configuration of the system, by improving workplace for operators. Therefore, it is necessary to evaluate the workload of each HF aspect. In an assembly system, configuring the workstation based on the operators could maximize the productivity of workers by reducing their physical fatigue [START_REF] Battini | New methodological framework to improve productivity and ergonomics in assembly system design[END_REF]. In an order picking warehouse, there exists different storage assignment methods that could minimize the total travel distance and time [START_REF] Calzavara | Analysis of economic and ergonomic performance measures of different rack[END_REF]. Thus, NMQ could address the physical fatigue in anatomical areas were the musculoskeletal symptoms are common. Also, in an assembly process, the workers have a great need for cognitive demand due to the physical layout of the workstation and also due to the high level of complexities associated with assembling the products. However, these issues could be met with when there exists a low time pressure. On the other side, high time pressure can increase the workload on operators [START_REF] Brolin | Experimental study of cognitive aspects affecting human performance in manual assembly[END_REF]. In this scenario, SWAT and NASA TLX are the most appropriate tool to measure the mental workload on operators. SWAT could evaluate mental effort load, psychosocial stress load and time load, but TLX measures additional dimension i.e. physical demand. Finally, worker's satisfaction is an important factor as it could impact workers' productivity, quality, and health [START_REF] Grosse | Incorporating human factors in order picking planning models: framework and research opportunities[END_REF]. Thus, it is important to analyze the work quality in a system setting. JCQ is the most optimum tool to analyze the work quality. JCQ mainly measures the ability of workers to take decisions in their settings, the emotional difficulties the workers face in their work and also evaluate the support operators receive from their superiors and coworkers.

Technology: With the advancement of technologies, it is believed that technologies could improve efficiency, reduces workload and human error but these aptitudes were not satisfied. Automation fails because the role of operators in performing the work is often misjudged. Although automation is able to perform the task, they lack the flexibility of workers [START_REF] Lee | Human Factors in Automoation Design[END_REF]. Thanks to the introduction of collaborative robots, which works in close collaboration with operators. However, there is a need to determine the proportion of activities the operator should perform. This could be achieved by analyzing the 3 HF aspects. In a manual assembly, operators and cobots share the work tasks. The main challenges are associated with the right distribution of workload among operators and cobots. Allotting fewer ergonomics tasks to cobots could reduce the physical fatigue of operators [START_REF] Malik | Complexity-based task allocation in human-robot collaborative assembly[END_REF]. Therefore, NMQ could analyze the musculoskeletal symptoms of an operator during the work and thereby delegates less ergonomic tasks to cobots. Automation could also influence the cognitive demand of operators; automation could change the task from direct involvement to monitoring. However, all operators are not passive with monitoring the task, further reducing their feedback from the system [START_REF] Lee | Human Factors in Automoation Design[END_REF]. To analyze the cognitive demand of the operator, tools like SWAT and NASA TLX could be used. Where SWAT could be more adaptable in this situation because of its three-dimensional scale provides a psychological model of workload judgment. Also, Automation creates a work environment in which demand for work is high but the decision latitudes decline, this could affect the physical aspects of the operator ranging from heart disease to depression [START_REF] Lee | Human Factors in Automoation Design[END_REF]. JCQ could predict the psychosocial characteristics of the work. JCQ's scale measures the demand for work and the decision latitudes from an operator's perspective.

Conclusion

Which is the best tool? The answer to this question is neither one tool could be placed in the first position. Each tool is assessing different aspects of HF. Hence, this paper introduced three aspects of humans such as mental, physical and psychosocial. Thus, NMQ was introduced to assess the physical aspects, NASA TLX and SWAT were introduced to assess the mental aspects and JCQ was introduced to analyze the psychosocial aspects. Thereafter, the contribution of the tools toward the PLS is studied, by classifying PLS into three segments such as Process, System Settings, and Technologies. Finally, this paper would assist the managers to choose the appropriate tool in evaluating PLS. The extension of this paper would suggest conducting an empirical study on evaluating the PLS segments by choosing an appropriate tool.


