Shu Wu

Philippe Castagliola
email: philippe.castagliola@univ-nantes.fr

Athanasios C Rakitzis

Petros E Maravelakis

Design of Attribute EWMA Type Control Charts with Reliable Run Length Performance

Keywords: Statistical Process Monitoring, Attribute EWMA chart, Markov chain, Normal mixture, Average Run Length, Poisson, Binomial

Attribute control charts assuming a Poisson (c chart) or a binomial distribution (np chart) are usually used when the quality characteristic cannot be measured on a continuous scale. For equivalent sample sizes, Shewhart type attribute control charts are known to be less efficient than their measurement counterparts (like the X chart) and, for this reason, practitioners often compensate it by supplementing them with an EWMA (Exponentially Weighted Moving Average) scheme. However, because of the discrete nature of count data, it is unfortunately impossible to compute exactly and accurately (by means of Markov chain of integral equation methods) the run length (RL) properties, such as its mean (ARL) and its standard deviation (SDRL) of these EWMA attribute control charts and, consequently, it is impossible to efficiently design them in order to minimize some out-of-control characteristics. For this reason, we propose in this paper a dedicated approach called "continuousify" method which, coupled with a classical Markov chain technique, allows to compute the RL properties of any EWMA attribute control chart in a reliable way. A numerical comparison shows that the RL properties obtained by using the proposed "continuousify" approach are very much alike to the ones calculated via simulation and without the "continuousify" approach. Illustrative examples are also provided to show how the proposed method can be implemented in practice.

Introduction

Statistical Process Monitoring (SPM) is a collection of statistical techniques providing a rational management of manufacturing processes, which allows high quality final products to be produced. Among SPM tools, the control charts are undeniably the ones that are the most used for identifying changes in processes. When the quality characteristic of interest can be measured, measurement type control charts (usually based on the normality assumption) are used (like the X chart for the mean or the S chart for the standard-deviation) but, when this quality characteristic cannot be measured (and only defective / not defective products or number of defects can be observed), attribute type control charts (such as the c and np charts) are commonly used.

For comparable sample sizes, Shewhart type attribute control charts are known to be less efficient than their measurement counterparts and, for this reason, practitioners often compensate it by supplementing them with the EWMA (Exponentially Weighted Moving Average) scheme, introduced by [START_REF] Roberts | Control Chart Tests Based on Geometric Moving Averages[END_REF], which combines current and previous observations and is known to be one of the most effective tools for monitoring sustained changes in a process. Although EWMA type control charts have been massively studied in the case of normaly distributed measurement data (among the most recent ones, we refer, for instance, to [START_REF] Khoo | A Study on EWMA Charts with Runs Rules -the Markov Chain Approach[END_REF], [START_REF] Riaz | Nonparametric Double EWMA Control Chart for Process Monitoring[END_REF], [START_REF] Zwetsloot | Robust Point Location Estimators for the EWMA Control Chart[END_REF], [START_REF] Arshad | Simultaneous Use of Runs Rules and Auxiliary Information with Exponentially Weighted Moving Average Control Charts[END_REF], [START_REF] Bilen | Dual-Monitoring Scheme for Multivariate Autocorrelated Cascade Processes with EWMA and MEWMA Charts[END_REF], [START_REF] Haridy | EWMA Chart with Curtailment for Monitoring Fraction Nonconforming[END_REF], [START_REF] Lu | Statistically Constrained Economic Design of Maximum Double EWMA Control Charts Based on Loss Functions[END_REF], [START_REF] Raji | On Designing a Robust Double Exponentially Weighted Moving Average Control Chart for Process Monitoring[END_REF], [START_REF] Maravelakis | Run Length Properties of Run Rules EWMA Chart using Integral Equations[END_REF] and [START_REF] Tang | Optimal Design of the Adaptive EWMA Chart for the Mean Based on Median Run Length and Expected Median Run Length[END_REF]), they have received less attention in the case of attribute data. For example, Gan (1990a,b) proposed modified EWMA control charts for monitoring Poisson and binomial counts, and he demonstrated that, based on the ARL (Average Run Length) criterion, these control charts are generally superior compared to the corresponding Shewhart control charts. [START_REF] Borror | Poisson EWMA Control Charts[END_REF] presented a methodology to design and evaluate the performance of the EWMA control chart for monitoring Poisson data and they found it more efficient than the Shewhart c chart and Gan (1990a)'s modified EWMA charts. Furthermore, other works concerning attribute EWMA control chart have been discussed by [START_REF] Somerville | Filtering and Smoothing Methods for Mixed Particle Count Distributions[END_REF], [START_REF] Zhang | Poisson DEWMA Control Chart[END_REF], [START_REF] Yeh | EWMA Control Charts for Monitoring High-Yield Processes Based on Non-Transformed Observations[END_REF], [START_REF] Szarka | A Review and Perspective on Surveillance of Bernoulli Processes[END_REF], [START_REF] Saghir | A Flexible and Generalized Exponentially Weighted Moving Average Control Chart for Count Data[END_REF], [START_REF] Aslam | A Hybrid Exponentially Weighted Moving Average Chart for COM-Poisson Distribution[END_REF] and [START_REF] Alevizakos | A Double Exponentially Weighted Moving Average Control Chart for Monitoring COM-Poisson Attributes[END_REF].

In order to investigate the performance of a control chart, some in-and outof-control RL (Run Length) properties have to be evaluated (for example, the ARL, MRL and SDRL) and accurate evaluation of these quantities is of great importance as they are often used in the design phase of a control chart. For example, for the design of an EWMA chart, the researcher has to find the optimal parameters λ * ∈ (0, 1] (smoothing parameter) and K * > 0 (control limit parameter) such that, for a particular shift in the process, the out-of-control ARL is minimized subject to the constraint that the in-control ARL equals ARL 0 , a predefined value. If the evaluation of these RL properties is not reliable, it is therefore hopeless to correctly design such a control chart. In the case of measurement data (usually, under the normality assumption) the RL properties of EWMA type control charts are obtained, in general, using either Markov chain or integral equation methods which are both based on a discretization of the control limits interval into m subintervals. When the number m of subintervals increases and becomes large enough (say m ≥ 200) these methods yield fairly accurate approximations. On the other hand, in the case of attribute / count data, due to the discrete nature of these data, it is impossible to accurately compute any RL properties using Markov chain or integral equation methods as the results will heavily fluctuate depending on the value of the selected number m of subintervals (this will be clearly highlighted in the paper through some examples). Of course, it is always possible to obtain these values using Monte-Carlo simulations but, even in this case, if it is quite easy to compute small ARL or MRL values with some precision, it becomes just impossible to obtain reliable results when these values become very large. In addition, the design phase becomes significantly time consuming.

Therefore, in an attempt to extend the preliminary work by [START_REF] Wu | A Distribution-Free EWMA Control Chart for Monitoring Time-Between-Events-and-Amplitude Data[END_REF], the goal of this paper is i) to propose a method, called "continuousify", that allows to compute the RL properties of any attribute EWMA type control charts in a reliable way, ii) to demonstrate the use of this method in the case of the Poisson and binomial distributions, iii) to design the corresponding PEWMA (for Poisson EWMA) and BEWMA (for Binomial EWMA) and iv) to compare the results with the ones obtained through Monte Carlo simulations as well as with the ones presented in the very recent contribution of [START_REF] Morais | Improving the ARL Profile and the Accuracy of its Calculation for Poisson EWMA Charts[END_REF] in which the authors tackle the same problem (for the PEWMA control chart only) but in a different way.

The paper is structured as follows. Section 2 introduces the "continuousify" technique as a general framework for discrete type distributions. Then, Section 3 focuses on two particular cases, i.e. the PEWMA control chart and the BEWMA control chart. In both cases, comparisons show how the "continuousify" technique allows to obtain stable and reliable ARL values. Moreover, we also provide the optimal values for the chart's constants. In Section 4, two illustrative examples clarify how the "continuousify" approach can be used to monitor nonconformities or nonconforming items using EWMA. Finally, Section 5 summarizes the main findings and suggest potential new research directions.

2 The "continuousify" method for attribute type EWMA control charts Let X i , i = 1, 2, . . ., be a sequence of i.i.d. (independent and identically distributed) discrete r.v. (random variables), defined on Ω = {ω 1 , ω 2 , . . .} and having f X (ω|θ) = P(X i = ω) as p.m.f. (probability mass function), where θ is a vector of parameters. In practice, it is actually possible to define and implement an attribute EWMA type control chart directly monitoring a process (via the observed X i values) by using the statistic

Z i = λX i + (1 -λ)Z i-1 , i = 1, 2, .
. ., where λ ∈ (0, 1] is some smoothing parameter to be fixed and Z 0 is some initial value. As already explained in the Introduction section of this paper, the problem of this approach is that, because of the discrete nature of the r.v. X i , i = 1, 2, . . ., it is impossible to accurately compute the RL properties of such a control chart (using Markov chain or integral equation methods) and, therefore, it is impossible to tune the chart parameters in order to obtain a predefined in-control performance. If, for instance, the Markov chain approach, (as detailed hereafter), is used in order to compute the ARL or the SDRL, the results will i) heavily fluctuate depending on the value of the selected number m of subintervals and ii) not exhibit any monotonic convergence when m increases, making useless such an approach. This point will be highlighted in Section 3.

Since the Markov chain and integral equation methods give good results in the case of continuous r.v., (and, more particularly in the case of the normal distribution, which is an unbounded one), we therefore suggest to transform each discrete r.v. X i , i = 1, 2, . . ., into a new continuous one, denoted as X * i , (say that we "continuousify" the r.v. X i ), defined on (-∞, +∞), and to monitor the process by using a traditional EWMA scheme. More precisely, we suggest to simply define X * i as a mixture of the r.v. Y i,ω1 , Y i,ω2 , . . . where, for each ω ∈ Ω, Y i,ω ∼ N(ω, σ), i.e.