Alessia Napoleone

Ann-Louise Andersen

Thomas Ditlev Brunoe

Kjeld Nielsen

Simon Boldt

Carin Rösiö

David Grube Hansen

Towards an Industry-Applicable Design Methodology for Developing Reconfigurable Manufacturing

Keywords: Reconfigurable manufacturing system, Changeable manufacturing, Reconfigurability, Design methodology

The concept of the Reconfigurable Manufacturing System (RMS) was introduced for the first time almost 20 years ago as a new manufacturing system concept with functionality and capacity being dynamically changeable through modularity, integrability, diagnosability, and customization. Since its introduction, the RMS concept has been extensively researched from various perspectives and new trends are today increasing its relevance. This research revisits the current status of both RMS research -in terms of research domains and trends -and RMS practice -in terms of potentialities and limitations towards broad industry application. Based on this, a design methodology in four steps is proposed and, to ensure its industry-applicability, the existence or lack of tools for each step is summarized as a basis for future research developments.

Introduction

The concept of the Reconfigurable Manufacturing System (RMS) was initially introduced by Koren in the late 1990's with the aim of providing capacity and functionality on demand and as an intermediate paradigm between the Dedicated Manufacturing System (DMS) with rigid structures and high efficiency and the Flexible Manufacturing System (FMS) with high in-built a-priori flexibility [START_REF] Koren | Reconfigurable Manufacturing Systems[END_REF]. Today, the concept both maintains and increases its relevance in education, research and practice [START_REF] Andersen | Towards a Generic Design Method for Reconfigurable Manufacturing Systems -Analysis and Synthesis of Current Design Methods and Evaluation of Supportive Tools[END_REF][START_REF] Bortolini | Reconfigurable Manufacturing Systems: Literature Review and Research Trend[END_REF][START_REF] Russo Spena | Requirements for the Design of Flexible and Changeable Manufacturing and Assembly Systems: A SME-Survey[END_REF]. Furthermore, new and additional competitive factors going beyond rapid responsiveness and lower cost motivates RMS in today's manufacturing environment [START_REF] Koren | Reconfigurable Manufacturing Systems: Principles, Design, and Future Trends[END_REF], e.g. in relation to environmental performance aspects such as recycling, de-manufacturing and re-manufacturing [START_REF] Tolio | Design, Management and Control of Demanufacturing and Remanufacturing Systems[END_REF][START_REF] Dubey | Explaining the Impact of Reconfigurable Manufacturing Systems on Environmental Performance: The Role of Top Management and Organizational Culture[END_REF] and in regard to implementation of smart manufacturing and industry 4.0 related technologies [START_REF] Bortolini | Reconfigurable Manufacturing Systems: Literature Review and Research Trend[END_REF][START_REF] Napoleone | Core Characteristics of Reconfigurability and their Influencing Elements[END_REF].Therefore, this research revisits the RMS paradigm from both academic and practice perspectives. In particular, the contribution focuses on: (i) outlining dominant research domains and trends in relation to reconfigurability, (ii) evaluating current relevance and limitations of industry application, and based on this (iii) proposing a four-step RMS design methodology to aid the wider industry transition and provide a basis for future research developments.

RMS in Research: Dominant Research Domains and Trends

Since the RMS concept was coined, RMS and "reconfigurability"or equivalently "changeability" at factory and firm levels -have received broad attention in research.