X * i =      Y i,ω1 if X i = ω 1 , Y i,ω2 if X i = ω 2 , . . . . . . ,
where N(ω, σ) stands for the Normal distribution with mean ω and standard deviation σ. Concretely speaking, this means that if, at i = 1, 2, . . ., we have X i = ω ∈ Ω then, in order to obtain X * i , we just have to generate a number from the N(ω, σ) distribution. The "continuousify" parameter σ > 0 has to be fixed and, as it will be shown later, its value does not significantly affect the performance of the control chart as long as it is neither too small nor too large. Since X * i , i = 1, 2, . . ., is defined as a mixture of normal distributions, its p.d.f. f X * (x|θ, σ) and c.d.f. F X * (x|θ, σ) are

f X * (x|θ, σ) = ω∈Ω f X (ω|θ)f N (x|ω, σ), (1)
F X * (x|θ, σ) = ω∈Ω f X (ω|θ)F N (x|ω, σ), (2)
where f N (x|ω, σ) and F N (x|ω, σ) are the p.d.f. and c.d.f. of the N(ω, σ) distribution, respectively. If µ = E(X) and V(X) are the mean and variance of X i , i = 1, 2, . . ., respectively, then it is not difficult to prove that the mean µ * = E(X * ) and the variance V(X * ) of X * i , i = 1, 2, . . ., are equal to (see Appendix for details):

E(X * ) = E(X), (3) V(X * ) = V(X) + σ 2 . ( 4
)
In this paper, we will focus on an upper-sided EWMA control chart for process monitoring based on the statistic Z * i defined as

Z * i = max(0, λX * i + (1 -λ)Z * i-1 ), (5)
with initial value Z * 0 = µ * 0 (corresponding to the in-control situation). Note that a similar approach can be used to design a lower-sided EWMA control chart properly modified. By definition, the asymptotic upper control limit UCL * of this chart (i.e. with "continuousify") is equal to

UCL * = E(X * ) + K λ 2 -λ V(X * ),
where K > 0 is a control chart parameter to be fixed. Using ( 3) and ( 4), the previous formula simplifies to

UCL * = E(X) + K λ(V(X) + σ 2 ) 2 -λ . (6)
In order to obtain the zero-state ARL and SDRL of the proposed EWMA control chart, we suggest to use the standard approach proposed by [START_REF] Brook | An Approach to the Probability Distribution of CUSUM Run Length[END_REF]. This approach assumes that the operation of this control chart can be well represented by a discrete-time Markov chain with m + 2 states. States k ∈ {0, 1, . . . , m} are transient and state m + 1 is an absorbing one. The transition probability matrix P of this discrete-time Markov chain is

P =   Q r 0 1   =        Q 0,0 Q 0,1 • • • Q 0,m r 0 Q 1,0 Q 1,1 • • • Q 1,m r 1 . . . . . . . . . . . . Q m,0 Q p,1 • • • Q m,m r m 0 0 • • • 0 1       
, where Q is the (m+1, m+1) matrix of transient probabilities, 0 = (0, 0, . . . , 0) and the (m + 1, 1) vector r satisfies r = 1 -Q1 (i.e. row probabilities must sum to 1) with 1 = (1, 1, . . . , 1) . The transient states k ∈ {1, . . . , m} are obtained by dividing the interval [0, UCL * ] into m subintervals of width 2∆, where ∆ = UCL * 2m and UCL * is the upper control limit as defined in (6). By definition, the midpoint of the k-th subinterval (representing state k) is equal to H k = (2k -1)∆. The transient state k = 0 corresponds to the "restart state" feature of the upper-sided EWMA chart (due to the presence of the max(. . . ) in ( 5)). This state is represented by the value H 0 = 0. It can be easily proven that the generic element Q k,j , k, j = 0, 1, . . . , m, of the matrix Q is equal to:

• if j = 0, Q k,0 = F X * - (1 -λ)H k λ θ, σ , • if j = 1, 2, . . . , m, Q k,j = F X * H j + ∆ -(1 -λ)H k λ θ, σ -F X * H j -∆ -(1 -λ)H k λ θ, σ ,
where

F X * (. . . |θ, σ) is the c.d.f. of X * i , i = 1, 2, .
. ., as defined in (2). Let q = (q 0 , q 1 , . . . , q m ) be the (m + 1, 1) vector of initial probabilities associated with the m + 1 transient states. In our case, we assume q = (1, 0, . . . , 0) , i.e. the initial state corresponds to the "restart state". When the number m of subintervals is sufficiently large (say m ≥ 200), this approach provides an effective method that allows the ARL and SDRL to be accurately evaluated by using the following classical formulas from the theory of Markov chains (see, for instance [START_REF] Neuts | Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach[END_REF] or [START_REF] Latouche | Introduction to Matrix Analytic Methods in Stochastic Modeling[END_REF])

ARL = q (I -Q) -1 1, SDRL = 2q (I -Q) -2 Q1 + ARL(1 -ARL).
Of course, steady-state ARL and SDRL values can also be obtained by adequaly modifying the vector q of initial probabilities defined above. However, this topic is beyond the scope of this paper and we will not consider further the steady-state properties.

It has to be noted that the choice of the normal N(ω, σ) distribution in ( 1) and (2) for defining the mixture is just a possible choice among many others. In fact, in [START_REF] Perdikis | An EWMA Signed Ranks Control Chart with Reliable Run Length Performances[END_REF], in a similar context, several symmetrical distributions / kernels (Normal, Parabolic, Biweight, Triweight, Cosine) have been tested and the conclusion was that the choice of the distribution / kernel clearly has almost no impact on the computation of the ARL and, therefore, the user is totally free to use the kernel / distribution of his / her choice (including the uniform distribution, for instance) without having to worry much about the reliability of the result.

Applications of the "continuousify" method

The goal of this section is to demonstrate the use of the "continuousify" method in the computation of the RL properties and the design of two upper-sided attribute EWMA control charts: the upper-sided PEWMA chart (assuming a Poisson distribution) and the upper-sided BEWMA chart (assuming a binomial distribution).

The "continuousified" upper-sided PEWMA chart

Let us assume that X i , i = 1, 2, . . . is a sequence of i.i.d. Poisson r.v. with parameter θ > 0. When the process is in-control, we have θ = θ 0 and when the process is out-of-control, we have θ = θ 1 > θ 0 . In this case, we simply have

θ = θ, Ω = {0, 1, 2, . . .} = N, f X (ω|θ) = e -θ θ ω ω! , E(X * ) = θ, V(X * ) = θ + σ 2 and the c.d.f. of X * is equal to F X * (x|θ, σ) = +∞ ω=0 f X (ω|θ)F N (x|ω, σ).
The upper control limit UCL * of the upper-sided PEWMA chart with the "continuousify" method is equal to

UCL * = θ 0 + K λ(θ 0 + σ 2 ) 2 -λ .
First of all, in order to have some insights about how to select the value of σ, we plotted in Figure 1 the p.d.f. f X * (x|θ, σ) (i.e. using (1)) of the r.v. X * for θ = 3 and for several values of σ ∈ {0.05, 0.1, 0.125, 0.15, 0.2, 0.4}. As it can be seen, if σ = 0.05, the resulting p.d.f. is too "peaky" and too close to the original discrete distribution, probably making the ARL values to still fluctuate a lot when the number of sub-intervals m varies. On the other hand, if σ = 0.4, the resulting p.d.f. is clearly too "over-smoothed". In this case, the ARL values will not fluctuate any longer but, due to the term "• • • + σ 2 " in ( 6) these values will be larger. This will be confirmed in the following investigations. Therefore a good tradeoff seems to be a value for σ

∈ [0.1, 0.2].
In Table 1 a comparison of the out-of-control ARL for the upper-sided PEWMA chart obtained without (left part) and with (right part) the "continuousify" method for a number of subintervals m ∈ {100, 110, . . . , 400} and for the following out-of-control situations (θ 0 , θ 1 ) ∈ {(1, 2), (1, 1.5), (2, 3), (4, 5), (4, 6)} is given. For illustrative purposes, the control chart parameters are K = 3 and λ = 0.2 and the "continuousify" parameter is set at σ = 0.125 (this value has also been used in [START_REF] Wu | A Distribution-Free EWMA Control Chart for Monitoring Time-Between-Events-and-Amplitude Data[END_REF]). As it can be seen (and as it was mentioned earlier):

• the ARL values obtained without "continuousify" heavily fluctuate (and they are sometimes even negative) depending on the value of m. Clearly, they do not exhibit any monotonic convergence when the number of subintervals m increases. For instance, in the case (θ 0 , θ 1 ) = (1, 1.5), the ARL values obtained without "continuousify" fluctuate from -82.8 (negative values may happen when the Markov chain approach does not converge) to 221.7.

• on the contrary, for m ≥ 100, the ARL values obtained with the "continuousify" method exhibit a strong stability and seem to converge rapidly to a reliable value. Even for m = 100 the results obtained with the "continuousify" approach are very reliable. This fact will reduce the time needed for the required computations for obtaining the RL distribution. For instance, using the same case (θ 0 , θ 1 ) = (1, 1.5), the ARL values obtained with "continuousify" converge rapidly to 28.4.

As a matter of comparison, in Table 1, we have also computed the ARL values of the upper-sided PEWMA chart using 10 6 Monte-Carlo simulation runs (see bottom of Table 1) for the without "continuousify" case only. What can be seen is that the out-of-control ARL values obtained with the "continuousify" method (see for example the case m = 400), i.e. 9.9, 28.4, 17.3, 33.4, 10.2 are almost the same or just a bit larger (in a negligeable way) to the ones obtained using simulations without the "continuousify" method, i.e. 9.9, 28.3, 17.2, 33.4, 10.2.

It is interesting to note that this topic (efficient ARL computation for EWMA control charts monitoring discrete data) has also grasped the attention of other researchers and, very recently, [START_REF] Morais | Improving the ARL Profile and the Accuracy of its Calculation for Poisson EWMA Charts[END_REF] have proposed an alternative method called the "Splitting Markov Chain". This approach has been applied to a two-sided PEWMA control chart (while, in this paper, we investigate a max-type upper sided PEWMA control chart). In order to compare the two methods, we have reproduced the three plots in Figure 3 and the same scale, but using the "continuousify" approach instead of the "Splitting Markov Chain" approach. The corresponding plots are in Figure 2 where the three leftmost ones have been plotted for σ = 0.125 while the three rightmost ones have been plotted for σ = 0.2. From these plots, we can conclude that

• the results are really comparable to the ones in [START_REF] Morais | Improving the ARL Profile and the Accuracy of its Calculation for Poisson EWMA Charts[END_REF].

Fluctuations happen for some values m < 100 but, rapidly, when m ≥ 100 the ARL values becomes stable.

• As expected, there are less fluctuations when σ = 0.2 rather than when σ = 0.125 but, as we already mentioned, the price to pay for this higher stability are larger ARL values as it can be noticed in the rightmost plots. Therefore a tradeoff has to be found and the value σ = 0.125 seems to be a good compromise.

As the "continuousify" method now provides stable values for the ARL it is therefore possible to design the upper-sided PEWMA chart in order to find the optimal parameters λ * and K * such that, for a particular shift from θ 0 to θ 1 = τ θ 0 , τ > 1, the out-of-control ARL is minimized subject to the constraint that the in-control ARL equals the desired ARL 0 value, e.g. ARL 0 = 370.4. The optimal combinations (λ * , K * ) are listed in Table 2 with the corresponding out-of-control ARL values for θ 0 ∈ {1, 2, 5, 10}, τ ∈ {1.1, 1.2, 1.5, 2, 3} and σ ∈ {0.1, 0.125, 0.15, 0.2}. For instance, when θ 0 = 1, τ = 2 (i.e. θ 1 = 2) and σ = 0.1, the optimal chart parameters are (λ * , K * ) = (0.115, 2.728) and the corresponding out-of-control ARL = 9.5. From Table 2, we can draw the following conclusions:

• No matter the value of σ, the out-of-control ARL values monotonically decrease when τ increases.

• The choice of the parameter σ ∈ {0.1, 0.125, 0.15, 0.2} does not significantly impact the performance of the upper-sided PEWMA chart with "continuousify" as the difference in term of out-of-control ARL is almost negligible for a specified combination (θ 0 , θ 1 ). For example, if θ 0 = 2 and τ = 1.1 (i.e. θ 1 = 2.2), the optimal parameters (λ * , K * ) are approximately equal to (0.03, 1.96), and the ARL values are approximately equal to 92, for the four possible choices of σ ∈ {0.1, 0.125, 0.15, 0.2}.

The upper-sided BEWMA chart with "continuousify"

Let us assume that X i , i = 1, 2, . . . is a sequence of i.i.d. binomial r.v. with parameters n and p. When the process is in-control, we have p = p 0 and when the process is out-of-control, we have p = p 1 > p 0 . In this case, we have The upper control limit UCL * of the upper-sided BEWMA chart with the "continuousify" method is equal to

θ = (n, p), Ω = {0, 1, 2, . . . , n}, f X (ω|θ) = n ω p ω (1 -p) n-ω , E(X * ) = np, V(X * ) = np(1 -p) + σ 2 and the c.d.f. of X * is equal to F X * (x|θ, σ) = n ω=0 f X (ω|θ)F N (x|ω, σ).
UCL * = np 0 + K λ(np 0 (1 -p 0 ) + σ 2 ) 2 -λ .
Similarly to Table 1, Table 3 compares the out-of-control ARL of the upper-sided BEWMA chart obtained without (left part) and with (right part) the "continuousify" method for a number of subintervals m ∈ {100, 110, . . . , 400} and for the following out-of-control situations (n, p 0 , p 1 ) = {(40, 0.05, 0.06), (20, 0.1, 0.12), (10, 0.1, 0.15), (20, 0.15, 0.18), (10, 0.15, 0.2)}. As for Table 1, the control chart parameters are K = 3 and λ = 0.2 and the "continuousify" parameter is set at σ = 0.125. As it can be seen, the conclusions drawn for the upper-sided PEWMA chart can also be drawn for the upper-sided BEWMA chart, i.e. i) the ARL values obtained without "continuousify" heavily fluctuate and they clearly do not exhibit any monotonic convergence when the number of subintervals m increases, ii) on the contrary, for m ≥ 100, the ARL values obtained with the "continuousify" method exhibit a strong stability and seem to converge rapidly to a reliable value, iii) the ARL values obtained with the "continuousify" method (see for example the case m = 400 with 74.1, 74.5, 27.8, 57.2, 39.9) are a bit larger (again in a negligeable way) than the ones obtained by simulations without the "continuousify" method, i.e. 73.6, 73.5, 27.4, 56.6, 39.4.

As the "continuousify" method also provides stable and reliable ARL values for the upper-sided BEWMA chart, it is therefore possible to design it in or- der to find the optimal parameters λ * and K * such that, for a particular shift from p 0 to p 1 = τ p 0 , τ ∈ (1, 1/p 0 ), the out-of-control ARL is minimized under the constraint that the in-control ARL = ARL 0 = 370.4. The optimal combinations (λ * , K * ) are listed in Table 4 with the corresponding out-ofcontrol ARL values for p 0 ∈ {0.05, 0.1, 0.15, 0.2}, τ ∈ {1.1, 1.2, 1.5, 2, 3} and σ ∈ {0.1, 0.125, 0.15, 0.2}. For instance, when p 0 = 0.1, τ = 1.2 (i.e. p 1 = 0.12), n = 20 and σ = 0.15, the optimal chart parameters are (λ * , K * ) = (0.030, 1.954) and the corresponding out-of-control ARL = 40.7. As for the upper-sided PEWMA chart, similar conclusions can be drawn from Table 4 for the the upper-sided BEWMA chart, i.e. i) no matter the value of σ, the out-of-control ARL values monotonically decrease when τ increases and ii) for a specified combination (p 0 , p 1 ), the choice of σ ∈ {0.1, 0.125, 0.15, 0.2} does not significantly impact the performance of the upper-sided BEWMA chart with "continuousify" in terms of the out-of-control ARL. For instance, for the four possible values of σ ∈ {0.1, 0.125, 0.15, 0.2}, if p 0 = 0.1, p 1 = 0.12 and n = 20, then the optimal parameters (λ * , K * ) are approximately equal to (0.03, 1.96) and the out-of-control ARL values range from 39.9 to 40.8.

4 Illustrative examples 4.1 Example of an upper-sided PEWMA chart with "continuousify"

This example is based on the data provided in Chapter 7 of Montgomery (2013) (Tables 7.7 and 7.8). These tables present the number of nonconformities observed in 44 (initially, they were 46 samples but two of them have been proved to be out-of-control) successive samples of 100 printed circuit boards. The data set is divided into two subsets: the first 24 samples are used as a Phase I data set and the remaning 20 samples are used as a Phase II data set (see column X i in Table 5).

Using the Phase I data, the in-control number of nonconformities per sample is estimated as θ0 = 472 24 = 19.67. Assuming K = 3, λ = 0.2 and σ = 0.125, the upper control limit UCL * of the upper-sided PEWMA chart with the "continuousify" method is equal to UCL * = 19.67 + 3 × 0.2 × (19.67 + 0.125 2 ) 2 -0.2 = 24.103.

Note that without the "continuousify" method (i.e. σ = 0) this upper control limit would have been UCL = 24.101 which is almost the same as for the with "continuousify" approach.

In addition to the number of nonconformities X i , Table 5 also lists the values of X * i , Z i (based on X i ) and Z * i for Phases I and II. These Phases have been treated separately with initial values Z 0 = Z * 0 = 19.67. As it can be seen the values of Z i (without "continuousify") are actually different but nevertheless very close to those of Z * i (with "continuousify"). This highlights the fact that the "continuousify" technique has a very small impact in the computation of the EWMA statistic itself (but it has a positive strong impact in the computation of the RL properties as we have seen before). In any case, the upper-sided PEWMA chart with "continuousify" does not detect any out-of-control situation as it also visible in the control chart plotted in Figure 3.

I Phase II i X i X * i Z i Z * i i X i X * i Z i Z * i 4.
2 Example of an upper-sided BEWMA chart with "continuousify"

This example is also based on the data provided in Chapter 7 of Montgomery (2013) (Tables 7.2 and 7.3). In these tables the author providesthe number of nonconforming frozen orange juice concentrate 6-oz cardboard cans in samples of n = 50 cans. This data set has been divided into two subsets: the first 24 samples are used as a Phase I data set and the remaning 40 samples are used as a Phase II data set (see column X i in Table 6).

Using the Phase I data, the estimate of the in-control proportion of nonconforming cans is p0 = 133 24×50 = 0.1108. Using Table 4 for p 0 = 0.1, τ = 1.2 (20% Note again that, without the "continuousify" method (i.e. σ = 0), this upper control limit would have been UCL = 6.320 which is almost the same as for the with "continuousify" approach.

In addition to the number of nonconforming cans X i , Table 6 also lists the values of X * i , Z i (based on X i ) and Z * i for Phases I and II. As for the previous example, these Phases have been treated separately with initial values Z 0 = Z * 0 = 50 × 0.1108 = 5.54. As it can be seen the values of Z i (without "continuousify") are different but they are nevertheless very close to those of Z * i (with "continuousify"). One more time, this highlights the fact that the "continuousify" technique has a very small impact in the computation of the EWMA statistic itself along with a positive impact in the computation of the RL properties). The upper-sided EWMA np chart with "continuousify" does not detect an out-of-control situation as it can be seen in the control chart plotted in Figure 4.

Conclusions

In this paper, we propose a new method to obtain reliable RL properties of an EWMA type chart for count data. Due to the discrete nature of count data, the traditional approaches that are used to obtain the exact RL properties of an EWMA chart show an unstable (and thus unreliable) performance. Of course the problem can be partially confronted by the use of Monte Carlo simulation. However, this approach can be very time consuming when the need is to appropriately design the chart (i.e. determine the optimal values for its chart's constants).