The two Domains of Research on RMS

Overall, there are two domains covered by literature; these are: (i) the design and (ii) the operations (and management) of such systems. Examples of research on RMS design include design methodologies [START_REF] Andersen | A Participatory Systems Design Methodology for Changeable Manufacturing Systems[END_REF][START_REF] Abdi | A Design Strategy for Reconfigurable Manufacturing Systems (RMSs) using Analytical Hierarchical Process (AHP): A Case Study[END_REF], economic evaluation of reconfigurability [START_REF] Andersen | Evaluating the Investment Feasibility and Industrial Implementation of Changeable and Reconfigurable Manufacturing Systems[END_REF][START_REF] Kuzgunkaya | Economic and Strategic Perspectives on Investing in RMS and FMS[END_REF], design of reconfigurable machines [START_REF] Bi | Development of Reconfigurable Machines[END_REF], reconfigurability characteristics and their implementation [START_REF] Rösiö | Supporting the Design of Reconfigurable Production Systems[END_REF], and identification and modelling of platforms as a foundation for reconfigurability [START_REF] Brunoe | Framework for Integrating Production System Models and Product Family Models[END_REF][START_REF] Gedell | Integrated Model for Co-Development of Products and Production Systems-a Systems Theory Approach[END_REF]. Operational issues include process plan generation for reconfigurable systems [START_REF] Touzout | Multi-Objective Sustainable Process Plan Generation in a Reconfigurable Manufacturing Environment: Exact and Adapted Evolutionary Approaches[END_REF][START_REF] Chaube | An Adapted NSGA-2 Algorithm Based Dynamic Process Plan Generation for a Reconfigurable Manufacturing System[END_REF], configuration selection problems [START_REF] Koren | Choosing the System Configuration for High-Volume Manufacturing[END_REF][START_REF] Xiaobo | A Stochastic Model of a Reconfigurable Manufacturing System Part 2: Optimal Configurations[END_REF], reconfiguration management [START_REF] Kurniadi | Digital Twin Approach for Solving Reconfiguration Planning Problems in RMS[END_REF] and scalability planning [START_REF] Deif | Investigating Optimal Capacity Scalability Scheduling in a Reconfigurable Manufacturing System[END_REF][START_REF] Wang | Scalability Planning for Reconfigurable Manufacturing Systems[END_REF].

In their traditional interpretation, RMSs are highly automated systems composed of CNC machines and/or Reconfigurable Machine Tools. More recently, RMSs have been more extensively interpreted as systems enabled by a multitude of aspects that can be designed and operationalized in a vast array of company and context specific forms [START_REF] Russo Spena | Requirements for the Design of Flexible and Changeable Manufacturing and Assembly Systems: A SME-Survey[END_REF][START_REF] Napoleone | Reconfigurable Manufacturing: A Classification of Elements Enabling Scalability and Convertibility[END_REF].

Literature reviews have also been conducted (e.g. [START_REF] Bortolini | Reconfigurable Manufacturing Systems: Literature Review and Research Trend[END_REF][START_REF] Khanna | Reconfigurable Manufacturing System: A State-of-the-Art Review[END_REF]) remarking that both the design and operation domains need further research. The main conclusion similar in all reviews is the need for practical guidelines driving industrial companies in the transition toward the RMS paradigm. Indeed, the implementation of RMS is still an open issue to the manufacturing professionals [START_REF] Khanna | Reconfigurable Manufacturing System: A State-of-the-Art Review[END_REF]. Case studies and best practices efficiently driving the transition of modern industrial companies toward RMS are needed [START_REF] Bortolini | Reconfigurable Manufacturing Systems: Literature Review and Research Trend[END_REF].

Reasons for the Increase of Interest in RMS

There are two additional reasons for RMS to keep and increase its theoretical importance around 20 years since the introduction: (i) its relevance to deal with the paramount need for manufacturing companies to consider sustainability issues [START_REF] Dubey | Explaining the Impact of Reconfigurable Manufacturing Systems on Environmental Performance: The Role of Top Management and Organizational Culture[END_REF][START_REF] Khezri | A Sustainable Reconfigurable Manufacturing System Designing with Focus on Environmental Hazardous Wastes[END_REF] and (ii) the potentialities represented by the inclusion of more recent digital technologies into the RMS paradigm [START_REF] Ribeiro | Transitioning from Standard Automation Solutions to Cyber-Physical Production Systems: An Assessment of Critical Conceptual and Technical Challenges[END_REF][START_REF] Lanza | Ad-Hoc Rescheduling and Innovative Business Models for Shock-Robust Production Systems[END_REF]. Regarding sustainability, higher reconfigurability of manufacturing systems leads to better environmental and economic performance, as well as to reduce the energy consumption [START_REF] Bortolini | Reconfigurable Manufacturing Systems: Literature Review and Research Trend[END_REF]. RMSs are emerging as one of the most popular manufacturing strategies to achieve sustainable manufacturing [START_REF] Dubey | Explaining the Impact of Reconfigurable Manufacturing Systems on Environmental Performance: The Role of Top Management and Organizational Culture[END_REF]. Indeed, RMSs can achieve high system sustainability thanks to their capability of producing multiple generations of products [START_REF] Koren | Reconfigurable Manufacturing Systems: Principles, Design, and Future Trends[END_REF].