The main findings from our study are i) the "continuousify" technique is a simple approach that allows to compute the RL properties of any attribute EWMA type control charts in a reliable way by obtaining stable values no matter the number of sub-intervals m > 100 used in the Markov chain method. Even for m = 100, the results are very stable and reliable and, obviously, less computational effort is needed. ii) the ARL values obtained using the "continuousify" method are negligibly larger than the ones obtained without the "continuousify" method, iii) the choice of the "continuousify" parameter σ has a negligeable influence on the evaluation of the RL properties as long as this one is not too small nor too big. A good compromise seems to be a value σ ∈ [0.1, 0.2], iv) the fact that the ARL values obtained using the "continuousify" method are stable makes possible to optimize any kind of EWMA chart for discrete distributions

Figure 1

 1 Figure 1: p.d.f. f X * (x|θ, σ) of X * for θ = 3 and σ ∈ {0.05, 0.1, 0.125, 0.15, 0.2, 0.4} 8

Figure 2 :

 2 Figure 2: Comparison with the plots in Figure 3 page 880 in Morais and Knoth (2020) 11

Figure 3 :

 3 Figure 3: Example (printed circuit boards) of an upper-sided PEWMA chart with "continuousify"

Figure 4 :

 4 Figure 4: Example (orange juice) of an upper-sided BEWMA chart with "continuousify"

Table 1 :

 1 Comparison of out-of-control ARL values for the upper-sided PEWMA chart obtained with and without "continuousify" when K = 3 and λ = 0.2

			without "continuousify"			with "continuousify" (σ = 0.125)
	θ 0 =	1	1	2	4	4	θ 0 =	1	1	2	4	4
	θ 1 =	2	1.5	3	5	6	θ 1 =	2	1.5	3	5	6
	100 10.9	56	17.2 105.8 11.9		9.9 28.4 17.3 33.5	10.2
	110	7.5	11.1	16	34.2	10.4		9.9 28.4 17.3 33.4	10.2
	120	7.1	10.3 17.5 -35.2	19		9.9 28.4 17.3 33.5	10.3
	130	8.3	14.7 18.5	27.1	10		9.9 28.4 17.3 33.5	10.2
	140	8.9	18.6 18.2	18.8	8.5		9.9 28.4 17.4 33.4	10.2
	150	8.3	17	18.1	10.5	6.9		9.9 28.4 17.3 33.5	10.3
	160 15.6 -82.8	17	20.1	8.8		9.9 28.4 17.3 33.5	10.2
	170	8.2	15.2	18	26.5	9.8		9.9 28.4 17.3 33.4	10.2
	180	8.9	18.5 16.4	19.9	9.1		9.9 28.4 17.3 33.5	10.3
	190 11.6 46.3 17.5	73.8	11.7		9.9 28.4 17.3 33.4	10.2
	200	7.2	12.1 16.3	22	8.9		9.9 28.4 17.3 33.4	10.2
	m 210	9	19.8 18.3	21.1	9		9.9 28.4 17.3 33.5	10.3
	220 15.8 -52.8 17.6	33.8	10.3		9.9 28.4 17.3 33.4	10.2
	230	7.3	12.8 17.3	63.8	11.1		9.9 28.4 17.3 33.4	10.2
	240	9.4	20.2 16.4 -45.1 15.2		9.9 28.4 17.3 33.5	10.2
	250	9.8	28	17.7	18	8.6		9.9 28.4 17.3 33.4	10.2
	260	9.2	21.1 16.5	32.8	9.8		9.9 28.4 17.3 33.4	10.2
	270	13	254	18	24.8	9.8		9.9 28.4 17.3 33.5	10.2
	280	8.3	19.3 17.7	432	13		9.9 28.4 17.3 33.4	10.2
	290	6.6	11.6 16.9	20.6	8.5		9.9 28.4 17.3 33.4	10.2
	300 15.4 -83.9 17.1	12.1	7.5		9.9 28.4 17.3 33.5	10.2
	310	8.2	14.2 17.3	41.7	10.6		9.9 28.4 17.3 33.4	10.2
	320 15.3 -50.1 15.2 345.7 12.8		9.9 28.4 17.3 33.4	10.2
	330	5.7	9.4	16.6	14.1	7.1		9.9 28.4 17.3 33.5	10.2
	340	9.2	21.2 16.9	30	9.9		9.9 28.4 17.3 33.4	10.2
	350 12.2 221.7 16.8 -298.7	13		9.9 28.4 17.3 33.4	10.2
	360	8.9	23.9 16.3 -16.4 26.2		9.9 28.4 17.3 33.5	10.2
	370	8.4	15.2 17.2	74.8	11.2		9.9 28.4 17.3 33.4	10.2
	380	6.3	10.3 18.3	20.4	8.8		9.9 28.4 17.3 33.4	10.2
	390	9.3	20.8 16.6	27.2	9.5		9.9 28.4 17.3 33.5	10.2
	400	7.2	10.7 17.5 122.1 12.1		9.9 28.4 17.3 33.4	10.2
	sim	9.9	28.3 17.2	33.4	10.2

Table 2 :

 2 Optimal combinations (λ * , K * ) and corresponding out-of-control ARL

	values for θ 0 ∈ {1, 2, 5, 10}, τ ∈ {1.1, 1.2, 1.5, 2, 3} and σ ∈ {0.1, 0.125, 0.15, 0.2}
	for the upper-sided PEWMA chart with "continuousify"
	θ 0	τ	θ 1	σ = 0.1	σ = 0.125	σ = 0.15	σ = 0.2
	1 1.1 1.1 (0.030, 1.986, 126.6) (0.030, 1.985, 126.8) (0.030, 1.984, 127.0) (0.030, 1.983, 127.6)
		1.2 1.2 (0.030, 1.986, 64.2)	(0.030, 1.985, 64.3)	(0.030, 1.984, 64.4)	(0.030, 1.983, 64.8)
		1.5 1.5 (0.035, 2.077, 22.4)	(0.035, 2.076, 22.4)	(0.035, 2.075, 22.5)	(0.035, 2.072, 22.6)
		2	2	(0.115, 2.728, 9.5)	(0.115, 2.725, 9.6)	(0.115, 2.722, 9.5)	(0.120, 2.737, 9.6)
		3	3	(0.260, 3.144, 4.0)	(0.245, 3.108, 4.0)	(0.265, 3.147, 4.0)	(0.250, 3.106, 4.0)
	2 1.1 2.2 (0.030, 1.965, 92.0)	(0.030, 1.964, 92.1)	(0.030, 1.964, 92.2)	(0.030, 1.963, 92.5)
		1.2 2.4 (0.030, 1.965, 43.0)	(0.030, 1.964, 43.1)	(0.030, 1.964, 43.1)	(0.030, 1.963, 43.2)
		1.5	3	(0.075, 2.446, 14.2)	(0.080, 2.476, 14.3)	(0.080, 2.475, 14.3)	(0.080, 2.473, 14.4)
		2	4	(0.210, 2.920, 5.8)	(0.215, 2.929, 5.8)	(0.215, 2.927, 5.8)	(0.185, 2.856, 5.8)
		3	6	(0.375, 3.163, 2.4)	(0.385, 3.173, 2.4)	(0.385, 3.171, 2.4)	(0.395, 3.177, 2.4)
	5 1.1 5.5 (0.030, 2.060, 55.3)	(0.030, 2.044, 55.6)	(0.030, 2.027, 55.9)	(0.030, 2.004, 55.1)
		1.2	6	(0.035, 2.031, 24.7)	(0.035, 2.031, 24.7)	(0.035, 2.031, 24.7)	(0.030, 2.004, 24.7)
		1.5 7.5	(0.150, 2.683, 7.5)	(0.150, 2.682, 7.5)	(0.150, 2.682, 7.5)	(0.155, 2.693, 7.6)
		2	10	(0.345, 2.978, 3.0)	(0.355, 2.987, 3.0)	(0.355, 2.987, 3.0)	(0.355, 2.986, 3.0)
		3	15	(0.585, 3.143, 1.3)	(0.595, 3.147, 1.3)	(0.610, 3.152, 1.3)	(0.640, 3.163, 1.3)
	10 1.1 11	(0.035, 2.025, 37.3)	(0.035, 2.025, 37.4)	(0.035, 2.025, 37.4)	(0.035, 2.025, 37.4)
		1.2 12	(0.055, 2.199, 15.7)	(0.055, 2.203, 15.7)	(0.075, 2.372, 15.8)	(0.075, 2.371, 15.8)
		1.5 15	(0.285, 2.846, 4.6)	(0.275, 2.836, 4.6)	(0.275, 2.836, 4.6)	(0.275, 2.835, 4.6)
		2	20	(0.530, 3.013, 1.8)	(0.530, 3.012, 1.8)	(0.545, 3.018, 1.8)	(0.545, 3.018, 1.8)
		3	30	(0.890, 3.113, 1.0)	(0.890, 3.112, 1.0)	(0.890, 3.112, 1.0)	(0.890, 3.110, 1.0)

Table 3 :

 3 Comparison of out-of-control ARL values for the upper-sided BEWMA chart obtained with and without "continuousify" when K = 3 and λ = 0.2

			without "continuousify"			with "continuousify" (σ = 0.125)
	n =	40	20	10	20	10	n =	40	20	10	20	10
	p 0 = 0.05	0.1	0.1 0.15 0.15 p 0 = 0.05 0.1	0.1 0.15	0.15
	p 1 = 0.06	0.12 0.15 0.18 0.2 p 1 = 0.06 0.12 0.15 0.18	0.2
		94.3	52.4 25.2 52.8 55.9		74.0 74.4 27.8 57.0	39.9
		146.4 51.6 30.4 68.9 44.9		74.0 74.4 27.8 57.2	39.9
		43.6	71.2 46.8 56.1 32.3		74.2 74.4 27.8 57.1	39.8
		81.7 114.1 23.7 52.6 30.4		74.1 74.4 27.8 57.2	39.9
		49.0	70.6 22.1 54.1 59.6		74.1 74.4 27.9 57.1	39.9
		59.3 100.7 26.7 63.0 38.5		74.1 74.4 27.9 57.2	39.8
		78.4 189.0 19.6 60.8 31.9		74.1 74.4 27.9 57.1	39.8
		79.6	87.8 42.6 53.0 62.8		74.1 74.4 27.9 57.2	39.9
		67.0	74.3 24.5 51.1 39.3		74.1 74.4 27.9 57.1	39.9
		58.4	62.7 32.8 52.9 32.2		74.1 74.4 27.8 57.2	39.8
		83.4	70.3 35.4 54.9 53.2		74.1 74.4 27.8 57.1	39.9
		62.1	68.2 29.0 59.5 47.8		74.1 74.4 27.8 57.2	39.9
		70.3	60.7 21.1 59.8 30.6		74.1 74.4 27.8 57.1	39.8
		84.1	67.3 35.7 50.9 38.1		74.1 74.4 27.9 57.2	39.9
		45.1	85.4 60.0 51.3 32.2		74.1 74.4 27.9 57.2	39.9
	m	60.6 121.3 48.6 55.0 29.2		74.1 74.4 27.9 57.2	39.8
		52.4	75.6 33.0 63.1 34.0		74.1 74.4 27.9 57.2	39.9
		78.0	55.7 24.5 62.4 29.8		74.1 74.4 27.9 57.1	39.9
		85.2	73.9 40.5 60.0 34.3		74.1 74.4 27.8 57.2	39.9
		82.5	63.9 23.8 52.0 38.5		74.1 74.4 27.8 57.1	39.8
		57.9	90.6 27.5 60.5 61.5		74.1 74.4 27.8 57.2	39.9
		191.6 61.4 19.8 62.3 30.3		74.1 74.4 27.8 57.1	39.9
		129.5 127.7 24.6 60.6 88.1		74.1 74.4 27.8 57.2	39.8
		81.3	62.9 29.9 64.9 29.0		74.1 74.4 27.9 57.1	39.9
		90.5	72.5 37.8 55.7 28.1		74.1 74.4 27.9 57.2	39.9
		70.1	80.2 27.6 54.3 38.4		74.1 74.4 27.9 57.1	39.9
		61.8	68.1 42.3 59.8 29.7		74.1 74.4 27.9 57.2	39.9
		74.9	72.5 34.7 51.2 56.6		74.1 74.4 27.8 57.1	39.9
		51.6 118.7 32.8 60.0 50.6		74.1 74.4 27.8 57.2	39.9
		60.0	74.9 20.1 57.5 33.4		74.1 74.4 27.8 57.1	39.9
		83.4	63.4 25.4 56.0 38.8		74.1 74.5 27.8 57.2	39.9
	sim	73.6	73.5 27.4 56.6 39.4

Table 5 :

 5 Example (printed circuit boards) of an upper-sided PEWMA chart with "continuousify"

	Phase

Table 4: Optimal combinations (λ * , K * ) and corresponding out-of-control ARL values for p 0 ∈ {0.05, 0.1, 0.15, 0.2}, τ ∈ {1.1, 1.2, 1.5, 2} and σ ∈ {0.1, 0.125, 0.15, 0.2} for the upper-sided BEWMA chart with "continuousify" [START_REF] Morais | Improving the ARL Profile and the Accuracy of its Calculation for Poisson EWMA Charts[END_REF] has shown that our "continuousify" method performs in a similar way in the case of the PEWMA control chart.