Regarding digital technologies, recent development of advanced diagnostics and cyber physical manufacturing systems can facilitate the design and operations of RMS [START_REF] Koren | Reconfigurable Manufacturing Systems: Principles, Design, and Future Trends[END_REF]; Cyber-Physical Systems have been proposed in the manufacturing area as suitable technologies for supporting rapid reconfiguration and system evolution at shop floor level [START_REF] Ribeiro | Transitioning from Standard Automation Solutions to Cyber-Physical Production Systems: An Assessment of Critical Conceptual and Technical Challenges[END_REF]. Despite the interesting insights, there is still a lack of literature focusing on the synergic relationship between RMS and Cyber-Physical Systems [START_REF] Bortolini | Reconfigurable Manufacturing Systems: Literature Review and Research Trend[END_REF].

RMS in Practice: Potentials and Limitations

The authors of this paper have previously conducted a survey [START_REF] Andersen | Prerequisites and Barriers for the Development of Reconfigurable Manufacturing Systems for High Speed Ramp-up[END_REF] of manufacturing companies, analyzing the potential of implementing RMS, as well as the barriers towards this. The general conclusion of this study was that few of the companies had skills and competences in place for developing RMS. Some differences were identified in terms of the readiness for developing reconfigurability in the different cases, which seemed to relate to which industry they were operating within, where electronics manufacturing seemed to have the highest degree of coordination between product design and manufacturing system design, which is essential when developing RMS. Furthermore, applying a long-term view when planning manufacturing system development and economically justifying them was observed in some companies but not others. The presence of enablers of reconfigurability was also analyzed. Generally, most companies had only few or none enablers present. Also, the companies generally recognized that their current level of the enablers was lower than what was actually needed to achieve the desired level of reconfigurability. The presence of specific enablers depended somewhat on different characteristics of the companies.

In another more recent study from the same authors, eight different cases were analyzed, addressing companies that were currently doing development or implementation of reconfigurability [START_REF] Andersen | Tailored Reconfigurability: A Comparative Study of Eight Industrial Cases with Reconfigurability as a Key to Manufacturing Competitiveness[END_REF]. This study analyzed the potentials in doing reconfigurability across the cases. In all cases it was found that reconfigurability did in fact hold a potential, however, on different levels and time frames. For example, low volume, high variety tended to have more potential in operational reconfigurations on short term, whereas higher volumes tended to have more potential on tactical and strategic level for reconfigurability. Due to these differences, the enablers that would enable reconfigurability in the different cases were also widely different. This clearly influences how reconfigurability should be developed in these companies, and contributes to the point that not one generic set of enablers can be applied to all companies, and thus not one single RMS development methodology can be applied to all companies, and therefore a context dependent method should be introduced.

Towards a Design Methodology

As both research and practice suggest the need for practical and context dependent guidelines driving industrial companies in the transition toward the RMS paradigm, a RMS design methodology is proposed in this section.

The suggested methodology is based on a generic design methodology provided by Andersen et al. [START_REF] Andersen | Towards a Generic Design Method for Reconfigurable Manufacturing Systems -Analysis and Synthesis of Current Design Methods and Evaluation of Supportive Tools[END_REF], which has been chosen for two reasons: (i) it is academically relevant, as it is built based on literature and (ii) also it attempts to support practitioners, providing a mapping of the different design challenges, suggested procedures, and applicable tools. The suggested RMS design methodology involves four steps consisting in several activities as shown in the figure below (Fig. 1); to ensure its applicability in industry, in each step, specific tools should be provided. In this section, some representative tools available in literature have been associated to individual steps and those not covered by specific tools have been identified as needing further research. The methodology is described in the remainder.