This method can be used for any discrete distributions like the Zero-Inflated Poisson, Binomial or Negative Binomial ones. Moreover, other smoothing kernels (in place of the normal kernel used in this paper) can also be investigated like the Epanechnikov, Biweight and Triweight (see [START_REF] Wand | Kernel Smoothing[END_REF]).

Appendix By definition, using (1), we have

Exchanging the integral and the sum gives

Using the same approach, we have E(X * 2 ) = ω∈Ω +∞ -∞

x 2 f N (x|ω, σ)dx

and since E(X * ) = E(X) we also have V(X * ) = V(X) + σ 2 .


Dr Niki Baccile
email: niki.baccile@sorbonne-universite.fr

Are microbial biosurfactants actually only surfactants?

The term biosurfactants refers to a complex mixture of metabolites with surface-active properties produced by specific microorganisms. However, nowadays trends moves towards isolation, screening and purifying single biocompatible, biodegradable biosurfactants with high commercialization potential. Current legislation limiting petrochemicals combined with environmentally concerned consumers did not only stimulate research and development but it also promoted large-scale production of this class of molecules. However, recent data recorded on single congeners question the actual pertinence of using the word "biosurfactant" associated to these molecules. By evaluating the accepted characteristics of surfactants and comparing them to the actual self-assembly and bulk properties in water of molecules traditionally called "biosurfactants", this opinion paper aims at showing that the term "biosurfactant" can be somewhat reductive when applied to specific individual compounds produced by fermentation.

The use of a more generic term, like bioamphiphile could probably be more pertinent and appropriate for consideration in the future.

Biosurfactants today

Fully bio-based surfactants are generally intended to describe molecules either prepared by chemical synthesis from bio-based synthons (e.g., alkylpolyglucoside esters) or by biological approaches such as extraction from plants (e.g., saponins), biocatalysis (alkylpolyglucosides from enzymatic catalysis) or fermentation using selected microorganisms, such as yeasts or bacteria. [START_REF] Baccile | Self-Assembly, Interfacial Properties, Interactions with Macromolecules and Molecular Modelling and Simulation of Microbial Bio-Based Amphiphiles (Biosurfactants)[END_REF] Decades of research in this field has continued on the tradition of using the term "biosurfactant" solely associated with the products obtained using the fermentation method. [START_REF] Desai | Microbial Production of Surfactants and Their Commercial Potential[END_REF][START_REF] Banat | Potential Commercial Applications of Microbial Surfactants[END_REF][START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF][START_REF] Van Bogaert | Microbial Production and Application of Sophorolipids[END_REF] The term "biosurfactant" is then associated to the surface active properties of complex mixtures of metabolites produced by some fungi and bacteria. According to the most plausible hypotheses, microorganisms use them to solubilize hydrophobic substrates as carbon source, to adhere and release to/from surfaces and enhance interfacial mobility, for their antiseptic properties or as energy reserve. [START_REF] Da Silva | Fungal Biosurfactants, from Nature to Biotechnological Product: Bioprospection, Production and Potential Applications[END_REF] More recently, these compounds have been more and more topic of research in the field of industrial microbiology, which has the goal of controlling their production, extraction, purification and, overall, industrialization through a fermentative process.

From a historical perspective, biosurfactants are described in the literature since the mid of the twentieth century: sophorolipids, obtained from the yeast Starmerella bombicola, were first described in 1968 by Tulloch, [START_REF] Tulloch | Structure and Reactions of Lactonic and Acidic Sophorosides of 17-Hydroxyoctadecanoic Acid[END_REF][START_REF] Tulloch | A New Hydroxy Fatty Acid Sophoroside from Candida Bogoriensis[END_REF] cellobioselipids, produced by the fungus Ustilago maydis and known as ustilagic acid, were reported first in 1951 [START_REF] Haskins | Biochemistry of the Ustilaginales: VII. Antibiotic Activity of Ustilagic Acid[END_REF] while rhamnolipids, produced by the bacteria Pseudomonas aeruginosa, are known since 1949. [START_REF] Edwards | Structure of a Rhamnolipid from Pseudomonas Aeruginosa[END_REF][START_REF] Jarvis | A Glyco-Lipide Produced by Pseudomonas Acruginosa[END_REF][START_REF] Hisatsuka | Formation of Rhamnolipid by Pseudomonas Aerugznosa and Its Function in Hydrocarbon Fermentation[END_REF] Other typical biosurfactants are mannosylerythritol lipids, produced by Candida antarctica. [START_REF] Kitamoto | Extracellular Accumulation of Mannosylerythritol Lipids by a Strain of Candida Antarctica[END_REF] All of these are grouped under the category of glycolipids (Figure 1), as they all contain a mono or disaccharide as hydrophilic headgroup and a lipid, generally a fatty acid backbone as the hydrophobic moiety. Another category of such molecules is composed of lipopeptides, like surfactin (Figure 1) and iturin A, produced by Bacillus subtilis. [START_REF] Arima | a Crystalline Peptidelipid Surfactant Produced by Bacillus Subtilis: Isolation, Characterization and Its Inhibition of Fibrin Clot Formation[END_REF][START_REF] Maget-Dana | Surfactin/Iturin A Interactions May Explain the Synergistic Effect of Surfactin on the Biological Properties of Iturin A[END_REF] Interestingly, although some phospholipids and polymeric emulsifiers (e.g. emulsan) have historically been classified as biosurfactants in most related literature in this area, [START_REF] Desai | Microbial Production of Surfactants and Their Commercial Potential[END_REF][START_REF] Banat | Potential Commercial Applications of Microbial Surfactants[END_REF][START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF][START_REF] Healy | Microbial Production of Biosurfactants[END_REF][START_REF] Sarubbo | Production, Properties, Applications, Trends, and General Perspectives[END_REF][START_REF] Manga | Biosurfactants: Opportunities for the Development of a Sustainable Future[END_REF] the majority of research share in this field is mostly focused on glycolipids and lipopeptides.

Fermentation-produced glycolipids and lipopeptides is certainly justified by their foaming properties, [START_REF] Lemieux | Biochemistry of the Ustilaginales II. Isolation and Partial Characterization of Ustilagic Acid[END_REF][START_REF] Razafindralambo | Foaming Properties of Surfactin, a Lipopeptide Biosurfactant from Bacillus Subtilis. JAOCS[END_REF][START_REF] Deleu | Interfacial and Emulsifying Properties of Lipopeptides from Bacillus Subtilis[END_REF][START_REF] Van Renterghem | From Lab to Market: An Integrated Bioprocess Design Approach for New-to-Nature Biosurfactants Produced by Starmerella Bombicola[END_REF][START_REF] Wang | Starmerella Bombicola: Recent Advances on Sophorolipid Production and Prospects of Waste Stream Utilization[END_REF][START_REF] Baccile | Development of a Cradleto-Grave Approach for Acetylated Acidic Sophorolipid Biosurfactants[END_REF] which is often observed during the biosynthesis process, as a consequence of their ability to lower the surface tension of water solutions which is described in most related studies in literature over the years. [START_REF] Baccile | Self-Assembly, Interfacial Properties, Interactions with Macromolecules and Molecular Modelling and Simulation of Microbial Bio-Based Amphiphiles (Biosurfactants)[END_REF][START_REF] Banat | Potential Commercial Applications of Microbial Surfactants[END_REF][START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF][START_REF] Rosenberg | High-and Low-Molecular-Mass Microbial Surfactants[END_REF] The existence of a critical micelle concentration (CMC) is also a typical surfactant feature and equally reported in a number of scientific papers. [START_REF] Baccile | Self-Assembly, Interfacial Properties, Interactions with Macromolecules and Molecular Modelling and Simulation of Microbial Bio-Based Amphiphiles (Biosurfactants)[END_REF][START_REF] Banat | Potential Commercial Applications of Microbial Surfactants[END_REF][START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF][START_REF] Rosenberg | High-and Low-Molecular-Mass Microbial Surfactants[END_REF] Their emulsification properties and actual use in existing commercial formulations of green This article is published within the context of the Special Issue on Biosurfactants at Current Opinion in Colloid and Surface Science 3 detergents further justifies the use of the term "biosurfactant". [START_REF] Banat | Potential Commercial Applications of Microbial Surfactants[END_REF][START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF][START_REF] Healy | Microbial Production of Biosurfactants[END_REF][START_REF] Sarubbo | Production, Properties, Applications, Trends, and General Perspectives[END_REF][START_REF] Cameotra | Synthesis of Biosurfactants and Their Advantages to Microorganisms and Mankind[END_REF][START_REF] Makkar | Advances in Utilization of Renewable Substrates for Biosurfactant Production[END_REF] Last but not least, antimicrobial effects, generally explained by their membrane lysis action similarly to many other positively and negatively-charged surfactants, were long described. [START_REF] Soberón-Chávez | New Insights in Biosurfactants Research[END_REF] However, recent research in this field carried out in the past decade shows evidence that the aqueous phase behaviour of fermentation-produced glycolipids and lipopeptides is more complex if taken as a whole [START_REF] Baccile | Self-Assembly, Interfacial Properties, Interactions with Macromolecules and Molecular Modelling and Simulation of Microbial Bio-Based Amphiphiles (Biosurfactants)[END_REF] and it may provoke a more thoughtful reasoning on the actual pertinence of employing the term "biosurfactant", which defines a well-specific property, that is of being a surface active agent.

Surfactants versus amphiphiles

As outlined in the Encyclopedia of Colloid and Interface Science (edition 2013, Page 1165), [START_REF] Tadros | Encyclopedia of Colloid and Interface Science[END_REF] surfactants are defined by four concomitant properties: 1-the ability to lower the surface tension of water, 2-the association into spherical aggregates, called micelles, beyond a 3-critical micellar concentration, CMC and finally, 4-micellization as a dynamic equilibrium process driven by the hydrophobic effect, generating a hydrophobic internal core and a hydrophilic shell containing the surfactant headgroups. A molecule characterized by a hydrophilic headgroup, and a lipophilic tail is typically given the general definition of an amphiphile, or amphipathic, and in many cases displays the characteristic behaviour of a surfactant in water above its Krafft point. The Krafft point, or Krafft temperature, is the temperature above which the surfactant shows a rapid increase in solubility with further increase of temperature (at the Krafft temperature, the solubility is equal to the CMC and an equilibrium occurs between hydrated surfactant solid, micelles, and monomers). [START_REF] Tadros | Encyclopedia of Colloid and Interface Science[END_REF] This is the case for many alkylammonium and alkyl sulfate salts, with a Krafft point broadly contained at around room temperature, between between 15°C and 30°C (e.g., 27-28°C for CTAB and 17°C for SDS). [START_REF] Tadros | Encyclopedia of Colloid and Interface Science[END_REF] A more pertinent definition of surfactants is the ability to foam and/or form oil-inwater emulsions.

However, not all amphiphiles share the features of surface active agents. Phospholipids are natural amphiphiles being the major constituent of the cell membrane. If they have a surface activity, they normally do not form micelles in the surrounding of room temperature, but stable bilayers, with reduced exchange dynamics. For these reasons, one of the most important works in the field of colloids science, Theory of Self-Assembly of Hydrocarbon Amphiphiles into Micelles and Bilayers by Israelachvili, Mitchell and Ninham 30 employs the term amphiphile in relationship to micelles-and bilayer-forming molecules. Nevertheless, one must acknowledge a certain ambiguity, as bilayer-forming molecules like phospholipids can also be considered as a subclass of zwitterionic surfactants. [START_REF] Tadros | Encyclopedia of Colloid and Interface Science[END_REF] More peculiar are the case of amphiphiles which have a rigid steroid (bile salts, like lithocholic acid), steroid-like (e.g., glycyrrhizic acid), or more complex, or exotic (e.g., fluorenylmethyloxycarbonyl, FMOC), hydrophobic segments. The acidic form of this class of amphiphiles has a tendency to fibrillate, a trait often associated to gelation. The fibres are semicrystalline and have an infinite aspect ratio. [31][START_REF] Goldshleger | Supramolecular Gels Based on Bile Acid Salts[END_REF][START_REF] Okesola | Applying Low-Molecular Weight Supramolecular Gelators in an Environmental Setting-Self-Assembled Gels as Smart Materials for Pollutant Removal[END_REF] Similarly, oleic acid, stearic acid or hydroxy stearic acid, just to cite some, do not belong to the domain of surfactants, but rather to the domain of oils and fats. Nevertheless, the solubility, micelle-forming and overall surfactant properties of all the molecules cited above may appear under alkaline conditions, when their corresponding salt is formed, as typically known for oleate and stearate metal salts.

Recent finding on the self-assembly properties of microbial "biosurfactants"

The study of the self-assembly behaviour of microbial biosurfactants is a relatively recent field of research. The first report was published in 1987 and showed the dual micellar (high pH, sodium salt) and vesicular (low pH, acidic form) nature of an aqueous solution of a mono-and di-rhamnolipids mixture (Figure 1). [START_REF] Ishigami | The PH-Sensitive Conversion of Molecular Aggregates of Rhamnolipid Biosurfactant[END_REF] This result has been confirmed by a number of additional studies. [START_REF] Shin | Rhamnolipid Morphology and Phenanthrene Solubility at Different PH Values[END_REF][START_REF] Dahrazma | Effects of Additives on the Structure of Rhamnolipid (Biosurfactant): A Small-Angle Neutron Scattering (SANS) Study[END_REF][START_REF] Champion | Electron Microscopy of Rhamnolipid (Biosurfactant) Morphology: Effects of PH, Cadmium, and Octadecane[END_REF][START_REF] Chen | Solution Self-Assembly and Adsorption at the Air-Water Interface of the Monorhamnose and Dirhamnose Rhamnolipids and Their Mixtures[END_REF] Similar results were also reported for the lipopeptide surfactin (Figure 1), assembling into membranes at acidic pH [START_REF] Shen | Surfactin Structures at Interfaces and in Solution: The Effect of PH and Cations[END_REF][START_REF] Shen | Aggregation of the Naturally Occurring Lipopeptide, Surfactin, at Interfaces and in Solution: An Unusual Type of Surfactant?[END_REF] or mycosubtilin, assembling into fibers. [START_REF] Hamley | Self-Assembly of Three Bacterially-Derived Bioactive Lipopeptides[END_REF] Sophorolipids, another well-known biosurfactant (Figure 1) typically obtained as a mixture of acetylated and non-acetylated acidic and lactonic forms bearing a C18:1 (oleil) backbone, can This article is published within the context of the Special Issue on Biosurfactants at Current Opinion in Colloid and Surface Science easily be converted into a non-acetylated acidic form. [START_REF] Baccile | Practical Methods to Reduce Impurities for Gram-Scale Amounts of Acidic Sophorolipid Biosurfactants[END_REF] The latter tends to form stable micellar solutions at a broad pH range. 43,[START_REF] Baccile | Self-Assembly Mechanism of PH-Responsive Glycolipids: Micelles, Fibers, Vesicles, and Bilayers[END_REF] As a matter of fact, this behaviour is probably related to the presence of minor amounts of other congeners, the complete removal of which is usually too tedious and expensive. [START_REF] Dhasaiyan | Micelles versus Ribbons: How Congeners Drive the Self-Assembly of Acidic Sophorolipid Biosurfactants[END_REF] On the other hand, the behaviour of the non-acetylated acidic saturated (C18:0, Figure 1) form of sophorolipids is completely different than that all other biosurfactants. The acidic form of C18:0 sophorolipids fibrillates at low pH, [START_REF] Baccile | Self-Assembly Mechanism of PH-Responsive Glycolipids: Micelles, Fibers, Vesicles, and Bilayers[END_REF][START_REF] Cuvier | PH-Triggered Formation of Nanoribbons from Yeast-Derived Glycolipid Biosurfactants[END_REF][START_REF] Cuvier | Synthesis of Uniform, Monodisperse, Sophorolipid Twisted Ribbons[END_REF] similarly to bile salts, while its sodium salt forms micellar solutions. [START_REF] Baccile | Self-Assembly Mechanism of PH-Responsive Glycolipids: Micelles, Fibers, Vesicles, and Bilayers[END_REF][START_REF] Cuvier | PH-Triggered Formation of Nanoribbons from Yeast-Derived Glycolipid Biosurfactants[END_REF] Similar results were observed for other microbially-produced glycolipids, like palmitic acid (C16:0) sophorolipids [START_REF] Baccile | Palmitic Acid Sophorolipid Biosurfactant: From Self-Assembled Fibrillar Network (SAFiN) To Hydrogels with Fast Recovery[END_REF] and cellobioselipids. [START_REF] Baccile | PH-Driven Self-Assembly of Acidic Microbial Glycolipids[END_REF] The behaviour of bolaform sophorosides is also worth noting, as these molecules fibrillate and form hydrogels at room temperature, with a transition to micelles above about 30°C to 35°C. [START_REF] Baccile | Bio-Based Glyco-Bolaamphiphile Forms a Temperature-Responsive Hydrogel with Tunable Elastic Properties[END_REF][START_REF] Poirier | Energy Landscape of Sugar Conformation Controls the Sol-to-Gel Transition in Self-Assembled Bola Glycolipid Hydrogels[END_REF] Figure 2 -Cryo-TEM images recorded on diluted aqueous solutions of glucolipid C18:1 (Figure 1) prepared under different physicochemical conditions: 1) micelles, basic pH (counterion: sodium); 2) unilamellar vesicles, acidic pH; 3) lamellar precipitate, pH below 4; 4) complex coacervates, basic pH in the presence of a polycation; 5) multilamellar wall vesicles, acidic pH in the presence of a polycation after reducing the pH from basic to acidic; 6) aggregated vesicles, acidic pH in the presence of a polycation; 7) fibers, basic pH (counterion, Ag + ); 8) wormlike micelles, basic pH (counterion, Fe 2+ ); 9) wormlike micelles, pH between 6 and 7; 10) multilamellar wall vesicles, acidic pH after increasing pH from below 4. Reprinted from Ref. [START_REF] Baccile | Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of Water[END_REF] , with permission from Elsevier.