Step 1. Classification of potential and goals. This step covers planning and making strategic decisions about the development project. A major challenge in this step is the life-cycle perspective and the uncertain nature of predictions [START_REF] Andersen | Towards a Generic Design Method for Reconfigurable Manufacturing Systems -Analysis and Synthesis of Current Design Methods and Evaluation of Supportive Tools[END_REF]. Scenario analysis should allow for getting insight into the real-market uncertainties, by considering market demand stochastically and not deterministically [START_REF] Khanna | Reconfigurable Manufacturing System: A State-of-the-Art Review[END_REF]. The definition and monitoring of KPIs based on measures of characteristics of reconfigurability could support the evaluation of system structural performance [START_REF] Rösiö | Towards an Assessment Criterion of Reconfigurable Manufacturing Systems within the Automotive Industry[END_REF][START_REF] Maganha | Understanding Reconfigurability of Manufacturing Systems: An Empirical Analysis[END_REF]. To quantify the potential in RMS compared to conventional manufacturing systems, scenario analysis and participatory approaches should aim at involving firms in the definition of future production requirements and the corresponding most suitable manufacturing paradigms [START_REF] Andersen | A Participatory Systems Design Methodology for Changeable Manufacturing Systems[END_REF][START_REF] Rösiö | Supporting the Design of Reconfigurable Production Systems[END_REF]. For each consistent scenario of the future, the projected factory should be analysed to draw conclusions about the evolution of the objects in the factory and their cost-effectiveness [START_REF] Wiendahl | Justifying Changeability. A Methodical Approach to Achieving Cost Effectiveness[END_REF]. Requirements should be well analysed and described, also recurring to conventional planning tools (e.g. requirements list) [START_REF] Tracht | Decision Making During Design and Reconfiguration of Modular Assembly Lines[END_REF]. Finally, to prioritize objectives and requirements, the Analytical Hierarchy Processing method could be used [START_REF] Abdi | A Design Strategy for Reconfigurable Manufacturing Systems (RMSs) using Analytical Hierarchical Process (AHP): A Case Study[END_REF][START_REF] Park | Improved Methodology for RMS Adaptability Evaluation[END_REF].

Step 2. Development of concept for reconfigurable production. At this stage, decisions should ensure that both product and manufacturing processes are designed for reconfigurability and that the two designs are adequately related with each other. In recent years, digitalization throughout the entire Product Lifecycle has allowed manufacturing companies for significant coordination gains between product and production system design, holding the potential of elevating companies to reach new levels of reconfigurability. One of the main challenges related to that is the effort in promptly updating the information when the system needs to be reconfigured. To ensure the adequate design of products, products should be classified based on either their morphological or technological features, clustering techniques or visual inspection can be used to this end [START_REF] Bejlegaard | Methodology for Reconfigurable Fixture Architecture Design[END_REF]. Bill-of-Materials trees and a tree dissimilarity measure can be used to allow matching integer programming model to address commonality of products and form product families [START_REF] Kashkoush | Product Family Formation for Reconfigurable Assembly Systems[END_REF]. To ensure reconfigurability, products should then be grouped based on production flow features. Techniques such as the Rank Order Clustering algorithm or the use of similarity coefficients [START_REF] Prasad | Assessment of a Reconfigurable Manufacturing System. Benchmarking[END_REF][START_REF] Goyal | A Comprehensive Approach to Operation Sequence Similarity Based Part Family Formation in the Reconfigurable Manufacturing System[END_REF] are the most widespread to this end. Based on company-specific data and following an iterative (and collaborative) approach, the relationship between the manufacturing system and its respective products can be studied and integrated product-process (object-oriented/functional) modelling can be achieved [START_REF] Michaelis | Function and Process Modeling for Integrated Product and Manufacturing System Platforms[END_REF][START_REF] Brunoe | Integrated Product-Process Modelling for Platform-Based Co-Development[END_REF]. The use of tools such as axiomatic design tech-niques and function-means formalism allow for achieving maximum possible decoupling of functional requirements based on existing functional requirements and the future functional requirements in order to enable a minimum of functional change when changes occur [START_REF] Bejlegaard | Methodology for Reconfigurable Fixture Architecture Design[END_REF]. In this step, the definition of production platforms, i.e. sets of manufacturing subsystems developed to form a common structure from which a stream of products can be efficiently produced [START_REF] Michaelis | Function and Process Modeling for Integrated Product and Manufacturing System Platforms[END_REF], is an important enabler of reconfigurability [START_REF] Brunoe | Integrated Product-Process Modelling for Platform-Based Co-Development[END_REF]. To define such platforms, a classification scheme of manufacturing systems [START_REF] Sorensen | Classification Coding of Production Systems for Identification of Platform Candidates[END_REF] and functional modeling [START_REF] Michaelis | Function and Process Modeling for Integrated Product and Manufacturing System Platforms[END_REF] are appropriate tools. Decisions on manufacturing modules and granularity level can be made using tools such as Design Structuring Matrix and Cladistics [START_REF] Bejlegaard | Methodology for Reconfigurable Fixture Architecture Design[END_REF][START_REF] Lameche | Modularity in the Design of Reconfigurable Manufacturing Systems[END_REF]. Finally, the definition of KPIs and the use of visualization tools allow for evaluating different design concepts [START_REF] Park | Improved Methodology for RMS Adaptability Evaluation[END_REF].