Even more peculiar is the behaviour of single-glucose lipids, both saturated, C18:0, and monounsaturated, C18:1 (Figure 1). Their acidic form at low pH assembles into single-layer membranes (may them be flat or curved) [START_REF] Baccile | Self-Assembly Mechanism of PH-Responsive Glycolipids: Micelles, Fibers, Vesicles, and Bilayers[END_REF][START_REF] Baccile | PH-Driven Self-Assembly of Acidic Microbial Glycolipids[END_REF]53,[START_REF] Renterghem | Easy Formation of Functional Liposomes in Water Using a PH-Responsive Microbial Glycolipid: Encapsulation of Magnetic and Upconverting Nanoparticles[END_REF] while their sodium salt forms a water soluble micellar solution. [START_REF] Baccile | Self-Assembly Mechanism of PH-Responsive Glycolipids: Micelles, Fibers, Vesicles, and Bilayers[END_REF][START_REF] Baccile | PH-Driven Self-Assembly of Acidic Microbial Glycolipids[END_REF] The deprotonated C18:1 glucolipid can even show a unique fibrillation and metallogelation behaviour in the presence of some monovalent and divalent cations (Ca 2+ , Ag + ), [START_REF] Poirier | Metallogels from Glycolipid Biosurfactant[END_REF] where cation-induced fibrillation is known for low-molecular weight gelators [START_REF] Shi | Calcium Ions to Cross-Link Supramolecular Nanofibers to Tune the Elasticity of Hydrogels over Orders of Magnitude[END_REF][START_REF] Chen | Salt-Induced Hydrogels from Functionalised-Dipeptides[END_REF] but not for surfactants, which are known since long time to grow into large wormlike micelles when This article is published within the context of the Special Issue on Biosurfactants at Current Opinion in Colloid and Surface Science exposed to an excess of counterions in water. [START_REF] Lequeux | Structure and Rheology of Wormlike Micelles[END_REF][START_REF] Jensen | Monitoring the Transition from Spherical to Polymer-like Surfactant Micelles Using Small-Angle X-Ray Scattering[END_REF] In fact, the C18:1 glucolipid can display nine different phases in a limited span of pH at room temperature, depending on pH but also on the type of counterion and sense of varying pH. [START_REF] Baccile | Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of Water[END_REF] The peculiar case of C18:1 glucolipid is sketched in Figure 2 and Figure 3, which show the typical cryogenic transmission electron micrographs and small angle X-ray scattering fingerprint of each phase, respectively. The overall selfassembly behaviour of most microbial glycolipids and lipopeptides has been reviewed by us recently, [START_REF] Baccile | Self-Assembly, Interfacial Properties, Interactions with Macromolecules and Molecular Modelling and Simulation of Microbial Bio-Based Amphiphiles (Biosurfactants)[END_REF] while the specific behaviour of glucolipid C18:1 [START_REF] Baccile | Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of Water[END_REF] and of a number of chemo derivatives of sophorolipids has been reported elsewhere. [START_REF] Baccile | Asymmetrical, Symmetrical, Divalent and Y-Shaped (Bola)Amphiphiles: The Relationship between Molecular Structure and Self-Assembly in Amino Derivatives of Sophorolipid Biosurfactants[END_REF] Figure 3 -Typical SAXS profiles recorded on glucolipid C18:1 (Figure 1) samples shown in Figure 2. Numbers indicate the same samples as in Figure 2, where 7a and 7b contain Ca 2+ and Ag + , respectively, as counterions. Reprinted from Ref. [START_REF] Baccile | Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of Water[END_REF] , with permission from Elsevier.

Bioamphiphiles

The complex, parametric-dependent, self-assembly behaviour of most microbial glycolipids and lipopeptides is at the interface of surfactants, phospholipids but also bile salts, fatty acids and low-molecular weight gelators. In the best case scenario, the triple micelle-, membrane-and fibre-forming ability is observed across microbial glycolipids and lipopeptides, while in the most complex case, such variety is observed for a single molecule under 0.1 1 10

q / nm -1 1 2 3 4 5 7a 7b comparable physicochemical conditions. This evidence casts doubt on the proper use of the word "biosurfactant". According to the definitions and descriptions given in the Encyclopedia of Colloid and Interface Science, when microbial glycolipids and lipopeptides assemble into vesicles or lamellae, the term "biosurfactant" is mildly appropriate, while in the case of fibrous assemblies, it would not be appropriate at all.

On the one hand, such multiphasic behaviour is generally observed for the neutral, generally acidic, form of this class of compounds, while the surfactant-like behaviour is generally observed for the corresponding deprotonated sodium salt. [START_REF] Baccile | Self-Assembly, Interfacial Properties, Interactions with Macromolecules and Molecular Modelling and Simulation of Microbial Bio-Based Amphiphiles (Biosurfactants)[END_REF] However on the other hand, it is precisely the acidic form that is represented and addressed to as being a "biosurfactant" in the dedicated literature. [START_REF] Desai | Microbial Production of Surfactants and Their Commercial Potential[END_REF][START_REF] Banat | Potential Commercial Applications of Microbial Surfactants[END_REF][START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF][START_REF] Healy | Microbial Production of Biosurfactants[END_REF][START_REF] Sarubbo | Production, Properties, Applications, Trends, and General Perspectives[END_REF][START_REF] Manga | Biosurfactants: Opportunities for the Development of a Sustainable Future[END_REF] The question after all is quite simple: why does one employ the term oil, or fat or gelator, for families of molecules like fatty acids and bile salts in their uncharged, generally acidic, form, and the term surfactant for the uncharged acidic form of sophorolipids, rhamnolipids, glucolipids, cellobiose lipids or surfactin, when most of the times these fermentation-produced molecules behave as lipids and gelators, particularly at acidic pH values? Would the term bioamphiphile, be more generally appropriate and less restrictive?

The term bioamphiphile could refer to an amphiphile of biological origin, microbial in this case, and it has the advantage of qualifying a molecule for its structural features rather than its properties, for instance a surfactant or a gelator. The expression microbial bioamphiphile would then include microbial glycolipids and lipopeptides on the sole basis of their microbiallyfermented origin and amphiphilic structures.

When addressing microbially-produced glycolipids and lipopeptides, the use of the term amphiphile is seldom used, 1,4,61-63 if compared to the term "microbial biosurfactants". One major reason is undoubtedly historical, as surface activity is classically associated to the mixture of metabolites produced by microorganisms; surface activity is also the property that is sought for when screening the production of new compounds by microorganisms. [START_REF] Da Silva | Fungal Biosurfactants, from Nature to Biotechnological Product: Bioprospection, Production and Potential Applications[END_REF] Of course, surface activity is also the property that it has a major interest in view of replacing synthetic surfactants.

Nevertheless, one of the first reviews in the field, published in 2002 by Prof. S. Lang, a microbiologist pioneer in the field of biosurfactants, was appropriately titled Biological amphiphiles (microbial biosurfactants). [START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF] It seems that this review employs the term "amphiphile" specifically and explicitly towards microbially-produced glycolipids and lipopeptides for the first time, despite the fact that the field had already witnessed at least four other previously published review articles, which promoted the term "microbial surfactants". [START_REF] Desai | Microbial Production of Surfactants and Their Commercial Potential[END_REF][START_REF] Banat | Potential Commercial Applications of Microbial Surfactants[END_REF][START_REF] Healy | Microbial Production of Biosurfactants[END_REF][START_REF] Rosenberg | High-and Low-Molecular-Mass Microbial Surfactants[END_REF] However, the use of the term "amphiphile" was not specifically justified with respect to the specific properties of this class of molecules. The use of the term "amphiphile", and more precisely the expression microbial bioamphiphile, would be more accurate, following the example of Prof. Lang. The expression microbial bioamphiphile may also be less misleading and more appropriate, in view of the recent advances in the field.

It is generally accepted the fact that the surface activity of microbial metabolites is never associated to a single compound, but to a naturally-selected formulation of molecules. [START_REF] Nitschke | Rhamnolipid Surfactants: An Update on the General Aspects of These Remarkable Biomolecules[END_REF][START_REF] Bogaert | Biosurfactants and Bioengineering of Production[END_REF] Individual molecules in the formulation may actually have poor, if not any surface activity at all. This is the case of lactonic sophorolipids, the most abundant component (about 60-80%) in the classical raw mixture of lactonic/open acidic forms of this family of compounds. [START_REF] Bogaert | Biosurfactants and Bioengineering of Production[END_REF] Lactonic sophorolipids are more insoluble and with a tendency to form membrane structures, instead of micelles. [START_REF] Penfold | Solution Self-Assembly of the Sophorolipid Biosurfactant and Its Mixture with Anionic Surfactant Sodium Dodecyl Benzene Sulfonate[END_REF] In this regard, its surfactant properties are arguable and use in surfactant-classics applications unlikely. [START_REF] Hirata | Natural Synergism of Acid and Lactone Type Mixed Sophorolipids in Interfacial Activities and Cytotoxicities[END_REF] On the other hand, the open acidic form of sophorolipids can certainly be considered as a surfactant, but its interfacial performances are poor. [START_REF] Koh | A Versatile Family of Sophorolipid Esters: Engineering Surfactant Structure for Stabilization of Lemon Oil-Water Interfaces[END_REF] Interestingly, it was shown that synergy between the lactonic and open acidic forms have better foaming and interfacial properties, [START_REF] Hirata | Natural Synergism of Acid and Lactone Type Mixed Sophorolipids in Interfacial Activities and Cytotoxicities[END_REF] a fact, which could help understanding the reason why mixtures of these congeners are produced in nature. A similar observation could be done for rhamnolipids, of which the natural formulation, about 50% of mono-and di-ramnose lipids, has optimized surface activity. [START_REF] Nitschke | Rhamnolipid Surfactants: An Update on the General Aspects of These Remarkable Biomolecules[END_REF] At neutral/acidic pH, mono-rhamnolipids tend to associate into vesicles, while di-rhamnolipids associate into into micelles, before precipitating into lamellar structures at pH below its pKa. [START_REF] Baccile | Self-Assembly of Rhamnolipid Bioamphiphiles: Understanding the Structure-Property Relationship Using Small-Angle X-ray Scattering[END_REF][START_REF] Esposito | Rhamnolipid Self-Aggregation in Aqueous Media: A Long Journey toward the Definition of Structure-Property Relationships[END_REF] Finally, it is worth saying that many metabolites produced in much lower amounts, like the C18:0 or C16:0 forms of sophorolipids, do not have surface activity at all, as they are insoluble and fibrillate in their neutral form. [START_REF] Cuvier | PH-Triggered Formation of Nanoribbons from Yeast-Derived Glycolipid Biosurfactants[END_REF][START_REF] Baccile | Palmitic Acid Sophorolipid Biosurfactant: From Self-Assembled Fibrillar Network (SAFiN) To Hydrogels with Fast Recovery[END_REF] These facts suggest that biological evolution has engineered optimal mixtures of metabolites to enhance surface activity and to promote properties like emulsification, adhesion or solubilisation, relevant for microbial survival.

Considering the fact that different congeners have different properties, a lot of research has been recently devoted to generate molecular variation, either through a genetic [START_REF] Lodens | Transformation of an Exotic Yeast Species into a Platform Organism: A Case Study for Engineering Glycolipid Production in the Yeast Starmerella Bombicola[END_REF] or chemical [START_REF] Pala | Unlocking the Potential of a Glycolipid Platform through Chemical Modification[END_REF][START_REF] Delbeke | Chemical and Enzymatic Modification of Sophorolipids[END_REF] engineering approach, sometimes combined. [START_REF] Pala | Unlocking the Potential of a Glycolipid Platform through Chemical Modification[END_REF] Engineered strains are able to produce both natural compounds, like acetylated or acidic sophorolipids, [START_REF] Baccile | Development of a Cradleto-Grave Approach for Acetylated Acidic Sophorolipid Biosurfactants[END_REF] and new-to-nature compounds, like glucolipids or sophorosides. [START_REF] Lodens | Transformation of an Exotic Yeast Species into a Platform Organism: A Case Study for Engineering Glycolipid Production in the Yeast Starmerella Bombicola[END_REF][START_REF] Van Bogaert | Synthesis of Bolaform Biosurfactants by an Engineered Starmerella Bombicola Yeast[END_REF] In the best case scenarios, yields are very good and selectivity high.

Although not yet developed at industrial scales, a number of (chemically or genetically) engineered microbial metabolites certainly cannot be catalogued as surfactants. Nevertheless, they still have both scientific significance, especially in the space colloids and surface science, [START_REF] Baccile | Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of Water[END_REF] and industrial interest. [START_REF] Roelants | Taking Biosurfactants from the Lab to the Market: Hurdles and How to Take Them by Applying an Integrated Process Design Approach[END_REF] This article is published within the context of the Special Issue on Biosurfactants at Current Opinion in Colloid and Surface Science

If one acknowledges that the field of microbial biosurfactants is evolving and rapidly expanding, it could be interesting, and most likely important, to improve the sharing of information and communication across communities, which for this field span from microbiology and genetic engineering to physical chemistry, colloids science, organic chemistry, process engineering and all the way to the private sector. The term "biosurfactant" implicitly implies that this class of compounds are surfactants of biological origin, with the obvious aim of replacing chemical surfactants in all applications where surfactants are employed, from detergency to cosmetics. However, as some single-molecule microbial glycolipids and lipopeptides, may them be obtained from wild type or genetically-modified microorganisms, may not display the expected surfactant-like properties under some conditions of use (e.g., low pH, room temperature, presence of specific salts), the expectations generated by the end-user (e.g., a private company) may be highly affected. Alternatively, an expression such as microbial bioamphiphile forces the specification of the actual properties, many of which are surface activity, gelling or membrane-forming, thus targeting whatever possible endapplication in a clearer, direct, and faster manner. In this regard, most microbial glycolipids (e.g., sophorolipids, rhamnolipids, cellobioselipids) have a double amphiphilic, referred to as bolaform, molecular structure. In the literature, such molecules are historically defined as bolaamphiphiles, the solution self-assembly properties of which are also quite atypical when compared to classical amphiphiles. [START_REF] Fuhrhop | Bolaamphiphiles Form Ultrathin, Porous, and Unsymmetric Monolayer Lipid Membranes[END_REF] Interestingly, the self-assembly of microbial glycolipids was discussed before within the context of bolaamphiphiles. [START_REF] Dhasaiyan | Self-Assembly of Bolaamphiphilic Molecules[END_REF] It goes without saying that the proposition above has to face at least three decades of the use of the term "biosurfactant" in hundreds of research and review articles. It is then obviously unrealistic to achieve a change in the terminology in the immediate future. However, it is still important to raise the debate, broaden the perspectives and share the idea that this class of microbially-produced amphiphiles are not just surfactants, but they are a whole new class of molecules, some of which with unique all-in-one properties (e.g., glucolipid C18:1) [START_REF] Baccile | Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of Water[END_REF] when compared to the wide range of existing natural and synthetic amphiphiles. It is then worth studying microbial bioamphiphiles not only in the market perspective of replacing synthetic chemicals with bio-based chemicals, but also of pushing forward the fundamental knowledge in the field of colloids and interface science.

Conclusions

Microbially-produced glycolipid and lipopeptide mixtures have historically been addressed as biosurfactants, for their ability to lower the surface tension of water, to form

This article is published within the context of the Special Issue on Biosurfactants at Current Opinion in Colloid and Surface Science 10 micelles and generate foam and oil-water emulsions. However, recent research in the field of physical chemistry applied to this class of compounds, especially to newer (genetically or chemically) engineered single molecules, shows a more complex behaviour, in particular by their neutral acidic form, thus questioning/assessing the use of the term surfactant. Just like phospholipids, bile salts and other complex organic acids (e.g., glycyrrhizic acid), microbial glycolipids and lipopeptides assemble into uni-and multilamellar vesicles and also into semicrystalline fibres, with sometimes unexpected gelling properties. This behaviour does not match the traditional definition of a surfactant, known to lower the surface tension of water but also to form micelles in a dynamic equilibrium beyond the CMC. Their behaviour in solution is then much richer and, by the systematic employment of the term surfactant, is underestimated. For this reason, and with the goal of improving the communication across communities and to broaden the application potential, this communication proposes to employ a more general expression describing these molecules as microbial bioamphiphiles. The expression biological amphiphiles was employed already twenty years ago by Prof. S. Lang,[START_REF] Lang | Biological Amphiphiles (Microbial Biosurfactants)[END_REF] therefore the goal of this commentary is to broaden the perspectives in using the word amphiphile, much more appropriately to molecules which have a surface activity, but also gelling and membrane-forming abilities.

Figure 1 -

 1 Figure 1 -Classical set of microbially produced glycolipids and lipopeptides traditionally referred to in literature as "biosurfactants".

Ref. 6 -

 6 **-Fungal Biosurfactants, from Nature to Biotechnological Product: Bioprospection, Production and Potential Applications -A recent review article in which biosurfactants are discussed from both a biological and functional perspective, with an emphasis on the techniques of characterization of the surfactant properties.---------------------Ref. 18 -**-Biosurfactants: Opportunities for the development of a sustainable future -Authors critically review the field of biosurfactants in terms of their Strengths and opportunities but also their weaknesses and Threats. 18 --------------------Ref. 22 -**-From Lab to Market: An Integrated Bioprocess Design Approach for New-to-Nature Biosurfactants Produced by Starmerella Bombicola A complete work spanning from genetic engineering giving rise to new sophorolipid derivatives to the evaluation of the application potential, passing through self-assembly ---------------------Ref. 23 -**-Starmerella Bombicola: Recent Advances on Sophorolipid Production and Prospects of Waste Stream Utilization -23 Authors review the recent advances in the sophorolipid production in view of their commercialization. Literature is analysed from the point of view of recent achievements in metabolic engineering, bioprocessing and employment of renewable feedstock----------------------Ref. 24 -**-Development of a Cradle-to-Grave Approach for Acetylated Acidic Sophorolipid Biosurfactants Probably the first multidisciplinary and multicollaborative work spanning from genetic engineering to life cycle analysis of a sophorolipid congener. This work also studies selfassembly and it correlates it to bulk properties like emulsification and foaming.---------------------Ref. 31 -*-Glycyrrhizic Acid: Self-Assembly and Applications in Multiphase Food Systems -Authors review the self-assembly and applications in O/W emulsions, gel emulsions, foams of glycyrrhizic acid, a biobased glycolipid saponin similar in structure to biosurfactants31 ---------------------Ref. 38 -**-Solution Self-Assembly and Adsorption at the Air-Water Interface of the Monorhamnose and Dirhamnose Rhamnolipids and Their Mixtures -Authors study the self-assembly properties of rhamnolipids in bulk water and at interfaces using neutron scattering and reflectivity and they identify important structural parameters like areaper-molecule, micellar radius, aggregation number. 38 ---------------------Ref. 40 -*-Aggregation of the Naturally Occurring Lipopeptide, Surfactin, at Interfaces and in Solution: An Unusual Type of Surfactant? Using neutron reflectivity and scattering data recorded on deuterium-modified surfactin biosurfactant, authors propose a new model of surfactin aggregation. 40 --------------------Ref. 52 -**-Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of WaterAuthors show that a so-called glycolipid biosurfactant in water is able to assemble into a broad range of morphologies (micelles, fibers, membranes, giant micelles, multilamellar wall vesicles, onions…) within relatively restraint physicochemical conditions.[START_REF] Baccile | Chameleonic Amphiphile: The Unique Multiple Self-Assembly Properties of a Natural Glycolipid in Excess of Water[END_REF] -----------------------Ref. 55 -**-Metallogels from Glycolipid Biosurfactant. The first study demonstrating metallogelation of biosurfactants and combining rheology, SAXS and cryo-TEM---------------------Ref. 66 -**-Solution Self-Assembly of the Sophorolipid Biosurfactant and Its Mixture with Anionic Surfactant Sodium Dodecyl Benzene Sulfonate Authors use neutron scattering to study the self-assembly of lactonic and acidic sophorolipids and their mixture. They suppose vesicular structures of lactonic sophorolipids. Ref. 67 -**-Natural Synergism of Acid and Lactone Type Mixed Sophorolipids in Interfacial Activities and CytotoxicitiesThis work suggests that the natural mixture of lactonic and acidic sophorolipids have improved interfacial properties than the single acidic and lactonic forms. This could help understanding the reason why wild type microorganisms produce mixtures of metabolites instead of pure compounds.---------------------Ref. 72 -**-Unlocking the Potential of a Glycolipid Platform through Chemical Modification Authors hypothesize that combination of genetic and chemical engineering is the strategy to develop new biobased performing bioamphophiles. 72 ---------------------Ref. 76 -**-Unlocking the Potential of a Glycolipid Platform through Chemical Modification An historical document illustrating the astonishing properties of amphiphiles with bolaform molecular structure. 76 ---------------------This article is published within the context of the Special Issue on Biosurfactants at Current Opinion in Colloid and Surface Science

Acknowledgements

My warmest acknowledgements to Dr. S. Roelants, Prof. I. Van Bogaert, Prof. W. Soetaert, Prof. C. Stevens at Gent University (Belgium), S. Ziebek at Fraunhofer Institute, Stuttgart (Germany), T. Tiso at RWTH Aachen (Germany), B. L.V. Prasad at NCL, Pune (India) for the fruitful discussions and providing me with microbial bioamphiphiles across the years. Acknowledgements also to Prof. Gloria Sobéron-Chavez (Universidad Autonoma de Mexico, Mexico) for commenting this work. A warm and thoughtful acknowledgement to Prof. I. M. Banat (Ulster University, UK) for reading, correcting and commenting this work. I am also very grateful to Prof. R. Hausmann (University of Hohenheim, Germany), Prof. Gloria Sobéron-Chavez and Prof. E. Deziel (University of Laval, Canada) for accepting this comment at the 2 nd

international Biosurfactant Conference held at the University of Hohenheim, Germany. I also thank the Reviewers of this manuscript, who strongly contributed to its present form.

References highlight

This article is published within the context of the Special Issue on Biosurfactants at Current Opinion in Colloid and Surface Science