Step 3. System design. This step is industry-specific and may need the definition of general tools. The implementation of a function-driven object-oriented methodology [START_REF] Gwangwava | A Methodology for Design and Reconfiguration of Reconfigurable Bending Press Machines (RBPMs)[END_REF] supports the following activities. First, the reconfigurability needs should be defined. Second, an evaluation of the existing manufacturing setup to identify primary functions, terminal functions and possible suited modules should be performed. The identified modules are to be included in the system design and should be modelled and simulated, to evaluate the design solution candidates through a trade-off analysis between reusing existing modules and designing new modules [START_REF] Gadalla | An Approach to Identify the Optimal Configurations and Reconfiguration Processes for Design of Reconfigurable Machine Tools[END_REF]. Third, existing modules -either internally developed at the company and/or commercially availableshould be classified through a module tree analysis [START_REF] Mpofu | A Morphology Proposal in Commercial-Off-the-Shelf Reconfigurable Machine Tools[END_REF]. A structural tree of the modules evaluates the constraints in reconfiguring the modules. The function tree, the module tree and the structural tree yield a module library from which industry specific RMS design prototyping is enabled. Finally, in this step, the right technology should be selected and the degree of automation for each operation should be defined.

Step 4. Anchoring/ Realisation. This step consists in the detailed design of the RMS. As the previous one, this step may need the definition of general tools. Moreover, issues such as the integration design of systems can be very case-specific, making the generalization of tools even more challenging. Overall, this step needs detailed specifications and documentation from the previous step, especially if the actual development of new modules is outsourced to technology suppliers. Furthermore, tools to optimize ramp-up procedures and quickly operationalise implemented changes are paramount to ensure cost-effective reconfigurations over time.

Conclusions and Future Work

This research proposes a methodology in four steps to design RMS. The methodology is based on available literature and considers the need of practitioners for context dependent guidelines. To ensure its applicability in industry, further research is needed. Indeed, the suggested methodology needs to be consolidated through the identification of relevant and industry-applicable tools for each individual activity (see Fig. 1) of the four steps and how they affect the characteristics of reconfigurability, as well as effect on sustainability and digitalization. Furthermore, many activities might require an adaptation of the already available tools or even the creation of new adequate and hopefully generalizable tools.

To pursue this objective, further research is needed. Above all, collaborative projects with companies might provide valuable insights to ensure the applicability of the method in industry.

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Fig. 1 .

 1 Fig. 1. Four-step methodology to design RMS Overall, the first two steps of the RMS design methodology proposed are well represented by tools already available in literature, the remaining two steps deserve further investigation to enable companies to use the proposed methodology.

Acknowledgements

This research was supported by the REKON project whose partners are Aalborg University, Jönköping University, the University of Southern Denmark, and The Danish Technological Institute. REKON is funded by The Danish Industry Foundation.